ICS 91.100.30

Tragwerke aus Beton, Stahlbeton und Spannbeton –Teil 1: Bemessung und Konstruktion

In keeping with current practice in standards published by the International Organization for Standardization(ISO), a comma has been used throughout as the decimal marker.

Contents

Ref. No. DIN 1045-1 : 2001-07English price group 32 Sales No. 0132

04.04

DEUTSCHE NORM July 2001

1045-1{

© No part of this translation may be reproduced without the prior permission ofDIN Deutsches Institut für Normung e.V., Berlin. Beuth Verlag GmbH, 10772 Berlin, Germany,has the exclusive right of sale for German Standards (DIN-Normen).

Translation by DIN-Sprachendienst.In case of doubt, the German-language original should be consulted as the authoritative text.

Plain, reinforced and prestressed concretestructures

Part 1: Design and construction

Continued on pages 2 to 122.

This standard, together withDIN 1045-2 to DIN 1045-4and DIN EN 206-1,July 2001 editions,supersedes DIN 1045,July 1988 edition,DIN 4219-2,December 1979 edition,DIN 4227-1,July 1988 edition,DIN 4227-2,May 1984 edition,and DIN 4227-4,February 1986 edition.

Page

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Normative references . . . . . . . . . . . . . . . . . 7

3 Concepts and symbols . . . . . . . . . . . . . . . . 83.1 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Quantities and symbols . . . . . . . . . . . . . . . 103.3 SI units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Documentation . . . . . . . . . . . . . . . . . . . . . . . 194.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2.1 General requirements . . . . . . . . . . . . . . . 194.2.2 Erection drawings for precast

elements . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2.3 Drawings of formwork and falsework . . 204.3 Design analysis . . . . . . . . . . . . . . . . . . . . . . 204.4 Specification of works . . . . . . . . . . . . . . . . 20

Page

5 Safety concept . . . . . . . . . . . . . . . . . . . . . . . 205.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Design resistance . . . . . . . . . . . . . . . . . . . . 205.3 Ultimate limit state . . . . . . . . . . . . . . . . . . . 215.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3.2 Ductility . . . . . . . . . . . . . . . . . . . . . . . . . . 215.3.3 Partial safety factors for actions and

resistances at ultimate limit state . . . . . 215.3.4 Combinations of actions and design

situations . . . . . . . . . . . . . . . . . . . . . . . . . 225.4 Serviceability limit states . . . . . . . . . . . . . . 225.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.4.2 Requirement classes . . . . . . . . . . . . . . . . 23

6 Durability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2 Exposure classes and minimum

concrete strength . . . . . . . . . . . . . . . . . . . . 236.3 Concrete cover . . . . . . . . . . . . . . . . . . . . . . 23

Page 2DIN 1045-1 : 2001-07

7 Basis of structural analysis . . . . . . . . . . . . 267.1 Requirements . . . . . . . . . . . . . . . . . . . . . . . 267.2 Imperfections . . . . . . . . . . . . . . . . . . . . . . . 277.3 Idealizations and simplifications . . . . . . . . 287.3.1 Effective flange width, load dispersal

and effective span . . . . . . . . . . . . . . . . . . 287.3.2 Other simplifications . . . . . . . . . . . . . . . . 31

8 Methods of analysis . . . . . . . . . . . . . . . . . . 328.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328.2 Linear-elastic analysis . . . . . . . . . . . . . . . . 328.3 Linear-elastic analysis with

redistribution of moments . . . . . . . . . . . . . 328.4 Plastic analysis . . . . . . . . . . . . . . . . . . . . . . 338.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.4.2 Simplified analysis of plastic rotation

for members predominately subjectedto bending . . . . . . . . . . . . . . . . . . . . . . . . 33

8.5 Non-linear analysis . . . . . . . . . . . . . . . . . . . 348.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 348.5.2 Basis of calculation for linear members

and one-way spanning slabs subjectedto bending with or without coexistentaxial force . . . . . . . . . . . . . . . . . . . . . . . . . 35

8.6 Linear members and walls in axialcompression . . . . . . . . . . . . . . . . . . . . . . . . 35

8.6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.6.2 Classification of structures and

members . . . . . . . . . . . . . . . . . . . . . . . . . 368.6.3 Methods of analysis . . . . . . . . . . . . . . . . 378.6.4 Imperfections . . . . . . . . . . . . . . . . . . . . . . 388.6.5 Model column method . . . . . . . . . . . . . . 388.6.6 Compression members with biaxial

eccentricity . . . . . . . . . . . . . . . . . . . . . . . . 408.6.7 Plain concrete compression members . . 418.6.8 Lateral buckling of slender beams . . . . . 428.7 Prestressed structures . . . . . . . . . . . . . . . . 428.7.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.7.2 Prestressing force . . . . . . . . . . . . . . . . . . 438.7.3 Loss of prestress . . . . . . . . . . . . . . . . . . . 448.7.4 Serviceability limit state . . . . . . . . . . . . . 458.7.5 Ultimate limit state . . . . . . . . . . . . . . . . . 458.7.6 Anchorage zones in pre-tensioned

members . . . . . . . . . . . . . . . . . . . . . . . . . 468.7.7 Anchorage zones of post-tensioned

or unbonded members . . . . . . . . . . . . . . 48

9 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.1 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 489.1.2 Strength . . . . . . . . . . . . . . . . . . . . . . . . . . 489.1.3 Deformation characteristics . . . . . . . . . . 499.1.4 Creep and shrinkage . . . . . . . . . . . . . . . . 499.1.5 Stress-strain curve for non-linear

methods of analysis and for straincalculations . . . . . . . . . . . . . . . . . . . . . . . 55

9.1.6 Stress-strain curve for section design . . 559.1.7 Characteristic values of concrete . . . . . 57

9.2 Reinforcement . . . . . . . . . . . . . . . . . . . . . . . 579.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 579.2.3 Stress-strain curve for global analysis . . 599.2.4 Stress-strain curve for section design . . 609.3 Prestressing steel . . . . . . . . . . . . . . . . . . . . 609.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 609.3.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 609.3.3 Stress-strain curve for design . . . . . . . . 61

10 Analyses for the ultimate limit state . . . 6210.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . 6210.2 Bending, axial force and coexistent

bending and axial force . . . . . . . . . . . . . . 6210.3 Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6310.3.1 Methods of analysis . . . . . . . . . . . . . . . 6310.3.2 Design shear force . . . . . . . . . . . . . . . . . 6310.3.3 Members not requiring design shear

reinforcement . . . . . . . . . . . . . . . . . . . . . 6410.3.4 Members requiring design shear

reinforcement . . . . . . . . . . . . . . . . . . . . . 6510.3.5 Shear between web and flange . . . . . . 6610.3.6 Shear transfer in joints . . . . . . . . . . . . 6710.3.7 Unreinforced members . . . . . . . . . . . . . 6910.4 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . 6910.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . 6910.4.2 Methods of analysis . . . . . . . . . . . . . . . 7010.4.3 Warping torsion . . . . . . . . . . . . . . . . . . . 7110.4.4 Unreinforced members . . . . . . . . . . . . . 7110.5 Punching shear . . . . . . . . . . . . . . . . . . . . . 7110.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . 7110.5.2 Loaded areas and sections used in

analyses . . . . . . . . . . . . . . . . . . . . . . . . . 7210.5.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 7510.5.4 Slabs or foundations without

punching shear reinforcement . . . . . . . 7610.5.5 Slabs or foundations containing

punching shear reinforcement . . . . . . . 7710.5.6 Minimum bending moments . . . . . . . . . 7810.6 Strut-and-tie models . . . . . . . . . . . . . . . . 7910.6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . 7910.6.2 Design of struts and ties . . . . . . . . . . . . 7910.6.3 Design of nodes . . . . . . . . . . . . . . . . . . . 8010.7 Partial area loading . . . . . . . . . . . . . . . . . . 8110.8 Fatigue analysis . . . . . . . . . . . . . . . . . . . . 8210.8.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . 8210.8.2 Internal forces and stresses at ultimate

limit state . . . . . . . . . . . . . . . . . . . . . . . . 8210.8.3 Methods of analysis . . . . . . . . . . . . . . . 8310.8.4 Simplified analyses . . . . . . . . . . . . . . . . 85

11 Analyses for the serviceability limitstate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

11.1 Limitation of stresses . . . . . . . . . . . . . . . . 8611.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . 8611.1.2 Limitation of compressive stresses

in concrete . . . . . . . . . . . . . . . . . . . . . . . 86

Page 3DIN 1045-1 : 2001-07

11.1.3 Limitation of stresses in thereinforcement . . . . . . . . . . . . . . . . . . . . 86

11.1.4 Limitation of stresses in prestressingsteel . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

11.2 Crack control and check fordecompression . . . . . . . . . . . . . . . . . . . . 86

11.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 8611.2.2 Minimum reinforcement for limitation

of crack width . . . . . . . . . . . . . . . . . . . 8811.2.3 Control of cracking without detailed

analysis . . . . . . . . . . . . . . . . . . . . . . . . . 8911.2.4 Calculation of crack width . . . . . . . . . 9111.3 Limitation of deformations . . . . . . . . . . 9211.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 9211.3.2 Simplified method to check control

of deformations in reinforced concretemembers . . . . . . . . . . . . . . . . . . . . . . . . 93

12 Detailing arrangements forreinforcement and tendons . . . . . . . . . . 93

12.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . 9312.2 Spacing of bars . . . . . . . . . . . . . . . . . . . 9312.3 Bending of steel . . . . . . . . . . . . . . . . . . . 9312.3.1 Mandrel diameter . . . . . . . . . . . . . . . . . 9312.3.2 Rebending . . . . . . . . . . . . . . . . . . . . . . 9412.4 Bonding conditions . . . . . . . . . . . . . . . . 9512.5 Design bond stress . . . . . . . . . . . . . . . . 9612.6 Anchorage of longitudinal

reinforcement . . . . . . . . . . . . . . . . . . . . . 9712.6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 9712.6.2 Anchorage length . . . . . . . . . . . . . . . . 9812.6.3 Shear reinforcement . . . . . . . . . . . . . . 9812.7 Anchorage of links and shear

reinforcement . . . . . . . . . . . . . . . . . . . . . 9912.8 Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9912.8.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 9912.8.2 Lap length . . . . . . . . . . . . . . . . . . . . . . 10112.8.3 Shear reinforcement . . . . . . . . . . . . . . 10212.8.4 Layered reinforcing fabric . . . . . . . . . . 10212.9 Bundles of bars . . . . . . . . . . . . . . . . . . . 10312.10 Tension members (tendons) . . . . . . . . 10512.10.1 General . . . . . . . . . . . . . . . . . . . . . . . . 10512.10.2 Pre-tensioned members . . . . . . . . . . 10512.10.3 Post-tensioned members . . . . . . . . . 10512.10.4 Unbonded members . . . . . . . . . . . . . 10512.10.5 Couplers . . . . . . . . . . . . . . . . . . . . . . . 106

13 Detailing arrangements for structuralmembers . . . . . . . . . . . . . . . . . . . . . . . . . . 106

13.1 Members predominately subjected tobending . . . . . . . . . . . . . . . . . . . . . . . . . . 106

13.1.1 Maximum and minimumreinforcement . . . . . . . . . . . . . . . . . . . . 106

13.1.2 Top reinforcement in prestressedmembers . . . . . . . . . . . . . . . . . . . . . . . . 106

13.2 Beams and T-beams . . . . . . . . . . . . . . . 10813.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 10813.2.2 Resistance to tension . . . . . . . . . . . . . 108

13.2.3 Shear reinforcement . . . . . . . . . . . . . . 10913.2.4 Torsion reinforcement . . . . . . . . . . . . . 11113.2.5 Top reinforcement in members with

large bar diameters . . . . . . . . . . . . . . . 11113.3 Solid slabs cast in-situ . . . . . . . . . . . . . 11113.3.1 Minimum thickness . . . . . . . . . . . . . . . 11113.3.2 Resistance to tension . . . . . . . . . . . . . 11113.3.3 Shear and punching shear

reinforcement . . . . . . . . . . . . . . . . . . . . 11213.4 Prefabricated floor systems . . . . . . . . . 11313.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 11313.4.2 Load dispersal . . . . . . . . . . . . . . . . . . . 11313.4.3 Floor decks with in-situ topping . . . . 11313.4.4 Plate action . . . . . . . . . . . . . . . . . . . . . 11613.5 Columns . . . . . . . . . . . . . . . . . . . . . . . . . 11613.5.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 11613.5.2 Limitations on area of longitudinal

reinforcement . . . . . . . . . . . . . . . . . . . . 11613.5.3 Shear reinforcement . . . . . . . . . . . . . . 11613.6 Deep beams . . . . . . . . . . . . . . . . . . . . . . 11713.7 Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11713.7.1 Reinforced concrete walls . . . . . . . . . 11713.7.2 Wall-to-ceiling connections in precast

construction . . . . . . . . . . . . . . . . . . . . . 11813.7.3 Sandwich panels . . . . . . . . . . . . . . . . . 11913.7.4 Unreinforced walls . . . . . . . . . . . . . . . . 11913.8 Connection and support of precast

elements . . . . . . . . . . . . . . . . . . . . . . . . . 11913.8.1 General . . . . . . . . . . . . . . . . . . . . . . . . . 11913.8.2 Compression joints . . . . . . . . . . . . . . . 11913.8.3 Rigid and high-strength

connections . . . . . . . . . . . . . . . . . . . . . 12013.8.4 Zones of support . . . . . . . . . . . . . . . . . 12013.9 Zones of load transfer . . . . . . . . . . . . . . 12013.9.1 Compression . . . . . . . . . . . . . . . . . . . . 12013.9.2 Tension . . . . . . . . . . . . . . . . . . . . . . . . . 12013.10 Forces associated with changes in

direction . . . . . . . . . . . . . . . . . . . . . . . . . 12113.11 Indirect support . . . . . . . . . . . . . . . . . . . 12113.12 Limitation of damage due to

accidental actions . . . . . . . . . . . . . . . . . 12113.12.1 General . . . . . . . . . . . . . . . . . . . . . . . . 12113.12.2 Peripheral ties . . . . . . . . . . . . . . . . . . 12113.12.3 Internal ties . . . . . . . . . . . . . . . . . . . . 12213.12.4 Horizontal wall and column ties . . . . 122

FiguresFigure 1: Application of equivalent geometricalimperfections . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 2: Effective flange width . . . . . . . . . . . . 28Figure 3: Approximate effective spans forcalculation of effective flange width . . . . . . . . 29Figure 4: Effective web width of T-beams ofvariable depth . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 5: Angle of spread of concentratedaxial loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Page 4DIN 1045-1 : 2001-07

Figure 6: T-beam with angle of spread ofprestressing forces . . . . . . . . . . . . . . . . . . . . . . 29Figure 7: Effective span of a beam or slab . . . 30

Figure 8: Direct and indirect support . . . . . . . 30Figure 9: Basic values of permitted plasticrotation for concrete of strength classesC12/16 to C50/60 and C100/115 . . . . . . . . . . . 34

Figure 10: Simplified moment-curvaturerelationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 11: Types of isolated column . . . . . . . . 36

Figure 12: Model column . . . . . . . . . . . . . . . . . 38Figure 13: Design model for calculation ofeffective eccentricity . . . . . . . . . . . . . . . . . . . . . 39

Figure 14: Scope of separate checks in thetwo principal planes . . . . . . . . . . . . . . . . . . . . . 40Figure 15: Reduced depth of cross section fora separate analysis in y-plane for e0z greaterthan 0,2 h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Figure 16: Simplified moment-curvaturerelationship for prestressed concretesections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 17: Steel stresses in anchorage zone ofpre-tensioned members . . . . . . . . . . . . . . . . . . 47

Figure 18: Final creep coefficient for normal-weight concrete in a dry indoor atmosphere . . 52Figure 19: Final creep coefficient for normal-weight concrete in an external humidatmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Figure 20: General shrinkage strain of normal-weight concrete for t approaching infinity . . . 54Figure 21: Drying shrinkage strain of normal-weight concrete for t approaching infinity . . . 54

Figure 22: Stress-strain curve for non-linearmethods of analysis and strain calculations . . 55Figure 23: Parabolic-rectangle stress-straincurve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Figure 24: Bilinear stress-strain curve . . . . . . 56Figure 25: Stress block . . . . . . . . . . . . . . . . . . 57

Figure 26: Stress-strain curve of reinforcementfor global analysis . . . . . . . . . . . . . . . . . . . . . . . 59Figure 27: Stress-strain curve of reinforcingsteel for section design . . . . . . . . . . . . . . . . . . . 60

Figure 28: Stress-strain curve of prestressingsteel for global analysis . . . . . . . . . . . . . . . . . . 61Figure 29: Theoretical stress-strain curve ofprestressing steel for section design . . . . . . . . 61

Figure 30: Possible distribution of strain inreinforcement and prestressing steel at theultimate limit state . . . . . . . . . . . . . . . . . . . . . . . 62Figure 31: Shear in cross sections of variableeffective depth . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Figure 32: Area of tension reinforcement, Asl,for determining longitudinal reinforcementratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 33: Truss system and notation formembers with shear reinforcement . . . . . . . . . 65

Figure 34: Connection of flange and web . . . 67Figure 35: Joint design . . . . . . . . . . . . . . . . . . 68

Figure 36: Torsion notation and equivalentsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 37: Design model for analysis ofpunching shear resistance . . . . . . . . . . . . . . . . 72Figure 38: Parts of critical section notsatisfying item (1) of subclause 10.5.2,relevant to punching shear . . . . . . . . . . . . . . . . 72Figure 39: Critical section around loadedareas located away from an unsupportededge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Figure 40: Critical section near an opening . . 73Figure 41: Critical section near anunsupported edge . . . . . . . . . . . . . . . . . . . . . . . 73Figure 42: Slab with column head with lH notgreater than 1,5 hH . . . . . . . . . . . . . . . . . . . . . . 74Figure 43: Slab with column head with lHgreater than 1,5 hH . . . . . . . . . . . . . . . . . . . . . . 75Figure 44: Approximate values of b . . . . . . . . 75Figure 45: Sections for analysis of punchingshear reinforcement . . . . . . . . . . . . . . . . . . . . . 76Figure 46: Effective zones for calculation ofminimum bending moments . . . . . . . . . . . . . . . 78Figure 47: Transverse tensile forces in acompression zone with confinement atconcentrated nodes . . . . . . . . . . . . . . . . . . . . . 80Figure 48: Nodal zone for analysis ofcompression nodes . . . . . . . . . . . . . . . . . . . . . . 80Figure 49: Nodal zone for analysis oftension-compression nodes . . . . . . . . . . . . . . . 81Figure 50: Node at reinforcement bend . . . . . 81Figure 51: Determining areas for partial arealoading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 52: Stress-number diagram forreinforcing and prestressing steel . . . . . . . . . . 84Figure 53: Zone in which reinforcement iseffective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Figure 54: Bonding conditions . . . . . . . . . . . . 95Figure 55: Shear reinforcement in theanchorage zone without transversecompression for bar diameters over 32 mm . . 99Figure 56: Anchorages and curtailment oflinks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 57: Overlapping bars . . . . . . . . . . . . . . 101Figure 58: Spacings s and s0 for calculationof a1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Figure 59: Shear reinforcement for lapjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Figure 60: Lap joints of welded reinforcingfabric (example) . . . . . . . . . . . . . . . . . . . . . . . . . 103Figure 61: Arrangement, minimum spacingand minimum concrete cover of bundles ofbars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 62: Anchorage of bundles of barswith widely spaced theoretical curtailmentpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 63: Anchorage of bundles of barswith closely spaced theoretical curtailmentpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 64: Lap joint in tension for bundles ofthree bars inclusive of a fourth bar . . . . . . . . . 105

Page 5DIN 1045-1 : 2001-07

Figure 65: Clear minimum spacing forpre-tensioned members . . . . . . . . . . . . . . . . . . 105Figure 66: Diagram of resisting tensile forces(tension envelope) and anchorage lengths inflexural members . . . . . . . . . . . . . . . . . . . . . . . . 108Figure 67: Combinations of stirrups and shearassemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109Figure 68: Permitted deviations from shearresistance diagram for conventionalbuildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Figure 69: Top reinforcement . . . . . . . . . . . . . 111Figure 70: Top and bottom cornerreinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 71: Edge reinforcement at the freeedges of slabs . . . . . . . . . . . . . . . . . . . . . . . . . . 112Figure 72: Arrangement of punching shearreinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 73: Floor joints for shear transmission(examples) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Figure 74: Loadbearing joints in two-wayspanning precast concrete floors incorporatingan in-situ topping (example) . . . . . . . . . . . . . . . 115Figure 75: Interlocking of panels . . . . . . . . . . . 116Figure 76: Support of floor slabs on precastconcrete walls . . . . . . . . . . . . . . . . . . . . . . . . . . 118Figure 77: Additional shear reinforcementat base of wall . . . . . . . . . . . . . . . . . . . . . . . . . . 118Figure 78: Transverse tensile stresses incompression joints . . . . . . . . . . . . . . . . . . . . . . 120Figure 79: Connection of secondary beams(plan view) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121Figure 80: Tying system providing foraccidental actions (floor plan) . . . . . . . . . . . . . 122

TablesTable 1: Partial safety factors for actions onstructures at ultimate limit state . . . . . . . . . . . 21Table 2: Partial safety factors to determine theresistance at ultimate limit state . . . . . . . . . . . 22Table 3: Exposure classes . . . . . . . . . . . . . . . . 24Table 4: Minimum concrete cover for corrosionprotection and tolerance on concrete cover . . 26Table 5: Maximum spacing of transverse ribsin combinations of ribbed floors/filler blockswithout topping . . . . . . . . . . . . . . . . . . . . . . . . . 31Table 6: Minimum concrete compressivestrength during prestressing with post-tensioned or unbonded members at time tj . . 44Table 7: Bond stress, as a function of actualconcrete compressive strength . . . . . . . . . . . . 46

Table 8: Density class, design dry densityand characteristic density of lightweightconcrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Table 9: Characteristic values of strengthand strain for normal-weight concrete . . . . . . 50Table 10: Characteristic values of strengthand strain for lightweight concrete . . . . . . . . . 51Table 11: Properties of reinforcement . . . . . . 58Table 12: Permitted welding processes, jointsand applications . . . . . . . . . . . . . . . . . . . . . . . . 59Table 13: Coefficients of friction androughness factors . . . . . . . . . . . . . . . . . . . . . . . 68Table 14: Moment coefficients and momentdistribution widths . . . . . . . . . . . . . . . . . . . . . . . 79Table 15: Ratio of bond strengths ofprestressing steel and reinforcing steel . . . . . 83Table 16: Parameters of stress-numberdiagrams for reinforcement . . . . . . . . . . . . . . . 83Table 17: Parameters of stress-numberdiagrams for prestressing steel . . . . . . . . . . . . 84Table 18: Requirements relating to crackcontrol and decompression . . . . . . . . . . . . . . . 87Table 19: Minimum requirement classes as afunction of exposure class . . . . . . . . . . . . . . . . 87Table 20: Limit reinforcing bar diameter as afunction of the design crack width . . . . . . . . . 89Table 21: Maximum reinforcing bar spacingas a function of the design crack width . . . . . 90Table 22: Coefficient to determine equivalentspan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94Table 23: Minimum mandrel diameters . . . . . 94Table 24: Minimum diameters of mandrels forwelded bent reinforcement . . . . . . . . . . . . . . . . 95Table 25: Design bond stress for reinforcingbars up to 32 mm in diameter with adequatebonding conditions . . . . . . . . . . . . . . . . . . . . . . 96Table 26: Coefficient aa as a function ofanchorage technique . . . . . . . . . . . . . . . . . . . . . 97Table 27: Coefficient a1 . . . . . . . . . . . . . . . . . . 101Table 28: Minimum lap lengths of transversebars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Table 29: Reinforcement ratio for calculationof minimum area of reinforcement . . . . . . . . . . 107Table 30: Minimum top reinforcement inprestressed members . . . . . . . . . . . . . . . . . . . . 107Table 31: Maximum longitudinal andtransverse spacing of hangers and shearassemblies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Table 32: Minimum thickness of loadbearingwalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

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ForewordThis standard has been prepared by Technical Committee Beton und Stahlbeton/Deutscher Ausschuss fürStahlbeton of the Normenausschuss Bauwesen (Building and Civil Engineering Standards Committee) on thebasis of the ENV 1992-1 *) series of standards, which has been adapted to produce this German Standard. Itcontains provisions relating to design and construction that deviate from ENV 1992-1, and includes provisionsfrom national standards superseded by this standard, where this has proved necessary to ensure the loadbearingcapacity, serviceability and durability of members designed and constructed according to this standard. **)The DIN 1045 standards series ***) comprises the following:Part 1: Design and constructionPart 2: Specification, properties, production and conformity of concrete (Application document for use withDIN EN 206-1) ***)Part 3: WorkmanshipPart 4: Supplementary specifications governing the production and conformity of precast elements

AmendmentsThis standard differs from the July 1988 edition of DIN 1045, December 1979 edition of DIN 4219-2, July 1988edition of DIN 4227-1, May 1984 edition of DIN 4227-2, and the February 1986 edition of DIN 4227-4 in thefollowing ways.

a) This standard has been developed as part of a restructuring programme involving standards on concrete.The standards listed above have been consolidated into the new 1045 standard series, each Part of whichdeals with a different aspect.b) The safety concept, structural analysis and design of concrete structures have been brought into line withthe current state of the art.c) Rules for the design of prestressed members and lightweight concrete members from DIN 4219-2 andthe DIN 4227 series are now incorporated here.d) A clear distinction is made between analysis for the ultimate limit state and analysis for the serviceabilitylimit state.e) The design procedures for shear have been revised.f) The standard now includes a fatigue analysis.g) Detailing specifications take into account the latest developments in construction materials technology.

Previous editionsDIN 1045: 1925-09, 1932-04, 1937-05, 1943xxx-04, 1959-11, 1972-01, 1978-12, 1988-07; DIN 4227: 1953xx-10,DIN 4227-1: 1979-12, 1988-07; DIN 4227-2: 1984-05; DIN 4227-4: 1986-02; DIN 4219-2: 1979-12.

IntroductionIn this standard, a distinction is made between principles and application rules, which differ from each otherin the following ways:

a) Principles contain general specifications, definitions and other details, that are of a binding nature, andrequirements and calculation models, deviations from which are not permitted unless explicitly stated.b) Application rules are generally recognized rules that obey the principles and meet the requirements theseformulate. Deviations from application rules are permitted provided they obey the principles and achieveserviceability, loadbearing capacity and durability equivalent to those required in this standard.In this standard, application rules are shaded grey to distinguish them from principles.

All dimensions are in millimetres.

1 Scope(1) This standard covers the design and construction of structures (including civil engineering structures)made from plain, reinforced and prestressed concrete with normal-weight and lightweight aggregate, ofstrength classes C12/15 to C100/115 and LC12/13 to LC60/66.The specifications of this standard apply to both normal-weight and lightweight concrete of the same strengthunless expressly stated that a specification covers lightweight concrete only.

*) European Prestandard.**) This English translation also includes subsequent amendments and addenda from a Corrigendum to

DIN 1045-1 (DIN 1045-1 Ber 1), issued in 2002.***) References to standards of the DIN 1045 series and to DIN EN 206-1 are to the July 2001 editions.

Page 7DIN 1045-1 : 2001-07

(2) This standard also applies to the design and construction of unreinforced walls in dwellings made fromlightweight concrete of strength class LC8/9. The concrete parameters required for their design shall then bederived from the provisions given in subclause 9.1.(3) This standard specifies requirements relating to the loadbearing capacity, serviceability and durability ofstructures. The serviceability analysis ensures the fitness-for-use, and to some extent the durability, of astructure. Theoretical limiting values relating to the durability of a structure are binding, the theoretical limitingvalues relating to its serviceability are given as guidance values.(4) This standard does not cover the following:

a) members made of no-fines lightweight concrete, aerated concrete, heavyweight concrete and memberscontaining reinforcement with a loadbearing function;b) special types of construction (e.g. mining shafts);c) thermal and sound insulation, and structural fire protection.

(5) For the design of certain civil engineering works made of concrete (e.g. bridges, segmental structures,dams, pressure vessels, offshore platforms, tanks), additional requirements may need to be taken into consid-eration.(6) Additional requirements and analyses are required in the design and construction of structures in earth-quake zones (e.g. paying particular attention to the ductility of members and reinforcement).(7) This standard does not contain provisions relating to the analysis of loadbearing capacity of lifting appli-ances, for which reference is made to Merkblatt ZH 1/17 Sicherheitsregeln für Transportanker und -systeme vonBetonfertigteilen

2 Normative referencesThis standard incorporates, by dated or undated reference, provisions from other publications. These norma-tive references are cited at the appropriate places in the text, and the titles of the publications are listed below.For dated references, subsequent amendment to or revisions of any of these publications apply to this standardonly when incorporated in it by amendment or revision. For undated references, the latest edition of thepublication referred to applies.

DIN 488-1 Reinforcement – Grades, properties and markingDIN 488-2 Reinforcing steel – Reinforcing steel bars – Dimensions and massDIN 488-3 Reinforcing steel – Reinforcing steel bars – TestingDIN 488-4 Reinforcing steel – Reinforcing steel fabric and wire – Design, dimensions and massDIN 488-5 Reinforcing steel – Reinforcing steel fabric and wire – TestingDIN 488-6 Reinforcing steel – InspectionDIN 488-7 Reinforcing steel – Verification of weldability of reinforcing steel bars – Test procedure and

evaluationDIN 1045-2 Plain, reinforced and prestressed concrete structures – Specification, performance, produc-

tion and conformity of concrete (Application document for use with DIN EN 206-1)DIN 1045-3 Plain, reinforced and prestressed concrete structures – WorkmanshipDIN 1045-4 Plain, reinforced and prestressed concrete structures – Supplementary specifications govern-

ing the production and conformity of precast concrete elementsDIN 1055-1 Action on structures – Mass and density of building materials and elements and materials to

be stockedDIN 1055-3 Action on structures – Self-weight and imposed loads for buildingsDIN 1055-8 Action on structures – Actions at erection stageDIN 1055-9 Action on structures – Accidental actionsDIN 1055-100 Actions on structures – Basis of design, safety concept and design rules *)DIN 4099-1 Welding of reinforcing steel – Execution of welding operationsDIN 4102-2 Fire behaviour of building materials and elements – Building elements – Concepts, require-

ments and testingDIN 4102-4 Fire behaviour of building materials and elements – Overview and design of classified building

materials, elements and membersDIN EN 206-1 Concrete – Part 1: Specification, performance, production and conformity

*) Currently at draft stage.

Page 8DIN 1045-1 : 2001-07

DIN EN ISO 4063 Welding and allied processes – Nomenclature of processes and reference numbers(ISO 4063 : 1998)

ENV 1992-1 Design of concrete structuresISO 1000 : 1992 SI units and recommendations for the use of their multiples and of certain other unitsISO 6707-1 : 1989 Building and civil engineering – Vocabulary – General termsISO 8930 : 1987 General principles on reliability for structures – List of equivalent termsDAfStb-Heft (DAfStb Code of practice) 525 Erläuterungen zur Reihe DIN 1045 – Tragwerke aus Beton, Stahl-beton und Spannbeton (Commentary on the DIN 1045 series – Plain, reinforced and prestressed concretestructures) *) 2)DAfStb-Richtlinie Belastungsversuche an Massivbauwerken (DAfStb Code of practice on loading tests on solidstructures) 2)DBV-Merkblätter (DBV Codes of practice) 3)

Abstandhalter (DBV Code of practice on spacers)Betondeckung und Bewehrung (DBV Code of practice on concrete cover and reinforcement)Rückbiegen von Betonstahl und Anforderungen an Verwahrkästen (DBV Code of practice on reinforcing steeland requirements relating to shelter boxes)

Merkblatt ZH 1/17 Sicherheitsregeln für Transportanker und -systeme von Betonfertigteilen (Safety rules oflifting appliances and systems used in the transport of precast concrete elements. 1)

3 Concepts and symbols3.1 ConceptsThe concepts in DIN 1055-100, ISO 6707-1 and ISO 8930 shall be applicable, together with the following.

3.1.1 Conventional buildingBuilding designed to sustain predominately static, uniformly distributed imposed loads up to 5 kN/m2, and insome cases concentrated loads up to 7 kN and loads from passenger cars.

3.1.2 Predominately static actionStatic action, or non-static action that may be regarded as static for the purposes of structural design (e.g.standard imposed load in multi-storey car parks, workshops, factories).

3.1.3 Predominately non-static actionSudden or recurrent action that often causes a change in internal forces and moments during the service lifeof a structure or member and that is not to be regarded as static for the purposes of structural design (e.g. loadsfrom cranes, crane tracks, forked-lift trucks, imposed loads on bridges).

3.1.4 Normal-weight concreteConcrete with a dry density over 2 000 kg/m³ but not more than 2 600 kg/m3.

3.1.5 Lightweight concreteConcrete of dense structure with a dry density between 800 kg/m3 and 2 000 kg/m3, produced using coarselightweight aggregate.

3.1.6 Heavyweight concreteConcrete with a dry density over 2 600 kg/m3.

3.1.7 Pre-tensioned memberTendon made from prestressing steel passed through a sheath cast in concrete and tensioned prior to concret-ing.

*) Currently at draft stage.1) Obtainable from the Hauptverband der gewerblichen Berufsgenossenschaften (Federation of German Indus-

trial Employers’ Liability Insurance Associations), Alte Heerstraße 111, 53757 Sankt Augustin, Germany.2) Obtainable from Deutscher Ausschuss für Stahlbeton im DIN (DIN Committee for Reinforced Concrete),

Burggrafenstr. 6, 10772 Berlin, Germany.3) Obtainable from Deutscher Beton- und Bautechnik-Verein e.V. (German Concrete and Concrete Technology

Association), Postfach 11 05 12, 10835 Berlin, Germany.

Page 9DIN 1045-1 : 2001-07

3.1.8 Post-tensioned memberTendon made from prestressing steel passed through a sheath cast in concrete, that is tensioned after theconcrete has hardened and is kept in position by anchorages. Hardening of grout injected into the sheathensures the bond between concrete and tendon.

3.1.9 Unbonded internal tendonTendon made from prestressing steel passed through a sheath cast in concrete, that is tensioned after theconcrete has hardened, is only connected to the structure by anchorages and redistributes loads in the concretein its curved regions.

3.1.10 Unbonded external tendonTendon made from prestressing steel, that is located outside the concrete but within the concrete envelope,is tensioned after the concrete has hardened and is connected to the structure by anchorages and deflectors.

3.1.11 Single strandCorrosion-protected steel strand, enclosed in a greased plastic sheath which allows its free movement in theaxial direction.

3.1.12 DeflectorElement with rounded end (e.g. concrete block, cross beam, steel member), over which an external tendon isdeflected.

3.1.13 Precast (concrete) elementElement that is produced in a factory or other location rather than in its final position. Precast elements mayneed to meet specific requirements (e.g. in respect of resistance to weathering).

3.1.14 Segmental structureStructure comprising individual precast elements (segments) in the direction of support and tensioned bymeans of tendons.

3.1.15 Sandwich panelPrecast element generally comprising a loadbearing layer and a facing layer, both of reinforced concrete, ateach side of a thermal insulating layer.

3.1.16 Composite elementElement comprising a precast element and in-situ topping combined with one or more other in-situ materials,with or without fasteners.

3.1.17 Unreinforced memberPlain concrete member or member with reinforcement in an amount below the required minimum.

3.1.18 Member predominately subjected to bendingMember which, at the ultimate limit state, has a relative eccentricity, ed/h, over 3,5.

3.1.19 Compression memberLinear or two-dimensional member subjected to compression, which, at the ultimate limit state, has a relativeeccentricity, ed/h, not greater than 3,5.

3.1.20 Beam (T-beam)Linear member predominately subjected to bending, with an effective span equal to not less than twice thedepth of section and a width of section or web width not more than four times the depth of section.

3.1.21 SlabPlane, two-dimensional member that is predominately subjected to bending at right angles to its surface, andwhose lesser span is equal to not less than twice its thickness while its width is not less than four times itsthickness.

3.1.22 ColumnLinear compression member whose greater cross-sectional dimension is not more than four times the smallerdimension.

3.1.23 Plate (wall)Plane, two-dimensional member whose greater cross-sectional dimension is not more than four times the lesserdimension and that is subjected to loads along its plane.

Page 10DIN 1045-1 : 2001-07

3.1.24 Deep beamPlane, plate-like member whose span is less than twice its depth of section and that is subjected to loads inits plane.

3.1.25 Concrete coverDistance between the surface of a reinforcing bar, pre-tensioned member or sheath of a post-tensionedmember and the nearest concrete surface.

3.1.26 DecompressionLimit state in which the cross section of the concrete is only just still in full compression when subjected to therelevant combination of actions.

3.2 Quantities and symbols3.2.1 Latin upper case symbols (without subscripts)A areaB flexural stiffnessC support reactionE modulus of elasticityF forceG shear modulusH horizontal forceI second moment of areaM momentN axial force; number of cycles (fatigue strength)P prestressing force; action due to prestressingQ variable actionR resistanceS first moment of area (section modulus)T torqueV shear force

C normal-weight concrete strength classD lightweight concrete bulk density classLC lightweight concrete strength class

3.2.2 Latin lower case letters (without subscripts)a distance; bearing widthb widthc concrete coverd effective depth; diameter; thicknesse eccentricityf strengthh member height, depth or thicknessi radius of gyrationk accidental angular displacement of tendons; number of elements in a storey; factor taking into account

uneven distribution of tensile stresses in concretel length; spanm moment per unit length or width; number of storeysn axial force per unit length; numberp transverse compressionr radiuss distance; bar spacingt time; thickness; ageu circumference, perimeterv shear force per unit lengthx depth (of compression zone); neutral axis depth; distance between tendon and end of componentz internal lever arm; height of wall

Page 11DIN 1045-1 : 2001-07

3.2.3 Greek symbols (without subscripts)a reduction factor taking into account long-term effects on concrete strength and for mutual conversion of

cylinder compressive strength and unconfined concrete compressive strength; angle of shear or punchingshear reinforcement to member axis; coefficient of linear thermal expansion; coefficient to determineequivalent span

b angle of spread of concentrated axial forces; shear component due to concentrated loads at supports;coefficient taking into account the effects of eccentricity

g partial safety factord ratio of redistributed to initial forces and momentse strainh correction factor for lightweight concrete, moment coefficientθ overall design angular displacement of tendons; angle of inclination of strutsf creep coefficient; factor taking into account second order effects on the loadbearing capacity of plain

concrete membersl slenderness ratio; ratio of distance between point of zero moment and maximum moment (after redistri-

bution to effective depth)m relative moment; coefficient of frictiony relative axial forcej ratio of bond strengths of prestressing steel and reinforcing steel; reduction factorr geometrical reinforcement ratio; densitys normal stressDs stress amplitudet shear stressD differential; gradient

3.2.4 Subscriptsb bondc concrete; compression; creepd design valuee eccentricityf flangeg permanent actionh member height or depthi idealized; control variablej control variablek characteristicl axial; longitudinallc lightweight concretem average, meanp prestressing; prestressing steelq variable actionr crack; relaxations reinforcement; shrinkaget tension; transverse; shearu limiting valuev actual concrete cover; verticalw web; wally yield strengthact actualcal calculatedcol columndir directeff effectivefat fatigue valueges overall valueind indirect

Page 12DIN 1045-1 : 2001-07

inf lower valuemax maximummin minimumnom nominal valuepl plasticred reduced valuereq requiredsup upper valuesurf surfacetot overall valueE stressEd design stressF action (force)G permanent actionL axialP prestressing force; action due to prestressingQ variable actionR system resistance; theoreticalRd design resistanceT transverse; torsiond redistribution (of forces)f creepm frictionI uncracked state of cross section (state I)II cracked state of cross section (state II)

3.2.5 Latin upper case symbols (with subscripts)AA upper deviation (area) from shear resistance diagramAc (sectional) area of a concrete memberAc,eff (sectional) area of zone of reinforcement of concrete section in which reinforcement is effectiveAc0 loaded areaAc1 theoretical area to accommodate resistance partial area loadAcrit area enveloped by critical section (punching shear)Act (sectional) area of tension zone in the concreteAct,ext area of concrete cover over stirrupsAE lower deviation (area) from shear resistance diagramAk (sectional) area enveloped by centrelines of wallsAload loaded area (punching shear)Ap (sectional) area of prestressing steelAs (sectional) area of reinforcementAs,act actual (sectional) area of reinforcementAsl (sectional) area of longitudinal tension or torsion reinforcementAs,req theoretically required (sectional) area of reinforcementAs,surf (sectional) area of top reinforcementAst (sectional) area of shear reinforcement parallel to member surfaceAsv (sectional) area of shear reinforcement normal to member surfaceAsw (sectional) area of shear or torsion reinforcementCEd design support reactionEc secant modulus of elasticity of normal-weight concreteEc0 tangent modulus of elasticity of normal-weight concrete at the origin of the stress-strain curve at

28 daysEcm mean secant modulus of elasticity of normal-weight concreteEd design stress; design strain; design internal force or momentElc secant modulus of elasticity of lightweight concreteElcm mean secant modulus of elasticity of lightweight concrete

Page 13DIN 1045-1 : 2001-07

Ep modulus of elasticity of prestressing steelEs modulus of elasticity of reinforcementFcd design concrete compressive strength; design axial force in flange due to bendingFcdj design axial force to be transferred via a jointFd design axial forceDFd variation in axial force acting in a one-sided flange sectionFEd sum of design vertical loadsFpd design strength of tendonFRdu resistance partial area loadFsd design tensile force in the reinforcementDFsd component of tensile force in longitudinal reinforcement due to shearFtd design transverse tensile forceGcm mean shear modulus of concreteHfd additional horizontal force used when designing bracing elementsIc second moment of area of a concrete sectionIT torsional moment of inertia of a concrete sectionIv warping torsional moment of inertia of a concrete sectionK1 factor used for calculating eccentricity e2 by second order theoryK2 factor used for calculating curvature in the critical sectionMEd design bending moment at supportMEds design applied moment referred to centroidal axis of reinforcementMp,dir statically determinate component of moment due to prestressingMp,ind hyperstatic component of moment due to prestressingMRd design resistance momentMy flow momentMu failure momentNba design axial force to take account imperfections in wallsNbal resistance axial compression at maximum moment capacity of sectionNbc design axial force to take account imperfections in columnsNEd design axial force in a concrete sectionNEd,m mean design axial force in a concrete sectionNEd,x mean design axial force due to prestressing or other actions that act in section of area Ac,x

NEd,y mean design axial force due to prestressing or other actions that act in section of area Ac,y

NRd design resistance axial forceNud design ultimate loadbearing capacity of a cross section subjected to axial compressionP0 prestressing force from pretensioned tendonPd design prestressing forcePk characteristic prestressing forcePk,inf minimum characteristic prestressing forcePk,sup maximum characteristic prestressing forcePm0 mean prestressing force immediately after stressing or transfer of load to concretePmt mean prestressing forceDPm loss of prestress due to frictionRd design resistanceTEd design applied torqueTRd design resistance torqueVccd design shear component in compression zoneVEd design shear forceVEd,max design maximum shear forceVEd,min design minimum shear forceVEd0 basic design shear force effective in the cross sectionVEd,T shear force due to torsionVEd,T+V shear force in wall due to coexistent torsion and shearVpd design shear component in prestressing steel at ultimate limit state

Page 14DIN 1045-1 : 2001-07

VRd design shear resistance

VRd,ct design shear resistance of a member without shear reinforcement

VRd,max design shear resistance, limited by strength of struts

VRd,sy design shear resistance, limited by the capacity of shear reinforcement

Vtd design shear component in tension zone (reinforcement)EcmIc nominal flexural stiffness

EcmIv warping stiffness

GcmIT torsional stiffness (see St. Venant’s theorem)

VEd,i/Zi shear flow

3.2.6 Latin lower case symbols (with subscripts)a l magnitude of shift of tension envelope

as (sectional) area of reinforcement passing through a joint

as,req required area of reinforcement in the section considered

as,act actual area of reinforcement in the section considered

asw area of additional shear reinforcement at base of wall

bc width of rectangular loaded area

beff effective flange width of a T-beam

bi actual flange width

bf flange width

bw web width/thickness

bv structurally relevant web width of T-beams of variable depth

cmin minimum concrete cover (to reinforcement)cnom nominal concrete cover

cv actual concrete cover

Dc margin to allow for accidental deviations from specified concrete cover

dbr mandrel diameter

dg maximum aggregate particle size (denoted Dmax in DIN EN 206-1)

dh external sheath diameter

dH overall depth of column head and slab

dp nominal or equivalent diameter of tendon, strand or wire

ds reinforcing bar diameter

ds* limiting diameter of reinforcing bars

dsm mean reinforcing bar diameter

dsq diameter of transverse reinforcing barsdsV equivalent diameter of bundles of bars

dx effective depth along x-axis

dy effective depth along y-axis

e0 first order design eccentricity

e1 sum of design and accidental eccentricities

e2 additional eccentricity due to second order effects

ea additional accidental eccentricity

ed design eccentricity

etot overall eccentricity

ef eccentricity due to creep

f0,2k characteristic 0,2 % proof stress of reinforcing steel

fbd design bond stressfbp bond stress along the transmission length of pre-tensioned members

fc concrete compressive strength

fcd,fat design unconfined concrete compressive strength for fatigue analysis

fcd design unconfined concrete compressive strength

Page 15DIN 1045-1 : 2001-07

fck,cube characteristic concrete compressive strength at 28 days determined by testing cubes **)fck,cyl characteristic concrete compressive strength at 28 days determined by testing cylinders *) (simplified

to fck in this standard)fckj cylinder concrete compressive strength at the time the creep-inducing stress is appliedfcm mean cylinder compressive strengthfcmj actual cylinder compressive strength during prestressingfcmj,min minimum cylinder compressive strength during prestressingfcR theoretical mean cylinder compressive strength (non-linear analysis)fct axial tensile strength of concretefct,0 tensile strength of concrete to which ds* values are relatedfct,eff effective tensile strength of concretefct,sp splitting tensile strength of concrete (denoted ftk in DIN EN 206-1)fctk;0,05 characteristic 5 % quantile of axial tensile strength of concretefctk;0,95 characteristic 95 % quantile of axial tensile strength of concretefctm mean axial tensile strength of concreteffatk characteristic fatigue strength of concreteflck characteristic cylinder compressive strength of lightweight concrete at 28 daysflck,cube characteristic cube compressive strength of lightweight concrete at 28 daysflckt;0,05 characteristic 5 % quantile of axial tensile strength of lightweight concreteflckt;0,95 characteristic 95 % quantile of axial tensile strength of lightweight concreteflcm mean cylinder strength of lightweight concreteflctm mean axial tensile strength of lightweight concretefp actual tensile strength of prestressing steelfp0,1 0,1 % proof stress of prestressing steelfp0,1k characteristic 0,1 % proof stress of prestressing steelfp0,1R theoretical mean 0,1 % proof stress of prestressing steel (non-linear analysis)fpk characteristic tensile strength of prestressing steelfpR theoretical mean tensile strength of prestressing steel (non-linear analysis)fR projected rib factorft actual tensile strengthftk characteristic tensile strength of reinforcing steelftk,cal characteristic tensile strength of reinforcing steel for design purposesftR theoretical mean tensile strength of reinforcing steel (non-linear analysis)fy actual yield strength of reinforcing steelfyd design yield strength of reinforcing steelfyk characteristic yield strength of reinforcing steelfyR theoretical yield strength of reinforcing steel (non-linear analysis)h0 effective depth of sectionhc depth of rectangular loaded areahf flange thicknesshges overall height of structurehH depth of column headhred reduced heightht depth of tension zone in section or subsection prior to initial crackingk1 factor used when determining minimum reinforcement to limit crack widthk1, k2 stress exponentska factor taking into account the effects at transition from column to the slabkc factor taking into account the effect of stress distribution in tension zonel0 effective span; equivalent height of compression memberl0t length of compression flange between lateral supportslb basic anchorage length of reinforcementlb,ind required anchorage length for conditions of indirect support

*) ‘cylinder compressive strength’, for short.**) ‘cube compressive strength’, for short.

Page 16DIN 1045-1 : 2001-07

lb,min minimum anchorage length of reinforcementlb,net required anchorage length of reinforcementlba anchorage length of pre-tensioned memberslbp transmission length of pre-tensioned memberslbpd design transmission length of pre-tensioned memberslc diameter of circular loaded arealcol height of column/tendon/compression member between idealized points of restraintleff effective spanleff,1, leff,2 effective span of floor slabs at each side of a jointleff,i effective length of end bay normal to a tielH distance of column face to edge of column headli equivalent spanln clear distance between faces of supportslp,eff dispersion length of a pre-tensioned memberlr transmission length of a pre-tensioned member at which first crack occursls lap lengthls,min minimum lap lengthlw width of zone containing punching shear reinforcementlx distance of section from beginning of anchorage length of tendonmEd design bending moment resistance per unit lengthnEd design axial force normal to joint, per unit lengthrcrit distance of critical section from centroid of loaded arearinf minimum coefficient taking into account scatter of prestressing forcersup maximum coefficient taking into account scatter of prestressing forcerj distance of column from centroid of systems0 distance of reinforcement from edge of membersL centre-to-centre distance of longitudinal ribssl spacing of longitudinal reinforcing barssmax maximum spacing of links, bent-up bars, shear inserts, etc.sq spacing of welded-on transverse reinforcing barssr,max maximum final crack spacingsr,max,x maximum final crack spacing along x-axissr,max,y maximum final crack spacing along y-axissT spacing of transverse ribssw spacing of shear, punching shear or torsion reinforcement along member axist0 age of concrete at beginning of loadingteff effective wall thicknesstj time of prestressingucrit critical perimeteruk perimeter of area Ak

vEd design shear resistancevRd design shear resistance per unit lengthvRd,c component of design shear resistance due to concretevRd,ct design shear resistance in joints of composite membersvrd,ct design shear resistance along critical perimeter of slab without punching shear reinforcementvrd,ct,a design shear resistance along the perimeter of slab outside zone containing punching shear rein-

forcementvRd,max maximum design shear resistance per unit length of critical perimetervRd,sy design shear resistance per unit length of inner sections for slab with punching shear reinforcementwk design crack widthxd depth of compression zone after redistribution of forceszcp distance between centroid of concrete section and tendons(1/r) curvature

Page 17DIN 1045-1 : 2001-07

3.2.7 Greek symbols (with subscripts)a1 coefficient for calculation of lap length of reinforcementa2 coefficient taking into account the area of reinforcing fabricaa angle of inclination to target axis; coefficient allowing for the anchorage techniqueac reduction factor for calculation of concrete compressive strength due to transverse tensile stressae ratio of elastic moduli of concrete and steel (ES/Ecm)al coefficient for calculation of transmission length of pre-tensioned membersan reduction factor for inclination taking into account combined action of adjacent, vertical loadbearing

membersap ratio of moduli of elasticity of concrete and prestressing steelb roughness factorbcc(t0) coefficient taking into account subsequent hardening of concretegc partial safety factor for concretegcq partial safety factor allowing for the scatter in material properties for concrete of strength class

C55/67 or LC 55/60 and highergEd,fat partial safety factor taking into account uncertainty in fatigue analysisgF partial safety factor for actionsgF,fat partial safety factor taking into account actions in fatigue analysisgG partial safety factor for permanent actionsgP partial safety factor for actions associated with prestressinggQ partial safety factor for variable actionsgR partial safety factor for the resistance of the system (non-linear analysis)gs partial safety factor for reinforcing or prestressing steelgs,fat partial safety factor for reinforcing and prestressing steel in fatigue analysisec compressive strain in the concreteec1 compressive strain in concrete on reaching fc

ec1u compressive strain in concrete after reaching fc

ec2 compressive strain in concrete on reaching limit strengthec2u maximum compressive strain in concreteecas general shrinkage strain in concreteecc creep strain in the concreteecds drying shrinkage strain in concreteecm mean strain in concreteecs shrinkage strain in concreteecsr final shrinkage strain in the concreteecu theoretical ultimate compressive strain in concreteelc compressive strain in lightweight concreteelcu theoretical ultimate compressive strain in lightweight concreteep strain in prestressing steelep

(0) initial pre-strain in prestressing steeles strain in the reinforcementesm mean strain in the reinforcementesu theoretical ultimate strain in the reinforcementeuk characteristic strain in the reinforcement at maximum loadeyd design yield strain in the reinforcementhE reduction factorh1 correction factorhp correction factorhx moment coefficient along x-axishy moment coefficient along y-axisθE angular displacement due to plastic deformationθfat angle of inclination of struts in fatigue analysisθpl,d design permitted angular displacement due to plastic deformation¡a factor taking into the effects at the transition from a column to a slab¡s factor taking into account the effect of depth of member on the effectiveness of the reinforcement

Page 18DIN 1045-1 : 2001-07

lmax slenderness ratio from which a compression member counts as being slenderlcrit critical slenderness ratioreff effective reinforcement ratiorl longitudinal reinforcement ratiorlx ratio of longitudinal reinforcement along x-axisrly ratio of longitudinal reinforcement along y-axisrtot geometric reinforcement ratiorw shear, punching shear or torsion reinforcement ratiosc stress in the concretescd design axial stress in concrete sectionscd,max design maximum axial stress in concretescd,max,equ maximum equivalent stress amplitudescd,min design minimum axial stress in concretescd,min,equ minimum equivalent stress amplitudescd,x design axial stress in concrete along x-axisscd,y design axial stress in concrete along y-axisscg stress in the concrete at the tendons due to quasi-permanent combinations of actionsscp0 initial stress in the concrete during prestressingsNd stress normal to jointsp stress in prestressing steelsp0 maximum stress transferred to steel during prestressingspd design tension in prestressing steelspg0 initial stress due to prestressing and permanent actionspm0 stress in the steel after transmission of prestressing force to the concretespt actual tension in prestressing steel at lr

sRd,max maximum design compressive strength of strutss stress in the reinforcementss2 tension in reinforcing steel or increase in tension in prestressing steel in state IIsst stress in reinforcement calculated on the basis of a cracked sectionDsp change in stress in prestressing steelDsp,c+s+r loss of prestress due to concrete creep and shrinkage and relaxation of prestressing steelDspd change in design tension in prestressing steelDspk change in characteristic tension in prestressing steelDspr reduction in stress in prestressing steel due to relaxationDsRsk stress amplitude for (reinforcing or prestressing) steelDsRsk(N*) stress amplitude for N* load cyclesDss,equ equivalent stress amplitudeDss,max maximum stress amplitude when member is subjected to a combination of actions affecting fatiguej1, j2 reduction factors (fatigue analysis)j1 ratio of bond strengths of reinforcing and prestressing steelf (r, t0) final creep coefficient of concretef (t, t0) creep coefficient of concrete

Page 19DIN 1045-1 : 2001-07

3.3 SI units(1) SI units shall be used as specified in ISO 1000.(2) The following units should be used in calculations:

a) for length: m, mm;b) for area: cm2, mm2;c) for forces, loads and actions: kN, kN/m, kN/m2;d) for mass: kN/m3;e) for stresses and strength: N/mm2;f) for moments: kNm.

4 Documentation4.1 Scope(1) The documentation is to include fully detailed construction drawings, the design analysis plus, if required,a project specification, general building inspectorate approvals (‘agréments’, for short) and test certificates.(2) The documentation shall also include details of any prestressing work (i.e. the type of work and its sched-uled commencing date, the technique used and the stressing programme and schedule.

4.2 Drawings4.2.1 General requirements(1) There shall be clear and succinct drawings of members, reinforcement, tendons and any embedded parts,together with all relevant dimensions. The information shown on the drawings shall correspond to that in thestructural analysis and shall cover all dimensions necessary for the construction of the members and forchecking design calculations.(2) Reference shall be made to other relevant drawings. If a drawing is subsequently altered, all drawingsaffected shall be amended correspondingly.(3) Reinforcement drawings shall particularly include the following information:

a) the strength class of the concrete, exposure classes and other requirements as specified in subclause6.2 of the present standard and of DIN 1045-2;b) the grades of reinforcing steel and prestressing steel (cf. subclauses 9.2 and 9.3);c) the number, nominal size, type and position of reinforcing bars, spacing of bars, overlaps and anchoragelengths, position, dimensions and types of welds (with details of filler materials), types and location offasteners, vibration channels, location of access doors;d) details of prestressing procedures, number, type and position of tendons, number, type and position ofanchorages and anchorage and coupler members, number, nominal size, type and position of the relevantreinforcement, type and nominal size of sheaths, details of grout;e) the required diameter of mandrels for bending reinforcing bars;f) the means of keeping reinforcement and tendons in place (e.g. type and arrangement of spacers) andarrangement, dimensions and workmanship of chairs for the top layer of reinforcement and tendons;g) the actual concrete cover, cv, derived from the nominal concrete cover, cnom, and the tolerance onconcrete cover, Dc (cf. subclause 6.3);h) joint design;i) any special measures with regard to quality assurance. 4)

(4) If precast elements are used, the following information shall also be given:a) the type of elements;b) the type or item reference number and the self-weight of the elements;c) the minimum compressive strength of the concrete during transport and erection;d) the type, position and permitted direction of action of lifting appliances for transport and erection, thepoints of support, and bearings;e) any structural detailing for impact protection;f) additional in-situ reinforcement, shown separately.

4.2.2 Erection drawings for precast elementsFor structures that include precast elements, layout drawings of the elements shall be made available for on-site work, with the relevant reference numbers and an item reference list. The layout drawings shall also includedetails of the bearing depths required for assembly purposes and the supports required by the precast ele-ments.

4) See, for example, DBV-Merkbatt Betondeckung und Bewehrung.

Page 20DIN 1045-1 : 2001-07

4.2.3 Drawings of formwork and falseworkErection drawings shall be provided for falsework and formwork requiring structural analysis or for formworkwhose sides are required to sustain high lateral pressure from fresh concrete.

4.3 Design analysis(1) In the design analysis, the structure and the load transfer mechanisms shall be described. The loadbearingcapacity and the serviceability of the structure and its members shall be set out in a clear and easily verifiablemanner. Results obtained by computation (e.g. action-effects, deformations) shall be represented in schematicform.(2) It is left to discretion whether action-effects are determined by elastic analysis (see clause 8). Design shallbe carried out in accordance with the principles of this standard. Any provisions deviating from this standard,and any unusual formulae that are not given in this standard, shall be accompanied by an indication of theirsource if generally available; otherwise, the derivations shall be given in sufficient detail to enable verificationthat these are correct.(3) For structures that include precast elements, details shall be given of the transport and erection proceduresfor the elements.

4.4 Specification of works(1) Information that is required for construction work or for checking drawings or design calculations, butwhich cannot be directly taken from the documents covered by subclauses 4.2 and 4.3, shall be provided inthe specification of works. Should fair-faced concrete be stipulated, appropriate directions shall be included.(2) For structures that include precast elements, information shall be provided on erection procedures (includ-ing any temporary propping and hangers), on their alignment and any temporary situations affecting load-bearing capacity and serviceability during erection. Special requirements relating to storage of precast ele-ments shall be specified in the drawings and erection instructions.

5 Safety concept5.1 General(1) The rationale behind this standard is the safety concept specified in DIN 1055-100. Subclauses 5.2 to 5.4specify additional, more specific, requirements. Information on actions is given in the DIN 1055 series ofstandards.(2) In order to ensure adequate reliability, the structure shall be analysed for ultimate limit state and service-ability limit state (see subclauses 5.3 and 5.4) and shall be designed in accordance with the specifications givenin clauses 12 and 13, taking into account the provisions of clause 6 to ensure durability.(3) The analyses for ultimate limit state and serviceability limit state shall take into consideration the antici-pated loads both during construction work and after its completion. Where precast elements are used, loadsfrom storage, transport and erection shall be included.

5.2 Design resistance(1) The characteristic values of material properties used in this standard are given in clause 9.(2) The design resistance, Rd, shall be obtained by means of equation (1) or (2), depending on which methodis used to determine action-effects.

a) Where the action-effects are determined by the linear-elastic method of analysis, as specified insubclauses 8.2 and 8.3, or by plastic analysis, as specified in subclause 8.4, Rd shall be obtained as follows:

fck fyk ftk,cal fp0,1k fpk Rd = RŒa ; ; ; ; › (1) gc gs gs gs gs

wherefck is the cylinder compressive strength of concrete;fp0,1k is the characteristic 0,1 % proof stress of prestressing steel;fpk is the characteristic tensile strength of prestressing steel;ftk,cal is the characteristic tensile strength of reinforcing steel for design purposes;fyk is the characteristic yield strength of reinforcing steel;a is a reduction factor (from subclause 9.1.6);gc and gs are the partial safety factors for concrete and reinforcing or prestressing steel, respectively (from

subclause 5.3.3).

Page 21DIN 1045-1 : 2001-07

b) Where action-effects are determined by the non-linear method described in subclause 8.5, the followingshall apply:

1 Rd = RŒfcR; fyR; ftR; fp0,1R; fpR› (2) gR

wherefcR, fpR, ftR, fyR, and fp0,1R are the theoretical mean strengths or 0,1 % proof stress of concrete, reinforcing

steel and prestressing steel;gR is the partial safety factor for the resistance of the system.

(3) In non-linear analysis, fcR, fyR, fpR, and gR shall be given the values from subclause 8.5.1.(4) When analysing existing structures, design resistance may also be determined empirically. 5)

5.3 Ultimate limit state5.3.1 General(1) Ultimate limit states are limit states corresponding to the maximum loadbearing capacity of a structure,i.e. states beyond which collapse or other forms of structural failure will occur.(2) The specifications of this standard apply to the analysis of the capacity of a structure to resist failure dueto rupture or strain beyond the specified limits in a cross section, joint or connection, or general failure of thesystem.(3) DIN 1055-100 shall apply with regard to analysis of the positional stability of structures (e.g. capacity toresist uplift, tilting or floating).

5.3.2 Ductility(1) Failure of a member due to the occurrence of initial cracking in the absence of prior warning shall beprecluded.(2) The requirement in item (1) above shall be deemed met by reinforced and prestressed concrete if thesecontain a minimum reinforcement as specified in 13.1.1.(3) Alternatively, prestressed concrete in structures subject to regular inspection shall also be deemed tosatisfy the requirement in item (1) above if the tendons can be readily accessed to check for their integrity usinga suitable non-destructive test or monitoring method.(4) Unreinforced linear members of rectangular cross section shall be deemed to satisfy the requirement initem (1) above if the ed/h ratio for the relevant combination of actions at the ultimate limit state is less than 0,4.

5.3.3 Partial safety factors for actions and resistances at ultimate limit state(1) The partial safety factors specified in DIN 1055-100 for actions on buildings shall also apply in thisstandard. They are given in table 1.

Table 1: Partial safety factors for actions on structures at ultimate limit state

5) Cf. DAfStb-Richtlinie für Belastungsversuche an Massivbauwerken.

Effects

1 2 3

Partial safety factor for

permanent actions, variable actions, prestressing,gG gQ gP

a) b)

1 Favourable 1,05 0,5 1,0

2 Unfavourable 1,35 1,5 1,0

a) This applies on condition that prestressing is considered as an action due to anchorageforces and forces associated with changes in direction or as an action-effect (cf. sub-clause 8.7.1).

b) See subclause 8.7.5 for the partial safety coefficient taking into account the increase instress in the prestressing steel of unbonded tendons.

(2) For the fatigue analysis as specified in subclause 10.8, partial safety factors gF,fat and gEd,fat shall be takento be equal to unity.(3) In linear-elastic analysis using the stiffness of uncracked sections and the mean secant modulus of elas-ticity, Ecm, a partial safety factor, gQ, equal to unity may be assumed for restraint.

Page 22DIN 1045-1 : 2001-07

(4) In the analysis of precast elements for the ultimate limit state, the partial safety factors for permanent andvariable actions, gG and gQ, respectively, for coexistent bending and axial force, shall both be taken to be 1,15,taking into due account any effects of crane transport and formwork adhesion.(5) For one and the same independent permanent action (e.g. self-weight), either the upper or lower value shallbe used for gG for all the spans of continuous beams and slabs. This does not apply to the analysis of positionalstability as specified in DIN 1055-100.(6) Partial safety factors for determining resistance shall be taken from table 2.

Table 2: Partial safety factors to determine the resistance at ultimate limit state

Design situation

1 2 3

Partial safety factor for

concrete,reinforcing or system

gca) b)

prestressing resistance,steel, gs or gs,fat gR

1 Permanent and 1,5 1,15temporary

2 Accidental 1,3 1,05 See

3Fatigue analysis as

1,5 1,15

subclause 8.5.1.

in subclause 10

a) See item (9) below for concrete of strength classes C55/67 and LC55/60 orhigher.

b) See item (8) below for unreinforced members.

(7) Where precast elements are subject to factory production control and third-party inspection, the partialsafety factor for the concrete, gc, may be reduced to 1,35 if the elements once cast are checked for concretestrength and those found below grade are rejected. The procedures in such cases shall be established by theresponsible inspection authorities.(8) Due to the poor deformation capacity of plain concrete, unreinforced members shall be assumed to havea gc value of 1,8 for permanent and variable and 1,55 for accidental design situations. These values apply forboth compression and tension.(9) To allow for the scatter of material properties, the factor gcq shall be multiplied by partial safety factor gcfor concrete of strength classes C55/67 and LC55/60 or higher, which is given by equation (3): 1gcq = ” 1,0 (3)

fck

1,1 –

500

with fck in N/mm2.

5.3.4 Combinations of actions and design situations(1) The design situations to be assumed for ultimate limit state analyses are specified in DIN 1055-100. Theindependent actions on the structure shall be deemed to be combined, depending on the design situation,making reference to DIN 1055-100 where combinations of actions are to be considered.(2) Subclause 10.8.3 specifies the combinations of actions to be assumed in the fatigue analysis.

5.4 Serviceability limit states5.4.1 General(1) Serviceability limit states are states beyond which specified service requirements are no longer met or theloadbearing capacity in the context of this standard is no longer ensured.(2) The analyses for the serviceability limit state cover stress limitation as specified in subclause 11.1, crackcontrol as specified in subclause 11.2 and control of deformation as specified in subclause 11.3.(3) Other limit states (e.g. for vibration or impact) may be of importance for certain structures but are not dealtwith in this standard.(4) The concept of analysis for the serviceability limit state, and the relevant design situations and combina-tions of actions for this state shall be that specified in DIN 1055-100.

Page 23DIN 1045-1 : 2001-07

(5) In analyses for the serviceability limit state, gF shall generally be assumed to be equal to unity (i.e. therepresentative value of an action or action-effect is also the design value used in calculations).

5.4.2 Requirement classes(1) As a basis for serviceability limit state analyses, minimum requirement classes shall be specified for theconstitutive members of the structure as a function of the combinations of environmental conditions andmembers given in table 3. The minimum requirement classes, to be specified by the client or building authorities,shall be those given in table 19.(2) The requirement classes specified for structures at the erection stage may deviate from those specifiedfor completed structures provided the durability of the members is not adversely affected.

6 Durability6.1 General(1) A structure is deemed to be of adequate durability if, over its projected period of use and if suitablymaintained over this period, it continues to fulfil its function with regard to loadbearing capacity and service-ability without this having a major negative effect on the properties relevant to its use.(2) The structure is deemed to be of adequate durability if, as well as meeting the requirements associatedwith the ultimate limit state and serviceability limit state analyses and the specifications of clauses 12 and 13,it satisfies the requirements of clause 6, the requirements relating to the composition and properties of concreteas specified in DIN EN 206-1 and DIN 1045-2, and the workmanship requirements as specified in DIN 1045-3.

6.2 Exposure classes and minimum concrete strength(1) Environmental conditions within the meaning of this standard are determined by chemical and physicalinfluences to which the structure as a whole, isolated members, the reinforcing or prestressing steel, and theconcrete itself are exposed, and that are not taken into direct account in the ultimate limit state and service-ability limit state analyses.(2) Each member shall be classified as a function of the environmental conditions to which it is directlyexposed (cf. table 3). Members may be exposed to more than one environmental condition, in which case theexposure classes shall be combined.(3) Table 3 shows the minimum concrete strength classes associated with each exposure class. The design ofmembers shall be based on the highest permitted minimum concrete strength class associated with each expo-sure class, established taking into consideration item (2). For post-tensioned or unbonded members, however,concrete of a strength class below C25/30 for normal-weight concrete and LC25/28 for lightweight concrete shallnot be used, while for pre-tensioned members the strength class shall not be below C30/37 or LC30/33.To ensure the durability of the concrete, additional requirements relating to its composition and properties, asspecified in DIN EN 206-1 and DIN 1045-2, may have to be considered.(4) Additional measures may be required to protect against other aggressive influences occurring during use(cf. DIN EN 206-1 and DIN 1045-2).

6.3 Concrete cover(1) Reinforcement shall have a minimum concrete cover to ensure its protection against corrosion and thetransfer of bonding forces.Specifications for fire-resistant members shall be taken from DIN 4102-2 and DIN 4102-4.(2) Reinforcement in members classed as unreinforced and top reinforcement as specified in subclause 13.2.5shall satisfy the requirements relating to concrete cover even if the reinforcement is not taken into considerationin the ultimate limit state and serviceability limit state analyses.(3) To ensure corrosion protection, the concrete cover shall be not less than the minimum cover, cmin, givenin table 4 as a function of the exposure class (cf. table 3). Where more than one type of environmental conditionsapply, the exposure class with the most stringent requirements shall be taken.(4) To ensure an adequate bond, cmin shall be equal to not less than the following:

– the bar diameter, ds, or the equivalent diameter of a bundle of bars, dsV;– the 2,5-fold nominal diameter of a pre-tensioned strand, dp, or the three-fold diameter of a pre-tensionedribbed wire;– the external diameter of a sheath for a post-tensioned member.

(5) Where prestressed concrete members have internal unbonded tendons, the minimum concrete cover inanchorage zones and in the vicinity of the free length of the enclosed tendon shall be taken from the relevantagrément.(6) Lightweight concrete members (except those in exposure class XC1) shall have a minimum concrete coverat least 5 mm more than the maximum particle size of the aggregate. The minimum cover to ensure adequatecorrosion protection and bond shall be as given in table 4 and item (4) above, respectively.

Page 24DIN 1045-1 : 2001-07

Table 3: Exposure classes

1 2 3 4

Minimum

ClassDescription of

Examples where exposure classes may occurconcrete

environment strengthclass

1 No risk of corrosion or attack

Members without reinforcement in a non-aggressiveX0 No risk of attack environment (e.g. unreinforced foundations not subjected

C12/15 or

to freeze/thaw attack, unreinforced interior members)LC12/13

2 Corrosion of reinforcement induced by carbonation a)

Members in rooms with normal air humidity (includingXC1 Dry or permanently wet kitchens, bathrooms, laundries in residential buildings); C16/20 or

members permanently submerged in water LC16/18

XC2 Wet, rarely dry Parts of water tanks; foundation membersC16/20 orLC16/18

Members to which outside air constantly or frequentlyhas access (e.g. open shed-type buildings); rooms with

XC3 Moderately humid a highly humid atmosphere (e.g. commercial kitchens, C20/25 or

baths, laundries, damp rooms of indoor swimming LC20/22

pools, byres)

XC4 Cyclic wet and dryExternal concrete members directly exposed to rain; C25/30 ormembers in fresh water tidal zone LC25/28

3 Corrosion of reinforcement, induced by chlorides other than from sea water

XD1 Moderately humid Spray zones of trafficked areas; private garages C30/37 c) orLC30/33

XD2 Wet, rarely drySwimming baths and salt water baths; members C35/45 c) orexposed to industrial waters containing chlorides LC35/38

XD3 Cyclic wet and dryMembers exposed to spray from traffic areas treated C35/45 c) orwith de-icing agents; parking decks without topping b) LC35/38

4 Corrosion of reinforcement, induced by chlorides from sea water

Exposure to airborneXS1 salt but no direct External members near to the coast

C30/37 c) or

contact with sea waterC30/33

XS2 Submerged Permanently submerged members in harbours C35/45 c) orLC35/38

XS3 Tidal, splash and spray C35/45 c) orzones

Quay wallsLC35/38

For a), b) and c), see page 25.

(continued)

Page 25DIN 1045-1 : 2001-07

1 2 3 4

Minimum

ClassDescription of

Examples where exposure classes may occurconcrete

environment strengthclass

5 Freeze/thaw attack, with or without de-icing agents

XF1 Moderate water satura- External members C25/30 ortion, no de-icing agent LC25/28

Moderate water satura- Spray and splash zones of traffic areas, with de-icing C25/30 orXF2 tion, with de-icing agent agent (other than XF4); sea water spray zone LC25/28or sea water

XF3High water saturation,

Open water tanks; members in fresh water tidal zoneC25/30 or

no de-icing agent LC25/28

Traffic areas treated with de-icing agents; predominatelyHigh water saturation, horizontal members exposed to spray from traffic areas

C30/37 orXF4 with de-icing agent or treated with de-icing agents, parking decks withoutLC30/33sea water topping b); members in sea tidal zone; scraper raceways

in sewage treatment plant

6 Exposure of concrete to chemical attack d)

XA1Slightly aggressive Tanks in sewage treatment plant; liquid manure C25/30 orchemical environment containers LC 25/28

Moderately aggressiveXA2 chemical environment,

Concrete members in contact with sea water; members C35/45 c) or

marine structuresin aggressive soil LC35/38

Highly aggressive Chemically aggressive industrial waste water treatment C35/45 c) orXA3 chemical environment plant; silage containers and animal feeding troughs; LC35/38cooling towers with flue gas disposal

7 Exposure of concrete to wear

XM1 Moderate wearMembers of industrial structures, subjected to traffic C30/37 c) orfrom vehicles with pneumatic tyres LC30/33

Members of industrial structures subjected to trafficC30/37 c) orXM2 Considerable wear from fork-lift trucks with pneumatic tyres or solid rubberLC30/33wheels

Members of industrial structures subjected to trafficfrom fork-lift trucks with rubber or steel wheels;

XM3 Extreme wear hydraulic structures in agitated waters (e.g. stillingC35/45 c) or

basins); surfaces subjected to frequent traffic fromLC35/38

tracklaying vehicles

a) Data on moisture relate to the condition within the concrete cover. It may generally be assumed that theconditions in the concrete cover are the same as the ambient conditions to which the member is exposed.However, this is not necessarily the case if there is a vapour barrier between the concrete and its environ-ment.

b) Such decks require additional surface protection, such as a crack-covering coating.c) One strength class down if aerated concrete is selected to cater for class XF.d) See DIN 206-1 and DIN 1045-2 for the limiting values of concrete composition and properties.

Table 3 (concluded)

Page 26DIN 1045-1 : 2001-07

Table 4: Minimum concrete cover for corrosion protection and tolerance on concrete cover

1 2 3

Minimum concrete cover, cmin, in mm a) b) for Tolerance onExposure

pre-tensioned and concreteclass

reinforcement post-tensioned cover,members c) Dc, in mm

1 XC1 10 20 10

XC2 20 30

2 XC3 20 30

XC4 25 35

XD1

3 XD2 40 50 15

XD3 d)

XS1

4 XS2 40 50

XS3

a) Where members are of concrete two strength classes higher than the mini-mum strength class specified in table 3, the concrete cover may be reducedby 5 mm. This does not, however, apply to members of exposure class XC1.

b) If in-situ concrete is bonded to a precast element, the concrete cover at jointedges may be reduced to 5 mm in the precast element and to 10 mm in thein-situ concrete. However, the provisions specified in item (4) below forensuring the bond shall be met if the reinforcement is to be fully utilized atthe erection stage.

c) The minimum concrete cover for post-tensioned members is measured inrelation to the surface of the sheath.

d) In some cases, reinforcement may need special corrosion protection.

(7) Concrete exposed to severe mechanical action shall satisfy the additional requirements specified inDIN 1045-2. Alternatively, resistance to wear may be improved by increasing the concrete cover, in whichcase cmin should be raised by about 5 mm for exposure class XM1, 10 mm for XM2 and 15 mm for XM3.(8) To take into account accidental deviations from the specified concrete cover, a value equal to Dc given intable 4 shall be added to cmin to give the nominal concrete cover, cnom.(9) Dc may be reduced by 5 mm if quality control during design, production and execution so permits.6)(10) A greater Dc value shall be used for reinforced members made from concrete that is placed againstuneven surfaces. The increase should generally be equal to the magnitude of the unevenness but not less than20 mm, and shall be approximately 50 mm when concrete is placed directly on the ground. Surfaces intendedto have a particular architectural effect (such as structured surfaces or coarse pebbledash) also require a greatermargin.(11) The actual concrete cover to reinforcement, cv, used for determining the equivalent height, is based onthe requirement that each reinforced member has the full nominal concrete cover.

7 Basis of structural analysis7.1 Requirements(1) It shall be ensured that all methods of analysis satisfy equilibrium.(2) If the compatibility conditions are not specifically verified as being met, it shall be ensured that the structurehas adequate deformation capacity up to the ultimate limit state and that, in the serviceability limit state, thereis no possibility of the structure behaving in a manner other than permitted.

6) See DBV-Merkblatt Betondeckung und Bewehrung and DBV-Merkblatt Abstandhalter.

Page 27DIN 1045-1 : 2001-07

(3) Equilibrium shall generally be checked on the basis of non-deformed structures (first-order theory). If,however, there is a considerable increase in internal forces and moments due to deflections, equilibrium willhave to be verified on the basis of the deformed structure (second-order theory).(4) The effects of actions occurring over time (e.g. creep, shrinkage) on internal forces and moments shall betaken into account if of relevance.(5) For members in conventional buildings, a simplified approach may be adopted as follows.

a) Second-order effects may be neglected if these cause a reduction in loadbearing capacity of less than 10 %.b) The effect on internal forces and moments of deformations due to shear and axial forces may be ne-glected if this effect is anticipated to be less than 10 %.

(6) For structures subjected to predominately static loading, the load history may generally be neglected anda constant increase in load may be assumed.

7.2 Imperfections(1) In the analysis for the ultimate limit state, with the exception of accidental design situations, considerationshall be given to any potentially unfavourable effects of imperfections in the geometry of the unloaded structure.(2) Individual bracing members shall be designed for the internal forces and moments determined in a globalanalysis, in which the effects of actions and imperfections of the structure as a whole are assumed to becombined.(3) The effect of imperfections may be taken into account by treating them as equivalent geometrical imper-fections.(4) When determining internal forces and moments in the structure as a whole, the effect of imperfections maybe considered by assuming the structure is inclined at an angle aa1, in radians, to the target axis: 1aa1 = ß 1/200 (4) 100 1hges

where hges is the overall height of the structure, in m.

(5) If there are several adjacent loadbearing members, aa1 may be reduced by the factor an, as follows: 1 +1/nan = 1 (5) 2

where n is the number of adjacent vertical, loadbearing members in a storey.Vertical members are classed as loadbearing if they can sustain at least 70 % of a design mean axial force, NEd,m,equal to FEd/n, with FEd being the total design axial force of all adjacent members in the considered storey.(6) As an alternative to the procedure described in item (4) above, the effect of equivalent horizontal forcesmay be substituted in the overall analysis of the structure and in the analysis of bracing members, bearings andany peripheral ties (see figures 1b) and 1d)).(7) Members that transfer stabilizing forces from the elements of a structure to be braced to the bracingelements should be designed to sustain an additional horizontal force, Hfd (see figure 1e)).

Hfd = (Nbc + Nba) ë aa2 (6)

where

aa2 = 0,008 / 12k , in radians (7)

wherek is the number of elements to be braced in the storey considered;Nbc and Nba are the design axial forces to take into account imperfections in columns and walls, respectively,

that are adjacent to the horizontal member responsible for load transfer (see figure 1e)).The horizontal forces, Hfd, shall be regarded as independent actions and shall not be further reduced bycombination factors since these are already taken into account in the vertical axial forces. Hfd does not needto be taken into consideration when designing vertical bracing members.

Page 28DIN 1045-1 : 2001-07

Key:a) and c): imperfections taken into account by use of deviations from the vertical;b) and d): imperfections taken into account by use of equivalent horizontal forces;e) imperfections taken into account by use of additional horizontal forces in members as specified in

item (7).

Figure 1: Application of equivalent geometrical imperfections

7.3 Idealizations and simplifications7.3.1 Effective flange width, load dispersal and effective span(1) The effective flange width of T-beams is a function of the flange and web dimensions, type of loading, span,conditions of support and shear reinforcement. The following specifications are applicable in all analyses forthe serviceability limit state, and are generally an adequate means of carrying out a rough analysis for theultimate limit state.(2) In bending analysis, the effective flange width of T-beams, beff, due to evenly distributed loads may beassumed to be as follows:

beff = 8 beff,i + bw (8)

where

beff,i = 0,2bi + 0,1 l0 ß 0,2 l0 (9)

ß bi

In the above:l0 is the effective span (distance between points of zero moment);bi is the actual flange width;bw is the web width.

n

DHj = 8 Vji ë aa1 i = 1

Figure 2: Effective flange width (notation)

Page 29DIN 1045-1 : 2001-07

(3) Assuming a fairly equal load distribution and constant bracing conditions in individual bays, l0 may betaken from figure 3.

Figure 3: Approximate effective spans for calculation of effective flange width

(4) Where beams are of variable depth, the web width, bw, in equation (8) may be increased by dimension bv(see figure 4).

Figure 4: Effective web width of T-beams of variable depth

(5) In the zone over which concentrated axial forces spread, the effective width may be determined by elastictheory. Alternatively, an angle of spread, b, of 35° may be assumed (see figure 5). The angle of spread ofanchorage forces in post-tensioned or unbonded members may also be assumed to be 35° (see figure 6).

Figure 5: Angle of spread of concentrated axial loads

Key:a) Plan viewb) Elevation1 Tendon axis

Figure 6: T-beam, with angle of spread of prestressing forces

Page 30DIN 1045-1 : 2001-07

(6) The effective span of a beam or slab, leff , may be determined as follows:

leff = ln + a1 + a2 (10)

whereln is the clear distance between the faces of supports;a1 and a2 are the distance between the faces of supports and the theoretical lines of support of the considered

span, respectively.

Appropriate values for a1 and a2 shall be determined as a function of the support and restraint conditions of themember (see figure 7) 7).

Key:

a aa) Non-continuous members Œ ß ai ß › 3 2

ab) Continuous members Œai = › 2

Figure 7: Effective span of a beam or slab

(7) Where beams or slabs are directly supported, the load from the supported member is counteracted bycompression at its lower edge. In monolithic members, the same support reaction may be assumed if thedistance of the lower edge of the supported member to the lower edge of the supporting member is greater thanthe depth of the supported member. Otherwise, indirect support shall be assumed (see figure 8).

Key:1 Supporting member2 Supported member

(h1 – h2) ö h2 for direct support(h1 – h2) < h2 for indirect support

Figure 8: Direct and indirect support

7) See DAfStb-Heft 525 for further information.

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7.3.2 Other simplifications(1) Continuous beams and slabs in conventional buildings may be analysed on the assumption that thesupports do not provide rotational restraint.(2) Regardless of the method of analysis used, where a beam or slab is continuous over a support which maybe considered to provide no restraint to rotation, the design bending moment at the support, MEd, calculated onthe basis of a span equal to the centre-to-centre distance between supports, may be reduced by a factor DMEd:

DMEd = CEd · a/8 (11)

whereCEd is the design support reaction;a is the bearing width.

(3) If a beam or slab continues over a support and is cast so as to connect with it monolithically, the criticaldesign moment at the support may be taken as that at the face of the support but not less than the minimumvalue specified in subclause 8.2 (5).(4) When determining the loads applied to supporting members as a result of the reactions from one-wayspanning slabs, ribbed floors and beams (including T-beams), free rotation (but not continuity) may be as-sumed. However, continuity should always be assumed for the first inner support and also for inner supportswhere the span ratio of adjacent bays, leff,1/leff,2, of approximately equal stiffness is between 0,5 and 2.(5) In conventional buildings, the shear force may be determined assuming the beam or slab is fully loadedif the ratio of spans of adjacent bays, leff,1/leff,2, of approximately equal stiffness is between 0,5 and 2.(6) In conventional framed structures, in which all horizontal forces are resisted by plates and internal columnsare rigidly connected to beams or slabs, bending moments due to vertical loading of the frame may be neglectedif the ratio of spans of adjacent bays, leff,1/leff,2, of approximately equal stiffness is between 0,5 and 2.(7) Edge supports in frames shall always be treated as vertical members rigidly connected to beams or slabs.This also applies to reinforced concrete walls connected to slabs.(8) In the analyses specified in subclauses 8.2 and 8.3, ribbed or waffle floors may be treated as solid slabsif the combination of flange and ribs are of adequate torsional stiffness. This may be assumed to be the caseif the following conditions apply:

a) ribs are spaced not more than 1 500 mm apart;b) the depth of ribs (measured from the flange bottom surface) is equal to not more than four times theirwidth;c) the thickness of the flange is not less than 1/10 of the mean clear spacing of the ribs or not less than50 mm (whichever is greater);d) there are transverse ribs whose mean clear spacing is equal to not more than ten times the overall floorthickness.

(9) In the analyses specified in subclauses 8.2 and 8.3, a ribbed floor/filler block combination without toppingmay be treated as a solid slab if transverse ribs are provided a distance sT apart, with sT not less than specifiedin table 5.

Table 5: Maximum spacing of transverse ribs in combinations of ribbed floors/filler blockswithout topping

Spacing of transverse ribs, sT, for a centre-to-centre distance of longitudinal ribs, sL,Building type

up to leff/8 over leff/8

Residential – 12 h

Other buildings 10 h 8 h

leff effective span of longitudinal ribsh overall thickness of ribbed floor

Page 32DIN 1045-1 : 2001-07

8 Methods of analysis8.1 General(1) Each calculation method shall demonstrate the level of reliability due in the particular context. This require-ment may be deemed satisfied if the specifications of subclauses 8.1 to 8.7 are met.(2) Where relevant, the effect of torsional stiffness shall be taken into account in the analysis.(3) The linear-elastic method is based on a linear relationship of action-effects to deformations.(4) Plastic methods of analysis are generally based on idealized elasto-plastic or idealized rigid-plastic rela-tionships of action-effects to deformations.(5) The term ‘non-linear’ refers to methods of analysis that consider non-linear relationships between action-effects and deformations. Methods involving verification of equilibrium taking into account deformations of thestructure are designated ‘second order theory’ methods.

8.2 Linear-elastic analysis(1) The linear-elastic method of analysis is based on the stiffness of cross sections in their uncracked state(state I). However, the stiffness of cross sections after cracking (state II) may also be used.(2) Application of the linear-elastic method does not usually require special action to ensure adequatedeformability provided members have at least the minimum reinforcement specified in this standard, withoutthis necessitating that critical sections contain an extremely high proportion of reinforcement.(3) For continuous beams in which the span ratio of adjacent spans, leff,1/leff,2, of T-beams of approximatelythe same stiffness, is more than 0,5 but less than 2, used in frames and in other members predominatelysubjected to bending (including continuous slabs with continuous support in transverse direction), the x/d ratioshould be not more than 0,45 for concrete up to strength class C50/60 and 0,35 for lightweight concrete ofstrength class C55/67 or higher, unless adequate ductility is provided by structural means (see item (5) ofsubclause 13.1.1). The neutral axis depth, x, shall be calculated using as a basis the design values of actionsand the material strength.(4) An analysis of non-prestressed continuous beams and slabs in conventional buildings is not necessary fordesign situations with favourable permanent actions if such beams and slabs contain the minimum reinforce-ment specified in this standard. This does not apply to the analysis of positional stability as in DIN 1055-100.(5) To take into account the idealization of the structure and the possibility of accidental deviations in thesystem at the erection stage, the design moment at the inner faces of supports of continuous beams shouldbe not less than 65 % of the bending moment, assuming fixity at the support faces.(6) Conventional methods of analysis for slabs assuming equal stiffness in both directions are only applicableif the distance of the longitudinal reinforcement to its associated shear reinforcement is not greater than 50 mmor d/10, whichever is higher.

8.3 Linear-elastic analysis with redistribution of moments(1) Redistribution of moments determined by the linear-elastic method specified in subclause 8.2 may be usedfor the analysis for the ultimate limit state provided equilibrium is satisfied.(2) The effects of moment redistribution shall be taken into account at all stages of analysis. This applies tothe design for bending or bending with coexistent axial force, design for shear, the provisions with respect toanchorages and for curtailment of reinforcement.(3) For continuous beams in which the ratio of adjacent spans, leff,1/leff,2, of beams of approximately the samestiffness is between 0,5 and 2, used in rigid frames and in other members predominately subjected to bending(including continuous slabs with continuous support in transverse direction), the ratio of the redistributedmoment to the original moment prior to redistribution, d, shall be within the following limits:

a) steel of high ductility:from 0,64 + 0,8 xd/d up to 0,7 for concrete of strength class up to C50/60 (12)orfrom 0,72 + 0,8 xd/d up to 0,8 for concrete of strength class from C 55/67 upwards, (13)and for lightweight concrete

b) steel of normal ductility:from 0,64 + 0,8 xd/d up to 0,85 for concrete of strength class up to C50/60 (14)

where xd/d is the relative depth of the compression zone at the ultimate limit state after redistribution, deter-mined on the basis of the design values of actions and material strengths.Where steel of normal ductility is used, no moment redistribution is permitted for concrete of strength class fromC 55/67 upwards and for lightweight concrete.d shall not exceed 0,9 at the nodes of rigid frames.

Page 33DIN 1045-1 : 2001-07

(4) Redistribution is not permitted in sway frames.(5) Redistribution is not permitted for unreinforced structures and structures made of precast segments withunreinforced contact joints.(6) To analyse slabs for shear, torque and support reactions, a linear interpolation shall be made between thestresses in a slab with fixed edges and those in a slab with pinned edges.

8.4 Plastic analysis8.4.1 General(1) Plastic analysis shall generally be used for members predominately subjected to bending under conditionsof ultimate limit state, unless these are made of lightweight concrete.(2) Without a specific check for compatibility, methods based on plastic theory may only be used for structurescomprising members of good deformation capacity.(3) In the case of two-way spanning slabs, methods of plastic analysis that do not include a specific checkof the rotation capacity of plastic hinges shall not be used unless the x/d ratio at the joints, at any point andin either direction, is not more than 0,25 for concrete of strength class up to C50/60 and 0,15 for concrete ofstrength class C55/67 and over, and (in the case of continuous slabs) the ratio of support moment to spanmoment is between 0,5 and 2. The depth of the compression zone, x, shall be determined on the basis of thedesign values of the actions and the material properties. If the above limits are exceeded, the rotation capacityshall be verified as specified in subclause 8.4.2.(4) Steel of normal ductility (see table 11) shall not be used for linear members and slabs designed on the basisof plastic theory.(5) Plates, even when of steel of normal ductility, may be designed on the basis of plastic theory withoutspecific verification of their rotation capacity being required.(6) Methods relying on plastic theory also form the basis for the analysis of trusses. Such methods may beapplied in the design of the unaffected parts of beams and slabs after cracking (see subclause 10.2 to 10.4) andin the design and detailing of areas of discontinuity (see subclause 10.6).

8.4.2 Simplified analysis of plastic rotation for members predominately subjected to bending(1) The simplified method of analysing linear members including one-way spanning slabs is based on ananalysis of the rotation capacity of designated linear member sections approximately 1,2 h long on the assump-tion that these are the first to undergo plastic deformation (i.e. to form plastic hinges), under combinations ofactions, enabling them to be treated as a cross section. Proof of adequate plastic rotation (angular displace-ment due to plastic deformation) at ultimate limit state may be deemed provided if it is proved that the actualangular displacement, θE, is less than the design permitted displacement, θpl,d, i.e.

θE ß θpl,d (15)

(2) At plastic hinges, the x/d ratio shall be not more than 0,45 for concrete of strength class up to C50/60 and0,35 for concrete of strength class C55/67 upwards.(3) θE shall be determined on the basis of the design values of actions and mean material strength values fromsubclause 8.5.1, together with the mean prestressing force at the point in time considered.(4) If the actual angular displacement of a plastic hinge is calculated by integration of the curvatures betweenthe hinges, it is sufficient to apply a simplified moment-curvature relationship as specified in subclause 8.5.2(3), taking into account the moments due to prestressing.(5) By way of simplification, the permitted angular displacement may be determined by multiplying the basicvalue of permitted displacement by a correction factor, kl, to take into account slenderness. The basic valueof permitted displacement for reinforcing steel of high ductility and concrete of strength classes up to C50/60and C100/115 shall be taken from figure 9. Values for concrete strength classes C55/67 to C90/105 may beobtained by linear interpolation. The values given are based on a shear slenderness ratio, l, of 3 and shall bemultiplied by kl in cases where l has a value other than 3.

kl = 1l/3 (16)

where l is the ratio of the distance between the point of zero moment and the maximum moment after redis-tribution to the effective depth, d.By way of simplification, l may be calculated from the design values of the bending moment and the respectiveshear force:

λ = MEd/(VEd · d) (17)

DAfStb-Heft 525 describes a more precise method to determine permitted plastic rotation.

Page 34DIN 1045-1 : 2001-07

Key:1 Concrete of strength classes C12/16 to C50/602 Concrete of strength class C100/115

Figure 9: Basic values of permitted plastic rotation for concrete of strength classes C12/16to C50/60 and C100/115 (based on a slenderness ratio of 3)

8.5 Non-linear analysis8.5.1 General(1) Non-linear methods of analysis may be used for both the ultimate limit state and serviceability limit state(provided that equilibrium and compatibility conditions are satisfied).(2) By specifying the size and position of reinforcement, non-linear methods also cover design for bendingwith or without coexistent axial force as specified in subclause 10.2.(3) Dimensional changes and changes in action-effects in a structure shall be calculated on the basis of stress-strain curves for concrete (see figure 22), reinforcing steel (see figure 26) and prestressing steel (see figure 28),using the mean values of material strength.(4) The following mean values of material strength shall be assumed in the calculations:

fyR = 1,1 fyk (18)

ftR = 1,08 fyR (for reinforcing steel of high ductility) (19)

ftR = 1,05 fyR (for reinforcing steel of normal ductility) (20)

fp0,1R = 1,1 fp0,1k (21)

fpR = 1,1 fpk (22)

fcR = 0,85 afck (for concrete of strength classes up to C50/60) (23)

fcR = 0,85 afck/gcq (for concrete of strength class C55/67 and over) (24)

with a as in items (2) and (4) of subclause 9.1.6 and gcq in item (9) of subclause 5.3.3.In the above, a partial safety factor, gR, of 1,3 (for permanent and temporary design situations and fatigueanalysis) or 1,1 (for accidental design situations for the design resistance) shall be taken into account.(5) The design resistance shall not be less than the design value of the associated combination of actions.(6) Ultimate limit state is assumed to have been reached if the steel or concrete reaches critical strain or thecritical state of indifferent equilibrium is reached in the whole system or parts of the system.(7) In the calculation, the theoretical ultimate strain in reinforcing steel, esu, should be taken to be 0,025, usingthe critical compressive strain in the concrete, ec1u, as in tables 9 and 10.(8) The contribution made by the concrete towards resisting tension between cracks (tension stiffening) shallbe taken into account unless conservative, in which case it may be neglected.

Page 35DIN 1045-1 : 2001-07

8) See DAfStb-Heft 525.

(9) Account should be taken of the contribution of the concrete in resisting tension by a method suited to thedesign situation 8).

8.5.2 Basis of calculation for linear members and one-way spanning slabs subjected to bending with orwithout coexistent axial force

(1) The action-effects for linear members and one-way spanning slabs may be calculated assuming theoreticalmoment-curvature relationships based on uniform cross sections.(2) This provides the basis by which to determine the action-effects for analyses for the ultimate limit stateand the serviceability limit state. Subclause 8.5.1 shall be applicable for analyses for the ultimate limit state.(3) By way of simplification, the moment-curvature relationship shown in figure 10 may be used, with thecurvatures (1/r)y and (1/r)u determined taking into account tension stiffening.

In the figure,BI or BII is the flexural stiffness in state I (uncracked) or state II (cracked) (= dM/d(1/r)MI or MII is the moment at transition from state I to state II;My is the flow moment;Mu is the failure moment;(1/r)I,II is the curvature associated with MI,II (i.e. MI,II/BI).

Figure 10: Simplified moment-curvature relationship

(4) For members subjected to axial force, the simplified moment-curvature relationship shown in figure 10 isonly applicable if the reference line by which to determine the moment coincides with the line of action of theaxial force. Otherwise (e.g. when the prestressing is along the centroidal axis), the effect of the resultingprecurvature shall be taken into account.

8.6 Linear members and walls in axial compression (second order theory)8.6.1 General(1) The equilibrium of structures comprising linear members or walls subjected to axial compression, and inparticular the equilibrium of the members themselves shall be verified taking into account the effects of defor-mations of the member if these reduce the loadbearing capacity by more than 10 %. Analysis shall be made ineach direction in which failure according to second order theory is liable to occur.(2) At the ultimate limit state it shall be verified for the most unfavourable combination of actions that in criticalsections the design value of actions according to second order theory does not exceed the design resistanceand both local and overall equilibrium are ensured.(3) The specifications of subclause 8.6 also apply to other structures (e.g. shells) and members in whichdeformations (including local ones, such as at the points of support of deep beams) considerably affectloadbearing capacity or where there is a risk of loss of equilibrium (e.g. due to lateral displacement of slenderbeams; cf. subclause 8.6.8).(4) The design values of the action-effects applied shall be determined taking into account dimensionalinaccuracies and uncertainties in respect of the position and direction of axial forces. In the absence of otherassumptions being made, these effects shall be taken into account by assuming geometrical imperfections (seesubclause 8.6.4).(5) Non-linear methods dealing with the overall structure as specified in subclause 8.5 may be used taking intoaccount plastic hinges as specified in subclause 8.4. However, plastic hinges (where (1/r)m is greater than(1/r)y, as shown in figure 10) are not permitted for members in axial compression as defined in item (1) above.

Page 36DIN 1045-1 : 2001-07

(6) If the action-effects according to first order theory are determined by the methods specified in subclause8.2, 8.3 or 8.4, the applied second-order action-effects or additional second-order action-effects due todeformations shall be determined by the method specified in subclause 8.5.(7) As a departure from the specifications of item (6) above, dimensional changes may be determined fromdesign values based on the mean values of material characteristics (e.g. fcm/gc, Ecm/gc). In order to determinethe ultimate loadbearing capacity for the critical cross section, however, the design values of material strength(e.g. a · fck/gc) are to be used.(8) The effect of tension stiffening may be neglected.

8.6.2 Classification of structures and members(1) For calculation purposes, the structures or structural members may be classified as braced or unbraceddepending on whether bracing elements are provided and as non-sway or sway depending on their sensitivityto second order effects as set out in item (1) of subclause 8.6.1 or, in the case of isolated members, due to lateraldisplacement.(2) A bracing element or a system of bracing elements requires to be sufficiently stiff to attract and transmitto the foundations all horizontal loads acting on the structure and to ensure the loadbearing capacity of thebraced subassembly.(3) Members within the meaning of item (1) above include compression members with an equivalent height,l0. These may be isolated compression members (see figures 11 a) and 11 b)) or compression members formingpart of a structure but which may be classed as isolated compression members for the purpose of analysis asspecified in items (1), (2) and (3) of subclause 8.6.1 (see figures 11 c) and 11 d)).

Key:a) Isolated columnb) Hinged columns in braced sway structure or non-sway structurec) Slender bracing element in a sway structure, regarded as an isolated columnd) Compression members with restrained ends in a braced sway or non-sway structure, regarded as isolated

columns

Figure 11: Types of isolated column

(4) The equivalent height of isolated compression members, equal to b × lcol (lcol being the height of themember between the idealized points of restraint), shall be a function of the degree of rigidity of restraint at theends of the member and the freedom of the ends of the member to move along its plane. DAfStb-Heft 525 givesdetails regarding the calculation of the equivalent height of isolated compression members. The slendernessratio of isolated compression members shall be calculated as l0/i, where i is the radius of gyration of the section.(5) In the absence of a more precise analysis, structures braced by vertical members (such as solid shear wallsor core structures) may be regarded as non-sway within the meaning of item (1) if the following conditions aremet:

a) The vertical bracing members are arranged more or less symmetrically and permit only negligible rotationabout the axis of the structure, in which case the lateral stiffness in both directions shall satisfy equation (25):

1 EcmIc 1 ö 1/(0,2 + 0,1 m) where m is not more than 3 (25)hges FEd

ö 1/0,6 where m is not less than 4

Page 37DIN 1045-1 : 2001-07

b) The vertical bracing members are not arranged more or less symmetrically or do not permit negligiblerotation, in which case the rigidity resulting from the combination of warping stiffness, EcmIv and torsionalstiffness, GcmIT, shall satisfy equation (26):

1 EcmIv 1 GcmIT 1 + 1 ö 1/(0,2 + 0,1 m) where m is not more than 3 (26)hges 7FEd,jërj

2 2,28 7 FEd,jërj2

ö 1/0,6 where m is not less than 4

wherem is the number of storeys;hges is the overall height of the structure (from the upper edge of the foundation or a rigid reference plane);rj is the distance of the column (j) from the centroid of the system;FEd is the sum of the design vertical loads with gF equal to unity;FEd,j is the sum of the design vertical load from the column with gF equal to unity;EcmIc is the total nominal flexural stiffness of all vertical bracing members that act in the direction con-

sidered and meet the requirements specified in item (2) of subclause 8.6.2 (the concrete tensilestress in the bracing members should be not greater than the values of fctm from table 9 or table 10;equivalent stiffness should be used in cases where the stiffness of the bracing members is liable tovary over their length);

Ecm Iv is the total nominal warping stiffness of all members providing torsional rigidity;GcmIT is the total torsional stiffness of all members (see Saint Venant’s theorem).

8.6.3 Methods of analysis(1) In the case of isolated compression members, a comparison of slenderness on the basis of limiting valuesshall be made on the basis of which it shall be decided whether second order effects need to be taken intoaccount.

(2) Isolated compression members are considered slender if their slenderness ratio exceeds a maximumslenderness ratio, lmax, of 25 for |yEd| not less than 0,41 (27) or for 16/1|yEd| for |yEd| less than 0,41 (28).

In the above,

NEdyEd = (29) Acëfcd

whereNEd is the mean design axial force of the compression member;Ac is the (sectional) area of the compression member;fcd is the design concrete compressive strength as defined in item (2) of subclause 9.1.6.

(3) Non-sway structures or compression members that are not considered slender do not need to be designedaccording to second order theory.(4) Even when considered slender, isolated compression members in braced non-sway structures need notbe checked for second order effects if their slenderness does not exceed the critical slenderness ratio obtainedby means of equation (30). This only applies in cases where the column is not subjected to transverse loads ormoments between its two ends and the axial force can be assumed constant over the length of the column.

lcrit = 25 (2 – e01/e02) (30)

where e01/e02 is the ratio of the respective eccentricities of axial force at the column ends (see figure 13) where|e01| ß |e02|.

In the case of a column being hinged at both ends, lcrit shall be equal to 25.

Item (9) below shall apply to the design of the column ends.(5) The effects due to creep may usually be neglected if both ends of a column are connected monolithicallywith the members that transmit the load or if, in the case of sway structures, the slenderness ratio of thecompression member is less than 50 and at the same time the e0/h ratio is greater than 2.(6) A simplified way of determining the second-order effects in isolated slender compression members is bythe model column method as set out in subclause 8.6.5.(7) See DAfStb-Heft 525 for second order analyses of whole structures.(8) If, when considering sway structures, it is assumed that the ends of the compression member are re-strained by connecting members (e.g. by a horizontal member), such connecting members shall also be de-signed to sustain this extra load.

j j

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(9) If the provisions of item (4) above apply, both ends of isolated compression members should be designed

to satisfy in the following equations:

MRd ö |NEd| · h/20 (31)

NRd ö |NEd| (32)

where h is the depth of cross section considered.

8.6.4 Imperfections(1) In the case of isolated compression members, geometrical equivalent imperfections may be accounted forby increasing the eccentricity of the axial force by an additional accidental eccentricity, ea, acting in the mostunfavourable direction:

ea = aa1 · l0/2 (33)

wherel0 is the equivalent height of the compression member as in subclause 8.6.2 (4);aa1 is the angle of inclination to the target axis as calculated using equation (4), taking hges to be equal to the

height of the compression member.

If the member bracing the compression member forms part of a structure of the type shown in figure 11 b), itshall be checked whether the assumption of an inclination of the structure (including members with and withouta bracing function) to the target axis as specified in subclause 7.2 results in a value of ea for the bracing memberthat is greater than that obtained by means of equation (33). The most unfavourable value shall be assumed.(2) The imperfections as in item (1) above need only be assumed where a check is made for second ordereffects.

8.6.5 Model column method(1) The method described in the following shall apply to compression members of rectangular or circular crosssection, with a first order design eccentricity, e0, of not less than 0,1 h (h being the depth of section in the planeconsidered).(2) The method may also be used for other forms of cross section and load eccentricities less than 0,1 h. Otherapproximate methods are more suitable, however. 8)(3) A model column is a cantilever column of a height, l, equal to l0/2, that is restrained at the base and freeto rotate at the top (see figure 12) and which, under the effects of loads and moments, is bent in simple curvaturewith the maximum moment occurring at the base.

Key:1 Design straight line of member axis2 Second order line of bending3 Line of action of resultant of NEd und HEd

Figure 12: Model column (notation)

For 8), see page 35.

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(4) Verification of equilibrium takes the form of a check of the second order effects in a critical cross sectionat the base of the model column, using as a basis the curvature (1/r) of the cross section when the columnundergoes maximum second order deflection.(5) Where compression members are of constant cross section (with regard to both concrete and reinforce-ment, neglecting joints), the total eccentricity for the model column, etot, shall be calculated for the criticalsection as follows:

etot = e1 + e2 (34)

with

e1 = e0 + ea. (35)

In the above:e0 is the design first order eccentricity (equal to MEd0/Ned);

whereMEd0 is the design bending moment (first order theory);NEd is the design axial force;

ea is the additional accidental eccentricity from equation (33);e2 is the additional eccentricity due to second order effects.

(6) For compression members in non-sway frames, that are of uniform cross section and are subjected tomoments varying over their height, with equal (cf. figure 13 a)) or different (cf. figure 13 b) and c)) eccentricitiesat both ends, eccentricity e0 in the critical cross section, as calculated by equation (36) or (37), shall be assumed,taking the larger of the two values.

e0 = 0,6 e02 + 0,4 e01 (36)

e0 = 0,4 e02 (37)

where e01 and e02 are the design eccentricities of the axial force according to first order theory at both ends ofthe column, with |e02| not less than |e01|.

Key:a) compression member with equal eccentricities at both ends;b) compression member with different eccentricities with the same operational signs at both ends;c) compression members with different eccentricities with different operational signs at both ends.

Figure 13: Design model for calculation of effective eccentricity

(7) e2 shall be determined on the basis of subclause 8.6.1.(8) By way of simplification, it is possible to calculate the maximum deflection, which is equal to the additionalsecond order eccentricity, e2, as follows:

e2 = K1 · (1/r) · l02/10 (38)

wherel0 is the equivalent height as in subclause 8.6.2 (4);(1/r) is the curvature in the critical cross section;K1 is equal to l/10 – 2,5 for values of l not less than 25 but not greater than 35, and equal to unity for values

of l greater than 35.(9) The curvature in the critical section, 1/r, may be approximated as follows:

(1/r) = 2 K2 · eyd/(0,9 d) (39)

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where

K2 = (Nud – NEd) / (Nud – Nbal) ß 1 (40)

In the above:eyd is the design yield strain in the reinforcement (equal to fyd/Es);d is the effective depth of the cross section in the anticipated direction of failure;Ned is the design applied axial force (negative in compression);Nud is the design ultimate loadbearing capacity of the cross section subjected only to axial compression

(it may be assumed to be equal to – (fcd · Ac + fyd · As));Nbal is the resistance axial compression at the maximum moment capacity of the section (in rectangular cross

sections with symmetrical reinforcement, it may be approximated to – (0,4 fcd · Ac)).

An assumption that K2 equals unity will always give conservative results.

8.6.6 Compression members with biaxial eccentricity(1) If it is necessary to consider the loadbearing behaviour in each of the two principal planes, there shall beseparate checks of the critical section, taking into due account the fact that the structural situation may differat the two ends of the member.(2) For compression members of rectangular cross section, separate checks are permitted in the y- andz-planes (see figure 14) if the e0y/b and e0z/h ratios satisfy either of the following conditions:

(e0z/h) / (e0y/b) ß 0,2 (41)

or

(e0y/b) / (e0z/h) ß 0,2 (42)

where e0y and e0z are the first order design eccentricities along section dimensions b and h, respectively.

Thus, the point of application of Ned is located within the hatched area in figure 14. A more accurate analysiswill be required if the conditions in equations (41) and (42) are not met.

Figure 14: Scope of separate checks in the two principal planes

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(3) Where compression members are of rectangular cross section and e0z is greater than 0,2 h, separateanalyses may only be made if the analysis of bending about the z-plane of the section is based on the reduceddepth, hred, from figure 15, which may be obtained assuming linear stress distribution by means of the followingequation:

h hhred = Œ1 + › ß h (43) 2 6 (e0z + eaz)

whereh is the larger of the two section dimensions;eaz is the additional eccentricity taking into account geometrical equivalent imperfections in the z-plane,

obtained by means of equation (33);e0z is the eccentricity as in equation (42).

Figure 15: Reduced depth of cross section for a separate analysis in the y-plane for e0zgreater than 0,2 h

8.6.7 Plain concrete compression members(1) Irrespective of the slenderness ratio, compression members made from plain concrete shall be regardedas slender. However, a second-order analysis is not required for such members if the lcol/h ratio is less than 2,5.(2) The slenderness ratio of plain concrete walls or columns cast in situ should generally be not greater than 85.(3) The axial compression capable of being sustained by a plain concrete slender column or wall in a non-swaystructure may be approximated as follows:

NRd = – (b · h · fcd · f) (44)

with

f = 1,14 (1 – 2 etot/h) – 0,02 l0/h and f between zero and 1 – 2 etot/h (45)

whereNRd is the design resistance axial compression;b is the width of cross section;h is the depth of cross section;f is the coefficient taking into consideration second order effects on the loadbearing capacity of plain

concrete compression members in non-sway structures;etot is the overall eccentricity (equal to e0 + ea + ef);

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e0 is the first order design eccentricity (to be used where actions from connecting floors (e.g. bendingmoments due to restraint, that are transmitted from a slab to a wall) and effects of horizontal wind loadare to be considered);

ea is the accidental additional eccentricity due to geometric imperfections (may be taken to be equal to0,5 l0/200) in the absence of more precise information;

ef is the eccentricity due to creep (may usually be neglected).Further details are given in DAfStb-Heft 525.

8.6.8 Lateral buckling of slender beams(1) The safety of slender beams against lateral buckling shall be checked.(2) Safety against lateral buckling may be deemed adequate if the requirements in equation (46) are satisfied.Otherwise, a more detailed analysis shall be made.

4 l0t

3

b ö 1Œ ›ëh (46) 50

whereb is the width of the compression flange;h is the depth of the beam;l0t is the length of the compression flange between lateral supports.

(3) Slender precast T-beams shall have adequate protection from lateral buckling during lifting, transport anderection.(4) The analysis of slender beams in their final position including their means of support shall take into accountthe possibility of accidental eccentricity of support.(5) Unless more exact instructions are given, the supporting structure shall be designed to sustain at least thetorque from the beam, Ted (cf. equation (47)).

TEd = VEd · leff/300 (47)

whereleff is the effective span of the beam;VEd is the design shear force (normal to beam axis).

(6) In detailed analyses of safety against lateral buckling, the action-effects in the deformed beam should bedetermined as specified in subclause 8.6.1 (7), taking imperfections (e.g. geometrical equivalent imperfections)into due account. Unless other information is available, ea shall be assumed to be equal to leff/300.

8.7 Prestressed structures8.7.1 General(1) Prestressing with the aid of tendons can be treated as an action due to anchorage forces and forcesassociated with a change in direction, or as an applied action-effect.(2) Alternatively, prestressing may also be considered as a strain condition with precurvature, in which casepre-strain is to be taken into account when considering the resistance of the member cross section.(3) The methods described in items (1) and (2) above give the same design result (cf. figure 16) if the stressesand strain in the prestressing bench at time t are designated as prestress or pre-strain respectively and theprestressing steel does not plasticize in the ultimate limit state.(4) Any of the methods specified in subclause 8.1 may be used in the analysis of prestressed structures.(5) When applying linear-elastic methods, the structurally indeterminate effect of prestressing shall be takeninto account. When applying non-linear methods and when using plastic methods to determine the requiredrotation, prestressing shall be treated as pre-strain with corresponding precurvature. In such cases, there is noneed to determine the structurally indeterminate moment due to prestressing, since the action-effects due toprestressing cannot be distinguished from those due to loading.(6) A check of rotation capacity as specified in subclause 8.4.2 is required in all cases where a plastic analysisof prestressed linear members at ultimate limit state is made.(7) In the case of unbonded tendons, the analysis should be carried out as specified in item (1) above, takinginto account the fact that, owing to deformations in the structure, the tendons located in the concrete undergoan increase in stress which is greater than that suffered in the prestressing bench.(8) In an analysis of bonded tendons, the prestressing steel shall be assumed to be in a rigid bond with theconcrete, ignoring the increase in force in the tendons as a result of deformation of the structure before the bondwas produced (i.e. at the erection stage).(9) External tendons may be assumed to be straight along their free length between deflectors.

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In the figure,BI and BII is the flexural stiffness in state I (uncracked) or state II (cracked) (see figure 10);(1/r)0 is the curvature as a result of prestressing;Mp,dir is the statically determinate component of the moment due to prestressing;Mp,ind is the hyperstatic component of the moment due to prestressing;MI,II is the moment at transition from state I to state II;My is the flow moment;Mu is the failure moment;(1/r)I,II is the curvature associated with MI,II (i.e MI,II/BI).a) Moments in action where item (1) is applicable;b) Moments in action where item (2) is applicable.

Figure 16: Simplified moment-curvature relationship for prestressed concrete sections

(10) Where tendons are located externally, the strain between two consecutive points of contact with thestructure is to be assumed constant. This strain shall be determined taking structural deformations into ac-count.(11) If, to simplify the analysis of structures with external tendons, linear-elastic theory is used for a globalanalysis, the increase in stress in the prestressing steel due to structural deformations may be neglected.

8.7.2 Prestressing force(1) The maximum force applied at the tendon (i.e. the force at the active end during tensioning), P0, shall benot more than the lesser of the following values:

Ap · 0,8 fpk or Ap · 0,9 fp0,1k (48)

(2) Overstressing is permitted on condition that the prestressing jack guarantees an uncertainty of measure-ment of the applied prestressing force remaining within t 5 % of the final prestressing force, in which case P0may be increased to become equal to AP 0,95 fp0,1k during pre-tensioning. 9)(3) The mean prestressing force at time t0 immediately after release of the pressure on the anchorages (post-tensioning or unbonded tensioning) or after release of the anchorages (pre-tensioning), Pm0, shall at no pointbe more than the lesser of the following values:

Ap · 0,75 fpk or Ap · 0,85 fp0,1k (49)

(4) In the calculation of Pm0, the following factors shall be taken into consideration, depending on the pre-stressing method used:

a) elastic deformation;b) short-term relaxation of the prestressing steel;c) friction loss;d) anchorage slip.

(5) Where unbonded internal tendons are used, temperature gradients between the tendon and the adjacentconcrete need not generally be taken into account.

9) This increase in stress may prove inadequate in cases where the friction coefficient is high (see DAfStb-Heft 525).

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(6) The mean prestressing force at a moment in time later than t0, Pmt, shall be determined as a function of theprestressing method. In addition to the factors listed in item (4) above, the losses in prestressing force due tocreep and shrinkage of the concrete and the long-term relaxatíon of the prestressing steel shall be taken intoaccount.(7) The concrete in post-tensioned or unbonded members shall have a minimum compressive strength at timetj, fcmj, during tensioning. Table 6 gives the minimum compressive strength for partial prestressing and finalprestressing as a function of the strength class of the concrete as required by the agrément. The values in thethird column of table 6 foresee a prestressing force in each tendon that does not exceed that permitted in theagrément by more than 30 %. If, at the time of prestressing, the concrete compressive strength (as determinedin hardening tests) is between the values in columns 3 and 4, the actual prestressing force may be obtained bylinear interpolation for values between 30 % and 100 %.(8) The actual loss of prestress during tensioning operations shall be checked by measuring the prestressingforce and the associated elongation.

Table 6: Minimum concrete compressive strength during prestressing with post-tensionedor unbonded members at time tj

ConcreteMinimum concrete cylinder compressive

strength class a)strength, fcmj,min in N/mm2 b)

Partial prestressing Final prestressing

1 C25/30 13 26

2 C30/37 15 30

3 C35/45 17 34

4 C40/50 19 38

5 C45/55 21 42

6 C50/60 23 46

7 C55/67 25 50

8 C60/75 27 54

9 C70/85 31 62

10 C80/95 35 70

11 C90/105 39 78

12 C100/115 43 86

a) Table applies by analogy to lightweight concrete of strengthclasses LC25/28 to LC60/66.

b) Conversion is necessary when using cubes for testing.

8.7.3 Loss of prestress(1) For calculation of the loss of prestress, the specifications of this sublause shall apply in combination withthe provisions of items (4) and (6) of subclause 8.7.2.(2) Compliance with the provisions set out in the agrément relevant to the prestressing method is required.(3) The loss of prestress due to friction at a distance, x, from the end of the member, DPm(x), in tendons maybe estimated as follows:

DPm(x) = P0 · (1 – e–m · (θ + k · x)) (50)

whereθ is the overall design angular displacement of tendons over a length x (independent of direction and algebraic

sign);k is the accidental angular displacement (per unit length);m is the coefficient of friction between tendon and duct (depends on surface texture of tendons and ducts,

change in length of tendons and surface texture of prestressing steel).

(4) Loss of prestress due to accidental angular displacement may be neglected in the case of external tendonsconsisting of parallel wires or strands.

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(5) In the case of unbonded tendons, friction need only be taken into account when determining the effectivemean prestressing force, Pmt, and the resulting moments due to transfer of the prestressing force.(6) The change in stress in the steel due to creep and shrinkage of the concrete and relaxation of the steel ata distance x up to time t = r, Dsp,c+s+r, may, when using single-strand bonded tendons, be calculated as follows:

ecsrëEP + Dspr + apëf (r, t0)ë(scg + scp0)Dsp,c+s+r = (51) AP Ac 1 + apë Œ1 + ëzcp

2 › [1 + 0,8 f (r, t0)] Ac Ic

whereecsr is the final shrinkage strain as specified in subclause 9.1.4;ap is the ratio of Ep/Ecm;EP is the modulus of elasticity of prestressing steel, as specified in subclause 9.3;Ecm is the mean modulus of elasticity of concrete from table 9 or table 10;Dspr is the reduction in stress in the steel at a distance x due to relaxation, with a value less than 0.

NOTE: Dspr may be determined using the data given in the prestressing steel agrément relating tothe ratio of the initial stress to the characteristic tensile strength (sp0/fpk), taking sp0 to be equal tospg0 – 0,3 Dsp,c+s+r, (where spg0 is the initial stress in the steel due to prestressing and permanentactions). As a simplification, sp0 may conservatively be assumed equal to spg0. It may be given avalue of 0,95 spg0 for conventional buildings. If data are not available from agréments, Dspr shallbe determined by iteration.

f (r, t0) is the final creep coefficient of the concrete as specified in subclause 9.1.4;scg is the stress in the concrete at the tendons due to the quasi-permanent combination of actions;scp0 is the initial stress in the concrete due to prestressing;Ic is the second moment of area of the concrete section;zcp is the distance between the centroid of the concrete section and the tendons.

Compressive stresses shall be given negative signs in equation (51).

(7) Equation (51) may be used to determine the loss of prestress in an unbonded tendon if, for creep andshrinkage, the concrete strain over the length of the tendon is assumed to occur in straight sections betweenthe idealized points of sign change or anchorage points (in the case of external tendons) or along the entirelength of the tendon (in the case of internal tendons).

8.7.4 Serviceability limit state(1) The possibility of scatter of the prestressing force shall be taken into account as follows:

Pk,sup = rsup · Pmt (52)

Pk,inf = rinf · Pmt (53)

wherePmt is the mean prestressing force;rinf minimum coefficient taking into account scatter of prestressing force;rsup maximum coefficient taking into account scatter of prestressing force.

(2) Coefficients rsup and rinf may generally be assumed as follows:rsup shall be equal to 1,05 and rinf equal to 0,95 for pre-tensioned or unbonded members and equal to 1,10 andrinf equal to 0,90, respectively, for post-tensioned members.

8.7.5 Ultimate limit state(1) The design prestressing force, Pd , equal to gP · Pmt, may generally be calculated using a value of gP equalto unity.(2) Any scatter of the prestressing force may generally be neglected in ultimate limit state analyses.(3) If the increase in tension in the prestressing steel is taken into account in unbonded tendons, the charac-teristic value of this increase, Dspk, shall be determined using the mean values of the material characteristics.In linear-elastic analysis, the design value of Dspk, Dspd (equal to gP · Dspk), shall be calculated with gP equal tounity.In non-linear analysis, gP shall be represented by an upper limiting value, equal to 1,2, or lower limiting value,equal to 0,83 (whichever is least favourable), taking into account cracking or any gaping joints (in the case ofsegmental structures).

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8.7.6 Anchorage zones of pre-tensioned members(1) Use of smooth wires is not permitted for pre-tensioning purposes.(2) A distinction shall be made between the following:

– the transmission length, over which the prestressing force from a pre-tensioned member, P0, is fullytransmitted to the concrete, lbp;– the dispersion length, over which the concrete stress is uniformly distributed over the concrete section,lp,eff;– the anchorage length, required for full anchorage of the maximum force in the tendon at ultimate limitstate, lba.

(3) Assuming that the prestressing force is transferred to the concrete via a constant bond stress, fbp, thetransmission length, lbp, may be determined as follows:

Ap spm0lbp = al ë ë (54) p ëdp fbpëh1

whereal is equal to unity where the tensioning is effected as a gradual process and equal to 1,25 where the

tensioning is effected abruptly;Ap is the nominal cross section of the strand or wire;dp is the nominal diameter of the strand or wire;spm0 is the stress in the steel after transmission of the prestressing force to the concrete;h1 is a correction factor (equal to unity for normal-weight concrete and to be taken from table 10 for

lightweight concrete).

(4) For normal (non-compacted) strands with an area of cross section, Ap, of not more than 100 mm2 and forindented wire with a diameter of not more than 8 mm, prestressed as specified in subclause 8.7.2, the bondstress, fbp, may be assumed to have the values specified in table 7, depending on the compressive strength ofthe concrete at the moment at which the prestressing force is transmitted to the concrete. When using ribbedwires not more than 12 mm in diameter, fbp should be determined experimentally. The values given in table 7may be used as an approximation.Where bonding conditions are moderate (cf. subclause 12.4), the values of bond stress in table 7 shall bereduced by the factor 0,7.

Table 7: Bond stress, as a function of actual concrete compressive strength

Actual concrete cylinder compressivestrength during prestressing, Strands and

fcmj, in N/mm2 a) b) indented wire Ribbed wire

1 25 2,9 3,8

2 30 3,3 4,3

3 35 3,7 4,8

4 40 4,0 5,2

5 45 4,3 5,6

6 50 4,6 6,0

7 60 5,0 6,5

8 70 5,3 6,9

9 80 5,5 7,2

10 90 or more 5,7 7,4

a) Intermediate values may be obtained by linear interpolation.b) Conversion is required if cubes are used for testing.

Bond stress, fbp, in N/mm2

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(5) It may be assumed that the prestressing force increases steadily from the end of the member along thetransmission length.(6) A design value of 0,8 lbp or 1,2 lbp shall be assumed for the transmission length, depending on which is lessfavourable in the particular context.(7) When determining the stress in the transfer zone, concrete stresses over the cross section may be assumedto be uniformly distributed at the end of the dispersion length, lp,eff.(8) lp,eff may be determined for rectangular cross sections with tendons in the vicinity of the base of the crosssection as follows:

lp,eff = 1lbpd2 + d2 (55)

wherelbpd is the design transmission length;d is the effective depth.

For other cross-sectional shapes, the dispersion length and the respective local stress distribution should beestablished applying the principles of elastic theory.(9) The anchorage of members subjected to bending is greatly affected by cracking .The anchorage zone maybe considered uncracked if, at ultimate limit state, the tensile stresses in the concrete taking into account theprestressing force are not greater than concrete tensile strength fctk;0,05. In this case, it may be assumed thatthere is sufficient anchorage over lbpd, without need for further investigation.(10) If the stress in the concrete is greater than fctk;0,05, it shall be verified that the envelope line of the appliedtensile force is not greater than that of the resisting tensile force in the prestressing steel and reinforcing steel(cf. figure 66). The tensile strength of the reinforcing steel shall be determined as shown in figure 17. Outsidethe transmission length or on appearance of the first crack (i.e. when x is not less than lr), the fbp values fromtable 7 shall be reduced to take into account the deterioration in the bonding conditions. The anchorage length,lba, may be determined as follows:

a) where cracking occurs outside length lbpd (cf. figure 17a):

Ap spd – spmtlba = lbpd ë ë (56) p ëdp fbpëh1ëhp

b) where cracking occurs within length lbpd (cf. figure 17b):

Ap spd – spt (x = lr)lba = lr + ë (57) p ëdp fbpëh1ëhp

wherehp is equal to 0,5 for strands and indented wires;hp is equal to 0,7 for ribbed wires.

Key:a) On introduction of prestressing force (1); at ultimate limit state without cracking along transmission length (2)b) With cracking along transmission length (3); point at which first crack occurs due to bending (4)

Figure 17: Steel stresses in anchorage zone of pre-tensioned members

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(11) The axial force to be accommodated at a distance, x, of the strand/tendon from the end of the member,FEd(x), is given by equation (58):

MEd(x) 1FEd(x) = + VEd(x)ë(cot θ – cot a) (58) z 2

whereMEd(x) is the design value of the applied bending moment at distance x;z is the internal lever arm (cf. subclause 10.3.4);VEd(x) is the design shear force at distance x;θ is the angle between the concrete struts and the longitudinal axis of the member (cf. subclause 10.3.4)

(for members without shear reinforcement, cot θ shall be equal to 3 and for members without shearreinforcement, cot a shall be zero);

a is the angle between the shear reinforcement and the member axis (as in subclause 10.3.4).

Cracking shall be taken into account when determining the anchorage force to be sustained by the prestressingsteel (cf. figure 17).

8.7.7 Anchorage zones of post-tensioned or unbonded membersThe required splitting tensile and supplementary reinforcement shall be taken from the agrément covering theprestressing method. The capacity of the structure to sustain loads and transfer them internally shall be verifiedusing a suitable method (e.g. using a strut-and-tie model as specified in subclause 10.6).

9 Materials9.1 Concrete9.1.1 General(1) The specifications of this subclause refer to normal-weight and lightweight concrete conforming toDIN EN 206-1 and DIN 1045-2.(2) The characteristic values relative to strength and strain specified in this subclause apply to both normal-weight and lightweight concrete members unless otherwise stated.(3) Strength classes of normal-weight concrete are identified by the prefix C and of lightweight concrete, bythe prefix LC, the first number designating the cylinder compressive strength, and the second number the cubecompressive strength (e.g. C20/25).(4) Lightweight concrete is classified according to its dry density into density classes as specified inDIN 1045-2 and DIN EN 206-1. The design values of the dry density, r, and the characteristic values shall betaken from table 8.

Table 8: Density class, design dry density and characteristic density of lightweight concrete

Density parameterDensity class

D1,0 D1,2 D1,4 D1,6 D1,8 D2,0

Design dry density (to determine 801 to 1 001 to 1 201 to 1 401 to 1 601 to 1 801 tomaterial characteristics), in kg/m³ 1 000 1 200 1 400 1 600 1 800 2 000

Plain1 050 1 250 1 450 1 650 1 850 2 050concrete

Reinforcedconcrete

1 150 1 350 1 550 1 750 1 950 2 150

Characteristic density(to determine load),in kg/m³

9.1.2 Strength(1) The strength classes specified in this standard are based on the characteristic cylinder compressivestrength, fck, after 28 days (cf. tables 9 and 10).(2) In certain cases (e.g. for prestressing or special treatment such as heat treatment), it may be necessary todetermine the compressive strength before or after the 28 day deadline.(3) The tensile strength, fct, refers to the maximum stress that can be reached under axial tension.(4) The axial tensile strength, fct, may be derived by approximation from the splitting tensile strength,fct,sp, using the following equation:

fct = 0,9 fct,sp (59)

Page 49DIN 1045-1 : 2001-07

9.1.3 Deformation characteristics(1) Elastic deformations in the concrete are principally a function of its composition (particularly that of theaggregate). The following is thus given for guidance only. Elastic deformations shall be determined separatelyin cases where the structure reacts sensitively to deviations.(2) Guideline values for the moduli of elasticity, Ecm and Elcm, may be taken from tables 9 and 10.(3) The Poisson’s ratio covering elastic strain may be assumed to be approximately equal to zero.(4) In most cases, a coefficient of linear thermal expansion equal to 10 · 10–6 K–1 may be taken for normal-weight concrete and 8 · 10–6 K–1 for lightweight concrete.(5) The difference between the coefficients of thermal expansion of steel and lightweight concrete may beneglected in the design calculations.

9.1.4 Creep and shrinkage(1) Creep and shrinkage in the concrete mainly depend on the ambient humidity, the dimensions of themember and the composition of the concrete. Creep is also heavily influenced by the maturity of the concreteat the time the initial load is applied, and the duration of loading and the load level. These influences shall betaken into account when determining the creep coefficient, f(t, t0), and the shrinkage strain, ecs.(2) The final creep coefficients, f(r, t0), and final shrinkage strain, ecsr, determined as specified in this sub-clause, may be regarded as the likely average values, with a possible scatter of approximately 30 % needingto be taken into account when designing structures sensitive to creep and shrinkage. The values given arebased on a concrete compressive strength equal to not more than 0,45fckj, with fckj being the cylinder compres-sive strength of the concrete at the time the creep-inducing stress is applied.(3) If the compressive strength is greater than 0,45fckj, the dependence of creep on the creep-inducing stress(which is not linear) shall be taken into account, particularly when pre-tensioned members are involved.(4) In cases such as in item (3), or should a more precise calculation be required, the creep coefficients maybe calculated by other suitable methods.(5) The final creep coefficients and the final shrinkage strain relate to concrete that is moist-cured over a periodnot longer than two weeks and exposed to normal ambient conditions with a mean relative humidity between40 % and 100 % and mean temperatures between 10 °C and 30 °C.(6) The percentage creep strain in the concrete at a time t

r, ecc(r, t0), may be calculated as follows:

scecc(r, t0) = f (r, t0) ë (60) Ec0

wheref(r, t0) is the final creep coefficient, which, for the sake of simplification, may be taken from figure 18 or 19

as a function of the relative humidity; it may be determined by linear extrapolation or interpolationwhere the mean relative humidity is under 50 % or between 50 % and 80 %;

Ec0 is the modulus of elasticity of the concrete as a tangent at the origin of the stress-strain curve after28 days (as an approximation, Ec0 may be taken to be equal to 1,1 Ecm, with Ecm as in table 9 or table 10);

sc is the (constant) creep-inducing stress in the concrete;t0 is the age of the concrete at the beginning of loading, in days.

(7) In the absence of test results for lightweight concrete, f(r, t0) from figure 18 or 19 may be taken as a basisif multiplied by factor hE from table 10. For concrete of strength classes LC12/13 and LC16/18, the finalcoefficient of creep thus calculated shall additionally be multiplied by a reduction factor h2 equal to 1,3.(8) See DAfStb-Heft 525 for guidance on calculating the coefficient of creep at a given time and for concretewhose stresses vary over time.

Page 50DIN 1045-1 : 2001-07

Pa

ram

-1

23

45

67

89

1011

1213

1415

No

tes

eter

Str

eng

th c

lass

12/1

516

/20

20/2

525

/30

30/3

735

/45

40/5

045

/55

50/6

055

/67

60/7

570

/85

80/9

590

/105

100/

115

1f c

k12

a )16

2025

3035

4045

5055

6070

8090

100

in N

/mm

2

2f ck

,cu

be

1520

2530

3745

5055

6067

7585

9510

511

5in

N/m

m2

3f c

m20

2428

3338

4348

5358

6368

7888

98 1

08

f cm

= f

ck +

8,

in N

/mm

2

4f c

tm1

,61

,92

,22

,62

,93

,23

,53

,84

,14

,24

,44

,64

,85

5,2

f ctm

= 0

,30

fck

(2/3

) ; u

p t

o c

lass

C5

0/6

0

f ctm

= 2

,12

ln

(1

+ f

cm/1

0);

fro

m c

lass

C5

5/6

7

5f c

tk;0

,05

1,1

1,3

1,5

1,8

22

,22

,52

,72

,93

3,1

3,2

3,4

3,5

3,7

f ctk

;0,0

5 =

0,7

fct

m

6f c

tk;0

,95

22

,52

,93

,33

,84

,24

,64

,95

,35

,55

,76

6,3

6,6

6,8

f ctk

;0,9

5 =

1,3

fct

m

7E

cmb)

2580

027

400

2880

030

500

3190

033

300

3450

035

700

3680

037

800

3880

040

600

4230

043

800

4520

0E

cm =

950

0 (f

ck +

8)1

/3;

in N

/mm

2

8e c1

–1,8

–1,9

–2,1

–2,2

–2,3

–2,4

–2,5

–2,5

5–2

,6–2

,65

–2,7

–2,8

–2,9

–2,9

5–3

,0A

s p

er t

ho

usa

nd

; se

e fig

ure

22

.9

e c1

u–3

,5–3

,4–3

,3–3

,2–3

,1–3

,0–3

,0

10n

2,0

2,0

1,9

1,8

1,7

1,6

1,5

5

11e c

2–2

,0–2

,03

–2,0

6–2

,1–2

,14

–2,1

7–2

,2A

s p

er t

ho

usa

nd

; se

e fig

ure

23

.12

e c2u

–3,5

–3,1

–2,7

–2,5

–2,4

–2,3

–2,2

13e c

3–1

,35

–1,3

5–1

,4–1

,5–1

,6–1

,65

–1,7

As

per

th

ou

san

d;

see

figu

re 2

4.

14e c3

u–3

,5–3

,1–2

,7–2

,5–2

,4–2

,3–2

,2

a )C

on

cret

e st

ren

gth

cla

ss C

12

/15

may

on

ly b

e u

sed

in

co

nju

nct

ion

wit

h p

red

om

inat

ely

stat

ic l

oad

ing

.b)

Rep

rese

nti

ng

a m

ean

sec

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mo

du

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of

elas

tici

ty a

t |o

c|R

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f cm.

Ta

ble

9:

Ch

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igh

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te

Page 51DIN 1045-1 : 2001-07

12

34

56

78

910

11

No

tes

Str

eng

th c

lass

12/1

316

/18

20/2

225

/28

30/3

335

/38

40/4

445

/50

50/5

555

/60

60/6

6

1f l

ck1

2a )

1620

2530

3540

4550

5560

in N

/mm

2

2f l

ck,c

ub

e13

1822

2833

3844

5055

6066

3f lc

m20

2428

3338

4348

5358

6368

f lcm

= f

lck +

8,

in N

/mm

2

4f lc

tmh 1

· f ct

mf c

tm a

s in

tab

le 9

, in

N/m

m2

h 1 =

0,4

0 +

0,6

0 r

/220

0c )

5f lc

tk;0

,05

h 1 ·

fct

k;0

,05

f ctk

;0,0

5 a

s in

tab

le 9

6f lc

tk;0

,95

h 1 ·

f ctk

;0,9

5f ct

k;0

,95 a

s in

tab

le 9

7E

lcm

b)

h E ·

Ecm

Ecm

as

in t

able

9

h E =

(r/

2200

)2c )

8e lc

1

–k ·

flc

m/E

lcm

k =

1,1

fo

r lig

htw

eig

ht

con

cret

e w

ith

lig

ht

san

d a

nd

= 1

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or

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con

cret

e w

ith

nat

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l sa

nd

. A

s p

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ho

usa

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; se

e fig

ure

22

.

9e lc

1u

e lc1

As

per

th

ou

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d;

see

figu

re 2

2.

10n

2

,01

,9

11e lc

2

–2

,0–2

,03

–2,0

6A

s p

er t

ho

usa

nd

; se

e fig

ure

23

.

12e lc

2u

–

3,5

h1ö

e c2u

e c2u

as

in t

able

9A

s p

er t

ho

usa

nd

; se

e fig

ure

23

.

13e lc

3

–1

,8A

s p

er t

ho

usa

nd

; se

e fig

ure

24

.

14e lc

3u

–

3,5

h1ö

e c3u

e c3u

as

in t

able

9A

s p

er t

ho

usa

nd

; se

e fig

ure

24

.

a )C

on

cret

e o

f st

ren

gth

cla

ss C

12

/13

may

on

ly b

e u

sed

in

co

nju

nct

ion

wit

h p

red

om

inat

ely

stat

ic l

oad

ing

.b)

Rep

rese

nti

ng

a m

ean

sec

ant

mo

du

lus

of

elas

tici

ty a

t |s

c| R

0,4

flc

m.

c )W

ith

r i

n k

g/m

3.

Ta

ble

10:

Ch

ara

cte

rist

ic v

alu

es

of

stre

ng

th a

nd

str

ain

fo

r lig

htw

eig

ht

co

nc

rete

Pa

ram

-et

er

Page 52DIN 1045-1 : 2001-07

Key:1 Cement strength class: 32,5N 10)2 Cement strength class: 32,5R or 42,5N 10)3 Cement strength class: 42,5R, 52,5N or 52,5R 10)

In the figure,h0 is the effective depth of section (equal to 2 Ac/u), in cm;Ac is the sectional area;u is the perimeter of the section (including 50 % of the inner perimeter in the case of box beams).

Figure 18: Final creep coefficient for normal-weight concrete in a dry indoor atmosphere(with a relative humidity of 50 %)

10) See DAfStb-Heft 525 for further examples of classification of cement types.

Page 53DIN 1045-1 : 2001-07

Key:1 Cement strength class: 32,5N 10)2 Cement strength class: 32,5R or 42,5N 10)3 Cement strength class: 42,5R, 52,5N or 52,5R 10)

In the figure,h0 is the effective depth of section (equal to 2 Ac/u), in cm;Ac is the sectional area;u is the perimeter of the section (including 50 % of the inner perimeter in the case of box beams).

Figure 19: Final creep coefficient for normal-weight concrete in an external humid atmosphere(with a relative humidity of 80 %)

(9) The final shrinkage strain in the concrete, ecsr, may be calculated as follows:

ecsr = ecasr + ecdsr(61)

whereecasr is the general shrinkage strain at t approaching infinity (from figure 20);ecdsr is the drying shrinkage strain at t approaching infinity (from figure 21).

(10) In the absence of test results, ecsr values obtained by means of equation (61) shall be used for lightweightconcrete, multiplied by a factor, h3, of 1,5 for concrete of strength classes LC12/13 and LC16/18 or 1,2 forconcrete of strength class LC20/22 and over.(11) See DAfStb-Heft 525 to calculate the shrinkage strain at a particular time.

For 10), see page 52.

Page 54DIN 1045-1 : 2001-07

Key:1 Cement strength class: 32,5N 10)2 Cement strength class: 32,5R or 42,5N 10)3 Cement strength class: 42,5R, 52,5N or 52,5R 10)

In the figure,h0 is the effective depth of section (equal to 2 Ac/u), in cm;Ac is the sectional area;u is the perimeter of the section (including 50 % of the inner perimeter in the case of box beams).

Figure 20: General shrinkage strain of normal-weight concrete for t approaching infinity

For 10), see page 52.

Key:1 Cement strength class: 32,5N 10)2 Cement strength class: 32,5R or 42,5N 10)3 Cement strength class: 42,5R, 52,5N or 52,5R 10)

In the figure,h0 is the effective depth of section (equal to 2 Ac/u), in cm;Ac is the sectional area;u is the perimeter of the section (including 50 % of the inner perimeter in the case of box beams).

Figure 21: Drying shrinkage strain of normal-weight concrete for t approaching infinity

Page 55DIN 1045-1 : 2001-07

9.1.5 Stress-strain curve for non-linear methods of analysis and for strain calculations(1) The stress-strain curve as shown in figure 22, to be used for non-linear methods of analysis and forcalculations of strain in concrete subjected to short-term stresses and uniaxial compression, is described asfollows:

sc k ëh – h2

= –Œ › (62)fc 1 + (k – 2)h

where

h = ec/ec1 (63)

and

k = –1,1 Ecm · ec1/fc (64)

In the above,Ecm is the modulus of elasticity (from table 9 or 10);ec1 is the compressive strain in the concrete on reaching the maximum of fc (from table 9 or 10);fc is the maximum concrete compressive strength; in non-linear methods of analysis, it may be assumed

equal to fcR as specified in subclause 8.5.1, and for strain calculations, equal to fcm.

Equation (62) shall apply for values of ec not greater than zero and not less than ec1u, with ec1u being thecompressive strain on reaching the limits of strength (see table 9 or 10).(2) Other idealized curves may be used if equivalent in approach to that described in item (1).

Figure 22: Stress-strain curve for non-linear methods of analysis and strain calculations

9.1.6 Stress-strain curve for section design(1) The stress-strain curve in figure 23 shall be used for section design. It is obtained by means of equations(65) and (66):

ec n

sc = –fcdë¯1–Œ1– › × with ec not greater than zero and not less than ec2 (65) ec2

sc = –fcd with ec not greater than ec2 and not less than ec2u (66)

wheren is the exponent of the parabola;ec2 is the compressive strain on reaching the limit of strength;ec2u is the maximum strain in the concrete.

Values shall be taken from table 9 or 10.

Page 56DIN 1045-1 : 2001-07

Figure 23: Parabolic-rectangle stress-strain curve

(2) The design unconfined concrete compressive strength, fcd, obtained by means of equation (67), shall beused as the basis for design for the ultimate limit state:

fcd = a · fck/gc (67)

wherea is a reduction factor taking into account long-term effects on the concrete strength and for mutual conver-

sion of the cylinder compressive strength and unconfined compressive strength (it shall be assumed equalto 0,85 for normal-weight concrete; in some cases (e.g. in short-term loading), higher values may beassumed (with a, however, not greater than unity); for lightweight concrete, a shall be selected as specifiedin item (4) below);

gc is the partial safety factor for concrete taken from table 2 or, for unreinforced members, as specified insubclause 5.3.3 (8); it shall be multiplied by gcq where concrete of strength class C55/67 or LC55/60 is used(see item (9) of subclause 5.3.3).

(3) Other idealized stress-strain curves (e.g. the bilinear curve in figure 24 with the values from table 9 ortable 10) may be used if equivalent to the parabolic-rectangle curve in respect of the distribution of compressivestresses. If the line of zero strain is located in the cross section, the stress block from figure 25 may also be usedas the pattern of stress in the concrete under the given conditions.

Figure 24: Bilinear stress-strain curve

(4) For lightweight concrete, a shall be assumed equal to 0,75 when using the curve in figure 23 or the diagramin figure 25, and 0,8 when using that in figure 24.

Page 57DIN 1045-1 : 2001-07

Key:x R 0,95 for fck equal to or less than 50 N/mm2

x = 1,05 – fck/500 for fck above 50 N/mm2

k = 0,80 for fck equal to or less than 50 N/mm2

k = 1,0 – fck/250 for fck above 50 N/mm2

NOTE: If the cross-sectional width decreases towards the edge in compression, fcd shall be reduced by a factorof 0,9.

Figure 25: Stress block

9.1.7 Characteristic values of concrete(1) The characteristic values of concrete are given in tables 9 and 10.(2) fcd calculated as specified in item (2) of subclause 9.1.6 is the design unconfined compressive strength ofthe concrete in its uncracked state. In the presence of transverse tensile stresses or transverse cracking, theresulting reduction in compressive strength shall be taken into account.(3) For simplification, this reduction may be taken to be equal to ac · fcd (with ac from subclause 10.3.4).(4) Higher strengths may be assumed where confined compressive stress is to be taken into account.

9.2 Reinforcement9.2.1 General(1) This clause is applicable to reinforcing bars and reinforcing fabric in their as-delivered condition, conform-ing to the DIN 488 series of standards and the relevant agréments. For coiled reinforcement, the requirementsshall apply to their condition following straightening.(2) Reinforcement covered by an agrément may only be used for concrete of strength class C70/85 if this isprovided for in the agrément.

9.2.2 Properties(1) The specifications given in this standard relate to weldable, ribbed reinforcing steel with a characteristicyield strength, fyk, of 500 N/mm2 and the other properties specified in table 11. Where relevant, these propertiesalso apply with regard to tensile and compressive stresses. For steel with properties deviating from those givenin table 11, specifications other than those given in this standard may be necessary.(2) Normally, the ductility requirements for reinforcing steel covered by an agrément (i.e. whether it is to beof normal or high ductility) are governed by the agrément. Otherwise, reinforcing steel shall be classed as beingof normal ductility.(3) The characteristic yield strength, fyk (denoted Re in the DIN 488 series of standards), and the characteristictensile strength, ftk (denoted Rm in the DIN 488 series of standards), are obtained as the ratio of the load at whichthe steel reaches its characteristic yield strength or the maximum load to the nominal cross section.(4) The parameter f0,2k may be used (see figure 26) for products without a marked characteristic yield strength.(5) All types of reinforcing steel are designed for temperatures ranging between –60 °C and 200 °C. Shouldthe temperature of the steel only once go beyond these limits, the material properties may be expected tochange significantly (see also item (3) of subclause 12.3.2).(6) A bend test is to be used to identify the suitability of steel for bending. The values given in table 11 applyto steel of a temperature higher than –10 °C.(7) Reinforcing steel shall be suitable for welding commensurate with the anticipated weld and welding pro-cess specified in table 12. Welding operations shall be executed as specified in DIN 4099-1.

Page 58DIN 1045-1 : 2001-07

Table 11: Properties of reinforcement

1 2 3 4 5

Quantile,Bars Fabric Bars Fabric as a

BSt 500 S BSt 500 M BSt 500 S BSt 500 M percentage

Ductility Normal High

1 Characteristic yield strength, fyk,in N/mm2 500 5

2 (ft/fy)k ratio 1,05 or greater 1,08 or greater 10 min.

3 (fy/fyk)0,90 ratio – Up to 1,3 10 max.

Characteristic strain in reinforce-4 ment at maximum load, euk, 25 50 10

as per thousand

Characteristic fatiguestrength, ffatk, in N/mm2, Up to 190 100 190 100

5at N equal to 2 · 106, 28 days

10(with the maximum notexceeding 0,6 fy)

a) for a Above – – 150 –bar diameter, ds, in mm 28 days

Projected rib factor, fR,for ds, in mm, from

65,0 to 6,0 0,039

5 min.6,5 to 8,5 0,0459,0 to 10,5 0,052

11,0 to 40,0 0,056

8 Undersize of nominal cross section,as a percentage

4 5 max.

Mandrel diameter in bendingtest for ds, in mm, from

9 6 to 12 5 ds14 to 16 6 ds

1 min.

20 to 25 8 ds28 to 40 10 ds

a) If a greater number of cycles is established empirically, the N* values from table 16 may be changedaccordingly.

Characteristic Product form and grade

Page 59DIN 1045-1 : 2001-07

Table 12: Permitted welding processes, joints and applications

Type of loading

1 2 3

Welding process, with reference number Type of joint and application

from DIN EN ISO 4063 Tension bars a) Compression bars a)

1 Flash welding 24 Butt joints

2Manual metal-arc welding 111 Butt joints where ds ö 20 mm, strap joints,Self-shielded tubular cored arc lap joints, cross joints c), joints with otherwelding

114steel components

3 135 Strap joints, lap joints, cross joints c),connections to other steel components

4 136 —Butt joints whereds ö 20 mm

5 Friction welding 42Butt joints, joints with other steelcomponents

6 Spot welding 21 Lap joints d) or strap joints b) d)

7 Flash welding 24 Butt joints

8 Manual metal-arc welding 111 — Butt joints whereds ö 16 mm

9 Metal active gas welding 135 — Butt joints where136 ds ö 20 mm

a) Bars of the same nominal diameter or of adjacent sizes may be welded together.b) The permitted ratio of nominal diameters of intersecting bars shall be not less than 0,57.c) With ds not exceeding 16 mm for loadbearing joints.d) With ds not exceeding 28 mm for loadbearing joints.

Predominatelystatic

Notpredominatelystatic

Metal active gas welding b)

9.2.3 Stress-strain curve for global analysis(1) In non-linear methods of analysis, a stress-strain curve as shown in figure 26 (continuous line) with es notgreater than euk shall be assumed.(2) By way of simplification, an idealized stress-strain curve (thick dashed line in figure 26) may be assumed,substituting fyR (from subclause 8.5.1) for fy.

Key:1 Idealized curve

Figure 26: Stress-strain curve of reinforcement for global analysis

Page 60DIN 1045-1 : 2001-07

9.2.4 Stress-strain curve for section design(1) The section design shall be carried out on the basis of the nominal area of cross section and the nominaldiameter, using the idealized stress-strain curve from figure 27 (thick dashed line).(2) Alternative 3 from figure 27 (dashed dotted line) may be used for simplified analysis.

Key:1 Idealized curve2 Curve for section design3 Simplified curve

Figure 27: Stress-strain curve of reinforcing steel for section design

(3) In the design, ftk,cal shall be assumed to be 525 N/mm2 and the strain in the steel, es shall be limited to avalue of esu equal to 0,025.(4) Unless otherwise specified in the DIN 488 standards series or in agréments, reinforcing steel may, fordesign purposes, be taken to have a coefficient of linear thermal expansion, a, of 10 · 10–6 K–1 and a modulusof elasticity, Es, of 200 000 N/mm2.

9.3 Prestressing steel9.3.1 General(1) This subclause covers wires, strands and bars used as prestressing steel in concrete structures.(2) The requirements of this subclause apply to the products in their as-delivered condition.(3) The provisions of the agréments shall apply with regard to product groups, production methods, proper-ties, test methods and conformity assessment.

9.3.2 Properties(1) The behaviour of prestressing steel can be identified by virtue of the following:

a) characteristic 0,1 % proof stress, fp0,1k;b) characteristic tensile strength, fpk;c) characteristic strain at maximum load, euk;d) modulus of elasticity, Ep;e) cross-sectional tolerances;f) fatigue strength;g) ductility;h) relaxation;i) surface texture (bonding properties).

(2) Prestressing steel shall have the properties specified in this standard.(3) The required tolerances on cross section and provisions relating to the surface texture of prestressing steelshall be taken from agréments.(4) In non-linear methods of analysis, a stress-strain curve as given in figure 28 (continuous thick line) shallbe used.(5) By way of simplification, an idealized curve (thick dashed line in figure 28) may be assumed, substitutingfp0,1R and fpR from subclause 8.5.1 for fp0,1 and fp, respectively.

Page 61DIN 1045-1 : 2001-07

Key:1 Idealized curve

Figure 28: Stress-strain curve of prestressing steel for global analysis

(6) It may generally be assumed that post-tensioned members and unbonded tendons are of high ductility andpre-tensioned members are of normal ductility.

9.3.3 Stress-strain curve for section design(1) Section design shall be carried out on the basis of the nominal area of cross section of the steel, using thetheoretical curve from figure 29.(2) Alternative 3 from figure 29 (dashed dotted line) may be used for simplification.

Key:1 Idealized curve2 Curve for section design3 Simplified curve

Figure 29: Theoretical stress-strain curve of prestressing steel for section esign

(3) The strain in the steel, ep, shall be limited to a value equal to (ep(0) + 0,025).

(4) The curve in figure 29 shall apply for temperatures between –20 °C and +200 °C.(5) Unless otherwise specified in agréments, prestressing steel may, for design purposes, be taken to havea coefficient of linear thermal expansion, a, of 10 · 10– 6 K–1 and a modulus of elasticity, Ep, of 195 000 N/mm2

(for strands) and 205 000 N/mm2 (for bars and wire).These values may be taken as characteristic values for theabove temperature range.

Page 62DIN 1045-1 : 2001-07

10 Analyses for the ultimate limit state10.1 GeneralSubclauses 10.2 to 10.4 contain provisions for beams, slabs and similar members whose cross sections areassumed to remain plane. Subclause 10.6 contains provisions for the design and construction of members likelyto be affected by stressing, and deep beams and other members that are not anticipated to remain plane.

10.2 Bending, axial force and coexistent bending and axial force(1) The following assumptions shall be made when determining the ultimate loadbearing capacity of reinforcedcross sections:

a) cross sections remain plane;b) there is a rigid bond between concrete and reinforcement;c) the tensile strength of the concrete is neglected;d) compression in the concrete is derived from the design stress-strain curves in subclause 9.1.6;e) the stress-strain curve for reinforcing steel is as specified in subclause 9.2.4 and for prestressing steelas specified in subclause 9.3.3;f) initial pre-strain, ep

(0), is taken into account when establishing the stress in the tendons.

(2) In unreinforced cross sections, the following assumptions and principles shall be made:a) cross sections remain plane;b) tension is not usually taken into consideration 9);c) the compression in the concrete may, optionally, be derived from the stress-strain curves in subclause9.1.6;d) calculations use a concrete strength class not higher than C35/45 or LC20/22.

(3) The compressive strain in the concrete shall be not more than ec2u from table 9 or (for lightweight concrete)elc2u from table 10, the strain in the reinforcing steel and in the prestressing steel, esu and ep

(0), respectively, shallnot exceed +0,025 (cf. figure 30).

Figure 30: Possible distribution of strain in reinforcement and prestressingsteel at the ultimate limit state

(4) In cross sections remaining in compression, the compressive strain at point C shall be not greater than ec2or elc2 from table 9 or 10.(5) Where there are slight eccentricities in normal-weight concrete (i.e. the ed/h ratio is not greater than 0,1),the favourable effect due to creep may, for simplification, be taken into account by giving ec2 a value of –0,002 2.(6) In the flanges of T-beams, box beams or members of similar cross section, that remain in full compression,strain in the centre of the slab shall not exceed the ec2 or elc2 values from table 9 or 10. The loadbearing capacityof the overall cross section does not need to be assumed to be less than that of the webs of height h and witha strain distribution as shown in figure 30.(7) Where structures have eccentric internal unbonded tendons, the increase in stress in these tendons,Dsp,may, for simplification, be assumed to be 100 N/mm2.(8) Tension reinforcement shall be desgined taking into account item (9) of subclause 10.3.4.

9) Exceptions are, for example, foundations, which are to be designed using fctk;0,05/gc (taking the value of gc forplain concrete as specified in item (8) of subclause 5.3.3 (cf. item (2) of subclause 10.3.3)).

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10.3 Shear10.3.1 Methods of analysis(1) Shear resistance is limited by various failure mechanisms. This is taken into account by use of the followingdesign values:VRd,ct as the design shear resistance of members without shear reinforcement;VRd,sy as the design shear resistance, limited by the capacity of the shear reinforcement;VRd,max as the design shear resistance, limited by the strength of the struts.

(2) According to design calculations, sections in which the design shear force, VEd, is not greater than VRd,ctdo not require shear reinforcement (cf. subclause 10.3.3). Beams and one-way spanning slabs with a b/h ratioof less than 5, however, always require provision of a minimum amount of shear reinforcement as specified insubclause 13.2.3 or 13.3.3.(3) Where VEd is greater than VRd,ct, shear reinforcement shall be provided so that VEd is not greater than VRd,sy(cf. subclause 10.3.4) and the specifications for the required minimum shear reinforcement specified insubclauses 13.2.3 and 13.3.3 are observed.(4) At no point in the cross section shall VEd exceed VRd,max (cf. subclause 10.3.4).

10.3.2 Design shear force(1) Where there is an evenly distributed load with conditions of direct support (cf. item (7) of subclause 7.3.1),the design shear force, VEd, used for calculation of the shear reinforcement may be assumed to act at a distanced from the edge of the support, in order to take into account the direct transmission to the support of compo-nents of the load close to the support.(2) In conditions of direct support, the shear component of a concentrated load acting at a distance, x, equalto 2,5 d or less from the edge of the support may be reduced by a factor b calculated using equation (68):

xb = (68) 2,5 d

(3) Items (1) and (2) above do not apply when determining VRd,max.(4) In members that are of variable effective depth or have inclined tendons, the design shear force, taking intoaccount the shear component in the tension and compression flanges normal to the member axis (cf. figure 31),shall be obtained by means of equation (69):

VEd = VEd0 – Vccd – Vtd – Vpd (69)

Key:1 Line of action of compressive force in the concrete2 Line of zero action3 Centroid of tendons4 Centroid of reinforcement

In the figure,VEd is the design shear force;VEd0 is the basic design shear force effective in the cross section;Vccd is the design shear component in the compression zone;Vtd is the design shear component in the tension zone (reinforcement);Vpd is the design shear component in the prestressing steel at the ultimate limit state (cf. subclause 8.7.5);

however, the condition Pmt ß Ap · fp0,1k/gs is to be satisfied).

Figure 31: Shear in cross sections of variable effective depth

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10.3.3 Members not requiring design shear reinforcement(1) The design shear resistance of members designed for bending without there being a need for the reinforce-ment to make a contribution to shear resistance, VRd,ct, shall generally be obtained by means of equation (70):

VRd,ct = [0,10 ¡ · h1 · (100 r1 · fck)1/3 – 0,12 scd] · bw · d (70)

with

200¡ = 1 + 1 ß 2,0 (71) d

In the above:h1 is equal to unity for normal-weight concrete and is to be taken from table 10 for lightweight concrete;r1 is the longitudinal reinforcement ratio, calculated as Asl/bwëd = 0,02;Asl is the area of tension reinforcement that extends not less than a distance d beyond the section considered

and is effectively anchored at that point (cf. figure 32); in the case of pre-tensioned members, the full areaof prestressing steel may be added to Asl;

bw is the minimum width of cross section in the tension zone, in mm;d is the effective depth of the tension reinforcement in the section considered, in mm;fck is the characteristic concrete compressive strength, in N/mm2;scd is the design axial stress in the concrete at the centroid of the section, obtained as Ned/Ac, in N/mm2;NEd is the design axial force in the section considered due to prestressing or other actions (having a value less

than zero when in compression).

Key:1 Section considered

Figure 32: Area of tension reinforcement, Asl, for determining longitudinal reinforcement ratio

(2) If it is shown that the tensile stresses in the concrete at the ultimate limit state are at all times less thanfctk;0,05/gc (using the values of gc for plain concrete as specified in item (8) of subclause 5.3.3), the shear resistancein those regions of reinforced and prestressed concrete members that are near to supports and are subjectedto predominately static loading shall be obtained by means of equation (72).

I ë bw fctk;0,05

2 fctk;0,05VRd,ct = ë1Œ › – alëscdë (72)

S gc gc

whereI is the second moment of area of the section;S is the first moment of area abour its centroid;al is a coefficient equal to lx/lbpd but not greater than unity (in pre-tensioned members) and equal to unity

in other cases;lx is the distance of the cross section considered from the beginning of the anchorage length of the

tendon;lbpd is the upper design transmission length of the tendon as specified in item (6) of subclause 8.7.6;fctk;0,05 is the characteristic 5 % quantile of axial tensile strength of concrete from table 9 or table 10 (to be not

greater than 2,7 N/mm2);gc is the partial safety factor for plain concrete as specified in item (8) of subclause 5.3.3;bw is the minimum width of cross section;scd is the design axial stress as in equation (71).

This analysis does not need to be performed for sections located less than h/2 from the front edge of the support.

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10.3.4 Members requiring design shear reinforcement(1) Shear design of members designed for coexistent bending and shear is based on truss analogy (seefigure 33). The inclination of the struts, θ, shall be as specified in item (3).(2) In the analysis of shear resistance, the internal lever arm, z, may normally be taken to be approximatelyequal to 0,9 d, assuming that in the case of members with inclined tendons in the precompressed tension zonethere is longitudinal reinforcement sufficient to sustain axial tension due to shear and that the links as specifiedin item (2) of subclause 12.7 are anchored in the compression zone. However, z shall be not greater thand – 2 cnom (cnom being the nominal concrete cover of the longitudinal reinforcement in the compression zone).

(3) The inclination of struts, θ, shall be limited as follows:

1,2 – 1,4 scd/fcd ß 3,0 for normal-weight concrete0,58 ß cot θ ß œ (73) 1 – VRd,c/VEd ß 2,0 for lightweight concrete

with

scdVRd,c = bctë0,10ëh1ëfck1/3ëŒ1 + 1,2 ›ëbwëz (74)

fcd

In the above:bct is a roughness factor (here, bct = 2,4);h1 is equal to unity for normal-weight concrete; see table 10 for lightweight concrete;scd is the design axial stress in the concrete at the centroid of the section (equal to Ned/Ac), in N/mm2;NEd is the design axial force in the cross section due to external action or prestressing (required to have a value

less than zero when in compression).

Key:1 Strut2 Compression flange3 Tie; shear reinforcement4 Tension flange; longitudinal reinforcement

In the figure,a is the angle of shear reinforcement with the longitudinal axis;θ is the angle of concrete struts with the longitudinal axis;Fsd is the design tensile force in the longitudinal reinforcement;Fcd is the design concrete compression in the axial plane;bw is the minimum width of the web;z is the internal lever arm in the part of the member considered;DFsd is the component of tensile force due to shear, equal to 0,5 |VEd| (cot θ – cot a)

Figure 33: Truss system and notation for members with shear reinforcement

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(4) VRd,sy shall be obtained by means of equation (75) for members with shear reinforcement:

AswVRd,sy = ëfydëzëcot θ (75) sw

where sw is the reinforcement spacing normal to the longitudinal axis, measured axially.

(5) By way of simplification, cot θ in equation (75) may be assumed to be equal to 1,2 for bending alone andfor coexistent bending and axial compression, and equal to unity for coexistent bending and axial tension.(6) The design shear resistance limited by the strength of struts, VRd,max, shall be obtained by means ofequation (76) for members with shear reinforcement normal to the (longitudinal) axis:

bwëzëacëfckVRd,max = (76) cot θ + tan θ

where ac is a reduction factor for calculation of concrete compressive strength; equal to 0,75 h1 (with h1 givena value equal to unity for normal-weight concrete and to be taken from table 10 for lightweight concrete).(7) In members with inclined shear reinforcement, VRd,sy and VRd,max shall be calculated taking into account theangle of shear reinforcement to the (longitudinal) axis of the member, a, by means of equations (77) and (78):

AswVRd,sy = ëfydëzë(cot θ + cot a)ësin a (77) sw

cot θ + cot aVRd,max = bwëzëacëfcdë (78) 1 + cot2 θ

whereac is a reduction factor as in item (6) above; equal to 0,75 h1, (with h1 given a value equal to unity for normal-

weight concrete and to be taken from table 10 for lightweight concrete);sw is the inclined shear reinforcement spacing, measured axially.

(8) If the section considered contains a row of adjacent grouted tendons with an overall diameter, 7dh, greaterthan bw/8, VRd,max, obtained by means of equation (76) or (78), shall be calculated on the basis of the nominalweb thickness, bw,nom, with the tendons in their most unfavourable position:

bw,nom = bw – 0,5 7dh for strength classes up to class C50/60 or LC50/55 (79)

or

bw,nom = bw – 1,0 7dh for strength class C55/67 or LC55/60, or higher (80)

where dh is the external sheath diameter.

The following shall apply for bw,nom where there are adjacent ungrouted or unbonded tendons:

bw,nom = bw – 1,3 7dh (81)

(9) The component of the tensile force due to shear, DFsd, shall be taken into account as shown in figure 33.Alternatively, the provisions of item (3) of subclause 13.2.2 may be applied.

10.3.5 Shear between web and flange(1) The connection of compression and tension flanges shall be analysed by truss analogy.(2) The design shear force, VEd, may be obtained as follows:

VEd = DFd (82)

where DFd is the variation in the axial force acting in a one-sided flange section of length av, in which the axialshear may be assumed constant (cf. figure 34).av shall be taken to be not more than half the distance between the points of zero moment and the maximummoment. Where there are high concentrated loads, the length of each section shall not extend beyond the pointsat which the shear force rises sharply.(3) Shear analysis may be carried out as specified in subclause 10.3.4, substituting hf for bw and av for z inequations (75) to (81). The mean axial stress in the concrete in the adjacent section of flange assumed to be oflength av shall be substituted for scd. By way of simplification, cot θ shall be assumed equal to unity in the tensionflange and equal to 1,2 in the compression flange.

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(4) In the absence of a more detailed analysis of coexistent stresses due to shear between web and flange andtransverse bending, the greater of the areas of steel required for each side, as calculated in the above or in thedesign for transverse bending, shall be selected. In the calculations, the zones of flexural compression andflexural tension shall be given separate treatment, using for each half of the shear reinforcement required forshear alone.

Key:1 Struts2 Point from which longitudinal reinforcement in the flange is to be anchored

Figure 34: Connection of flange and web (notation)

10.3.6 Shear transfer in joints(1) The transfer of shear in the joints between adjacent precast elements, between in-situ concrete and aprecast element, or between runs of in-situ concrete is dependent on the surface condition in the joint. Thefollowing terminology is used to describe that condition:

a) ‘very smooth’: the surface has been produced by concreting against steel or smooth formwork;b) ‘smooth’: the surface has been smoothed or made by slipforming or extrusion, or has not been treatedfollowing compaction;c) ‘rough’: the surface satisfies the criteria defining roughness 10);d) ‘corrugated’: the geometry of the surface is as shown in figure 35 a) or the grain structure is exposed.

(2) The design shear resistance per unit length, vEd, to be transferred across the contact faces between in-situconcrete and precast elements or between two concreting sections may be obtained by means of equation (83).

Fcdj VEdvEd = ë (83) Fcd z

whereFcdj is the design axial force to be transferred via the joint;Fcd is the design axial force in the flange due to bending of the section considered, calculated as MEd/z.

(3) In the absence of joint reinforcement, the design shear resistance in joints of composite members, vRd,ct,including joints between floor and walling units, shall be calculated as follows:

vRd,ct = [0,042 · h1 · bct · fck1/3 – m · sNd] · b (84)

whereh1 is equal to unity for normal-weight concrete and is to be taken from table (10) for lightweight concrete;bct is the roughness factor from table 13 and item (4) below;fck is the characteristic compressive strength of the in-situ concrete or precast element (whichever is less),

in N/mm2;m is the coefficient of friction from table 13;sNd is the stress normal to the joint (less than zero if compression), in N/mm2, calculated as follows:

nEdsNd = ö –0,6 fcd b

wherenEd is the lower design axial force normal to the joint per unit length (cf. figure 35 a);b is the width of the contact face (e.g. width of a horizontal joint).

10) See DAfStb-Heft 525 for definition of ‘surface roughness’.

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Table 13: Coefficients of friction and roughness factors

Surface condition Roughness factor, bct Coefficient of friction, m

1 Corrugated 2,4 1

2 Rough 2,a) 0,7

3 Smooth 1,4a) 0,6

4 Very smooth 0 0,5

a) See item (4) below.

Key:a) Corrugated surfaceb) Shear resistance diagram for distribution of required joint reinforcement

1 1st concreting section2 2nd concreting section3 Internal tie4 Joint

Figure 35: Joint design

(4) bct shall be equal to zero in cases where rough or smooth joints are subjected to tension acting normal totheir faces.(5) Where reinforcement passes through joints in composite members, including joints between floor andwalling units, the design shear resistance limited by the capacity of shear reinforcement, vRd,sy, shall be calcu-lated as follows:

vRd,sy = as · fyd · (cot θ + cot a) · sin a – m · sNd · b (85)

whereas is the area of the reinforcement passing through the joint, per unit length;a is the angle of the reinforcement passing through the joint, to the member axis (cf. figure 35 a)), with a value

between 45° and 90°.

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The inclination of struts shall satisfy the following condition:

1,2 m – 1,4 scd/fcd ß 3 for normal-weight concrete1,0 ß cot θ ß œ (86) 1 – vRd,c/vEd ß 2 for lightweight concrete

with vRd,ct taken from equation (84).

scd shall be given a value equal to the design axial stress in the adjacent section (less than zero if compression).

(6) Determination of the reinforcement necessary shall be based on equation (85). Where members are sub-jected to bending, the reinforcement may be arranged incrementally to correspond with the line of shear force(cf. figure 35 b)). Members acting as plates may have a concentration of reinforcement at the ends of the joint.The shear reinforcement for the joint shall be anchored as specified in this standard at both sides of the contactface.(7) Where wall plates have peripheral ties and internal ties as specified in item (4) of subclause 13.12.3, jointsshall be checked applying the coefficients bct and m from table 13. However, for slabs without interlocking joints,vRd should be given a value, in N/mm2, not greater than 0,15 b (with b in m).(8) If precast slabs with an in-situ topping are to be designed to withstand permanent suspended loads, thebond shall be verified in the region in which the load is directly introduced.

10.3.7 Unreinforced members(1) The tensile strength of plain concrete at the ultimate limit state shall be taken into account when consid-ering shear if it can be proved that it retains its tensile strength even if cracking occurs.(2) An unreinforced member may be considered free of cracks if, at the ultimate limit state, it is in compressionin all relevant design situations or if the principal tensile stress in the concrete is not greater than 1 N/mm2.(3) If cracking cannot be ruled out, the design shear resistance, VRd, shall be determined in the uncracked partof the member, with calculations based on the stress condition in the most unfavourable design situation.(4) If the conditions set out in item (2) of subclause 10.3.3 apply, the shear resistance of unreinforced memberssubjected to coexistent shear, bending and axial stress may be obtained by means of equation (72), with a1assumed equal to unity.

10.4 Torsion10.4.1 General(1) Where the static equilibrium of a structure depends upon the torsional resistance of its elements, full designfor torsion, covering both the ultimate and serviceability limit states, will be necessary.(2) Where, in hyperstatic structures, torsion occurs only when the compatibility conditions are met, it willusually be unnecessary to take torsional rigidity into account in the analysis. However, links and longitudinalbars shall be provided to avoid excessive cracking. Compliance with the requirements specified in sub-clauses 11.2 and 13.2.4 will normally be sufficient for this purpose.(3) The torsional resistance of sections is to be calculated assuming a thin-walled closed section, in which theequilibrium conditions are satisfied by the presence of the corresponding shear flow. Solid sections may bereplaced by equivalent thin-walled sections (cf. figure 36 b)). The equivalent wall thickness of hollow sectionsshall not exceed the actual wall thickness. Sections of complex shape, such as ‘T’ sections, may be dividedinto a series of subsections. The overall torsional resistance shall then be taken as the sum of the resistancesof the subsections.(4) The distribution of torque over the series of subsections may generally be assumed to be in the samerelationship as the stiffnesses of the uncracked subsections.(5) Calculations may be carried out separately for each subsection.(6) In solid sections that are approximately rectangular in shape, no shear or torsion reinforcement other thanthe minimum reinforcement as specified in item (5) of subclause 13.2.3 is necessary provided the design torque,TEd, satisfies the following conditions:

VEd ë bwTEd ß (87) 4,5

with

4,5 TEdVEd ¯1 + × ß VRd,ct (88) VEd ë bw

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10.4.2 Methods of analysis(1) The shear force due to torsion, VEd,T, that acts in a wall of the section considered shall be obtained bymeans of equation (89):

TEd ë zVEd,T = (89) 2 Ak

whereTEd is the design torque;Ak is the area enveloped by the centrelines of the walls;z is the height of the wall, defined as the distance of the points of intersection of the centreline of the wall

with the centrelines of adjacent walls.

The centrelines of the walls are defined by lines connecting the longitudinal bars in the corners (cf. figure 36 b)).

(2) The torsion reinforcement in a wall of the section considered shall be designed by truss analogy(cf. figure 36 b)), with the struts inclined at an angle, θ, as specified in item (3) of subclause 10.3.4. Any shearforce in the wall due to coexistent torsion and shear, VEd,T+V, obtained by means of equation (90) shallbe substituted for VEd in equation (73), and the effective wall thickness, teff, shall be substituted for bw inequation (74). Angle θ shall be used for separate analysis of shear and torsion, with the reinforcement require-ment for each being subsequently combined.

VEd ë teffVEd,T+V = VEd,T + (90) bw

whereVEd is the design shear force from subclause 10.3.2;teff is the effective wall thickness, equal to twice the distance between centreline and external face but not

more than the actual wall thickness (cf. figure 36).

For simplification, the reinforcement requirement for torsion only may be calculated (assuming a θ value of 45°)and added to the shear reinforcement determined as specified in subclause 10.3.4.

Key:1 Link2 Longitudinal bars3 Centreline of wall (i)4 Shear flow, VEd,i/zi

a) Shear flow (notation)b) Equivalent hollow section and equivalent wall for truss analogy

Figure 36: Torsion (notation) and equivalent systems

Detail X

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(3) The design resistance torque of the section or subsection, TRd,sy, shall be obtained by means of equations(91) and (92), taking as a basis the minimum centreline length of the section considered.

AswTRd,sy = ëfydë2 Akëcot θ (91) sw

or

AslTRd,sy = ëfydë2 Akëtan θ (92) uk

whereAsw is the (sectional) area of torsion reinforcement normal to the member axis;sw is the spacing of the torsion reinforcement along the member axis;fyd is the design yield strength of steel;Asl is the (sectional) area of the longitudinal torsion reinforcement;Ak is the (sectional) area enclosed by the centreline;θ is the angle between struts and member axis;uk is the perimeter of area Ak.

For compression flanges, the longitudinal torsion reinforcement may be reduced in relation to the compressiveforces that are in action; for tension flanges, it shall be added to the other longitudinal reinforcement.

(4) When considering torsion alone, the maximum design resistance torque in the section or subsection,TRd,max, shall be obtained by means of equation (93), taking as a basis the minimum centreline length of thesection considered:

ac,redëfcdë2 AkëteffTRd,max = (93) cot θ + tan θ

with ac,red equal to ac in box sections with reinforcement at the inner and outer faces of the walls, and equal to0,7 ac in other cases (cf. item (6) of subclause 10.3.4 for ac).

(5) Torsional resistance of sections or subsections is limited by the torsional resistance of the struts. This mustbe assumed where there is coexistent shear and torsion and the section or each subsection meets the followingconditions:

a) solid sections:

TEd 2 VEd

2

¯ × + ¯ × ß 1 (94) TRd,max VRd,max

b) box sections:

TEd VEd + ß 1 (95)

TRd,max VRd,max

10.4.3 Warping torsion(1) In the ultimate limit state, it is usually safe to ignore stresses due to restrained warping of a section.(2) When considering closed thin-walled sections and solid sections, warping may normally be neglected.

10.4.4 Unreinforced members(1) Subclause 10.3.7 will apply by analogy for unreinforced members subjected to coexistent torsion andshear.(2) A cracked member shall not be assumed to have the capacity to resist torque unless it can be verified ashaving such.

10.5 Punching shear10.5.1 General(1) The principles and provisions set out in this subclause supplement those of subclause 10.3. They deal withpunching shear in slabs containing bending reinforcement determined in accordance with subclause 10.2 andpunching shear in foundations and ribbed slabs with a solid section around the loaded area (see item (2)) if thesolid section is not less than 1,5 d further than the critical section.

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(2) Punching shear may be due to concentrated loads or bearing reactions acting over a relatively small area,known as the loaded area, Aload.(3) Figure 37 shows an appropriate design model for checking punching shear failure at the ultimate limit state,with the critical area, Acrit, assumed to be parallel to Aload.(4) The analysis of punching shear resistance shall be carried out along defined perimeters, outside of whichthe member is to meet the requirements of subclause 10.3.

Key:a) Cross sectionb) Plan view of slab (br = 33,7°)

1 Slab2 Foundation slab

Figure 37: Design model for analysis of punching shear resistance

10.5.2 Loaded areas and critical sections used in analyses(1) The specifications of this clause may be used when considering loaded areas of the following shapes:

a) circular, with a diameter up to 3,5 d (d being the mean effective depth of the member considered);b) rectangular, with a perimeter not exceeding 11 d and a ratio of length to breadth of 2 or less;c) any shape, with dimensions similar to the above.

The critical sections of adjacent loaded areas shall not overlap.

(2) If the conditions of item (1) above are not satisfied by slabs supported on walls or columns due to the shearforce being concentrated in the corners of the bearing surfaces, the critical sections in figure 38 shall be usedunless a more detailed analysis is made.

Key:1 Loaded area, Aload

2 Parts of critical sections relevant to punching shear

Figure 38: Parts of critical section not satisfying item (1) of subclause 10.5.2,relevant to punching shear

3 Loaded area, Aload

4 Critical section5 Critical radius6 Critical area, Acrit

7 Critical perimeter, ucrit

a1 ß a, 2b, or 5,6d – b1

b1 ß b or 2,8d

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(3) The critical section around a circular or rectangular loaded area located away from unsupported edges isto be assumed situated at a distance of 1,5 d from the loaded area (cf. figure 39).

Key:1 Loaded area, Aload

See figure 37 for d.

Figure 39: Critical section around loaded areas located away from an unsupported edge

(4) The critical area, Acrit, is the area enveloped by the critical section.(5) Other sections within and outside the critical area shall be assumed to be affine in shape to the criticalsection.(6) For loaded areas whose perimeters are located not more than 6 d from openings, the part of the criticalsection contained between two tangents drawn to the outline of the opening from the centre of the loaded areais considered to be ineffective (cf. figure 40).

Key:1 Loaded area, Aload

2 OpeningIf l1 is greater than l2, then l2 is equal to 1l1 · l2. See figure 37 for d.

Figure 40: Critical section near an opening

(7) For loaded areas situated near an unsupported edge or an unsupported corner, the critical section shallbe assumed to be as shown in figure 41 if this gives a perimeter (not including the unsupported edges) less thanthat obtained from items (3) and (6) above.

Key:1 Loaded area, Aload

2 Unsupported edgeSee figure 37 for d.

Figure 41: Critical section near an unsupported edge

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(8) Where the edge of a loaded area is more than 3 d from an unsupported edge, the shear resistance shallbe determined on the basis of a critical section as shown in figure 39.(9) Where loaded areas are located at or near (i.e. at a distance less than d from) an unsupported edge orcorner), special edge reinforcement as specified in item (10) of subclause 13.3.2, with a link spacing, sw, of notmore than 100 mm, is required along the unsupported edge.(10) For columns with inclined heads with a distance from the column face to the edge of the column head,lH, not greater than 1,5 hH (cf. figure 42), an analysis is only required in the part of the critical section not locatedwithin the column head. The distance of this section from the centroid of the loaded area, rcrit, is given byequation (96).

rcrit = 1,5 d + lH + 0,5 lc (96)

wherelH is the distance from the column face to the edge of the column head;lc is the diameter of a circular loaded area.

In the case of rectangular columns with a rectangular head with lH not greater than 1,5 hH (cf. figure 42) and thewidth of the loaded area, bc, not greater than its depth, hc, the smaller of the following values shall be takenas rcrit:

1,5 d + 0,56 1bcëhcrcrit = œ (97) 1,5 d + 0,64 bc

For columns with stepped heads with lH not greater than 1,5 hH, the full area of the column head shall beassumed as the loaded area.

Key:1 Critical section2 Loaded area, Aload

Figure 42: Slab with column head with lH not greater than 1,5 hH

(11) For slabs with column heads where lH is greater than 1,5 hH (cf. figure 43), the analysis of the criticalsection outside the column head shall be accompanied by an analysis of the critical sections within the columnhead.(12) The outer and inner distances from the centre of the loaded area to the critical sections in figure 43, rcrit,exand rcrit,in, may be assumed as follows:

rcrit,ex = 1,5 d + lH + 0,5 lc (98)

rcrit,in = 1,5 (d + hH) + 0,5 lc (99)

(13) The provisions of subclause 10.5.3 shall also apply for checks of points within the column head, butsubstituting dH from figure 43 for d.

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Key:1 Critical section2 Loaded area Aload,in

3 Loaded area Aload,ex

Figure 43: Slab with column head with lH greater than 1,5 hH

10.5.3 Analysis(1) Design for punching shear is by truss analogy, using a design system based on the following values ofdesign shear resistance per unit length of the critical perimeter, with sections that are affine in shape to thecritical section (cf. figure 45):vRd,ct design shear resistance per unit length of the critical perimeter, for a slab without punching shear

reinforcement;vRd,ct,a design shear resistance per unit length of the outer critical perimeter outside the zone containing

punching shear reinforcement; it characterizes the transition from design shear resistance withoutshear reinforcement to shear resistance as specified in 10.3.3 as a function of the width of the zonecontaining punching shear reinforcement, lw (cf. figure 45);

vRd,sy design shear resistance per unit length of inner sections, for a slab with punching shear reinforcement;vRd,max maximum design shear resistance per unit length of the critical perimeter.

(2) The design shear resistance per unit length, vEd, in the section considered shall be as follows:

bëVEdvEd = (100) u

whereVEd is the overall design shear force;u is the critical perimeter considered, as shown in figure 45;b is a coefficient taking into account the effects of eccentricity in the critical section of edge and corner

columns, and in internal columns forming part of irregular systems (when considering non-sway systems,the values in figure 44 may be assumed unless a more precise analysis is carried out, which is usuallyrequired for sway systems).

Key:1 Corner column2 Edge column3 Internal column

Figure 44: Approximate values of b

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Key:1 Section2 Effective spacing of rows of links, sw

Figure 45: Sections for analysis of punching shear reinforcement

(3) A reduction in the applied shear force from concentrated loads near supports as specified in subclause10.3.2 is not permitted.(4) When analysing foundation slabs, VEd may be reduced by an amount corresponding to the favourableeffect of the ground pressure over the critical area. Not more than 50 % of the critical area specified in item (4)of subclause 10.5.2 may be used when determining the ground reaction forces, however.(5) The design shear component from inclined tendons, Vpd, acting parallel to VEd and located within thesection considered, may be taken into consideration as specified in subclause 10.3.2.(6) In slabs without punching shear reinforcement, it shall be checked that vEd is not greater than vRd,ct alongthe critical perimeter as specified in subclause 10.5.2. (101)(7) In slabs with punching shear reinforcement, the following checks shall be made.

a) The shear force along the critical perimeter, vEd, obtained by means of equation (100) shall not be greaterthan the maximum design shear resistance, vRd,max. (102)b) It shall be checked that vEd is not greater than vRd,sy at each inner critical perimeter, as shown in figure 45.

(103)c) To avoid failure outside the punching shear reinforcement zone, it shall be checked that vEd is not greaterthan vRd,ct,a along the outer critical perimeter. (104)

10.5.4 Slabs or foundations without punching shear reinforcementFor slabs or foundations without punching shear reinforcement, the design shear resistance along the criticalperimeter, vRd,ct, as specified in subclause 10.5.2 shall be obtained by means of equation (105):

vRd,ct = [0,14 h1 ¡ · (100 · rl · fck)1/3 – 0,12 scd] · d (105)

where

200¡ = 1 +1 ß 2,0 (106) d

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In the above:h1 is a factor equal to unity for normal-weight concrete, and is to be taken from table 10 for

lightweight concrete;d is the mean effective depth, in mm, equal to (dx + dy)/2, with dx and dy the effective depth of

the slab in x and y directions, respectively, in the critical section considered;r1 is the mean longitudinal reinforcement ratio within the section considered, obtained as fol-

lows:

ß 0,40 fcd/fydr1 = 1r1xër1y œ ß 0,02

wherer1x and r1y are the ratios of the longitudinal reinforcement in x and y directions, respectively,

which is bonded within the section considered and is anchored outside it; seeitem (9) of subclause 10.5.2 for corner and edge columns;

scd is the design axial stress in the concrete at the centroid of the section considered, in N/mm2,with

scd,x + scd,yscd = 2

wherescd,x is the design axial stress in the concrete within the section considered in x-direction,

obtained as

scd,y is the design axial stress in the concrete within the section considered in y-direction,

obtained as

NEd,x and NEd,y are the design mean axial forces that act in sections Ac,x and Ac,y, respectively.

10.5.5 Slabs or foundations containing punching shear reinforcement(1) The maximum design shear resistance, vRd,max, in slabs containing punching shear reinforcement within thecritical section shall be obtained by means of equation (107):

vRd,max = 1,5 vRd,ct (107)

(2) Where the reinforcement is arranged normal to the plane of the slab, the reinforcement required in eachrow considered (cf. figure 45) shall be calculated by means of equations (108) and (109) and assumed to beevenly distributed (cf. subclause 13.3.3).

a) For the first row of reinforcement at a distance equal to 0,5 d from the column edge, vRd,sy (cf. subclause10.5.3) shall be as follows:

¡s · Asw · fydvRd,sy = vRd,c + (108) u

b) The following shall apply for the other rows of reinforcement, to be spaced not more than 0,75 d apart:

¡s · Asw · fyd · dvRd,sy = vRd,c + (109) u · sw

wherevRd,c is the component of the design shear resistance due to the concrete, assumed to be equal to

vRd,ct from equation (105);¡s · Asw · fyd is the design strength of the punching shear reinforcement in the direction of shear, for each

row of reinforcement;u is the perimeter of the critical section;sw is the effective spacing of rows of reinforcement from figure 45, equal to 0,75 d or less;¡s is a factor taking into account the effect of the depth of the member on the effectiveness of

the reinforcement, and satisfying the following condition:

d – 400 ö 0,7¡s = 0,7 + 0,3 œ (with d in mm) (110) 400 ß 1,0

NEd,y ; Ac,y

NEd,x ; Ac,x

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(3) If inclined bars are used as punching shear reinforcement, these shall be inclined at an angle of between45° and 60° to the plane of the slab. If only inclined bars are used, these may only be arranged at a distanceof 1,5 d around the support (see figure 72), with d denoting the effective depth of the slab or foundation.The required reinforcement shall be calculated using equation (111) in a perimeter at a distance 0,5 d from theedge of the column, with d as above.

1,3 As ë sin a · fydvRd,sy = vRd,c + (111) u

where1,3 As · sin a · fyd is the design strength of the punching shear reinforcement in the direction of shear action;a is the angle of the punching shear reinforcement to the plane of the slab (cf. figure 45).

(4) The outer section shall be located at a distance 1,5 d from the last row of reinforcement (cf. figure 45). Thedesign shear resistance outside the zone containing punching shear reinforcement shall be calculated asfollows:

vRd,ct,a = ¡a · vRd,ct (112)

wherevRd,ct is the design shear resistance obtained from equation (105), but taking into consideration the longitu-

dinal reinforcement ratio, r1, in the outer critical section;¡a is a factor taking into account the effects at the transition from the column to the slab with a design shear

resistance as determined in accordance with subclause 10.3.3, and is obtained as follows:

0,29 lw¡a = 1 – ö 0,71 (113) 3,5 d

with lw equal to the width of the zone containing punching shear reinforcement, outside the loaded area(cf. figure 45).

(5) Detailing requirements for punching shear reinforcement are specified in subclause 13.3.3. The requiredratio of punching shear reinforcement in the inner sections shall be not less than the following:

Aswrw = ö rw,min (114) sw · u

or, in the case of inclined punching shear reinforcement,

As ë sin arw = ö rw,min sw · u

with sw equal to d and rw,min as specified in item (5) of subclause 13.2.3.

10.5.6 Minimum bending moments(1) Shear resistance shall be ensured by designing the slabs near the columns to withstand minimum bendingmoments, mEd, unless the structural analysis shows higher values of moment resistance to be required(cf. figure 46).

Key:1 Edge y2 Edge x

Figure 46: Effective zones for calculation of minimum bending moments

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(2) In the absence of other provisions, the following minimum moments per unit length shall be assumed:

mEd,x = hx · VEd and mEd,y = hy · VEd (115)

whereVEd is the design shear force;hx and hy are the moment coefficients from table 14 in x and y directions, respectively (cf. figure 46).

The minimum moments should be assumed to act over the width given in table 14 (cf. figure 46).

Table 14: Moment coefficients and moment distribution widths

Position of column Tension at Tension atEffective

Tension at Tension atEffective

top of slab bottom ofwidth b)

top of slab c) bottom ofwidth b)

slab slab

1 Internal column 0,125 0 0,3 ly 0,125 0 0,3 lx

(per metre2 Column at edge x a) 0,25 0 0,15 ly 0,125 0,125 of edge

length)

(per metre3 Column at edge y a) 0,125 0,125 of edge 0,25 0 0,15 lx

length)

(per metre (per metre4 Corner column 0,5 0,5 of edge 0,5 0,5 of edge

length) length)

a) See figure 46 for designation of edges and notation of column spacing.b) See figure 46.c) The bottom of the slab is the side facing the loaded area, the top is the side facing away from the loaded

area.

Moment coefficient inx-direction, hx

Moment coefficient iny-direction, hy

10.6 Strut-and-tie models10.6.1 General(1) Strut-and-tie models consist of struts, ties and the connecting nodes. These shall be analysed assumingconditions of equilibrium at the ultimate limit state, on the basis of which they shall be designed as specifiedin subclauses 10.6.2 and 10.6.3.(2) The position and direction of ties in such models shall agree with that of the actual reinforcement.(3) For optimum compatibility, strut-and-tie models (and in particular the position and direction of principalstruts) should be designed for the distribution of stress by linear elastic theory.(4) Hyperstatic strut-and-tie models may be used if the geometry and loading configurations are modifiedaccordingly.(5) When determining the linear force in hyperstatic strut-and-tie models, the different E moduli of the strutsand ties may be considered in approximation. For simplification, individual hyperstatic linear forces may beselected by analogy with the forces assumed for the purpose of the linear-elastic method of analysis.(6) Superposition of the results from more than one strut-and-tie model is not generally permitted but isallowed in the case of the models being largely the same for each action.

10.6.2 Design of struts and ties(1) Struts shall be designed for compression and for tension in transverse direction (cf. figure 47). In the caseof plane strut-and-tie models, they shall also be designed for tension normal to the plane of the model.Transverse tension in the compression zone as a result of narrowing at a node may be determined using a localstrut-and-tie model.(2) The maximum design compressive strength of struts, sRd,max, shall be equal to 1,0 h1ëfcd for uncrackedconcrete compression zones and 0,75 h1ëfcd for struts parallel to cracks, with h1 equal to unity for normal-weightconcrete and to be taken from table 10 for lightweight concrete.Struts with intersecting cracks may require smaller values (cf. DAfStb-Heft 525).(3) The design stress in the tie reinforcement and in the strut reinforcement designed to resist transversetensile forces shall be not more than fyd for the reinforcing steel and fp0,1k/gs for prestressing steel.

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Figure 47: Transverse tensile forces in a compression zone with confinement at concentrated nodes

(4) The reinforcement shall extend to the concentrated nodes and shall be of constant diameter along itslength. Where smeared nodes extend over the greater part of the structure, the reinforcement in the nodal zonemay have staggered ends provided these do not affect its ability to deflect compression forces as required.(5) The anchorage length of the reinforcement in the tension-compression nodes shall commence at thebeginning of the node, where initial compressive stresses from the struts meet, and are deflected by, theanchored reinforcement (cf. figure 49).(6) In parts of members with parallel tension and compression flanges, the depth of the compression zone orthe depth of the stress block shall, for the sake of compatibility, be limited to the dimensions that would berequired were a linear strain distribution to be assumed.

Figure 48: Nodal zone for analysis of compression nodes

(7) For struts whose compression zones become considerably more confined towards concentrated nodes,no further analysis of compressive stresses is required if the adjacent nodes are verified as specified in sub-clause 10.6.3.

10.6.3 Design of nodes(1) The provisions of this clause also apply in areas of concentrated load transmission in structures in whichno part is analysed using strut-and-tie models.(2) Unless a more detailed analysis is made, the design compressive stresses in concentrated nodes shall belimited by using a value of sRd,max equal to 1,1 h1ëfcd in compression nodes without anchored ties (e.g. as infigure 48), and 0,75 h1ëfcd in tension-compression nodes with anchored ties if all angles between tension andcompression struts are at least 45° (e.g. as in figure 49). In the above, h1 shall be equal to unity for normal-weightconcrete and is to be taken from table 10 for lightweight concrete. 7)Higher values may be assumed in more detailed analyses (cf. subclause 10.7).

For 7), see page 30.

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(3) Nodes at which there are bends in the reinforcement (e.g. as in figure 50) require proof that the bendingroller diameter is as specified in subclause 12.3.1.

Figure 49: Nodal zone for analysis of tension-compression nodes

Figure 50: Node at reinforcement bend

10.7 Partial area loading(1) When considering partial loading of an area Ac0 (cf. figure 51), the resistance partial area load, FRdu, shallbe determined as follows:

FRdu = Ac0 · fcd · 1Ac1/Ac0 ß 3,0 fcd · Ac0 for normal-weight concrete (116)

FRdu = Ac0 · flcd · (Ac1/Ac0)r/4 800 ß r/800 flcd · Ac0 for lightweight concrete (117)

whereAc0 is the loaded area;Ac1 is the theoretical area to accommodate the resistance partial area load (cf. figure 51);r is the design dry (bulk) density of lightweight concrete, in kg/m3.

(2) The theoretical area to accommodate the resistance partial area load, Ac1, shall meet the following con-ditions.

a) Ac1 shall have a geometry similar to Ac0.b) In the direction of loading, the centroids of Ac1 and Ac0 shall coincide.c) In each direction, Ac1 shall be not more than three times the dimensions of the loaded area.d) If a number of compressive forces act on the concrete cross section, the Ac1 areas shall not overlap overthe section depth, h.

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Key:1 Axis in loading direction

Figure 51: Determining areas for partial area loading

In addition, the depth over which the load is to be distributed in the direction of loading shall satisfy theconditions given in figure 51.The value of FRdu shall be reduced if the local loads are not equally distributed over Ac0 or if high shear forcesare present.(3) Items (1) and (2) above shall not be applied when analysing those parts of members containing tendonanchorages, which should be verified using suitable strut-and-tie models.(4) Reinforcement shall be provided to withstand transverse tensile forces occurring in the loaded area(cf. subclause 13.9).

10.8 Fatigue analysis10.8.1 General(1) Loadbearing members undergoing considerable changes in stress due to predominately non-static load-ing shall be designed for fatigue, with separate analyses performed for concrete and steel.(2) A fatigue analysis is not normally required for conventional buildings.(3) Special studies are required for lightweight concrete.

10.8.2 Internal forces and stresses at ultimate limit state(1) Stress analyses of sections in tension shall assume cracking, neglecting the tensile strength of the concretebut assuming compatibility of strain in the concrete and steel.(2) By way of approximation when determining the internal forces and stresses, a ratio of the elastic moduliof steel and concrete, ae, equal to ten may be assumed.(3) The differences in the bonding behaviour of reinforcing and prestressing steel shall be taken into accountby increasing the stresses in the reinforcing steel by the factor h, to be calculated as follows.

As + Aph = (118) As + Ap 1 j (ds/dp)

whereAs is the (sectional) area of the reinforcement;AP is the (sectional) area of prestressing steel;ds is the maximum diameter of reinforcing bars;dP is the nominal diameter or equivalent diameter of prestressing steel tendons or wire, equal to 1,6 1AP for

bundles of tendons, 1,20 dwire for single strands comprising three wires and 1,75 dwire for single strandscomprising seven wires;

j is the ratio of the bond strengths of bonded tendons and reinforcing bars located in the concrete, to betaken from table 15.

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Bond strength ratio, j

Post-tensioned members of concretePre-tensioned

up to class C50/60 from class C55/67 members

or LC50/55 or LC55/60

1 Smooth bars — 0,3 0,15

2 Strands 0,6 0,5 0,25

3 Profiled wires 0,7 0,6 0,3

4 Ribbed bars 0,8 0,7 0,35

(4) Where members are provided with shear reinforcement, the forces in the reinforcement and in the concreteshall be determined by truss analogy.(5) Unless a more detailed fatigue analysis is required for shear reinforcement, stress amplitudes may bedetermined assuming a strut angle, tan θfat, equal to 1tan θ, with θ taken from subclause 10.3.4.

10.8.3 Methods of analysis(1) If a simplified analysis as described in subclause 10.8.4 is not possible, a special analysis of structuraldurability shall be performed to check that the cumulative damage will not be greater than unity.(2) The Palmgren-Miner rule shall be used to determine the cumulative damage. For the calculations, thestress-number diagrams for reinforcement and prestressing steel from figure 52 (based on the parameters fromtables 16 and 17), shall be used, dividing Dr by gs,fat. The values given in table 16 apply to steel conforming tothe DIN 488 series of standards, and to other types of steel unless otherwise specified in the agrément.(3) Analysis shall be carried out for steel and concrete in general taking into account the following combina-tions of actions:

a) permanent actions;b) the characteristic prestressing force, Pk;c) the probable value of settlement, if unfavourable;d) the frequent value of thermal action, if unfavourable;e) action due to imposed loads.

Table 16: Parameters of stress-number diagrams for reinforcement

Table 15: Ratio of bond strengths of prestressing steel and reinforcing steel

Type ofreinforcement

Stress exponentStress

Type of reinforcement Number of amplitude,cycles, N* DsRsk,

k1 k2 in N/mm2

1 Straight and bent bars a) 106 5 9 d) 195

2Welded bars (including joints),

107 3 5 58couplings b) c)

a) For values of dbr less than 25 ds, DsRsk shall be multiplied by a reduction factor,j1, equal to 0,35 + 0,026 dbr/ds, where ds is the bar diameter and dbr is the mandreldiameter. For bars with ds over 28 mm diameter, DsRsk shall be multiplied by areduction factor, j2, equal to 0,8.

b) Unless other stress-number diagrams are specified in agréments or by individualagreement.

c) For welded bars and couplings, the stress-number diagram applies to DsRskvalues up to 380 N/mm2 (at N* = 0,036 · 106). Above this value, the diagram forbent and straight bars with the parameters in line 1 shall apply.

d) Applies to non-corrosive environments (cf. class XC1 in table 3). In all othercases, k2 shall be equal to 5.

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Table 17: Parameters of stress-number diagrams for prestressing steel

Stress exponentStress

Type of prestressing steel a) Number of amplitude,cycles, N* DsRsk,

k1 k2 in N/mm2

1 Pre-tensioned

2Single strandsin plastic ducts

Straight tendons;3 bent tendons in 10 150

plastic ducts

4Bent tendons in

7 120steel ducts 3

5 Couplings 5 80

a) Unless other stress-number diagrams are specified in agréments or by individualagreement.

Post-ten-sioned

106

5

9 185

(4) Instead of a special analysis of structural durability (cf. item (1) above), fatigue analysis may use equivalentstress amplitudes for steel (cf. item (5) below) and equivalent compressive stresses for concrete (cf. item (6)below) if enough is known about the nature and consequences of their action in standard cases.

Figure 52: Stress-number diagram for reinforcing and prestressing steel

(5) Reinforcing and prestressing steel shall be deemed to have adequate fatigue resistance if the followingcondition is met:

DsRsk (N*)gF,fatëgEd,fat ëDss,equ ß (119) gs,fat

whereDsRsk (N*) is the stress amplitude for N* load cycles from the stress-number diagram in figure 52 (see tables 16

and 17 for parameters);Dss,equ is the equivalent stress amplitude; this may be approximated to Dss,max for conventional buildings;Dss,max is the maximum stress amplitude when subjected to the combination of actions affecting fatigue;gF,fat is a partial safety factor taking into account actions in fatigue analysis, specified in item (2) of

subclause 5.3.3;gEd,fat is a partial safety factor taking into account uncertainty in fatigue analysis, to be taken from item

(2) of subclause 5.3.3;gs,fat is a partial safety factor for reinforcing and prestressing in fatigue analysis, to be taken from table 2.

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(6) Concrete in compression may be deemed to have adequate resistance to fatigue if the following conditionis met:

Ecd,max,equ + 0,43 11 – Requ ß 0,1 (120)

with

scd,min,equRequ = (121) scd,max,equ

and

|scd,max,equ|Ecd,max,equ = (122) fcd,fat

In the above, scd,max,equ and scd,min,equ are the maximum and minimum stresses, respectively, from the equivalentstress amplitude for 106 cycles.

10.8.4 Simplified analyses(1) Simplified analyses shall be performed using the combinations of actions for the serviceability limit stateas specified in DIN 1055-100.(2) Non-welded reinforcing bars in tension may be deemed to have adequate fatigue strength if the stressamplitude is not more than 70 N/mm2 when subjected to the frequent combination of stresses.(3) Reinforcing and prestressing steel at welds or couplings shall be deemed to be of adequate fatiguestrength if the cross section of the concrete in these regions is in compression when subjected to the frequentcombination of stresses, but taking into account a reduction factor of 0,75 for the mean prestressing force, Pmt.(4) The fatigue strength of concrete subjected to compression shall be deemed adequate if the followingcondition is met:

|scd,max| |scd,min| ß 0,9 up to class C50/60 or LC50/55 ß 0,5 + 0,45 œ (123) fcd,fat| fcd,fat ß 0,8 from class C55/67 or LC55/60

where

fckfcd,fat = bcc(t0)ëfcdëŒ1 – › (124) 250

In the above,scd,max is the design maximum axial stress under the frequent combination of actions;scd,min is the design minimum axial stress in the section in which scd,max occurs (scd,min to be equal to zero where

tension is present);bcc(t0) is a coefficient taking into account subsequent hardening of concrete, equal to e0,2(1–128/t0);t0 is the age of the concrete at the beginning of loading, in days.fck is the characteristic compressive strength, in N/mm2.

(5) Equation (123) also applies to the struts of members with shear reinforcement that are subjected to shear,in which case fcd,fat shall be reduced by factor ac from subclause 10.3.4.(6) In the case of members without shear reinforcement, the concrete may be deemed to have adequate fatiguestrength in shear if the following conditions are met:

VEd,mina) ö 0: VEd,max

|VEd,max| |VEd,min| ß 0,9 up to class C50/60 or LC50/55; ß 0,5 + 0,45 œ (125) |VRd,ct| |VRd,ct| ß 0,8 from class C55/67 or LC55/60

VEd,minb) < 0: VEd,max

|VEd,max| |VEd,min| ß 0,5 – (126) |VRd,ct| |VRd,ct|

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whereVEd,max is the design maximum shear force under the frequent combination of actions;VEd,min is the design minimum shear force under the frequent combination of actions in the section in which

VEd,max occurs;VRd,ct is the design shear resistance as in equation (70).

11 Analyses for the serviceability limit state11.1 Limitation of stresses11.1.1 General(1) To ensure the proper functioning of the structure and ensure an adequate service life, inordinate damageto the structure of the concrete and non-elastic deformations in the reinforcing and prestressing steel shall beprecluded by keeping within the limits of stress, as specified in subclause 11.1.2, 11.1.3 and 11.1.4.(2) Stress analyses shall be performed separately for the erection stage and the completed structure.(3) Non-prestressed conventional buildings designed as specified in clause 10 do not usually require stressanalyses as specified in subclauses 11.1.2 and 11.1.3 if both of the following conditions apply:

a) the action-effects are determined by elastic theory and are redistributed by not more than 15 % at theultimate limit state;b) detailing is as specified in clause 13, in particular, with the minimum reinforcement complying with thespecifications of subclause 13.1.1.

11.1.2 Limitation of compressive stresses in concrete(1) In members exposed to exposure classes XD1 to XD3, XF1 to XF4, and XS1 to XS3 (cf. table 3) with no otherprotection (such as increased concrete cover in the compression zone or confinement of the compression zoneby shear reinforcement), longitudinal cracking shall be precluded by limiting the compressive stresses in theconcrete to a value equal to 0,6 fck under the rare combination of actions.(2) If the serviceability, loadbearing capacity or durability of a structure is greatly affected by creep, thecompressive stresses due to quasi-permanent combinations of actions shall be limited to a value equal to0,45 fck to preclude inordinate deformation due to creep.(3) At anchorages and bearings, the above analyses are not required if the specifications of subclause 8.7.7and clause 13 are met.

11.1.3 Limitation of stresses in the reinforcementThe tensile stresses in reinforcement subjected to direct loading due to the rare combination of actions shallbe limited to a value equal to 0,8 fyk. Stresses may be equal to 1,0 fyk if they are solely due to imposeddeformations.

11.1.4 Limitation of stresses in prestressing steel(1) Tensile stresses in the prestressing steel making up the tendons shall be determined in each cross section,basing calculations on the prestressing force due to the quasi-permanent combination of actions after deduc-tion of the losses in prestressing force as specified in subclause 8.7.3. They shall be not greater than 0,65 fpk.(2) After release of the prestressing force or loosening of the anchorage, the mean stress in the prestressingsteel due to the rare combination of actions in any part and at any time shall be not greater than 0,9 fp0,1k or0,8 fpk, whichever is less.

11.2 Crack control and check for decompression11.2.1 General(1) Cracking cannot be precluded in the tension zones of concrete. The crack width shall be limited to ensurethat the proper functioning of the structure and its appearance and durability are not adversely affected bycracking.(2) Cracks in the concrete may also occur for other reasons (e.g. due to plastic shrinkage or chemical reactionscausing changes in volume). This standard does not specify how to preclude such cracks or limit their width.(3) When verifying that crack widths are within the specified limits, a distinction shall be made between thephase during which cracking occurs and the phase following completion of cracking. The analysis and calcu-lation methods may be used as approximate methods in both phases if minimum reinforcement as specifiedin subclause 11.2.2 is provided for distribution of cracks.(4) Since the methods specified in this clause do not permit an exact forecast or control of crack width, thevalues calculated shall be regarded as being for guidance only, and the occurrence of higher values in practicecannot be ruled out. This, however, should not present a risk if the provisions of this clause are met.(5) The methods stated in subclause 11.2.3 and 11.2.4 enable calculation of the crack width and its controlwhere reinforcement is bonded (i.e. in the zone in which the reinforcement is effective). Wider cracks may occuroutside this zone.

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(6) A member may be deemed to comply with the specifications of this clause if it meets the requirements givenin tables 18 and 19 in respect of its durability and appearance. Crack widths of members that are to meet specialrequirements (e.g. water-retaining structures) may be subjected to more stringent requirements. However, thisis not dealt with in this standard.

Table 18: Requirements relating to crack control and decompression

Requirementclass decompression

limitation ofDesign crack width,

crack widthwk, in mm

1 A Rare –

2 B Frequent Rare0,2

3 C Quasi-frequentFrequent

4 D –

5 E –Quasi-frequent

0,3

6 F – 0,4

Combination of actions for check of

(7) When considering the erection stage, the client may lay down other requirement classes than thoserequired after completion, given in table 19. However, these shall be not lower than the minimum requirementclasses in table 19.

Table 19: Minimum requirement classes as a function of exposure class

Minimum requirement class for

Exposure class(es) unbondedreinforced

post-tensioning pre-tensioningtensioning

concretemembers

1 XC1 D D F F

2 XC2, XC3 and XC4 C a) C E E

3 XD1, XD2, XD3 b), C a) B E EXS1, XS2 and XS3

a) Class D may be used if corrosion protection is ensured by other means. Details of such are setout in the agréments relating to the individual tensioning methods.

b) Additional corrosion protection may be required in certain cases.

(8) The requirements relating to crack control and decompression for members with a combination of bondedtendons and unbonded tendons shall be the same as those for bonded tendons.(9) The decompression limit requires that the section subjected to a combination of actions at the edge of thetension zone that is precompressed during the erection stage due to prestressing is completely under compres-sion on completion of the construction work.(10) Crack control is to comprise the following checks:

a) a check for minimum reinforcement as specified in subclause 11.2.2;b) a check for limitation of crack width under the combination of actions from subclause 11.2.3 or 11.2.4.

(11) Where strut-and-tie models are based on elastic theory, the stresses in the steel calculated from the linearforces may be used in the check for limitation of crack width. Tensile forces may also occur in places notrequiring reinforcement according to the strut-and-tie model. Such tensile forces may need to be accommo-dated by suitable structural reinforcement (e.g. for deep beams as specified in subclause 13.6).(12) Slabs of exposure class XC1 subjected to bending without any notable axial tension do not require acheck for limitation of crack width if the overall slab thickness is not greater than 200 mm, the specificationsof subclause 13.3 are met and more stringent requirements relating to crack width (cf. item (6) above) do notapply.(13) If reinforcing fabric with a cross section, as, of not less than 6 cm2/m is layered (cf. subclause 12.8.4),limitation of crack width shall be checked at the joint, assuming a 25 % increase in steel stress.

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11.2.2 Minimum reinforcement for limitation of crack width(1) Restraint due to imposed deformations or internal stresses shall be accommodated by minimum reinforce-ment, designed to take into account the requirements relating to crack limitation for the combination of action-effects resulting in initial cracking of the member.(2) Non-prestressed members and unbonded tendons may be of reduced cross section if the imposed defor-mation is sufficiently minor for it to be unlikely to cause cracking. In such cases, the minimum reinforcementmay be designed by considering the cross section subjected to the imposed deformation, taking into accountthe requirements relating to the limitation of crack width.(3) In pretensioned bonded members, the minimum reinforcement required to ensure controlled cracking isnot required in concrete subjected to a rare combination of actions and pretensioning as expressed by therelevant characteristic values, with compressive stresses not more than 1 N/mm2 occurring in the concrete atthe edge of the section.(4) In the case of profiled cross sections, such as hollow boxes or T-beams, the minimum reinforcement foreach segment of a cross section (comprising webs and flanges) shall be identified separately.(5) Unless a more precise analysis shows reinforcement of smaller cross section to be sufficient, the minimumarea of reinforcement in the tension zone of the section or subsection considered, required to limit crack widths,As, shall be obtained by means of equation (127):

As = kc · k · fct,eff · Act/ss (127)

wherekc is a factor taking into account the effect of stress distribution within the tension zone prior to initial

cracking, and also taking into account the change in the inner lever arm at the transition to state II, tobe calculated from:

sckc = 0,4 ¯1 + × ß 1 (128) k1 · fct,eff

sc is the stress in the concrete at the level of the centroidal axis of the section or subsection in the uncrackedcondition under the combination of actions resulting in initial cracking of the overall cross section (lessthan zero in the case of compression);

k1 is a factor equal to 1,5 h/hq for axial compressive force and 2/3 for axial tensile force;h is the depth of the section or subsection;hq is equal to h where h is less than 1 m and equal to 1 m where h is not less than 1 m;k is a factor taking into account unevenly distributed tensile stresses in the concrete; its value depends on

the type of tensile stress to which it is assigned, as follows:

a) where tensile stresses are the result of restraint due to internal stresses (e.g. caused by early thermalcontraction), k shall be equal to 0,8 where h is not greater than 300 and equal to 0,5 where h is not lessthan 800 mm (intermediate values may be obtained by linear interpolation), using as h the depth or widthof the section or subsection, whichever is smaller);b) where tensile stresses are the result of restraint due to imposed deformations (e.g. caused by settle-ment of supports), k shall be equal to unity.

Act is the area of the tension zone in the concrete located in the section or subsection considered (the tensionzone being the part of the section or subsection that is calculated as being in tension prior to theoccurrence of cracking when subjected to the combination of actions which will result in initial crackingof the overall cross section);

fct,eff is the axial tensile strength of the concrete effective at a particular point in time (it is equal to the meantensile strength of concrete, fctm, of the strength class in which cracking is to be expected. In many cases(e.g. if restraint is due to early thermal contraction), the concrete may crack during the first three to fivedays after placing, depending on the environmental conditions, the shape of the member and the typeof formwork; in such cases, unless a more detailed analysis is carried out, fct,eff shall be set at 50 % of themean tensile strength after 28 days. If it cannot be assumed that cracking will occur within the first28 days, a tensile strength of at least 3 N/mm2 for normal-weight concrete and at least 2,5 N/mm2 forlightweight concrete should be assumed);

ss is the stress in the reinforcement required to limit the crack width, and is a function of the limit diameter,ds*, from table 20.

The reinforcement is mainly to be located at the edge of the section in tension, although a suitable percentageshall be distributed over the tension zone to prevent cracks converging to form wider cracks.

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Table 20: Limiting reinforcing bar diameter as a function of the design crack width

Steel stress,Limiting reinforcing bar diameter, ds*, in mm,

ss, in N/mm2for a design crack width, wk, in mm, of

0,4 0,3 0,2

1 160 56 42 28

2 200 36 28 18

3 240 25 19 13

4 280 18 14 9

5 320 14 11 7

6 360 11 8 6

7 400 9 7 5

8 450 7 5 4

(6) Cracking shall be assumed sufficiently limited if the following applies with regard to bar diameters:

kc · k · ht fct,eff fct,effds = ds*· · ö ds*· (129) 4(h – d) fct,0 fct,0

whereds* is the limit diameter of the reinforcement (from table 20);h is the depth of the member;d is the effective depth;ht is the depth of the tension zone in the section or subsection prior to initial cracking;fct,0 is the tensile strength of the concrete to which the ds* values in table 20 relate (here, equal to 3 N/mm2).

(7) The required minimum reinforcement in a square of 300 mm sides around a pre-tensioned or post-tensioned member may be reduced in the magnitude of j1 · Ap, with Ap the area of prestressing steel in thetendon and j1 the ratio of bond strengths of prestressing steel and reinforcing steel (taking into account thedifference in diameter), to be calculated as follows:

dsj1 =1j · (130) dp

wherej is the ratio of mean bond strengths of prestressing steel and reinforcing steel, from table 15;ds is the maximum actual diameter of the reinforcing bars;dp is the equivalent diameter of the prestressing steel tendons or wires (cf. equation (118)).

11.2.3 Control of cracking without detailed analysis(1) Crack widths are limited to acceptable values by setting limits to bar diameters or bar spacing as a functionof the anticipated stresses.(2) The maximum values given in tables 20 and 21 usually ensure that cracks are kept within the accepted limitsprovided the following conditions are met:

a) for cracking mainly due to restraint, the bar diameters do not exceed those given in table 20;b) for cracking mainly due to loading, either the bar diameters are those given in table 20 or the bar spacingis as given in table 21.

(3) The steel stresses given in tables 20 and 21 shall be determined for a cracked section (state II) and therelevant combination of actions, and in the case of prestressed members using the relevant characteristicprestressing force.

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(4) The reinforcing bar diameters, ds, given in table 20 may be modified as a function of the height of themember and shall be modified as a function of the effective tensile strength of the concrete, fct,eff, as follows:

ss · As fct,effds = ds* · ö ds* · (131) 4(h – d) · b · fct,0 fct,0

whereds* is the bar diameter from table 20;ss is the stress in the reinforcing steel in state II, with item (5) below taken into account for members with

bonded tendons;As is the area of the reinforcement (cf. equation (127));h is the height of the member;d is the effective depth;b is the width of the tension zone;fct,0 is the tensile strength of the concrete (here, equal to 3 N/mm2), to which the values in table 20 are referred.

(5) Tension in the reinforcement of pre-tensioned bonded members shall be obtained for the consideredcombination of actions by means of equation (132), taking into account the difference in bonding behaviourbetween the reinforcing steel and prestressing steel.

1 1ss = ss2 + 0,4 fct,eff Œ – › (132) reff rtot

wheress2 is the tension in the reinforcing steel or the increase in tension in the prestressing steel in state II for the

considered combination of actions, assuming a rigid bond;reff is the effective reinforcement ratio, taking into account the variation in bond strength, to be calculated

as follows:

As + j12 ë Apreff = (133)

Ac,eff

rtot is the geometric reinforcement ratio, calculated as follows:

As + Apr tot = (134) Ac,eff

whereAs is the (sectional) area of the reinforcement (see notation of equation (127));AP is the (sectional) area of the tendons located in the effective zone of the reinforcement (Ac,eff);Ac,eff is the (sectional) area of the zone of the concrete section in which the reinforcement is effective

(cf. figure 53);j1 is the ratio of bond strengths obtained from equation (130);fct,eff is the effective tensile strength of the concrete as specified in item (5) of subclause 11.2.2.

Table 21: Maximum reinforcing bar spacing as a function of the design crack width

Steel stress,Maximum reinforcing bar spacing, in mm,

ss, in N/mm2for a design crack width, wk, in mm, of

0,4 0,3 0,2

1 160300

300 200

2 200 250 150

3 240 250 200 100

4 280 200 150 50

5 320 150 100 –

6 360 100 50 –

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Key:a) Beamb) Slab (with x the depth of the compression zone in state I)c) Member in tension

1 Zone in which reinforcement is effective2 Centroidal axis of reinforcement

Figure 53: Zone in which reinforcement is effective

(6) If bars with different diameters are used in a section, a mean bar diameter, dsm, equal to 7ds,i2/7ds,i shall

be assumed.(7) Where there are bundles of bars, their equivalent diameter shall be used instead of the diameter of theindividual bars (cf. item (2) of subclause 12.9).(8) In the case of welded fabric with double bars, the diameter of a single bar shall be used in calculations.(9) The crack width may be assumed satisfactory from the point of view of shear if the specifications ofsubclauses 13.2.3 and 13.3.3 are met.

11.2.4 Calculation of crack width(1) The limit of crack width may also be obtained by calculation. The design crack width, wk, shall be obtainedas follows:

wk = sr,max · (esm – ecm) (135)

wheresr,max is the maximum final crack spacing;esm is the mean strain in the reinforcement under the relevant combination of loads allowing for the contri-

bution of concrete to tension stiffening;ecm is the mean strain in the concrete.

(2) The difference in mean strain in the concrete and in the reinforcement may be calculated as follows:

fct,eff ss – 0,4 · (1 + ae · reff) reff ssesm – ecm = ö 0,6 (136) Es Es

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whereae is the ratio of elastic moduli of concrete and steel;reff is the effective reinforcement ratio obtained by means of equation (133);fct,eff is the effective tensile strength of the concrete, as specified in item (5) of subclause 11.2.2;ss is the stress in the reinforcement in a cracked section. Where pre-tensioned bonded members are used,

the specifications of item (5) of subclause 11.2.3 shall be met.

(3) For members subjected only to restraint due to internal stresses (e.g. due to early thermal contraction),equation (136) may be solved substituting sst for ss, with sst being the stress in the reinforcement calculated onthe basis of a cracked section under the loading conditions causing initial cracking.(4) The maximum final crack spacing may be obtained as follows:

ds ss · dssr,max = ß (137) 3,6 reff 3,6 fct,eff

wherereff is the effective reinforcement ratio obtained from equation (133);ds is the reinforcing bar diameter (if a section contains bars with different diameters, or bundles of bars or

double bars, the provisions specified in items (6), (7) and (8) of subclause 11.2.3 shall apply).Where welded fabric is used, the crack spacing shall be assumed to be not more than double the mesh size.

(5) Where cracks occur at an angle of more than 15° to the direction of the reinforcement in members rein-forced in two orthogonal directions, the crack spacing may be obtained by means of equation (138):

whereθ is the angle between the reinforcement along the x-axis and in the direction of the principal

tensile stress;sr,max,x and sr,max,y are the maximum final crack spacings along the x and y axes, respectively, obtained by

means of equation (137).

(6) Where crack widths are to be calculated for design situations in which tension is due to a combination ofrestraint and loading, the equations in this subclause may be used but the strain due to loading, calculated onthe basis of a cracked section, should be increased by the strain due to restraint.(7) If the strain due to restraint is not more than 0,8 o/oo, it is generally sufficient to base calculations of crackwidth on the stress due to either restraint or loading, whichever is greater.(8) If there is no, or only insufficient, bonded reinforcement in the areas where cracking is to be checked, anupper limit shall be set for the crack width. In such cases, the maximum crack spacing may be assumed to beequal to twice the depth of the cracks. Examples where this may apply are walls undergoing early thermalcontraction where the bottom of the wall is restrained by a previously cast base. In this case, sr,max may beassumed to be twice the height of the wall.

11.3 Limitation of deformations11.3.1 General(1) Deformations of a member or structure shall not adversely affect the proper functioning or appearance ofthe member itself or of adjacent members (e.g. partitions, glazing, cladding, services).(2) Deformations shall not adversely affect the proper functioning of machinery or apparatus supported by thestructure. However, this aspect is not dealt with in this standard.(3) This subclause only deals with deformations occurring in vertical direction in members subjected tobending, with a distinction made between the following:

a) sag (i.e. vertical deformation of a member relative to the shortest line connecting the points of support);b) deflection (i.e. vertical deformation of a member relative to the system line of the member (e.g. relativeto the precamber if upward deflection is incorporated in the formwork).

(4) In some cases it may be necessary to consider deformation other than that due to bending (e.g. deforma-tion due to shear, torsion or the partial reduction in size of vertical members). These can usually be neglected,however.(5) Deformations due to dynamic actions are not dealt with in this standard.

1sr,max =

(138) cos θ sin θ + sr,max,x sr,max,y

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(6) The limits of deformation shall be established as a function of the type and purpose of the structure, andthe surface structure of adjacent members. The values given below will be used for guidance unless other (morestringent or less stringent) requirements need to be taken into account.(7) The following specifications should ensure the satisfactory performance of buildings such as dwellings,offices, public buildings and factories. Limits shall be agreed with the client if there are special circumstancesrestricting the use of the guideline values given below.(8) It may be assumed that the appearance and serviceability of a structure will not be affected unless the sagof a beam, slab or cantilever subjected to the quasi-permanent combination of actions exceeds 1/250 of thespan (with the span of cantilevers assumed to be 2,5 times their length). This value may be increased in caseswhere the sag has no effect on serviceability and no special requirements need to be met regarding theappearance of the structure.(9) Precamber may be used to compensate for some or all of the deflection, but any upward deflectionincorporated in the formwork should not generally exceed 1/250 of the span.(10) Damage to adjacent elements (e.g. partitions) may occur if the deflection (including time-dependentdeformations) following installation of these members is excessive. As a guide, a limit of 1/500 may be placedon the span. This limit may be relaxed in cases where the element which might suffer damage has been designedto accommodate greater deflections or is known to be capable of sustaining greater deformation withoutdamage.

11.3.2 Simplified method to check control of deformations in reinforced concrete members(1) A simplified method of checking the control of deflection in reinforced concrete members is by limiting thespan to depth ratio, li/d.(2) It is usually sufficient to limit to 35 or less the span-to-depth ratio of floor slabs in conventional buildingsmade of normal-weight concrete. The span-to-depth ratio shall be not greater than 150/li (with li in m) for floorslabs in conventional buildings with more stringent deflection requirements (cf. item (10) of subclause 11.3.1).These values shall be multiplied by a reduction factor of hE

0,15 (taking hE from table 10) to obtain the limit forlightweight concrete.(3) Where members are subjected to bending, with deflection predominately caused by the load acting in thebay under consideration, an equivalent span, l1, equal to aëleff may be used. a may be taken from table 22 forcommon systems. In the case of rectangular slabs with linear support, the smaller of the two equivalent spansshall be used as a basis; for flat slabs, the larger of the two equivalent spans shall be used.(4) Edge and internal spans of continuous members may be treated as in line 2 or 3 of table 22 provided theratio of adjacent effective spans, leff,1/leff,2 is greater than 0,8 but not greater than 1,25.

12 Detailing arrangements for reinforcement and tendons12.1 General(1) The specifications of this clause are applicable to steel bar reinforcement, tendons and, unless otherwisespecified, reinforcing fabric subjected to predominately static or non-static loading. Special provisions forbundles of bars are given in subclause 12.9. Reinforcing fabric with double bars shall be treated as bundles of bars.(2) Anchorages of reinforcement required for the ultimate and serviceability limit states and any lap joints shallbe in accordance with the provisions of this standard.(3) Bar diameters, ds, of more than 32 mm shall only be used in members of a thickness equal to not less than15 ds.

12.2 Spacing of bars(1) Bar spacing shall be such as to allow the concrete to be placed and satisfactorily compacted and to ensurean adequate bond between reinforcement and concrete.(2) The clear horizontal and vertical spacing between individual parallel bars or layers of parallel bars shall benot less than the diameter of the thickest bar, or 20 mm. Bar spacing in concrete of maximum aggregate particlesize, dg, greater than 16 mm shall be not less than dg + 5 mm, unless there are special measures to place andcompact the concrete.(3) Where there are several horizontal layers of bars, the bars in each layer should be located vertically aboveeach other and there should be sufficient gaps to permit the passage of an internal vibrator.(4) The provisions of subclause 12.8.1 shall be taken into account for overlapping bars.

12.3 Bending of steel12.3.1 Mandrel diameter(1) The minimum diameter to which a bar is bent shall be such as to preclude spalling or splitting of theconcrete inside the bend of the bar, and to prevent bending cracks in the bar.

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System Coefficient a = li/leff

1 1

2 0,8

3 0,6

4 Inner spans: 0,7 a)Edge spans: 0,9 a)

5 2,4

a) Where slabs are made of concrete of strength class C30/37 orhigher, these values may be multiplied by a reduction factorof 0,1.

Table 22: Coefficient to determine equivalent span

Table 23: Minimum mandrel diameters

Minimum mandrel diameter, dbr, in mm, for a

bar diameter, ds, in mm, ofminimum concrete cover perpendicular

to plane of curvature

less than 20 20 or greater greater than greater than up to 50100 or 7 ds 50 or 3 ds or 3 ds

Hooks, bendsand loops 4 ds 7 ds – – –

Bent-up bars orother curved bars – – 10 ds 15 ds 20 ds

(2) The diameter of the mandrel used shall be not less than the values given in table 23. These values shall beincreased by 30 % for lightweight concrete.(3) For welded bars and reinforcing fabric, the minimum mandrel diameters given in table 24 shall also apply.

12.3.2 Rebending(1) Rebending of reinforcing steel is an extra stress factor, both for the steel and the surrounding concrete.(2) Cold bending shall be subject to the following conditions:

a) The bar diameter, ds, shall be not more than 14 mm. It is not permitted to bend reinforcing steel to andfro at the same place.b) In the case of predominately static loading, the mandrel diameter shall be equal to not less than 6 ds whenthe steel is bent.c) In the case of predominately non-static loading, the mandrel diameter shall be equal to not less than 15 dswhen the bar is bent. The stress amplitude of the steel shall be not less than 50 N/mm2.

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Table 24: Minimum diameters of mandrels for welded bent reinforcement

Minimum mandrel diameter, dbr, for

predominately static loading predominately non-static loading

Welds outside Welds inside Welds on Welds onbends bends outside of bends inside of bends

For a *) < 4 ds 20 ds

20 ds 100 ds 500 dsFor a *) ö 4 ds

Values asin table 23.

*) Distance between start of bend and weld.

d) Reinforcement boxes for connections shall be designed so as not to adversely affect the loadbearingcapacity of the concrete section or the corrosion protection of the reinforcement.e) The shear force shall not exceed 0,6 VRd,max (with VRd,max as specified in subclause 10.3.4) in the vicinityof the bend.

(3) Hot bending and rebending shall be subject to the following conditions:a) Reinforcing steel subjected to hotbending (i.e. at a temperature of 500 °C or over) shall have a charac-teristic yield strength of 250 N/mm2.b) In the case of predominately non-static loads, the amplitude of stress in the steel shall be not more than50 N/mm2.

(4) Technical details are given in DBV-Merkblatt Rückbiegen von Betonstahl und Anforderungen an Verwahr-kästen.

12.4 Bonding conditions(1) The quality of the bond mainly depends on the surface pattern of the bar, the dimensions of the memberand the position and inclination of the reinforcement during concreting.(2) The bonding conditions shall be considered adequate where the following bar types are used:

a) all bars at an angle of inclination, a, between 45° and 90° to the horizontal during concreting (cf. figure 54 a));b) all bars at an angle of inclination between 0° and 45° to the horizontal during concreting that are embed-ded in members not less than 300 mm thick in the direction of concreting (cf. figure 54 b));c) all bars at an angle of inclination between 0° and 45° to the horizontal during concreting that are embed-ded in members more than 300 mm thick and either not more than 300 mm above the bottom surface of thefresh concrete (cf. figure 54 c)) or not less than 300 mm below the top face of the member or section afterconcreting (cf. figure 54 d));d) vertical members fabricated in horizontal position (e.g. columns) that are compacted using an externalvibrator and whose external cross-sectional dimensions are not greater than 500 mm.

(3) All other conditions shall be considered poor.(4) Bonding conditions in members made using travelling formwork shall be considered poor.

Key:a) and b) Adequate bonding conditions for all barsc) and d) Bars outside of hatched area: adequate bonding conditions;

Bars in hatched area: poor bonding conditions

1 Direction of concreting

Figure 54: Bonding conditions

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12.5 Design bond stress(1) By specifying the ultimate bond stress it is ensured that there is an adequate margin of safety against bondfailure at the ultimate limit state and that no significant relative displacement occurs between the steel andconcrete at the serviceability limit state.(2) In conditions of adequate bond, the design bond stress, fbd, is given as a function of the characteristicconcrete compressive strength in table 25. This has been obtained as follows:

fctk;0,05fbd = 2,25 (139) gc

with gc as specified in subclause 5.3.3.

Table 25: Design bond stress for reinforcing bars up to 32 mm in diameter with adequatebonding conditions

Characteristic compressive Design bond stress,strength of concrete, fbd, in N/mm2fck, in N/mm2

12 1,6

16 2

20 2,3

25 2,7

30 3

35 3,4

40 3,7

45 4

50 4,3

55 4,4

60 4,5

70 4,7

80 4,8

90 4,9

100 4,9

(3) In conditions of poor bond, the values from table 25 shall be multiplied by the factor h1 from table 10.(4) Where bar diameters are greater than 32 mm, the values of fbd from table 25 shall be multiplied by the factor(132 – ds)/100 (with ds in mm). In the case of lightweight concrete, the use of such bars shall be justified on thebasis of experience or tests. The values of fbd from table 25 shall then be multiplied by the factor h1 (132 – ds)/100(with h1 from table 10, last column).(5) The values from table 25 may be increased, choosing either of the following options.

a) If there is compression transverse to the plane of the reinforcement, the values from table 25 may beincreased by the factor 1/(1 – 0,04 p) ß 1,5, with p being the mean transverse compression in the anchorageor lap zone, in N/mm2.b) The values from table 25 may be increased by 50 % if there is concrete cover of equal to at least 10 dson all sides with adequate reinforcement. This does not apply to overlaps with a centre-to-centre spacing,s, equal to 10 ds, as in figure 57.

(6) The values in table 25 shall be reduced by one-third if at right angles to the plane of the reinforcement,transverse tension is anticipated to cause cracking parallel to the axis of the reinforcing bars in the anchoragezone 12). No reduction need be made if under predominately static loading, the design width of the cracksparallel to the bars, wk, is limited to 0,2 mm or less.

12) See DAfSb-Heft 525 for examples.

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12.6 Anchorage of longitudinal reinforcement12.6.1 General(1) Reinforcing bars or welded fabric shall be anchored to enable the internal forces to which they are sub-jected to be transmitted to the concrete and to avoid longitudinal cracking or spalling of the concrete in theanchorage zone. Shear reinforcement may be required in certain circumstances (cf. subclause 12.6.3).(2) The permitted anchorage techniques are given in table 26.

Table 26: Coefficient aa as a function of anchorage technique

Type of anchorage

Coefficient aa

Tension Com-

bars a) pressionbars

1 a) Straight bar 1,0 c) 1,0 c)

2

b) Hook c) Bend d) Loop

0,7 b) (1,0) –

3e) Straight bar with at least one welded cross wire

0,7 0,7or transverse bar within length lb,net

4

f) Hook g) Bend h) Loop(plan view)

0,5 (0,7) –

Illustrations show reinforcing feature with at least one welded cross wire ortransverse bar within length lb,net at start of bend.

5

i) Straight bars with at least two welded cross wires ortransverse bars within lb,net (with the spacing, s, less than100 mm, but not less than 5 ds or 50 mm); only permitted for 0,5 0,5single bars with ds of 16 mm or less and double bars withds of 12 mm or less.

a) The bracketed values in this column shall apply where the concrete cover normal to the plane of curvatureis equal to less than 3 ds, where there is no transverse compression acting in the bend or where there isno concentration of links in the anchorage zone.

b) aa may be reduced to 0,5 for loops with a mandrel diameter, dbr, equal to 15 ds or more.c) Loadbearing connections shall be used for welded transverse bars where the ratio of the diameters of the

longitudinal and shear reinforcement is not less than 0,7.

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(3) Bends, hooks or loops are not to be used as compression reinforcement.(4) Bars greater than 32 mm in diameter shall be anchored as straight bars or using mechanical devices.(5) See subclause 13.9 for provisions relating to the transmission of concentrated anchorage forces in theconcrete.(6) Mechanical devices shall be covered by agréments unless their effectiveness can be proved by calculation.

12.6.2 Anchorage length(1) The basic anchorage length is the straight length required for anchoring the design tensile force, Fsd, equalto Asëfyd (with fyd equal to fyk/gs), assuming a constant bond stress over the anchorage length and bar circum-ference, as specified in subclause 12.5.(2) The basic anchorage length required for the anchorage of an individual bar, lb, is as follows:

ds fydlb = ë (140) 4 fbd

(3) The required anchorage length, lb,net, may be calculated as follows:

As,reqlb,net = aaëlbë ö lb,min (141) As,act

whereAs,req is the theoretically required (sectional) area of reinforcement;As,act is the actual (sectional) area of reinforcement;lb,min is the minimum anchorage length (equal to 0,3 aaëlb ö 10 ds for anchorages of tension members and

0,6 lb ö 10 ds for anchorages of compression members);aa is a coefficient taking ínto account the effectiveness of the anchorage techniques, from table 26.

(4) The anchorage lengths required to accommodate tensile forces are specified in subclause 13.2.2.(5) Subclause 8.7.6 shall apply with regard to the anchoring of pretensioned members.

12.6.3 Shear reinforcement(1) In the anchorage zone of reinforcing bars, the local tensile stresses occurring in the concrete as a resultof transverse tensile forces shall be accommodated by shear reinforcement.(2) The requirements of item (1) above shall be deemed satisfied if the following conditions are met:

a) detailing arrangements or other favourable factors (e.g. transverse compression) preclude splitting of theconcrete;b) the minimum number of links specified in clause 13 (for beams or supports) or shear reinforcement (forslabs or walls) are provided.

(3) Bar diameters over 32 mm that are not subjected to transverse compression require additional shearreinforcement as shown in figure 55. This shall be not less than specified below.

a) Parallel to the member surface:

Ast = n1 · 0,25 As (142)

b) Normal to the member surface:

Asv = n2 · 0,25 As (143)

whereAs is the area of section of one anchored bar;n1 is the number of layers of reinforcement with bars anchored in that section;n2 is the number of reinforcing bars anchored in each layer.

Shear reinforcement shall be equally distributed over the anchorage zone. Bars shall be evenly distributed witha centre-to-centre spacing approximately five times the bar diameter. 13)

13) Further detailing information is given in DAfStb-Heft 525.

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Key:1 Anchored reinforcing bars2 Continuous reinforcing bar

Figure 55: Shear reinforcement in the anchorage zone without transverse compressionfor bar diameters over 32 mm

12.7 Anchorage of links and shear reinforcement(1) Links and shear reinforcement shall be anchored using hooks or bends or welded-on shear reinforcementas shown in figures 56 a) to figures 56 d).(2) The anchorage shall be situated in the compression zone between the centroid and the compression edge,as is generally the case when the shear reinforcement extends over the total depth of section. In the tensionzone, the anchorage devices shall be arranged as close as possible to the compression edge. The tensilereinforcement shall be held in place by links.(3) Anchorages in the tension or compression zone with welded-on transverse bars as shown in figures 56 c)and 56 d) are only permitted if there is adequate concrete cover to prevent spalling. This shall be deemed thecase if the minimum cover at the side of the links in the anchorage zone is equal to not less than 3 ds (ds beingthe link diameter), but not less than 50 mm. In cases where the concrete cover is less than this, it shall bechecked empirically whether adequate protection is provided.(4) In the case of beams, the curtailment of links in the compression zone shall be as shown in figure 56 e) orfigure 56 f) and in the tension zone, as shown in figure 56 g) or 56 h).(5) In the case of T-beams, the curtailment of links required to contribute to shear strength in the flange shallbe by means of continuous shear reinforcement as shown in figure 56 i) if the design shear force, VEd, is not morethan two-thirds of the maximum shear resistance, VRd,max, as specified in subclause 10.3.4.

12.8 Joints12.8.1 General(1) Lap joints shall be formed by joining reinforcement using mechanical fasteners or welding (for end-to-endjoints) or by overlapping the reinforcement.(2) Lap joints shall be executed in a way so as to ensure that forces are transmitted from one bar to the other,that no spalling of the concrete occurs at the joints, and that the crack width at the end of the joint is not greaterthan the values specified in subclause 11.2.(3) Lap joints of bars greater than 32 mm in diameter are only permitted in members that are predominatelysubjected to bending.(4) Lap joints should be staggered, where possible, and at no point in areas of major stress shall all barsoverlap.(5) In plastic analysis as specified in subclause 8.4 or 8.5, there shall be no laps permitted at plastic hinges.(6) The clear space between two lapped bars and the distance between centrelines of laps shall be as infigure 57. Lapped joints are considered to be staggered in the longitudinal direction when the distance betweenthe centrelines of the laps is not less than 1,3 times the lap length, ls, obtained by means of equation (144).(7) Mechanical end-to-end joints shall be covered by an agrément.

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Key:1 Hook or bend2 Link3 Compression zone4 Tension zone5 Top shear reinforcement6 Bottom reinforcement of

connected flange

Key:a) Hookb) Bendc) Straight bar with two welded-on transverse barsd) Straight bar with one welded-on transverse bare) and f) Curtailment in compression zoneg) and h) Curtailment in tension zone (ls as in subclause 12.8.2, with aa equal to 0,7 if there are hooks or bends

at ends of links)i) Curtailment of T-beams at the flange

Figure 56: Anchorages and curtailment of links

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Figure 57: Overlapping bars

12.8.2 Lap length(1) The lap length. ls, shall be not less than the following:

ls = lb,net · a1 ö ls,min (144)

wherelb,net is the anchorage length obtained by means of equation (141);a1 is the coefficient from table 27;ls,min is the minimum lap length, equal to 0,3 aa · a1 · lb, but equal to not less than 15 ds or 200 mm;aa is the coefficient from line 1 or 2 of table 26 (i.e. the effect of welded-on transverse bars is neglected);lb is the basic anchorage length obtained by means of equation (140).

(2) If the clear spacing of the lapped bars is greater than or equal to 4 ds (cf. figure 57), the lap length shall beincreased by the difference between the actual clear spacing and 4 ds.

Table 27: Coefficient a1

Key1 Centrelines of laps2 Distance between centrelines

of laps3 Centreline along lap4 Edge of member

Coefficient a1 for a percentage of spliced barsin the cross section of reinforcement layer

up to 30 % above 30 %

Tension lap fords < 16 mm 1,2 a) 1,4 a)

ds ö 16 mm 1,4 a) 2,b)

Compression lap 1, 1,

a) a1 shall be equal to unity where s is not less than 10 ds and s0 is not less than 5 ds.b) a1 shall be equal to 1,4 where s is not less than 10 ds and s0 is not less than 5 ds.

Figure 58: Spacings s and s0 for calculation of a1

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12.8.3 Shear reinforcement(1) Shear reinforcement shall be provided as follows at lap jonts:

a) The total area of the shear reinforcement, 7Ast, shall be not less than the total area of cross section ofone spliced bar, As (cf. figure 59). This provision applies to each spliced bar where the clear spacing of barsis over 4 ds, as specified in item (2) of subclause 12.8.2.b) The shear reinforcement shall be made of links if s is equal to not more than 12 ds (cf. figure 58) and bestraight in other cases.c) The shear reinforcement in members predominately subjected to bending shall be located between thelongitudinal reinforcement and the surface of the concrete (cf. figure 59).

Key:a) Bars in tensionb) Bars in compression

Figure 59: Shear reinforcement for lap joints

(2) If the diameter of the spliced bars is less than 16 mm in concrete up to strength class C55/67 or LC45/50 andless than 12 mm in concrete of class C60/75 or LC50/55 or higher, or if the percentage of lapped bars in anyone section is not more than 20 %, the actual shear reinforcement as specified in clause 13 shall be consideredsufficient.(3) In members of concrete of strength class C70/85 or higher that are predominately subjected to bending,the laps shall be enclosed by links, with the total area of cross section of the vertical legs equal to the requiredarea of cross section of the spliced longitudinal reinforcement.(4) In the case of multi-layer reinforcement where more than 50% of the cross section of the individual layersis spliced in one section, lap joints shall be enclosed by links designed to adequately support all the lapped bars.

12.8.4 Layered reinforcing fabric(1) Layered reinforcing fabric with a sectional area, as, of not more than 12 cm2/m need not be staggered.Staggering of laps of fabric of larger cross section is only permitted in the inner layer of multilayer reinforcement,with not more than 60 % of the sectional area of the required reinforcement made up of the laps.(2) The lap length (cf. figure 60 a) shall be not less than the following:

as,reqls = lbëa2ë ö ls,min (145) as,act

wherelb is the basic anchorage length obtained by means of equation (140);a2 is a coefficient taking into account the sectional area of the fabric, equal to 0,4 + as,act/8, but not less than

unity and not more than 2;as,req is the required area of reinforcement in the section considered, in cm2/m;as,act is the actual area of reinforcement in the section considered, in cm2/m;ls,min is the minimum lap length, equal to 0,3 a2ëlb, but not less than the spacing of the welded-on transverse

bars, sq, or 200 mm.

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Key:a) Longitudinal directionb) Transverse direction

Figure 60: Lap joints of welded reinforcing fabric (example)

(3) In the case of multilayer reinforcement, the laps of the individual layers shall be staggered at not less than1,3 times the lap length in longitudinal direction.(4) Additional shear reinforcement is not necessary where the laps are located.(5) Shear reinforcement in slabs and walls as specified in item (2) of subclause 13.3.2 and item (5) of subclause13.7.1 may be lapped in a line. The minimum lap lengths are given in table 28. There shall be at least twolongitudinal bars within each lap length, as shown in figure 60 b).

Table 28: Minimum lap lengths of transverse bars

Bar diameter of transverse bars, Minimum lap length of transverse bars,ds, in mm ls,min

Up to 6 Above or equal to sl or 150 mm

Above 6 up to 8,5 Above or equal to sl or 250 mm

Above 8,5 up to 12 Above or equal to sl or 350 mm

Above 12 Above or equal to sl or 550 mm

1) sl denotes spacing of longitudinal bars.

12.9 Bundles of bars(1) Bundles of bars comprise two or three bars individually measuring not more than 28 mm in diameter, thatare in contact with each other and are held together by suitable means during fixing and concreting.(2) Unless otherwise specified below, the requirements of subclauses 12.1 to 12.8 shall apply, substitutingthe equivalent diameter, dsV, for the diameter of an individual bar, ds, in all calculations featuring the bardiameter. dsV is the diameter of a bar with the same sectional area as the bundle, and is obtained as follows fora bundle of n bars all having the same diameter, ds.

dsV = dsë1n (146)

(3) dsV shall be not greater than 36 mm in members subjected predominately to tension (i.e. with the line ofzero strain outside the section).(4) For concrete of strength class C70/85 or higher, dsV shall be not greater then 28 mm unless a more rigorousanalysis is made.(5) Figure 61 shall apply with regard to the arrangement of bars in a bundle and the minimum concrete coverand clear spacing of bundles. The nominal concrete cover shall be as specified in subclause 6.3.(6) For the anchorage of bundles, the ends of the bars shall be staggered (cf. figures 62 and 63) except in thecase of bundles in tension which, irrespective of dsV, may be curtailed directly over end and intermediatesupports, and bundles in tension of diameter below 28 mm , which may also be curtailed before supports in oneplace without staggering of individual bars.(7) Where bars are anchored as shown in figure 62, the diameter of the individual bar shall be used in thecalculation of anchorage length.

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a shall be not less than dsV or 20 mm. See also item (2) of subclause 12.2.

Figure 61: Arrangement, minimum spacing and minimum concrete cover of bundles of bars

(8) Where bars are anchored as shown in figure 63, dsV shall be used in the calculation of anchorage length.(9) Bars making up bundles in compression need not be staggered. For an equivalent diameter over 28 mm,at least four links of 12 mm diameter shall be provided at the end of the bundle unless other precautions havebeen taken to ensure resistance to maximum compression (e.g. by the suitable arrangement of cut-off barswithin a loadbearing slab), in which case it is sufficient to provide a link outside the anchorage zone.

Key:1 to 3 Individual bars in a bundleE Theoretical curtailment point

Figure 62: Anchorage of bundles of bars with widely spaced theoretical curtailment points

Key:1 to 3 Individual bars in a bundle

Figure 63: Anchorage of bundles of bars with closely spaced theoretical curtailment points

(10) The lap length, ls, shall be determined as specified in subclause 12.8.2. Bundles of bars comprising twobars with an equivalent diameter not greater than 28 mm may be spliced without staggering of the bars, in whichcase dsV is to be used to calculate ls.(11) For bundles comprising two bars with an equivalent diameter greater than 28 mm and for bundles com-prising three bars, the individual bars shall be staggered by not less than 1,3 ls in longitudinal direction(cf. figure 64), however there shall be no more than four bars in any lap cross section. In this case, the diameterof a single bar shall be used to calculate ls.(12) Bundles of bars shall not be used in lightweight concrete unless their suitability is corroborated byexperience or tests. In such cases, the diameter of individual bars shall be not greater than 20 mm.

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Key:1 to 3 Individual bars in a bundle4 Fourth bar

Figure 64: Lap joint in tension for bundles of three bars inclusive of a fourth bar

12.10 Tension members (tendons)12.10.1 General(1) The following provisions shall apply unless the tendons are covered by an agrément containing otherrequirements.(2) The spacing of tendons shall be such as to enable the concrete to be placed and compacted satisfactorily.(3) There shall be a minimum of 20 mm of concrete between bonded tendons and zinc-coated embeddedparts or reinforcement. Connections between metals are not permitted.

12.10.2 Pre-tensioned members(1) The use of smooth wires is not permitted for pre-tensioned members.(2) The horizontal and vertical clear minimum spacing between pre-tensioned members is shown in figure 65.

Figure 65: Clear minimum spacing for pre-tensioned members

(3) Tendons may be bundled outside the anchorage zone provided this permits the satisfactory placing andcompaction of the concrete.(4) Tendons comprising extruded wires or strands may be deflected after tensioning or may be pretensionedafter deflection if they are not susceptible to movement at bends and the ratio of bending radius and tendondiameter is not less than 15.(5) In the anchorage zone, close shear reinforcement (cf. subclause 13.9) shall be provided to take into accountsplitting tensile forces from the anchorages. In simple types of construction (e.g. prestressed concrete hollowslabs), this is not necessary if the splitting tensile stress is not greater than fctk;0,05/gc (with gc as specified in item(8) of subclause 5.3.3).(6) Subclause 8.7.6 shall apply with the regard to the anchorage of pre-tensioned members.

12.10.3 Post-tensioned membersThe minimum clear spacing of sheaths shall be 0,8 times the external diameter (round sheaths) or 40 mm invertical and 50 mm in horizontal direction (round and rectangular sheaths).

12.10.4 Unbonded members(1) External tendons shall be such as to ensure that these can be satisfactorily inspected and replaced.Subclause 12.10.3 shall apply with regard to internal tendons.(2) Bundling of internal tendons is permitted outside the anchorage zones if placing and compaction of theconcrete can be carried out satisfactorily and forces associated with a change in direction can be accommo-dated.(3) The minimum bending radius of single strands 13 mm in diameter shall be 1,7 m and of single strands15 mm in diameter, 2,5 m.(4) Unless specified otherwise, those parts of external tendons involved in anchorage and deflection shall bedesigned to enable replacement of the tendon without causing damage to other parts of the construction.

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(5) Precautions shall be taken to exclude critical cross vibration of external tendons due to imposed loads,wind or other causes.(6) Design deflections of a tendon up to an angle of 0,017 5 rad may be effected without a deviator unless useof the latter is specified in the agrément covering the prestressing technique.(7) Where slabs not more than 450 mm in height contain single-strand tendons and fixed upper and lowerlayers of reinforcement, it is sufficient to connect each of the single strands with one of the layers of reinforce-ment at no less than two places, provided the following conditions are satisfied:

a) the distance between fixings at the support is between 300 mm and 1 000 mm;b) the distance between the tendon anchorage and the connection with the upper reinforcement layer is notmore than 1 500 mm;c) the distance between between the tendon anchorage and the connection with the lower reinforcement,or between the connections of the top and bottom reinforcement, is not less than 3 000 mm.

12.10.5 Couplers(1) Couplers shall be positioned so as not to adversely affect the loadbearing capacity of the member and, inthe case of fixed couplers, so as to enable any temporary anchorage that may be needed during the erectionstage to be properly introduced.(2) Couplers should usually be located outside intermediate anchorages, preferably in areas of low stress.(3) The placing of couplers on more than 70 % of the tendons in any one section should be avoided whereloading is of the predominately non-static type.

13 Detailing arrangements for structural members13.1 Members predominately subjected to bending13.1.1 Maximum and minimum reinforcement(1) The minimum reinforcement to ensure ductility of members as specified in subclause 5.3.2 to controlcracking (neglecting the prestressing force in prestressed members) shall be calculated using the mean axialtensile strength of the concrete, fctm, from tables 9 and 10 and taking the stress in the steel, ss, to be equal to fyk.(2) In prestressed member sections containing two or more bonded tendons, the tendons may be countedtowards the minimum reinforcement (see item (1) above) if these are located at a distance equal to not more than0,2 h *) or 250 mm (whichever is less) from the reinforcement. The calculations shall assume a stress in theprestressing steel not greater than fyk.(3) The minimum reinforcement shall be evenly distributed over the width and proportionately over the depthof the tension zone. To improve ductility (irrespective of the provisions to accommodate tension), the bottomminimum span reinforcement shall be continuous between and above the supports. Tendons and reinforcementcontinued upwards shall not be taken into account. Above internal supports, a top minimum reinforcement shallbe provided that extends over a length of at least a quarter of each of the two adjacent spans. The top minimumreinforcement shall be continued over the full length of cantilevers. The reinforcement shall be anchored at theend support and at the internal support over a length not less than that specified in item (8) of subclause 13.2.2;joints shall be designed to withstand the full tensile force.(4) The proportional area of reinforcement in any cross section (including at lap joints) shall be equal to notmore than 0,08 Ac.(5) Where the x/d ratio in densely reinforced T-beams is greater than specified in item (3) of subclause 8.2, theminimum reinforcement should consist of links with a diameter not less than 10 mm and taking the maximumspacing, smax, from line 3 of table 31, unless confinement of the flexural compression zone is achieved by othermeans.

13.1.2 Top reinforcement in prestressed members(1) All prestressed members require top reinforcement, with minimum values of the reinforcement ratio beinggiven in table 29.(2) Pre-tensioned members located where the top reinforcement has twice the minimum concrete coverspecified in subclause 6.3 may be counted with their full surface area as part of the top reinforcement.(3) The top reinforcement shall be located in the tension and compression zone of slabs and shall take the formof two layers of reinforcement located approximately at right angles to each other with the bars in each layerspaced not more than 200 mm apart.(4) Slabs, flanges and wide beams subjected to a class XC1 environment do not require the minimum topreinforcement at the outer compression edge from table 30.(5) Precast slabs less than 1,2 m in width do not require the top reinforcement in transverse direction fromline 2 of table 30.

*) h – depth or thickness of member.

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Table 29: Reinforcement ratio for calculation of minimum area of reinforcement

Characteristic compressiveMinimum reinforcement ratio,strength of concrete, r, as per thousand a)fck, in N/mm2

12 0,51

16 0,61

20 0,70

25 0,83

30 0,93

35 1,02

40 1,12

45 1,21

50 1,31

55 1,34

60 1,41

70 1,47

80 1,54

90 1,60

100 1,66

a) r is calculated as 0,16 fctm/fyk. For lightweight concrete, r maybe multiplied by h1 from table 10, with h1 taken to be 0,85 ormore.

Table 30: Minimum top reinforcement in prestressed members

Minimum area of reinforcement, in cm2

Slabs, flanges and beams Beams with bw ß h, and webswith bw > h of T-beams and box beams

Members of ex- Members of Members of ex- Members ofposure classes other expo- posure classes other expo-

XC1 to XC4 sure classes XC1 to XC4 sure classes

Each side of beams

Each supported or unsupported0,5 r h or r h or r hf 0,5 r bw r bw

edge of beams of depth 1 m and0,5 r hf per metre per metre per metre

over a)per metre

Compression zone at outer edgeof beams and slabs b) 0,5 r h or r h or r hfPrecompressed tension zone of 0,5 r hf per metre – r h bw

slabs a) b) per metre

Compression flanges over 12 cmhigh (with top and bottom – r hf per metre – –considered separately) a)

a) Top reinforcement over 3,4 cm2/m in each direction is not required.b) See items (4) and (5) of subclause 13.1.2h beam depth or slab thicknesshf thickness of compression or tension flange of profiled sectionsbw beam web widthr reinforcement ratio from table 29

Location of reinforcement

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(6) The longitudinal reinforcement specified in item (1) of this subclause and in subclauses 11.2.2 and 13.1.1need not be added together. Instead, the larger value shall be taken into consideration in each case.(7) In all calculations at the ultimate limit state and serviceability limit state, the top reinforcement may countas part of the required reinforcement if it is anchored and positioned to comply with the specifications.

13.2 Beams and T-beams13.2.1 General(1) Detailing of beams at end supports shall take into consideration any restraint that has not been accountedfor in the calculations. Assuming free rotation, the cross sections of end supports shall be designed to sustaina support moment that is not less than 25 % of the moment in the adjacent span. The reinforcement shall extendinto the end span over a length equal to one quarter of the span, measured from the edge of the support.(2) In slabs of T-beams and box beams, the tension reinforcement shall be provided over a width equal to notmore than half the effective span as specified in subclause 7.3.1.

13.2.2 Resistance to tension(1) Adequate resistance to tension shall be demonstrated for the ultimate and serviceability limit states.(2) Where an analysis is carried out as specified in subclause 8.2 or 8.3, demonstration of adequate resistanceto tension as specified in item (1) above is not generally required for the serviceability limit state.(3) The tension envelope on which the reinforcement requirement depends may be determined by shifting theenvelope representing the axial tensile force in the reinforcement by an amount, al (cf. figure 66), obtained asfollows:

zal = ë (cot θ – cot a) ö 0 (147) 2

whereθ is the angle of the concrete struts to the member axis, as specified in subclause 10.3.4;a is the angle of the shear reinforcement to the member axis;z is the internal lever arm, generally taken to be equal to 0,9 d (cf. subclause 10.3.4).

Key:1 Envelope for

Fsd = MEds/z + NEd

whereMEds is the design applied moment referred to the centroidal axis of the reinforcement

2 Envelope after shift by an amount al

3 Tension envelope

Figure 66: Diagram of resisting tensile forces (tension envelope) and anchorage lengthsin flexural members

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(4) Where the tension reinforcement in the flange outside the web is positioned as specified in item (2) ofsubclause 13.2.1, al shall be increased by the distance of the individual bars from the web edge.(5) Cut-off bars shall be anchored over a length, lb,net (cf. figure 66), obtained by means of equation (141).Anchorages at end supports shall be as specified in item (8) below.(6) At least one quarter of the span reinforcement shall extend, and be anchored, to the support.(7) Reinforcement anchorages at end supports shall be capable of withstanding the following tensile force, Fsd:

alFsd = VEdë + NEd ö VEd/2 (148) z

(8) The required anchorage length at end supports shall be calculated as follows.a) For conditions of direct support:

2lb,dir = lb,net ö 6 ds (149)

3

(A bond stress, fbd, higher than specified in item (5) a) of subclause 12.5 shall not be used in the calculation.)

b) For conditions of indirect support:

lb,ind = lb,net ö 10 ds (150)

wherelb,net is the anchorage length as specified in equation (141);ds is the bar diameter of the longitudinal reinforcement to be anchored.

Although the anchorage length shall be assumed to begin at the front edge of the support, the reinforcementshall extend at least beyond the theoretical line of effective support (cf. item (6) of subclause 7.3.1). Subclause8.7.6 shall apply with regard to pre-tensioned members.(9) At intermediate supports of continuous members, the required reinforcement shall extend by not less than6 ds beyond the edge of the support.(10) The bottom reinforcement at intermediate supports should be capable of withstanding positive momentsas a result of accidental actions (e.g. settlement of supports, explosions, etc.).(11) To achieve adequate lateral rigidity, precast elements with an leff/b ratio greater than 20 shall include aproportion of longitudinal reinforcement that is concentrated at the sides of the tension and compression zones.

13.2.3 Shear reinforcement(1) Shear reinforcement should be at an angle of 45° to 90° to the centroidal axis of the member. It maycomprise a combination of the following types of reinforcement:

a) stirrups enclosing longitudinal tension reinforcement and the compression zone;b) bent-up bars;c) shear assemblies in the form of cages, ladders, etc. that do not enclose the longitudinal reinforcement(cf. figure 67).

Key:1 Stirrup2 Stirrup cage as shear assembly3 Parallel bars as shear assembly

Figure 67: Combinations of stirrups and shear assemblies

(2) Bent-up bars and shear assemblies may only be used as shear reinforcement in beams when used inconjunction with stirrups, which shall make up at least 50 % of the required shear reinforcement.

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(3) Where slender precast concrete beams are used (e.g. box, T or I sections with a web width of not more than80 mm), shear assemblies in which only one bar is visible in a section may be used on their own as shearreinforcement provided the compression zone and the tension reinforcement are enclosed separately by stir-rups.(4) The shear reinforcement ratio, rw, shall be obtained by means of equation (151):

Aswrw = (151) swëbwësin a

whereAsw is the (sectional) area of an element of the shear reinforcement (e.g. stirrup);sw is the spacing of the elements of shear reinforcement (along the beam axis);bw is the web width;a is the angle of the shear reinforcement to the beam axis (i.e. equal to 90° for links normal to the beam axis).

(5) The minimum shear reinforcement ratio, rw,min, shall be equal to r except in the case of slender crosssections with prestressed tension flanges, where it shall be equal to 1,6 r (with values of r taken from table 29).(6) The maximum spacing of hangers or shear assemblies in longitudinal and transverse directions shall beas given in table 31.

Table 31: Maximum longitudinal and transverse spacing of hangers and shear assemblies

Maximum spacing, smax

For concrete of strength class

up to C50/60 over C50/60 up to C50/60 over C50/60or LC50/55 or LC50/55 or LC50/55 or LC50/55

Longitudinal spacing Transverse spacing

Up to 0,30 VRd,max 0,7 h or 300 mm 0,7 h or 200 mm h or 800 mm h or 600 mm

Above 0,30 VRd,maxup to 0,60 VRd,max

0,5 h or 300 mm 0,5 h or 200 mmh or 600 mm h or 400 mm

Above 0,60 VRd,max 0,25 h or 200 mm

a) See subclauses 10.3.2 and 10.3.4 for VEd and VRd,max. VRd,max may also be calculated approximately,using a value of θ equal to 40°.

Design shear force, VEda)

(7) The maximum longitudinal spacing of bent-up bars, smax, shall be equal to 0,5 hë(1 + cot a) (152).The maximum spacing of bent-up bars in transverse direction shall be as given in table 31.(8) The shear reinforcement shall be arranged along the longitudinal axis of the member so as to have therequired design shear resistance at every point.(9) In the case of conventional buildings, the shear reinforcement may be distributed along the longitudinalaxis as shown in figure 68.

Key:1 Upper deviation (area), AA

2 Lower deviation (area), AE

(AA ß AE)

Figure 68: Permitted deviations from shear resistance diagram for conventional buildings

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13.2.4 Torsion reinforcement(1) A system consisting of links and longitudinal reinforcement at right angles to one another shall be used asthe torsion reinforcement specified in subclause 10.4.2. Links in beams and webs of T-beams shall be curtailedas shown in figures 56 g) and h).(2) The spacing of torsion links in longitudinal direction shall be not more than the maximum spacing specifiedin table 31 and not more than a value equal to uk/8 (cf. item (3) of subclause 10.4.2 for uk)(3) Longitudinal bars should be evenly distributed around the inner periphery of the links. Where cross sec-tions are polygonal in shape, there shall be at least one longitudinal bar in each corner. If longitudinal reinforce-ment is provided in the corners, the longitudinal bars shall be spaced not more than 350 mm apart.

13.2.5 Top reinforcement in members with large bar diameters(1) In order to avoid spalling of the concrete cover and to control cracking, top reinforcement is required inmembers containing individual bars with a bar diameter, ds, over 32 mm or bundles of bars with an equivalentdiameter, dsV, over 32 mm.(2) The top reinforcement should comprise reinforcing fabric or individual bars not more that 10 mm in diam-eter, and should be located outside the links.(3) The minimum concrete cover of the top reinforcement shall be as specified in subclause 6.3.(4) The position of the top reinforcement shall be on the lines of figure 69, with a sectional area, As,surf, notgreater than 0,02 Act,ext (with Act,ext denoting the area of concrete cover over the stirrups).(5) The longitudinal top reinforcement may count towards the tension reinforcement and the transverse barstowards the shear reinforcement provided these are correctly positioned and anchored.

Key:1 Individual bars or bundles of bars, with ds or dsV over 32 mm2 Top reinforcement (As,surf )

Figure 69: Top reinforcement

13.3 Solid slabs cast in-situ13.3.1 Minimum thicknessThe minimum thickness of solid slabs cast in situ shall be 70 mm, except for slabs with bent-up bars as shearreinforcement, where it is to be 160 mm and those with stirrups, where it is to be 200 mm.

13.3.2 Resistance to tension(1) The provisions of subclause 13.2.1 and 13.2.2 shall apply by analogy for the resistance to tension of solidslabs cast in situ.As a departure from the provisions of item (6) of subclause 13.2.2, however, at least half of the span reinforce-ment shall be continued to and anchored at the support. al shall be equal to d for reinforced concrete slabswithout shear reinforcement. Slabs with pre-tensioned members shall meet the specifications of item (11) ofsubclause 8.7.6.(2) Shear reinforcement in one-way slabs shall be not less than 20 % of the tension reinforcement. Theminimum diameter of reinforcing bars shall be 5 mm.(3) In two-way slabs, the reinforcement in the direction subjected to the minor load shall be not less than 20 %of that in the direction subjected to the major load.(4) The maximum bar spacing of tension reinforcement shall be 250 mm for slabs not less than 250 mm thickand 150 mm for slabs not more than 150 mm thick, with intermediate values being obtained by linear interpo-lation. The spacing of bars making up the shear reinforcement or the reinforcement in the direction with theminor load shall be not more than 250 mm.(5) If torsional rigidity is considered in the slab analysis, the reinforcement in the corners of the slab shall bedesigned to take torque into account.

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(6) Instead of torsion reinforcement, fabric or bars in a rectangular grid, not less than 0,3 leff,min in length(cf. figure 70) may be arranged at both the top and bottom corners of the slab, parallel to its sides, the bars orthe fabric having the same area of cross section as the span reinforcement. 14)

Key:1 Torsion reinforcement

Figure 70: Top and bottom corner reinforcement

(7) At slab corners at which a freely supported and a restrained edge meet, half of the corner reinforcementspecified in item (2) above shall be placed at right angles to the free edge.(8) Where analysis of slabs supported on four sides is made on the assumption of a single-span beam orneglecting torsional rigidity, corner reinforcement as specified in item (6) above shall be provided for crackcontrol.(9) If slabs are rigidly connected to edge beams or to slabs in adjacent bays, there is no need to verify torqueand torsion reinforcement may be dispensed with.(10) Longitudinal and shear reinforcement shall be provided along free edges, as shown in figure 71.

Key:1 Free edge2 Link3 Longitudinal reinforcement

Figure 71: Edge reinforcement at the free edges of slabs

(11) The reinforcement specified in item (10) above is not required for foundations and internal members inconventional buildings.(12) To avoid progressive damage to flat slabs, part of the reinforcement in the span shall be continued pastthe column strips at internal and edge columns, or, alternatively, shall be anchored there. The sectional areaof the reinforcement required shall be not less than that obtained from equation (153) and shall be located inthe area of zone of load transfer. The provisions allowing for a reduction in VEd are not applicable in this case.

VEdAs = (153) fyk

13.3.3 Shear and punching shear reinforcement(1) Detailing of reinforcement to accommodate shear and punching shear shall be as specified in subclause13.2.3 unless otherwise specified below.(2) A minimum shear reinforcement is not necessary in slabs calculated as not requiring shear reinforcement(i.e. where the design shear force, VEd, is less than the design shear resistance, VRd,ct) with a ratio of width toheight greater than 5. Slabs with a ratio of width to height of less than 4 shall be treated as beams.

14) See DAfStb-Heft 525 for other reinforcement geometry.

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Where the ratio of width to height is between 4 and 5, a minimum amount of reinforcement is required. This maybe obtained by linear interpolation between zero and unity for slabs calculated as not requiring shear reinforce-ment, and between 0,6 and unity for slabs requiring shear reinforcement (i.e. where VEd is greater than VRd,ct).(3) In slabs where VEd is equal to 0,3 VRd,max or less, the shear reinforcement shall consist completely of bent-up bars or shear assemblies. The specifications of item (2) of subclause 13.2.3 shall apply to slabs with VEdgreater than 0,3 VRd,max.(4) The maximum spacing of stirrups in longitudinal and transverse direction, smax, shall be as follows:

a) In longitudinal direction:– for VEd up to 0,3 VRd,max: 0,7 h;– for VEd greater than 0,3 VRd,max up to 0,6 VRd,max: 0,5 h;– for VEd greater than 0,6 VRd,max: 0,25 h.

b) In transverse direction: h.The maximum spacing of bent-up bars in longitudinal direction shall be equal to h.(5) The position of any punching shear reinforcement shall be as in figure 72.(6) The bar diameter of punching shear reinforcement shall be adjusted to suit the actual mean effective depthof the slab. It shall not exceed 0,05 d (where d is the slab depth). (154)(7) If only a single row of stirrups is required as punching shear reinforcement, a second row, constituted bythe minimum reinforcement calculated as in equation (114) with a bar spacing equal to 0,75 d shall be provided.

13.4 Prefabricated floor systems13.4.1 GeneralUnless contrary to the following, the specifications of subclause 13.3 shall also apply to prefabricated floorsystems.

13.4.2 Load dispersal(1) Load dispersal between adjacent floor elements shall be ensured by using a suitable means of connectionto enable shear transmission.(2) The following means of connection are permitted:

a) grouted joints, with or without shear reinforcement (cf. figure 73);b) welded or bolted connections;c) reinforced concrete topping.

(3) The dispersal of concentrated and line loads may need to be verified empirically or by calculation.(4) Where floors are designed on the basis of evenly distributed imposed loads, the analysis of shear trans-mission by suitable connections shall be made assuming a shear force acting along the joint with the samemagnitude as an imposed load with a transmission width of 0,5 m. There is generally no need to verify that theforce is subsequently resisted by the adjacent members. If the joint is in the flange of T-beams, it shall be verifiedthat the cantilever moment due to the force in the joint is greater than the moment occurring under full loading.(5) Decks made of precast elements of width not more than 1,2 m do not require shear reinforcement asspecified in item (2) of subclause 13.2.2.

13.4.3 Floor decks with in-situ topping(1) Where there is a loadbearing in-situ topping over precast concrete panels, combining to form a composite flooras described in subclause 10.3.6 for design purposes, the topping shall be at least 50 mm thick. The shear reinforce-ment may be located in either the panel or the topping. Item (6) of subclause 8.2 shall be taken into account.(2) When considering loading normal to the joint in decks that span in two directions, only continuous rein-forcement or such with joints as shown in figure 74 may be taken into account, and only if it meets the followingconditions: The diameter of the reinforcing bars shall be not more than 14 mm, the area of reinforcement, as,shall be not more than 10 cm2/m and the design shear force, VEd, shall be not more than 0,5 VRd,max (cf. sub-clause 10.3.4 for VEd and VRd,max). In addition, the joint shall be secured by rigid reinforcement (such as latticegirders) spaced not more than twice the floor thickness apart. The cross section of the part of the rigidreinforcement connecting with the longitudinal reinforcement in the panel shall be designed to resist the tensionfrom this longitudinal reinforcement.(3) In the analysis, the effect of torsional rigidity may only be taken into account if, there is no joint betweenpanels within the torsion zone of width equal to 0,3 l from the corner of the slab, or if the joint is secured byreinforcement at a distance of not more than 100 mm, from its edge. It shall be verified that the floor has adequateresistance to torque.(4) Verification for adequate resistance to torque is not required if the slab is rigidly connected to the edgebeam or adjacent bays.(5) In the case of end supports that do not need to carry loads from walls, reinforcement shall be providedalong the line of effective support, using 6 cm2 of reinforcement per metre of line of support over a width of0,75 m, to connect the topping to the precast deck.

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Key:a) Punching shear reinforcement with hangersb) Punching shear reinforcement with bent-up bars

1 Loaded area

Figure 72: Arrangement of punching shear reinforcement

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Key:a) Grouted jointb) Welded joint

1 Grouted recessFigure 73: Floor joints for shear transmission (examples)

Key:a) Joint in shear reinforcementb) Joint in longitudinal reinforcement

1 Precast panel2 In-situ concrete3 Longitudinal reinforcement4 Loadbearing shear reinforcement (in precast panel)5 Additional loadbearing shear reinforcement6 Lattice girder7 Additional longitudinal reinforcement

Figure 74: Loadbearing joints in two-way spanning precast concrete floors incorporating anin-situ topping (example)

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13.4.4 Plate action(1) A floor composed of precast elements is regarded as a loadbearing plate if, in its final state, it forms acontiguous even surface, the individual parts of the floor are connected to each other so as to have adequateresistance to compression, and any lateral stresses (e.g. due to wind action or deviations of the support fromthe vertical) can be accommodated due to arch or truss action and the presence of ties (see subclause 13.12.2for peripheral ties).(2) Ties acting as a truss shall take the form of reinforcement positioned in the joints between the panels or,alternatively, in the in-situ topping. They shall be anchored in the end members as specified in subclause 12.6and have joints as specified in subclause 12.8. The reinforcement in the end members and ties shall be analysed.(3) Joints intersected by struts forming part of the equivalent arch or truss system shall be analysed asspecified in subclause 10.3.6. If, as a result, there is a need for interlocking in horizontal direction, this shall beas shown in figure 75.

Key:a) For horizontal shearb) For coexistent horizontal and vertical shear

Figure 75: Interlocking of panels

13.5 Columns13.5.1 General(1) The minimum cross-sectional size (side length or diameter) of columns of full cross section, cast in situ,shall be 200 mm and 120 mm for precast elements fabricated horizontally.(2) The diameter of longitudinal reinforcing bars shall be not less than 12 mm.(3) Longitudinal bars shall be spaced not more than 300 mm apart. Columns of polygonal cross section shallbe provided with at least one bar in each corner. Columns of circular cross section shall contain not less thansix bars. For column sections up to 400 mm in width and of a depth not greater than their width, it is sufficientto have one reinforcing bar in each corner.

13.5.2 Limitations on area of longitudinal reinforcement(1) The minimum total area of longitudinal reinforcement, As,min, shall be obtained by means of equation (155):

As,min = 0,15 |NEd|/fyd with fyd = fyk/gs (155)

(2) The total area of reinforcement in any cross section shall be not more than 0,09 Ac. This also applies at lapjoints.

13.5.3 Shear reinforcement(1) Longitudinal reinforcement in columns shall be enclosed by shear reinforcement (i.e. links, hoops orhelices) the diameter of which shall be not less than a quarter of the maximum diameter of the longitudinalreinforcement, but not less than 6 mm. Cross wires in reinforcing fabric shall be not less than 5 mm in diameter.

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(2) When using bundles of bars as compression reinforcement, with dsV greater than 28 mm, the minimum bardiameter of individual links and helices shall, as a departure from item (1) above, be 12 mm.(3) Shear reinforcement shall be suitably anchored. Links shall be as shown in figure 56 e).(4) The link spacing shall not exceed the smallest of the following dimensions:

a) 12 times the minimum diameter of the longitudinal bars;b) the smaller cross-sectional size of the column;c) 300 mm.

(5) The link spacing specified in item (4) above shall be reduced by a factor of 0,6:a) directly above and below beams or slabs over a height equal to the larger cross-sectional size;b) at lapped joints if the greatest diameter of the longitudinal bars is more than 14 mm.

(6) Shear reinforcement as specified in items (1) to (5) above shall be used to secure a maximum of fivelongitudinal bars at each corner against buckling.(7) Further longitudinal bars than the ones in item (6) above and bars located further than 15 times the linkdiameter away from the corner, shall be secured by additional shear reinforcement as specified in item (3) above.This shall be no more than twice the distance from the shear reinforcement specified in items (1) and (4) above.

13.6 Deep beams(1) The minimum wall thickness of deep beams shall be the same as that for walls as specified in sub-clause 13.7.(2) Deep beams shall be provided with reinforcing fabric at each side, with the main reinforcement anddistributed reinforcement each having an area not less than 1,5 cm2/m and 0,075 % of the area of concrete.(3) The mesh size of the reinforcing fabric shall be not greater than twice the wall thickness and not more than300 mm.

13.7 Walls13.7.1 Reinforced concrete walls(1) The provisions of this subclause apply to reinforced concrete walls whose reinforcement is taken intoaccount in the analysis at the ultimate limit state. The specifications for slabs (cf. subclause 13.3) shall applywhere walls are predominately subjected to out-of-plane bending. Semi-finished products shall comply withthe specifications in the relevant agréments.(2) See table 32 for minimum wall thicknesses.

Table 32: Minimum thickness of loadbearing walls

Minimum thickness of loadbearing wall, in cm

Unreinforced wall under Reinforced wall under

non-continu- continuous non-continu- continuousous floors floors ous floors floors

C12/15In-situ concrete 20 14 — —or LC12/13

From C16/20or LC16/18

In-situ concrete 14 12 12 10

Precast 12 10 10 8

Type of concrete (strength class)

(3) The area of vertical reinforcement shall be not less than 0,003 Ac, but not more than 0,04 Ac in the caseof slender walls (cf. subclause 8.6.3) or walls with a value of |NEd| not less than 0,3 fcd · Ac. It shall be not lessthan 0,001 5 Ac in all other cases. Half of the vertical reinforcement should be located at each face of the wall.(4) The reinforcement ratio should be the same at both faces of the wall.(5) The area of shear reinforcement shall be not less than 20 % of that of the vertical reinforcement. Laterallyloaded walls, slender walls and walls with |NEd| not less than 0,3 fcd · Ac shall be provided with shear reinforce-ment with an area not less than 50 % of that of the vertical reinforcement. The horizontal reinforcement parallelto the outside of the wall and to the free edges should be in the outermost position.(6) The diameter of horizontal reinforcement shall be not less than one quarter of the diameter of the verticalbars.(7) Any two adjacent horizontal bars shall be spaced not more than 350 mm apart.

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(8) For concrete of strength class C70/85 or higher, the distance between two adjacent vertical bars shall benot more than twice the wall thickness or 300 mm (whichever is less).(9) Any loadbearing vertical reinforcement with an area greater than 0,02 Ac shall be enclosed by links asspecified in subclause 13.5.3.(10) At the free edges of walls with reinforcement with a sectional area, As, not less than 0,003 Ac at each side,the corner bars shall be secured by links (cf. figure 71).(11) The outermost bars on both faces of the wall shall be interconnected at four or more places per squaremetre of wall, all at different heights. This interconnection may, for example, be by means of S-hooks, in thecase of thick walls, by using links to anchor the bars (inside the wall). The free ends of the links shall have ananchorage length equal to 0,5 lb, (cf. subclause 12.6.2 for lb). If the diameter of loadbearing bars is not morethan 16 mm and these are covered by a layer of concrete not less than 2 ds, use of S-hooks is not required andthe reinforcing bars in compression may be located on the outside. This is obligatory in the case of compressionbars in fabric.

13.7.2 Wall-to-ceiling connections in precast construction(1) If a precast concrete wall is to be installed under a joint between two floor slabs or under a floor slab thatis fully connected to an external wall (cf. figure 76) and no other protective measures are taken, not more than50 % of the loadbearing section of the wall shall be regarded as making a structural contribution in the designcalculations and the connection is to be suitably designed.

Key:a) Intermediate supportb) End support

1 Precast walls2 Precast floor slabs3 Grouted joint

Figure 76: Support of floor slabs on precast concrete walls

(2) As a departure from item (1) above, not more than 60 % of the loadbearing section of a wall may be takeninto account if the following applies:

a) the area of additional shear reinforcement at the base of the wall, asw (cf. figure 77), is not less thanh/8 (with asw in cm2/m and h in cm); (156)b) the spacing of this reinforcement in longitudinal direction is not more than h or 200 mm, whichever isless; (157)c) the diameter of bars of the longitudinal reinforcement of area Asl at the base of the wall is not less than6 mm.

Key:1 Precast concrete wall2 Floor

Figure 77: Additional shear reinforcement at base of wall

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13.7.3 Sandwich panels(1) The design of sandwich panels shall take into account the effects of temperature, humidity, drying andshrinkage as a function of time.(2) Only corrosion-resistant materials that are covered by an agrément shall be used to connect the differentlayers forming sandwich panels.(3) The minimum reinforcement of the loadbearing layer should be not less than 1,3 cm2/m at either side inhorizontal and vertical directions. It is not usually necessary to provide edge reinforcement (cf. figure 71).(4) A single layer of reinforcement may be provided in the facing of sandwich panels.(5) Facing and core shall together be not less than 70 mm thick.

13.7.4 Unreinforced walls(1) Unreinforced walls shall be designed as specified in item (2) of subclause 10.2.(2) See table 32 for the the minimum thickness of unreinforced walls.(3) Walls shall be designed taking into account all recesses, chases, penetrations and cavities, except forvertical chases and recesses in walls and vertical chases at wall junctions that can be made by mortising at anysubsequent stage. This is the case if chases and recesses are of a depth up to 30 mm or not more than 1/6 ofthe wall thickness, and a width no more than the wall thickness, they are spaced at least 2 m apart and the wallis not less than 120 mm thick.

13.8 Connection and support of precast elements13.8.1 General(1) Connections for use in precast construction shall be designed for adequate strength, based on the as-sumptions used in the analysis of the structure as a whole and in the design of the individual elements requiringconnection. The connections shall be designed to withstand relative displacement in order to ensure theirloadbearing capacity and the loadbearing capacity of the structure as a whole.(2) Effects of imperfections resulting from the standard of workmanship shall be taken into consideration.(3) Connections shall, furthermore, be designed to avoid the premature cracking or spalling of the concreteat the ends of the members.(4) Connections shall be designed taking into due account tolerances, erection requirements, and ease ofexecution and monitoring.

13.8.2 Compression joints(1) Compression joints are joints that remain in full compression when subjected to the most unfavourablecombination of stresses.(2) Bed joints grouted with mortar, concrete or hardened polymer may be used if precautions are taken toexclude relative movement on either side of the joint during hardening of the grout.(3) Dry joints should only be used if the mean compressive stress in the concrete is not greater than 0,4 fcdand the required standard of workmanship is achieved in the factory and on site.(4) Compression joints cause considerable transverse tensile stresses in the adjacent members (see fig-ure 78). Support conditions are regarded as ‘hard’ if the modulus of elasticity of the joint material is higher than70 % of the modulus of elasticity of the adjacent members. Hard joints include joints made of cement mortar(cf. figure 78 c)), in which transverse tensile stresses occur as a result of deflection of the loadbearing sectionsof the reinforcement and concrete.(5) Transverse tensile stresses at the joints shall be resisted by suitable reinforcement in the adjacent mem-bers. This reinforcement may be concentrated in the zone in which the stresses occur.(6) Where resilient material is used in joints (cf. figure 78 a)), it may also be necessary to reinforce the joint itselfunless other means are used to prevent the material in the joint from giving way.(7) The design resistance axial force for compression joints should be calculated using accepted designmodels, with the resistance axial force of centrically loaded joints between supports as follows:

NRd = –¡ · (Ac · fcd + As · fyd) (158)

where ¡ takes into account the reinforcement ratio of the support and the joint thickness. 15)(8) For coexistent shear and axial force in the joint, the design shear force, VEd, may be neglected if this is lessthan 0,1 NEd (NEd being the design axial compression). See subclause 10.3.6 for shear alone.

15) See DAfStb-Heft 525 for ¡.

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13.8.3 Rigid and high-strength connections(1) It shall be ensured that the bending moments and tensile forces from the joints are transferred to theconnecting members.(2) The transfer of tensile forces at joints can be achieved with the following types of connections:

a) lap joints (e.g. straight bars, bent-up bars, hoops);b) welded connections;c) screwed or grouted sockets;d) clamping (e.g. using a socket).

Other types of connection may be used if of proven suitability.

13.8.4 Zones of supportZones of support are the parts of the supporting and supported members directly at the point of support. Theyshall be designed and constructed to fulfil their function while keeping within the specified manufacturing anderection tolerances.

13.9 Zones of load transfer13.9.1 Compression(1) Where one or more concentrated forces are transferred to a member, local supplementary reinforcementshall be provided that is capable of resisting the splitting tensile forces that will occur as a result.(2) This secondary reinforcement may comprise links or layers of reinforcement bent in the shape of hair pins.In long walls, straight bars of adequate length may be used as an alternative.

13.9.2 TensionTo accommodate tensile forces, reinforcing steel anchorages with the required anchorage length, lb,net, (whereappropriate, taking into account item (6) of subclause 12.5) may be anchored in the part of the cross sectionfacing away from the load, as specified in subclause 12.6.2, or else they may be joined as specified in sub-clause 12.8.

Compression

TensionKey:a) Tensile force at faces of jointb) Splitting tensile forcec) Transverse tensile force

1 Precast elements2 Resilient material3 Hard material, partial joint4 Hard material, full joint

Figure 78: Transverse tensile stresses in compression joints

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13.10 Forces associated with changes in directionIt shall be ensured that forces occurring as a result of changes in direction of tensile or compressive forces areresisted.

13.11 Indirect support(1) Where members are supported indirectly, suspension reinforcement shall be provided at member intersec-tions. This shall take the full load of the support reactions occurring in the intersecting members.(2) Suspension reinforcement should preferably be made of links that enclose the main reinforcement in thesupporting member. A number of these links may be located outside the immediate area of intersection of thetwo members (i.e. in the hatched area in figure 79) if horizontal reinforcement with the same overall cross sectionas that of the links is distributed over the depth of the member.

Key:1 Supported beam2 Supporting beam

h1 depth of supporting beamh2 depth of supported beam (not greater than h1)

Figure 79: Connection of secondary beams (plan view)

(3) Where the supporting members are wide beams or slabs, the suspension reinforcement provided in thesebeams or slabs shall be not wider than the effective depth of the beams they support.

13.12 Limitation of damage due to accidental actions13.12.1 General(1) In the event of accidental actions, damage shall be prevented from occurring to the building to an extentdisproportionate to the original cause (cf. DIN 1055-100).(2) If the provisions of this clause and the other provisions of this standard are met, it may be assumed thatthe chance failure of an individual member or a part of the structure or the occurrence of acceptable localdamage will not result in the failure of the whole structure.(3) In conventional buildings, peripheral ties shall be used to limit damage in the event of accidental actions.In precast construction, internal ties, and horizontal column and wall ties may additionally be used.(4) If structures are divided by expansion joints into structurally independent sections, each section shall havea separate tying system.(5) When designing tying systems, it may be assumed that the reinforcement is fully utilized up to its charac-teristic strength and has the tensile strength specified in subclauses 13.12.2 to 13.12.4. Reinforcement pro-vided for other purposes may also constitute, or form part of, the tying system.(6) In the design of tying systems, action-effects other than those caused directly by accidental actions or asa consequence of local damage may be neglected.(7) Reinforcement used for ties may overlap as specified in subclause 12.8. Secure mechanical anchoragedevices should be used if the joints between precast elements are too narrow.(8) Ties may be completely immersed in the topping or may be located at connections. If ties are interruptedin any plane, the effect of eccentricity shall be taken into consideration.(9) Ties may be post-tensioned.

13.12.2 Peripheral ties(1) If plate action is required to ensure overall lateral stability of conventional buildings, peripheral ties shallbe used to secure each floor and roof. These ties may include reinforcement that are part of the internal ties orreinforcement to subclause 13.1, 13.2, 13.3 and 13.4.4 and that are designed or constructed as specified initems (2) and (3) below.

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(2) Continuity may be achieved by overlapping the longitudinal reinforcement over a length equal to twice lb.The splice shall be enclosed by links (cf. figure 71) or helices spaced not more than 100 mm apart. Alternatively,welding or fasteners may be used.(3) Peripheral ties should be capable of resisting a design tensile force, FEd, in kN, equal to 10 leff,i, but not morethan 70 kN (with leff,i the effective span of the end bay normal to the tie, in m).

13.12.3 Internal ties(1) If internal ties are provided, these shall be located in each floor and roof plane in two directions approxi-mately normal to each other. They shall be continuous over their whole length and should be anchored with theirfree ends in the peripheral ties or else shall be continued as horizontal ties to walls or columns.(2) The ties may be evenly distributed in slabs, beams, walls or other suitable members. In walls, they shouldbe located within a region 0,5 m above or below the floor slab.(3) The ties shall be capable of sustaining a design tensile force, FEd, of 20 kN/m in each direction.(4) Where floor slabs are without a topping and cannot have ties distributed over their direction of span, theties may be concentrated in the joints between the elements, in which case each joint shall be capable ofwithstanding at least the following:

leff,1 + leff,2FEd = ë 20 kN ß 70 kN (159) 2

where leff,1 and leff,2 are the effective spans of the floor slabs at each side of the joint normal to the joint, in m(cf. figure 80).

13.12.4 Horizontal wall and column ties(1) End supports and loadbearing and stiffening external walls should be horizontally anchored at the top inconventional buildings and at the top and bottom in high-rise buildings.(2) The ties should be capable of resisting a design tensile force, FEd, of 10 kN per metre of external wall orcolumn. The tensile force assumed to act in columns shall be not more than 150 kN.(3) Corner columns shall be anchored in two directions. The reinforcement used for peripheral ties may alsobe used for column ties.(4) Where, in high-rise buildings, there are no joints in external wall panels between the walls that stiffen themand the length of the panels between these walls is not more than double their height, the connections at thebottom of the panels may be replaced by connections with the same overall tensile force provided in the bottomhalf of the vertical jonts between the external walls and their stiffening walls.

Key:1 Columns2 Joists/walls3 Peripheral ties4 Internal ties5 Horizontal wall/column ties

Figure 80: Tying system providing for accidental actions (floor plan)

(5) At the top edge of loadbearing internal wall panels, at least 0,7 cm2 of reinforcement per metre of panelshall reach into the cavity between the floor panels. This reinforcement may be joined at two points for panels2,5 m or over in length. Where wall panels are up to 2,5 m in long, one connection, in the centre of the panel,will be sufficient. Alternatively, other suitable means may be used instead of this reinforcement.