SAFE Design Manual Title - Springerextras.springer.com/1995/978-3-642-78702-7/INFORMATION-MANUALS...SAFE Design Manual iv Determine Factored Moments for the Strip 4-15 Design Flexural

Computers and Structures, Inc.Berkeley, California, USA

Version 8.0.0August 2004

SAFE

Integrated Analysis and Design of Slab Systems

Design Manual

Copyright Computers and Structures, Inc., 1978-2004.The CSI Logo is a registered trademark of Computers and Structures, Inc.

SAFE and CSiDETAILER are trademarks of Computers and Structures, Inc.“Watch & Learn” is a trademark of Computers and Structures, Inc.

Windows is a registered trademark of Microsoft Corporation.Adobe and Acrobat are registered trademarks of Adobe Systems Incorporated.

Copyright

The computer program SAFE and all associated documentation are proprietary andcopyrighted products. Worldwide rights of ownership rest with Computers andStructures, Inc. Unlicensed use of the program or reproduction of the documentation inany form, without prior written authorization from Computers and Structures, Inc., isexplicitly prohibited.

Further information and copies of this documentation may be obtained from:

Computers and Structures, Inc.1995 University Avenue

Berkeley, California 94704 USA

Phone: (510) 845-2177FAX: (510) 845-4096

e-mail: [email protected] (for general questions)e-mail: [email protected] (for technical support questions)

web: www.csiberkeley.com

DISCLAIMER

CONSIDERABLE TIME, EFFORT AND EXPENSE HAVE GONE INTO THEDEVELOPMENT AND DOCUMENTATION OF SAFE. THE PROGRAM HAS BEENTHOROUGHLY TESTED AND USED. IN USING THE PROGRAM, HOWEVER,THE USER ACCEPTS AND UNDERSTANDS THAT NO WARRANTY ISEXPRESSED OR IMPLIED BY THE DEVELOPERS OR THE DISTRIBUTORS ONTHE ACCURACY OR THE RELIABILITY OF THE PROGRAM.

THE USER MUST EXPLICITLY UNDERSTAND THE ASSUMPTIONS OF THEPROGRAM AND MUST INDEPENDENTLY VERIFY THE RESULTS.

i

Contents

Design Manual

1 Introduction 1-1

2 Design for ACI 318-02

Design Load Combinations 2-4

Strength Reduction Factors 2-4

Beam Design 2-5

Design Flexural Reinforcement 2-5

Determine Factored Moments 2-5

Determine Required FlexuralReinforcement 2-6

Design for Rectangular Beam 2-6

Design for T-Beam 2-9

Design Beam Shear Reinforcement 2-13

Determine Shear Force 2-13

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SAFE Design Manual

ii

Determine Concrete Shear Capacity 2-14

Determine Required ShearReinforcement 2-14

Slab Design 2-15

Design for Flexure 2-15

Determine Factored Moments forthe Strip 2-16

Design Flexural Reinforcement forthe Strip 2-16

Check for Punching Shear 2-17

Critical Section for Punching Shear 2-17

Transfer of Unbalanced Moment 2-17

Determination of Concrete Capacity 2-17

Determination of Capacity Ratio 2-18

3 Design for CSA A23.3-94



Beam Design 3-5

Design Beam Flexural Reinforcement 3-5



Design for Flexure of a RectangularBeam 3-6

Design for Flexure of a T-Beam 3-9


Determine Shear Force and Moment 3-14

Contents

iii



Slab Design 3-15









4 Design for BS 8110-85


Design Strength 4-4

Beam Design 4-5




Design of a Rectangular Beam 4-6

Design of a T-Beam 4-8


Slab Design 4-14


SAFE Design Manual

iv







5 Design for Eurocode 2


Design Strength 5-5

Beam Design 5-5




Design as a Rectangular Beam 5-8

Design as a T-Beam 5-10


Slab Design 5-18







Contents

v


6 Design for NZ 3101-95



Beam Design 6-5




Design for Flexure of a RectangularBeam 6-7

Design for Flexure of a T-Beam 6-9


Determine Shear Force and Moment 6-14



Slab Design 6-15








SAFE Design Manual

vi


7 Design for IS 456-78 (R1996)


Design Strength 7-5

Beam Design 7-5



Determine Required Flexural Reinforcement 7-6

Design as a Rectangular Beam 7-9

Design as a T-Beam 7-11


Slab Design 7-18









References

1 - 1

Chapter 1

Introduction

SAFE automates several slab and mat design tasks. Specifically, it inte-grates slab design moments across design strips and designs the requiredreinforcement; it checks slab punching shears around column supportsand concentrated loads; and it designs beam flexural and shear rein-forcements. The design procedures are described in the chapter entitled"SAFE Design Techniques” in the Welcome to SAFE Manual. The actualdesign algorithms vary based on the specific Design Code chosen by theuser. This manual describes the algorithms used for the various codes.

It is noted that the design of reinforced concrete slabs is a complex sub-ject and the Design Codes cover many aspects of this process. SAFE is atool to help the user in this process. Only the aspects of design docu-mented in this manual are automated by SAFE design. The user mustcheck the results produced and address other aspects not covered bySAFE design.

SAFE™SAFE™

Design Load Combinations 2 - 1

Chapter 2

Design for ACI 318-02

This chapter describes in detail the various aspects of the concrete designprocedure that is used by SAFE when the user selects the American codeACI 318-02 (ACI 2002). Various notations used in this chapter are listedin Table 1-1. For referencing to the pertinent sections of the ACI code inthis chapter, a prefix “ACI” followed by the section number is used.

The design is based on user-specified loading combinations, although theprogram provides a set of default load combinations that should satisfyrequirements for the design of most building type structures.

English as well as SI and MKS metric units can be used for input. Thecode is based on Inch-Pound-Second units. For simplicity, all equationsand descriptions presented in this chapter correspond to Inch-Pound-Second units unless otherwise noted.

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SAFE Design Manual

2 - 2 Design Load Combinations

Table 2-1 List of Symbols Used in the ACI Code

Ag Gross area of concrete, sq-in

As Area of tension reinforcement, sq-in

A's Area of compression reinforcement, sq-in

As(required) Area of steel required for tension reinforcement, sq-in

Av Area of shear reinforcement, sq-in

Av /s Area of shear reinforcement per unit length of member, sq-in/in

a Depth of compression block, in

ab Depth of compression block at balanced condition, in

amax Maximum allowed depth of compression block, in

b Width of member, in

bf Effective width of flange (T-Beam section), in

bw Width of web (T-Beam section), in

b0 Perimeter of the punching critical section, in

b1 Width of the punching critical section in the direction ofbending, in

b2 Width of the punching critical section perpendicular to thedirection of bending, in

c Depth to neutral axis, in

cb Depth to neutral axis at balanced conditions, in

d Distance from compression face to tension reinforcement, in

d' Concrete cover to center of reinforcing, in

ds Thickness of slab (T-Beam section), in

Ec Modulus of elasticity of concrete, psi

Chapter 2 - Design Load Combinations

Design Load Combinations 2- 3

Table 2-1 List of Symbols Used in the ACI Code

Es Modulus of elasticity of reinforcement, assumed as29,000,000 psi (ACI 8.5.2)

f 'c Specified compressive strength of concrete, psi

fy Specified yield strength of flexural reinforcement, psi

fys Specified yield strength of shear reinforcement, psi

h Overall depth of a section, in

Mu Factored moment at section, lb-in

Pu Factored axial load at section, lb

s Spacing of the shear reinforcement along the length of thebeam, in

Vc Shear force resisted by concrete, lb

Vmax Maximum permitted total factored shear force at a section, lb

Vu Factored shear force at a section, lb

Vs Shear force resisted by steel, lb

β1 Factor for obtaining depth of compression block in concrete

βc Ratio of the maximum to the minimum dimensions of thepunching critical section

εc Strain in concrete

εc, max Maximum usable compression strain allowed in extreme con-crete fiber, (0.003 in/in)

εs Strain in reinforcing steel

εs,min Minimum tensile strain allowed in steel rebar at nominalstrength for tension controlled behavior (0.005 in/in)

ϕ Strength reduction factor

γf Fraction of unbalanced moment transferred by flexure

γv Fraction of unbalanced moment transferred by eccentricity ofshear

SAFE Design Manual


Design Load CombinationsThe design load combinations are the various combinations of the pre-scribed load cases for which the structure needs to be checked. For thiscode, if a structure is subjected to dead load (DL), live load (LL), patternlive load (PLL), wind (WL), and earthquake (EL) loads, and consideringthat wind and earthquake forces are reversible, the following load com-binations must be considered (ACI 9.2.1).

1.4 DL1.2 DL + 1.6 LL (ACI 9.2.1)

1.2 DL + 1.6 * 0.75 PLL (ACI 13.7.6.3)

0.9 DL ± 1.6 WL1.2 DL + 1.0 LL ± 1.6 WL (ACI 9.2.1)

0.9 DL ± 1.0 EL1.2 DL + 1.0 LL ± 1.0 EL (ACI 9.2.1)

The IBC 2003 basic load combinations (Section 1605.2.1) are the same.These are also the default design load combinations in SAFE when theACI 318-02 code is used. The user should use other appropriate loadingcombinations if roof live load is separately treated, or other types ofloads are present.

Strength Reduction FactorsThe strength reduction factors, ϕ, are applied on the specified strength toobtain the design strength provided by a member. The ϕ factors for flex-ure and shear are as follows:

ϕ = 0.90 for flexure (tension controlled) and (ACI 9.3.2.1)

ϕ = 0.75 for shear. (ACI 9.3.2.3)

The user is allowed to overwrite these values. However, caution is ad-vised.

Chapter 2 - Beam Design

Beam Design 2- 5

Beam DesignIn the design of concrete beams, SAFE calculates and reports the re-quired areas of steel for flexure and shear based on the beam moments,shear forces, load combination factors, and other criteria described in thissection. The reinforcement requirements are calculated at the ends of thebeam elements.

All of the beams are designed for major direction flexure and shear only. Effects resulting from any axial forces, minor direction bending, andtorsion that may exist in the beams must be investigated independentlyby the user.

The beam design procedure involves the following steps:

Design flexural reinforcement

Design shear reinforcement

Design Flexural ReinforcementThe beam top and bottom flexural steel is designed at the two stations atthe ends of the beam elements. In designing the flexural reinforcementfor the major moment of a particular beam for a particular station, thefollowing steps are involved:

Determine factored moments

Determine required flexural reinforcement

Determine Factored MomentsIn the design of flexural reinforcement of concrete beams, the factoredmoments for each load combination at a particular beam section are ob-tained by factoring the corresponding moments for different load caseswith the corresponding load factors.

The beam section is then designed for the maximum positive and maxi-mum negative factored moments obtained from all of the load combina-tions. Positive beam moments produce bottom steel. In such cases thebeam may be designed as a Rectangular or a T-beam. Negative beam

SAFE Design Manual

2 - 6 Beam Design

moments produce top steel. In such cases the beam may be designed as arectangular or inverted T-beam.

Determine Required Flexural ReinforcementIn the flexural reinforcement design process, the program calculates boththe tension and compression reinforcement. Compression reinforcementis added when the applied design moment exceeds the maximum mo-ment capacity of a singly reinforced section. The user has the option ofavoiding the compression reinforcement by increasing the effectivedepth, the width, or the grade of concrete.

The design procedure is based on the simplified rectangular stress block,as shown in Figure 2-1 (ACI 10.2). Furthermore, it is assumed that thenet tensile strain of the reinforcing steel shall not be less than 0.005 (ten-sion controlled) (ACI 10.3.4). When the applied moment exceeds themoment capacity at this design condition, the area of compression rein-forcement is calculated on the assumption that the additional momentwill be carried by compression and additional tension reinforcement.

The design procedure used by SAFE, for both rectangular and flangedsections (L- and T-beams), is summarized in the following subsections. Itis assumed that the design ultimate axial force does not exceed ϕ(0.1f'

c

Ag) (ACI 10.3.5); hence, all of the beams are designed for major direc-tion flexure and shear only.

Design for Rectangular Beam

In designing for a factored negative or positive moment, Mu (i.e., design-ing top or bottom steel), the depth of the compression block is given by a(see Figure 2-1), where,

a =bf

Mdd

c

u

ϕ−−

'85.0

22 , (ACI 10.2)

where, the value of ϕ is taken as that for a tension controlled section,which is 0.90 (ACI 9.3.2.1) in the above and the following equations.


Beam Design 2- 7

Figure 2-1 Rectangular Beam Design

The maximum depth of the compression zone, cmax, is calculated based onthe limitation that the tensile steel tension shall not be less than εsmin,which is equal to 0.005 for tension controlled behavior (ACI 10.3.4):

cmax = minsmaxc

maxc

ε+εε

(ACI 10.2.2)

where,

εcmax = 0.003 (ACI 10.2.3)

εsmin = 0.005 (ACI 10.3.4)

The maximum allowable depth of the rectangular compression block,amax, is given by

amax =β1cmax (ACI 10.2.7.1)

where β1 is calculated as follows:

SAFE Design Manual

2 - 8 Beam Design

β1 =0.85 − 0.05

−1000

4000'cf , 0.65 ≤ β1 ≤ 0.85 (ACI 10.2.7.3)

If a ≤ amax (ACI 10.3.4), the area of tensile steel reinforcement is thengiven by

As =

−ϕ

2

adf

M

y

u .

This steel is to be placed at the bottom if Mu is positive, or at the topif Mu is negative.

If a > amax, compression reinforcement is required (ACI 10.3.5) and iscalculated as follows:

− The compressive force developed in concrete alone is given by

C = 0.85f'

c bamax , and (ACI 10.2.7.1)

the moment resisted by concrete compression and tensile steel is

Muc = C

−

2maxa

d ϕ .

− Therefore the moment resisted by compression steel and tensilesteel is

Mus = Mu − Muc.

− So the required compression steel is given by

A'

s = ( )( )ϕ−− '85.0 '' ddff

M

cs

us , where

f's = Esεcmax

−

max

max '

c

dc ≤ fy. (ACI 10.2.2, 10.2.3, and ACI 10.2.4)

− The required tensile steel for balancing the compression in con-crete is


Beam Design 2- 9

As1 = ϕ

−

2maxa

df

M

y

us , and

the tensile steel for balancing the compression in steel is given by

As2 = ( )ϕ− 'ddf

M

y

us .

− Therefore, the total tensile reinforcement is As = As1 + As2, and thetotal compression reinforcement is A's. As is to be placed at thebottom and A's is to be placed at the top if Mu is positive, and A's isto be placed at the bottom and As is to be placed at the top if Mu isnegative.

Design for T-Beam

(i) Flanged Beam Under Negative Moment

In designing for a factored negative moment, Mu (i.e., designing topsteel), the calculation of the steel area is exactly the same as describedfor a rectangular beam, i.e., no T-Beam data is used.

(ii) Flanged Beam Under Positive Moment

If Mu > 0 , the depth of the compression block is given by

a = d – fc

u

bf

Md

ϕ−

'85.0

22 , (ACI 10.2)

where, the value of ϕ is taken as that for a tension controlled section,which is 0.90 (ACI 9.3.2.1) in the above and the following equations.

The maximum depth of the compression zone, cmax, is calculated based onthe limitation that the tensile steel tension shall not be less than εsmin,which is equal to 0.005 for tension controlled behavior (ACI 10.3.4):

cmax = minmax

max

sc

c

ε+εε

(ACI 10.2.2)

SAFE Design Manual

2 - 10 Beam Design

where,

εcmax = 0.003 (ACI 10.2.3)

εsmin = 0.005 (ACI 10.3.4)

The maximum allowable depth of the rectangular compression block,amax, is given by

amax =β1cmax (ACI 10.2.7.1)

where β1 is calculated as follows:

β1 =0.85 − 0.05

−

1000

4000'cf, 0.65 ≤ β1 ≤ 0.85 (ACI 10.2.7.3)

If a ≤ ds, the subsequent calculations for As are exactly the same aspreviously defined for the rectangular section design. However, inthis case, the width of the beam is taken as bf. Compression rein-forcement is required if a > amax.

If a > ds, calculation for As has two parts. The first part is for balanc-ing the compressive force from the flange, Cf, and the second part isfor balancing the compressive force from the web, Cw, as shown inFigure 2-2. Cf is given by

Cf = 0.85f'

c(bf − bw) min(ds , amax).

Therefore, As1 = y

f

f

C and the portion of Mu that is resisted by the

flange is given by

Muf = Cf ( )

−

2

,min maxadd s ϕ.

Again, the value for ϕ is 0.90. Therefore, the balance of the moment, Mu

to be carried by the web is given by

Muw = Mu − Muf .


Beam Design 2- 11

Figure 2-2 T-Beam Design

The web is a rectangular section of dimensions bw and d, for which thedesign depth of the compression block is recalculated as

a1 = d − wc

uw

bf

Md

ϕ−

'2

85.0

2. (ACI 10.2)

If a1 ≤ amax (ACI 10.3.5), the area of tensile steel reinforcement isthen given by

As2 =

−ϕ

21a

df

M

y

uw , and

As = As1 + As2 .

This steel is to be placed at the bottom of the T-beam.

If a1 > amax , compression reinforcement is required (ACI 10.3.5) andis calculated as follows:

− The compressive force in the web concrete alone is given by

SAFE Design Manual

2 - 12 Beam Design

C = 0.85f '

cbwamax. (ACI 10.2.7.1)

− Therefore the moment resisted by the concrete web and tensilesteel is

Muc = C

−

2maxa

d ϕ, and

the moment resisted by compression steel and tensile steel is

Mus = Muw − Muc.

− Therefore, the compression steel is computed as

A's = ( )( )ϕ−− '85.0 '' ddff

M

cs

us , where

f '

s = εsεcmax

−

max

max

c

dc '≤ fy. (ACI 10.2.2, 10.2.3 and ACI 10.2.4)

− The tensile steel for balancing compression in web concrete is

As2 =

ϕ

−

2max

y

uc

adf

M, and

the tensile steel for balancing compression in steel is

As3 = ( )ϕ− 'ddf

M

y

us .

− The total tensile reinforcement is As = As1 + As2 + As3, and the totalcompression reinforcement is A's. As is to be placed at the bottomand A's is to be placed at the top.

Minimum and Maximum Tensile ReinforcementThe minimum flexural tensile steel required in a beam section is given bythe minimum of the following two limits:


Beam Design 2- 13

As ≥ max

y

c

f

f '3 bwd and db

f wy

200 or (ACI 10.5.1)

As ≥ 3

4 As(required) (ACI 10.5.3)

An upper limit of 0.04 times the gross web area on both the tension rein-forcement and the compression reinforcement is imposed upon request asfollows:

0.04 bd Rectangular beam

As ≤ 0.04 bwd T-beam

0.04 bd Rectangular beamA'

s ≤0.04 bwd T-beam

Design Beam Shear ReinforcementThe shear reinforcement is designed for each load combination at twostations at the ends of each beam element. In designing the shear rein-forcement for a particular beam for a particular loading combination at aparticular station resulting from beam major shear, the following stepsare involved:

Determine the factored shear force, Vu.

Determine the shear force, Vc, that can be resisted by the concrete.

Determine the reinforcement steel required to carry the balance.

The following three sections describe in detail the algorithms associatedwith the above-mentioned steps.

Determine Shear ForceIn the design of the beam shear reinforcement of a concrete beam, theshear forces for a particular load combination at a particular beam sec-

SAFE Design Manual

2 - 14 Beam Design

tion are obtained by factoring the associated shear forces and momentswith the corresponding load combination factors.

Determine Concrete Shear CapacityThe shear force carried by the concrete, Vc, is calculated as follows:

Vc = 2 cf ' bwd . (ACI 11.3.1.1)

A limit is imposed on the value of cf ' as cf ' ≤ 100. (ACI 11.1.2)

Determine Required Shear Reinforcement The shear force is limited to a maximum of

Vmax = Vc + (8 cf ' ) bwd. (ACI 11.5.6.9)

Given Vu, Vc and Vmax , the required shear reinforcement is calculatedas follows, where ϕ, the strength reduction factor, is 0.75 (ACI9.3.2.3).

If Vu ≤ (Vc / 2) ϕ ,

s

Av = 0 , (ACI 11.5.5.1)

else if (Vc / 2) ϕ < Vu ≤ ϕVmax ,

s

Av = ( )

df

VV

ys

cu

ϕϕ−

, (ACI 11.5.6.2)

s

Av ≥ max

w

yw

y

cb

fb

f

f 50,

75.0 '

(ACI 11.5.5.3)

else if Vu > ϕVmax,

a failure condition is declared. (ACI 11.5.6.9)

Chapter 2 - Slab Design

Slab Design 2- 15

The maximum of all the calculated Av /s values, obtained from each loadcombination, is reported along with the controlling shear force and asso-ciated load combination number.

The beam shear reinforcement requirements displayed by the programare based purely upon shear strength considerations. Any minimum stir-rup requirements to satisfy spacing and volumetric considerations mustbe investigated independently of the program by the user.

Slab Design Similar to conventional design, the SAFE slab design procedure in-volves defining sets of strips in two mutually perpendicular directions.The locations of the strips are usually governed by the locations of theslab supports. The moments for a particular strip are recovered from theanalysis and a flexural design is completed based on the ultimate strengthdesign method (ACI 318-02) for reinforced concrete as described in thefollowing sections. To learn more about the design strips, refer to thesection entitled "SAFE Design Techniques" in the Welcome to SAFEmanual.

Design for FlexureSAFE designs the slab on a strip-by-strip basis. The moments used forthe design of the slab elements are the nodal reactive moments, whichare obtained by multiplying the slab element stiffness matrices by theelement nodal displacement vectors. These moments will always be instatic equilibrium with the applied loads, irrespective of the refinementof the finite element mesh.

The design of the slab reinforcement for a particular strip is completed atspecific locations along the length of the strip. Those locations corre-spond to the element boundaries. Controlling reinforcement is computedon either side of those element boundaries. The slab flexural design pro-cedure for each load combination involves the following:

Determine factored moments for each slab strip.

Design flexural reinforcement for the strip.

SAFE Design Manual

2 - 16 Slab Design

These two steps, which are described in the next two subsections, are re-peated for every load combination. The maximum reinforcement calcu-lated for the top and bottom of the slab within each design strip, alongwith the corresponding controlling load combination numbers, is ob-tained and reported.

Determine Factored Moments for the StripFor each element within the design strip, the program calculates thenodal reactive moments for each load combination. The nodal momentsare then added to get the strip moments.

Design Flexural Reinforcement for the StripThe reinforcement computation for each slab design strip, given thebending moment, is identical to the design of rectangular beam sectionsdescribed earlier (or to the T-beam if the slab is ribbed). When the slabproperties (depth, etc.) vary over the width of the strip, the programautomatically designs slab widths of each property separately for thebending moment to which they are subjected and then sums the rein-forcement for the full width. Where openings occur, the slab width isadjusted accordingly.

Minimum and Maximum Slab Reinforcement

The minimum flexural tensile reinforcement required for each directionof a slab is given by the following limits (ACI 7.12.2):

As ≥ 0.0018 bh yf

60000(ACI 7.12.2.1)

0.0014 bh ≤ As ≤ 0.0020 bh (ACI 7.12.2.1)

In addition, an upper limit on both the tension reinforcement and com-pression reinforcement has been imposed to be 0.04 times the grosscross-sectional area.


Slab Design 2- 17

Check for Punching ShearThe algorithm for checking punching shear is detailed in the section enti-tled “Slab Punching Shear Check” in the Welcome to SAFE manual.Only the code specific items are described in the following subsections.

Critical Section for Punching ShearThe punching shear is checked on a critical section at a distance of d/2from the face of the support (ACI 11.12.1.2). For rectangular columnsand concentrated loads, the critical area is taken as a rectangular area,with the sides parallel to the sides of the columns or the point loads (ACI11.12.1.3).

Transfer of Unbalanced MomentThe fraction of unbalanced moment transferred by flexure is taken to beγ f Mu and the fraction of unbalanced moment transferred by eccentricityof shear is taken to be γ v Mu ,

γ f = ( ) 21321

1

bb+, and (ACI 13.5.3.2)

γ v = 1 − γ f, (ACI 13.5.3.1)

where b1 is the width of the critical section measured in the direction ofthe span and b2 is the width of the critical section measured in the direc-tion perpendicular to the span.

Determination of Concrete CapacityThe concrete punching shear stress capacity is taken as the minimum ofthe following three limits:

SAFE Design Manual

2 - 18 Slab Design

ϕ c

c

f '42

β

+

ϕ

α+

0

2b

dscf 'vc = min

ϕ 4 cf '

(ACI 11.12.2.1)

where, βc is the ratio of the minimum to the maximum dimensions of thecritical section, b0 is the perimeter of the critical section, and αs is a scalefactor based on the location of the critical section.

40 for interior columns,30 for edge columns, andαs =20 for corner columns.

(ACI 11.12.2.1)

A limit is imposed on the value of cf ' as

cf ' ≤ 100 . (ACI 11.1.2)

Determination of Capacity RatioGiven the punching shear force and the fractions of moments transferredby eccentricity of shear about the two axes, the shear stress is computedassuming linear variation along the perimeter of the critical section. Theratio of the maximum shear stress and the concrete punching shear stresscapacity is reported by SAFE.


Chapter 3

Design for CSA A23.3-94

This chapter describes in detail the various aspects of the concrete designprocedure that is used by SAFE when the user selects the Canadian code,CSA A23.3-94 (CSA 1994). Various notations used in this chapter arelisted in Table 3-1. For referencing to the pertinent sections of the Cana-dian code in this chapter, a prefix “CSA” followed by the section numberis used.


English as well as SI and MKS metric units can be used for input. Thecode is based on Newton-Millimeter-Second units. For simplicity, allequations and descriptions presented in this chapter correspond to New-ton-Millimeter-Second units unless otherwise noted.

SAFE™SAFE™

SAFE Design Manual


Table 3-1 List of Symbols Used in the Canadian Code

As Area of tension reinforcement, sq-mm

A's Area of compression reinforcement, sq-mm

As(required) Area of steel required for tension reinforcement, sq-mm

Av Area of shear reinforcement, sq-mm

Av / s Area of shear reinforcement per unit length of the member,sq-mm/mm

a Depth of compression block, mm

ab Depth of compression block at balanced condition, mm

b Width of member, mm

bf Effective width of flange (T-Beam section), mm

bw Width of web (T-Beam section), mm

b0 Perimeter of the punching critical section, mm

b1 Width of the punching critical section in the direction ofbending, mm

b2 Width of the punching critical section perpendicular to thedirection of bending, mm

c Depth to neutral axis, mm

cb Depth to neutral axis at balanced conditions, mm

d Distance from compression face to tension reinforcement,mm

d' Concrete cover to center of reinforcing, mm

ds Thickness of slab (T-Beam section), mm

Ec Modulus of elasticity of concrete, MPa

Es Modulus of elasticity of reinforcement, assumed as 200,000MPa

f '

c Specified compressive strength of concrete, MPa



Table 3-1 List of Symbols Used in the Canadian Code

fy Specified yield strength of flexural reinforcement, MPa

fys Specified yield strength of shear reinforcement, MPa

h Overall depth of a section, mm

Mf Factored moment at section, N-mm

s Spacing of the shear reinforcement along the length of thebeam, in

Vc Shear resisted by concrete, N


Vf Factored shear force at a section, N

Vs Shear force at a section resisted by steel, N

α1 Ratio of average stress in rectangular stress block to thespecified concrete strength





ϕc Strength reduction factor for concrete

ϕs Strength reduction factor for steel

ϕm Strength reduction factor for member



λ Shear strength factor

SAFE Design Manual


Design Load CombinationsThe design load combinations are the various combinations of the pre-scribed load cases for which the structure needs to be checked. For thiscode, if a structure is subjected to dead load (DL), live load (LL), patternlive load (PLL), wind (WL), and earthquake (EL) loads, and consideringthat wind and earthquake forces are reversible, the following load com-binations should be considered (CSA 8.3):

1.25 DL1.25 DL + 1.50 LL (CSA 8.3.2)

1.25 DL + 1.50 *0.75 PLL (CSA 13.9.4.3)

1.25 DL ± 1.50 WL0.85 DL ± 1.50 WL1.25 DL + 0.7 (1.50 LL ± 1.50 WL) (CSA 8.3.2)

1.00 DL ±1.00 EL1.00 DL + (0.50 LL ± 1.00 EL) (CSA 8.3.2)

These are also the default design load combinations in SAFE when theCSA A23.3-94 code is used. The user should use other appropriate load-ing combinations if roof live load is separately treated, or other types of loads are present.

Strength Reduction FactorsThe strength reduction factor, ϕ, is material dependent and is defined asfollows:

ϕ = 0.60 for concrete and (CSA 8.4.2)

ϕ = 0.85 for steel. (CSA 8.4.3)



Beam Design 3- 5

Beam DesignIn the design of concrete beams, SAFE calculates and reports the re-quired areas of steel for flexure and shear based on the beam moments,shear forces, load combination factors, and other criteria described in thissection. The reinforcement requirements are calculated at the end of thebeam elements.

All of the beams are designed for major direction flexure and shear only.Effects resulting from any axial forces, minor direction bending, and tor-sion that may exist in the beams must be investigated independently bythe user.


Design beam flexural reinforcement

Design beam shear reinforcement

Design Beam Flexural ReinforcementThe beam top and bottom flexural steel is designed at the two stations atthe end of the beam elements. In designing the flexural reinforcement forthe major moment of a particular beam for a particular station, the fol-lowing steps are involved:

Determine the maximum factored moments

Determine the reinforcing steel


The beam section is then designed for the maximum positive and maxi-mum negative factored moments obtained from all of the load combina-tions. Positive beam moments produce bottom steel. In such cases the

SAFE Design Manual

3 - 6 Beam Design

beam may be designed as a Rectangular or a T-beam. Negative beammoments produce top steel. In such cases the beam is always designed asa rectangular section.


The design procedure is based on the simplified rectangular stress block,as shown in Figure 3-1 (CSA 10.1.7). Furthermore, it is assumed that thecompression carried by concrete is less than or equal to that which can becarried at the balanced condition (CSA 10.1.4). When the applied mo-ment exceeds the moment capacity at the balanced condition, the area ofcompression reinforcement is calculated assuming that the additionalmoment will be carried by compression and additional tension rein-forcement.

In designing the beam flexural reinforcement, the following limits areimposed on the steel tensile strength and the concrete compressivestrength:

fy ≤ 500 MPa (CSA 8.5.1)

f '

c ≤ 80 MPa (CSA 8.6.1.1)

The design procedure used by SAFE for both rectangular and flangedsections (L- and T-beams) is summarized in the next two subsections. Itis assumed that the design ultimate axial force in a beam is negligible;hence, all of the beams are designed for major direction flexure and shearonly.


Beam Design 3- 7

Design for Flexure of a Rectangular BeamIn designing for a factored negative or positive moment, Mf

(i.e., designing top or bottom steel), the depth of the compression blockis given by a, as shown in Figure 3-1, where,

Figure 3-1 Design of a Rectangular Beam Section

a = d − bf

Md

cc

f

ϕα '1

22

− , (CSA 10.1)

where the value of ϕc is 0.60 (CSA 9.4.2) in the above and followingequations. See Figure 3-1. Also α1, β1, and cb are calculated as follows:

α1 = 0.85 – 0.0015 f '

c ≥ 0.67, (CSA 10.1.7)

β1 = 0.97 – 0.0025 f '

c ≥ 0.67, and (CSA 10.1.7)

cb = yf+700

700d. (CSA 10.5.2)

The balanced depth of the compression block is given by

SAFE Design Manual

3 - 8 Beam Design

ab = β1 cb. (CSA 10.1.7)

If a ≤ ab (CSA 10.5.2), the area of tensile steel reinforcement is thengiven by

As =

−

2a

df

M

ys

f

ϕ.

This steel is to be placed at the bottom if Mf is positive, or at the topif Mf is negative.

If a > ab (CSA 10.5.2), compression reinforcement is required and iscalculated as follows:

− The factored compressive force developed in concrete alone isgiven by

C = ϕc α1

'cf bab , and (CSA 10.1.7)

the factored moment resisted by concrete and bottom steel is

Mfc = C

−

2ba

d .

− The moment resisted by compression steel and tensile steel is

Mfs = Mf − Mfc.


A'

s = ( )( )''1

' ddff

M

ccss

fs

−− αϕϕ, where

'sf = 0.0035 Es

−

c

dc ' ≤ fy. (CSA 10.1.2 and CSA 10.1.3)



Beam Design 3- 9

As1 =

sb

y

fc

s

adf

M

ϕ

−

, and

the tensile steel for balancing the compression in steel is

As2 = ( ) sy

fc

ddf

M

ϕ'−.

− Therefore, the total tensile reinforcement is As = As1 + As2, and thetotal compression reinforcement is A's. As is to be placed at thebottom and A's is to be placed at the top if Mf is positive, and A's isto be placed at the bottom and As is to be placed at the top if Mf isnegative.

Design for Flexure of a T-Beam

(i) Flanged Beam Under Negative MomentIn designing for a factored negative moment, Mf (i.e., designing topsteel), the calculation of the steel area is exactly the same as for a rectan-gular beam, i.e., no T-Beam data is used.

(ii) Flanged Beam Under Positive MomentIf Mf > 0, the depth of the compression block is given by (see Figure 3-2).

a = d − fcc

f

bf

Md

ϕα '1

2 2− . (CSA 10.1)

where the value of ϕc is 0.60 (CSA 9.4.2) in the above and followingequations. See Figure 3-2. Also α1, β1, and cb are calculated as follows:

α1 = 0.85 – 0.0015 'cf ≥ 0.67, (CSA 10.1.7)

β1 = 0.97 – 0.0025 'cf ≥ 0.67 , and (CSA 10.1.7)

SAFE Design Manual

3 - 10 Beam Design

cb =yf+700

700 d. (CSA 10.5.2)

Figure 3-2 Design of a T-Beam Section

The depth of compression block under balanced condition is givenby

ab = β1cb . (CSA 10.1.4)

If a ≤ ds, the subsequent calculations for As are exactly the same asthose for the rectangular section design. However, in this case thewidth of the beam is taken as bf. Compression reinforcement is re-quired if a > ab.

If a > ds, calculation for As has two parts. The first part is for balanc-ing the compressive force from the flange, Cf, and the second part isfor balancing the compressive force from the web, Cw. As shown inFigure 3-2,

Cf = α1'

cf (bf – bw) min(ds , amax) . (CSA 10.1.7)


Beam Design 3- 11

Therefore, As1 = sy

cf

f

C

ϕϕ

and the portion of Mf that is resisted by the

flange is given by

Mff = Cf ( )

−

2

,min maxs add ϕc .

Therefore, the balance of the moment, Mf, to be carried by the web isgiven by

Mfw = Mf − Mff.

The web is a rectangular section of dimensions bw and d, for which thedepth of the compression block is recalculated as

a1 = d − wcc

fw

bf

Md

ϕα '1

2 2− . (CSA 10.1)

If a1 ≤ ab (CSA 10.5.2), the area of tensile steel reinforcement is thengiven by

As2 =

−

21a

df

M

ys

fw

ϕ, and

As = As1 + As2 .


If a1 > ab (CSA 10.5.2), compression reinforcement is required and iscalculated as follows:

− The compressive force in the concrete web alone is given by

C = α 'cf bw ab , and (CSA 10.1.7)

the moment resisted by the concrete web and tensile steel is

SAFE Design Manual

3 - 12 Beam Design

Mfc = C

−

2ba

d ϕc .


Mfs = Mfw − Mfc .


A'

s = ( )( )''1

' ddff

M

cccs

fs

−− αϕϕ, where

'sf = 0.0035Es

−

c

dc '≤ fy . (CSA 10.1.2 and CSA 10.1.3)


As2 =

sb

y

fc

adf

M

ϕ

−

2

, and


As3 = ( ) sy

fs

ddf

M

ϕ'−.

− Total tensile reinforcement is As = As1 + As2 + As3, and the totalcompression reinforcement is A'

s. As is to be placed at the bottomand A'

s is to be placed at the top.

Minimum and Maximum Tensile Reinforcement

The minimum flexural tensile steel required for a beam section is givenby the minimum of the two limits:

As ≥ y

c

f

f '2.0 bw h, or (CSA 10.5.1.2)


Beam Design 3- 13

As ≥ 3

4 As(required). (CSA 10.5.1.3)

In addition, the minimum flexural tensile steel provided in a T-sectionwith flange under tension in an ordinary moment resisting frame is givenby the limit:

As ≥ 0.004 (b − bw) ds. (CSA 10.5.3.1)


0.04 b d Rectangular beam


0.04 b d Rectangular beamA'

s ≤0.04 bwd T-beam

Design Beam Shear ReinforcementThe shear reinforcement is designed for each load combination at the twostations at the ends of the beam elements. In designing the shear rein-forcement for a particular beam for a particular loading combination at aparticular station resulting from beam major shear, the following stepsare involved:

Determine the factored shear force, Vf.



In designing the beam shear reinforcement, the following limits are im-posed on the steel tensile strength and the concrete compressivestrength:

fys ≤ 500 MPa (CSA 8.5.1)

SAFE Design Manual

3 - 14 Beam Design

'cf ≤ 80 MPa (CSA 8.6.1.1)

The following three subsections describe the algorithms associated withthe above-mentioned steps.

Determine Shear Force and MomentIn the design of the beam shear reinforcement of a concrete beam, theshear forces and moments for a particular load combination at a particu-lar beam section are obtained by factoring the associated shear forces andmoments with the corresponding load combination factors.


Vc = 0.2ϕcλ'

cf bwd, if d ≤ 300

(CSA 11.3.5.1)

Vc = d+1000

260ϕcλ

'cf bwd ≥ 0.1 ϕcλ

'cf bwd, if d > 300

(CSA 11.3.5.2)

where λ is taken as one for normal weight concrete.

Determine Required Shear Reinforcement The shear force is limited to a maximum limit of

Vmax = Vc + 0.8 ϕcλ'

cf bwd . (CSA 11.3.4)

Given Vu, Vc and Vmax, the required shear reinforcement in area/unitlength is calculated as follows:

If Vf ≤ (Vc / 2),

s

Av = 0, (CSA 11.2.8.1)


Slab Design 3- 15

else if (Vc / 2) < Vf ≤ [ ( )dbfV wcsc'06.0ϕ+ ],

s

Av = ys

wc

f

bf '06.0, (CSA 11.2.8.4)

else if [ ( )dbfV wcsc'06.0ϕ+ ] < Vf ≤ Vmax ,

s

Av = ( )

df

VV

yss

cf

ϕ−

, (CSA 11.3.7)

else if Vf > Vmax ,

a failure condition is declared. (CSA 11.3.4)



Slab DesignSimilar to conventional design, the SAFE slab design procedure involvesdefining sets of strips in two mutually perpendicular directions. The loca-tions of the strips are usually governed by the locations of the slab sup-ports. The moments for a particular strip are recovered from the analysisand a flexural design is completed based on the ultimate strength designmethod for reinforced concrete as described in the following sections. Tolearn more about the design strips, refer to the section entitled "SAFEDesign Techniques" in the Welcome to SAFE manual.

SAFE Design Manual

3 - 16 Slab Design

Design for FlexureSAFE designs the slab on a strip-by-strip basis. The moments used forthe design of the slab elements are the nodal reactive moments, whichare obtained by multiplying the slab element stiffness matrices by theelement nodal displacement vectors. Those moments will always be instatic equilibrium with the applied loads, irrespective of the refinementof the finite element mesh.

The design of the slab reinforcement for a particular strip is completed at specific locations along the length of the strip. Those locations corre-spond to the element boundaries. Controlling reinforcement is computedon either side of those element boundaries. The slab flexural design pro-cedure for each load combination involves the following:





Design Flexural Reinforcement for the StripThe reinforcement computation for each slab design strip, given thebending moment, is identical to the design of rectangular beam sectionsdescribed earlier. When the slab properties (depth, etc.) vary over thewidth of the strip, the program automatically designs slab widths of eachproperty separately for the bending moment to which they are subjectedand then sums the reinforcement for the full width. Where openings oc-cur, the slab width is adjusted accordingly.


Slab Design 3- 17

Minimum and Maximum Slab ReinforcementThe minimum flexural tensile reinforcement provided in each directionof a slab is given by the following limit (CSA 13.11.1):

As ≥ 0.0020 bh (CSA 7.8.1)



Critical Section for Punching ShearThe punching shear is checked on a critical section at a distance of d/2from the face of the support (CSA 13.4.3.1 and CSA 13.4.3.2). For rec-tangular columns and concentrated loads, the critical area is taken as arectangular area with the sides parallel to the sides of the columns or thepoint loads (CSA 13.4.3.3).

Transfer of Unbalanced MomentThe fraction of unbalanced moment transferred by flexure is taken to beγf Mu and the fraction of unbalanced moment transferred by eccentricityof shear is taken to be γv Mu, where

γf = ( ) 21321

1

bb+ , and (CSA 13.11.2)

γv = 1− ( ) 21321

1

bb+ , (CSA 13.4.5.3)


SAFE Design Manual

3 - 18 Slab Design

Determination of Concrete CapacityThe concrete punching shear factored strength is taken as the minimumof the following three limits:

ϕc

+

cβ2

1 0.2λ 'cf

vc = minϕc

+

0

2.0b

dsα λ 'cf

ϕc 0.4 λ 'cf

(CSA 13.4.4)


4 for interior columns,

αs = 3 for edge columns, and

2 for corner columns.

(CSA 13.4.4)

Also the following limits are imposed on the steel and concretestrengths:

f y ≤ 500 MPa (CSA 8.5.1)

'cf ≤ 80 MPa (CSA 8.6.1.1)



Chapter 4

Design for BS 8110-85

This chapter describes in detail the various aspects of the concrete designprocedure that is used by SAFE when the user selects the British limitstate design code BS 8110 (BSI 1989). Various notations used in thischapter are listed in Table 4-1. For referencing to the pertinent sectionsof the British code in this chapter, a prefix “BS” followed by the sectionnumber is used.



SAFE™SAFE™

SAFE Design Manual


Table 4-1 List of Symbols Used in the BS 8110-85 Code

Acv Area of section for shear resistance, mm2

Ag Gross area of cross-section, mm2

As Area of tension reinforcement, mm2

A's Area of compression reinforcement, mm2

Asv Total cross-sectional area of links at the neutral axis, mm2

Asv / sv Area of shear reinforcement per unit length of the member,mm2/mm


b Width or effective width of the section in the compressionzone, mm

bf Width or effective width of flange, mm

bw Average web width of a flanged beam, mm

d Effective depth of tension reinforcement, mm

d' Depth to center of compression reinforcement, mm



fcu Characteristic cube strength at 28 days, MPa

'sf

Compressive stress in a beam compression steel, MPa

fy Characteristic strength reinforcement, MPa

fyv Characteristic strength of link reinforcement, MPa (<460MPA)

h Overall depth of a section in the plane of bending, mm

hf Flange thickness, mm

K Normalized design moment, M/bd2fcu



Table 4-1 List of Symbols Used in the BS 8110-85 Code

K' Limiting normalized moment for a singly reinforced concretesection taken as 0.156

k1 Shear strength enhancement factor for support compression

k2 Concrete shear strength factor, [ ] 31

25cuf

M Design moment at a section, MPa

Msingle Limiting moment capacity as a singly reinforced beam, MPa

sv Spacing of the links along the length of the beam, in

T Tension force, N

V Design shear force at ultimate design load, N

u Perimeter of the punch critical section, mm

v Design shear stress at a beam cross-section or at a punchcritical section, MPa

vc Design ultimate shear stress resistance of a concrete beam,MPa

vmax Maximum permitted design factored shear stress at a beamsection or at the punch critical section, MPa

x Neutral axis depth, mm

xbal Depth of neutral axis in a balanced section, mm

z Lever arm, mm

βb Moment redistribution factor in a member

γf Partial safety factor for load

γm Partial safety factor for material strength

εc Maximum concrete strain, 0.0035

εs Strain in tension steel

ε'

s Strain in compression steel

SAFE Design Manual


Design Load CombinationsThe design load combinations are the various combinations of the pre-scribed load cases for which the structure needs to be checked. For thiscode, if a structure is subjected to dead load (DL), live load (LL), patternlive load (PLL), wind (WL), and earthquake (EL) loads, and consideringthat wind and earthquake forces are reversible, the following load com-binations must be considered (BS 2.4.3):

1.4 DL1.4 DL + 1.6 LL (BS 2.4.3.1.1)

1.4 DL + 1.6 PLL

1.0 DL ±1.4 WL1.4 DL ± 1.4 WL1.2 DL + 1.2 LL ± 1.2 WL (BS 2.4.3.1.1)

1.0 DL ± 1.4 EL1.4 DL ± 1.4 EL1.2 DL + 1.2 LL ±1.2 EL

These are also the default design load combinations in SAFE when theBS 8110-85 code is used. The user should use other appropriate loadingcombinations if roof live load is separately treated, or other types ofloads are present.

Design StrengthThe design strength for concrete and steel are obtained by dividing thecharacteristic strength of the material by a partial factor of safety, γm. Thevalues of γm used in the program are listed below, which are taken fromBS Table 2.2 (BS 2.4.4.1):

Values of γγγγm for the ultimate limit state

Reinforcement 1.15

Concrete in flexure and axial load 1.50

Shear strength without shear reinforcement 1.25


Beam Design 4- 5

These factors are already incorporated in the design equations and tablesin the code. SAFE does not allow them to be overwritten.

Beam DesignIn the design of concrete beams, SAFE calculates and reports the re-quired areas of steel for flexure and shear based on beam moments, shearforces, load combination factors, and other criteria described in this sec-tion. The reinforcement requirements are calculated at two check stationsat the ends of the beam elements.

All of the beams are designed for major direction flexure and shear only.Effects resulting from any axial forces, minor direction bending, and tor-sion that may exist in the beams must be investigated independently bythe user.




Design Beam Flexural ReinforcementThe beam top and bottom flexural steel is designed at the two stations atthe ends of the beam elements. In designing the flexural reinforcementfor the major moment of a particular beam for a particular station, thefollowing steps are involved:




SAFE Design Manual

4 - 6 Beam Design

The beam section is then designed for the maximum positive and maxi-mum negative factored moments obtained from all of the load combina-tions at that section. Positive beam moments produce bottom steel. Insuch cases, the beam may be designed as a Rectangular or a T-beam.Negative beam moments produce top steel. In such cases, the beam isalways designed as a rectangular section.


The design procedure is based on the simplified rectangular stress block,as shown in Figure 4-1. Furthermore, it is assumed that moment redistribution in the member does not exceed 10% (i.e., βb ≥ 0.9) (BS3.4.4.4). The code also places a limitation on the neutral axis depth, x/d ≤0.5, to safeguard against non-ductile failures (BS 3.4.4.4). In addition,the area of compression reinforcement is calculated assuming that theneutral axis depth remains at the maximum permitted value.

The design procedure used by SAFE, for both rectangular and flangedsections (L- and T-beams), is summarized in the next two subsections. Itis assumed that the design ultimate axial force does not exceed 0.1 fcu Ag

(BS 3.4.4.1); hence, all of the beams are designed for major directionflexure and shear only.

Design of a Rectangular Beam

For rectangular beams, the limiting moment capacity as a singly rein-forced beam, Msingle, is obtained first for a section. The reinforcing steelarea is determined based on whether M is greater than, less than, or equalto Msingle. See Figure 4-1.


Beam Design 4- 7

Figure 4-1 Design of Rectangular Beam Section

Calculate the ultimate limiting moment of resistance of the section assingly reinforced.

Msingle = K'fcu bd2, where (BS 3.4.4.4)

K' = 0.156.

If M ≤ Msingle the area of tension reinforcement, As, is obtained from

As = ( )zf

M

y87.0, where (BS 3.4.4.4)

z = d

−+9.0

25.05.0K

≤ 0.95d, and (BS 3.4.4.4)

K = 2bdf

M

cu

. (BS 3.4.4.4)

SAFE Design Manual

4 - 8 Beam Design

This is the top steel if the section is under negative moment and thebottom steel if the section is under positive moment.

If M > Msingle, the area of compression reinforcement, A's, is given by

A'

s = ( )'' ddf

MM

s

single

−−

, (BS 3.4.4.4)

where d' is the depth of the compression steel from the concretecompression face, and

'sf = Esεc

−

maxx

d'1 ≤ 0.87 fy (BS 3.4.4.4, 2.5.3)

This is the bottom steel if the section is under negative moment.From equilibrium, the area of tension reinforcement is calculated as

As = ( )zf 0.87

M

y

single+ ( )( )'ddf 0.87

MM

y

single

−−

, where (BS 3.4.4.4)

z = d

−+9.0

'25.05.0

K= 0.777d , (BS 3.4.4.4)

xmax = ( ) 45.0zd − . (BS 3.4.4.4)

Design as a T-Beam

(i) Flanged Beam Under Negative MomentThe contribution of the flange to the strength of the beam is ignored. Thedesign procedure is therefore identical to the one used for rectangularbeams, except that in the corresponding equations, b is replaced by bw.

(ii) Flanged Beam Under Positive MomentWith the flange in compression, the program analyzes the section byconsidering alternative locations of the neutral axis. Initially the neutralaxis is assumed to be located in the flange. On the basis of this assump-


Beam Design 4- 9

tion, the program calculates the exact depth of the neutral axis. If thestress block does not extend beyond the flange thickness, the section isdesigned as a rectangular beam of width bf. If the stress block extendsbeyond the flange width, the contribution of the web to the flexuralstrength of the beam is taken into account. See Figure 4-2.

Assuming the neutral axis to lie in the flange, the normalized moment isgiven by

K = 2dbf

M

fcu

. (BS 3.4.4.4)

Then the moment arm is computed as

z = d

−+9.0

25.05.0K

≤ 0.95d, (BS 3.4.4.4)

the depth of neutral axis is computed as

x = 45.0

1 (d − z), and (BS 3.4.4.4)

the depth of compression block is given by

a = 0.9x . (BS 3.4.4.4)

If a ≤ hf, the subsequent calculations for As are exactly the same aspreviously defined for the rectangular section design. However, inthis case, the width of the beam is taken as bf. Compression rein-forcement is required if K > K'.

If a > hf, calculation for As has two parts. The first part is for balanc-ing the compressive force from the flange, Cf, and the second part isfor balancing the compressive force from the web, Cw, as shown inFigure 4-2.

SAFE Design Manual

4 - 10 Beam Design


In this case, the ultimate resistance moment of the flange is given by

Mf = 0.45 fcu (bf – bw) min(hf, amax) [d – 0.5min(hf, amax)], (BS 3.4.4.5)

the moment taken by the web is computed as

Mw = M – Mf, and

the normalized moment resisted by the web is given by

Kw = 2dbf

Mw

wcu

. (BS 3.4.4.4)

− If Kw ≤ K' (BS 3.4.4.4), the beam is designed as a singly rein-forced concrete beam. The area of steel is calculated as the sumof two parts, one to balance compression in the flange and one tobalance compression in the web.

As = ( )[ ]max,min5.087.0 ahdf

M

fy

f

− +

zf

M

y

w

87.0, where


Beam Design 4- 11

z = d

−+9.0

25.05.0 wK ≤ 0.95d.

− If Kw > K' (BS 3.4.4.4), compression reinforcement is requiredand is calculated as follows:

The ultimate moment of resistance of the web only is given by

Muw = K' fcu bw d2 . (BS 3.4.4.4)

The compression reinforcement is required to resist a moment ofmagnitude Mw − Muw. The compression reinforcement is com-puted as

A'

s = ( )'' ddf

MM

s

uww

−−

,

where, d' is the depth of the compression steel from the concretecompression face, and

f's = Esεc

−

maxx

d'1 ≤ 0.87 fy. (BS 3.4.4.4, 2.5.3)

The area of tension reinforcement is obtained from equilibrium

As = ( )fy

f

hdf

M

5.087.0 − + ( )df

M

y

uw

777.087.0 + ( )'87.0 ddf

MM

y

uww

−−

.

Minimum and Maximum Tensile ReinforcementThe minimum flexural tensile steel required for a beam section is givenby the following table, which is taken from BS Table 3.27 (BS 3.12.5.3)with interpolation for reinforcement of intermediate strength:

SAFE Design Manual

4 - 12 Beam Design

Minimum percentage

Section SituationDefinition ofpercentage fy = 250 MPa fy = 460 MPa

Rectangular 100 bh

As0.24 0.13

f

w

b

b< 0.4 100

hb

A

w

s0.32 0.18

T-Beam with web in

tension

f

w

b

b≥ 0.4 100

hb

A

w

s0.24 0.13

T-Beam with web in

compression 100

hb

A

w

s0.48 0.26

The minimum flexural compression steel, if it is required at all, providedin a rectangular or T-beam section is given by the following table, whichis taken from BS Table 3.27 (BS 3.12.5.3) with interpolation for rein-forcement of intermediate strength:

Section SituationDefinition ofpercentage

Minimumpercentage

Rectangular 100 bh

As'

0.20

Web in tension 100 ff

s

hb

A '

0.40

T-BeamWeb in compression 100

hb

A

w

s'

0.20

In addition, an upper limit on both the tension reinforcement and com-pression reinforcement has been imposed to be 0.04 times the grosscross-sectional area (BS 3.12.6.1).


Beam Design 4- 13

Design Beam Shear ReinforcementThe shear reinforcement is designed for each loading combination in themajor direction of the beam. In designing the shear reinforcement for aparticular beam for a particular loading combination, the following stepsare involved (BS 3.4.5):

Calculate the design shear stress as

v = cvA

V , Acv = bwd, where (BS 3.4.5.2)

v ≤ vmax, and (BS 3.4.5.2)

vmax = min (0.8 cuf , 5 MPa). (BS 3.4.5.2)

Calculate the design concrete shear stress from

vc = m

kk

γ2179.0

31

100

bd

As

41

400

d, (BS 3.4.5.4)

where,

k1 is the enhancement factor for support compression, and isconservatively taken as 1, (BS 3.4.5.8)

k2 = 3

1

25

cuf

≥ 1, and (BS 3.4.5.4)

γm = 1.25. (BS 3.4.5.2)

However, the following limitations also apply:

0.15 ≤ bd

As100 ≤ 3, (BS 3.4.5.4)

d

400 ≥ 1, and (BS 3.4.5.4)

fcu ≤ 40 MPa (for calculation purpose only). (BS 3.4.5.4)

SAFE Design Manual

4 - 14 Slab Design

As is the area of tensile steel.

Given v, vc and vmax , the required shear reinforcement in area/unitlength is calculated as follows (BS Table 3.8, BS 3.4.5.3):

If v ≤ vc + 0.4,

s

sv

s

A =

yv

w

f

b

87.04.0

, (BS 3.4.5.3)

else if (vc + 0.4) < v ≤ vmax ,

s

sv

s

A=

( )yv

wc

f

bvv

87.0

−. (BS 3.4.5.3)

else if v > vmax ,

a failure condition is declared. (BS 3.4.5.2)

In the above expressions, a limit is imposed on the fyv as

fyv ≥ 460 MPa. (BS 3.4.5.1)

The maximum of all the calculated Asv /sv values, obtained from each loadcombination, is reported along with the controlling shear force and asso-ciated load combination number.


Slab DesignSimilar to conventional design, the SAFE slab design procedure involvesdefining sets of strips in two mutually perpendicular directions. The lo-cations of the strips are usually governed by the locations of the slabsupports. The moments for a particular strip are recovered from theanalysis, and a flexural design is completed based on the ultimatestrength design method for reinforced concrete (BS 8110-85) as de-


Slab Design 4- 15

scribed in the following subsections. To learn more about the designstrips, refer to the section entitled "SAFE Design Techniques" in theWelcome to SAFE manual.





These two steps, which are described in the next two sections, are re-peated for every load combination. The maximum reinforcement calcu-lated for the top and bottom of the slab within each design strip, alongwith the corresponding controlling load combination numbers, is ob-tained and reported.


Design Flexural Reinforcement for the StripThe reinforcement computation for each slab design strip, given thebending moment, is identical to the design of rectangular beam sectionsdescribed earlier. When the slab properties (depth, etc.) vary over the

SAFE Design Manual

4 - 16 Slab Design

width of the strip, the program automatically designs slab widths of eachproperty separately for the bending moment to which they are subjectedand then sums the reinforcement for the full width. Where openings oc-cur, the slab width is adjusted accordingly.

Minimum and Maximum Slab ReinforcementThe minimum flexural tensile reinforcement required in each direction ofa slab is given by the following limit (BS 3.12.5.3, BS Table 3.27) withinterpolation for reinforcement of intermediate strength:

0.0024 bh if fy ≤ 250 MPa

As ≥ 0.0013 bh if fy ≥ 460 MPa(BS 3.12.5.3)

In addition, an upper limit on both the tension reinforcement and com-pression reinforcement has been imposed to be 0.04 times the grosscross-sectional area (BS 3.12.6.1).


Critical Section for Punching ShearThe punching shear is checked on a critical section at a distance of 1.5dfrom the face of the support (BS 3.7.7.4). For rectangular columns andconcentrated loads, the critical area is taken as a rectangular area, withthe sides parallel to the sides of the columns or the point loads (BS3.7.7.1).

Determination of Concrete CapacityThe concrete punching shear factored strength is taken as follows (BS3.7.7.4):


Slab Design 4- 17

vc = m

kk

γ2179.0

31

100

bd

As

41

400

d, where, (BS 3.4.5.4)

k1 is the enhancement factor for support compression, and is conser-vatively taken as 1, (BS 3.4.5.8)

k2 =

31

25

cuf

≥ 1, and (BS 3.4.5.4)

γm = 1.25. (BS 3.4.5.2)

However, the following limitations also apply:

0.15 ≤ bd

As100 ≤ 3, (BS 3.4.5.4)

d

400 ≥ 1, (BS 3.4.5.4)

v ≤ min (0.8 cuf , 5MPa), and (BS 3.4.5.2)

fcu ≤ 40 MPa (for calculation purpose only). (BS 3.4.5.4)

As = area of tensile steel, which is taken as zero in current implemen-tation.

Determination of Capacity RatioGiven the punching shear force and the fractions of moments transferredby eccentricity of shear about the two axes, the nominal design shearstress, v, is calculated from the following equation:

v = du

Veff, where (BS 3.7.7.3)

Veff = V

++yV

M

xV

Mf xy 5.15.1 , (BS 3.7.6.2 and BS 3.7.6.3)

SAFE Design Manual

4 - 18 Slab Design

u is the perimeter of the critical section,

x and y are the length of the side of the critical section parallel to theaxis of bending,

Mx and My are the design moment transmitted from the slab to thecolumn at connection,

V is the total punching shear force, and

f is a factor to consider the eccentricity of punching shear force andis taken as

1.00 for interior columns,

f = 1.25 for edge columns, and

1.25 for corner columns.

(BS 3.7.6.2 andBS 3.7.6.3)

The ratio of the maximum shear stress and the concrete punching shearstress capacity is reported by SAFE.


Chapter 5

Design for Eurocode 2

This chapter describes in detail the various aspects of the concrete designprocedure that is used by SAFE when the user selects the European con-crete design code, 1992 Eurocode 2 (CEN 1992). Various notations usedin this chapter are listed in Table 5-1. For referencing to the pertinentsections of the Eurocode in this chapter, a prefix “EC2” followed by thesection number is used.



SAFE™SAFE™

SAFE Design Manual


Table 5-1 List of Symbols Used in the Eurocode 2

Ac Area of concrete section, mm2


A's Area of compression reinforcement, mm2

Asw Total cross-sectional area of links at the neutral axis, mm2

Asw /sv Area of shear reinforcement per unit length of the member,mm2






d' Effective depth of compression reinforcement, mm



fcd Design concrete strength = fck / γc , MPa

fck Characteristic compressive concrete cylinder strength at 28days, MPa

fyd Design yield strength of reinforcing steel = fyk /γs, MPa

fyk Characteristic strength of shear reinforcement, MPa

'sf

Compressive stress in a beam compression steel, MPa

fywd Design strength of shear reinforcement = fywk /γs, MPa

fywk Characteristic strength of shear reinforcement, MPa

h Overall thickness of slab, mm




hf Flange thickness, mm

M Design moment at a section, N-mm

m Normalized design moment, M/bd2αfcd

mlim Limiting normalized moment capacity as a singly reinforcedbeam

sv Spacing of the shear reinforcement along the length of thebeam, mm

u Perimeter of the punch critical section, mm

VRd1 Design shear resistance from concrete alone, N

VRd2 Design limiting shear resistance of a cross-section, N

Vsd Shear force at ultimate design load, N

x Depth of neutral axis, mm

xlim Limiting depth of neutral axis, mm

α Concrete strength reduction factor for sustained loading andstress-block

β Enhancement factor of shear resistance for concentrated load;also the coefficient that takes account of the eccentricity ofloading in determining punching shear stress; factor for thedepth of compressive stress block

γf Partial safety factor for load

γc Partial safety factor for concrete strength


γs Partial safety factor for steel strength

δ Redistribution factor

εc Concrete strain


SAFE Design Manual



ν Effectiveness factor for shear resistance without concretecrushing

ρ Tension reinforcement ratio, As /bd

ω Normalized tensile steel ratio, As fyd /αfcd bd

ω' Normalized compression steel ratio, A'

s fyd γs/α f'

s bd

ωlim Normalized limiting tensile steel ratio

Design Load CombinationsThe design load combinations are the various combinations of the pre-scribed load cases for which the structure needs to be checked. For thiscode, if a structure is subjected to dead load (DL), live load (LL), patternlive load (PLL), wind (WL), and earthquake (EL) loads, and consideringthat wind and earthquake forces are reversible, the following load com-binations must be considered (EC2 2.3.3):

1.35 DL1.35 DL + 1.50 LL (EC2 2.3.3.1)

1.35 DL + 1.50 PLL

1.35 DL ± 1.50 WL1.00 DL ± 1.50 WL1.35 DL + 1.35 LL ± 1.35 WL (EC2 2.3.3.1)

1.00 DL ± 1.00 EL1.00 DL + 1.5*0.3 LL ± 1.0 EL (EC2 2.3.3.1)

These are also the default design load combinations in SAFE when theEurocode is used. The user should use other appropriate loading combi-nations if roof live load is separately treated, or other types of loads arepresent.

Chapter 5 - Design Strength

Design Strength 5- 5

Design StrengthThe design strength for concrete and steel are obtained by dividing thecharacteristic strength of the material by a partial factor of safety, γm. Thevalues of γm used in the program are listed below. The values are recom-mended by the code to give an acceptable level of safety for normalstructures under regular design situations (EC2 2.3.3.2). For accidentaland earthquake situations, the recommended values are less than thetabulated value. The user should consider those separately.

The partial safety factors for the materials, the design strengths of con-crete and steel are given as follows:

Partial safety factor for steel, γs = 1.15, and (EC2 2.3.3.2)

Partial safety factor for concrete, γc = 1.15. (EC2 2.3.3.2)


Beam Design In the design of concrete beams, SAFE calculates and reports the re-quired areas of steel for flexure and shear based on the beam moments,shears, load combination factors, and other criteria described in this sec-tion. The reinforcement requirements are calculated at two check stationsat the ends of the beam elements. All of the beams are designed for ma-jor direction flexure and shear only. Effects resulting from any axialforces, minor direction bending, and torsion that may exist in the beamsmust be investigated independently by the user.




SAFE Design Manual

5 - 6 Beam Design





The beam section is then designed for the maximum positive and maxi-mum negative factored moments obtained from all the of the load com-binations. Positive beam moments produce bottom steel. In such casesthe beam may be designed as a Rectangular or a T-beam. Negative beammoments produce top steel. In such cases the beam is always designed asa rectangular section.

Determine Required Flexural Reinforcement In the flexural reinforcement design process, the program calculatesboth the tension and compression reinforcement. Compression rein-forcement is added when the applied design moment exceeds the maxi-mum moment capacity of a singly reinforced section. The user has theoption of avoiding the compression reinforcement by increasing the ef-fective depth, the width, or the grade of concrete.

The design procedure is based on the simplified rectangular stress block,as shown in Figure 5-1. The area of the stress block and the depth of thecenter of the compressive force from the most compressed fiber are takenas


Beam Design 5- 7


C = α fcd a d and

a = β x,

where x is the depth of the neutral axis, and α and β are taken respec-tively as

α = 0.8, and (EC2 4.2.1.3.3)

β = 0.8. (EC2 4.2.1.3.3)

α is the reduction factor to account for the sustained compression andrectangular stress block. α is generally assumed to be 0.80 for the as-sumed rectangular stress block (EC2 4.2.1.3.3). β factor considers thedepth of the stress block and it is assumed to be 0.8 (EC2 4.2.1.3.3).

Furthermore, it is assumed that moment redistribution in the memberdoes not exceed the code specified limiting value. The code also places alimitation on the neutral axis depth, to safeguard against non-ductile fail-ures (EC2 2.5.3.4.2). When the applied moment exceeds the limiting

SAFE Design Manual

5 - 8 Beam Design

moment capacity as a singly reinforced beam, the area of compressionreinforcement is calculated assuming that the neutral axis depth remainsat the maximum permitted value.

The design procedure used by SAFE, for both rectangular and flangedsections (L- and T-beams), is summarized in the next two subsections. Itis assumed that the design ultimate axial force does not exceed 0.08 fck Ag

(EC2 4.3.1.2); hence all of the beams are designed for major directionflexure and shear only.

Design as a Rectangular BeamFor rectangular beams, the normalized moment, m, and the normalizedsection capacity as a singly reinforce beam, mlim, are obtained first. Thereinforcing steel area is determined based on whether m is greater than,less than, or equal to mlim.

Calculate the normalized design moment, m

m = cdfbd

M

α2, where

α is the reduction factor to account for sustained compression and βfactor considers the depth of the neutral axis. α is generally assumedto be 0.80 for the assumed rectangular stress block, (EC2 4.2.1.3.3).α is also generally assumed to be 0.80 for the assumed rectangularstress block, (EC2 4.2.1.3.3). The concrete compression stress blockis assumed to be rectangular (see Figure 5-1), with a stress value of αfcd, where fcd is the design concrete strength and is equal to cckf γ .

Calculate the normalized concrete moment capacity as a singly rein-forced beam, mlim.

mlim = βlimd

x

−

limd

x

21

β,

where the limiting value of the ratio of the neutral axis depth at theultimate limit state to the effective depth, [ ]limdx , is expressed as a


Beam Design 5- 9

function of the ratio of the redistributed moment to the moment be-fore redistribution, δ, as follows:

limd

x

= 25.1

44.0−δ if fck ≤ 35 , (EC2 2.5.3.4.2)

limd

x

= 25.1

56.0−δ if fck > 35 , and (EC2 2.5.3.4.2)

δ is assumed to be 1.

If m ≤ mlim, a singly reinforced beam will suffice. Calculate the nor-malized steel ratio,

ω = 1− m21 − .

Calculate the area of tension reinforcement, As, from

As = ω

yd

cd

f

bdfα.

This is the top steel if the section is under negative moment and thebottom steel if the section is under positive moment.

If m > mlim, the beam will not suffice as a singly reinforced beam.Both top and bottom steel are required.

− Calculate the normalized steel ratios ω', ωlim, and ω.

ωlim = βlim

d

x= 1 − lim21 m−

ω' = dd

mm'

lim

1 −−

, and

ω = ωlim + ω'

where, d' is the depth of the compression steel from the con-crete compression face.

SAFE Design Manual

5 - 10 Beam Design

− Calculate the area of compression and tension reinforcement, A'

s

and As, as follows:

A'

s = ω'

'

s

cd

f

bdfα, and

As = ω

yd

cd

f

bdfα,

where, 'sf is the stress in the compression steel, and is given by

'sf = Esεc

−

limx

d'1 ≤ fyd. (EC2 4.2.2.3.2)

Design as a T-Beam


The contribution of the flange to the strength of the beam is ignored ifthe flange is in the tension side. See Figure 5-2. The design procedure istherefore identical to the one used for rectangular beams. However, thewidth of the web, bw, is taken as the width of the beam.

(ii) Flanged Beam Under Positive MomentWith the flange in compression, the program analyzes the section byconsidering alternative locations of the neutral axis. Initially the neutralaxis is assumed to be located within the flange. On the basis of this as-sumption, the program calculates the depth of the neutral axis. If thestress block does not extend beyond the flange thickness, the section isdesigned as a rectangular beam of width bf. If the stress block extendsbeyond the flange, additional calculation is required. See Figure 5-2.

Calculate the normalized design moment, m.

m = cdf fdb

M

α2, where


Beam Design 5- 11


α is the reduction factor to account for sustained compression. α is gen-erally assumed to be 0.80 for assumed rectangular stress block, (EC24.2.1.3). See also page 5-7 for α. The concrete compression stress blockis assumed to be rectangular, with a stress value of αfcd.

Calculate the limiting value of the ratio of the neutral axis depth at theultimate limit state to the effective depth, [ ]limdx , which is expressed as

a function of the ratio of the redistributed moment to the moment beforeredistribution, δ, as follows:

limd

x

= 25.1

44.0−δ, if fck ≤ 35, (EC2 2.5.3.4.1)

limd

x

= 25.1

56.0−δ, if fck > 35, (EC2 2.5.3.4.1)

δ is assumed to be 1.

Calculate the limiting values:

SAFE Design Manual

5 - 12 Beam Design

mlim = βlimd

x

−

limd

x

21

β,

ωlim = βlimd

x

,

amax = ωlimd,

Calculate ω, a, and d

x as follows:

ω = 1 − m21 − , and

a = ωd ≤ amax.

d

x =

βω

.

If a ≤ hf , the neutral axis lies within the flange. Calculate the area oftension reinforcement, As, as follows:

− If m ≤ mlim ,

ω = 1 − m21 − , and

As = ω

cd

wcd

f

dbfα.

− If m > mlim ,

ω' = dd'-1

mm lim−,

ωlim = βlimd

x

,

ω = ωlim + ω',


Beam Design 5- 13

As = ω,

yd

wcd

f

dbfα and

A'

s = ω'

'

s

cd

f

dbfα, where

'sf = Esεc

−

limx

d'1 ≤ fyd. (EC2 4.2.2.3.2)

If a > hf , the neutral axis lies below the flange. Calculate the steelarea required for equilibrating the flange compression, As2.

As2 = ( )

yd

cdfwf

f

fhbb α−,

and the corresponding resistive moment is given by

M2 = As2 fyd

−

2fh

d .

Calculate the steel area required for a rectangular section of width bw

to resist moment, M1 = M − M2 , as follows:

m1 = cdw fdb

M

α21 , and

− If m1 ≤ mlim,

ω1 = 1 − 121 m− , and

As1 = ω1

yd

wcd

f

dbfα .

− If m1 > mlim,

SAFE Design Manual

5 - 14 Beam Design

ω' = dd

mm

'1lim1

−−

,

ωlim = βlim

d

x,

ω1 = ωlim + ω',

A'

s = ω'

'

s

cd

f

bdfα , and

As1 = ω1

yd

wcd

f

dbfα,

where, 'sf is given by

'sf = Esεc

−

limx

d'1 ≤ fyd . (EC2 4.2.2.3.2)

− Calculate the total steel area required for the tension side.

As = As1 + As2

Minimum and Maximum Tensile ReinforcementThe minimum flexural tensile steel required for a beam section is givenby the following equation (EC2 5.4.2.1.1):

ykf

6.0bd

Rectangular beam

As ≥

ykf

6.0bwd

T-beam(EC2 5.4.2.1.1)

In no case in the above equation should the factor vkf6.0 be taken as

less than 0.0015.


Beam Design 5- 15

ykf

6.0≥ 0.0015 (EC2 5.4.2.1.1)

The minimum flexural tension reinforcement required for control ofcracking (EC2 4.4.2) should be investigated independently by the user.

An upper limit on the tension reinforcement and compression reinforce-ment has been imposed to be 0.04 times the gross cross-sectional area(EC 5.4.2.1.1).

Design Beam Shear ReinforcementThe shear reinforcement is designed for each loading combination at thetwo stations at the ends of the beam elements. The assumptions in de-signing the shear reinforcement are as follows:

The beam sections are assumed to be prismatic. The effect of anyvariation of width in the beam section on the concrete shear capacityis neglected.

The effect on the concrete shear capacity of any concentrated or dis-tributed load in the span of the beam between two columns is ig-nored. Also, the effect of the direct support on the beams providedby the columns is ignored.

All shear reinforcement is assumed to be perpendicular to the longi-tudinal reinforcement.

The effect of any torsion is neglected for the design of shear rein-forcement.

In designing the shear reinforcement for a particular beam for a particu-lar loading combination, the following steps of the standard method areinvolved.

Obtain the design value of the applied shear force VSd from theSAFE analysis results.

Calculate the design shear resistance of the member without shearreinforcement.

VRd1 = β[τRd k(1.2 + 40 ρ1)] (bwd), where (EC2 4.3.2.3)

SAFE Design Manual

5 - 16 Beam Design

β = enhancement factor for shear resistance for members withconcentrated loads located near the face of the support. β is takenas 1.

τRd = basic design shear strength of concrete = 0.25fctk 0.05 / γc ,

fctk 0.05 = 0.7 fctm , (EC2 3.1.2.3)

fctm = 0.3 fck 2/3 , (EC2 3.1.2.3)

k = strength magnification factor for curtailment of longitudinalreinforcement and is considered to be 1,

ρ1 = tension reinforcement ratio = db

A

w

s1 ≤ 0.02, and

As1 = area of tension reinforcement.

Calculate the maximum design shear force that can be carried with-out crushing the notional concrete compressive struts, VRd2.

VRd2 = 2

1ν

c

ckf

γ(0.9bwd), where (EC2 4.3.2.3)

ν is the effectiveness factor = 0.7 − 200

cdf≥ 0.5. (EC2 4.3.2.3)

Given VSd, VRd1, VRd2,red, the required shear reinforcement in area/unitlength is calculated as follows:

If VSd ≤ VRd1 ,

v

sw

s

A = 0 , (EC2 4.3.2.3)

else if VRd1,< VSd ≤ VRd2 ,

v

sw

s

A =

( )ywk

sRdSd

fd

VV

9.01 γ−

(EC2 4.3.2.4.3)


Beam Design 5- 17

else if VSd > VRd2 ,

a failure condition is declared. (EC2 4.3.2.2)

An upper limit is imposed on the steel tensile strength:

fywk / γs ≤ MPa (EC2 4.3.2.2)

The maximum of all of the calculated Asw /sv values, obtained from eachload combination, is reported along with the controlling shear force andassociated load combination number.

A lower limit is imposed on Asw /s:

v

sw

s

A ≥ ρw,minbw (EC2 5.4.2.2)

where ρw,min is obtained from the following table (EC2 Table 5.5):

Minimum Values of Shear Stress Ratio, ρρρρw,min

(EC2 5.4.2.2, EC2 Table 5.5)

Concrete Strength fy ≤ 220 220 < fy ≤ 400 fy > 400

'cf ≤ 20 0.0016 0.0009 0.0007

20 < 'cf ≤ 35 0.0024 0.0013 0.0011

'cf > 35 0.0030 0.0016 0.0013


SAFE Design Manual

5 - 18 Slab Design

Slab DesignSimilar to conventional design, the SAFE slab design procedure involvesdefining sets of strips in two mutually perpendicular directions. The lo-cations of the strips are usually governed by the locations of the slabsupports. The moments for a particular strip are recovered from theanalysis, and a flexural design is completed based on the ultimatestrength design method (Eurocode 2) for reinforced concrete as describedin the following sections. To learn more about the design strips, refer tothe section entitled "SAFE Design Techniques" in the Welcome to SAFEmanual.





These two steps, which are described in the next two subsection, are re-peated for every load combination. The maximum reinforcement calcu-lated for the top and bottom of the slab within each design strip, alongwith the corresponding controlling load combination numbers, is ob-tained and reported.


Slab Design 5- 19

Determine Factored Moments for StripFor each element within the design strip, the program calculates thenodal reactive moments for each load combination. The nodal momentsare then added to get the strip moments.

Design Flexural Reinforcement for the StripThe reinforcement computation for each slab design strip, given thebending moment, is identical to the design of rectangular beam sectionsdescribed earlier. When the slab properties (depth, etc.) vary over thewidth of the strip, the program automatically designs slab widths of eachproperty separately for the bending moment to which they are subjectedand then sums the reinforcement for the full width. Where openings oc-cur, the slab width is adjusted accordingly.

Minimum and Maximum Slab Reinforcement

The minimum flexural tensile reinforcement required in each direction ofa slab is given by the following limits:

yf

6.0bd

As ≥

0.0015 bd

(EC2 5.4.2.1.1)

In addition, an upper limit on both the tension reinforcement and com-pression reinforcement has been imposed to be 0.04 times the grosscross-sectional area (EC2 5.4.2.1.1).


Critical Section for Punching ShearThe punching shear is checked on a critical section at a distance of 1.5dfrom the face of the support (EC2 4.3.4.2.2).

SAFE Design Manual

5 - 20 Slab Design

Determination of Concrete CapacityThe factored concrete punching shear strength is taken as the designshear resistance per unit length without shear reinforcement.

νRd1 = [τRd k(1.2 + 40 ρ1)] d, where (EC2 4.3.4.5.1)

τRd = basic design shear strength = c

ctkf

γ05.025.0

, (EC2 4.3.2.3)

fctk0.05 = 0.7 fctm, (EC2 3.1.2.3)

fctm = 0.3 32

ckf , (EC2 3.1.2.3)

k = 1.6 – 1000

d ≥ 1.0 , d in mm (EC2 4.3.4.5.1)

ρ1 = yx 11 ρρ ≤ 0.015

d = 2

yx dd +,

ρ1x and ρ1y are the reinforcement ratios in the X and Y directions re-spectively, conservatively taken as zeros, and

dx and dy are the effective depths of the slab at the points of intersec-tion between the design failure surface and the longitudinal rein-forcement, in the X and Y directions respectively.

Determination of Capacity RatioGiven the punching shear force and the fractions of moments transferredby eccentricity of shear about the two axes, the factored punching shearforce per unit length is taken as follows:

νSd = u

VSd β, where (EC2 4.3.4.3)

νSd is the total design shear force developed,


Slab Design 5- 21

u is the perimeter of the critical section, and

β is the coefficient that account for the effects of eccentricity of load-ing

1.15 for interior columns,

β = 1.40 for edge columns, and (EC2 4.3.4.3)

1.50 for corner columns.

The ratio of the maximum factored shear force and the concrete punch-ing shear resistance is reported by SAFE.


Chapter 6

Design for NZS 3101-95

This chapter describes in detail the various aspects of the concrete designprocedure that is used by SAFE when the user selects the New Zealandcode, NZS 3101-95 (NZS 1995). Various notations used in this chapterare listed in Table 6-1. For referencing to the pertinent sections of theNew Zealand code in this chapter, a prefix “NZS” followed by the sec-tion number is used.



SAFE™SAFE™

SAFE Design Manual


Table 6-1 List of Symbols Used in the New Zealand Code

Acv Area of concrete used to determine shear stress, sq-mm

Ag Gross area of concrete, sq-mm

As Area of tension reinforcement, sq-mm

A'

s Area of compression reinforcement, sq-mm

As(required) Area of steel required for tension reinforcement, sq-mm

Av Area of shear reinforcement, sq-mm

Av /s Area of shear reinforcement per unit length of the member,sq-mm/mm

s Spacing of shear reinforcement along the length of the beam,mm


ab Depth of compression block at balanced condition, mm

amax Maximum allowed depth of compression block, mm

b Width of member, mm

bf Effective width of flange (T-Beam section), mm

bw Width of web (T-Beam section), mm

b0 Perimeter of the punching critical section, mm

b1 Width of the punching critical section in the direction ofbending, mm

b2 Width of the punching critical section perpendicular to thedirection of bending, mm

c Depth to neutral axis, mm

cb Depth to neutral axis at balanced conditions, mm

d Distance from compression face to tension reinforcement,mm

d' Concrete cover to center of reinforcing, mm




ds Thickness of slab (T-Beam section), mm



'cf Specified compressive strength of concrete, PMa (17.5 ≤ '

cf

≤ 100)

fy Specified yield strength of flexural reinforcement, MPa (fy ≤500)

fyt Specified yield strength of shear reinforcement, MPa (ft ≤800)

h Overall thickness of slab or overall depth of a beam, mm

M* Factored moment of section, N-mm

Vc Shear resisted by concrete, N


V* Factored shear force at a section, N

Vs Shear force at a section resisted by steel, N

v Average design shear stress at a section, MPa

vb Basic design shear stress resisted by concrete, MPa

vc Design shear stress resisted by concrete, MPa

vmax Maximum design shear stress permitted at a section, MPa

α1 Concrete strength factor to account for sustained loading andequivalent stress block




SAFE Design Manual




ϕb Strength reduction factor for bending

ϕs Strength reduction factor for shear



Design Load CombinationsThe design load combinations are the various combinations of the pre-scribed load cases for which the structure needs to be checked. For thiscode, if a structure is subjected to dead load (DL), live load (LL), patternlive load (PLL), wind (WL), and earthquake (EL) loads, and consideringthat wind and earthquake forces are reversible, the following load com-binations must be considered (NZS 4203-92 2.4.3):

1.4 DL1.2 DL + 1.6 LL (NZS 4203-92 2.4.3.3)

1.2 DL + 1.6*0.75 PLL (NZS 3101-95 14.9.6.3)

1.2 DL ± 1.0 WL0.9 DL ±1.0 WL1.2 DL + 0.4 LL ± 1.0 WL (NZS 4203-92 2.4.3.3)

1.0 DL ± 1.0 EL1.0 DL + 0.4 LL ± 1.0 EL (NZS 4203-92 2.4.3.3)

These are also the default design load combinations in SAFE wheneverthe NZS 3101-95 code is used. The user should use other appropriateloading combinations if roof live load is separately treated, or other typesof loads are present.

Chapter 6 - Strength Reduction Factors

Strength Reduction Factors 6- 5

Strength Reduction FactorsThe default strength reduction factor, ϕ, is taken as

ϕb = 0.85 for bending, (NZS 3.4.2.2)

ϕs = 0.75for shear. (NZS 3.4.2.2)


Beam DesignIn the design of concrete beams, SAFE calculates and reports the re-quired areas of steel for flexure and shear based on the beam moments,shear forces, load combination factors and other criteria described in thissection. The reinforcement requirements are calculated at two check sta-tions at the ends of the beam elements. All the beams are designed formajor direction flexure and shear only. Effects resulting from any axialforces, minor direction bending, and torsion that may exist in the beamsmust be investigated independently by the user.







SAFE Design Manual

6 - 6 Beam Design


The beam section is then designed for the maximum positive and maxi-mum negative factored moments obtained from all the of the load com-binations. Positive beam moments produce bottom steel. In such cases,the beam may be designed as a Rectangular or a T-beam. Negative beammoments produce top steel. In such cases, the beam is always designedas a rectangular section.


The design procedure is based on the simplified rectangular stress blockas shown in Figure 6-1 (NZS 8.3.1.6). Furthermore, it is assumed that thecompression carried by concrete is 0.75 times that which can be carriedat the balanced condition (NZS 8.4.2). When the applied moment ex-ceeds the moment capacity at the balanced condition, the area of com-pression reinforcement is calculated on the assumption that the additionalmoment will be carried by compression and additional tension rein-forcement.

In designing the beam flexural reinforcement, the following limits areimposed on the steel tensile strength and the concrete compressivestrength:

fy ≤ 500 MPa (NZS 3.8.2.1)

'cf ≤ 100 MPa (NZS 3.8.1.1)


Beam Design 6- 7

The design procedure used by SAFE, for both rectangular and flangedsections (L- and T-beams) is summarized below. All the beams are de-signed only for major direction flexure and shear.

Design for Flexure of a Rectangular Beam

In designing for a factored negative or positive moment, M*

(i.e., designing top or bottom steel), the depth of the compression block,a (see Figure 6-1), is computed as

a = d − bf

Md

bcϕα '1

*

22

− , (NZS 8.3.1)

where the default value of ϕb is 0.85 (NZS 3.4.2.2) in the above and fol-lowing equations. Also α1 is calculated as follows:

α1 = 0.85 − 0.004 ( 'cf − 55), 0.75 ≤ α1≤ 0.85. (NZS 8.3.1.7)

Also β1 and cb are calculated as follows:

β1 = 0.85 − 0.008 ( 'cf − 30), 0.65 ≤ β1 ≤ 0.85, and (NZS 8.3.1.7)

cb = yf+600

600d. (NZS 8.4.1.2)

The maximum allowed depth of the compression block is given by

amax = 0.75β1cb. (NZS 8.4.2 and NZS 8.3.1.7)

If a ≤ amax (NZS 8.4.2), the area of tensile steel reinforcement is thengiven by

As =

−

2

*

adf

M

ybϕ.

This steel is to be placed at the bottom if M* is positive, or at the topif M* is negative.

SAFE Design Manual

6 - 8 Beam Design


If a > amax (NZS 8.4.2), compression reinforcement is required (NZS8.4.1.3) and is calculated as follows:

− The compressive force developed in concrete alone is given by

C = α1

'cf bamax, and (NZS 8.3.1.7)

the moment resisted by concrete and bottom steel is

M*

c = C

−

2maxa

d ϕb.


M*

s = M* − M*

c .


Beam Design 6- 9


A'

s = ( )( ) bcs

s

ddff

M

ϕα ''1

'

*

−−, where

'sf = 0.003Es

−

c

dc ' ≤ fy. (NZS 8.3.1.2 and NZS 8.3.1.3)


As1 =

bmax

y

c

adf

M

ϕ

−

2

*

, and

the tensile steel for balancing the compression in steel is

As2 = ( ) by

s

ddf

M

ϕ'

*

−.

− Therefore, the total tensile reinforcement, As = As1 + As2, and totalcompression reinforcement is A'

s. A s is to be placed at the bot-tom and A'

s is to be placed at the top if M* is positive, and A'

s is tobe placed at the bottom and A s is to be placed at the top if M* isnegative.

Design for Flexure of a T-Beam


In designing for a factored negative moment, M* (i.e., designing topsteel), the calculation of the steel area is exactly the same as above, i.e.,no T-Beam data is to be used.

SAFE Design Manual

6 - 10 Beam Design



If M* > 0, the depth of the compression block is given by (see Figure 6-2).

a = d − fbc bf

Md

ϕα '1

*

22

− , (NZS 8.3.1)

The maximum allowed depth of the compression block is given by

amax = 0.75 β1cb . (NZS 8.4.2 and NZS 8.3.1.7)

If a ≤ ds (NZS 8.4.2), the subsequent calculations for As are exactlythe same as described previously for the rectangular section design.However, in this case, the width of the beam is taken as bf. Compres-sion reinforcement is required if a > amax.

If a > ds (NZS 8.4.2), calculation for As has two parts. The first partis for balancing the compressive force from the flange, Cf, and the


Beam Design 6- 11

second part is for balancing the compressive force from the web, Cw.As shown in Figure 6-2,

Cf = α1 'cf (bf − bw) min(ds, amax). (NZS 8.3.1.7)

Therefore, As1 = y

f

f

C and the portion of M* that is resisted by the

flange is given by

M*

f = Cf

( )

−

2

add maxs ,min

ϕb .

Therefore, the balance of the moment, M* to be carried by the web isgiven by

M*

w = M* − M*

f.

The web is a rectangular section of dimensions bw and d, for whichthe depth of the compression block is recalculated as

a1 = d − wbc

w

bf

Md

ϕα '1

*2 2

− . (NZS 8.3.1)

If a1 ≤ amax (NZS 8.4.2), the area of tensile steel reinforcement is thengiven by

As2 =

−

21

*

adf

M

yb

w

ϕ, and

As = As1 + As2 .


If a1 > amax (NZS 8.4.2), compression reinforcement is required andis calculated as follows:

− The compressive force in the concrete web alone is given by

SAFE Design Manual

6 - 12 Beam Design

Cw = α1

'cf bwamax , and (NZS 8.3.1.7)

the moment resisted by the concrete web and tensile steel is

M*

c = Cw

−

2maxa

d ϕb.


M*

s = M*

w − M*

c .


A'

s = ( )( ) bcs

s

ddff

M

ϕα ''1

'

*

−−, where

'sf = 0.003Es

−

c

dc ' ≤ fy . (NZS 8.3.1.2 and NZS 8.3.1.3)


As2 =

bmax

y

c

adf

M

ϕ

−

2

*

, and


As3 = ( ) by

s

ddf

M

ϕ'

*

− .

− Total tensile reinforcement, As = As1 + As2 + As3, and total com-pression reinforcement is A's. As is to be placed at the bottom andA's is to be placed at the top.

Minimum and Maximum Tensile ReinforcementThe minimum flexural tensile steel required for a beam section is givenby the minimum of the two limits:


Beam Design 6- 13

As ≥ y

c

f

f

4

'

bwd, or (NZS 8.4.3.1)

As ≥ 3

4As(required). (NZS 8.4.3.3)




0.04 bd Rectangular beamA's ≤

0.04 bwd T-beam

Design Beam Shear ReinforcementThe shear reinforcement is designed for each load combination at twostations at the ends of each beam element. In designing the shear rein-forcement of a particular beam for a particular loading combination at aparticular station resulting from beam major shear, the following stepsare involved:

Determine the factored shear force, V*.



In designing the beam shear reinforcement, the following limits are im-posed on the steel tensile strength and the concrete compressivestrength:

fyt ≤ 500 MPa (NZS 3.8.2.1 and NZS 9.3.6.1)

'cf ≤ 100 MPa (NZS 3.8.1.1)

SAFE Design Manual

6 - 14 Beam Design

The following three sections describe the algorithms associated with theabove steps.

Determine Shear Force and MomentIn the design of the beam shear reinforcement of concrete frame, theshear forces and moments for a particular load combination at a particu-lar beam section are obtained by factoring the associated shear forces andmoments with the corresponding load combination factors.


The basic shear strength for rectangular section is computed as

νb =

+

db

A

w

s1007.0 'cf , where (NZS 9.3.2.1)

'cf ≤ 70 , and (NZS 9.3.2.1)

0.08 'cf ≤ νb ≤ 0.2 '

cf . (NZS 9.3.2.1)

The allowable shear capacity is given by,

νc = νb. (NZS 9.3.2.1)

Determine Required Shear Reinforcement The average shear stress is computed for a rectangular section as

ν* = db

V

w

*

. (NZS 9.3.1.1)

The average shear stress is limited to a maximum limit of

vmax = min{ MPa9,2.0,1.1 ''cc ff }. (NZS 9.3.1.8)


Slab Design 6- 15

The shear reinforcement is computed as follows:

If ν* ≤ ϕs ( )2cv ,

s

Av = 0, (NZS 9.3.4.1)

else if ϕs ( )2cv < ν* ≤ ϕs (νc + 0.35),

s

Av = ytf

b35.0, (NZS 9.3.4.3)

else if ϕs (νc + 0.35) < ν* ≤ ϕs νmax, (NZS 9.3.6.3)

s

Av = ( )

yts

wcs

f

bvv

ϕϕ−

*

else if ν* > νmax,

a failure condition is declared. (NZS 9.3.1.8)


The beam shear reinforcement requirements displayed by the programare based purely on shear strength considerations. Any minimum stirruprequirements to satisfy spacing and volumetric considerations must beinvestigated independently of the program by the user.

Slab DesignSimilar to conventional design, the SAFE slab design procedure involvesdefining sets of strips in two mutually perpendicular directions. The lo-cations of the strips are usually governed by the locations of the slabsupports. The moments for a particular strip are recovered from theanalysis, and a flexural design is completed based on the ultimatestrength design method for reinforced concrete (NZS 3101-95) as de-

SAFE Design Manual

6 - 16 Slab Design

scribed in the following subsections. To learn more about the designstrips, refer to the section entitled "SAFE Design Techniques" in theWelcome to SAFE manual.

Design for FlexureSAFE designs the slab on a strip-by-strip basis. The moments used forthe design of the slab elements are the nodal reactive moments, whichare obtained by multiplying the slab element stiffness matrices by theelement nodal displacement vectors. These moments will always be instatic equilibrium with the applied loads, irrespective of the refinementof the finite element mesh.





Determine Factored Moments for the StripFor each element within the design strip, for each load combination theprogram calculates the nodal reactive moments. The nodal moments arethen added to get the strip moments.

Design Flexural Reinforcement for the StripThe reinforcement computation for each slab design strip, given thebending moment, is identical to the design of rectangular beam sectionsdescribed earlier. When the slab properties (depth, etc.) vary over the


Slab Design 6- 17

width of the strip, the program automatically designs slab widths of eachproperty separately for the bending moment to which they are subjectedand then sums the reinforcement for the full width. Where openings oc-cur, the slab width is adjusted accordingly.

Minimum Slab ReinforcementThe minimum flexural tensile reinforcement required for each directionof a slab is given by the following limit (NZS 8.4.3.4):

yf

7.0 bh

if fy < 500 MPa

As ≥

0.0014 bh if fy ≥ 500 MPa

(NZS 7.3.30.1)



Critical Section for Punching ShearThe punching shear is checked on a critical section at a distance of d/2from the face of the support (NZS 9.3.15.1). For rectangular columns andconcentrated loads, the critical area is taken as a rectangular area with thesides parallel to the sides of the columns or the point loads (NZS9.3.15.1).

Transfer of Unbalanced MomentThe fraction of unbalanced moment transferred by flexure is taken to beγf M

* and the fraction of unbalanced moment transferred by eccentricityof shear is taken to be γv M

*, where

SAFE Design Manual

6 - 18 Slab Design

γf = ( ) 21321

1

bb+ , and (NZS 14.3.5)

γv = 1 − ( ) 21321

1

bb+, (NZS 9.3.16.2)


Determination of Concrete CapacityThe concrete punching shear factored strength is taken as the minimumof the following three limits:

ϕs ( )cβ21 + 0.17 'cf

vc = min ϕs

+

021

b

dsα 0.17 '

cf (NZS 9.3.15.2)

ϕs 0.33 'cf


40 for interior columns,

αs = 30 for edge columns, and (NZS 9.3.15.2)

20 for corner columns.

A limit on 'cf is imposed as follows:

'cf ≤ 70 (NZS 9.3.2.1)


Slab Design 6- 19



Chapter 7

Design for IS 456-78 (R1996)

This chapter describes in detail the various aspects of the concrete designprocedure that is used by SAFE when the user selects the Indian Code IS345-78 Revision 1996 (IS 1996). Various notations used in this chapterare listed in Table 7-1. For referencing to the pertinent sections of the In-dian code in this chapter, a prefix “IS” followed by the section number isused.



SAFE™SAFE™

SAFE Design Manual


Table 7-1 List of Symbols Used in the Indian Code

Ac Area of concrete, mm2

Acv Area of section for shear resistance, mm2

Ag Gross cross-sectional area of a frame member, mm2


A'

s Area of compression reinforcement, mm2

Asv Total cross-sectional area of links at the neutral axis, mm2

Asv /sv Area of shear reinforcement per unit length of the member,mm2/mm

a1 Width of the punching critical section in the direction ofbending, mm

a2 Width of the punching critical section perpendicular to thedirection of bending, mm





d' Effective depth of compression reinforcement, mm

dcompression Depth of center of compression block from most compressedface, mm

D Overall depth of a beam or slab, mm

Df Flange thickness in a T-beam, mm



fcd Design concrete strength = fck / γc, MPa




fck Characteristic compressive strength of concrete, MPa

'sf Compressive stress in a beam compression steel, MPa

fyd Design yield strength of reinforcing steel = fy / γs, MPa

fy Characteristic strength of reinforcement, MPa

fys Characteristic strength of shear reinforcement, MPa

h Overall thickness of slab, mm

k Enhancement factor of shear strength for depth of the beam

Msingle Design moment resistance of a section as a singly reinforcedsection, N-mm

Mu Ultimate factored design moment at a section objected, N-mm

m Normalized design moment, M / db2αfck

sv Spacing of the shear reinforcement along the length of thebeam, mm

Vu Shear force of ultimate design load, N

vc Allowable shear stress in punching shear mode, N

xu Depth of neutral axis, mm

xu,max Maximum permitted depth of neutral axis, mm

Z Lever arm, mm

α Concrete strength reduction factor for sustained loading;also fraction of moment to be transferred by flexure in aslab-column joint

β Factor for the depth of compressive force resultant of theconcrete stress block

βc Ratio of the maximum to minimum dimensions of thepunching critical section

γc Partial safety factor for concrete strength

SAFE Design Manual



γf Partial safety factor for load, and fraction of unbalancedmoment transferred by flexure


γs Partial safety factor for steel strength

γv Fraction of unbalanced moment transferred by eccentricityof shear

δ Enhancement factor of shear strength for compression

εc,max Maximum concrete strain in the beam and slab (=0.0035)


εs' Strain in compression steel

ρ Tension reinforcement ratio, As /bd

τv Average design shear stress resisted by concrete, MPa

τc Basic design shear stress resisted by concrete, MPa

τc,max Maximum possible design shear stress permitted at a sec-tion, MPa

τcd Design shear stress resisted by concrete, MPa

Design Load CombinationsThe design load combinations are the various combinations of the pre-scribed load cases for which the structure needs to be checked. For thiscode, if a structure is subjected to dead load (DL), live load (LL), patternlive load (PLL), wind (WL), and earthquake (EL) loads, and consideringthat wind and earthquake forces are reversible, the following load com-binations have to be considered (IS 35.4):

1.5 DL1.5 DL + 1.5 LL (IS 35.4.1)

Chapter 7 - Design Strength

Design Strength 7- 5

1.5 DL + 1.5*0.75 PLL (IS 30.5.2.3)

1.5 DL ± 1.5 WL0.9 DL ± 1.5 WL1.2 DL + 1.2 LL ± 1.2 WL (IS 35.4.1)

1.5 DL ±1.5 EL0.9 DL ± 1.5 EL1.2 DL + 1.2 LL ±1.2 EL (IS 35.4.1)

These are also the default design load combinations in SAFE wheneverthe Indian Code is used. The user should use other appropriate loadingcombinations if roof live load is separately treated, or other types ofloads are present.

Design StrengthThe design strength for concrete and steel are obtained by dividing thecharacteristic strength of the material by a partial factor of safety, γm. Thevalues of γm used in the program are as follows:

Partial safety factor for steel, γs = 1.15 , and (IS 35.4.2.1)

Partial safety factor for concrete, γc = 1.5. (IS 35.4.2.1)

These factors are incorporated in the design equations and tables in thecode. SAFE does not allow them to be overwritten.

Beam DesignIn the design of concrete beams, SAFE calculates and reports the re-quired areas of steel for flexure and shear based on the beam moments,shears, load combination factors, and other criteria described in this sec-tion. The reinforcement requirements are calculated at two check stationsat the ends of the beam elements.

All the beams are only designed for major direction flexure and shear. Effects resulting from any axial forces, minor direction bending, and

SAFE Design Manual

7 - 6 Beam Design

torsion that may exist in the beams must be investigated independentlyby the user.








The beam section is then designed for the maximum positive and maxi-mum negative factored moments obtained from all the of the load com-binations. Positive beam moments produce bottom steel. In such casesthe beam may be designed as a Rectangular or a T-beam. Negative beammoments produce top steel. In such cases the beam is always designed asa rectangular section.

Determine Required Flexural ReinforcementIn the flexural reinforcement design process, the program calculates boththe tension and compression reinforcement. Compression reinforcementis added when the applied design moment exceeds the maximum mo-ment capacity of a singly reinforced section. The user has the option of


Beam Design 7- 7

avoiding the compression reinforcement by increasing the effectivedepth, the width, or the grade of concrete.

The design procedure is based on the simplified parabolic stress block, asshown in Figure 7-1 (IS 37.1). The area of the stress block, C, and thedepth of the center of the compressive force from the most compressedfiber, d, are taken as

C = α fck xu and (IS 37.1)

dcompression = β xu, (IS 37.1)

where xu is the depth of the compression block, and α and β are taken re-spectively as

α = 0.36, and (IS 37.1)

β = 0.42. (IS 37.1)

α is the reduction factor to account for sustained compression and thepartial safety factor for concrete. α is generally assumed to be 0.36 forthe assumed parabolic stress block (IS 37.1). β factor considers thedepth of the neutral axis.

Furthermore, it is assumed that moment redistribution in the memberdoes not exceed the code specified limiting value. The code also places alimitation on the neutral axis depth as shown in the following table, tosafeguard against non-ductile failures (IS 37.1).

fy xu,max /d

250 0.53

415 0.48

500 0.46

SAFE Design Manual

7 - 8 Beam Design

Figure 1 Design of a Rectangular Beam Section

SAFE uses interpolation between the three discrete points given in thecode.

0.53 if fy ≤ 250

0.53 − 0.05 165

250−yf if 250 < fy ≤ 415

0.48 − 0.02 85

415−yf if 415 < fy ≤ 500d

xu max, =

0.46 if fy ≥ 500

(IS 37.1)

When the applied moment exceeds the capacity of the beam as a singlyreinforced beam, the area of compression reinforcement is calculated as-


Beam Design 7- 9

suming that the neutral axis depth remains at the maximum permittedvalue. The maximum fiber compression is taken as

εc,max = 0.0035, (IS 37.1)

and the modulus of elasticity of steel is taken to be

Es = 200,000 MPa . (IS 37.1)

The design procedure used by SAFE, for both rectangular and flangedsections (L- and T-beams), is summarized in the next two subsections. Itis assumed that the design ultimate axial force can be neglected; hence,all of the beams are designed for major direction flexure and shear only.

Design as a Rectangular Beam

For rectangular beams, the limiting depth of neutral axis, xu,max, and themoment capacity as a singly reinforced beam, Msingle, are obtained first forthe section. The reinforcing steel area is determined based on whether Mu

is greater than, less than, or equal to Msingle. See Figure 7-1.

Calculate the limiting depth of the neutral axis.

0.53 if fy ≤ 250

0.53 − 0.05 165

250−yf if 250 < fy ≤ 415

0.48 − 0.02 85

415−yf if 415 < fy ≤ 500d

xu max, =

0.46 if fy ≥ 500

(IS 37.1)

Calculate the limiting ultimate moment of resistance as a singly rein-forced beam.

SAFE Design Manual

7 - 10 Beam Design

Msingle = αfckbd2

−

d

x

d

x uu max,max, 1 β , where (IS E-1.1)

α = 0.36 , and (IS E-1.1)

β = 0.42 . (IS E-1.1)

Calculate the depth of neutral axis xu as

ββ

2

411 m

d

xu −−= ,

where the normalized design moment, m, is given by

m = ck

u

fbd

M

α2.

If Mu ≤ Msingle, the area of tension reinforcement, As, is obtained from

As = ( )zf

M

sy

u

γ/, where (IS E-1.1)

z =

−

d

xd uβ1 . (IS 37.1)

This is the top steel if the section is under negative moment and the bot-tom steel if the section is under positive moment.

If Mu > Msingle, the area of compression reinforcement, A'

s, is given by

A'

s = ( )''sin

ddf

MM

s

gleu

−−

, (IS E-1.2)

where d' is the depth of the compression steel from the concrete com-pression face, and

'sf = εc,maxEs

−

max,

'1

ux

d≤

s

yf

γ. (IS E-1.2)


Beam Design 7- 11

This is the bottom steel if the section is under negative moment. Fromequilibrium, the area of tension reinforcement is calculated as

As = ( ) ( )( )'ddf

MM

zf

M

sy

singleu

sy

single

−−

+γγ

, where (IS E-1.2)

z =

−d

xd u max,1 β . (IS 37.1)

Design as a T-Beam

(i) Flanged Beam Under Negative MomentThe contribution of the flange to the strength of the beam is ignored ifthe flange is in the tension side. See Figure 7-2. The design procedure istherefore identical to the one used for rectangular beams. However, thewidth of the web, bw, is taken as the width of the beam.


With the flange in compression, the program analyzes the section byconsidering alternative locations of the neutral axis. Initially, the neutralaxis is assumed to be located within the flange. On the basis of this as-sumption, the program calculates the depth of the neutral axis. If thestress block does not extend beyond the flange thickness, the section isdesigned as a rectangular beam of width bf. If the stress block extends be-yond the flange, additional calculation is required. See Figure 7-2.

Assuming the neutral axis to lie in the flange, calculate the depth ofneutral axis, xu, as

ββ

2

411 m

d

xu −−= ,

where the normalized design moment, m, is given by

m = ckf

u

fdb

M

α2.

SAFE Design Manual

7 - 12 Beam Design

If

≤

d

D

d

x fu , the neutral axis lies within the flange. The subse-

quent calculations for As are exactly the same as previously defined forthe rectangular section design (IS E-2.1). However, in this case thewidth of the compression flange, bf, is taken as the width of the beam,b, for analysis. Compression reinforcement is required if Mu > Msingle.

If

>

d

D

d

x fu , the neutral axis lies below the flange. Then cal-

culation for As has two parts. The first part is for balancing the com-pressive force from the flange, Cf, and the second part is for balancingthe compressive force from the web, Cw, as shown in Figure 7-2.


− Calculate the ultimate resistance moment of the flange as

Mf = 0.45 fck (bf − bw)yf (d − 0.5 yf) , (IS E-2.2)

where yf is taken as follows:

Df if Df ≤ 0.2dyf =

0.15xu + 0.65Df if Df > 0.2d(IS E-2.2)


Beam Design 7- 13

− Calculate the moment taken by the web as

Mw = Mu − Mf.

− Calculate the limiting ultimate moment of resistance of the web fortension only reinforcement.

Mw,single = αfckbwd2

−

d

x

d

x uu max,max, 1 β where (IS E-1.1)

0.53 if fy ≤ 250

0.53 − 0.05 165

250−yf if 250 < fy ≤ 415

0.48 − 0.02 85

415−yf if 415 < fy ≤ 500d

xu max, =

0.46 if fy ≥ 500

(IS 37.1)

α = 36 , and (IS 37.1)

β= 42. (IS 37.1)

• If Mw ≤ Mw,single, the beam is designed as a singly reinforced concretebeam. The area of steel is calculated as the sum of two parts, one tobalance compression in the flange and one to balance compression inthe web.

As = ( )( ) ( ) zf

M

ydf

M

sy

w

fsy

f

γλ+

− 5.0 , where

z =

−

d

xd uβ1 ,

ββ

2

411 m

d

xu −−= , and

SAFE Design Manual

7 - 14 Beam Design

m = ckw

w

fdb

M

α2.

• If Mw > Mw,single, the area of compression reinforcement, A's, is given by

A's = ( )''

singl,

ddf

MM

s

eww

−−

,

where d' is the depth of the compression steel from the concretecompression face, and

'sf = εc,maxEs .

−

maxux

d

,

'1 ≤

s

yf

γ. (IS E-1.2)

This is the bottom steel if the section is under negative moment.From equilibrium, the area of tension reinforcement is calculatedas

As = ( )( ) ( ) ( )( )'5.0,,

ddf(

MM

zf

M

ydf

M

sy

singleww

sy

singlew

fsy

f

−−

++− γγγ

,

where

z =

−

d

xd maxu,1 β .

Minimum Tensile ReinforcementThe minimum flexural tensile steel required for a beam section is givenby the following equation (IS 25.5.1.1):

bdfy

85.0 Rectangular beam

As ≥db

f wy

85.0 T-beam(IS 25.5.1.1)


Beam Design 7- 15

An upper limit on the tension reinforcement (IS 25.5.1.1) and compres-sion reinforcement (IS 25.5.1.2) has been imposed to be 0.04 times thegross web area.


As ≤ 0.04 bwd T-beam(IS 25.5.1.1)

0.04 bd Rectangular beamA's ≤

0.04 bwd T-beam(IS 25.5.1.2)

Design Beam Shear ReinforcementThe shear reinforcement is designed for each loading combination at twostations at the ends of each beam element. The assumptions in designingthe shear reinforcement are as follows:

The beam sections are assumed to be prismatic. The effect of anyvariation of width in the beam section on the concrete shear capacity isneglected.

The effect on the concrete shear capacity of any concentrated or dis-tributed load in the span of the beam between two columns is ignored.Also, the effect of the direct support on the beams provided by the col-umns is ignored.

All shear reinforcement is assumed to be perpendicular to the longitu-dinal reinforcement.

The effect of any torsion is neglected for the design of shear rein-forcement.

The shear reinforcement is designed for each loading combination in themajor direction of the beam. In designing the shear reinforcement for aparticular beam for a particular loading combination, the following stepsare involved (IS 39.2):

Calculate the design nominal shear stress as

τv = cv

u

A

V, Acv = bwd, where (IS 39.1)

SAFE Design Manual

7 - 16 Beam Design

τv ≤ τc,max, and (IS 39.2.3)

the maximum nominal shear stress, τc,max is given in the IS Table 14 asfollows:

Maximum Shear Stress, ττττc,max (MPa)(IS 39.2.3, IS Table 14)

Concrete Grade M15 M20 M25 M30 M35 M40

τc,max (MPa) 2.5 2.8 3.1 3.5 3.7 4.0

The maximum nominal shear stress, τc,max, is computed by the followingequation, which matches the IS Table 14 exactly.

2.5 if fck < 15

2.5+0.35

15−ckf if15 ≤ fck<20

2.8+0.35


3.1+0.45


3.5+0.25


3.7+0.35


τc,max=

4.0 if fck≥ 40

(IS 39.2.3)

Calculate the design shear strength of concrete from

τcd = kδτc, (IS 39.2)

where k is the enhancement factor for the depth of the beam sectionand is computed by


Beam Design 7- 17

k = 1.6 – 0.002d, 1.0 ≤ k ≤ 1.3. (IS 39.2.1.1)

The above expression represents the table given in IS 39.2.1.1, whichis shown below:

The Value of the Enhancement Factor, k(IS 39.2.1.1)

Overall depth of slab, d (mm) ≥300 275 250 225 200 175 ≤150

Factor, k 1.00 1.05 1.10 1.15 1.20 1.25 1.30

δ is the enhancement factor for compression and is given by

1+3 ckg

u

fA

P ≤ 1.5

if Pu > 0, Under Compression

δ =1 if Pu ≤ 0, Under Tension

(IS 39.2.2)

δ is always taken as 1, and

τc is the basic design shear strength for concrete, which is given by

τc = 0.64

41

31

25100

cks f

bd

A. (IS 39.2.1)

The above expression tries to represent the IS Table 13 approximately.It should be mentioned that the value of γc has already been incorpo-rated in the IS Table 13 (see Note in IS 35.4.2.1). The following limi-tations are enforced in the determination of the design shear strength asis done in the table.

0.25 ≤ bd

As100 ≤ 3, (IS 39.2.1)

fck ≤ 40 MPa (for calculation purpose only). (IS 39.2.1)

The shear reinforcement is computed as follows:

If τv ≤ τcd + 0.4, provide minimum links defined by

SAFE Design Manual

7 - 18 Slab Design

ys

w

v

sv

f

b

s

A

87.04.0≥ , (IS 39.3 and IS 25.5.1.6)

else if τcd + 0.4 < τv ≤ τc,max, provide links given by

( )ys

wcdv

v

sv

f

b

s

A

87.0

ττ −≥ , (IS 39.4)

else if τv > τc,max,

a failure condition is declared. (IS 39.2.3)

In calculating the shear reinforcement, a limit was imposed on the fyv as

fyv ≤ 415 MPa. (IS 39.4)

The maximum of all of the calculated Asv /sv values, obtained from eachload combination, is reported along with the controlling shear forceand associated load combination number.

The beam shear reinforcement requirements displayed by the programare based purely on shear strength considerations. Any minimum stir-rup requirements to satisfy spacing and volumetric considerations mustbe investigated independently of the program by the user.

Slab DesignSimilar to conventional design, the SAFE slab design procedure involvesdefining sets of strips in two mutually perpendicular directions. The lo-cations of the strips are usually governed by the locations of the slabsupports. The moments for a particular strip are recovered from theanalysis, and a flexural design is completed based on the limit state ofcollapse for reinforced concrete (IS 37, as described in the followingsubsections. To learn more about the design strips, refer to the sectionentitled "SAFE Design Techniques" in the Welcome to SAFE manual.


Slab Design 7- 19







Design Flexural Reinforcement for the StripThe reinforcement computation for each slab design strip, given thebending moment, is identical to the design of rectangular beam sectionsdescribed earlier. When the slab properties (depth, etc.) vary over thewidth of the strip, the program automatically designs slab widths of eachproperty separately for the bending moment to which they are subjectedand sums the reinforcement for the full width. Where openings occur, theslab width is adjusted accordingly.

SAFE Design Manual

7 - 20 Slab Design

Minimum Slab ReinforcementThe minimum flexural tensile reinforcement required for each directionof a slab is given by the following limits (IS 25.5.2):

0.0015 bh if fy < 500 MPaAs ≤

0.0012 bh if fy ≥ 500 MPa(IS 25.5.2.1)

In addition, an upper limit on both the tension reinforcement and com-pression reinforcement has been imposed to be 0.04 times the grosscross-sectional area (IS 25.5.1.1).


Critical Section for Punching ShearThe punching shear is checked on a critical section at a distance of d/2from the face of the support (IS 30.6.1). For rectangular columns andconcentrated loads, the critical area is taken as a rectangular area, withthe sides parallel to the sides of the columns or the point loads (IS30.6.1).

Transfer of Unbalanced MomentThe fraction of unbalanced moment transferred by flexure is taken to beαMu and the fraction of unbalanced moment transferred by eccentricityof shear is taken to be (1 − α) Mu (IS 30.6.2.2) , where

α = ( ) 21321

1

aa+, and (IS 30.3.3)

where a1 is the width of the critical section measured in the direction ofthe span and a2 is the width of the critical section measured in the direc-tion perpendicular to the span.

Chapter 7 -

7- 21

Determination of Concrete CapacityThe concrete punching shear factored strength is taken as the following.

vc = ks τc, where (IS 30.6.3.1)

ks = 0.5 + βc ≤ 1.0, (IS 30.6.3.1)

τc = 0.25 ckf , and (IS 30.6.3.1)

β c = ratio of the minimum to the maximum dimensions of the sup-port section.


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References

ACI, 2002. Building Code Requirements for Reinforced Concrete (ACI318-02) and Commentary (ACI 318R-02), American ConcreteInstitute, Detroit, Michigan.

BSI, 1989. BS 8110: Part 1, Structural Use of Concrete, Part 1, Code ofPractice for Design and Construction, British Standards Institute,London, UK. Issue 2.

CEN, 1992. ENV 1992-1-1, Eurocode 2: Design of Concrete Structures,Part 1, General Rules and Rules for Buildings, European Com-mittee for Standardization, Brussels, Belgium.

CEN, 1994. ENV 1991-1, Eurocode 1: Basis of Design and Action onStructures – Part 1, Basis of Design, European Committee forStandardization, Brussels, Belgium.

CSA, 1994. A23.3-94, Design of Concrete Structures, Structures Design,Canadian Standards Associated, Rexdale, Ontario, Canada.

IS, 1996. Code of Practice for Plan and Reinforced Concrete, Third Edi-tion, Twentieth Reprint, March 1996, Bureau of Indian Stan-dards, Nanak Bhavan, 9 Bahadur Shah Zafar Marg, New Delhi110002, India.

SAFE™SAFE™

SAFE Design Manual

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NZS, 1995. Concrete Structures Standard, Part 1 – Design of ConcreteStructures, Standards New Zealand, Private Bag 2439, Welling-ton, New Zealand.

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SAFE Design Manual Title - Springerextras.springer.com/1995/978-3-642-78702-7/INFORMATION-MANUALS...SAFE Design Manual iv Determine Factored Moments for the Strip 4-15 Design Flexural