DESIGN OF END PLATE JOINTS SUBJECT TO MOMENT AND NORMAL FORCE

Zdeněk Sokol1, Franti�ek Wald1, Vincent Delabre2, Jean-Pierre Muzeau2, Marek �varc1

ABSTRACT The presented work describes design model of end plate joints loaded by combination of bending moment and normal force. Two sets of tests were performed to check the prediction - tests at University in Coimbra and tests prepared at Czech Technical University in Prague. The procedure for the interaction is already available for base plates. The paper presents application of this model to beam-to-column connections and beam splices and verification of the model to available tests. Key Words: steel structures, structural connections, end plate connection, beam-to-column connection, moment � normal force interaction, design model, component method. 1. INTRODUCTION

The European practice of design of structural frames was significantly improved by introducing knowledge of the joint behaviour [1]. Modelling of the joints requires models for evaluation of moment-rotation diagrams of the joints by mechanical engineering methods [2] supported by experimental investigation. The prediction of behaviour of typical joints made from open sections reached application level [3]. The mechanical model for the joints is using component method [4] and can be used in many applications including multi-storey building frames [1], [5]. Prediction of joint behaviour in portal frames is in most cases limited by interaction of normal force and bending moment. Application of component method extended to M-N interaction was published by Jaspart et al. [6]. Simple procedure solving the M-N interaction by linear interpolation between normal force and bending moment resistances for beam-to-column joints was adopted in the current European design procedure. This is safe procedure but it may be too conservative for non-symmetrical joints. Simple design procedure for prediction of base plate resistance and stiffness was developed by Steenhuis [7]. This procedure is based on component method [9] using typical components for base plates [10]. _______________________________________________

1Czech Technical University in Prague, CZ 166 29 Praha 6, Czech Republic 2Université Blaise Pascal, CUST, BP206, F 63174 Aubière Cedex, France

1

2. EXPERIMENTS

Two sets of experiments are available to evaluate the behaviour of end plate joints loaded by bending moment and normal force. The experiments performed in Structural Laboratory of Coimbra University are designed to validate model of flush end-plate beam-to-column joints [11]. The experiments carried out in laboratory of the Czech Technical University in Prague [12] are used to model behaviour of beam splices and beam-to-column end plate joints, see Fig. 1 [13]. The sets were designed to continue the experimental work carried by Zoetemeijer [14]. Data of both sets are available at server of Structural Laboratory of Coimbra University at http://www.dec.uc.pt/~luisss/eccs/eccspreview.htm.

v v

45º

45º

60º

60º

F SdFSd

Fig. 1 - Tests SN and NN carried out at CTU in Prague

3. COMPONENT METHOD

The application of component method to joint modelling requires three basic steps: identification of the components of the joint, see Fig. 2, evaluation of force-deflection diagram of individual components (in terms of initial stiffness, resistance and deformation capacity) and assembly of the components to obtain behaviour of the joint. For simplicity, the assembling procedures for stiffness, resistance and deformation capacity are carried out separately. The model of behaviour of the components of end plate beam-to-column joints is based on work of Zoetemeijer [3]. The model was improved by introduction of the column web panel in shear [15].

!

!

"!#

$%

&

$%

&

% %

'

components in tension

components in compresion

"! &'

"&'

"&'

&'

& &'

(

(

(

(

(

(

(

(

Fig. 2 - List of components of end plate beam-to-column joints and beam splices:

# column web in shear, $ column web in compression, % beam flange in compression, ! column web in tension, " column flange in bending, 'bolt row in tension, &end plate in bending, ( beam web in tension

2

The beam-to-column joints and beam splices are loaded by combination of shear force, normal force and bending moment. The normal force has significant influence on the bending moment resistance of the joint. Two typical loading paths may be distinguished. In case of non-proportional loading, the normal force is applied to the end plate connection in first step followed by application of the moment. In case of proportional loading, the normal force and the bending moment are applied simultaneously with constant ratio between the moment and normal force. Fig. 3 shows moment-rotation curve for typical connection loaded by proportional and non-proportional loading. In case of non-proportional loading, the stiffness of the joint is higher than for non-proportional loading. This effect is caused by presence of the normal force, which keeps the end plate in the contact at low bending moments, therefore only the components in compression contribute to deformation of the joint.

Moment

Rotation

MRd

Sd

S j.ini

Plasticity of the weakest component

Bolt rows in tension, beam flange in compression

Nonlinear part of the curve

Non-proportional loading

Proportional loading

e N0

Normal force

Moment

Non-proportional

Proportional

Joint resistance

loading

loading

Both beam flanges in compression0

0

Fig. 3 - Moment - rotation curve for the proportional loading, the non-proportional and proportional loading paths on the moment - normal force interaction diagram

The size and shape of the contact area between the end plate and the column flange are

based on effective rigid area [8]. Position of the neutral axis can be evaluated from equilibrium equations taking into account resistance of the tension and compression parts Ft.Rd and Fc.Rd respectively, and the applied normal force NSd and bending moment MSd. Plastic distribution of internal forces is assumed for the calculation, see Fig. 4.

NSdM Sd

Ft.Rd

z

zt

zc

Fc.RdActive part

Neutral axis

Centre of the part in tension

Centre of the part in compression

Fig. 4 - The equilibrium in the joint

3

The prediction model for the bending resistance and the rotational stiffness may take into account only the effective area at beam flanges [7] and the effective area at the beam web is neglected, as shown in Fig. 5. It is assumed the compression force acts at the centre of the flange in compression also in cases of limited size of outstand of the plate. The tension force is located in the bolt row in tension. In case of two or more bolt rows in tension part, the resistance of the part in tension is obtained as the resulting force of the active bolt rows.

NSd

MSd

NSd

MSdz

Ft.Rd

z

zt

zc

Fc.Rd

Fc.t.Rd

zc.t

zc.b

Fc.b.Rd

a) b)

Fig. 5 - Model with the effective area at the flanges only; a) one bolt row in tension; b) no bolts in tension

The forces represent resistances of the components in tension Ft.Rd, and in compression

Fctl.Rd, Fc.b.Rd resistance of which is calculated separately [1]. For simplicity, the model will be derived for proportional loading only. Using equilibrium equations, the following formulas

may be derived from Fig. 5a) assuming the eccentricity cSd

Sd zNMe −≤= .

tcSdSd F

zzN

zM

≤+ (1)

and

ctSdSd F

zzN

zM

−≤− . (2)

Since constNM

NMe

Rd

Rd

Sd

Sd === for proportional loading, equations (1) and (2) may be

rearranged to

−

+

=

ez1

zF

1ez

zF

minM

l.t

c

c

t

Rd . (3)

When the eccentricity cSd

Sd zNMe −≥= , see Fig. 5b), there is no tension force in the bolt

row, but both parts of the connection are loaded in compression. In this case, the equation (3) needs to be modified to

4

−

−

+

−

=

1e

zzF

1e

zzF

minM

t.c

b.c

b.c

t.c

Rd . (4)

The rotational stiffness of the connection is based on the deformation stiffness of the

components, see [1].

NSd

MSd

NSd

M Sd

t

c

c.t

c.bδ

δ δ

δa) b)

zz

zt

zc

zc.t

z c.b

φ

φ

Fig. 6 - The mechanical model of the end plate

The elastic deformation of the components in tension and compression parts, see Fig. 6a), may be expressed as

t

cSdSd

t

cSdSd

l.t kzEzNM

kEz

zNz

M+

=+

=δ , (5)

c

tSdSd

r.c

tSdSd

r.c kzEzNM

kEz

zNz

M−

=−

=δ , (6)

and the joint rotation is calculated using deformation of these components

−+

+=

+=

c

tSdSd

t

cSdSd2

ct

kzNM

kzNM

zE1

zδδφ . (7)

The rotational stiffness of the joint depends on the bending moment which is induced by the normal force applied with constant eccentricity e

φSd

inijM

S =. . (8)

The stiffness is derived by substituting the rotation of the joint (7) into equation (8)

∑+=

+

+=

k1

zEee

e

k1

k1

zEeNM

MS2

0

tc

2

0SdSd

Sdini;j , (9)

where the eccentricity e0 is defined as follows

5

tc

ttcc0 kk

kzkze+−

= . (10)

The non-linear part of the moment-rotation curve may be modelled by introducing the shape factor µ, which depends on ratio γ of the acting forces and their capacities

( ) 15,1 7,2 ≥= γµ . (11) Assuming the lever arms zt and zc of the components are approximately equal to h/2, i.e. one half oh the height of the connected beam, the factor γ can be defined as

SdRd

SdSd

Nh5.0MNh5.0M

++

=γ , (12)

and by substituting the eccentricity e, this can be simplified to

2he

MM

2he

Sd

Rd +

+=γ . (13)

Using the above described factor µ allows modelling of the moment-rotation curve of the joint, which is loaded by proportional loading, in the form [7]

∑+=

k1

zEee

eS2

0j

µ. (14)

-20 -10 10 20 30 Moment, kNm

100

200

300

-200

-300

-100

Normal force, kN

SN 1500

SN 1000

0

0

Method according to EN 1993-1-8

Fig. 7a) - Moment-normal force interaction diagram for SN joints and loading path of the experiments

4. EVALUATION ON TESTS

The proposed model was evaluated on tests carried out at Czech Technical University in Prague [12]. There were performed two tests of SN (beam-to-column) type and three tests of NN (beam splices). All five tests were loaded by bending moment with constant eccentricity. The eccentricity at the joint varied according to the total height v of the specimen, see Fig. 1. The eccentricities were 730 and 1083 mm for SN1000 and SN1500 joints and 244, 344 and 511 mm for NN750, NN1000 and NN1500 tests.

6

0

-400

-200

200

400

600

800

-40 0 20 40 Moment, kNm

Normal force, kN

NN 1000

Test NN 1500

Proposed prediction

-20

Test NN 750Method according to EN 1993-1-8

Fig. 7b) - Moment-normal force interaction diagram for NN joints

and loading path of the experiments

The bending moment and normal force were evaluated from the loading force FSd, including second order effect, see Fig. 7. The rotation was calculated from change of the total height v, which was measured during the test. The calculation takes into account axial shortening and bending of the elements, but the shear deformation was neglected.

The following pictures show comparison of experimental and calculated moment-rotation curves for the tests. The agreement of the calculated and measured resistance is very good in all cases. The SN joints failed by buckling of the unstifened column web in compression. This failure mode was proved by the calculation, showing the column web in compression as the weakest point of the joint. The NN joints failed by buckling of the beam flange in compression. This was also proved by the calculation of test NN750 when combination of forces from bending and compression lead to high force in flange, which limits the bending moment resistance. For the other NN tests, the calculation gives bending moment resistance limited by end plate in bending. This is caused by higher eccentricity of these tests. However, the bending moment resistance limited by beam flange in compression is very close.

0

5

10

15

20

25

30

35

0 0,01 0,02 0,03 0,04

Moment, kNm

Rotation, rad

Test SN 1000

Prediction

0

5

10

15

20

25

30

0 0,01 0,02 0,03 0,04

Moment, kNm

Rotation, rad

Test SN 1500

Prediction

35

Fig. 8 - Experimental results and calculated moment rotation curves for SN tests

7

0

10

20

30

40

50

60

0 0,01 0,02 0,03 0,04

Moment, kNm

Rotation, rad

Test NN 750

Prediction

0

10

20

30

40

50

60

0 0,01 0,02 0,03 0,04

Moment, kNm

Rotation, rad

Test NN 1000

Prediction

0

10

20

30

40

50

60

0 0,01 0,02 0,03 0,04

Moment, kNm

Rotation, rad

Test NN 1500

Prediction

Fig. 9 - Experimental results and calculated moment rotation curves for NN tests

5. SENSITIVITY STUDY

The following pictures show influence of the main parameters on moment-normal force interaction diagram. Both joints have different resistance of the tension part, when they are loaded by positive and negative bending moments resulting in non-symmetrical M-N diagrams. This is more significant for SN joints, which have only one bolt row at the bottom flange.

-30 -20 -10 10 20 30Moment, kNm

100

200

300

400

500

-200

-300

-100

Normal force, kN

t = 5,3 mmw

t = 6 mmw

t = 8 mmw

Fig. 10 - Influence of column web thickness, SN joints

8

The column web thickness tw has influence on resistance of the compression zone of the SN type joint, see Fig. 10, which gives higher bending moment resistance especially in cases when bending and compression is applied to the joint. On the contrary, change of the end plate thickness tp significantly changes only resistance in tension, see Fig. 11. Very small increase in compression is caused by bigger effective width of the column web in compression and has almost no effect on overall behaviour of the joint. Similar conclusions can be made from study on NN type joint, see Fig. 12.

-20 -10 10 20 30-100

100

200

300

400

-300

-400

-200

Normal force, kN

t = 8 mmp

t = 10 mmp

t = 20 mmp

Moment, kNm

0

0

Fig. 11 - Influence of end plate thickness, SN joints

-800

-600

-400

-200

0200

400

600

800

-40 -20 0 10 20 30 40 50 60Moment, kNm

Normal force, kN

t = 15 mmp

t = 10 mmp

t = 5 mmp

Fig. 12 - Influence of end plate thickness, NN joints 6. CONCLUSIONS Procedure in EN 1993-1-8 (2002) brings safe assumption of M-N interaction diagram of end plate beam-to-column connections. The procedure developed for base plates is applicable for beam-to-column connections with good accuracy of prediction of resistance and stiffness.

9

Check of resistance of beam flange in compression, which is included in beam-to-column joint resistance calculation, is convenient for practical application. In certain cases, it is more efficient to provide a separate check of section loaded by combination of bending and normal force. Acknowledgements The authors wish to dedicate this work to Mr. Martin Steenhuis, our good friend, who worked with us in the field of structural connections for years, and who died tragically last summer. This research has been supported by Grant COST C12.01 and Grant J01-98:210000001 of the Czech Ministry of Education, Youth and Sports. References [1] ENV - 1993-1-1/A2, Design of Steel Structures - General rules and roles for buildings,

Annex J, Joints in Building frames, European Prenorm, CEN, Brussels 1998. [2] Zoetemeijer P., Summary of the Research on Bolted Beam-to-Column Connections

(period 1978 - 1983), Rep. No. 6-85-M, Steven Laboratory, Delft 1983. [3] Zoetemeijer P., Proposal for Standardisation of Extended End Plate Connection based on

Test results and Analysis, Rep. No. 6-83-23, Steven Laboratory, Delft 1983. [4] Jaspart J. P., Etude de la semi-rigidite des noeudes poutre-colone et son influence sur la

resistance et la stabilite des ossatures en acier, MSMT, Liège 1991. [5] Wald F., Steenhuis M., The Beam-to-Column Bolted Joint Stiffness According to

Eurocode 3, in Proceedings of Workshop 1992 - COST C1, Strasbourg 1993, pp. 503 - 516.

[6] Jaspart J.P., Mac B., Cerfontaine F., Strength of joints subject to combined action of bending moments and axial forces, in Proceedings of the Conference Eurosteel ´99, Vol. 2, CTU Prague, Prague 1999, pp. 465 - 468.

[7] Steenhuis, C. M., Assembly Procedure for Base Plates, TNO Building and Construction Research Report 98-R-0477, Delft 1998.

[8] Wald F., Column Bases. CTU Prague, Prague 1995, p. 137. [9] COST C1, Column Bases in Steel Building Frames, ed. K. Weynand, Brussels 1999. [10] Wald F., Steenhuis C. M., Jaspart J. P., Brown D., Component method for base plates,

Journal of Constructional Steel Research, in printing. [11] Lima L., Da Silva L. Simões, Vellasco P., Andrade S., Experimental behaviour of flush

end-plate beam-to-column joints under bending and axial force, ECCS TC 10 report No. 516, Universidade De Coimbra, Coimbra 2001.

[12] �varc M., Wald F., Experiments with end plate joints, research report, COST C12, CTU Prague, Prague 2000.

[13] Wald F., �varc M., Experiments with end plate joints subject to moment and normal force, in Contributions to Experimental Investigation of Engineering Materials and Structures, CTU Reports No: 2-3, Prague 2001, pp. 1 - 13.

[14] Zoetemeijer P., A Design Method for the Tension Side of Statically Loaded Bolted Beam-to-Column Connections, Heron, No. 20 (1), Delft 1974.

[15] Tschemmernegg F., Huber K., Rahmentragwerke in Stahl under besonderer Berüchsichtigung den steinlosen Bauweise, Österreichisher Stahlbauverband, Wien 1987, p. 167.

10