Mofid_2001_On the Analytical Model of Beam-To-column Semi-rigid Connections

Thin-Walled Structures 39 (2001) 307–325www.elsevier.com/locate/tws

On the analytical model of beam-to-columnsemi-rigid connections, using plate theory

M. Mofid 1,a,*, M. Ghorbani Asl2,b, S.L. McCabe3,a

a Civil Engineering Department, University of Kansas, Lawrence, KS 66045, USAb Civil Engineering Department, Sharif University of Technology, Tehran, Iran

Received 10 October 2000; received in revised form 30 November 2000; accepted 22 December 2000

Abstract

An analytical method is presented that can be used to determine the behavior of a particularsteel beam-to-column extended end plate connection, in both linear and non-linear regions.This article demonstrates a closed form solution of the equations of deformation, for this typeof connection. Besides, a step-by-step analytical procedure for establishment of the linear partof M-q curve of this form of connection is developed. However, this technique can properlybe extended to the non-linear regions, which is not considered in this article. The correctnessof the results has been ascertained by a comparison, using non-linear finite element modelsas well as experimental approach; and very good agreement has been obtained. Furthermore,the writers believe that this method will efficiently serve design engineers in real design con-ditions. 2001 Elsevier Science Ltd. All rights reserved.

Keywords:Steel connection; Semi-rigid connection; End plate

1. Introduction

The problem of flexibility of beam to column connections and its effect onbehavior of steel structures has been of interest to numerous engineers and scientistsfor more than 80 years. At the University of Illinois in 1917, Young performed the

* Corresponding author. Fax:11-785-864-5631.E-mail address:[email protected] (M. Mofid).

1 Associate Professor.2 Research Affiliate.3 Professor & Chair.

0263-8231/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved.PII: S0263 -8231(01 )00002-7

308 M. Mofid et al. / Thin-Walled Structures 39 (2001) 307–325

Nomenclature

q Intensity of distributed loadq0 Intensity of linear loadE Modulus of elasticityG Shear modulusn Poisson’s ratioMt, M Total moment applied to connectionP Load applied to each boltPy Ultimate external load applied to end plateV0 Pure plastic shear force capacity per unit lengthM0 Pure plastic bending moment capacityFyb Yielding stress of boltsmt Distributed torsional momentmp Plastic moment capacity per unit lengthCW Warping constantw Transverse deflectionJ Torsional constant of plateg Angular strainn Number of bolts in each rowt Thickness of end platem Odd integer numbersq Rotation of plateA, B, C Constant coefficientsd Distance between centerline of beam top and bottom flangeslb Length of boltsAb Area of each boltMp, Mpy Combined plastic bending moment and shear forceVp, Vpy Combined plastic shear force

first experiment to estimate the rigidity of steel beam-to-column connections [1].Apparently this was the first investigation on semi-rigid joints.

In practical analysis and design of steel structures, two types of connections areconsidered; “completely rigid connections” and “simple connections”. However,according to the investigation on real behavior of the connections, any structuralbeam-to-column joint can be called semi-rigid. This means that even the rigid con-nections have a relative flexibility, which should properly be taken into account. In-plane bending moment, applied to semi-rigid connection, causes relative rotation ofbeam and column. The effect of this deformation has a significant destabilizinginfluence on the frame stability, since additional drift will occur as a result of thedecrease in effective stiffness of the members to which the connections are attached.An increase in the frame drift will intensify theP-D effect and hence, the overall

309M. Mofid et al. / Thin-Walled Structures 39 (2001) 307–325

stability of the frame will be affected. Therefore, in an exact analysis and designprocedure, effect of connection flexibility on the behavior and strength of the framemust be taken into account.

In recent studies, Jenkins et al. [1] conducted experiments on extended end plateconnections. Popov [2], experimented rigid connections to study the panel zonebehavior. Azzizinamini et al. [3] experimentally studied the behavior of the connectionswith bolted extended end plate, accompanied with web angles. He analytically calcu-lated the initial stiffness of the top and seat angle connections. As another analyticalapproach to this problem, Kishi and Chen [4] worked on connection with angle. Also,they were followed by Bahaari and Sherbourne [5], who produced a 2D plane-stressfinite element model for extended end plate connections. In 1994, they developed anapproximate 3D finite element model, using a computer program [6]. Additionally,some other researchers who have worked in this area are given in Refs. [7–12].

Experimental study of the connections is very expensive, and it is also not practicalto experiment all different types and sizes of connections. On the other hand, the3D finite element models are highly complicated and cannot be used easily to simu-late the connection as well as the whole structure. What is needed is a reliableapproximate model that can easily be constructed and be simple enough to be usedas a connecting element in non-linear analysis of steel structures in most computerprograms [13]. The objectives of this investigation are:

O To study the behavior of extended end plate connection, which is widely used inhigh-rise buildings and industrial structures, and to develop a step-by-step analyti-cal technique to prepare a bi-linear moment-rotation curve for this type of connec-tions.

O To verify the presented analytical solution by applying it to some test problems,and comparing the results with finite element models as well as experimental data.

O To inspect the important factors in the behavior of this type of connections throughan extensive parametric study.

2. Theory of the analytical model

The results of an experiment or an analytical model cannot be used directly inthe analysis and design of structures. Therefore, some numerical models are requiredto simulate these results. Usually, for this purpose, engineers use modeling tech-niques, such as bi-linear, piecewise linear, cubic spline, exponential, and power mod-els. The bi-linear model, as shown in Fig. 1, is found to be relatively satisfactoryfor simulation of theM-q curve of steel connections. Besides, it is simple enoughto be used by non-linear finite-element programs. It has to be noted that this modelcan be introduced through three parameters. These constants areKi, My andKs wherethey are defined as the initial stiffness, yielding moment and the secondary stiffnessof the connection, respectively.

In this article, the behavior of an extended end plate connection, as shown in Fig. 2,


Fig. 1. Bi-linear model for simulation of theM-q curve.

Fig. 2. Extended end plate connection.

has been studied. Besides, an extensive parametric study, based on bi-linear moment-rotation curve, is newly performed for this type of connection. The followingassumptions are made through this investigation:

O Beam and column are rigid.O Center of rotation of the connection, coincides with the middle of beam lower

flange.O Applied force from the beam upper flange is considered for the total plate width.O Center line of the bolts are assumed as a clamped edge for the plate.O Deformation of the bolts in the beam compression flange vicinity, is neglected.O Applied forces are equally divided between bolts, above and below of beam ten-

sional flange.


Fig. 3. Different modes of end plate deformation. (I): Transverse loading mode; (II): Torsional deflec-tion mode.

2.1. Calculation of the initial stiffness

To calculate the initial stiffness of the connectionKi the end plate deformationshould be considered. However, because of the complexity of this deformation, ithas been separated into the two simple parts, as shown in Fig. 3; these parts containdeformation caused by transverse loading and torsional deflection of the end plate,accordingly.

The idealized conditions for calculation of the transverse loading deformation isshown in Fig. 4. In this part, a simply supported plate with distributed load of inten-sity of q on its center is considered, Fig. 5. Solving the governing differential equ-ation of plates with the proper loading, Eq. (1), and imposing the clamped conditionfor y5±b/2 edges, results the maximum displacement of the plate’s center which ispresented in Eq. (2).

d4wdx4 12

d4wdx2dy21

d4wdy4 5

qD

(1)

Fig. 4. Idealized condition for transverse deformation.


Fig. 5. Simulation of transverse loading “simply supported plate with central loading”.

wmax5O`m51

(21)m21

2 (Km1Am)1a2

2p2DO`m51

(21)m21

2 am

tanh(am)m2cosh(am)

Em (2)

ParametersAm and Em can be found in Refs. [13] and [14]. Besides, applying theappropriate limits, Eqs. (9) and (10), the maximum deflection may be concluded asshown in Fig. 4.

Km54a4q

m5p5D(21)

m212 sinSmpu

2a D (3)

am5mpb2a

(4)

D5Et3

12(12n2)(5)

gm5mpv4a

(6)

Am5F(am,gm,Km) (7)

Em5G(am,gm,Km) (8)

5Lim wmax

ua

JP0 v JP

qo5qv

0(9)


5Lim wmax

ba

JP 0(10)

The first limit, Eq. (9), changes the square distributed load of intensityq, into alinear load and the second limit, Eq. (10), makes “ru” and “st” edges free. Theresult is:

wmax5b3qo

pDO`m51

(21)m21

2

48m55.231023b3

qo

D(11)

Comparison of the results of the above mentioned analytical approach, with approxi-mate beam model, reveals almost 10% difference. Besides, the shear deformation,consideringn50.30 and shear area55/6 of the net area, can be calculated as follows:

ws5Eb2

0

gdx50.78qob

Et(12)

Therefore, the total deflection of the plate center for the first part loading is:

5wt55.231023qob3

Et5qoF

F55.231023b3

D10.78

bEt

(13)

The idealized conditions for calculation of the second part loading, caused by tor-sional effect, is considered in Fig. 6. In this idealization, only half of the plate hasbeen taken into account. According to the theory, the governing differential equationof bending, caused by torsion for semi-thick plates, can be presented in the follow-ing form:

d4F

dx4 1GJ

ECw

d2F

dx2 5mt (14)

where the boundary conditions are:


Fig. 6. Idealized condition for torsional deformation.

d2F

dx2 50 @x5g

ECw

d3F

dx3 2GJdF

dx52M @x5g

dF

dx50 @x50

F50 @x50

(15)

General solution for Eq. (14) is

F5Asinh1! GJECw

x21Bcosh1! GJECw

x21Cx1D (16)

Satisfying all the boundary conditions and simplification of results reveals the differ-ence between deflection of two edges of the plate as follows:

5ww5q.d5S1

2D3MgGt3 SScosh(S)2sinh(S)

Scosh(S) D5MZ

S54.3gd

(n50.3)

Z53g

2Gt3SScosh(S)2sinh(S)Scosh(S) D

(17)


Deflection of the bolts is also calculated, using theory of solid mechanics:

wb5plbAbE

5Mtlb

2dnAbE(18)

Based on the virtual work theory, the initial stiffness of the connection “Ki”, canaccordingly be determined. For this calculation the stiffness of all the parts includingtraverse loading, warping action and bolt’s displacements have been combined andthe final value of “Ki” is:

Ki51

S 2k11k2

D1lb

2nd2AbE

(19)

where,

k15d2l

Fandk2

dZ

2.2. Calculation of the yielding moment

To calculate the yielding moment of the connection,My, the yield line theorybased on virtual work procedure is used [14,15]. The two most critical mechanismsand their appropriate interacted forcesVp andMp are investigated and are shown inFigs. 7–10, accordingly. For the first mechanism, the internal and external works aredetermined and the result is written in Eq. (22):

Fig. 7. Critical mechanism I, for the calculationMy1.


Fig. 8. Critical mechanism II, for the calculationMy2.

Fig. 9. Demonstration of the interacted forcesVp and Mp, for the mechanism I.

Fig. 10. Demonstration of the interacted forcesVp and Mp, for the mechanism II.

1IW54mp

l

b2

1gd1

d2b2

g 2 (20)

EW5Pyd5My

dD (21)


Therefore, theMy1 for first mechanism can be concluded as follows:

My154mpF2ldb

1g1d(2d2b)

2g G (22)

Similarly, using the same procedure from the second critical mechanism, Fig. 8, Eq.(23) can easily be concluded:

My25mp12l1ld9

b2

1N21nAbFyb1 Nd9

N1b2

1d92b22 (23)

2.3. Secondary stiffness of the connection

There is no analytical method for calculation of the secondary stiffness of connec-tions and usually test results are used to estimate the value ofKs. For example, theratio of Ks/Ki<0.15, is suggested for the top and seat angle connections by Marleyand Stelmack [1]. This ratio is resulted by test and includes the effects of strainhardening and changes in geometry of the connection. Therefore, in this article, thesuitable ratio will be proposed according to the test results.

2.4. Example problem

An extended end plate connection shown on Fig. 11, with various thickness of12 mm, through, 25 mm is considered. The parameters ofKi, My andKs are calculated

Fig. 11. Example problem, end plate with various thicknesses.


Table 1Results of the parametric study on the example problem, Fig. 11

t (mm) D3105 F31026 Z31027 K13108 K23106 Ki My1My2

12 3.3 30.7 14.7 5.55 20 2.47 7.1 11.315 6.5 16.8 7.5 10.15 39 4.82 11.1 11.720 15.4 8.1 3.2 21.2 92 9.06 19.7 12.525 30.1 4.8 1.6 35.8 180 13.26 30.8 13.6

accordingly. A complete parametric study against end plate thicknesses is presented.Results are shown in Table 1. Comparison of the results with Jenkins experimentaldata [1] and also analytical model of Bahaari and Sherboume [6] are shown in Figs.12–19. Besides, it has revealed an excellent agreement between two results.

The graphs show that values ofR5Ks/Ki<0.1 through 0.12, for thick plates; andR<0.12 through 0.15 for thin plates, are suitable. The effect of shear deformationon the behavior of plate is shown in Figs. 20 and 21. Also, Fig. 22 reveals thecontribution of each component of the connection in the total deformation; and Fig.23 shows yielding moments of both yielding lines mechanism, versus the plate thick-ness.

Fig. 12. Comparison of bi-linear model and experimental [1].






Fig. 16. Comparison of bi-linear model and 3D F.E model [6].


Fig. 17. Comparison of bi-linear model and 3D F.E model [6].

Fig. 18. Comparison of bi-linear model and 3D F.E. model [6].


Fig. 19. Comparison of bi-linear model and 3D F.E. model [6].

Fig. 20. Effect of shear deformation on total deflacation of the plate.


Fig. 21. Effect of shear deformation on initial stiffness of connection.

Fig. 22. Contribution of plate and bolt to rotation of end-plate connection.

3. Conclusion

In this study an analytical procedure for estimation of the behavior of extendedend plate connections has been presented. The theory in this method is mainly basedon the theory of plates. Substitution of the values of material property and dimensions


Fig. 23. Yielding moment of mechanisms I and II vs. plate thickness.

of the connection in the proposed formulas leads to formation of a bi-linear moment-rotation curve, which demonstrates the connection behavior and can be used by scien-tists and engineers through non-linear programs. Comparison of the results with theexperiment as well as the exact finite-element models reveals very good agreementbetween them. The important results of this investigation can be summarized as fol-lows:

O Effect of shear deformation on the stiffness of the end plate connection is insig-nificant and can be fairly neglected.

O Ration ofR5Ks/Ki varies between 0.1 and 0.12 for the thick plates and 0.12 and0.15 for the thin plates

O For thin plates, the first mechanism is critical and also the second mechanismgoverns the thick plates. This conclusion exactly complies with the study ofBahaari and Sherbourne [6].

O The initial stiffness of the connection is mostly affected by the plate thicknessand width, beam depth and the distance between centerline of the bolts.

O The yielding moment of the connection depends mainly on the plate thickness,distance between centerline bolts, the bolt area and materials.

References

[1] SSRC. Guide to stability design criteria for metal structures. 4th ed. New York: John Wiley andSons Publishers, 1998.


[2] Popov FP. Panel zone flexibility in seismic moment joints. Journal of Constructional Steel Research1987;8:91–118.

[3] Azizinamini A, Bradburn H, Radziminski B. Initial stiffness of semi-rigid steel beam-to-columnconnections. Journal of Constructional Steel Research 1987;8:71–90.

[4] Kishi N, Chen WF. Moment-rotation relations of semi-rigid connections with angles. Journal ofStructural Engineering, ASCE 1990;116(7):1813–34.

[5] Bahaari MR, Sherbourne AN. Computer modeling of an extended end-plate bolted connection. Com-puters and Structures 1994;52(5):879–94.

[6] Sherbourne AN, Bahaari MR. 3D simulation of end-plate bolted connections. Journal of StructuralEngineering, ASCE 1994;120(11):3122–36.

[7] Lui BM, Chen WF. Steel frame analysis with flexible joints. Journal of Constructional Steel Research1987;8:161–202.

[8] Chen WF, Kishi N. Semi-rigid steel beam-to-column connections: data base and modeling. Journalof Structural Engineering, ASCE 1989;115(1):105–19.

[9] Sibai WA, Frey F. New semi-rigid joint elements for non-linear analysis of flexibly connectedframes. Journal of Constructional steel Research 1993;25:185–99.

[10] Yau CY, Chan SL. Inelastic and stability analysis of flexibly connected steel frames by springs-in-series models. Journal of Structural Engineering, ASCE 1994;120(10):2803–19.

[11] Agha WY, Aktan HM, Olowokere OD. Seismic response of low- rise steel frames, Journal of Struc-tural Engineering, ASCE 1989;115(3):594–607.

[12] Chen WF, Liu EM. Stability design of steel frames. London: CRC Press, 1991.[13] Ghorbani Asl MH. Effect of joint flexibility on behavior of steel structures. Thesis presented in

Sharif University of Technology in partial fulfillment of the requirement for degree of master ofscience, Tehran, Iran, 1996.

[14] Timoshenko S, Krieger S. Theory of plates and shells. 2nd ed. New York: McGraw-Hill Pub-lishers, 1969.

[15] Timoshenko S, Gere J. Theory of elastic stability. 2nd ed. New York: McGraw-Hill Publishers, 1961.

Documents

Mofid_2001_On the Analytical Model of Beam-To-column Semi-rigid Connections