Nonlinear analysis of frames with flexible connections

Computers and Structures 79 (2001) 1097±1107

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Nonlinear analysis of frames with ¯exible connections Miodrag Sekulovic, Ratko Salatic * Faculty of Civil Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Yugoslavia Received 15 October 1999; accepted 16 October 2000

Abstract The e€ects of ¯exibility and eccentricity in the nodal connections of plane frames due to static loading are considered in this paper. A numerical model that includes both nonlinear connection behavior and geometric nonlinearity of the structure is developed. Two types of geometric nonlinear analysis are considered: with and without the bowing e€ect in¯uence. The sti€ness matrix for the beam with ¯exible eccentric connections is developed based on the analytical solution of the second-order analysis equations. The numerical model presented has the same number of degrees of freedom as the corresponding model in the conventional analysis used for the frames with fully rigid connections. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Steel structures; Plane frame; Semi-rigid connection; Eccentric connection; Nonlinear analysis; Bowing e€ect

1. Introduction Standard analyses procedures for frame structures are based on the assumption of the ideal nodal connections. Thus, two extreme idealizations for connections are used: perfectly rigid and ideally pinned. Models with ideal connections simplify the analysis procedure but often cannot represent the real structural behavior. This discrepancy is reported in numerous experimental investigations of steel frames with di€erent types of connections [1]. Making an ideal connection is very dicult, impractical and is not economically justi®ed. Hence, real connections are more or less ¯exible or semirigid. The ¯exibility of connections is the subject of many current building codes for steel structures such as British Standards, Eurocode 3, and the speci®cations of the American Institute of Steel Construction (AISC). In general, nodal connections of plane frames are subjected to the in¯uence of bending moments, axial forces and shear forces. The e€ects of axial and shear forces can usually be neglected, and only the in¯uence of

*

Corresponding author. E-mail address: [email protected] (R. Salatic).

bending moments is of practical interest. The constitutive moment±rotation relation, M±a, depends on the particular type of connection. Most experiments have shown that the curve M±a is nonlinear in the whole domain and for all types of connections. Therefore, modeling of the nodal connection is important for the design and accuracy in the frame structure analysis. There are several approaches on how to incorporate the ¯exibility of the nodal connections in the discrete analysis of the frames. The simplest and the most common are linear models. Often, many authors use the socalled corrective matrices to modify the conventional sti€ness matrices of the beams with full ®xity at both ends [2±7]. Elements of the corrective matrices are functions of the particular nondimensional parameters ± ®xity factors [2], or rigidity index [3]. In Ref. [8], such an approach is used in the context of the optimization of steel frames with ¯exible connections. In addition to the linear behavior, many papers focus on the nonlinear analysis of the static and dynamic behavior of the frames with ¯exible connections using di€erent models of geometric nonlinearity of elements and nodal connections [9±16]. In¯uence of ¯exibility and eccentricity in connections on the dynamic behavior of plane frames, within the linear theory, is investigated in Ref. [17].

0045-7949/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 1 ) 0 0 0 0 4 - 9

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M. Sekulovic, R. Salatic / Computers and Structures 79 (2001) 1097±1107

This paper deals with the static nonlinear behavior of plane frames with semi-rigid connections. Two types of nonlinearities are considered: geometric nonlinearity of the structure and material (constitutive) nonlinearity of the connections. These nonlinearities are interactive. The eccentricity of the connections is also considered. The sti€ness matrix and the vector of equivalent nodal forces for the prismatic beam with eccentric ¯exible connections are developed using the variational approach. The matrix of the interpolation functions is obtained applying analytical solutions of the governing di€erential equations second-order analysis, so that each beam represents one ®nite element. Therefore, an increase in the number of elements and degrees of freedom in the discrete analysis is avoided. Nodal displacements and rotations are chosen as the primary unknowns, while displacements and rotations of the element ends are eliminated. Thus, the number of degrees of freedom is the same as for the system with ideally rigid connections. The proposed sti€ness matrix is more general than the corresponding matrices obtained by other authors. Some of previously developed matrices are special cases of the herein developed matrix and can be obtained by its simpli®cation. Two models of the second-order analysis for the geometric nonlinear analysis of a member are considered: with and without the bowing e€ect. In many cases, such as orthogonal frames with rigid or ¯exible connections, di€erences in the accuracy between these two models are very small. In these cases, the simpler and more economical model without bowing is recommended to be used. However, in the case of other structures, such as shallow arches, the di€erence between these two models can be signi®cant. In that case, use of the model that includes the bowing e€ect is necessary. To describe the nonlinear behavior of the connections, di€erent models are used in research work. In this paper, the three-parameter power model is used [18,19]. Based on theoretical problem formulation, a computer program was written that covers the above mentioned models as well as the linear analysis of plane frames with ¯exible eccentric connections. The application of the program is illustrated by several numerical examples presented at the end of this paper. 2. Equilibrium equations for the beam with ®xed ends in the second-order theory Di€erential equations of the plane straight beam in the second-order analysis is developed from the principle of virtual work [11,20]. They can be expressed in the following form, with displacements as primary unknowns [21]: ‰EA…u;x ‡ 12v2;x

e0 †Š;x ˆ 0;

…1a†

‰EA…u;x ‡ 12v2;x

‰…EIv;xx ‡ j0 †Š;xx

e0 †v;x Š;x

p…x† ˆ 0; …1b†

where u and v are the axial and lateral displacements of the beam centerline, E is modulus of elasticity, A and I are cross-sectional area and moment of inertia, e0 and j0 are initial deformations (dilatation and ¯exure change), p…x† is lateral distributed load, while subscript … †;x denotes partial derivative with respect to x. Eqs. (1) with appropriate boundary conditions de®ne stress±strain state of the straight beam. These equations are nonlinear and coupled, and hence, an analytical solution cannot generally be obtained. Nonlinearity and coupling in Eqs. (1) is the consequence of the term 1/2v2;x in Eq. (1a), which introduces the in¯uence of bending to axial deformation (bowing e€ect). Relations between internal forces, axial forces N and bending moments M, with displacements u and v are given as: N ˆ EA…e

e0 † ˆ EA…u;x ‡ 12v2;x

M ˆ EI…j

j0 † ˆ

e0 †;

EI…v;xx ‡ j0 †;

…2a† …2b†

where: e ˆ u;x ‡ 12v2;x ; jˆ

…3a†

v;xx :

…3b†

From Eqs. (1a) and (2a) it can be concluded that the axial force N is constant along the beam axis, i.e., N ˆ EIk 2 ;

…4a†

where: k2 ˆ

A u;x ‡ 12v2;x I

 e0 :

…4b†

Substituting Eq. (4a) into Eq. (1b), in the case of prismatic beam …EI ˆ const† and compressive force, leads to: v;xxxx

k 2 v;xx ˆ

p…x† EI

…j0 †;xx :

…5†

As a result, simpli®ed Eqs. (4a) and (5) for the axial and lateral beam deformations are derived that are formally decoupled. Analytical solutions for these equations are readily obtained and can be found in the literature [11,21]. However, Eqs. (4a) and (5) are indeed coupled through the parameter k2 , which is the function of both displacement components. Based on Eqs. (4a) and (5), the sti€ness equations in the second-order analysis can be presented independently for the axial and ¯exural loads.

M. Sekulovic, R. Salatic / Computers and Structures 79 (2001) 1097±1107

1099

2.1. Axial sti€ness equations Integration of Eq. (2a) within the bounds of 0 and l leads to: Nˆ

EA …u2 l

u1 †

where: eb ˆ

D 1 ˆ l 2l

Z

EAe0

l

EAeb ;

…6†

v2;x dx:

0

…7†

In the above expressions, u1 and u2 are axial displacements of the beam ends, D is the shortening of the member due to bending, i.e. bowing e€ect [22], and eb is the axial strain due to bowing e€ect. From Eqs. (6) and (7), axial sti€ness equations become: Ra ˆ ka qa where:

Q0 

EA 2l

…8†

 1 ; 1

Q0 ˆ EAe0

Qb ˆ

Qb ;

Z 0

l

…9a† 

v2;x dx

1 1



 ˆ EAeb

 1 : 1

…9b†

In Eqs. (8) and (9) the following notation is used: Ra is the axial force vector, q is the vector of nodal displacements, ka is axial sti€ness matrix, Q0 and Qb are equivalent force vectors, from initial deformation and bowing e€ect, respectively. In the case when the bowing e€ect can be neglected, Eq. (8) reduces to the one in linear analysis. When the in¯uence of the bowing e€ect is signi®cant, it is necessary to determine equivalent compressive force vector Qb which, from Eqs. (7) and (9b) is the function of the unknown displacements v…x†. To solve this nonlinear problem, an iterative procedure is usually used. Di€erent models based on analytical and numerical solutions have been proposed [23]. 2.2. Flexural sti€ness equations Flexural sti€ness equations have the same form as in the linear theory: Rf ˆ kf qf

Qf ;

…10†

where subscript f denotes ¯exure. Eq. (10) can be expressed in several di€erent forms, some of which are of particular interest: the form obtained based on the analytical solution of Eq. (5) and the simpli®ed form obtained on the assumptions of the linearized second-order theory. In the former case, it is convenient to express force± displacement relations at the beam ends in the form given by Goto and Chen [11]:

Fig. 1. Nodal displacements and forces.

8 9 2 Vi > > > > < = 6 Mi ˆ6 4 Vk > > > > : ; Mk

12/1 6l/2 12/1 6l/2

6l/2 4l2 /3 6l/2 2l2 /4

12/1 6l/2 12/1 6l/2

38 9 6l/2 > vi > > < > = 2l2 /4 7 7 ui ; 6l/2 5> v > > : k> ; 4l2 /3 uk …11a†

where /i , i ˆ 1; . . . ; 4, are correction functions to multiply elements of the conventional sti€ness matrix. Those functions are trigonometric or hyperbolic depending on whether the axial force is compressive or tensile. Where N has zero value, all of these functions /i , i ˆ 1; . . . ; 4, reduce to 1 and the sti€ness matrix is the same as the one in the linear analysis. Analytical expressions for the functions /i and appropriate expansions in the power series form (convenient for the numerical analysis) can be found in Ref. [11]. The convention for the positive direction of the nodal displacements and forces is shown in Fig. 1. It is repeatedly pointed out, that the relations between forces and displacements at the end of the beam (Eq. (11a)) are nonlinear since functions /i depend on the axial force which is also a function of nodal displacements and rotations. In the case of linearized second-order analysis, the sti€ness matrix has the form: kf ˆ k0 ‡ kg ;

…11b†

where k0 is the conventional sti€ness matrix and kg is the geometric sti€ness matrix. The sti€ness matrix kf in the form of Eq. (11b) can be obtained if polynomial interpolation functions are used instead of the analytical solutions of Eq. (5) expressed in the form of trigonometric or hyperbolic functions.

3. Sti€ness equations for a member with ¯exible eccentric connections 3.1. Shape functions Fig. 2 shows straight beam with ¯exible nodal connections which are eccentric in the nodes 1 and 2 with eccentricities e1 and e2 , respectively. The relation be~ i and rotations of the nodes ui , as tween end rotations u shown in Fig. 2, can be expressed as:

1100

M. Sekulovic, R. Salatic / Computers and Structures 79 (2001) 1097±1107

where: D ˆ …1 ‡ 4g1 /3 †…1 ‡ 4g2 /3 † gi ˆ

EI ; lci

4g1 g2 /24 ;

i ˆ 1; 2:

…17a† …17b†

Substituting Eqs. (13) and (16) into Eq. (15b), the additional end rotation force vector a can be expressed through the end displacements vector ~ q: a ˆ S~ q; where: 2

0 16 s21 6 Sˆ 4 D 0 s41 Fig. 2. A beam with ¯exible and eccentric connections.

ai ;

i ˆ 1; 2;

…12†

where ai are the additional end rotations due to ¯exibility of connections which are modeled with rotational springs. Rotations ai depend on the springs sti€ness ci ~ i at the ends of the beam as: and bending moments M ai ˆ

ei M ; ci

i ˆ 1; 2:

…13†

Starting from Eq. (11a) and taking into account Eq. (12), end moments can be expressed in the following way: 8 9 v~1 > > >    > < = e1 EI 6/2 4l/3 6/2 2l/4 u1 a1 M ˆ e2 v~2 > l2 6/2 2l/4 6/2 4l/3 > M > > : ; u2 a2 ˆ H…~q

a†; …14†

where:  ~ qT ˆ v~1

u1

aT ˆ f 0 a1 (

v~2

u2 ;

0 a2 g:

Substituting Eq. (13) into Eq. (14) leads to: )   e1 2g2 /4 M EI 1 ‡ 4g2 /3 ˆ 2 e2 Dl 2g1 /4 1 ‡ 4g1 /3 M 8 9 v~ > > > > 1> > >  > < 6/2 4l/3 6/2 2l/4 u1 =  ; 6/2 2l/4 6/2 4l/3 > > v~2 > > > > > > : ; u2

…15a† …15b†

…16†

0 s22 0 s42

0 s23 0 s43

3 0 s24 7 7; 0 5 s44

…19†

whose nonzero elements are de®ned as: s21 ˆ

~ i ˆ ui u

…18†

s23 ˆ 6l ‰g1 ‡ 2g1 g2 …2/3

 s22 ˆ 4 g1 /3 ‡ g1 g2 …4/23

/4 †Š/2 ;

 /24 † ;

…20b†

s24 ˆ 2g1 /4 ; s41 ˆ

…20a†

…20c†

s43 ˆ 6l ‰g2 ‡ 2g1 g2 …2/3

/4 †Š/2 ;

s42 ˆ 2g2 /4 ;

…20d† …20e†

 s44 ˆ 4 g2 /3 ‡ g1 g2 …4/23

 /24 † :

…20f†

The function describing lateral displacement v…x†, for the element with ¯exible connections, can be written in the usual way using interpolation functions and the nodal displacements vector: v…x† ˆ N…x†…~ q

a† ˆ N…x†…I

S†~ q;

…21†

where: N…x† ˆ ‰ N1 …x† N2 …x† N3 …x† N4 …x† Š

…22†

denoting the matrix of interpolation functions obtained based on the analytical solutions of the second-order analysis, already presented in Ref. [21]. The e€ect of the connection eccentricity, due to the size of the nodal connections, is usually neglected in the analysis and design of the plane frames. However, it may in¯uence structural behavior. In Fig. 2, the eccentricity is modeled by short, in®nitely sti€ elements whose lengths are e1 and e2 . In the case of small rotations, relations between end displacements v~1 and v~2 and nodal displacements v1 and v2 , can be expressed in the matrix form as: ~ q ˆ …I ‡ E†q;

…23†

M. Sekulovic, R. Salatic / Computers and Structures 79 (2001) 1097±1107

where: 2

qT ˆ f v1

u1

3.2. Semi-rigid connection modeling

3

0 0 0 0

0 e1 60 0 Eˆ6 40 0 0 0

0 0 7 7; e2 5 0 v2

…24a†

u2 g:

…24b†

Substitution of Eq. (23) into Eq. (21) leads to: ~ v…x† ˆ N…x†…I ‡ G†q ˆ N…x†q; where

2

Gˆ… S‡E

0 16 g21 6 SE† ˆ 4 D 0 g41

…25†

De1 g22 0 g42

0 g23 0 g43

3 0 g24 7 7: De2 5 g44 …26†

Elements of the matrix G are de®ned as: g21 ˆ g22 ˆ

g24 ˆ g41 ˆ

g23 ˆ

6 ‰g1 l

‡ 2g1 g2 …2/3

/4 †Š/2 ;

…27a†

6e1 ‰g1 l

‡ 2g1 g2 …2/3 /4 †Š/2   4 g1 /3 ‡ g1 g2 …4/23 /24 † ;

6e2 ‰g1 l

‡ 2g1 g2 …2/3

g43 ˆ

g42 ˆ

6e1 ‰g2 l

g44 ˆ

6e2 ‰g2 l

6 ‰g2 l

/4 †Š/2

‡ 2g1 g2 …2/3

‡ 2g1 g2 …2/3

1101

…27b† 2g1 /4 ;

/4 †Š/2 ;

/4 †Š/2

‡ 2g1 g2 …2/3 /4 †Š/2   4 g2 /3 ‡ g1 g2 …4/23 /24 † :

2g2 /4 ;

…27c† …27d† …27e†

…27f†

If in Eqs. (27) functions /i , i ˆ 1; . . . ; 4, are replaced by 1, the corresponding expressions in the linear theory are obtained [17]. Eq. (25) treats lateral displacements v…x† of the element centerline, for the case with ¯exible connections, in the same manner as for the element with rigid connections. The only di€erence is in the matrix of the interpolation functions, which is in this case obtained by modi®cation of the matrix of the interpolation functions for the elements with rigid connections. Correction matrix G, de®ned by Eq. (26), contains the e€ects of connection ¯exibility and eccentricity: separately (®rst and second term) and coupled (third term). If in Eq. (26) S and E are replaced with 0, it follows that G ˆ 0, and Eq. (25) is reduced to the well-known equation for the member with rigid connections.

Based on numerous experimental results, it is recognized that the behavior of ¯exible connections, described through the M±a curve, is nonlinear in the whole domain for almost every connection type. Based on these results, a variety of models for analytical expressions of M±a line is proposed. In this paper, the three-parameter model proposed by Richard and Abbott [18] and Kishi et al. [19] is used. For di€erent commonly used semirigid connection, this model can be formulated as: ci a M ˆh  n i1=n ; 1 ‡ aa0

…28†

where ci is the initial connection sti€ness, n is the shape parameter, a0 ˆ Mu =ci is the reference plastic rotation (Fig. 3), and Mu is the ultimate moment capacity. From Eq. (28), tangent connection sti€ness ct and relative rotations a, can be directly obtained as: ct ˆ

dM ci ˆh  n i…n‡1†=n ; da a 1 ‡ a0

aˆ h ci 1

M  n i1=n : M Mu

…29a†

…29b†

Spring sti€ness, modeled by Eq. (28), is de®ned as the tangent slope to the M±a curve. It decreases with the increase in the moments (loads) starting from the initial value of ci and decreasing to 0 when M ˆ Mu . The values for the initial connection sti€ness and ultimate moment capacity Mu and for the shape parameter are usually obtained from empirical expressions that depend on the type of the connection. These expressions are developed using the results of the experimental investigations, curve ®tting techniques and the steel connection database developed by Chen and Kishi [24]. For the unloading phase, elastic behavior of the connection independent of the load level is assumed. The connection sti€ness for unloading phase is approximately equal to the tangent sti€ness for the loading phase (Fig. 3). Initial sti€ness is one of the essential parameters in the analysis of ¯exible connections, and its relation with ®xity factors ci is given by expression:   3EI ci : …29c† ci ˆ 1 ci l Since initial sti€ness varies from 0, in the case of pinned connections, to 1 for the case of fully rigid connections, it is more convenient to express the ¯exibility e€ect through the ®xity factors whose values are normalized from 0 to 1.

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M. Sekulovic, R. Salatic / Computers and Structures 79 (2001) 1097±1107

Fig. 3. Three-parameter power connection model.

3.3. Sti€ness matrix and equivalent nodal force vector Sti€ness matrix for the beam element with ¯exible connections through the total potential energy, can be written as: Z 2 X 1 l …EAe2 ‡ EIj2 † dx ‡ ci a2i : …30† Uˆ 2 0 iˆ1 The integral term in the above equation is the potential energy of the beam and the sum refers to the potential energy of the springs. Substituting Eq. (3) into Eq. (30), and taking into account Eq. (4a), the following can be obtained for the straight prismatic element: Z Z 2 X EI 2 4 l EI l 2 Uˆ k dx ‡ v;xx dx ‡ ci a2i 2 0 2A 0 iˆ1 ˆ Ua ‡ Uf ‡ Us ;

…31†

where: Ua ˆ

EI 2 4 k l; 2A

EI Uf ˆ 2 Us ˆ

Z 0

l

v2;xx dx;

2 X ci a2i ;

…32a†

…32b†

…32c†

iˆ1

denoting deformation energy of the beam, axial (Ua ) and ¯exural (Uf ), and potential energy of the springs (Us ).

Potential energies due to axial deformation and bending are coupled, since parameter k2 , from Eq. (4b), includes derivatives of both axial and lateral displacements. With the assumption that k 2 ˆ const:, internal energy of the beam with ¯exible connections corresponding to the bending, can be expressed independently of the energy corresponding to the axial deformation. Thus, after substituting Eq. (25) into Eq. (32b), the following can be obtained: 1 Uf ˆ qT …I ‡ G†T 2

Z 0

l

h i  T 00 00 EI …N …x†† N …x† dx

 …I ‡ G†q;

…33†

or in other form: Uf ˆ 12qT …kII ‡ kef †;

…34†

where matrices kII and kef are de®ned as: Z lh i …N00 …x††T N00 …x† dx; kII ˆ EI

…35a†

kef ˆ GT kII ‡ kII G ‡ GT kII G;

…35b†

0

denoting beam sti€ness matrix with the rigid connections according to the second-order analysis and correction matrix that accounts for the e€ects of ¯exibility and eccentricity respectively. Elements of the matrix kII , which is nonlinear, are computed from the analytical solutions of the di€erential equations of the second-order analysis, Eq. (5). In

M. Sekulovic, R. Salatic / Computers and Structures 79 (2001) 1097±1107

particular cases, the simpli®ed form of this matrix corresponding to the linearized second-order analysis and presented by Eq. (11b) can also be used. Potential energy of the springs, Eq. (32c), can be expressed in the following matrix form:

The equivalent generalized end force vector due to distributed loads along the beam, p…x† is obtained in the usual manner: Z l Z l Qˆ p…x†NT …x† dx ˆ …I ‡ G†T p…x†N…x† dx: …41† 0

Us ˆ 12 aT Ca; where: 2

0 0 6 0 c1 Cˆ6 40 0 0 0

…36†

3 0 0 0 07 7: 0 05 0 c2

…37†

Us ˆ

ˆ

1 T q ks q; 2

…38†

where: ~ T CG; ~ ks ˆ G 8 0 > >
9 > > =

…39a† 2

0 6 g21 1 1 ~ ˆ G ˆ 6 > 0 > D4 0 > : > ; g2 g41

0

Components of the vector Q, for some simple load distributions and temperature change are given in the closed form by Sekulovic and Malcevic [21]. In general case, elements of the vector Q are computed numerically. 4. Numerical examples

Substituting Eq. (18) into Eq. (36), and considering Eq. (23), leads to: 1 T ~T ~ q G CGq 2

1103

0 g22 0 g42

0 g23 0 g43

3 0 g24 7 7: 0 5 g44

…39b†

From Eqs. (34) and (38), the total potential energy due to the bending for the beam with ¯exible and eccentric connections can now be written as: U ˆ Uf ‡ Us ˆ 12qT …kII ‡ kef ‡ ks †q:

…40†

Based on theoretical observations presented in previous sections, a computer program has been developed and numerical analysis of plain single bay frames with di€erent number of storeys has been performed. For illustration, only the results of two simple cases are presented herein: a simple portal frame and a two-storey single bay frame (Fig. 4). In both cases, ideal connections (rigid and pinned) as well as two types of semi-rigid beam-to-column connections (DWA ± double web angle and TSDWA ± top and seat angle with double web angle) were considered. For the semi-rigid type of connections, database developed by Chen and Kishi [24] was used. Characteristic results for the horizontal displacements at the top and bending moments at the base of the frames corresponding to the ideal and semi-rigid connections according to the linear and three di€erent levels

Fig. 4. Simple portal frame and two-storey simple bay frame.

1104

M. Sekulovic, R. Salatic / Computers and Structures 79 (2001) 1097±1107

of approximations in the second-order theory are presented in Tables 1 and 2. In addition, the critical load of bifurcation stability is also given in the table. For the case of connection eccentricity, the corresponding results are presented in Table 3. From these tables, it is clear that there is a signi®cant di€erence between results for the frames with ideal and with semi-rigid (DWA and TSDWA) connections. In¯uence of the second-order theory can be seen from the di€erence between the results obtained using linear and second-order analysis. Also, it should be noted that di€erent levels of approximation in the second-order analysis do not produce the di€erence of practical interest. Bowing e€ect is negligible in both connection cases: ideal and semi-rigid. Change in horizontal displacement of node 2 and bending moment in node 1 as a function of ®xity factors between the column and the beam for the di€erent load levels are shown in Figs. 5 and 6, respectively. Displacements and moments are normalized by dividing their values with the corresponding values for the pinned type of connections. It can be concluded that both moment and displacement lines have the same character ± they have lower values for the higher values of ®xity

factors. In the case of linear analysis, the results decrease independently of the load level, while in the case of the second-order analysis the decrease is greater for the higher load levels. Fig. 7 shows the di€erence (in percent) in displacements and bending moments obtained by the secondorder analysis and linear analysis for two di€erent load levels (P ˆ 100 kN and P ˆ 400 kN). These di€erences are signi®cant for the case of the higher load (for DWA gv ˆ 30:86%, gm ˆ 26:67% for ®xed connections gv ˆ 25:86%, gm ˆ 22:3%) and it should be taken into account during the design of frame elements. Fig. 8 shows the change in the critical load of bifurcation stability for the portal frames with di€erent number of storeys as a function of ®xity factors. Critical force is normalized by dividing it with the corresponding results for the case of fully ®xed joints. It can be concluded that critical force increases with the increase in the ®xity factors almost linearly (for the portal frame from 0.33Pcr to 1.0Pcr , for the three-storey single bay frame from 0.05Pcr to 1.0Pcr , where Pcr is the critical load for the same frame with ®xed joints). It is clear that frames with semi-rigid connections can stand signi®cantly higher loads before they buckle than

Table 1 The simple portal frame (P ˆ 450 kN, H ˆ 0:005P , e1 ˆ e2 ˆ 0) Type of connection

Rigid TSDWA DWA Pinned

l n l n

Displacement of node 3 (10

4

m)

Bending moment of node 1 (kN m)

First order

Second order Simpli®ed

Without bowing

With bowing

25.79 28.70 28.77 30.95 34.25 75.73

36.38 42.34 42.59 47.41 61.16 868.69

36.42 42.39 42.65 47.49 61.36 936.21

36.42 42.39 42.65 47.49 61.36 957.19

Critical load

First order

Second order Simpli®ed

Without bowing

With bowing

2.524 2.639 2.642 2.728 2.863 4.503

3.377 3.665 3.678 3.910 4.575 43.591

3.376 3.663 3.675 3.908 4.575 46.629

3.376 3.663 3.675 3.908 4.575 47.674

1530 1395 1383 1289 1289 489

Table 2 The two-storey simple bay frame (P ˆ 100 kN, H ˆ 0:005P , e1 ˆ e2 ˆ 0) Type of connection

Rigid TSDWA DWA Pinned

l n l n

Displacement of node 5 (10

4

m)

Bending moment of node 1 (kN m)

First order

Second order Simpli®ed

Without bowing

With bowing

23.35 27.85 27.91 31.51 35.06 176.61

25.45 31.10 31.18 35.78 41.03 945.63

25.45 31.10 31.18 35.78 41.04 947.39

25.45 31.10 31.18 35.78 41.04 925.41

Critical load

First order

Second order Simpli®ed

Without bowing

With bowing

1.171 1.239 1.240 1.292 1.344 3.001

1.248 1.335 1.336 1.405 1.485 12.457

1.248 1.335 1.336 1.405 1.485 12.474

1.248 1.335 1.336 1.405 1.485 12.665

1115 921 984 806 802 122

M. Sekulovic, R. Salatic / Computers and Structures 79 (2001) 1097±1107

1105

Table 3 The simple portal frame (p ˆ 250 kN/m) Type of connection

Eccentricity

Rotation of node 3 (10 First order

3

m)

Bending moment of node 1 (kN m)

Second order

First order

Simpli®ed

Without bowing

Second order Simpli®ed

Without bowing

Rigid TSDWA-l DWA-l

0.0 0.0 0.0

146.94 132.81 122.63

165.55 148.67 136.89

166.46 149.83 137.64

232.09 209.78 193.69

278.15 249.79 230.00

281.87 253.12 233.08

Rigid TSDWA-l DWA-l

0.1 0.1 0.1

149.55 137.14 128.22

167.52 152.82 142.53

168.46 153.58 143.26

236.24 216.64 202.55

281.11 256.27 239.03

284.58 259.44 242.00

Rigid TSDWA-l DWA-l

0.2 0.2 0.2

151.32 140.62 132.96

168.70 155.95 147.15

169.47 156.67 147.84

239.06 222.16 210.06

282.38 261.04 246.30

285.59 264.02 249.13

Fig. 5. The in¯uence of connection ¯exibility on the horizontal displacement.

Fig. 6. The in¯uence of connection ¯exibility on the bending moment at the base.

Fig. 7. The di€erence in (a) displacements and (b) moments obtained by second-order and linear analyses.

1106

M. Sekulovic, R. Salatic / Computers and Structures 79 (2001) 1097±1107

the system signi®cantly decrease with the increase in ¯exibility of joints. As expected, the numerical results show that the bowing e€ect does not have a practical in¯uence in the analysis of orthogonal frames. However, the eccentricity of connections may have a practical in¯uence depending on the type and size of the connection.

References

Fig. 8. The in¯uence of connection ¯exibility on the critical load.

the frames with pinned connections, but also signi®cantly lower than the frames with fully rigid connections. 5. Conclusion This paper deals with the static nonlinear behavior of plane steel frames with ¯exible and eccentric connections. A numerical model that includes both nonlinear connection behavior and geometric nonlinearity of the structure is developed. The sti€ness matrix and the vector of the equivalent nodal force for the prismatic beam with eccentric ¯exible connections are developed using analytical solutions of the governing di€erential equations of the second-order analysis. The proposed sti€ness matrix is more general than the corresponding matrices previously obtained by other authors. To describe the nonlinear behavior of the connections, the three-parameter power model is used. A computer program is developed for the analysis and design of steel frames. Based on the above investigations and the numerical results it can be concluded that the connection ¯exibility has signi®cant in¯uence on the behavior analysis of the frames. Both the e€ects contribute to signi®cant increase in the point displacements and to change in the distribution of internal forces in the system. The in¯uence of the geometric nonlinearity increases with the load. It is higher when semi-rigid type of connections are used than in the case of fully rigid connections. It is also observed that the critical load and buckling capacity of

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