Paper 1994-Nonlinear Finite Element Analysis of Three Dimensional Frames

8/10/2019 Paper 1994-Nonlinear Finite Element Analysis of Three Dimensional Frames

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Nonlinear finite element analysis of three dimensional frames

M . S AVA R DD irec t io n r le s s t rl ic tl lr e s, Mir z is tt r e rl es T ~ N I Z S ~ O I . I S~ i ~ i t b e c , ~ ie D ec ,QC G I K 5Z 1, Crrrzrrdn

A N D

D . BEAULIEU N D M . FAFARDDiprrrterr~etztcle girzie civil, Utziversite Lnvnl,Saitzte-Fay, QC G I K 7P 4, Crrrzrrdn

Received March 25, 1991Revised manuscript accepted November 9, 1993

This paper presents a numerical model for the nonlinear analysis of three-dimensional frames using the finiteelement method. The model is based on a general variational formulation for thin-walled beams with open or closedsections; the mathematical development uses an updated Lagrangian description for shallow arches, taking intoaccount geometrical and material nonlinearities, residual stresses, member initial imperfections, warping, and con-nection flexibility. The model has been used to evaluate second-order effects and the influence of residual stressesand joint flexibility on the behavior of plane fram es. A se ~n irig id pace frame has been analyz ed and the gainin rigidityoffered by a horizontal rigid diaphragm has been evaluated.

K ey buorr1.s: analysis, connection, finite element, nonlinearity, residual stress, thin-walled beams, warping.

Cet article prCsente un modele numCrique d'analyse non IinCaire par ClCments finis permettant de simuler leco~n porte men t es cadres mktalliques tridimensionnels. Ce modkle est bas6 sur le dCveloppement d'une formulationvariationnelle gCnCrale des poutres parois minces sectio ns ouvertes ou fermkes, en description lagrangienneactualisee. De plus, le modele permet de tenir compte de la non-IinCaritC gComCtrique, de la plasticitC, des contraintesrCsiduelles, des dCfauts de rectitude des pieces, du gauchissement des sections et de la rigidit6 partielle des assem-blages. Ce modele aC t C utilisC pour Cvaluer I'importance des effets du deuxikme ordre et des contraintes rCsiduellesainsi que I ' influence de la rigidit6 imparfaite des assemblages pour des cadres plans. De plus, nous avons puapprCcier I'apport de rigidit6 qu'offrei l n diaphragme rigide horizontal dansu n cadre rigide simple tridimensionnel.

Mots cle s analy se, assem blag e, contr aite rCsiduelle, ClCment fini, gauchiss eme nt, non-IinCaritC, poutres paroisminces.

Can 1. Civ. Eng. 21. 461 170 1994)

ntroductionDesign considerations for inembers in three-dimensional

f rames inc lude a good knowledge o f how these memberswill behave under the applied loads. Thus, i t is essential toa d e q ~ ~ a t e l yvaluate the intensity of each m ember 's internalforces. For this purpose, a complete analysis of the structureshould take into account the following factors: combinationof axial and shear forces, along with bending and twistingmoments; P-h effects (geometric nonlinearity); elastoplasticbehav ior o f the mater ia l ; connec t ion f lex ib i l i ty ; res idua lstress distr ibution ; and contr ibution of r igid diap hragm s tothe structure 's overall r igidity.

To perform the analysis in question, the numerical modeldeveloped by Ako ussah e t a l . 1987) was genera l ized byintroducing new section geometries, i.e., rectangular, circular,open, and closed sections. Also, a new f inite element wasd e v e lo p e d to s im u la t e t h e n o n l in e ar b e h a v io r o f f l e x ib l econnections.

The numerical model was validated using several examples,each one covering a particular aspect of structural behavior.Example checks the p roposed model ' s accuracy fo r s im-~ ~ l a t i n glexibly connected frames. Example2 compares ournumerical results with experimental measuiemen ts taken ona plane frame. Example3 studies the influence of a horizontaldiaphragm in a one-story space frame.

Mathematical developmentUpon application of a load, the stress to which a solid is

su b je c t e d c a u se s a s t a t e o f e q u i l i b r iu m b e twe e n e x t e rn aa n d in t e r n a l f o r c e s . T h i s e q u i l i b r iu m c a n b e d e t e r m in e du s in g th e we ig h te d r e s idu a l m e th o d a n d th e p r in c ip l e ovirtual work. The result of integrating over the solid's voluma w eighting function +) multiplied by the differential equationsof equ i l ib r ium nus st equal zero .

Using a weighting function composed of vir t~ial isplacements kinematically admissible w ithin predefined boundarcondit ions, we can determine the variationalformulation ofthe elastic problem in terms of the solid's unknown configura t ion . The cho ice o f an updated Lagrang ian descr ip t ionmakes the resolution possible. Th e Lagrangian description itermed ~ipdatedwhen the integration domain is defined as thel as t eq u i li b riu m c o n f ig ~~r a t io ne termined .

We have used C to represent the known equilibrium con-f iguration (or current configuration) at the beginning of load step and w , o represent i ts corresponding variationalform. The weak integral form of the variational formulationof the elastic problem of the current configuration may beexpressed as (Fafard e t a l . 1989)

NOTE:Written discussion of this paper is welcomed and will be whe re ( *I) s the Green-Lagrange first variation deformationreceived by the Editor until October 31, 19 94 (addre ss inside tensor, (s') is the second Piola-Kirchhoff stress tensor,u * )front cover). is the virtual displac emen t vector, (fd) is the force per uniI r~olcd n C a n ; ~ d ; t Impilur6 ;m C:tn;td;t


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FIG.1 . Member cross section: ( a) typical cross section;b) displacements in the plane of the cross section.

of volume vector, and f,l) is the external force per unit of ssumpt ion 3 . Shear s t ra ins in the p lanes normalsur face vec to r. Equat ion [ l ] c o m p le te ly d e f in e s t h e e q u i - the middle surface of thin walls are negligible.l ibr ium condit ions for a solid to which surface and volume Using the equil ibr ium of the longitudinal forces on a

loa ds are applie d. ferential portion of the wall, we obtain (Nish ino et al. 1To evaluate the unknow n configurationc , or the increment av aw a @

of displacement { u : that occurs when the load is increased [6] ~ ( x , ) = u y- z- w-from configuration C 1 o configuration C2, limited Taylor ax ax axseries of the integral formW : was developed . The f i r s t two a @t e rm s o f t h is s e ri e s a r e [71 y , , - a - ax

[2] W W ; W ; I ... = 0 where @ is the angle of twist. The warping function,w, anthe coefficient can be evaluated using [8] and [9] for o

To solve this equation, the Newton-Raphson method w as se ct io n s a n d [ l o ] a n d [ 111 fo r closed sections.used wi th matr ix t r iangu lar iza t ion a t each i te ra t ion . Th isiterative method lends itself well to an elastoplastic analysis [8] w = dsrequ i r ing knowledge o f the change in s t resses and defor-m a ti on s a t e a ch l oa d in g s te p . [ 9 ] a = - 2 5

Kinematics of open th in -wal led beamsDetails on the developments in this section can be found

in Akoussah et al. (1987), Vlassov (1961), and Nishino et al.(1973).

The following relations describe the displacementsu, v, andw for open and closed sections along the X,Y a n d Z axesrespectively. Shear and axial forces, bending and twistingm o m e n t s , a n d c r o ss - se c t io n a l wa r p in g c a n b e d e v e lo p e dsimultaneously.

Based on the cross section of a thin-walled member shownin Fig . 1 , the fo l lowing can be assu med:

s su m p t io n 1. Cross sec t ions do no t d is to r t in the i rown plane.

A slight angle of twist , using the shear center of coordi-nates z and yo as the center of rotation, an d point o as theorigin of the coordinate system, results in

[41 v(x, s) = v,(x) (z z,)@(x)

To evaluate the axial displacement11 and the correspondingdeformat ion yn ,?, he fo l low ing must be assum ed:

ssumpt ion 2 . Shear s t ra ins due to shear s t resses inequilibrium with the changes of normal stresses are negligible.

T h e v e c t o r i a l f o r m o f t h e d i s p l a c e m e n t f i e l d m a y expressed as

Th e kinematic description of any point on the beam crsection can be obtained by projecting the Cartesian displaments in the curv i l inear base(x, s , 5 ) in o rder to use tbeam s properties and to introduce the Marguerre approto take in to accoun t the f la t tened beam theory (Akouse t a l . 1 9 8 7 ) . We h a v e c a l l e d v a n d v t h e d i sp l a c e m e nalong the s and 5 axis respectively (Fig. la) . The kinemadisplacements of point q can be described using the l imiTay lor se r ies deve loped around po in t p o f coord ina tes xand 5 = 0 on the middle surface of the cross section shoin Fig. 1 . Based on the Kirchhoff-Love assumption:


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FIG.2. External forces acting on a member.

I.1 [16] (d,) = ( u , v l W , @ , o l oZl @; u2 v2 w2 @2 0).2 0 : ~@With an updated lagrang ian fo rmula t ion , the express io

of the internal vir tual work evaluated for the configuratioC for each elementary subdomain becomes (Akoussah et a

Y+ 1 5 ) ~ Z + tn5)vv + o ~ r ; , ) @ , , ~ ] 1987)

mv, vv, r, +c @l l~, 77W, + ti,@

whereI; = m(y Y,,) l(z 2,)

b = l(y - y o ) m(z z,)

In these equations, 1 and are the cosin e directors betweenthe (y, z and (s, 5) coordinate systems. Equation 1131 com -pletely describes the real and vir tual kinematic f ields of anarbitrary point on the beam cross section in the curvil inearsystem (x, s , 5 .This equation is used in the development ofthe express ions o f the in te rna l v i r tua l work , the ex terna lvirtual work, and the tangent stiffness matrix necessary withthe i terative procedure chosen for the resolution. These aregiven in Appendix 2.

in i te e lements

Bent ~i le tnen tIf a one-dimensional cubic element is selected for bending

and torsion and a one-dimensionalC linear element is chosenf o r th e a x i a l f o r c e , t h e d i s p l a c e m e n t v e c t o r { u }c a n b eapprox imated by the fo l lowing func t ions (Akoussah e t a l .1987):

where { M f represen ts the resu l tan t fo rce vec tor in tegra teover the en t i re sec t ion , and [B,] represen ts the degrees of reedom-to-deformat ion t ransformat ion matr ix .

The elementary external virtual work in the current configura t ion may be expressed as

wh e r e

is the vector of the external forces acting on the thin-wallemember. F igure 2 i l lus t ra tes these fo rces in the i r posi t ivsense of action.

The rate of the internal vir tual work m ay be expressed a

with

[201 [K,l = [K,,I + [Kg]

where {d,} is the nodal displacement vector: [221 [Kg] = jL ([B,,, I ~ I D ~ ~ I [ B , , ~) d ~


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464 C A N . 1 C I V. ENG. VOL. 21 , 1994

The matrix [DO11 takes the constitutive elastoplastic law ofcombined s t resses (ax ia l and to rs iona l shear fo rces) in toaccount. The mixed formulation was used for the variationalprinciple to avoid membran e locking (c interpolation formemb rane displacements; Akoussah et al. 1986). The completeexpressions for [B,,,], [B,,,], [D o ll , and [DO21 can be found inAkoussah et al . (1987).

Equations [17] and [19] are expressed using the local coor-dinate system of an element for which the displacements are

assumed to be small (no large strain). This assumption allowsus to use the Eulerian equation for the virtual work in thelocal system (n o Lagrangian variable). T he dom ain of inte-gration used is the last known configuration in equilibrium.If a sufficiently large number of integration points spreadedover the cross section of the member is used, residual stressand shear lag will be adequately modeled. Of course, distor-s ions and re la ted e ffec ts canno t be s imula ted by a beamelement. To correctly account for these effects, a platelshellelement would be required.

To account fo r large displacements and large rotations, at ransformat ion matr ix ca lcu la ted wi th the updated nodalcoordinates of the element was applied to transform the localresidual vector in the global system.

There fore, the Newton-Raphson procedure yields (Bathe1981)

where l i )is the relative displacement vector of the connectio

[26] (li) = ii i iv 8 , 8, 8;)

The residual vector {R(ii)} epresents the unbalanced forcin the connection, i.e., the difference between external ainternal forces. Since the external loads applied to a connectiare generally nul, the residual vector may be interpreted as tvector of the internal forces at each end of the connectio

If K,, K,, and K, are the internal forces in the connectiocorresponding respectively to the unit relative displacemen2, C, and iv at both e nds of the con nection, and K,,,, KO,,andKO, are the bending momen ts corresponding to the unit relatirotations 8,, 8,, and 8, at the ends of the connection, the tangent stiffness inatrix of the connection may be expressed

Once again, a l imited Taylor ser ies development of thintegral form allows us to determine the unknown equilibriuconfiguration C from the previously determined configurt ion C1 ,giving us

In order to translate this last equation (written as a functiW ( ~ ~ ) ( [ K , (U ) ] { A U } { ~ ( u ) } )= 0 of the relative displacements) in terms of the nodal degre

of freedom, the following transformation must be performeT h e ve ct or ( ~ ( u ) )s the equil ibr ium residual vector of theentire structure for i teration i. [291 { f i l , ~ , [ B ~ ~ ] , x I ,~ 14 x 1

Th e s tandard Newton-Raphson method can be used to whereso lve [23] fo r {Ale }. T h e c o r r ec t io n i s t h e n a d d e d to t h ecorrect displacement vector evaluated at the previous iteration: [ - I 0 0 0 0 0 0 - 1 0 0 0 0 0 0 1

The calculations are continued as long as the ratio of thenorm of the correction vector to the norm of the displacementvector is less than a g iven value (generally 0 .001).

Fin ite elerrierzt app roa ch for flexible corlrzectiorlsTo take the f lexibil i ty of the connections into account, a

three-dimensional f inite element h aving as many degrees offreedom as the beam element presented above was developed.The rigidity matrix for this connection element is constructedbased on the s lope o f the fo rce-d isp lacement curves andmoment- ro ta tion curves represen ting the behav ior o f theconnection. These curves are required as model input data for

The rate of the integral form becomes

where

each one of the first six nodal degrees of freedom. The seventhdegree of freedom which accounts for cross-sectional warping The tangent matr ix [K,(u)l defined above can b e direc

is considered unrestrained in the connection model. assembled with the tangent inatr ix of the beam elemen

The mathematical approach used to simulate the behavior The g loba l tangen t matr ix i s therefo re in f luenced by t

o f t h e c o n n e c t i o n i s v e r y s i m i l a r t o t h e b e a m e l e m e n t presence of the con nections.

appro ach . By means o f the p r inc ip le o f v i r tua l work andthe weighted residual method, the equilibrium configuration onnect ion offse ts

can be established. The integral form is writ ten as (Savard A structure is generally represented by lines passing throuet al. 1989) the center of gravity of its members. It is assumed that t


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columns W250x73

Rotation, ( ~ 1 0 ' ~ad)

FIG. 3. Semirigid plane frame:(a) geometry and loading; 13) finite element model; c ) moment-rotation curve for the connection

structural properties of the members are valid over the entirelength of these l ines. Consequently, a beam whose ends areto a column flange is considered to have uniform propertiesover i ts entire length even though i t is recognized that theend portions, equal to half the depth of the respective columns,are much more r igid.

To p r e v e n t t h i s s l i g h t a n o m a ly wh ic h c o u ld a f f e c t t h ebehavior of the structure when columns are deep and heavy,we followed the recommendations of Nethercot et al. (1987),who suggested introducing a rigid panel, whose length equalshalf the column depth, at each end of the beam. The flexuralrigidity in these portions of a beam is closer to the flexuralrigidity of a rigid panel than to that of a beam.

umerical examplesThe beam f in i te e lement p resen ted here may be used to

analyze two- or three-dimensional structures. The element 'scharacter ist ics may be specif ied singly or simultaneously.Therefore, i t is relatively easy to adjust the model to thecharacter ist ics of other models used in com ~a ra ti ve tudies.

Two dimensional semirigid frame ex at~ ll~ leTo demonstrate the accuracy of the proposed finite elemen

m o d e l f o r e l a s t i c f r a m e s w i th se m i r ig id c o n n e c t io n s , wcompared our results with those of Brun and Picard (1976The geometry and loads acting on the structure are showin Fig. 30. The sam e f rame was ana lyzed by Adams (1968in his investigation of P-A effects.

A uniformly distributed load of 40.86 kN/m was applied the frame on each f loor along with three concentrated hoizontal loads, as shown. The finite element model is presentein Fig. 30 and includes three beam elements for each membeTh e connections are represented by spring s at the beam endRigid panels have been included in the analysis where beamare connec ted to co lumn f langes .

To represent the behavior of a connection, Brun and Picar(1976) used a standardized experimental moment-rotation curvand found the following equation to be representative of thtypical moment-rotation curve for the type of connection use

[33] = 1.04 X lo- (KM) 1.62 X 10- I ' (KM)

Th e first example is based on a three-story three-bay planef rame wi th f lex ib le connec t ions . A s imple p lane f rame is 4.62

X ~o- ' KM)'

analyzed in the second example and our numerical results In [33] is the rotation of the connection, M is the bendinare compared with experimental measu rements. In the third moment, and K is a factor accounting for geometr ic paramexample , we ana lyze a one-s to ry space f rame wi th a r igid eters of the connection. This function is illustrated in Fig.3c.horizontal diaphragm. In our numerical mod el, this curve is subdivided into a seri


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beam S 250 x 38

columns : W 130 x 28I

FIG. 6. Experimental plane frame: a ) geometry and loading;(b) finite element m odel;(c) residual stress distribution;(d ) stress-strainrelationship for steel.

no residual stressesFinite element results;

flexible connections;no residual stressesA Finite elemen t results;

flexible connections;

0 20 40 60 80 100 120 140 160 180

Horizontal displacement, rnm )

FIG. 7. Load-displacement relationship xperimental plane frame.

were introduced at each end of the beam . Since the moment-ro ta t ion re la t ionsh ips fo r the connec t ions were unknown,w e a s s u m e d t h a t t h e c u r v e u s e d i n e x a m p l e 1 c o u l d b e

app l ied to the beam- to-co lumn connec t ions and assumedperfectly r igid base connections for the columns.The residual stress distribution considered over the length

of each member is shown in Fig.6c The maximum residualstress was taken to equal 30 of the material yield stress, asis usua l ly recommended . The s tandard ized s t ress- s t ra inrelationship for steel used in the analysis is shown in Fig.6d

Th e load sequence m ay significantly affect the results. Th eload sequence we considered was essentially the same as theexperimental one and consisted in applying the four verticalloads and then gradually increasing the horizontal loadH.

The resu l t s ob ta ined fo r bo th geometr ica l and mater ia lnonlinearities are plotted in Fig.7 along with the experimental

measuremen ts. First , we observed that connection r igidit iinf luence the behavior of the wh ole structure. The ult imaload for the perfectly r igid fra me is8-10 higher than th

corresponding value for the semirigid frame. Residual stressseem to have very little effect on the frame s ove rall responsWi t h t h e f l e x i b l e c o n n e c t io n e l e m e n t s , o u r n u m e r i c

model suggests a f rame response very s imi la r to the onmeasured experimentally unti l the horizontal displacemenA reaches 80 mm. Past this point, the model seems to oveest imate the d isp lacement . Many fac to rs can exp la in thdifference. First, the horizontal displacement of80 mm correspon ds to the fram e s ult im ate load. Past this l imit , thhorizontal load must be reduced in order to maintain equl ib r ium. Plas t ic deformat ions may have occur red and thstress-deformation curve sho wn in Fig.6c1 as well as themoment-rotation curve considered for the conn ections a


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C A N . J. C I V. ENG. VOL. 21 994

P= 4 7 6 k N

beams: W250x73

columns: W200x46 1 048 1

FIG.8. One-story space frame: a) model overview; ( 0 ) plane view; c) elevation

5 loo

0 5 10 15 20 25 30 35

Displacem ent of node3 along the x axis mm

FI G. . Load-displacement relationship ne-story space frame.

the residual stress distribution, may prove inadequate. Also,rotations may have occurred in the base connection s, whichare considered perfectly rigid in the numerical model. Otherfactors, such as init ial geometr ic imperfections and out-of-plane displacements could also have affected the frame response.

One-s to ry spac e f i -a~ne sa lnp leT h e m a in o b j e c t iv e o f t h i s e x a m p le wa s t o v e r i f y t h e

model s validity an d performance in simulating the behaviorof three-dimensional structures composed of linear membersand r igid diaphragms . To do so, the space frame studied byWynhoven and Adams (1970) was reana lyzed .

As shown in Fig . 8 a , the f in i te e lement model inc ludesf lex ib le connec t ions and r ig id pane ls wherever a beam isc o n n e c t e d to a c o lu m n f l a n g e. Fo u r b e a m e l e m e n t s we r eused to model each member. The elastic limit,F,, consideredin analysis is equal to 238 M Pa .

A ver t ica l load , P, of 476 k N was app l ied a t the upperend o f each co lumn of the f rame. These loads were he ld

constant while two horizontal load s were gradually app lione a t node 3 and the o ther, four t imes lower, a t node 4The results are presented in Fig. 9 where the intensity,H, isplotted against the displacement of node3 along the .r axis

First , we performed a simplif ied analysis which assumall connections to be r igid, no residual stresses and perfee las top las t ic behav ior fo r the s tee l . These approx imat iowere a lso made by Wynhoven and Adams (1970) . In thcase, both models yielded very similar results. The init is l o p e s a r e t h e s a m e a n d t h e r e i s a m e r e2 d i f f e r e n c ebetween the ult imate loads evaluated by the models. Whthe residual stress distribution show n in Fig. 6c is consideralong with the moment-rotation curve shown in Fig.3 c forthe connection, the ult imate load is the same as in the simplif ied model, but the init ial slope is sl ightly lower.

Th e above results were obtained when nodes to 4 wefree to move with respect to one another. In order to evaluathe frame s behavior when these relative displacements aconstrained, we included a horizontal diaphragm simulati


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S AVA R D T AL. 469

a r i g i d r o o f i n t h e u p p e r p a r t o f t h e f r a m e . T h e r i g i ddiaphragm was modeled with two six-node tr iangular shelle l e m e n t s ( F a f a r d e t a l . 1 9 8 9 ) . T h e r e s u l t s a r e s h o w n i nFig. 9 . The load-displacement curves show an init ial slopethat is significantly steeper and an ultimate load that is 15h igher than in the p rev ious case . Th is expec ted behav iortends to validate the model and i l lustrates the importanceof considering horizontal rigid diaphragms in the analysis ofspace f rames.

ConclusionsThe results obtained with the new beam and connection

e lements p resen ted in th is paper show tha t the p roposedf in i te e l e m e n t m o d e l s a r e a d e q u a te f o r s im u la t i n g o f t h ebehav ior o f two- and th ree-d imensiona l f rames under theaction of an increasing load, taking into account geometricaland material nonlineari t ies, connection f lexibil i ty, residualstresses, and cross-section warping.

The lack of data on published experimental tests inay justifythe differences observed between numerical and experimentalresults. To determine a structure's ultimate state of loading,a good appro xima tion of the real mom ent-rotation curve ofthe connection and column bases is needed, along with initialmember imperfections and out-of-plane displacements. Theproposed numerical model can also take into account para-meters that are difficult to measure. As soon as precise mea-surements are available, i t will be possible to include themadequately in a numerical analysis.

Finally, the fact that the different character ist ics of thisnumerical model can be specif ied singly or simultaneouslyis very usefu l fo r compara t ive s tud ies . The model a l lowsevalua t ion o f the va l id i ty and per fo rmance o f s impl i f iedanalyses designed for special applications. I t can also be avaluable tool to improve actual design specifications or helpdevelop new ones .

AcknowledgmentsT h e a u t h o r s w i sh t o t h a n k t h e N a t u r al S c i e n c e s a n d

Engineer ing Research Counci l o f Canada , the Fonds pourla fo rmat ion de chercheurs e t l ' a ide la recherche o f theQ u e b e c M i n i s t r y o f E d u c a t i o n , a n d t h e C a n a d i a n S t e e lIndustries Construction Council for their financial contributionto this project .

Adam s, P.F. 19 68. The design of steel-beam co lumns. Can adianSteel Industries Construction Council, Willowdale, Ont.

Akoussah, E., Beaulieu, D., and Dhatt, G. 1986. Curved beamelement via penaltylmixed formulation for nonlinear in-planeanalysis. C om~nu nicationsn Applied Numerical Methods, :6 17-623.

Akoussah, E., Beaulieu, D., and Dhatt, G. 1987. Analyse no11IinCaire des structures parois minces par ClCments finis etson application aux b iti ~n en ts ndustriels. Report GCT-87-07,Department of Civil Engineering, UniversitC Laval, Sainte-Foy, Que.

Arn old, P., Adam s, P.F., and Lu , L.W. 1968. Strength and behav iorof an inelastic hybrid frame. ASCEJ o ~ ~ r n a lf the Struct~~ralDivision, 94(ST1): 243-266.

Bathe, K.J. 1981. Finite element proceduresi n engineeringanalysis. Prentice-Hall Inc., Eqglewood Cliffs, N.J.

Brun, P., and Picard, A. 1976. Etude d'un assemblage impar-faite~n ent igide et des effets de son utilisation dansun multi-CtagC. Report GCI-7 6-03 , Departmen t of Civil Engineering,UniversitC Laval, Sainte-Foy , Que.

Fafard, M., Dhatt. G., and Batoz, J.L. 1989. A new discreKirchhoff p la te lshe l l e lement wi th updated p rocedureComputer and Structures, 4: 591-602.

Malvern, L.E. 1969. Introduction to the mechanics of a continumedium. Prentice-Hall Inc., Englewood Cliffs, N.J.

Massicotte, B., and Beau lieu, D. 1 984. Effct dessystkmeentremises-diaphrag~nes ur la stabilitC des poteaux. RepGCT-84-05, Department of Civil Engineering, UniversitC LavSainte-Foy, Que.

Nethercot, D.A., Kirby, P.A., and Rifai, A.M. 1987. Co lu ~n nn

partially restrained construction: analytical stud ies. CanadJournal of Civil Engin eering, 14 (4): 485-497.Nishino, F., Kasemset, G., and Lee, S.L. 1973. Variational form

ation of stability problems for thin-walled members. IngenieArchives, 43: 58-68.

Savard, M., Beaulieu, D., and Fafard, M. 1989. Mod6lisatiopar ClCnlents finis des bfitiments ossa ture mCtallique. RepGCT-89-05, Department of Civil Engineering, UniversitC LavSainte-Foy, Que.

Vlassov, V.Z. 196 1. Thin walled e lastic beam s. Translated froRussian. Israel Program forScientific Translation, Jer~~ salIsrael.

Wynhove, J.H., and Adam s, P.F. 1970. Elastic-plastic analysof three-dimensional structures. Struct~~ralngineering RepoNo. 29, Departm ent of Civil Eng ineering, University of Albe

Edmonton, Alta.

Appendix 1 List of symbolsdegree s o f f reedom-to-deformat ion t ransfomation matr ixr e l at i v e d i s p l a c e i n e n t s - t o - n o d a l d e g r e e s freedom transformation matr ixnodal displacement vectordifferential element of surfacedifferential element of volumevir tual deformation rate tensorsur face fo rcesvo lume fo rcesdistr ibuted forces alongs y a n d a x e scenter of gravitytangent st iffness matr ix

171 dis t r ibu ted moments a longs y and axesin te rpo la t ion f i~nc t ionsshear cen terequ i l ib r ium res idua l vec to rsecon d Piola-Kirchhoff stress vectord isp lacements a long s , and axesv i r tua l d isp lacement vec to rincrement of displacement vectorrelative displacement vectorintegral form

internal vir tual workfirst derivative forxsecond derivative forcoordinates of the shear centerweigh t ing func t ionsincrease or variationGreen-Lagrange f i r s t var ia t ion defor ina t ivectoraxial strain along s and axesang le o f twis t a round lo ng i t~i d ina l x isshear strain in the s-5 planeCauchy s t ress tensorwarping function


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470 C A N . I. CIV. ENG. VOL. 21, 1994

Appendix

Virtual work theoryThe internal virtual work may be expressed as (Malvern

1969)

where [D: is the virtual deformation rate matrix and [u]

the Cauchy stress tensor. In order to introduce assumptions 1,2, and 3 in to the in ternal v i r tual work express ions , [Allmust be transformed in the curvilinear system x, s, 5) shownin Fig. 1. This transformation accou nts for the curva ture ofthe flattened beam. The Marguerre theory was used with theassumption that by,) , ?,) , (z,,y,,) 4 1. If we also assume thatthe section is composed of a series of assembled flat elements,the internal work may be expressed as follows after inte-gration on the cross section (Akoussah et al. 1987):

where

[A401 ( M ) = (MI,M, T,,, M,)

sect ion , and f and f; are appl ied a t a d is tance dy and dzfrom the shear center, as shown in Fig.2 .

Tangent stiffness matrixThe tangent stiffness matrix needed to be defined to solve

the nonl inear problem. This matr ix may be expressed a(Akoussah et al. 1987)

where

[ A 7 n ] q t , = v:Av,, + rv:Aw,,

[A7b] (q;) = (-(v:A@ + @*A ,,) (w;A@ + @'2Aw, , r ) )

@ + A@ @ * A @.I ) v A@ .r ))and

[A801 AN = jA AS, dA

[A861 (AM) = (AM? AM; AT,, AM,)

AM; = (y li)AS,,I

AM, = o ~ 5 ) A s . ~ . ~

M, = jA 5 A where u is evaluated using [9] for an open section and [ I lfor a closed section.

The resul t ing force N and moments (M ) are def ined by Also,[A4a] and [A4b] . The combined inf luence of tors ion andflexion on the virtual axial strain can be observed in [A301 [A91 (V)

= (V, V,)

and on the virtual flexural strain in [A3b].

E xte rn al v i r t ~ ~ a lork v z = - J l r r s ~ ; v1 = j A n r , , d ~Using the Cartesian system of coordinates as the kinematic

variables, the external virtual work, in the absence of volumeforces, may be expressed as [A101 = ((My z,N) (M, Y,N) q M P )

> ,v* + f - W + (f;dy fydz)@*[A51 X t j L ( f r < .. M, = JA[((y 15) yo)' ((z mi ) z012]u.I'.r d A

+ m,@* + m,w':, + nz-v;,,) dLLoadfr is applied at the center of gravity of the beam cross TI = I / ; L ~ X S

Paper 1994-Nonlinear Finite Element Analysis of Three Dimensional Frames

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