Advaced Steel Costructio Vol. 6, No. 1, pp. 567-577 (2010) 567 TOTAL INCREMENTAL ITERATIVE FORCE RECOVERY METHOD AND THE APPLICATION IN PLASTIC HINGE ANALYSIS OF STEEL FRAMES Fawu Wag 1,* ad Yaopeg Liu 2 1 Departmet of Civil Egieerig, Najig Uiversity of Aeroautics ad Astroautics, Yudao Street, Najig, Chia 2 Departmet of Civil ad Structural Egieerig, The Hog Kog Polytechic Uiversity, Hughom, Kowloo, Hog Kog *(Correspodig author: E-mail: fwwag@uaa.edu.c) Received: 15 Jauary 2008; Revised: 15 February 2008; Accepted: 20 October 2008 ABSTRACT: I this paper a total icremetal iterative force recovery method suitable to plastic hige aalysis is proposed ad it is foud that this icremetal iteratio force recovery procedure has a covergece rate comparable to total secat method. The icremetal force recovery maer makes this method be capable of recoverig member iteral forces ad deformatios correctly ad coveietly i the existece of plastic hige. The capability of the proposed total icremetal iterative force recovery method to cope with plastic hige is illustrated by two bechmark examples. Keywords: Noliear aalysis; steel frames; force recovery procedure; plastic hige 1. INTRODUCTION I the former paper [1] a ew total icremetal iterative force recovery method is proposed by the authors, which resembles icremetal iterative system solutio techique ad utilizes both the secat ad taget stiffess. It is foud through the umerical examples that this icremetal iteratio force recovery procedure has a covergece rate comparable to total secat iteratio method ad remais the icremetal force recovery maer. This paper provides a detailed descriptio about the applicatio of this ew icremetal force recovery method i the plastic hige aalysis of framed structures. Geometric oliearity, material plasticity ad semi-rigid coectios are the most importat oliear sources of framed structures. Whe material plasticity is ivolved, there are maily two types of methods: (1) plastic zoe method; or (2) plastic hige method. The plastic zoe method discretizes each member ito may fiber segmets logitudially ad trasversely ad the plastic hige method assumes plasticity lumped oly at specific locatio alog a elemet the other part of elemet remais elastic. Although the plastic zoe solutio may be cosidered exact, it is ot coducive to daily use i egieerig desig because it is too computatioally itesive. O the other had the refied plastic hige method ca approximate the effects of distributed plasticity alog the elemet legth ad hige gradual formatio, ad remais relatively efficiet ad ecoomical[2-5]. I practice the icremetal ad iterative solutio strategy is employed to trace the load vs. deflectio path of a structure, which ca be divided ito three stages: the predictor, corrector ad error-checkig phases. The corrector or force recovery procedure determies the accuracy of the solutio while the predictor affects oly the covergece speed ad directio of iteratio [6]. Differet solutio schemes such as arc-legth method ca be used to traverse the limit poits ad capture the post-bucklig aalysis behavior of the structures [7]. The Hog Kog Istitute of Steel Costructio www.hkisc.org
568 Total Icremetal Iterative Force Recovery Method ad the Applicatio i Plastic Hige Aalysis of Steel Frames The load icremet sizes cotrol techique has to be used to ispect the formig of plastic hige i plastic hige aalysis, which scales back the icremet size to the yield surface so that the elemet forces of specific cross-sectio comply exactly with those o the yield surface [8]. This techique works for plastic hige aalysis but is tedious ad low efficiecy as plastic hige is member based ad may occur frequetly for complex practical structures. Although a lower limit ca be specified to avoid small step legth i the case of frequet occurrece of yield higes, exact scalig to the yield surface is ot always possible [9,10]. I spite of rapidly icreased computed speed ad memory capacity of persoal computers, the computatioal efficiecy is crucial for the daily use of oliear or advaced aalysis which requires extesive computatioal time ad effort ivolvig hudreds of load cases ad thousads of iteratios, ot to say whe desig optimizatio is used where may desig parameters are chaged ad optimized cotiuously [11]. Actually i advaced aalysis ad desig the behavior of the whole structure is draw more attetio tha the exact time or load step whe specific plastic hige occurs. For the icremetal ad iterative solutio strategy the chage i the stiffess of practical complex structures caused by plastic hige has little differece with that caused by secod order effect. The equilibrium path of a structure ca be obtaied by icremetal ad iterative solutio strategy if the elemet behavior is correctly modeled o matter i elastic rage ad plastic rage. As the corrector determies the accuracy of the modelig, it is atural ad eat to ispect the formig of plastic hige ad recover the correct iteral forces ad deformatios accordigly i force recovery procedure. I the followig of this paper Sectio 2 summarizes the popular beam-colum elemets i co-rotatioal formulatio. A simple elastic-plastic hige model is adopted ad sectio sprig is used to model the plastic hige for cosistecy i Sectio 3. The approach of modifyig differet force recovery procedure for plastic hige aalysis is detailed I Sectio 4. The capability of the proposed total icremetal iteratio force recovery method to cope with plastic hige is illustrated by two classic umerical examples. 2. BEAM-COLUMN ELEMENT Co-rotatioal formulatio is suitable for the geometrically oliear aalysis of framed structure as the referece frame of every elemet ca be easily determied by the positios of two odes of the elemet. I co-rotatioal method the displacemet u of elemet ca be decomposed ito two parts: the rigid body displacemets u r ad the atural deformatios u l. The effect of the rigid body displacemets u r is to rotate the iitial odal forces from the last cofiguratio to the preset cofiguratio without geeratio of additioal iteral forces. The separatio of the rigid body displacemets from the elemet deformatio makes the formulatio of beam-colum elemet simplified as the large deflectio ad rotatio effect ca be cosidered i the deformatio extractio operatio. The simplest beam-colum elemet is the cubic elemet which is robust ad successful except for the ecessity of usig several elemets for each member to obtai sufficiet accuracy [12]. The stability fuctio elemet [13] ad PEP elemet [14] are capable of dealig with secod-order aalysis usig oe elemet per member i practical applicatio, which simplifies the aalysis ad desig work as there is o differece betwee the aalysis model ad the physical model. I geeral the secat stiffess relatioships of beam-colum elemet ca be expressed as EI M1 c1 c21 c1 c22 L (1)
Fawu Wag ad Yaopeg Liu 569 EI M2 c1 c21 c1 c22 L (2) e P EA b b L x, y 2 2 11 2 21 2 (3) M x 2 GJ Pr x (4) L Where = x, y are two pricipal directios of member; e = elogatio of the elemet; f = {M 1y, M 2y, M 1z, M 2z, P, M x } T is local iteral force vector ad u = {θ 1y, θ 2y, θ 1z, θ 2z, e, θ x } T is local deformatio vector; ad E = elasticity modulus; I = momet of iertia; A = sectio area; L = legth of the elemet; Ad c 1, c 2 ad b 1, b 2 are axial force relatig coefficiets ad has differet forms for differet types of elemet ad is ot repeated here. It should be oted that the last term of Eq. 3 represets the bowig effect which plays importat role i the oliear iteractio of momets ad axial force. The taget stiffess matrix ca be obtaied by takig a variatio of the secat stiffess with respect to the displacemet degrees of freedom ad the axial force by which the secod-order effect is cosidered. Ad the fial expressio of the icremetal stiffess relatioship ca be writte as e f k u (5) Where Δf ad Δu are icremetal iteral force ad deformatio vector; [k e ] is elemet taget stiffess matrix. The elemet taget stiffess matrix i global coordiates ad the global taget stiffess matrix of the whole structure ca be trasformed ad assembled usig the stadard procedure of co-rotatioal formulatio ad the fiite elemet aalysis. 3. ELASTIC-PLASTIC-HINGE MODEL As the mai poit of this paper is ot details of the plastic hige model, the simple elastic-plastic hige model is selected to illustrate the applicatio of the proposed force recovery method i plastic hige aalysis. I the elastic-plastic hige model the cross-sectio behavior is elastic-perfectly plastic ad the plasticity is accouted for by zero-legth plastic higes at two eds of each elemet. As the cross-sectio is assumed to be either ideally elastic or fully plastic, the iitial yield surface ad fully plastic yield surface are superposed which may be represeted by AISC LRFD biliear iteractio equatios P 8 M P 1.0 for 0.2 (6) P 9 M P y p y P M P 1.0 for 0.2 (7) P M P 2 y p y
570 Total Icremetal Iterative Force Recovery Method ad the Applicatio i Plastic Hige Aalysis of Steel Frames Where P y = the squash load of the cross sectio; M p = the plastic momet capacity for member uder pure bedig actio; ad P ad M = the secod-order axial force ad bedig momet at the cross sectio beig cosidered. Although the icremetal force-displacemet relatioships ca be deduced for differet situatios such as plastic higes at oe ed ad both eds, the sectio sprig cocept has the merits of simplicity ad cosistece. The mai idea of the sectio sprig model is the same as semi-rigid sprig model that a deformed beam-colum elemet with two sprigs each attached to the elemet ed ode, ad i fact the two kids of sprigs ca be combied as sprigs i series to simulate semi-rigid coectio ad plastic hige together [15,16]. The momet- rotatio relatioship of the sectio sprig ca be writte as M k ( ) k (8) s b s s 1 1 1 1 1 1 M k ( ) k (9) s b s s 2 2 2 2 2 2 Ad the sprig taget stiffess ca be expressed i the icremetal form as M k ( ) k (10) t s b t s s 1 1 1 1 1 1 M k ( ) k (11) t s b t s s 2 2 2 2 2 2 The total ad icremetal rotatioal stiffess relatioships of beam elemet ca be extracted from the secat stiffess relatioship i Eqs. 1 to 4 ad the taget stiffess matrix show i Eq. 5 respectively M k k (12) b b b b 1 11 1 12 2 M k k (13) b b b b 2 21 1 22 2 M k k (14) t b b t b b 1 11 1 12 2 M k k (15) t b b t b b 2 21 1 22 2 Through momet equilibrium coditio, the additioal iteral degree-of-freedoms ca be elimiated via the static codesatio procedure ad fially the total ad icremetal stiffess relatioships ca be writte as s s s b s s s b M 1 1 1 22 2 1 2 21 1 1 k k k k k k k k 1 M s s b s s s b s 2 k 2 1k2 k12 k 2 k2k2 k11 k 1 t s s s b s s s b 1 1 1 22 2 1 2 21 1 1 k k k k k k k k 1 t s s b s s s b s 2 2 1 2 12 2 2 2 11 1 M M k k k k k k k k (16) (17)
Fawu Wag ad Yaopeg Liu 571 Where k b 11 k s 1 k b 22 k s 2 k b b 12k21 k11 k1 k22 k2 k12 k21 t t b t s t b t s t b t b (18) (19) Where left superscript t represets taget quatity; K b ij are the relevat stiffess coefficiets of stiffess matrix; K s i are the stiffess of sectio sprigs at two eds of elemet. I computatioal practice the stiffess of sectio sprigs K s i ca be assiged a very large or a very small stiffess accordig to whether the plastic hige comes ito beig. This sectio sprig model ca be exteded to refied plastic hige model where the stiffess of the sprig is varied from ifiity to zero ad i this situatio the taget ad secat sprig stiffess ca be differet. 4. FORCE RECOVERY PROCEDURE IN PLASTIC HINGE ANALYSIS Before the corrector or force recovery procedure is exercised i the icremetal iterative aalysis, the elemet deformatio 0 [u] i last cofiguratio ad the preset elemet deformatio icremets [Δu] have bee extracted from the system icremetal displacemets which are computed by the predictor equatios. The task of the force recovery procedure is to recover the elemet iteral forces [f] at the curret cofiguratio. 4.1 Total Secat Iterative Method The elemet iteral forces ca be calculated directly by the secat stiffess i Eqs. 1 to 4 with the preset elemet deformatio [u] cumulated from the elemet deformatio 0 [u] i last cofiguratio ad the preset deformatio icremets [Δu]. The difficulty arises due to the fact that the expressio for member axial force as give by Eq. 3 ivolves bowig fuctios which i tur are fuctios of the axial force parameter q = PL 2 /EI. As a result the whole equatios are implicit fuctios of axial force P. To solve this problem, a axial force iteratio procedure has to be adopted which is based o first-order Taylor series expasio of Eq. 3. As the secat stiffess equatios are explicit ad exact, this force recovery method has the capability of good ad rapid covergece tha the force icremetal method i elastic rage. If plastic hige behavior is cosidered, there is abrupt poit i the elemet force-deformatio curve whe plastic hige comes ito beig. The explicit secat stiffess relatioship does exist i simple situatio but has differet forms for elastic ad plastic rage. The situatio is further complicated as axial force iteratio procedure must be adopted to obtai the exact axial force value which is the basis of determiig the exact positio of yield surface. So this force recovery method is recommeded oly for the elastic oliear aalysis of frames. 4.2 Force Icremetal Method The elemet recovery forces ca also be cumulated from the force icremets which ca be calculated by the product of elemet atural stiffess ad atural deformatios f ke u (20)
572 Total Icremetal Iterative Force Recovery Method ad the Applicatio i Plastic Hige Aalysis of Steel Frames i which k e is the elemet atural stiffess matrix. The iteral forces of the elemet ca be updated by addig the icremetal elemet forces to the elemet iteral forces i last cofiguratio 0 f f f (21) The accuracy of force icremets is oly approached i the first order of the displacemet icremets so the load icremet step should be small eough to obtai a sufficietly accurate solutio. The secat iterative method has obstacle to cope with the abrupt poit i plastic hige aalysis. I icremetal type force recovery methods thigs get simple ad atural: firstly scale back to the right positio where plastic hige occurs, the calculate the force icremet causig by deformatio icremet remaied usig the modified icremetal stiffess. The whole procedure ca be illustrated as follows e f f f (22) f k u (23) e (24) p e f f f M M p M (25) Where e f ad p f are elemet iteral forces i elastic ad plastic rage respectively; k e is the modified icremetal stiffess after plastic hige formed ad α is a scale factor that reverse the force to the right positio where plastic hige occurs. Whe the momet axial force iteractio equatios such as Eqs. 6-7 are itroduced as yield surface, a oe dimesioal search procedure should be adopted to fid the exact scale factor istead of Eq. 25 beig used directly. Actually the force-deformatio relatioship is ot a straight lie but a curve with abrupt poit as illustrated by Figure 1. Because of the taget approximatio, the force icremet method ca oly get force-deformatio path OA B, which has the equal iteral forces with the real path OAB but the deformatio where plastic hige occurs is differet. This is ot a problem as the load cycle step should be small eough i force icremetal method ad the itroduced error ca be eglected. 4.3 Total Icremetal Iterative Method The secat iterative method is more accurate ad rapid while the force icremetal method has the advatage of simplicity ad flexibility of copig with abrupt poit o force-deformatio curve. Their combied use is a sesible choice i improvig the computatioal accuracy ad efficiecy, just like the icremetal-iterative method used i the solutio of system equilibrium equatios. Simulatig the modified Newto-Raphso method, the total icremetal iteratio ca be writte as i i i f [ k] u f 1 f f u 1 u u (26) i i i (27) i i i (28)
Fawu Wag ad Yaopeg Liu 573 i i1 0 k i (29) u u u u k 1 f A B A B O u Figure 1. Illustratio of Differet Force Recovery Method I this iteratio the total deformatio i [u] ca be easily calculated by the reverse form of secat stiffess Eqs. 1-4. This iterative process of Eqs. 26-29 cotiues util the ubalaced deformatio i [Δu] is sufficietly small. Ad it should be oted that the taget matrix i Eq. 26 is kept uchaged durig the iteratio, which makes this iteratio use miimal computatioal effort compared with system equatio solutio. The same as the force icremetal method, a oe dimesioal search procedure should be adopted to scale member state right back to the yield surface ad the exact scale factor ca be obtaied. Ulike the force icremetal method, the exact poit A that plastic hige occurs ca be coverged by the iteratio procedure. If the explicit secat stiffess relatioships with plastic hige exist, aother iteratio as Eqs. 26-29 ca be cotiued to arrive the specified deformatio state. Otherwise Eq. 24 ca be utilized i the case of oly icremetal force-deformatio ca be obtaied coveietly whe refied plastic hige model is used. As the force recovery method ca simulate the member exactly i the existece of plastic hige, especial load icremet sizes cotrol techique to scale the whole structure back to plastic hige poit is ot eeded. 5. NUMERICAL EXAMPLES 5.1 Vogel Portal Frame Figure 2 shows a portal frame proposed ad solved by Vogel. The frame possesses a out-of-plumbess of 1/400 ad a member iitial imperfectio of 1/1000 alog the colum legth. The sectio of the beam is HEA-340 ad that of colums is HEB-300. This paper uses a sigle elemet ad elastic plastic hige model with the proposed force recovery method to model the beam ad the colum. The computed ultimate load factor is 1.00 which is close to the result of [17]. Load-displacemet curves by differet methods are showed i Figure 3. The total icremetal iterative method ad force icremetal method give almost same results, but the total icremetal
574 Total Icremetal Iterative Force Recovery Method ad the Applicatio i Plastic Hige Aalysis of Steel Frames iterative method ca still obtai approximate collapse load eve with 10 load cycles. It is viewed from umerical iteratio process that the total icremetal iterative method has early same performace i elastic ad plastic rage, with 2 or 3 more iteratios icreasig i presece of plastic hige. Figure 2. Vogel Portal Frame 1.2 1.0 Load Factor 0.8 0.6 0.4 Elastic aalysis Force icremetal method Total icremetal method with 100 load cycle Total icremetal method with 10 load cycle 0.2 5.2 Vogel Six Story Frame 0.0 0.000 0.005 0.010 0.015 0.020 Lateral Dispalcemet (m) Figure 3. Load-displacemet Curve of Vogel Portal Frame Figure 4 shows the well kow Vogel six story frame. The frame possesses a out-of-plumbess of 1/400 ad a member iitial imperfectio of 1/1000 alog the colum legth. The sectio of the beam is IPE-240 to IPE-400 ad that of colums is HEB-200 to HEB-260. This paper uses a sigle elemet ad elastic plastic hige model with the proposed force recovery method to model the beam ad the colum. The computed ultimate load factor is 1.1 which is which is close to the result of [17]. From Figure 5 it ca be see that almost same collapse load is obtaied by the total icremetal iterative method with differet load steps. As the exact load path is ot always eeded i desig, the proposed method is effective ad rapid, which is demaded ad welcomed by egieers. Figure 6 shows the bedig momet diagram ad plastic hige positio of Vogel six story frame ear collapse.
Fawu Wag ad Yaopeg Liu 575 Figure 4. Vogel Six Story frame 6. CONCLUSION This paper provides a detailed descriptio about the applicatio of the total icremetal iterative force recovery method i the plastic hige aalysis of framed structures. This method utilizes both the secat ad taget stiffess ad remais icremetal maer i the process of recoverig iteral forces, which is coveiet to the modelig of plastic hige. As the force recovery method ca simulate the member exactly i the existece of plastic hige, especial load icremet sizes cotrol techique to scale the whole structure back to plastic hige poit is ot eeded. Numerical examples show this method is useful i the plastic hige aalysis ad daily desig of steel frames. 1.2 1.0 Load Factor 0.8 0.6 0.4 0.2 Total icremetal method with 120 load cycle Total icremetal method with 12 load cycle 0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Lateral Displacemet (m) Figure 5. Load-displacemet Curve of Vogel Six Story Frame
576 Total Icremetal Iterative Force Recovery Method ad the Applicatio i Plastic Hige Aalysis of Steel Frames Figure 6. Bedig Momet Diagram ad Plastic Hige Positio of Vogel Six Story Frame REFERENCES [1] Wag, F.W. ad Liu, Y.P., Total ad Icremetal Iteratio Force Recovery Procedure for the Noliear Aalysis of Framed Structures, Advaced Steel Costructio, 2009, Vol. 5, pp.500-514. [2] Liew, J.Y.R., White, D.W. ad Che, W.F., Secod-order Refied Plastic-hige Aalysis for Frame Desig. Part I, Joural of Structural Egieerig, ASCE, 1993, Vol. 119, pp. 3196-3216. [3] Liew, J.Y.R., White, D.W. ad Che W.F., Secod-order Refied Plastic-hige Aalysis for Frame Desig. Part II, Joural of Structural Egieerig, ASCE, 1993, Vol. 119, pp. 3217-3237. [4] Zhou, Z.H. ad Cha, S.L., Elastoplastic ad Large Deflectio Aalysis of Steel Frames by Oe Elemet Per Member. I: Oe Hige Alog Member, Joural of Structural Egieerig, ASCE, 2004, Vol. 130, pp. 538-544. [5] Cha, S.L. ad Zhou, Z.H., Elastoplastic ad Large Deflectio Aalysis of Steel Frames by Oe Elemet Per Member. II: Three Higes Alog Member, Joural of Structural Egieerig, ASCE, 2004, Vol. 130, pp. 545-553. [6] Yag, Y.B., Li, S.P. ad Leu, L.J., "Solutio Strategy ad Rigid Elemet for Noliear Aalysis of Elastically Structures Based o Updated Lagragia formulatio", Egieerig Structures, 2007, Vol. 29, pp. 1189-1200. [7] Cha, S.L. ad Chui, P.P.T., No-liear Static ad Cyclic Aalysis of Steel Frames with Semi-rigid Coectios, Elsevier Sciece Ltd., 2000. [8] Che, H., Noliear Ielastic Aalysis of Steel-cocrete Composite Frames. PhD thesis, Dept. of Civil Egieerig, Natioal Uiv. of Sigapore, Sigapore, 2000. [9] Liew, J. Y. R., Che, H., Shamugam, N.E. ad Che, W.F., Improved Noliear Plastic Hige Aalysis of Space Frame Structures, Egieerig Structures, 2000, Vol. 22, No.. 10, pp. 1324 1338.
Fawu Wag ad Yaopeg Liu 577 [10] Liew, J.Y.R., ad Tag, L.K., Advaced Plastic Hige Aalysis For the Desig of Tubular Space Frames, Egieerig Structures, 2000, Vol. 22, No. 7, pp. 769 783. [11] Wag, F.W. ad Cha, S.L., Optimizatio ad Sesitivity Aalysis of Space Frames Allowig for Large Deflectio, Egieerig Structures, 2006, Vol. 28, pp. 1395-1406. [12] Cha, S.L. ad Kitiporchai, S., Geometric Noliear Aalysis of Asymmetric Thi-walled Beam-colums, Egieerig Structures, 1987, Vol.9, pp. 243-254. [13] Gu, J.X., Large Displacemet Elastic Aalysis of Space Frames Allowig for Flexural-torsioal Bucklig of Beams, Dept. of Civil ad Structural Egieerig, Hog Kog Polytechic Uiversity, Hog Kog, 2004. [14] Cha, S.L. ad Zhou, Z.H., Poitwise Equilibratig Polyomial Elemet for Noliear Aalysis of Frames, Joural of Structural Egieerig, 1994, Vol. 120, No. 6, pp. 1703-1717. [15] Iu, C.K. ad Cha, S.L., A Simulatio-based Large Deflectio ad Ielastic Aalysis of Steel Frames uder Fire. Joural of Costructioal Steel Research, 2004, Vol. 60, pp. 1495-524. [16] Iu, C.K., Ielastic Fiite Elemet Aalysis of Composite Beam o the Basis of Plastic Hige Approach, Egieerig Structures, 2008, Vol. 30, pp. 291-302. [17] Toma, S. ad Che, W.F., Europea Calibratio Frames for Secod-order Ielastic Aalysis, Egieerig Structures, 1992, Vol. 14, No. 1, pp. 7-14.