Capacity of Open Rectangular Shear Walls

Transcription

1 ISSN: ISO 900:008 Certfed Volume, Issue, January 0 Capacty of Open Rectangular Shear Walls Essam M. Awdy, Hlal A. M. Hassan Assst Professor, Structural Engneerng, Zagazg Unversty, Egypt Abstract A study was developed to assess the relatve nfluences of varous torsonal resstng on the capacty of open rectangular shear walls. The stablty s consdered a four system of loads whch produce zero dstorton on cross secton of shear wall. Fnte element software ANSYS s used to perform the bucklng analyss of open rectangular shear walls, also theoretcal analyss wll be presented for obtanng the crtcal bucklng loads of open rectangular shear walls. An extensve set of parameters s nvestgated ncludng dmensonal parameters (walls thckness, shape factor, monosymmtry, and proporton factor) and a dscusson of the results are llustrated.. Fnally, Conclusons whch may be useful for desgners, have been drawn, and represented. Index Terms Thn wall, Ansys, Torson, Bucklng, and Crtcal loads. I. INTRODUCTION As the heght of buldng ncreases, the lateral loads as well as the vertcal loads tends to control the desgn. The rgdty and stablty requrements become more mportant than the strength requrement. The frst way to satsfy these requrements s to ncrease the sze of the members whch may lead to ether mpractcal or uneconomcal members. The second s to change the form of the structure nto somethng more rgd and stable to confne the dsplacements and ncrease stablty. The core supported structure serves the man structural element for supportng loads. The core nvarably has openng for access nto buldng servces, therefore, ts cross secton can be consdered open. The core behaves as thn walled open secton connected by lntel beams or floor slabs, whch leads to large warpng deformaton throughout the heght of buldng, whch are depended on geometrcal characterstcs of core walls. Therefore, when the core undergoes warpng deformaton, the floor slab and lntel beam are forced to bend out of plane n resstng the warpng deformaton of the core, where the system are nterconnected. It s necessary n most cases to defne the geometry and loadng condtons by analytcal closed formulas to obtan the optmal practcal soluton and to defne the choce of the best cross secton characterstcs shear wall, whch offer a hgh degree of decreases out of plane bendng and twstng forces on floor slabs and lntel beam. Torson usually assumed to be secondary mportance and shear wall wear often desgned to resst axal, bendng and shear forces only. If the shear wall s restrant at the ends aganst warpng, axal stresses wll result as well as a redstrbuton of the stresses. Nader & Sad [] presented an analytcal soluton for the bucklng of moderately thck functonally graded sectoral plates. The stablty equatons were derved accordng to Mndln plate theory and the egnvalue problem for fndng the crtcal bucklng load was obtaned. Camotm et al.[] provde an overvew of the generalzed beam theory fundamentals and report the bucklng and post bucklng behavours of the elastc sotropc/orthotropc members. The lateral bucklng of beams of arbtrary cross secton takng nto account moderate large dsplacements s dscussed by Evangels & John [], the stablty crteron s based on the postve defnteness of the second varaton of the total potental energy and was establshed usng the analogy equaton method.. By adoptng the jont equlbrum for the angled frame (wth thn-walled I-beams) and the force-dsplacement relatons for the members defned at the bucked poston (rather than the ntal poston), the analytcal solutons for bucklng moments was presented by Jong [4]. -D second-order plastc-hnge analyss accountng for lateral torsonal bucklng was developed by Seung et al. [5]. A model consstng of unbraced length and cross secton shape was used for accountng the lateral torsonal bucklng. Also, effcent ways of assessng steel frame behavor ncludng gradual yeldng assocated wth resdual stresses and flexure, second-order effect, and geometrc mperfectons were presented by Seung [6]. Fnte element software, LUSAS.6, was used to study the warpng behavor of cantlever steel beam wth openngs subjected to couple torsonal force at the free end by Tan [7]. The analyss of the results showed that openng has a close relatonshp wth warpng snce openng can reduce web stffness. The crtcal bucklng loads are extremely senstve to the boundary condtons, shape and dmensons of ts cross secton of shear wall. Also the elastc and nelastc bucklng behavor for shear wall of unform symmetrc cross 75

2 ISSN: ISO 900:008 Certfed Volume, Issue, January 0 secton s dfferent than shear walls of monosymmetrc cross secton. A. Statement of the Problem Torson usually assumed to be secondary mportance and shear walls were often desgned to resst axal, bendng and shear forces only. If the shear wall s restrant at the ends aganst warpng, axal stresses wll result as well as a redstrbuton of the stresses. Hence, the statement of the problem n ths study s to fnd out the relatonshp between warpng affect and the shape of shear walls. Also, fnd out the capacty of the open shear walls to the crtcal loads B. Objectves The objectve of ths study was to carry out theoretcal parametrc studes obtaned by closed formulas and compare them wth the results of a numercal smulaton. The study also attempted to determne the optmum shear wall cross secton, whch has a major mpact on the structural behavour and desgn of hgh rse buldng. It s realzed that the fulflment of the followng sub-objectves would n turn fulfl the man objectve:. To analyze the effect of wall heght on capacty of renforced concrete open shear walls.. To analyze the effect of wall thckness on capacty of renforced concrete open shear walls.. To analyze the effect of monosymmetrcal of cross secton on capacty of renforced concrete open shear walls. 4. To analyze the effect of cross secton shape on capacty of renforced concrete open shear walls. 5. To analyze the effect of proporton of cross secton on capacty of renforced concrete open shear walls. C. Scope Theoretcal analyzes were developed to smulate the capacty of open shear walls, and then fnte element models were developed to check the stablty usng the ANSYS program. The analyss carred out s conducted on 8 renforced concrete open shear walls; the study s lmted to the followng scopes:. The shear wall s prsmatc.. The shear wall s long [(a\l), (b\l)] < 0.. Fve cases for shear wall heght effect are consdered (Ho,.5Ho,.50Ho,.75Ho, and.00ho). 4. Shear wall thckness s assumed 0, 0, 40, and 50 cm. 5. Symmetrc and monosymmetrc shear wall cross secton shape are consdered. 6. Two cases for cross secton shape are consdered. 7. Two cases for proportonal lmts s assumed.50, and.00. Conclusons from the current research and recommendatons for future studes are ncluded. II. THEORITICAL ANALYSIS A. Possble Bucklng The frst type s the torsonal bucklng, where the mddle part of shear wall rotates bodly relatve to the ends. It s lnked to low torsonal rgdty. The nstablty s possble through a combnaton of flexural and torsonal type accordng to the boundary condtons, dmensons and type of cross secton for long or ntermedate shear wall heght. Then, bucklng can be torsonal or flexural bucklng n the elastc range. The second type s the local bucklng whch appears as seres of waves along the heght of shear wall, whch s lmted only by the characterstc strength of materal due to local bucklng n the plastc or nonlnear range. B. Analytcal Analyss The second order theory s vald for shear walls wth arbtrary cross secton and boundary condtons. It can be appled to shear wall of large dmensons. The more mportant formulas repeated by the same appled notatons as n N. S. Trahar [8]. A crtcal forces can be calculated at whch the shear wall buckle out ts plane of ntal confguraton. It s also the hgher load at whch equlbrum postons wth zero dsplacement are possble. Smultaneously t s an magned load and a convenent reference load regarded as nstable one. Therefore, crtcal force can be expressed by closed mathematcal formulas depend on geometrc propertes of cross secton and Euler's loads. In ths way, crtcal force can be calculated separately or from combned loads. The equlbrum equatons for the stablty are proved from general forms of second order theory.. Crtcal longtudnal force P cr :- The crtcal longtudnal force s the smallest root obtaned from the followng general equaton:- 76

3 P Where ( p ISSN: ISO 900:008 Certfed Volume, Issue, January 0 A p A p A r A r A A ) P A A ( A r ( p p p p p p ) A r p p p 0 P EI A () P EI A () P r ( E A K s ), (4) r ( A) ( A) (5) n (9) H descrbe boundary condton of each dsplacement. Crtcal Moment Mcr :- Stablty condton s based on crtcal moment,mcr where crtcal moment s the smaller value from M cr and M cr for unform moment about η axs, M cr p (4p 4A r P P ) (A ) (0) By the same way crtcal moment about axs η can be obtaned as follows:- M cr A p (4A p 4A Ar P P ) (A ) (). Crtcal bmoment Accordng to nstablty condton, crtcal bmoment may be determned from loaded bmoment only as follows:- Bcr ( E A K ). () s H p () p (6) (7) (8) p ) III. NUMERICAL ANALYSIS In order to valdate the present formulatons, numercal analyss for the crtcal bucklng loads on open shear wall core fxed at the base are carred out usng the commercal fnte element program ANSYS (verson wth cvl FEM software) whch has been used for many analyses of structures n recent years. By usng ANSYS, there are two prmary means to perform a bucklng analyss: Egenvalue: Egenvalue bucklng analyss predcts the theoretcal bucklng strength of an deal elastc structure. It computes the structural egenvalues for the gven system loadng and constrants. Ths s known as classcal Euler bucklng analyss. Bucklng loads for several confguratons are readly avalable from tabulated solutons. However, n real-lfe, structural mperfectons and nonlneartes prevent most real- world structures from reachng ther egenvalue predcted bucklng strength;.e. t over-predcts the expected bucklng loads. Ths method s not recommended for accurate, real-world bucklng predcton analyss. Nonlnear: Nonlnear bucklng analyss s more accurate than egenvalue analyss because t employs non-lnear, large-deflecton; statc analyss to predct bucklng loads. Its mode of operaton s very smple: t gradually ncreases the appled load untl a load level s found whereby the structure becomes unstable (.e. suddenly a very small ncrease n the load wll cause very large deflectons). The true non-lnear nature of ths analyss thus permts the modellng of geometrc mperfectons, load perturbatons, materal nonlneartes and gaps. For ths type of analyss, note that small off-axs loads are necessary to ntate the desred bucklng mode. The nonlnear bucklng analyss procedure s used; the cvl FEM s adopted for the pre-processor whle the ANSYS s adopted for both the soluton and post-processor stage. 77

4 ISSN: ISO 900:008 Certfed Volume, Issue, January 0 For purposes of comparson, shear wall cores wth fve dfferent heghts are consdered, and for each heght of the cores, dfferent cross secton propertes are used to calculate the crtcal loads. The heghts adopted n the analyss are 60m, 75m, 90m, 05m, and 0m, and the secton propertes n all the analyses are showed n Table. All the shear wall cores have the same cross sectonal area for all the studed cases. Table Detals of the nvestgated shear wall cores. H (m) t a b c d e H t a b c d e a t e b d WALL CROSS SECTION c IV. ANALYSIS RESULTS AND DISCUSSION In total, fve parameters are nvestgated n the current study. These are: - Four values for shear wall cross secton thckness are consdered (0, 0, 40, and 50 cm.). - Fve values for shear wall heght effect are consdered (60, 75, 90, 05, and 0 m.). - Symmetrc and monosymmetrc shear wall cross secton are consdered. - Two cases for cross secton shape factor are consdered. - Two cases for proportonal lmts s assumed.5, and.0 Comparsons of the obtaned results from theoretcal and ANSYS were made and the convergence of the fve response parameters s shown n Table. Both procedures gave very smlar results. Table Crtcal bucklng load Pcr (/ 000 t) H m t cm F.B. ansys F.B. Theor. T.B. Theor

5 ISSN: ISO 900:008 Certfed Volume, Issue, January F.B. Flexural Bucklng T.B. Torson Bucklng The results obtaned from the analyss are presented and dscussed n such a way, that the effect of dfferent parameters can be separately nvestgated. The effect of each above mentoned parameters on crtcal bucklng forces was examned. A. Crtcal Bucklng Load Pcr The results of the models whch were nvestgated to study nfluence of core thckness on the bucklng load were presented n Fgures a, b. It has been already known a reducton n rotaton or warpng can be expected consequently as ncreasng the core walls thckness. The results show that the ncreasng the shear wall heght ( from H to H) leads to decrease the capacty by about 75% for wall thckness 0 cm. Increasng the wall thckness from 0 cm to 50 cm leads to ncreasng the flexural bucklng capacty by about 00%, whle the torson bucklng capacty ncreased by about 400% Fg Fg. -a Effect of thckness on P cr t = 0. m & Fg. -b Effect of thckness on P cr t = 0.5 m Nonsymmetry of shear wall affect on bucklng behavour under axal force was shown n Fgure. Nonsymmetry s assumed by cancellng the lpped stffener from one edge of the core cross sectonal area. Increasng the shear wall heght ( from H to H) leads to decrease the flexural bucklng capacty by about 0% for wall thckness 0 cm., on the other hand, the torsonal bucklng capacty decreases by about 40%. The shape of cross secton s assumed by cancellng the lpped stffener from the two edges of the core cross sectonal area, keepng the total cross sectonal area of the core s the same as all models. The affect of shape rato of shear wall cross secton on bucklng behavour under axal force was shown n Fgure. For shape rato of.5 of cross secton, the flexural bucklng capacty ncreased and the torsonal bucklng capacty decreased compared wth shape rato.0. 79

6 ISSN: ISO 900:008 Certfed Volume, Issue, January 0 Fg. Effect of monosymmtry on P cr Fg. Effect of shape on P cr The proporton (degree of rectangularty of cross secton rato) effect on the shear wall capacty s shown n Fgures 4a, b. The results ndcate that as the rato of rectangularty ncreases, the flexural bucklng capacty decrease by about 60%. Also, the results show that there s no effect for the proporton rato on the torsonal bucklng capacty. Fg. 4-a Effect of proporton (=.5) on P cr Fg. 4-b Effect of proporton (=.0) on Pcr 80

7 ISSN: ISO 900:008 Certfed Volume, Issue, January 0 B. Crtcal Bucklng Moment Mcr Also, the results obtaned from the analyss are presented and dscussed n such a way, that the effect of dfferent parameters can be separately nvestgated. The effect of each parameter on crtcal bucklng moment was examned. The results of the models whch were nvestgated to study nfluence of core thckness on the bucklng moment were presented n Fgures 5a, b. The results show that the ncreasng the shear wall heght ( from H to H) leads to decrease the bucklng moment capacty by about 65% for wall thckness 0 cm. Increasng the wall thckness from 0 cm to 50 cm leads to ncreasng the bucklng moment capacty by about 400%. Fg. 5-a Effect of thckness on M cr t = 0. m Fg. 5-b Effect of thckness on M cr t = 0.5 m Nonsymmetry of shear wall affect on bucklng behavour under moment was shown n Fgure 6.. Increasng the shear wall heght ( from H to H) leads to decrease the bucklng moment capacty by about 40% for wall thckness 0 cm. The affect of shape rato of shear wall cross secton on bucklng behavour under moment was shown n Fgure 7. Fg. 6 Effect of monosymmtry on M cr Fg. 7 Effect of shape on M cr The proporton (degree of rectangularty of cross secton rato) effect on bucklng moment capacty s shown n Fgures 8a, b. The results ndcate that as the rato of rectangularty ncreases, the bucklng moment capacty decrease by about 0%. 8

8 ISSN: ISO 900:008 Certfed Volume, Issue, January 0 Fg. 8-a Effect of proporton (=.5) on M cr Fg. 8-b Effect of proporton (=.0) on M cr V. CONCLUSION On the bass of the precedng study and lmtatons, the followng conclusons are drawn:- The absolute crtcal forces and type of bucklng of the shear wall have been shown to be nfluenced strongly by the thcknesses of shear wall cross secton. The capacty of shear wall has been shown to be nfluenced strongly by the lpped stffener length. The desgner should take nto account the effect of these crtcal forces and the expected type of bucklng at whch they act. In practce, thckness, geometrcal parameters of cross secton must be taken nto consderaton for obtanng the capacty of the open shear wall core. In spte of the fact that the fnte element method s well suted for establshng the crtcal forces accordng to characterstc and boundary condtons on shear wall stablty, t s better and easy to check the shear wall stablty by usng the obtaned closed formulas for rabd estmaton of crtcal forces. Because the relatve nfluence of changng the dmensons of the consdered parameters on core torsonal behavour have been presented to facltate a reasonable confguraton for cores subject to torson, the ratonal method of replacng central core by regular shape must be depended on the closed formulas for suggeston the optmal cross secton of shear wall to reduce the torsonal stresses and ts effect on capacty of shear wall. A dmensonal nvestgaton has been developed for a lnear structural torsonal analyss of shear wall usng analytcal and numercal analyss. The method of analyss dffers from prevously publshed technques; t allows recognzng forces and geometrcal propertes of core cross secton. Wthn the lmtatons of assumptons adopted and practcal range of varable examned n ths study. REFERENCES [] A.Nader, and A. R. Sad (0). An analytcal soluton for bucklng of moderately thck functonally graded sector and annular sector plates. Journal of Arch Appl. Mech. 8, [] D. Camotm, N. Slvestre, R. Goncalves, and P.B. Dns. (006). GBT Based structural analyss of thn walled members. Advances n engneerng structures, mechancs & constructon, 87-04, Netherlands. [] Evangelos J. S. and John A. D. Lateral bucklng analyss of arbtrary cross secton by BEM (009), Journal of comp. Mech. Vol. 45 September, -. [4] Jong Dar (009). Lateral bucklng analyss of angled frames wth thn-walled I-beams. Journal of Marne scence and technology, Vol. 7-, 9-, Tape, Tawan. 8

9 ISSN: ISO 900:008 Certfed Volume, Issue, January 0 [5] Seung E. K., Jaehong L., and Joo S. P. (00). -D second-order plastc-hnge analyss accountng for lateral torsonal bucklng, Internatonal Journal of solds and structures, Vol. 9, [6] Paul A. S. and Charles J. C. (00). Torsonal analyss of structural steel member, AISC, Steel desgn gude seres, October, Chcago, USA. [7] Tan Yu Cha (005). Warpng behavor of cantlever steel beam wth openng, Thess of master submtted at unversty of Technology Malaysa. [8] N. S. Trahar (00). Non-lnear elastc non-unform torson, Thess of PHD submtted at unversty of Sydney Australa. [9] Atzber L., Danny J.Y., and Mguel A. S. (006). Lateral torsonal bucklng of steel. Stablty and Ductlty of Steel Structures, September 6-8, Lsbon, Portugal. [0] Thue P. V. and Jaehong L (00). Free vbraton of axally loaded thn-walled composte Tmoshenko beams. Journal of Arch Appl. Mech. September, Vol 0, LIST OF SYMBOLS Cross secton area of shear wall. P The Euler flexural bucklng loads,, r Polar radus of gyraton Shear canter poston from the centred shear of wall cross secton, A A. A Young's modulus. Prncpal moment of nerta,. Crtcal bendng moment cr cr,cr. Monsymmetrc parameter, s H cr,. Torsonal rgdty for open shear wall secton. Total heght of shear wall. Descrbe boundary condton of each dsplacement. Warpng moment of nerta (prncpal sectaran moment) Crtcal bmomert 8