Modifying the Shear Buckling Loads of Metal Shear Walls for Improving Their Energy Absorption Capacity

Transcription

1 Modfyng the Shear Bucklng Loads of Metal Shear Walls for Improvng Ther Energy Absorpton Capacty S. Shahab, M. Mrtaher, R. Mrzaefa and H. Baha 3,* George W. Woodruff School of Mechancal Engneerng, Georga Insttute of Technology, Atlanta, GA 3033, USA Department of cvl Engneerng KNT Unversty, Tehran, Iran 3 School of Engneerng and Desgn, Brunel Unversty, Uxbrdge, UB8 3PH, UK (Receved: 3 June 00; Receved revsed form: 7 March 0; Accepted: 4 March 0) Abstract: In ths paper, an approxmate method s proposed for achevng predefned ncreases n the bucklng threshold of a metal shear wall n order to ncrease ts energy absorpton capacty. The frst and second-order dervatves of shear bucklng loads of a shear wall wth respect to the thckness n ts dfferent regons are calculated. Based on these egendervatves, and by usng the frst and second order Taylor expansons, the necessary change n the thckness of plate n varous regons s calculated for ncreasng the shear bucklng loads by a specfc value. The presented modfcaton algorthm s mplemented for shear walls wth dfferent aspect ratos, materal propertes and boundary condtons. An ntal senstvty analyss s carred out for fndng the regons wthn the shear wall where modfyng the thckness has the most nfluence on the bucklng loads. Based on the senstvty analyss results, approprate regons of plate are selected and the necessary modfcaton n thckness of these regons s calculated for achevng a relatvely large predefned change n shear bucklng load. By smulatng the post-bucklng response of both ntal and modfed plates n a case study, the mprovement n the energy absorpton capablty of the modfed plate s also studed. Key words: shear wall, shear bucklng, egenvalue, egenvector, senstvty analyss.. INTRODUCTION In earthquake engneerng, reducng the earthquake damage on buldngs has been one of the most mportant ssues n recent years. Passve energy dsspaton systems are consdered as one of the basc technologes used to protect buldngs from earthquake effects and mnmze sesmc damage. These systems nclude a range of materals and devces for enhancng dampng, stffness and strength whch mtgate the sesmc hazards (Soong and Dargush 997; Constantnou et al. 998; Soong and Spencer 00). Sesmc dampers are one of the passve systems whch are nstalled n the buldngs to absorb the energy of the ground moton n the buldngs. When sesmc energy s transmtted through the dampers, a porton of energy s dsspated and the moton of buldng s damped (Soong and Spencer 00). Among varous types of sesmc dampers, the metallc yeld dampers are effcent means to dsspate the earthquake energy through nelastc deformaton of metals. The metallc yeld dampers can be found n dfferent geometrc confguratons of X- shaped, E-shaped, honeycomb-shaped and shear walls. In earthquake engneerng, thn steel shear walls have had an mportant role because of ther unque performance characterstcs. Shear walls can easly provde hgh n-plane stran and stffness. In addton, * Correspondng author. Emal address: hamd.baha@brunel.ac.uk; Fax: ; Tel: Advances n Structural Engneerng Vol. 4 No

2 Modfyng the Shear Bucklng Loads of Metal Shear Walls for Improvng Ther Energy Absorpton Capacty unform shear stress dstrbuton throughout the plate makes t capable of dsspatng a large amount of energy due to the large sze of yeldng regon. The desgn of shear walls has been an actve feld of research snce the early work of Kulak (985), and Takahash et al. (973). Several theoretcal and practcal analyses related to desgn of shear walls have been presented n the lterature (Tromposch and Kulak 987; Elgaaly 997, 998; Alna and Dastfan 006; Mattes et al. 008). In usng shear walls as energy dsspatng systems, ncreasng the shear bucklng threshold s a maor challenge for desgners because the occurrence of shear bucklng mpedes development of pure shear mechansm nto plastc deformaton. As a consequence, the lower shear bucklng threshold leads to a lower energy absorpton capacty of the shear walls. In order to guarantee plastc deformatons before the occurrence of shear bucklng, low yeld shear walls (LYSW) were the frst proposed soluton. These shear walls were made by low yeld strength metals such as steel and alumnum alloys because of ther stable hysteretc behavor up to large deformatons (Nakagawa et al. 996; Ra and Wallace 998). In ths paper, a new method s ntroduced for modfyng the shear bucklng threshold. In ths method, the amount of change n the bucklng load s predefned and can vary accordng to desgn specfcatons. By changng the thckness n some specfc regons whch are dentfed by carryng out an ntal senstvty analyss, the shear bucklng load can be modfed by any desred value. Structural modfcaton by consderng the geometrc or physcal propertes as desgn parameters s a known process for structures and many efforts have been made by researchers to mplement dfferent applcatons of these methods. In desgn applcatons, calculatng the dervatves of egenvalues and egenvectors wth respect to arbtrary parameters for a structural system has mportant mplcatons. Based on Taylor expanson, the egendervatves can be used to approxmately formulate the drect and nverse egenvalue problems. Drect approxmaton for egenvalue problem allows the calculaton of the change of each egenvalue due to an arbtrary change n desgn parameters. In the nverse approxmate egenvalue problem formulaton, the requred structural modfcatons to acheve predefned modfcatons n egenvalues and egenvectors are computed. For structural responses that are formulated n egenvalue form (such as free vbraton and lnear bucklng) the tme consumng teratve methods may be avoded by defnng the problem as an nverse egenvalue problem (Aryana et al. 007; Mrzaefar et al. 007; Mrzaefar et al. 008a, b). Recently, Mrzaefar et al. (009) proposed an approxmate method for smultaneous modfcaton of natural frequences and bucklng loads of thn rectangular sotropc plates, and Shahab et al. (009) nvestgated a new method for performng predefned smultaneous modfcaton of natural frequences and bucklng loads of composte cylndrcal panels. In the present paper, the bucklng load and the bucklng mode dervatves of a shear wall wth respect to any geometrcal or materal property are calculated and used for formulatng the approxmate egenvalue problems. The drect egenvalue problem s defned as calculatng the correspondng changes n the plate bucklng loads due to changes n an arbtrary structural property and the nverse egenvalue problem s defned as the determnaton of the requred geometrcal and/or physcal changes n the plate for achevng the predefned shear bucklng loads. Modfcaton of the shear bucklng load can cause the shear wall to undergo plastc deformatons pror to the occurrence of shear bucklng whch can n turn lead to the energy absorpton capacty of the wall beng ncreased remarkably (Mattes et al. 008). In all the prevously reported methods of modfyng the shear bucklng threshold (e. g. addng stffeners to the plate or changng the materal propertes), the amount of change n the bucklng load cannot be specfed by the desgner. In the present study, a new applcaton of a prevously developed method (Mrzaefar et al. 009; Shahab et al. 009) s ntroduced whch allows the desgner capable toncrease the lateral bucklng load by a predefned value. Two steel and alumnum plates wth dfferent aspect ratos and boundary condtons are modeled and the proposed modfcaton algorthm s mplemented for these case studes. The thckness of plate n varous regons s consdered as the desgn varable and a senstvty analyss s performed for fndng the regons n whch changng the thckness has the most nfluence on ncreasng the bucklng loads. For both the case studes, the necessary modfcaton n desgn varables s calculated for achevng a predefned ncrease n the shear bucklng loads. In these cases, the fnte element method s used for calculatng the change of bucklng loads by mplementng the computed approxmate desgn parameters and t s shown that wth the proposed method, accurate modfcatons n the shear bucklng loads are obtaned, wthout performng numerous desgn teratons. Also, n a case study the effect of mplementng the proposed modfcaton on the postbucklng response of a shear wall s studed and t s 48 Advances n Structural Engneerng Vol. 4 No. 6 0

3 S. Shahab, M. Mrtaher, R. Mrzaefa and H. Baha shown that ncreasng the shear bucklng threshold has a sgnfcant effect on the energy absorpton capacty of plate.. THEORY In ths secton, the fnte element formulaton for lnear bucklng analyss of a rectangular plate under unform shear loads s derved by rewrtng the stablty equaton n the standard egenvalue form. The frst and second order dervatves of egenvalues whch are the shear bucklng loads are calculated and based on these egendervatves, and by usng frst and second-order Taylor expansons the frst and secondorder approxmaton for modfyng shear bucklng loads are formulated... Bucklng Loads In ths nvestgaton, the lnear bucklng egenvalue problem n the context of a fnte element formulaton s consdered. Therefore, usng nonlnear terms n the components of the stran s necessary. For a plate n x-y plane, the followng nonlnear stran feld should be consdered: w w w εxx = εyy γ xy () x = y = w,, x y where, ε xx, ε yy and γ xy are the n-plane nonlnear terms of stran and w s the out of plane dsplacement. As t s shown n Eqn (), for lnear bucklng analyss of plates, only one term n nonlnear components of stran s consdered and other terms can be neglected due to the flat geometry of the structure (Chen and Lew 004). The total potental energy functonal consstng of the stran energy and work done by ntal stresses due to nplane loads can be wrtten as: () The term U b stands for stran energy due to bendng (Mrzaefar et al. 009), and the term U m presents membrane stran energy assocated wth lateral deflecton, w = w(x, y), of the plate: U m = x y w x w y UT = Ub + Um T N N x xy N N (3) where, N x, N y are the n-plane forces n the x and y drectons and s the n-plane shear force whch s the only nonzero force n ths paper. By usng the well-known procedure n the fnte xy y w x dx dy w y element method (Reddy 993), the total potental energy can be expressed n terms of nodal varables as: e T e e e T e e UT = { d } [ K ]{ d } + { d } [ Kσ ]{ d }, (4) where, [K e ] and [K e σ] are the element structural stffness and geometrc stffness matrces, respectvely and {d e } s the generalzed nodal dsplacement vector. Mnmzaton of the functonal presented n Eqn 4 leads to the element governng equaton whch can be wrtten as: (5) Usng the fnte elements assembly procedure (Reddy 993) to obtan the global structural and geometrc stffness matrces, the governng equaton for the bucklng analyss of a plate s presented as: (6) where, [K], [K σ ] and {d} are the global stffness matrx, geometrcal stffness matrx and generalzed nodal dsplacement vector. For lnear bucklng analyss, the governng equaton can be rewrtten n the form of an egenvalue problem: [ K] d λ[ K σ ] d, (7) where, [K σ] * s the geometrc stffness matrx assocated wth a reference axal load.e. a unt shear load. Solvng the egenvalue problem expressed n Eqn 7 gves λ whch after multplcaton by the reference axal load specfes the crtcal shear bucklng load. Solvng ths equaton, also gves the egenvectors whch represent the bucklng modes assocated wth each shear bucklng load... The Frst and Second-Order Dervatves of Egenvalues By takng Eqn (7) nto consderaton and followng the procedure detaled n reference (Mrzaefar et al. 009), the rate of change of th egenvalue wth respect to an arbtrary parameter (b ) for bucklng loads s expressed as λ = K e e e e σ 0 [ K ]{ d } + [ K ]{ d } = { }. [ K]{ d} + [ K ]{} d = {}, mn * { }= { } d m dn σ 0 Kσ mn λ dm dn mn, =,,..., N (8) where N s the number of degrees of freedom for the system and the summaton conventon s used for the subscrpts. In addton, by usng only the frst term n Taylor expanson, the change of each bucklng load (as * Advances n Structural Engneerng Vol. 4 No

4 Modfyng the Shear Bucklng Loads of Metal Shear Walls for Improvng Ther Energy Absorpton Capacty the egenvalues) can be expressed as (9) To obtan the second order dervatve of egenvalues, Eqn 8 s dfferentated wth respect to an arbtrary parameter b k λ k whch shows the dependency of the second dervatve of egenvalue to the egenvector dervatve. For calculatng the egenvector dervatves, modal superposton assumpton whch expresses the egenvector dervatves as a lnear combnaton of all the egenvectors s used (Fox and Kapoor 968) () Followng the procedure explaned n (Mrzaefar et al. 007) the unknown coeffcents a kp n Eqn can be calculated. Usng the frst two terms of Taylor expanson, the second order approxmaton for shear bucklng loads can be expressed as () The necessary changes n the parameter b to acheve the desred egenvalues can be calculated based on the nverse method expressed n Eqn 9 for the frst order approxmaton or by solvng the system of equatons presented n Eqn for the second order approxmaton. For mplementng the frst order approxmaton, n order to solve the nverse problem (fndng the necessary changes n the parameter b to acheve the desred egenvalues) Eqn 8 and 9 are consdered and for the second order approxmaton, n the frst step, for each egenvector the coeffcents a k are calculated and the dervatve of each egenvecto r wth respect to b s calculated usng Eqn. Then, the values are calculated usng Eqn 0. For fndng λ k = d m k K K d λ λ λ = b + ( b) n k mn mn k λ λ = b λ Kσ = a p kp dn * mn dn + d K mn λ K λ σ * k k m * σ mn d n (0) the necessary modfcaton to acheve the desred changes n the egenvalues, quadratc system of equatons presented n Eqn are solved by settng the change of shear bucklng loads ( λ ) as known parameters and necessary changes n the structural propertes as the unknown parameters. The procedure for mplementng the ntroduced modfcaton algorthm ncludng nverse and drect approxmate methods s shown n the flowchart of Fgure. 3. NUMERICAL RESULTS 3.. Verfcaton of the Fnte Element Model To ensure the accuracy of the proposed fnte element model and examne the results of the developed code for shear bucklng analyss, a plate smply supported n all edges made of Alumnum EN-AW 050A wth materal propertes mentoned n Table s consdered as the case study. The plate length, and thckness are set to a =.5 m and h = 5 mm, respectvely and two dfferent values are consdered for the plate wdth. A total of 64 (8 8) four node elements are used n the FE modelng. The plate s subected to shear loads around all edges whch cause a unform dstrbuton of shear stresses n the plate. The shear bucklng loads of ths plate are calculated usng a code developed n ths work based on the presented fnte element formulaton. The results for the crtcal bucklng load are compared aganst the results obtaned from the commercal fnte element code ABAQUS n Table. 3.. Inverse Approxmate Egenvalue Problem In ths secton, shear bucklng modfcaton of two plates s carred out by usng the nverse egenvalue problem. By performng the structural modfcaton of ths secton, the necessary changes n the thckness of the structure are calculated n order to acheve desred modfcatons n shear bucklng loads. As wll be shown n the followng case studes, performng a modfcaton for shear bucklng loads wth a sgnfcant change from the ntal confguraton usually does not satsfy the requred precson n one step. To deal wth ths problem, a repeated procedure s used for achevng the desred modfcaton A repeated procedure for precse modfcaton of shear bucklng loads As the frst case study, consder a smply supported rectangular plate wth ntal thckness of 5 mm. All the geometrc and materal specfcatons are the same as those n the case study of Secton 3.. In ths case study, the plate s dvded nto eght dentcal regons as shown n Fgure. In the fnte element model, the elements n each regon are consdered as a group and 50 Advances n Structural Engneerng Vol. 4 No. 6 0

5 S. Shahab, M. Mrtaher, R. Mrzaefa and H. Baha Structure's egenvalues and egenvectors are calculated usng Equaton (7) Dervatves of structural stffness and geometrc stffness matrces are calculated The frst and second order dervatves of bucklng loads are calculated usng Equatons (8) and (0) For nverse approxmate method Equaton (9) and () for the frst and second order approxmatons are consdered respectvely by settng the parameters λ as known values and the parameters b as the unknowns For drect approxmate method New values of bucklng loads are obtaned usng Equaton (9) for the frst order and Equaton () for the second order approxmatons The necessary modfcaton n desgn parameters to acheve the desred changes n bucklng loads are found No Errors are acceptable Yes New desgn parameters gve the predefned bucklng loads Fgure The procedure for mplementng the modfcaton algorthm Table. Mechancal features of alumnum EN-AW 050A and low-yeld-strength (LYS) steel Materal E (GPa) ν b x Alumnum(EN-AW 050A) LYS Steel a b Table. Comparson of crtcal shear bucklng load (KN/m) for a smply supported rectangular plate under n-plane shear force b h Developed code ABAQUS the thckness of plate n each group s expressed as a desgn parameter b. A predefned ncrease of 30% for the crtcal shear bucklng load s consdered and the second order approxmaton s mplemented for fndng the requred change n the thckness of each a y Fgure The element groups for rectangular smply supported plate Advances n Structural Engneerng Vol. 4 No

6 Modfyng the Shear Bucklng Loads of Metal Shear Walls for Improvng Ther Energy Absorpton Capacty Table 3. Modfcaton of the crtcal shear bucklng load (KN/m) of a rectangular plate smply supported n all edges (see Fgure for element groups) λ Intal Modfed (exact) Modfed (S.O.) Thckness modfcaton (mm) Step λ 30% h 3 =.5 Step λ 0.5% h 7 =. Step 3 λ.3% h 4 = 0.3 S. O. Second order approxmaton. regon for modfyng the shear bucklng load by ths amount as shown n Table 3. In the frst step of modfcaton, thckness of plate n regon number 3 s consdered as the desgn parameter. Solvng the quadratc equaton presented n Eqn gves an ncrease n the thckness of ths group by amount of.5 mm. Implementng ths change on the thckness of plate reveals that the modfcaton error s not neglgble. So, n step the quadratc equaton for second order approxmaton s consdered wth new thcknesses (7.5 mm for group number 3 and 5 mm for all the other groups) as the ntal confguraton for the modfcaton. In ths step, predefned modfcaton s consdered to be equal to the errors of the second order approxmaton n step. Changng the thckness n group 7 s nvolved n modfyng the shear bucklng load and solvng the quadratc equaton gves an ncrease n the thckness of group 7 by amount of. mm. In step 3 whch s the last step, the algorthm starts wth the new thcknesses (7.5 mm and 6. mm for groups number 3 and 7, respectvely and 5 mm for all the other groups). The modfcaton value s the error of second order approxmaton obtaned n step, and changng the thckness n group 4 s chosen as the desgn parameter n ths step. Solvng the quadratc equaton gves the fnal thcknesses as 7.5 mm, 6. mm and 5.3 mm for group number 3, 7 and 4, respectvely, and 5 mm for the other groups to acheve the shear bucklng load wth neglgble dfference of 0.4% wth the desred shear bucklng load. In practce, after performng a lmted number of teratons (3 n the presented case study) the rate of modfcaton slows down and from the computatonal pont of vew t s not worthwhle to contnue the steps. However, the convergence crteron depends on the requred accuracy and the teratons can be contnued untl the requred accuracy s acheved, as t s shown n Table 3, the fnal modfed egenvalue s 7040 KN/m whch s n a good agreement n the case where a shear bucklng load of 7050 KN/m was desred. As another example, consder a LYS steel plate smply supported n all edges wth the mechancal propertes gven n Table. The ntal thckness of the plate s 5 mm and the plate has a square shape wth length a = m. In ths case study, a senstvty analyss s frst carred out usng second order approxmaton n order to fnd approprate regons wthn the structure n whch the thckness change has the most nfluence on the shear bucklng load. Usng Eqn 8 and 0, the rate of change of each egenvalue wth respect to change of the thckness n each element n the fnte element model s calculated. By modelng the plate wth 44 elements ( ), the contour plot for dstrbuton of the second dervatve of the crtcal bucklng load wth respect to each element s thckness s shown n Fgure 3. Note that, the frst dervatve has a smlar dstrbuton wth dfferent values. By consderng ths counter plot, t s obvous that changng the thckness n the regons near the dagonal have most nfluence on modfyng the crtcal bucklng load. By consderng the fact that n practce shear walls are subected to cyclc loadng n opposte drectons, t s obvous that the thckness n the regons near both dagonals have the same effect on the shear bucklng load n recprocal loadngs. Based on the performed senstvty analyss, the thckness n three Fgure 3 Dstrbuton of the second dervatve of the crtcal shear bucklng load wth respect to thckness for a square plate smply supported n all edges Advances n Structural Engneerng Vol. 4 No. 6 0

7 S. Shahab, M. Mrtaher, R. Mrzaefa and H. Baha regons as shown n Fgure 4 are chosen as the desgn parameters and the procedure s repeated for modfcaton of crtcal shear bucklng load. Table 4 shows the modfcaton procedure. An ncrease of 30% n the crtcal shear bucklng load s consdered as the goal and the nverse problem s solved usng the second order approxmaton. A repeated procedure s carred out and the change of shear bucklng load due to changng thckness n groups, and 3 s obtaned. By performng three steps, the crtcal shear bucklng load s modfed up to the predefned value wth an error of 0.35%. As t s shown n the experments on post-bucklng response of structures, by ncreasng the rate of loadng on the structure, hgher bucklng loads are actvated (Shaker et al. 007). So n the case of dynamc loadng (whch occurs n the applcatons of shear walls), hgher bucklng loads and bucklng modes rather than the crtcal bucklng load may affect the structural response. The proposed method n ths paper s capable of modfyng the hgher bucklng loads as well as the crtcal shear bucklng load. In the followng case study, modfcaton of the frst two bucklng loads s consdered. A square steel plate wth smply supported horzontal edges and free vertcal y Smply supported Fgure 4. The element groups for smply supported square plate selected based on senstvty analyss 3 x edges s taken nto consderaton whch s a common boundary condton n applcaton of shear walls n buldngs. Plate s ntal thckness s 5 mm whch wll be modfed n some regons. An ntal senstvty analyss s performed for the frst two bucklng loads by calculatng the frst and second order dervatves of each bucklng load wth respect to the plate thckness n dfferent regons. Dstrbuton of the second order dervatve of the crtcal shear bucklng load wth respect to thckness s depcted n Fgure 5(a). As t s shown the hghly senstve regons are located on the free edges and near the corners. Fgure 5(b) shows the second order dervatve of the second shear bucklng load wth respect to the plate thckness. It can be seen that the thckness n the regons n the corners and also n the md-span of the vertcal free edges have the most nfluence on the second shear bucklng load. It s worthy to note that, dstrbuton of the frst order dervatves has the same pattern as shown n Fgure 5 but wth dfferent values. Based on the senstvty analyss results, the thckness n four regons s selected as the desgn varable (see Fgure 6). Modfyng the frst two shear bucklng loads by amount of 0% s consdered as the desgn goal. As t s shown n Table 5, a two step modfcaton s performed and n each step a set of two smultaneous quadratc equatons (as presented n Eqn ()) s solved for fndng the necessary change n two regons. The modfcaton n the thckness of these four regons s shown n Table 5. In the fnal step the second bucklng load s modfed to a desred value exactly and the crtcal bucklng load s modfed to the predefned value wth an acceptable accuracy The Effect of Proposed Modfcaton on the Post-Bucklng Behavor As explaned prevously, the ultmate goal of ncreasng the lateral bucklng threshold n shear walls usng the proposed method n ths work s to ncrease the energy absorpton capablty of shear walls. The fact that ncreasng the shear bucklng loads mproves the energy absorpton capacty of shear walls s a well known phenomenon studed by many researches n the lterature and t s beyond the scope of ths paper to Table 4. Modfcaton of crtcal shear bucklng load (KN/m) for a square plate smply supported n all edges (see Fgure 4 for element groups) λ Intal Modfed (exact) Modfed (S.O.) Thckness modfcaton (mm) Step λ 30% h =.5 Step λ 5.4% h =.3 Step 3 λ.06% h 3 = 0. Advances n Structural Engneerng Vol. 4 No

8 Modfyng the Shear Bucklng Loads of Metal Shear Walls for Improvng Ther Energy Absorpton Capacty (a) Smply supported x Free (b) y Fgure 6. The element groups for square plate wth two smply supported and two free edges selected based on senstvty analyss Fgure 5. Dstrbuton of the second dervatve of (a) the crtcal shear bucklng load and (b) the second bucklng load wth respect to thckness expound on ths phenomenon. However, n order to study the effect of proposed modfcatons on the energy absorpton capacty of shear walls, the postbucklng behavor of a shear wall before and after mplementng the modfcaton for ncreasng the lateral bucklng threshold s studed n ths secton. The square LYS steel plate studed n secton 3.. s consdered as a case study. The plate s smply supported around all ts edges. The plate has a square shape wth length a = m and ts ntal thckness s 5 mm. The modfed thckness for ncreasng the crtcal shear bucklng load by an amount of 30% s gven n Table 4 (see Fgure 4 for the element groups). The nonlnear stress-stran materal response s consdered to be the same as UNS30403 alloy. These nonlnear materal propertes for ths alloy are gven n (Rasmussen 003) but for comparson purposes, the elastc modulus s consdered to be the same as prevous case studes for LYS steel (see Table ). The commercal fnte element software ABAQUS s used for smulatng the post-bucklng behavor of both ntal and modfed plates. An ntal lnear bucklng analyss s carred out for calculatng the shear bucklng loads and bucklng modes. These modes are mplemented as an mperfecton n the fnte element model and the Rks method s used for calculatng the post-bucklng response of the plate. Both the ntal plate (wth Table 5. Modfcaton of the frst and second shear bucklng loads (KN/m) for a square plate smply supported n two edges (see Fgure 6 for element groups) λ Intal Modfed (exact) Modfed (S.O.) Thckness modfcaton (mm) Step λ 0% h = 0.3, λ 0% h = 0.7 Step λ 4.7% h 3 = 0.4, λ 0.84% h 4 = Advances n Structural Engneerng Vol. 4 No. 6 0

9 S. Shahab, M. Mrtaher, R. Mrzaefa and H. Baha (a) (b) Uz +.50e e e e e 0 +.7e e e e e e e e 03 Z Y X Uz +.50e e e e e e e e e e e e e 03 Fgure 7. The out-of-plane deflectons n the post-bucklng analyss of (a) ntal; and (b) modfed plate (The ponts of maxmum deflecton are shown by A and B) unform thckness) and modfed plate (wth thckness dstrbuton gven n Table 4) are analyzed. For comparson purposes, n both cases the soluton s stopped when a maxmum out-of-plane deflecton of.5 cm s reached. Fgure 7 shows the deformed shape of (a) ntal and (b) modfed plate and the out-of-plane deflectons. The ponts at whch the deflecton s maxmum s shown by symbols A and B for the ntal and modfed plates, respectvely. As t s shown, due to change of thckness n dfferent regons n the modfed plate, the maxmum deflecton s correspondng to a dfferent locaton from the ntal plate. As t shown n Fgure 7, the proposed modfcaton for ncreasng the crtcal shear bucklng load, changes the post-bucklng response remarkably. The varaton of appled shear load wth the out-of-plane deflecton for ntal and modfed plates s shown n Fgure 8 (see Fgure 7 for the ponts at whch the out-of-plane deflecton s measured). As t s seen n ths fgure, the nonlnear response for the modfed plate starts at a hgher shear force compared wth the ntal plate. Ths ncrease of the maxmum force n the lnear response of modfed plate shows the ncrease of crtcal shear bucklng load that s mplemented on the plate usng the proposed modfcaton n secton 3... Fgure 8 shows the sgnfcant effect of ncreasng the shear bucklng load threshold on the appled shear force n the post-bucklng response. As ponted out prevously ths modfcaton n the shear force causes an ncrease n the absorbed energy durng post-bucklng response. Fgure 8 shows the varaton of plastc dsspated energy wth the out-of-plane deflecton for both the orgnal plate and the modfed plate. It s seen that n the same out-of-plane deflecton, the modfed plate absorbs 6% more energy compared to the orgnal plate. It s worth notng that ths ncrease n the energy absorpton capacty s acheved by ncreasng the weght of the structure by less than 7% and ncreasng the energy Shear force (N/m) Modfed plate 0. Intal plate Maxmum out-of-plane deflecton (m) Fgure 8. The appled shear load versus the out-of-plane deflecton for ntal and modfed plates Advances n Structural Engneerng Vol. 4 No

10 Modfyng the Shear Bucklng Loads of Metal Shear Walls for Improvng Ther Energy Absorpton Capacty Plastc dsspated energy (N.m) Modfed plate Intal plate Maxmum out-of-plane deflecton (m) Fgure 9. The plastc dsspated energy versus the out-of-plane deflecton for ntal and modfed plate. absorpton capacty can be controlled by settng the modfcaton n shear bucklng threshold usng the proposed method of ths paper. It s worth notng that studyng the dfferent possble methods (such as addng metal sheets usng bolted onts or machnng a thck plate) for changng the thckness n varous regons and consderng the effcency of these methods from an economc pont of vew s an open feld of research that may be consdered n future works. 4. CONCLUSIONS In ths study, an effcent formulaton has been developed for ncreasng the shear bucklng threshold of metal shear walls whch leads to an mprovement n ther energy absorpton capacty. For performng ths modfcaton, the frst and second order dervatves of bucklng loads wth respect to the shear wall thckness n dfferent regons are calculated. Based on these egendervatves and by usng the frst and second order Taylor expansons the modfcaton problem s formulated n the form of a system of algebrac equatons. A senstvty analyss s performed for fndng the regons wthn the shear wall n whch a change n thckness has the most nfluence on the shear bucklng loads. Approprate regons of plate are selected based on the senstvty analyss and shear bucklng loads are ncreased by relatvely large values (up to 30%) by fndng the necessary change n the thckness of these regons. Varous case studes are presented and the accuracy of the proposed method s shown for shear walls wth dfferent aspect ratos, materal propertes and boundary condtons. Also, n a case study the nfluence of mplementng the proposed modfcaton for ncreasng the bucklng threshold on the post-bucklng response of the shear wall s studed and t s shown that the proposed modfcaton sgnfcantly mproves the energy absorpton capablty of the shear wall. REFERENCES Alna, M.M. Dastfan, M. (006). Behavour of thn steel plate shear walls regardng frame members, Journal of Constructonal Steel Research, Vol. 6, No. 7, pp Aryana, F., Baha H., Mrzaefar, R., Yelagh, A. (007). Dynamc behavor modfcaton of FGM plate wth ntegrated pezoelectrc layers usng frst and second-order approxmatons, Internatonal Journal for Numercal Methods n Engneerng, Vol. 70, No., pp Chen, X.L. and Lew, K.M. (004). Bucklng of rectangular functonally graded materal plates subected to nonlnearly dstrbuted n-plane edge loads, Smart Materal and Structures, Vol. 3, No. 6, pp Constantnou, M.C., Soong, T.T. and Dargush, G.F. (998). Passve Energy Dsspaton Systems for Structural Desgn and Retroft, Multdscplnary Center for Earthquake Engneerng Research, Buffalo NY, USA. Elgaaly, M. and Lu, Y. (997). Analyss of thn steel plate shear walls, Journal of Structural Engneerng, ASCE, Vol. 3, No., pp Elgaaly, M. (998). Thn steel plate shear walls behavor and analyss, Thn-Walled Structures, Vol. 3, No. 3, pp Fox, R.L. and Kapoor, M.P. (968). Rates of change of egenvalues and egenvectors, AIAA Journal, Vol. 6, No., pp Kulak, G.L. (985). Behavor of Steel Plate Shears Walls, Amercan Insttute of Steel Constructon, Chcago, IL, USA. De Mattesa, G., Mazzolanb F.M. and Pancob S. (008). Expermental test on pure alumnum shear panels wth welded stffeners, Engneerng Structures, Vol. 30, No. 6, pp Mrzaefar, R., Baha, H., Aryana, F. and Yelagh, A. (007). Optmzaton of the dynamc characterstcs of composte plates usng an nverse approach, Journal of Composte Materals, Vol. 4, No. 6, pp Mrzaefar, R., Baha, H. and Shahab, S. (008a). Actve control of natural frequences of FGM plates by pezoelectrc sensor/actuator pars, Smart Materal and Structures, Vol. 7, No. 4, Mrzaefar, R., Baha, H. and Shahab, S. (008b). A new method for fndng the frst-and second-order dervatves of asymmetrc nonconservatve systems wth applcaton to an FGM plate actvely controlled by pezoelectrc sensor/actuators, Internatonal Journal for Numercal Methods n Engneerng, Vol. 75, No., pp Mrzaefar, R., Shahab, S., Baha, H. (009). An approxmate method for smultaneous modfcaton of natural frequences and 56 Advances n Structural Engneerng Vol. 4 No. 6 0

11 S. Shahab, M. Mrtaher, R. Mrzaefa and H. Baha bucklng loads of thn rectangular sotropc plates, Engneerng Structures, Vol. 3, No., pp Nakagawa, S., Khara, H., Tor, S., Nakata, Y., Matsuoka, Y., Fysawa, K. (996). Hysteretc behavor of low yeld strength steel panel shear walls expermental nvestgaton, Proceedngs of the th World Conference on Earthquake Engneerng, Acapulco, Méxco. (CD-ROM) Ra, D.C. and Wallace, B.J. (998). Alumnum shear-lnk for enhanced sesmc performance, Journal of Earthquake Engneerng and Structural Dynamcs, Vol. 7, No. 4, pp Rasmussen, K.J.R. (003). Full-range stress stran curves for stanless steel alloys, Journal of Constructonal Steel Research, Vol. 59, No., pp Reddy, J.N. (993). An Introducton to the Fnte Element Method, McGraw-Hll, New York, USA. Shahab, S., Mrzaefar, R. and Baha, H. (009). Coupled modfcaton of natural frequences and bucklng loads of composte cylndrcal panels, Internatonal Journal of Mechancal Scences, Vol. 5, No. 9 0, pp Shaker, M., Mrzaefar, R. and Salehghaffary, S. (007). New nsghts nto the collapsng of cylndrcal thn-walled tubes under axal mpact load, Journal of Mechancal Engneerng Scence, Vol., No. 8, pp Soong, T.T. and Dargush, G.F. (997). Passve Energy Dsspaton Systems n Structural Engneerng, Wley, London. Soong, T.T. and Spencer Jr, B.F. (00). Supplemental energy dsspaton: state-of-the-art and state-of-the practce, Engneerng Structures, Vol. 4, No. 3, pp Takahash, Y., Takemoto, Y., Takeda, T. and Takag, M. (973). Expermental study on thn steel shear walls and partcular bracngs under alternatve horzontal load, IABSE Symposum on Resstance and Ultmate Deformablty of Structures Acted on by Well-defned Repeated Loads, Lsbon, Portugal, pp Tromposch, E.W. and Kulak, G.L. (987). Cyclc and Statc Behavor of Thn Panel Steel Plate Shear Walls, Structural Engneerng Report 45, Department of Cvl Engneerng, Unversty of Alberta, Edmonton, Canada. Advances n Structural Engneerng Vol. 4 No

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