modifying the shear buckling loads of metal shear walls

1. INTRODUCTIONIn earthquake engineering, reducing the earthquakedamage on buildings has been one of the mostimportant issues in recent years. Passive energydissipation systems are considered as one of the basic technologies used to protect buildings fromearthquake effects and minimize seismic damage.These systems include a range of materials and devicesfor enhancing damping, stiffness and strength whichmitigate the seismic hazards (Soong and Dargush 1997;Constantinou et al. 1998; Soong and Spencer 2002).Seismic dampers are one of the passive systems whichare installed in the buildings to absorb the energy of the

Advances in Structural Engineering Vol. 14 No. 6 2011 1247

Modifying the Shear Buckling Loads of

Metal Shear Walls for Improving Their

Energy Absorption Capacity

S. Shahab1, M. Mirtaheri2, R. Mirzaeifa1 and H. Bahai3,*

1George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA2Department of civil Engineering KNT University, Tehran, Iran

3School of Engineering and Design, Brunel University, Uxbridge, UB8 3PH, UK

(Received: 23 June 2010; Received revised form: 7 March 2011; Accepted: 14 March 2011)

Abstract: In this paper, an approximate method is proposed for achieving predefinedincreases in the buckling threshold of a metal shear wall in order to increase its energyabsorption capacity. The first and second-order derivatives of shear buckling loads ofa shear wall with respect to the thickness in its different regions are calculated. Basedon these eigenderivatives, and by using the first and second order Taylor expansions,the necessary change in the thickness of plate in various regions is calculated forincreasing the shear buckling loads by a specific value. The presented modificationalgorithm is implemented for shear walls with different aspect ratios, materialproperties and boundary conditions. An initial sensitivity analysis is carried out forfinding the regions within the shear wall where modifying the thickness has the mostinfluence on the buckling loads. Based on the sensitivity analysis results, appropriateregions of plate are selected and the necessary modification in thickness of theseregions is calculated for achieving a relatively large predefined change in shearbuckling load. By simulating the post-buckling response of both initial and modifiedplates in a case study, the improvement in the energy absorption capability of themodified plate is also studied.

Key words: shear wall, shear buckling, eigenvalue, eigenvector, sensitivity analysis.

ground motion in the buildings. When seismic energy istransmitted through the dampers, a portion of energy isdissipated and the motion of building is damped (Soongand Spencer 2002). Among various types of seismicdampers, the metallic yield dampers are efficient meansto dissipate the earthquake energy through inelasticdeformation of metals. The metallic yield dampers canbe found in different geometric configurations of X-shaped, E-shaped, honeycomb-shaped and shear walls.

In earthquake engineering, thin steel shear wallshave had an important role because of their uniqueperformance characteristics. Shear walls can easilyprovide high in-plane strain and stiffness. In addition,

* Corresponding author. Email address: [email protected]; Fax: +44-1895-256392; Tel: +44-1895-265773.

uniform shear stress distribution throughout the platemakes it capable of dissipating a large amount ofenergy due to the large size of yielding region. Thedesign of shear walls has been an active field ofresearch since the early work of Kulak (1985), andTakahashi et al. (1973). Several theoretical andpractical analyses related to design of shear walls havebeen presented in the literature (Tromposch and Kulak1987; Elgaaly 1997, 1998; Alinia and Dastfan 2006;Matteis et al. 2008).

In using shear walls as energy dissipating systems,increasing the shear buckling threshold is a majorchallenge for designers because the occurrence of shearbuckling impedes development of pure shear mechanisminto plastic deformation. As a consequence, the lowershear buckling threshold leads to a lower energyabsorption capacity of the shear walls. In order toguarantee plastic deformations before the occurrence ofshear buckling, low yield shear walls (LYSW) were thefirst proposed solution. These shear walls were made bylow yield strength metals such as steel and aluminiumalloys because of their stable hysteretic behavior up tolarge deformations (Nakagawa et al. 1996; Rai andWallace 1998).

In this paper, a new method is introduced formodifying the shear buckling threshold. In this method,the amount of change in the buckling load is predefinedand can vary according to design specifications. Bychanging the thickness in some specific regions whichare identified by carrying out an initial sensitivityanalysis, the shear buckling load can be modified by anydesired value. Structural modification by considering thegeometric or physical properties as design parametersis a known process for structures and many efforts havebeen made by researchers to implement differentapplications of these methods. In design applications,calculating the derivatives of eigenvalues andeigenvectors with respect to arbitrary parameters for astructural system has important implications. Based onTaylor expansion, the eigenderivatives can be used toapproximately formulate the direct and inverseeigenvalue problems. Direct approximation foreigenvalue problem allows the calculation of thechange of each eigenvalue due to an arbitrary change indesign parameters. In the inverse approximateeigenvalue problem formulation, the required structuralmodifications to achieve predefined modifications ineigenvalues and eigenvectors are computed. Forstructural responses that are formulated in eigenvalueform (such as free vibration and linear buckling) thetime consuming iterative methods may be avoided by defining the problem as an inverse eigenvalueproblem (Aryana et al. 2007; Mirzaeifar et al. 2007;

Mirzaeifar et al. 2008a, b). Recently, Mirzaeifar et al.(2009) proposed an approximate method forsimultaneous modification of natural frequencies andbuckling loads of thin rectangular isotropic plates, andShahab et al. (2009) investigated a new method forperforming predefined simultaneous modification ofnatural frequencies and buckling loads of compositecylindrical panels.

In the present paper, the buckling load and thebuckling mode derivatives of a shear wall with respectto any geometrical or material property are calculatedand used for formulating the approximate eigenvalueproblems. The direct eigenvalue problem is defined ascalculating the corresponding changes in the platebuckling loads due to changes in an arbitrary structuralproperty and the inverse eigenvalue problem is definedas the determination of the required geometrical and/orphysical changes in the plate for achieving the pre-defined shear buckling loads. Modification of the shearbuckling load can cause the shear wall to undergoplastic deformations prior to the occurrence of shearbuckling which can in turn lead to the energyabsorption capacity of the wall being increasedremarkably (Matteis et al. 2008). In all the previouslyreported methods of modifying the shear bucklingthreshold (e. g. adding stiffeners to the plate orchanging the material properties), the amount ofchange in the buckling load cannot be specified by thedesigner. In the present study, a new application of apreviously developed method (Mirzaeifar et al. 2009;Shahab et al. 2009) is introduced which allows thedesigner capable toincrease the lateral buckling load bya predefined value.

Two steel and aluminum plates with differentaspect ratios and boundary conditions are modeledand the proposed modification algorithm isimplemented for these case studies. The thickness ofplate in various regions is considered as the designvariable and a sensitivity analysis is performed forfinding the regions in which changing the thicknesshas the most influence on increasing the bucklingloads. For both the case studies, the necessarymodification in design variables is calculated forachieving a predefined increase in the shear bucklingloads. In these cases, the finite element method isused for calculating the change of buckling loads byimplementing the computed approximate designparameters and it is shown that with the proposedmethod, accurate modifications in the shear bucklingloads are obtained, without performing numerousdesign iterations. Also, in a case study the effect ofimplementing the proposed modification on the post-buckling response of a shear wall is studied and it is

1248 Advances in Structural Engineering Vol. 14 No. 6 2011

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

shown that increasing the shear buckling thresholdhas a significant effect on the energy absorptioncapacity of plate.

2. THEORYIn this section, the finite element formulation forlinear buckling analysis of a rectangular plate underuniform shear loads is derived by rewriting thestability equation in the standard eigenvalue form.The first and second order derivatives of eigenvalueswhich are the shear buckling loads are calculated andbased on these eigenderivatives, and by using first andsecond-order Taylor expansions the first and second-order approximation for modifying shear bucklingloads are formulated.

2.1. Buckling LoadsIn this investigation, the linear buckling eigenvalueproblem in the context of a finite element formulationis considered. Therefore, using nonlinear terms in thecomponents of the strain is necessary. For a plate in x-y plane, the following nonlinear strain field shouldbe considered:

(1)

where, εxx, εyy and γxy are the in-plane nonlinear termsof strain and w is the out of plane displacement. As itis shown in Eqn (1), for linear buckling analysis ofplates, only one term in nonlinear components ofstrain is considered and other terms can be neglecteddue to the flat geometry of the structure (Chen andLiew 2004).

The total potential energy functional consisting of thestrain energy and work done by initial stresses due to in-plane loads can be written as:

(2)

The term Ub stands for strain energy due to bending(Mirzaeifar et al. 2009), and the term Um presentsmembrane strain energy associated with lateraldeflection, w = w(x, y), of the plate:

(3)

where, Nx, Ny are the in-plane forces in the x and ydirections and Nxy is the in-plane shear force which isthe only nonzero force in this paper.

By using the well-known procedure in the finite

U

w

x

w

y

myx

=

∂

∂

∂

∂

∫∫1

2

TT

x xy

xy y

N N

N N

w

x

w

y

∂

∂

∂

∂

dx dy

U U UT b m= +

ε ε γxx yy xyw

x

w

y

w=

∂

∂

=∂

∂

=

∂1

2

1

2

2 2

, ,∂∂

∂

∂

x

w

y

element method (Reddy 1993), the total potential energycan be expressed in terms of nodal variables as:

(4)

where, [Ke] and [Keσ] are the element structural stiffness

and geometric stiffness matrices, respectively and {de}is the generalized nodal displacement vector.

Minimization of the functional presented in Eqn 4leads to the element governing equation which can bewritten as:

(5)

Using the finite elements assembly procedure (Reddy1993) to obtain the global structural and geometricstiffness matrices, the governing equation for thebuckling analysis of a plate is presented as:

(6)

where, [K], [Kσ] and {d} are the global stiffness matrix,geometrical stiffness matrix and generalized nodaldisplacement vector. For linear buckling analysis, thegoverning equation can be rewritten in the form of aneigenvalue problem:

(7)

where, [K *σ] is the geometric stiffness matrix associated

with a reference axial load i.e. a unit shear load. Solvingthe eigenvalue problem expressed in Eqn 7 gives λ whichafter multiplication by the reference axial load specifiesthe critical shear buckling load. Solving this equation,also gives the eigenvectors which represent the bucklingmodes associated with each shear buckling load.

2.2. The First and Second-Order Derivatives ofEigenvalues

By taking Eqn (7) into consideration and following theprocedure detailed in reference (Mirzaeifar et al. 2009),the rate of change of ith eigenvalue with respect to anarbitrary parameter (bj) for buckling loads is expressed as

(8)

where N is the number of degrees of freedom for thesystem and the summation convention is used for thesubscripts. In addition, by using only the first term inTaylor expansion, the change of each buckling load (as

∂

∂=∂

∂−

∂

∂

=

λλ σ

i

j

mn

jm

i in

i mn

jm

i in

b

K

bd d

K

bd d

m n

*

, 1,, ,...,2 N

[ ] [ ] ,*K d K d{ } = { }λ σ

[ ]{ } [ ]{ } { },K d K d+ =σ 0

[ ]{ } [ ]{ } { }.K d K de e e e+ =σ 0

U d K d d K dTe T e e e T e e= +

1

2

1

2{ } [ ]{ } { } [ ]{ },σ


S. Shahab, M. Mirtaheri, R. Mirzaeifa and H. Bahai

the eigenvalues) can be expressed as

(9)

To obtain the second order derivative of eigenvalues,Eqn 8 is differentiated with respect to an arbitraryparameter bk

which shows the dependency of the second derivative ofeigenvalue to the eigenvector derivative. For calculatingthe eigenvector derivatives, modal superpositionassumption which expresses the eigenvector derivativesas a linear combination of all the eigenvectors is used(Fox and Kapoor 1968)

(11)

Following the procedure explained in (Mirzaeifar etal. 2007) the unknown coefficients aikp in Eqn 11 can becalculated. Using the first two terms of Taylorexpansion, the second order approximation for shearbuckling loads can be expressed as

(12)

The necessary changes in the parameter bj to achievethe desired eigenvalues can be calculated based on theinverse method expressed in Eqn 9 for the first orderapproximation or by solving the system of equationspresented in Eqn 12 for the second order approximation.For implementing the first order approximation, inorder to solve the inverse problem (finding thenecessary changes in the parameter bj to achieve thedesired eigenvalues) Eqn 8 and 9 are considered and forthe second order approximation, in the first step, foreach eigenvector the coefficients aijk are calculated and the derivative of each eigenvecto r with respect tobj is calculated using Eqn 11. Then, the values

are calculated using Eqn 10. For finding∂ ∂ ∂2 λi

j kb b�

∆ ∆ ∆λλ λii

jj

i

jjb

bb

b=∂

∂+

∂

∂

1

2

2

22( )

∂

∂=

d

ba dn

i

kikp

p

n

(10)∂

∂ ∂=

∂

∂

∂

∂−

∂

∂

2

2λ

λ σ

i

j k

m

i

k

mn

j

imn

jb b

d

b

K

b

K

b

*

+

∂

∂ ∂−

∂

∂ ∂−∂

∂

d d

K

b b

K

b b b

in

im

mn

j k

imn

j k

i

k

2 2

λλσ

* ∂∂

∂

K

bdmn

j

in

σ*

∆ ∆λλii

jjb

b=∂

∂

the necessary modification to achieve the desired changesin the eigenvalues, quadratic system of equationspresented in Eqn 12 are solved by setting the change of

shear buckling loads ( ) as known parameters andnecessary changes in the structural properties as theunknown parameters. The procedure for implementingthe introduced modification algorithm including inverseand direct approximate methods is shown in theflowchart of Figure 1.

3. NUMERICAL RESULTS3.1. Verification of the Finite Element ModelTo ensure the accuracy of the proposed finite elementmodel and examine the results of the developed codefor shear buckling analysis, a plate simply supportedin all edges made of Aluminium EN-AW 1050A withmaterial properties mentioned in Table 1 is consideredas the case study. The plate length, and thickness areset to a = 1.5 m and h = 5 mm, respectively and twodifferent values are considered for the plate width. Atotal of 64 (8 × 8) four node elements are used in theFE modeling.

The plate is subjected to shear loads around all edgeswhich cause a uniform distribution of shear stresses inthe plate. The shear buckling loads of this plate arecalculated using a code developed in this work based onthe presented finite element formulation. The results forthe critical buckling load are compared against theresults obtained from the commercial finite elementcode ABAQUS in Table 2.

3.2. Inverse Approximate Eigenvalue ProblemIn this section, shear buckling modification of twoplates is carried out by using the inverse eigenvalueproblem. By performing the structural modification ofthis section, the necessary changes in the thickness ofthe structure are calculated in order to achieve desiredmodifications in shear buckling loads. As will be shownin the following case studies, performing a modificationfor shear buckling loads with a significant change fromthe initial configuration usually does not satisfy therequired precision in one step. To deal with thisproblem, a repeated procedure is used for achieving thedesired modification.

3.2.1. A repeated procedure for precisemodification of shear buckling loads

As the first case study, consider a simply supportedrectangular plate with initial thickness of 5 mm. Allthe geometric and material specifications are the sameas those in the case study of Section 3.1. In this casestudy, the plate is divided into eight identical regionsas shown in Figure 2. In the finite element model, theelements in each region are considered as a group and

∆λi



the thickness of plate in each group is expressed as adesign parameter bj. A predefined increase of 30% forthe critical shear buckling load is considered and thesecond order approximation is implemented forfinding the required change in the thickness of each



Structure's eigenvalues and eigenvectors are calculated usingEquation (7)

Derivatives of structural stiffness and geometric stiffness matrices are calculated

The first and second order derivatives of bucklingloads are calculated using Equations (8) and (10)

For inverse approximate method For direct approximate method

No

Errors areacceptable

New design parameters give thepredefined buckling loads

Yes

Equation (9) and (12) for the first andsecond order approximations are

considered respectively by setting the

parameters ∆ i as known values and

the parameters ∆ bj as the unknowns

New values of buckling loads areobtained using Equation (9) for thefirst order and Equation (12) for the

second order approximations

The necessary modification indesign parameters to achieve the

desired changes in bucklingloads are found

λ

Figure 1 The procedure for implementing the modification algorithm

Table 1. Mechanical features of aluminium EN-AW

1050A and low-yield-strength (LYS) steel

Material E (GPa)

Aluminium(EN-AW 1050A) 70 0.33LYS Steel 210 0.3

νν

Table 2. Comparison of critical shear buckling load

(KN/m) for a simply supported rectangular plate

under in-plane shear force

Developed code ABAQUS

1.5 200 5.42 × 103 5.69 × 103

40 6.77 × 103 6.99 × 103

1 200 7.19 × 103 7.5 × 103

40 8.98 × 103 9.15 × 103

Nxy

y

x

3

4

7

a

b

Nxy

Nxy

Nxy

Figure 2 The element groups for rectangular simply supported plate

ab

bh



Table 3. Modification of the critical shear buckling load (KN/m) of a rectangular plate simply supported in all

edges (see Figure 2 for element groups)

Initial Modified (exact) Modified (S.O.) Thickness modification (mm)

Step 1 λ1 30% 5420 7050 6380 ∆h3 = 2.5Step 2 λ1 10.5% 6380 7050 6890 ∆h7 = 1.2Step 3 λ1 2.32% 6890 7050 7040 ∆h4 = 0.23

S. O. Second order approximation.

∆∆λλ

region for modifying the shear buckling load by thisamount as shown in Table 3. In the first step ofmodification, thickness of plate in region number 3 isconsidered as the design parameter. Solving thequadratic equation presented in Eqn 12 gives anincrease in the thickness of this group by amount of 2.5 mm. Implementing this change on the thickness ofplate reveals that the modification error is notnegligible. So, in step 2 the quadratic equation forsecond order approximation is considered with newthicknesses (7.5 mm for group number 3 and 5 mm forall the other groups) as the initial configuration for themodification. In this step, predefined modification isconsidered to be equal to the errors of the second orderapproximation in step 1. Changing the thickness ingroup 7 is involved in modifying the shear bucklingload and solving the quadratic equation gives anincrease in the thickness of group 7 by amount of 1.2 mm. In step 3 which is the last step, the algorithmstarts with the new thicknesses (7.5 mm and 6.2 mmfor groups number 3 and 7, respectively and 5 mm forall the other groups). The modification value is theerror of second order approximation obtained in step 2,and changing the thickness in group 4 is chosen as thedesign parameter in this step. Solving the quadraticequation gives the final thicknesses as 7.5 mm, 6.2 mm and 5.23 mm for group number 3, 7 and 4,respectively, and 5 mm for the other groups to achievethe shear buckling load with negligible difference of0.14% with the desired shear buckling load.

In practice, after performing a limited number ofiterations (3 in the presented case study) the rate ofmodification slows down and from the computationalpoint of view it is not worthwhile to continue the steps.However, the convergence criterion depends on therequired accuracy and the iterations can be continueduntil the required accuracy is achieved, as it is shown inTable 3, the final modified eigenvalue is 7040 KN/mwhich is in a good agreement in the case where a shearbuckling load of 7050 KN/m was desired.

As another example, consider a LYS steel platesimply supported in all edges with the mechanicalproperties given in Table 1. The initial thickness of the

plate is 5 mm and the plate has a square shape withlength a = 1 m. In this case study, a sensitivity analysisis first carried out using second order approximation inorder to find appropriate regions within the structure inwhich the thickness change has the most influence on theshear buckling load. Using Eqn 8 and 10, the rate ofchange of each eigenvalue with respect to change of thethickness in each element in the finite element model iscalculated. By modeling the plate with 441 elements (21 × 21), the contour plot for distribution of the secondderivative of the critical buckling load with respect toeach element’s thickness is shown in Figure 3. Note that,the first derivative has a similar distribution withdifferent values. By considering this counter plot, it isobvious that changing the thickness in the regions nearthe diagonal have most influence on modifying thecritical buckling load. By considering the fact that inpractice shear walls are subjected to cyclic loading inopposite directions, it is obvious that the thickness in theregions near both diagonals have the same effect on theshear buckling load in reciprocal loadings. Based onthe performed sensitivity analysis, the thickness in three

0.8

0.8

1

1

1

2.5

1.5

0.5

3.5

3

20.6

0.6

0.4

0.4

0.2

0.2

× 105

Figure 3 Distribution of the second derivative of the critical shear

buckling load with respect to thickness for a square plate simply

supported in all edges

regions as shown in Figure 4 are chosen as the designparameters and the procedure is repeated formodification of critical shear buckling load.

Table 4 shows the modification procedure. Anincrease of 30% in the critical shear buckling load isconsidered as the goal and the inverse problem is solvedusing the second order approximation. A repeatedprocedure is carried out and the change of shearbuckling load due to changing thickness in groups 1, 2and 3 is obtained. By performing three steps, the criticalshear buckling load is modified up to the predefinedvalue with an error of 0.35%.

As it is shown in the experiments on post-bucklingresponse of structures, by increasing the rate of loadingon the structure, higher buckling loads are activated(Shakeri et al. 2007). So in the case of dynamicloading (which occurs in the applications of shearwalls), higher buckling loads and buckling modesrather than the critical buckling load may affect thestructural response. The proposed method in this paperis capable of modifying the higher buckling loads aswell as the critical shear buckling load. In thefollowing case study, modification of the first twobuckling loads is considered. A square steel plate withsimply supported horizontal edges and free vertical

edges is taken into consideration which is a commonboundary condition in application of shear walls inbuildings. Plate’s initial thickness is 5 mm which willbe modified in some regions. An initial sensitivityanalysis is performed for the first two buckling loadsby calculating the first and second order derivatives ofeach buckling load with respect to the plate thicknessin different regions. Distribution of the second orderderivative of the critical shear buckling load withrespect to thickness is depicted in Figure 5(a). As it isshown the highly sensitive regions are located on thefree edges and near the corners. Figure 5(b) shows thesecond order derivative of the second shear bucklingload with respect to the plate thickness. It can be seenthat the thickness in the regions in the corners and alsoin the mid-span of the vertical free edges have the mostinfluence on the second shear buckling load. It isworthy to note that, distribution of the first orderderivatives has the same pattern as shown in Figure 5but with different values. Based on the sensitivityanalysis results, the thickness in four regions isselected as the design variable (see Figure 6).

Modifying the first two shear buckling loads byamount of 10% is considered as the design goal. As it isshown in Table 5, a two step modification is performedand in each step a set of two simultaneous quadraticequations (as presented in Eqn (12)) is solved forfinding the necessary change in two regions. Themodification in the thickness of these four regions isshown in Table 5. In the final step the second bucklingload is modified to a desired value exactly and thecritical buckling load is modified to the predefined valuewith an acceptable accuracy.

3.3. The Effect of Proposed Modification onthe Post-Buckling Behavior

As explained previously, the ultimate goal of increasingthe lateral buckling threshold in shear walls using theproposed method in this work is to increase the energyabsorption capability of shear walls. The fact thatincreasing the shear buckling loads improves theenergy absorption capacity of shear walls is a wellknown phenomenon studied by many researches in theliterature and it is beyond the scope of this paper to



Table 4. Modification of critical shear buckling load (KN/m) for a square plate simply supported in all edges

(see Figure 4 for element groups)


Step 1 λ1 30% 22000 28600 27200 ∆h1 = 1.5Step 2 λ1 5.14% 27200 28600 28300 ∆h2 = 1.3Step 3 λ1 1.06% 28300 28600 28500 ∆h3 = 0.1

∆∆λλ

Nxy

Nxy

Nxy

Nxy

y

Simply supported

3

1

2

x

Figure 4. The element groups for simply supported square plate

selected based on sensitivity analysis

expound on this phenomenon. However, in order tostudy the effect of proposed modifications on theenergy absorption capacity of shear walls, the post-buckling behavior of a shear wall before and afterimplementing the modification for increasing thelateral buckling threshold is studied in this section. The

square LYS steel plate studied in section 3.2.1 isconsidered as a case study. The plate is simplysupported around all its edges. The plate has a squareshape with length a = 1 m and its initial thickness is 5mm. The modified thickness for increasing the criticalshear buckling load by an amount of 30% is given inTable 4 (see Figure 4 for the element groups). Thenonlinear stress-strain material response is consideredto be the same as UNS30403 alloy. These nonlinearmaterial properties for this alloy are given in(Rasmussen 2003) but for comparison purposes, theelastic modulus is considered to be the same asprevious case studies for LYS steel (see Table 1). Thecommercial finite element software ABAQUS is usedfor simulating the post-buckling behavior of both initialand modified plates. An initial linear buckling analysisis carried out for calculating the shear buckling loadsand buckling modes. These modes are implemented asan imperfection in the finite element model and theRiks method is used for calculating the post-bucklingresponse of the plate. Both the initial plate (with



Table 5. Modification of the first and second shear buckling loads (KN/m) for a square plate simply supported

in two edges (see Figure 6 for element groups)


Step 1 λ1 10% 10100 11100 10600 ∆h1 = 0.3,λ2 10% 10600 11700 11800 ∆h2 = 0.7

Step 2 λ1 4.71% 10600 11100 10800 ∆h3 = 0.4,λ2 0.84% 11800 11700 11700 ∆h4 = −0.3

∆∆λλ

0.8

0.8

1

1

5

4

3

2

1

0.6

0.6

0.4

0.4

(b)

0.2

0.2

× 105

0.8

0.8

1

1

5

4

3

2

1

6

0.6

0.6

0.4

0.4

(a)

0.2

0.2

× 105

Figure 5. Distribution of the second derivative of (a) the critical

shear buckling load and (b) the second buckling load with respect

to thickness

Nxy

Nxy

Nxy

Nxy

y

Simply supported

3

4

1

2

x

Free

Figure 6. The element groups for square plate with two simply

supported and two free edges selected based on sensitivity analysis

uniform thickness) and modified plate (with thicknessdistribution given in Table 4) are analyzed. Forcomparison purposes, in both cases the solution isstopped when a maximum out-of-plane deflection of2.5 cm is reached. Figure 7 shows the deformed shapeof (a) initial and (b) modified plate and the out-of-planedeflections. The points at which the deflection ismaximum is shown by symbols A and B for the initialand modified plates, respectively. As it is shown, due tochange of thickness in different regions in the modifiedplate, the maximum deflection is corresponding to adifferent location from the initial plate.

As it shown in Figure 7, the proposed modificationfor increasing the critical shear buckling load, changesthe post-buckling response remarkably. The variationof applied shear load with the out-of-plane deflectionfor initial and modified plates is shown in Figure 8 (seeFigure 7 for the points at which the out-of-planedeflection is measured). As it is seen in this figure, thenonlinear response for the modified plate starts at ahigher shear force compared with the initial plate. Thisincrease of the maximum force in the linear responseof modified plate shows the increase of critical shearbuckling load that is implemented on the plate usingthe proposed modification in section 3.2.1. Figure 8shows the significant effect of increasing the shearbuckling load threshold on the applied shear force inthe post-buckling response. As pointed out previously

this modification in the shear force causes an increasein the absorbed energy during post-buckling response.

Figure 8 shows the variation of plastic dissipatedenergy with the out-of-plane deflection for both theoriginal plate and the modified plate. It is seen that in thesame out-of-plane deflection, the modified plate absorbs62% more energy compared to the original plate. It isworth noting that this increase in the energy absorptioncapacity is achieved by increasing the weight of thestructure by less than 7% and increasing the energy



Uz+2.501e−02+2.295e−02+2.039e−02+1.784e−02+1.528e−02+1.272e−02+1.016e−02+7.598e−03+5.039e−03+2.480e−03−7.973e−05−2.639e−03−5.198e−03

Uz

Y

XZ

(a)(b)

+2.501e−02+2.247e−02+1.943e−02+1.639e−02+1.335e−02+1.032e−02+7.275e−03+4.236e−03+1.196e−03−1.843e−03−4.883e−03−7.923e−03−1.096e−03

Figure 7. The out-of-plane deflections in the post-buckling analysis of (a) initial; and (b) modified plate (The points of maximum deflection

are shown by A and B)

5.4×105

4.8

4.2

3.6

3.0

2.4

1.8

0.12

0.6

00 0.005

Maximum out-of-plane deflection (m)

She

ar fo

rce

(N/m

)

0.015 0.0250.020.01

Initial plateModified plate

Figure 8. The applied shear load versus the out-of-plane deflection

for initial and modified plates



absorption capacity can be controlled by setting themodification in shear buckling threshold using theproposed method of this paper. It is worth noting thatstudying the different possible methods (such as addingmetal sheets using bolted joints or machining a thickplate) for changing the thickness in various regions andconsidering the efficiency of these methods from aneconomic point of view is an open field of research thatmay be considered in future works.

4. CONCLUSIONSIn this study, an efficient formulation has beendeveloped for increasing the shear buckling thresholdof metal shear walls which leads to an improvement intheir energy absorption capacity. For performing thismodification, the first and second order derivatives ofbuckling loads with respect to the shear wall thicknessin different regions are calculated. Based on theseeigenderivatives and by using the first and second orderTaylor expansions the modification problem isformulated in the form of a system of algebraicequations. A sensitivity analysis is performed forfinding the regions within the shear wall in which achange in thickness has the most influence on the shearbuckling loads. Appropriate regions of plate areselected based on the sensitivity analysis and shearbuckling loads are increased by relatively large values(up to 30%) by finding the necessary change in thethickness of these regions. Various case studies arepresented and the accuracy of the proposed method isshown for shear walls with different aspect ratios,material properties and boundary conditions. Also, in acase study the influence of implementing the proposed

modification for increasing the buckling threshold onthe post-buckling response of the shear wall is studiedand it is shown that the proposed modificationsignificantly improves the energy absorption capabilityof the shear wall.

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00

0.005

Pla

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ener

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N.m

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600

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Maximum out-of-plane deflection (m)0.015 0.0250.020.01

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modifying the shear buckling loads of metal shear walls

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