Research Article On the Shear Buckling of Clamped Narrow ...downloads.hindawi.com/journals/mpe/2015/569356.pdfaccurately predict the critical buckling load of a narrow rectangular

Research ArticleOn the Shear Buckling of Clamped NarrowRectangular Orthotropic Plates

Seyed Rasoul Atashipour and Ulf Arne Girhammar

Department of Civil, Environmental and Natural Resources Engineering, Division of Structural andConstruction Engineering-Timber Structures, Luleå University of Technology, 971 87 Luleå, Sweden

Correspondence should be addressed to Seyed Rasoul Atashipour; [email protected]

Received 19 October 2015; Accepted 29 October 2015

Academic Editor: Francesco Tornabene

Copyright © 2015 S. R. Atashipour and U. A. Girhammar. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

This paper deals with stability analysis of clamped rectangular orthotropic thin plates subjected to uniformly distributed shearload around the edges. Due to the nature of this problem, it is impossible to present mathematically exact analytical solution forthe governing differential equations. Consequently, all existing studies in the literature have been performed by means of differentnumerical approaches. Here, a closed-form approach is presented for simple and fast prediction of the critical buckling load ofclamped narrow rectangular orthotropic thin plates. Next, a practical modification factor is proposed to extend the validity of theobtained results for a wide range of plate aspect ratios. To demonstrate the efficiency and reliability of the proposed closed-formformulas, an accurate computational code is developed based on the classical plate theory (CPT) bymeans of differential quadraturemethod (DQM) for comparison purposes. Moreover, several finite element (FE) simulations are performed via ANSYS software.It is shown that simplicity, high accuracy, and rapid prediction of the critical load for different values of the plate aspect ratio andfor a wide range of effective geometric and mechanical parameters are the main advantages of the proposed closed-form formulasover other existing studies in the literature for the same problem.

1. Introduction

The shear buckling analysis of clamped composite plates isof great importance in design of many types of engineeringstructures. Unlike the problem of normal buckling of plates,the shear buckling problem of plates is mathematicallydescribed by differential equations having a term with odd-order of derivatives with respect to each of the planar spatialcoordinates. Therefore, their governing equations cannot besolved exactly. Such problems are almost always analysedand solved using different numerical approaches. Apart fromthe loading type, clamped boundary conditions at all plateedges make the problem more difficult for finding an exactanalytical solution.

During the past decades, many investigators have studiedthe shear buckling problem of rectangular plates. One of thefirst efforts dealing with shear buckling analysis of clampedisotropic plates with finite dimensions can be attributedto Budiansky and Conner [1] using Lagrangian multiplier

method. A useful review of the studies on the shear bucklingof both isotropic and orthotropic plates was presented byJohns [2]. Shear buckling analysis of antisymmetric cross ply,simply supported rectangular plates was carried out by Hui[3] using Galerkin procedure. Kosteletos [4] studied shearbuckling response of laminated composite rectangular plateswith clamped edges using Galerkin method. Biggers andPageau [5] computed shear buckling loads of both uniformand composite tailored plates using finite element method.Xiang et al. [6] employed pb-2 Rayleigh-Ritz approach toobtain critical shear loads of simply supported skew plates.Loughlan [7] studied the shear buckling of thin laminatedcomposite plates and examined the effect of bend-twistcoupling on their behaviour using a finite strip procedure.Lopatin and Korbut [8] utilized the finite difference methodto investigate the shear buckling of thin clamped orthotropicplates. Shufirn and Eisenberger [9] analysed the buckling ofthin plates under combined shear and normal compressiveloads using the multiterm extended Kantorovich method.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 569356, 11 pageshttp://dx.doi.org/10.1155/2015/569356

2 Mathematical Problems in Engineering

The shear buckling load of rectangular composite platesconsisting of concentric rectangular layups was investigatedby Papadopoulos and Kassapoglou [10] by means of aRayleigh-Ritz approach. Wu et al. [11] calculated the criticalshear buckling loads of rectangular plates by the extendedspline collocation method (SCM). Uymaz and Aydogdu [12]carried out the shear buckling analysis of functionally gradedplates for various boundary conditions based on the Ritzmethod. Shariyat and Asemi [13] performed a nonlinearelasticity-based analysis for the shear buckling of rectangularorthotropic functionally graded (FG) plates surrounded byelastic foundations using a cubic B-spline finite elementapproach.

Evidently, all the above-mentioned numerical studieshave some deficiencies like convergence difficulties andbeing time-consuming compared to analytical and closed-form solutions. Therefore, it is not easy and time-efficientto predict the critical shear buckling loads and investigatethe effect of various parameters by the use of numericalsolution approaches. To the best of authors’ knowledge, noclosed-form solution can be found in the literature for theshear buckling of composite rectangular plates with finitedimensions. To fill this apparent void, the present work iscarried out to provide efficient and reliable explicit formulasfor rapid prediction of the fundamental critical shear buck-ling loads of clamped orthotropic rectangular plates. Therange of validity of the proposed closed-form formulas isextended by introducing a practical modification factor. Also,in order to demonstrate the efficiency and reliability of theproposed closed-form formulas, an accurate computationalcode is developed bymeans of differential quadraturemethod(DQM) for comparison purposes. Moreover, several finiteelement (FE) simulations are performed via ANSYS software.

This paper is only devoted to a principle study of theshear buckling behavior and, for illustration, is applied to alaminated veneer lumber (LVL) panel. Other failure modes,such as the shear strength, are not included in the analysis.

2. Definition of the Problem andGoverning Equations

Consider a clamped narrow rectangular orthotropic plateof length 𝑎, width 𝑏, and thickness 𝑡, subjected to a uni-formly distributed shear load per length 𝑆

𝑥𝑦(Figure 1). The

coordinates system is shown in the figure. We employ theclassical plate theory (CPT) of Kirchhoff to study the shearbuckling of thin plates. The governing equation of CPT forthe orthotropic plates is expressed as

𝐷11

𝜕4

𝑤

𝜕𝑥4+ 2 (𝐷

12+ 2𝐷33)

𝜕4

𝑤

𝜕𝑥2𝜕𝑦2+ 𝐷22

𝜕4

𝑤

𝜕𝑦4

= 2𝑆𝑥𝑦

𝜕2

𝑤

𝜕𝑥𝜕𝑦

,

(1)

where𝑤 is transverse displacement, and𝐷𝑖𝑗are stiffness coef-

ficients of orthotropic materials and are defined as follows:

𝐷11=

𝐸1𝑡3

12 (1 − ]12]21)

,

𝐷12=

]21𝐸1𝑡3

12 (1 − ]12]21)

,

𝐷22=

𝐸2𝑡3

12 (1 − ]12]21)

,

𝐷33=

1

12

𝐺12𝑡3

(2)

in which 𝐸1and 𝐸

2are modulus of elasticity of orthotropic

material in𝑥 and𝑦directions, respectively;𝐺12is the in-plane

shear modulus and ]𝑖𝑗are the Poisson’s ratios.

The plate is assumed to be fully clamped. Thus, thefollowing boundary conditions should be considered at theplate edges:

𝑤|𝑥=0,𝑎

= 0,

𝑤|𝑦=0,𝑏

= 0,

(3a)

𝜕𝑤

𝜕𝑥

𝑥=0,𝑎

= 0,

𝜕𝑤

𝜕𝑦

𝑦=0,𝑏

= 0.

(3b)

3. An Efficient Closed-Form Solution

As mentioned earlier, no exact analytical approach exists forthe problem of shear buckling of rectangular orthotropicplates, not only due to the loading type, but also becauseof the fully clamped boundary conditions. The computa-tional numerical approaches are usually time-consumingfor obtaining the results with adequate accuracy. Therefore,it is reasonable to find an efficient method for predictingthe critical loads. Timoshenko and Gere [14] presented anapproximate solution for the shear buckling of narrow rectan-gular plates with the limitation of simply supported boundaryconditions and isotropic material. We start by extending themethod for the orthotropic narrow rectangular plates withclamped boundary conditions. To this end, we consider thefollowing expression for the transverse displacement of thebuckled plate:

𝑤 (𝑥, 𝑦) = [1 − cos(2𝜋𝑏

𝑦)] ⋅ sin [𝜋𝑠

(𝑥 − 𝛼𝑦)] , (4)

where 𝑠 and 𝛼 represent the length of half-waves of thebuckled plate and the slope of the nodal lines. Clearly, (4)satisfies the clamped edge conditions at the long edges 𝑦 =0, 𝑏. However, this approximate approach is not capable ofsatisfying the clamped boundary conditions at the two short

Mathematical Problems in Engineering 3

y

xa

b

tSxy

Sxy

Figure 1: Geometric configuration and coordinate system of a narrow rectangular orthotropic plate with fully clamped edges subjected to auniformly distributed shear load.

edges. The work done by the external forces and the strainenergy during the buckling of the plate are defined by

Δ𝑇 = −𝑆cr ∫2𝑠

0

∫

𝑏

0

𝜕𝑤

𝜕𝑥

𝜕𝑤

𝜕𝑦

𝑑𝑥 𝑑𝑦,

Δ𝑈 =

1

2

∫

2𝑠

0

∫

𝑏

0

[𝐷11(

𝜕2

𝑤

𝜕𝑥2)

2

+ 𝐷22(

𝜕2

𝑤

𝜕𝑦2)

2

+ 2𝐷12(

𝜕2

𝑤

𝜕𝑥2)(

𝜕2

𝑤

𝜕𝑦2)

+ 4𝐷33(

𝜕2

𝑤

𝜕𝑥𝜕𝑦

)

2

]𝑑𝑥𝑑𝑦.

(5)

By substituting the proposed form of the transverse displace-ment from (4) into (5) and equating the work of externalforces to the strain energy (i.e., Δ𝑇 = Δ𝑈), a closed-formformula is obtained for the critical buckling load as

𝑆cr =𝜋2

𝑠2

6𝑏4𝛼

{16𝐷22

+ 8 [(𝐷12+ 2𝐷33) + 3𝐷

22𝛼2

] (

𝑏

𝑠

)

2

+ 3 [𝐷11+ 2 (𝐷

12+ 2𝐷33) 𝛼2

+ 𝐷22𝛼4

] (

𝑏

𝑠

)

4

} .

(6)

The obtained formula for the critical buckling load should beminimized with respect to the unassigned parameters 𝑠 and𝛼. To this end, we differentiate (6) one time with respect to 𝑠and then with respect to 𝛼. It is easy to show that the resultedset of algebraic equations can be represented as

3𝐷11+ 4 (𝐷

12+ 2𝐷33) (

𝑠

𝑏

)

2

− 12𝐷22(

𝑠

𝑏

)

2

𝛼2

− 3𝐷22𝛼4

= 0,

3𝐷11+ 6 (𝐷

12+ 2𝐷33) 𝛼2

+ 3𝐷22𝛼4

− 16𝐷22(

𝑠

𝑏

)

4

= 0.

(7)

Exact solution of the above set of equations can be repre-sented in the form

𝛼 =

√2

2

⋅√√

𝛿2(7𝛿1− 19𝛿

2

2/3)

√𝛿3+ 𝛿4/ℓ + ℓ/3

+ 2𝛿3−

𝛿4

ℓ

−

ℓ

3

+ √𝛿3+

𝛿4

ℓ

+

ℓ

3

− 𝛿2,

(8)

where

ℓ =

3√𝛿5+ √𝛿2

5− 27𝛿

3

4.

(9)

Also, the parameter 𝑠 is expressed in terms of 𝛼 as follows:

𝑠 =

𝑏

2

4√3 (𝛿

1+ 2𝛿2𝛼2+ 𝛼4). (10)

In (8) through (10), the coefficients 𝛿𝑖(𝑖 = 1, 2, . . . , 5) are

defined as

𝛿1=

𝐷11

𝐷22

,

𝛿2=

𝐷12

𝐷22

,

𝛿3= −

5

3

(

𝐷11

𝐷22

) +

20

9

(

𝐷12

𝐷22

)

2

,

𝛿4=

1

12

(

𝐷11

𝐷22

)

2

−

7

18

(

𝐷11

𝐷22

)(

𝐷12

𝐷22

)

2

+

49

108

(

𝐷12

𝐷22

)

4

,

𝛿5=

485

8

(

𝐷11

𝐷22

)

3

−

587

8

(

𝐷11

𝐷22

)

2

(

𝐷12

𝐷22

)

2

+

293

24

(

𝐷11

𝐷22

)(

𝐷12

𝐷22

)

4

+

181

216

(

𝐷12

𝐷22

)

6

(11)

in which

𝐷12= 𝐷12+ 2𝐷33. (12)


For the isotropic case, 𝛿1= 𝛿2= 1 and, consequently, the

parameters 𝛼 and 𝑠 from (8)–(11) are reduced to

𝛼 =

1

4√3

,

𝑠 =

𝑏

2

√1 + √3.

(13)

Therefore, the critical buckling load from (6) is reduced to

𝑆cr =8

3

4√3 (√3 + 1)

𝜋2

𝐷

𝑏2. (14)

It is worth to rewrite the obtained closed-form formula in theconventional form as follows:

𝑆cr = 𝑘𝑠𝜋2

𝐷11

𝑏2

. (15)

Therefore, the dimensionless coefficient 𝑘𝑠is represented in

the form

𝑘𝑠=

4

3𝛿1𝛼

{√3 (𝛿1+ 2𝛿2𝛼2+ 𝛼4) + 𝛿2+ 3𝛼2

} (16)

in which the coefficients 𝛼, ℓ, and 𝛿𝑖(𝑖 = 1, 2, . . . , 5) are

defined by (8), (9), and (11).It will be shown that the obtained closed-form formulas

accurately predict the critical buckling load of a narrowrectangular orthotropic plate in shear with clamped edges.Apparently, the accuracy of the obtained closed-form for-mulas decreases when the plate aspect ratio decreases. Toenhance the validity range of the obtained formulas for lowervalues of the plate aspect ratio and generalize them, wepropose a simple practical modification factor (𝐶mf ) to bemultiplied by the dimensionless coefficient 𝑘

𝑠in the form:

𝐶mf = 1 +3/𝐸 + 1

4 [(𝑎/𝑏)4

+ 1]

. (17)

In the next section, a differential quadrature (DQ) code isdeveloped for comparison purposes to prove the high accu-racy of the proposed closed-form formulas for predicting thecritical buckling load of the rectangular orthotropic plates inshear with clamped edges. Evidently, fast and easy predictionof the critical buckling load is the main advantage of theobtained closed-form formulas over existing studies in theliterature based on time-consuming numerical approaches.

4. Differential Quadrature Solution

The differential quadrature method (DQM), as an appropri-ate method among various numerical solution approaches,has been mostly utilized by scientists for the eigen-bucklinganalysis of composite rectangular plates under in-plane nor-mal compressive loads (e.g., see [15–18]). Here, we employthis methodology to solve the problem of shear bucklingof clamped thin composite plates. To this end, we define

the transverse displacement 𝑤 as a multipolynomial throughdiscretized points𝑊

𝑖,𝑗= 𝑤(𝑥

𝑖, 𝑦𝑗) in the domain:

𝑤 (𝑥, 𝑦) =

𝑁𝑥

∑

𝑖=1

𝑁𝑦

∑

𝑗=1

𝑊𝑖,𝑗𝑓𝑖(𝑥) 𝑔𝑗(𝑦) , (18)

where 𝑁𝑥and 𝑁

𝑦are the number of grid points in 𝑥 and

𝑦 directions, respectively, and the Lagrange interpolationpolynomials 𝑓

𝑖(𝑥) and 𝑔

𝑗(𝑦) are defined in the form

𝑓𝑖(𝑥) =

𝑁𝑥

∏

𝑘=1,𝑘 ̸=𝑖

𝑥 − 𝑥𝑘

𝑥𝑖− 𝑥𝑘

,

𝑔𝑗(𝑦) =

𝑁𝑦

∏

𝑘=1,𝑘 ̸=𝑗

𝑦 − 𝑦𝑘

𝑦𝑖− 𝑦𝑘

.

(19)

It is assumed that the following equations are satisfied for thefunction 𝑤(𝑥, 𝑦) and its derivatives [19]:

𝑤(𝑛)

𝑥(𝑥𝑖, 𝑦𝑗) =

𝑁𝑥

∑

𝑘=1

𝑐(𝑛)

𝑖𝑘𝑊𝑘,𝑗,

𝑖 = 1, 2, . . . , 𝑁𝑥; 𝑛 = 1, 2, . . . , 𝑁

𝑥− 1,

𝑤(𝑚)

𝑦(𝑥𝑖, 𝑦𝑗) =

𝑁𝑦

∑

𝑘=1

𝑐(𝑚)

𝑗𝑘𝑊𝑖,𝑘,

𝑗 = 1, 2, . . . , 𝑁𝑦; 𝑚 = 1, 2, . . . , 𝑁

𝑦− 1,

(20)

where 𝑐 and 𝑐 are weighting coefficients in the DQM fordifferentiation of 𝑤 with respect to 𝑥 of order 𝑛 and 𝑦 oforder𝑚, respectively. Details on calculations of the weightingcoefficients, according to Shu’s general approach [19], aregiven in Appendix.

Substituting (20) into (1) results in the discretized govern-ing equation as follows:

𝐷11

𝑁𝑥

∑

𝑘=1

𝑐(4)

𝑖𝑘𝑊𝑘,𝑗+ 2 (𝐷

12+ 2𝐷33)

𝑁𝑥

∑

𝑘1=1

𝑁𝑦

∑

𝑘2=1

𝑐(2)

𝑖𝑘1

𝑐(2)

𝑗𝑘2

𝑊𝑘1 ,𝑘2

+ 𝐷22

𝑁𝑦

∑

𝑘=1

𝑐(4)

𝑗𝑘𝑊𝑖,𝑘= 2𝑆𝑥𝑦

𝑁𝑥

∑

𝑘1=1

𝑁𝑦

∑

𝑘2=1

𝑐(1)

𝑖𝑘1

𝑐(1)

𝑗𝑘2

𝑊𝑘1 ,𝑘2

.

(21)

To obtain accurate results, the distribution of the grid pointsshould be denser at the edge-zones.Thus, an appropriate gridpoint distribution pattern is used as

𝑥𝑖

𝑎

=

1

2

[1 − cos( 𝑖 − 1𝑁𝑥− 1

𝜋)] , 𝑖 = 1, 2, . . . , 𝑁𝑥,

𝑦𝑗

𝑏

=

1

2

[1 − cos(𝑗 − 1

𝑁𝑦− 1

𝜋)] , 𝑗 = 1, 2, . . . , 𝑁𝑦,

(22)

where 𝑥𝑖and 𝑦

𝑗are the coordinates of 𝑖th and 𝑗th grid

points, respectively. Figure 2 shows the mesh distribution ona narrow rectangular domain based on (22).


y

x23

j

Ny...

...

j+ 1

1 2 3 4 i Nx· · · · · ·i − 1 i + 1 i + 2

Figure 2: Illustration of the DQ meshed narrow rectangular plate.

The boundary conditions (3a) and (3b) can be rewrittenin the form

𝑊1,𝑗= 0, at 𝑥 = 0,

𝑊𝑁𝑥 ,𝑗

= 0, at 𝑥 = 𝑎,(23a)

𝑊𝑖,1= 0, at 𝑦 = 0,

𝑊𝑖,𝑁𝑦

= 0, at 𝑦 = 𝑏,(23b)

𝑁𝑥

∑

𝑘=1

𝑐(1)

1,𝑘𝑊𝑘,𝑗= 0, at 𝑥 = 0,

𝑁𝑥

∑

𝑘=1

𝑐(1)

𝑁𝑥,𝑘𝑊𝑘,𝑗= 0, at 𝑥 = 𝑎,

(24a)

𝑁𝑦

∑

𝑘=1

𝑐(1)

1,𝑘𝑊𝑖,𝑘= 0, at 𝑦 = 0,

𝑁𝑦

∑

𝑘=1

𝑐(1)

𝑁𝑦,𝑘𝑊𝑖,𝑘= 0, at 𝑦 = 𝑏.

(24b)

Clearly, (23a) and (23b) can be easily satisfied at the four edgesof the plate. However, (24a) and (24b) cannot be directlysubstituted into (21).This difficulty can be easily overcome byusing a simple method described by Shu and Du [20]. Basedon this approach, the two equations (24a) are coupled to givetwo solutions𝑊

2,𝑗and𝑊

𝑁𝑥−1,𝑗as follows:

𝑊2,𝑗=

1

𝛼0

𝑁𝑥−2

∑

𝑘=3

𝛽𝑘𝑊𝑘,𝑗,

𝑊𝑁𝑥−1,𝑗

=

1

𝛼0

𝑁𝑥−2

∑

𝑘=3

𝛾𝑘𝑊𝑘,𝑗,

𝑗 = 3, 4, . . . , 𝑁𝑦− 2.

(25)

Using a similar method, the two equations (24b) can becoupled to give two solutions for𝑊

𝑖,2and𝑊

𝑖,𝑁𝑦−1:

𝑊𝑖,2=

1

𝛼0

𝑁𝑦−2

∑

𝑘=3

𝛽𝑘𝑊𝑖,𝑘,

𝑊𝑖,𝑀−1

=

1

𝛼0

𝑁𝑦−2

∑

𝑘=3

𝛾𝑘𝑊𝑖,𝑘,

𝑖 = 3, 4, . . . , 𝑁𝑥− 2.

(26)

The coefficients 𝛼0, 𝛽𝑘, 𝛾𝑘, 𝛼0, 𝛽𝑘and 𝛾𝑘, in (25) and (26), are

defined in Appendix.For the points near the four corners, 𝑊

2,2, 𝑊𝑁𝑥−1,2

,𝑊2,𝑁𝑦−1

, and𝑊𝑁𝑥−1,𝑁𝑦−1

can be determined by coupling (24a)and (24b) in the following form:

𝑊2,2=

1

𝛼0𝛼0

𝑁𝑥−2

∑

𝑘1=3

𝑁𝑦−2

∑

𝑘2=3

𝛽𝑘𝛽𝑘𝑊𝑘1 ,𝑘2

,

𝑊𝑁𝑥−1,2

=

1

𝛼0𝛼0

𝑁𝑥−2

∑

𝑘1=3

𝑁𝑦−2

∑

𝑘2=3

𝛾𝑘𝛽𝑘𝑊𝑘1 ,𝑘2

,

𝑊2,𝑁𝑦−1

=

1

𝛼0𝛼0

𝑁𝑥−2

∑

𝑘1=3

𝑁𝑦−2

∑

𝑘2=3

𝛽𝑘𝛾𝑘𝑊𝑘1 ,𝑘2

,

𝑊𝑁𝑥−1,𝑁𝑦−1

=

1

𝛼0𝛼0

𝑁𝑥−2

∑

𝑘1=3

𝑁𝑦−2

∑

𝑘2=3

𝛾𝑘𝛾𝑘𝑊𝑘1 ,𝑘2

.

(27)

Hence, the discretized governing equation (21) should beapplied for the interior mesh points in which 3 ≤ 𝑖 ≤ 𝑁

𝑥− 2,

3 ≤ 𝑗 ≤ 𝑁𝑦− 2.

Applying the discretized governing equation (21) for allthe interior grid points (i.e., 3 ≤ 𝑖 ≤ 𝑁

𝑥− 2, 3 ≤ 𝑗 ≤

𝑁𝑦− 2) and satisfying the boundary conditions (23a) and

(23b) together with (25)–(27) will result in a set of algebraicequations in terms of𝑊

𝑖𝑗. These equations can be expressed

in the form of a matrix equation as follows:

AW − 𝑆crBW = 0, (28)


where A and B are two matrices of the coefficients, andW isthe deflection vector in terms of𝑊

𝑖,𝑗. Also, 𝑆cr is the critical

shear load per length (𝑆𝑥𝑦,cr). Generally, the set of algebraic

equations (28) should be of order (𝑁𝑥× 𝑁𝑦) by (𝑁

𝑥× 𝑁𝑦).

However, the use of (25) and (26) for 𝑊2,𝑗, 𝑊𝑁𝑥−1,𝑗

, 𝑊𝑖,2,

𝑊𝑖,𝑁𝑦−1

and (27) for 𝑊2,2, 𝑊𝑁𝑥−1,2

, 𝑊2,𝑁𝑦−1

and 𝑊𝑁𝑥−1,𝑁𝑦−1

reduces the order of the algebraic equations to [𝑁𝑥× 𝑁𝑦−

2(𝑁𝑥+ 𝑁𝑦) + 12] by [𝑁

𝑥× 𝑁𝑦− 2(𝑁

𝑥+ 𝑁𝑦) + 12].

By taking advantage of the reduced algebraic equations,the size of the analysis domain decreases and, consequently,a finer mesh can be used resulting in more accurate eigenval-ues.

5. Numerical Results and Discussion

A mathematical code is developed according to the abovedescribedDQ solution to obtain the accurate critical bucklingloads and their corresponding mode shapes for compari-son studies. Also, some simulations were performed usingANSYS software. It should be pointed out that the assump-tions of the classical plate theory (CPT) are incorporated intothe simulations performed by ANSYS. The results obtainedfrom the proposed closed-form formulas are compared tothose obtained by the DQ code and ANSYS simulations toshow the reliability of the formulas.

To generalize the numerical results, the following dimen-sionless parameters are introduced:

𝑘𝑠=

𝑏2

𝜋2𝐷11

𝑆cr =𝑏2

𝑡

𝜋2𝐷11

𝜏cr,

𝐸 =

𝐸2

𝐸1

,

𝐺 =

2 (1 + ]12)

𝐸1

𝐺12,

(29)

where 𝑘𝑠is dimensionless shear buckling parameter in

terms of the critical load 𝑆cr (𝑆𝑥𝑦,cr) and the correspondingcritical shear stress 𝜏cr, 𝑡 is the thickness, and 𝐸 and 𝐺are dimensionless elasticity and shear moduli of orthotropicmaterial, respectively. As a special case, the orthotropic plateis converted to an isotropic one when 𝐸 = 𝐺 = 1.

Table 1 shows a convergence study of the DQ code aswell as comparison of the dimensionless shear bucklingparameter 𝑘

𝑠obtained from the closed-form approach, DQ

solution, and those from [1] for fully clamped isotropic plates.The obtained buckling parameters are compared for variousvalues of the aspect ratio (𝜂 = 𝑎/𝑏) and two different typesof buckling modes: symmetric and antisymmetric. It can beobserved from Table 1 that the obtained critical loads fromboth closed-form formulas and the DQ solution are in a verygood agreement with the results of [1] for different values ofthe aspect ratio. Also, the results of this table confirm theconvergence and stability of the obtained critical loads for allcases.

Variations of dimensionless fundamental shear bucklingparameter 𝑘

𝑠versus the plate aspect ratio are depicted in

Figure 3 for different values of the dimensionless materialproperties: 𝐸 and 𝐺. It can be seen from Figure 3 that

Accurate DQ codeClosed-form approach

S

AS

S

S

S

A

A

A

A

A: antisymmetricS: symmetric

S

AS

S

S

A

S

E = 2, G = 1

E = 1, G = 2

E = 1, G = 0.5

E = 1, G = 1(Isotropic)

E = 0.5, G = 1

4

6

8

10

12

14

16

18

20

1 1.5 2a/b

2.5 3 3.5 4

ks

Figure 3: The critical buckling load coefficient for rectangularorthotropic plates under pure uniform shear load versus the aspectratio for different material properties (]

21= 0.3).

the results of the proposed closed-form approach are invery good agreement with those of the time-consumingcomputational DQ solution for all cases, even for squareplates. It is worth noting that the introduced closed-formformulas only predict the fundamental critical loads, eithersymmetric or antisymmetricmode. Although the plate aspectratio influences the type of fundamental shear bucklingmode, their critical loads are very close to each other. Sincethe curves of this figure are provided in dimensionless form,they can be used for estimating the critical shear loads ofclamped orthotropic plates with a wide range of geometricand material properties.

To study the shear buckling of clamped narrow rectan-gular orthotropic plates, a special engineering panel is con-sidered called laminated veneer lumber (LVL). This timbersheathing,with the commercial nameofKerto-Q, is subjectedto a distributed uniform shear load. The mechanical andgeometric properties of LVL are as follows:

𝐸1= 10.5GPa,

𝐸2= 2.4GPa,

𝐺12= 0.6GPa,

]21= 0.05,

𝑎 = 3.0m,

𝑏 = 0.7275m,

𝑡 = 0.027m.

(30)


Table 1: Convergence study and comparison of critical buckling load parameter (𝑘𝑠) of fully clamped isotropic rectangular plates with various

values of the aspect ratio (𝜂 = 𝑎/𝑏) under uniform distributed shear load.

𝑁𝑥× 𝑁𝑦

𝑘𝑠= 𝑏2

𝑆cr/ (𝜋2

𝐷11)

𝜂 = 1.00 𝜂 = 1.25 𝜂 = 1.50 𝜂 = 2.00 𝜂 = 3.00

Sym. Antisym. Sym. Antisym. Sym. Antisym. Sym. Antisym. Sym. Antisym.7 × 7 14.938 16.335 12.543 13.488 11.619 12.129 11.358 10.618 12.696 12.9019 × 9 14.801 17.307 12.487 13.979 11.602 12.056 10.774 10.418 9.816 10.00511 × 11 14.643 16.904 12.347 13.658 11.457 11.797 10.579 10.251 9.568 9.70913 × 13 14.642 16.920 12.347 13.671 11.459 11.805 10.583 10.249 9.535 9.62615 × 15 14.642 16.919 12.347 13.670 11.458 11.804 10.582 10.248 9.568 9.63217 × 17 14.642 16.919 12.347 13.670 11.458 11.804 10.582 10.248 9.569 9.631Closed-form approach 14.382 — 12.374 — 11.170 — — 10.152 9.705 —Reference [1] 14.64 — 12.35 — 11.45 11.79 10.58 10.32 9.57 9.64

Table 2: Convergence study and comparison of critical bucklingshear stress, 𝜏cr (MPa), of fully clamped orthotropic rectangular plate(LVL) with FEM simulation and the closed-form approach.

𝑁𝑥× 𝑁𝑦

𝜏cr (MPa)Symmetric mode Antisymmetric mode

7 × 7 42.788 43.7019 × 9 31.421 32.41011 × 11 30.484 31.10713 × 13 30.321 30.58215 × 15 30.334 30.62317 × 17 30.331 30.61719 × 19 30.332 30.618Closed-form approach 30.534 —ANSYS simulation 30.410 30.694

In order to ensure convergence and accuracy of thedeveloped DQ code for the shear buckling loads of theorthotropic narrowplates, a convergence study is presented inTable 2 for the LVL sheathing. Also, the results are comparedwith those obtained from an accurate FE simulation viaANSYS software as well as the results of the closed-formapproach. It is evident that the obtained critical in-planeshear stresses from the closed-form formulas are in verygood agreement with those achieved from the DQ code andANSYS simulation.

In order to get a better physical sense of the symmetricand antisymmetric modes, the mode shapes of a squareisotropic clamped plate are provided via both the devel-oped DQ code and ANSYS simulation and are illustratedin Figure 4. Also, the symmetric and antisymmetric modeshapes of the LVL narrow rectangular plate are presented inFigure 5 to show the influence of the geometric shape on themode shapes. It can be seen that the mode shapes obtainedby the DQ code are the same as those obtained by ANSYSsimulation. Also, comparison between the correspondingmode shapes of Figures 4 and 5 reveals the fact that thenumber of half-waves increases by increasing the aspect ratio.

In Figure 6, the mode shape of the LVL obtained fromthe closed-form approximation is shown. It can be observedthat the mode shape predicted by this approximate method,

except at the two short edges, is similar to those of othermethods.

Variations of the in-plane shear stress of the clampedLVL plate versus different geometric and mechanical param-eters are shown in Figures 7 and 8, respectively. It shouldbe pointed out that the results in these two figures arepresented in dimensional form to more directly study theinfluence of various parameters on the critical in-plane shearstress. In both figures, the results of DQ code, closed-formapproach and FE simulation via ANSYS are provided toshow the reliability and efficiency of the developed closed-form approach for a wide variety of different geometricand material properties. In Figure 7, 𝜆 denotes one of thegeometric properties relative to the reference value for theLVL panel and in Figure 8 𝛾 represents one of the materialproperties. The subscript “LVL” in the relative expressions𝜆𝑖/𝜆LVL and 𝛾𝑖/𝛾LVL refers to the reference values of the LVL

panel (see (30)). It can be observed from Figure 7 that thecritical buckling stress considerably decreases as the width 𝑏of the LVL plate increases, whereas decreasing the length 𝑎results in a small increase of the critical buckling stress. Also,it is evident that the critical stress significantly increases whenthe thickness of the plate increases. However, it should bementioned that for large thicknesses the classical plate theoryis no longer valid due to the neglect of the transverse sheardeformations.

Figure 8 reveals the fact that, by increasing any of thematerial properties, the critical buckling stress increases.It is also obvious that Young’s modulus 𝐸

2has the largest

effect and the shear modulus 𝐺12

the smallest effect on thecritical buckling stress of the clamped narrow rectangularorthotropic plate subjected to uniform in-plane shear load.

Influence of Poisson’s ratio ]21

on the critical in-planeshear stress of the clamped LVL is shown in Figure 9 based onthe obtained closed-form formulas, DQ solution, andANSYSsimulations. It can be observed that increasing Poisson’s ratiocan slightly increase the critical shear stress.

In Figures 7–9, very good agreement between the curvesobtained from the closed-form approach and those based ontheDQ code as well as ANSYS simulation shows the accuracyand reliability of the proposed efficient closed-form formulafor all cases.


DQ code ANSYS simulation

(a)

DQ code ANSYS simulation

(b)

Figure 4: Contour plot of themode shape corresponding to the critical buckling loads of a fully clamped isotropic square plate: (a) symmetricmode, (b) antisymmetric mode.

DQ code

ANSYS simulation(a)

DQ code

ANSYS simulation(b)

Figure 5: Contour plot of the mode shape corresponding to the critical buckling loads of a fully clamped orthotropic rectangular plate (LVL):(a) symmetric mode, (b) antisymmetric mode.


Figure 6: Contour plot of the mode shape corresponding to thecritical buckling load of the orthotropic LVL plate based on theclosed-form approach.

0.5 0.75 1 1.25 1.5 1.75 20

20

40

60

80

100

120

Closed-form approachDQ code

ANSYS simulation

𝜆3 = t

𝜆1 = a

𝜆2 = b

LVLref

𝜆i/𝜆LVL

𝜏cr

Figure 7: Effect of different geometric properties on the criticalshear stress of the orthotropic LVL panel. The horizontal axis showsthe relative value of the geometric parameter with reference to thevalue of the LVL panel.

6. Conclusions

In this paper, the shear buckling of clamped narrow rectan-gular orthotropic plates was investigated. An efficient closed-form approach was presented to easily and fastly predict thecritical shear buckling loads and correspondingmode-shapesof the clamped narrow rectangular orthotropic plates. Also,a practical modification factor was proposed to extend thevalidity range of the obtained explicit formulas. To prove theaccuracy and effectiveness of the closed-form approach, anaccurate DQ code was developed and the critical bucklingloads and their corresponding mode shapes were extracted.Also, several accurate FE simulations using ANSYS softwarewere performed. It was shown that the proposed closed-formapproach can predict the critical buckling loads with theacceptable accuracy for a wide range of effective parameterswithout any computational effort. The effect of various geo-metric andmechanical parameters was investigated bymeansof three different methods: closed-form approach, DQ code,and ANSYS simulations. It was observed that the criticalbuckling load considerably decreases by increasing the width𝑏 of the narrow plates whereas decreasing the length 𝑎 results

0.5 0.75 1 1.25 1.5 1.75 2

20

25

30

35

40

45

50


ANSYS simulation𝜏

cr

LVLref

𝛾2 = E2

𝛾1 = E1

𝛾3 = G12

𝛾i/𝛾LVL

Figure 8: Effect of different material properties on the critical shearstress of the orthotropic LVL panel. The horizontal axis shows therelative value of the material parameter with reference to the valueof the LVL panel.

0.5 0.75 1 1.25 1.5 1.75 2

20

25

30

35

40

45

50


ANSYS simulation

𝜏cr LVLref

�12/(�12)LVL

Figure 9: Effect of Poisson’s ratio on the critical shear stress of theorthotropic LVL panel. The horizontal axis shows the relative valueof the material parameter with reference to the value of the LVLpanel.


in a very small increase of the critical buckling load. Also itwas shown, among different material properties, that Young’smodulus 𝐸

2and the shear modulus 𝐺

12have the largest and

smallest effects on the critical buckling load, respectively.

Appendix

Calculation of Weighting andOther Coefficients

The weighting coefficients for the first-order derivatives areexpressed as

𝑐(1)

𝑖𝑗=

𝛼(1)

(𝑥𝑖)

(𝑥𝑖− 𝑥𝑗) 𝛼(1)(𝑥𝑗)

,

𝑐(1)

𝑖𝑖= −

𝑁𝑥

∑

𝑗=1,𝑗 ̸=𝑖

𝑐(1)

𝑖𝑗

𝑖, 𝑗 = 1, 2, . . . , 𝑁𝑥, 𝑖 ̸= 𝑗

𝑐(1)

𝑖𝑗=

𝛽(1)

(𝑦𝑖)

(𝑦𝑖− 𝑦𝑗) 𝛽(1)(𝑦𝑗)

,

𝑐(1)

𝑖𝑖= −

𝑁𝑦

∑

𝑗=1,𝑗 ̸=𝑖

𝑐(1)

𝑖𝑗

𝑖, 𝑗 = 1, 2, . . . , 𝑁𝑦, 𝑖 ̸= 𝑗

(A.1)

in which the functions 𝛼 and 𝛽 are represented in the form

𝛼(1)

(𝑥𝑖) =

𝑁𝑥

∏

𝑗=1,𝑗 ̸=𝑖

(𝑥𝑖− 𝑥𝑗) ,

𝛽(1)

(𝑦𝑖) =

𝑁𝑦

∏

𝑗=1,𝑗 ̸=𝑖

(𝑦𝑖− 𝑦𝑗) .

(A.2)

The higher-order weighting coefficients are expressed by thefollowing recursive relations:

𝑐(𝑛)

𝑖𝑗= 𝑛(𝑐

(1)

𝑖𝑗𝑐(𝑛−1)

𝑖𝑖−

𝑐(𝑛−1)

𝑖𝑗

𝑥𝑖− 𝑥𝑗

) ,

𝑐(𝑛)

𝑖𝑖= −

𝑁𝑥

∑

𝑗=1,𝑗 ̸=𝑖

𝑐(𝑛)

𝑖𝑗

𝑖, 𝑗 = 1, 2, . . . , 𝑁𝑥; 𝑛 = 2, 3, . . . , 𝑁

𝑥− 1, 𝑖 ̸= 𝑗,

𝑐(𝑚)

𝑖𝑗= 𝑚(𝑐

(1)

𝑖𝑗𝑐(𝑚−1)

𝑖𝑖−

𝑐(𝑚−1)

𝑖𝑗

𝑦𝑖− 𝑦𝑗

) ,

𝑐(𝑚)

𝑖𝑖= −

𝑁𝑦

∑

𝑗=1,𝑗 ̸=𝑖

𝑐(𝑚)

𝑖𝑗

𝑖, 𝑗 = 1, 2, . . . , 𝑁𝑦; 𝑚 = 2, 3, . . . , 𝑁

𝑦− 1, 𝑖 ̸= 𝑗.

(A.3)

The coefficients 𝛼0, 𝛽𝑘, 𝛾𝑘, 𝛼0, 𝛽𝑘, and 𝛾

𝑘, in (25) and (26),

are defined as

𝛼0= 𝑐(1)

𝑁𝑥,2⋅ 𝑐(1)

1,𝑁𝑥−1− 𝑐(1)

1,2⋅ 𝑐(1)

𝑁𝑥 ,𝑁𝑥−1,

𝛽𝑘= 𝑐(1)

1,𝑘⋅ 𝑐(1)

𝑁𝑥,𝑁𝑥−1− 𝑐(1)

1,𝑁𝑥−1⋅ 𝑐(1)

𝑁𝑥,𝑘,

𝛾𝑘= 𝑐(1)

1,2⋅ 𝑐(1)

𝑁𝑥,𝑘− 𝑐(1)

1,𝑘⋅ 𝑐(1)

𝑁𝑥,2,

𝛼0= 𝑐(1)

𝑁𝑦,2⋅ 𝑐(1)

1,𝑁𝑦−1− 𝑐(1)

1,2⋅ 𝑐(1)

𝑁𝑦,𝑁𝑦−1,

𝛽𝑘= 𝑐(1)

1,𝑘⋅ 𝑐(1)

𝑁𝑦 ,𝑁𝑦−1− 𝑐(1)

1,𝑁𝑦−1⋅ 𝑐(1)

𝑁𝑦,𝑘,

𝛾𝑘= 𝑐(1)

1,2⋅ 𝑐(1)

𝑁𝑦 ,𝑘− 𝑐(1)

1,𝑘⋅ 𝑐(1)

𝑁𝑦,2.

(A.4)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors would like to express their sincere appreciationfor the financial support from the Regional Council ofVästerbotten, the County Administrative Board in Nor-rbotten, and The European Union’s Structural Funds, TheRegional Fund.

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Research Article On the Shear Buckling of Clamped Narrow ...downloads.hindawi.com/journals/mpe/2015/569356.pdfaccurately predict the critical buckling load of a narrow rectangular