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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
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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
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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)
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4 Mathematical Problems in Engineering
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).
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Mathematical Problems in Engineering 5
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)
-
6 Mathematical Problems in Engineering
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)
-
Mathematical Problems in Engineering 7
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.
-
8 Mathematical Problems in Engineering
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.
-
Mathematical Problems in Engineering 9
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
Closed-form approachDQ code
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
Closed-form approachDQ code
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.
-
10 Mathematical Problems in Engineering
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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Mathematical Problems in Engineering
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Algebra
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