International Journal of Civil Engineering and Technology (IJCIET)
Volume 10, Issue 04, April 2019, pp. 1715-1721, Article ID: IJCIET_10_04_180
Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=10&IType=04
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
© IAEME Publication
Scopus Indexed
FURTHER ASSESSMENT OF BUCKLING
STABILITY OF STEEL PLATES
Farhad Riahi
Dept. of Civil Engineering, Mahabad Branch, Islamic Azad University, Mahabad, Iran
Dept. of Civil Engineering, Boukan Branch, Islamic Azad University, Boukan, Iran
Vahed Mam Ghaderi
Dept. of Civil Engineering, Mahabad Branch, Islamic Azad University, Mahabad, Iran
Tadeh Zirakian*
Dept. of Civil Engineering and Construction Management, California State University,
Northridge, CA, USA
Bijan Sanaati
Dept. of Civil Engineering, Boukan Branch, Islamic Azad University, Boukan, Iran
*
Corresponding Author; Tel.: +1-818-677-7718, E-mail: tadeh.zirakian@csun.edu
ABSTRACT:
Thin plates are commonly-used structural members with wide range of
applications. In spite of the proliferation of research on the buckling behavior of such
members, further detailed investigations are still required for the performance
assessment and improvement of such structural members. On this basis, the buckling
behavior of thin steel plates subjected to uniaxial compressive loading is further
investigated in this research endeavor via detailed numerical simulation by considering
some key factors including the support conditions, aspect ratio, and slenderness. The
results and findings of this study can provide further insight into the efficient design
and application of such important structural members.
Keywords: Plate, Steel, Buckling, Numerical simulation
Cite this Article: Farhad Riahi, Vahed Mamghaderi, Tadeh Zirakian and Bijan
Sanaati, Further Assessment of Buckling Stability of Steel Plates. International
Journal of Civil Engineering and Technology, 10(04), 2019, pp. 1715-1721
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Further Assessment of Buckling Stability of Steel Plates
1. INTRODUCTION
Plates and shells are important structural elements in engineering and their buckling is a major
aspect of structure damage. The buckling problem of a thin plate subjected to in-plane
compressive and/or shear loading is important in the construction industry. The buckling of
plates under distributed edge forces has been studied by numerous investigators, including
Timoshenko (1910), Leggett (1937), Zetlin (1955), White and Cottingham (1962), Bulson
(1969), Hopkins (1969), Khan and Walker (1972), Khan et al. (1977), and Young and Lui
(2005). The first work in this area was perhaps reported by Van der Neut (1958), which
considered a uniaxial compressive loading with a half sine distribution. Later, Benoy (1969)
considered a uniaxial compressive loading with a parabolic distribution and obtained an energy
solution. Shakerley and Brown (1996) investigated elastic buckling of simply supported and
fully fixed plates with eccentrically placed rectangular cutouts using conjugate method. Their
results were given for uniaxial compression and shear loading. Bert and Devarakonda (2003)
studied the buckling of rectangular plates subjected to non-linearly distributed in-plane loading
using analytical solution method. They considered a rectangular plate subjected to parabolic
loads along with the removal of some deficiencies of earlier reported work. Zhong and Gu
(2006) reported a study on the buckling analysis of simply supported rectangular plates
subjected to linearly varying edge loads. An analytical solution to buckling problem was
developed and the effect of load intensity variation on the critical load was investigated.
Recently, Mijušković et al. (2014) presented an accurate buckling analysis for thin rectangular
plates under locally-distributed compressive stresses. Interestingly, Wang et al.’s (2007) and
Wang’s (2015) studies have also shown that the differential quadrature method can yield
accurate buckling loads for rectangular plates under uniformly or non-uniformly distributed
edge compressions. It is also noteworthy that some other researchers have conducted
comprehensive research, most recently, on the buckling of shells and plates, e.g. McCann et al.
(2016), Riahi et al. (2017 and 2018), and Vu et al. (2019).
In this paper, the buckling behavior of rectangular steel plates subjected to uniaxial
compression is investigated by considering some key parameters including the support
conditions, aspect ratio, and slenderness. The current numerical study intends to provide a
better understanding of effects of the aforementioned significant geometrical properties on the
buckling stability performance of thin steel plates, which can consequently result in efficient
design and application of such commonly-used and important structural members.
2. GEOMETRICAL AND MATERIAL PROPERTIES OF THE PLATE
MODELS
The considered boundary conditions of the plates are illustrated in Fig. 1, which include CCCC:
clamped along all edges; SSSS: simply-supported along all edges; CFFS: clamped along the
free edges and simply-supported along the loaded edges; SFCC: simply-supported along the
free edges and clamped along the loaded edges. The letters S, C, and F respectively stand for
simple, clamped, and free edges. All plate models are subjected to uniaxial compressive
loading, as shown in Fig. 2.
Figure. 1. Boundary conditions of the plate models
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Figure. 2. Geometry and loading
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Farhad Riahi, Vahed Mamghaderi, Tadeh Zirakian and Bijan Sanaati
As it is seen in Fig. 2, the plate length (a) lies along the x-axis, the plate width (b) lies along
the y-axis, and the plate thickness (t) is constant along the z-axis. It was assumed that midplane of the plate is coinciding with the x-y plane. The plate length (a) was considered to vary
between 500 mm and 4000 mm. Moreover, the plate width (b) was considered to vary between
500 mm and 1000 mm, and also the plate thickness was taken as 1 mm and 2 mm. Thirty-two
plate models, in total, were considered for the purpose of this research. The resulting thirtytwo models were divided into four groups based on the considered boundary conditions. The
geometrical properties of the plate models are summarized in Table 1.
Table 1. Geometry of specimens
Plate model
Geometrical property
Plate width (b) [mm]
Plate length (a) [mm]
Plate thickness (t) [mm]
Length/width ratio (a/b)
P500
500
500
1000 1500
1
1
2
3
500
500
1
1
500
2000
1
4
P1000
1000 1000
2000 3000
2
2
2
3
1000
1000
2
1
1000
4000
2
4
The mechanical properties of the adopted steel material include: yield stress = 355 MPa,
Poisson’s ratio = 0.3, Young’s modulus = 2.1×105 MPa, and bulk density = 7850 kg/m3.
3. FORMULATION OF THE PROBLEM
As illustrated in Fig. 2, an isotropic thin rectangular plate with length (𝑎), width (𝑏), and
thickness (𝑡), subjected to uniaxial compressive loads 𝑃(𝑥) was considered for the purpose of
the current investigation. On this basis, the displacement field could be expressed as:
𝜕𝑤
𝑢̅(𝑥, 𝑦, 𝑧) = −𝑧 𝜕𝑥 ,
𝜕𝑤
𝑣̅ (𝑥, 𝑦, 𝑧) = −𝑧 𝜕𝑦 , 𝑤
̅(𝑥, 𝑦, 𝑧) = 𝑤(𝑥, 𝑦)
(1)
Where, (𝑢, 𝑣, 𝑤) are the displacement components along the x, y, z coordinate directions,
respectively, and 𝑤 is the transverse deflection of a point on the mid-plane, i.e. z = 0. Assuming
the material of the plate to be isotropic and to obey Hooke’s law, the stress-strain relations can
be given by
𝐸
𝐸
𝐸
𝜎𝑥𝑥 = 1−𝜈2 (𝜀𝑥𝑥 + 𝜈𝜀𝑦𝑦 ), 𝜎𝑦𝑦 = 1−𝜈2 (𝜀𝑦𝑦 + 𝜈𝜀𝑥𝑥 ), 𝜎𝑥𝑦 = 𝐺𝛾𝑥𝑦 = 2(1+𝜈) 𝛾𝑥𝑦
(2)
Where, 𝐸 = Young’s modulus, 𝐺 = shear modulus, and 𝜈 = Poisson’s ratio. Eqs. (1) to (3)
constitute the mathematical formulation of the problem under consideration. Timoshenko and
Gere (1961) investigated the buckling behavior of a simply supported rectangular plate
subjected to uniform loading. The elastic critical buckling load (𝑃𝑐𝑟 ) of such thin plates can be
calculated using Eq. (3):
𝑘𝜋2 𝐸𝑡 3 𝑙
𝑃𝑐𝑟 = 12(1−𝑣2 )ℎ2
(3)
Where, ℎ = depth of plate, 𝑘 = buckling coefficient of plate subjected to edge loading, 𝑙 =
plate length, and 𝑡 = plate thickness. The buckling coefficient is calculated against the plate
aspect ratio a/b. An approximate analytical solution was provided by Timoshenko and Gere
(1961), as given in Eq. (4).
𝑘 = 0.456 + (𝑏⁄𝑎)2
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(4)
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Further Assessment of Buckling Stability of Steel Plates
4. NUMERICAL ANALYSIS
ABAQUS (2016) software was used for numerical simulation of the plate models. Thirty-two
axisymmetric plate models with geometrical and material nonlinearities were developed. The
plates were modeled using the eight-node SHELL element S8R5 with six degrees of freedom
at each node including three translations and three rotations. An irregular mesh in finite element
modeling was employed for simulation of the plates in which the element size was selected to
be b/30 along the plate width. Linear and nonlinear buckling analyses were performed to obtain
the buckling mode shapes, capacities, and deformations of the plate models. Fig. 3 shows
typical buckling mode shapes of plates. Fig. 4, also, illustrates some typical analysis results on
the large deflection collapse of the plate models.
Figure. 3. Typical buckling modes of plates
Figure. 4. Large deflection collapse of the plates under uniaxial compression
5. DISCUSSION OF RESULTS
5.1. Effects of Boundary Conditions and Aspect Ratio on Buckling Capacity
The buckling capacities of plates with various aspect ratios and support conditions are
illustrated in Fig. 5. In general, these results demonstrate the increasing of the buckling capacity
of the plates as the aspect ratio increases beyond unity. This is indicative of effectiveness of
the aspect ratio on the capacity performance of the plates. It is observed that the amount of
fixity of the plate edges, as in CCCC and SFCC cases, plays an important role in improving
the buckling capacities of the plates. Plates with CCCC support conditions possess the highest
buckling strengths. In cases of SSSS and CFFS support conditions, it is evident that the aspect
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Farhad Riahi, Vahed Mamghaderi, Tadeh Zirakian and Bijan Sanaati
CCCC
SSSS
CFFS
SFCC
24
22
20
18
16
14
12
10
8
6
4
2
0
1
buckling load (*10^5 N/m)
buckling load (*10^5 N/m)
ratio parameter has a minimal effect on the buckling capacity of the plate. Plates with CFFS
support conditions have the lowest performance due to the possession of two free edges; this
can be quite important in the seismic applications of plates with free edges, for instance, the
beam-attached infill panels in steel plate shear wall (SPSW) systems which requires the
designer’s proper and adequate assessments.
2
a/b
3
24
22
20
18
16
14
12
10
8
6
4
2
0
4
CCCC
SSSS
CFFS
SFCC
1
2
a/b
(a) P500
3
4
(b) P1000
Figure. 5. Effects of plate aspect ratio and boundary conditions on buckling capacity
5.2. Effects of Boundary Conditions an Aspect Ratio on Maximum Displacement
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0.14
CCCC
SSSS
CFFS
SFCC
1
Maximum displacement-U3(m)
Maximum displacement-U3(m)
The maximum out-of-plane displacements of the models with various geometrical properties
were obtained from the nonlinear static analyses for further stability (and serviceability)
performance(s) assessment of the plates. These results are illustrated in Fig. 6. It is found that
by increasing of the a/b aspect ratio from 1 to 4, the plate maximum displacement increases as
well, consistent with all cases. The highest and the lowest maximum displacement values are
associated with the SFCC and CCCC support conditions, respectively. Despite the
scatteredness of the results, it may be concluded that fixity and proper arrangement of the plate
edges, especially the free edges by considering the loading conditions in structures, can
improve the out-of-plane displacement (and serviceability) performance(s) of plates.
2
a/b
3
4
(a) P500
CCCC
SSSS
CFFS
SFCC
0.12
0.10
0.08
0.06
0.04
0.02
0.00
1
2
a/b
3
(b) P1000
Figure. 6. Effects of plate aspect ratio and boundary conditions on maximum displacement
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Further Assessment of Buckling Stability of Steel Plates
6. CONCLUSION
In this paper, the buckling stability performance of steel plates with different support
conditions, aspect ratios, and slenderness values, subjected to uniaxial compression was
investigated through detailed numerical simulations. Linear and nonlinear buckling analyses
were performed and the buckling capacities and maximum out-of-plane displacements of the
plate models were obtained. It was shown that the edge fixity can be instrumental in improving
the buckling stability performance of plates. More importantly, accurate and careful assessment
of the loading conditions in structures and accordingly proper placement of plates with different
edge supports can improve the performance of the plate elements and indeed the main objective
of this study was to provide the engineers and designers with more insight into the efficient
design and application of plates as commonly-used structural members.
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