3On the use of finite strip method for buckling analysis of moderately thick plate by refined plate theory and using new types of functions

A numerical method is developed for the buckling analysis of moderately thick plate with different boundary conditions. The procedure use the finite strip method in conjunction with the refined plate theory (RPT). Various refined shear displacement models are employed and compared with each other. These models account for parabolic, hyperbolic, exponential, and sinusoidal distributions of transverse shear stress, and they satisfy the condition of no transverse shear stress at the top and bottom surfaces of the plates without using a shear correction factor. The number of independent unknown functions involved here is only four, as compared to five functions in the shear deformation theories of Mindlin and Reissner. The numerical results of present theory are compared with the results of the first-order and the other higherorder theories reported in the literature. From the obtained results, it can be concluded that the present study predicts the behavior of rectangular plates with good accuracy.


Latin American Journal of Solids and
overestimates the buckling loads and natural frequencies.To overcome this shortcoming of the CPT, many shear deformation plate theories, which account for the transverse shear deformation effects, have been developed including the first-order shear deformation theory (FSDT) developed by Reissner (1945); Mindlin (1951).The FSDT accounts for the transverse shear deformation effect, but requires a shear correction factor to satisfy the stress-free conditions at the top and bottom surfaces of the plate (Dawe and Roufaeil, 1978;Wang et al., 2001;Bui and Rondal, 2008).
Although the FSDT provides a sufficiently accurate description of response for thin to moderately-thick plates, it is not convenient to use due to the difficulty of determining an accurate shear correction factor.Thus, to avoid the use of a shear correction factor, many higher-order shear deformation plate theories (HSDTs) were proposed, including the theories of Reddy (1984); Ambartsumian (1958); Levinson (1980); Murthy (1981); Kaczkowski (1968); Panc (1975); Karama et al. (2009Karama et al. ( , 2003)); Mantari et al. (2012); Zenkour (2005); Mechab et al. (2012); Touratier (1991); Benyoucef et al. (2010); Atmane et al. (2010); Soldatos (1992).Although the HSDTs with five unknowns provided sufficiently accurate results for thin to thick plate, their equations of motion were more complicated than those of the FSDT and CPT.Therefore, Shimpi (2002) developed a two-variable refined plate theory (RPT) which is simple to use.The Shimpi's theory is based on the assumption that the in-plane and transverse displacements consist of bending and shear components, and that the bending components do not contribute to shear forces and, likewise, the shear components do not contribute to bending moments.The most significant feature of this theory is that it applies transverse shear strains across the thickness as a quadratic function and satisfies the zero stress boundary conditions at the top and bottom surfaces of the plate without using a shear correction factor.Also, by having fewer unknowns in the equations, this theory enjoys a simpler form which is close to that of the classical plate theory.Some of the most important papers written based on this theory are: Shimpi and Patel (2006a) extended the RPT to the vibration of isotropic plates.The RPT was applied to orthotropic plates by Shimpi and Patel (2006b) in the bending and vibration problems.Thai andKim (2012, 2011) derived the Levy solution of the RPT for the bending, buckling, and vibration of orthotropic plates.Kim et al. (2009) derived the Navier solution of the RPT for the buckling of orthotropic plates.Vo and Thai (2012) adopted the RPT for the buckling and vibration analyses of laminated beams.Recently, the RPT has been extend to nanobeams (2012), nanoplates (2013,2011), functionally graded sandwich plates (2011), and functionally graded plates (2012).Most of the studies based on the refined plate theory has been confined to the use of a particular function for the prediction of transverse shear deformation and have been conducted by using the Navier and Levy solutions.
In this paper, various simple higher-order shear deformation plate theories for the buckling of orthotropic and laminated composite plates are developed.These theories account for parabolic, hyperbolic, exponential, and sinusoidal distributions of transverse shear stress, and they satisfy the condition of no transverse shear stress at the top and bottom surfaces of the plates without using a shear correction factor.The number of unknown functions involved here is only four, compared to five functions in the case of shear deformation theories of Mindlin and Reissner which by removing this one unknown, we can save in the volume, time and cost of extra computations.The analysis employs the finite strip method.This method is applied to study the local instability of thick plates under compression with different boundary conditions.The numerical Latin American Journal of Solids and Structures 12 (2015) 561-582 results of present theory are compared with the results of the first-order and the other higherorder theories reported in the literature.This paper is organized into the following sections.In section 2, the different shear strain shape functions are presented and its application in the finite strip procedure is overviewed.Numerical results and discussions are presented in section 3.In section 4, some concluding remarks are highlighted.

Refined plate theory (Basic assumptions)
Consider the plate and a cartesian coordinate system as shown in Figure 1.The shear components s u and s v , in conjunction with s w , give rise to the ( ) i f z variations of shear strains xz γ , yz γ and hence to shear stresses xz σ , yz σ along the plate thickness h in such a way that shear stresses xz σ , yz σ are zero at the top and bottom surfaces of the plate.Consequently, s u and s v can be expressed as The objective of this paper is to develop various models to employ the new functions ( ) i f z for the buckling analysis of orthotropic and laminated composite plates under compression loading.These functions are shown in Table 1 and are depicted in Figure 2.
Hyperbolic shear deformation theory (HSDT) Sinusoidal shear deformation theory (SSDT) Table 1: Different shear strain shape functions.Functions ( ) i f z must be chosen to satisfy the following constraints: (3)

Kinematics
Based on the assumptions made in the preceding section and using equations (1) through (3b), the displacement field can be obtained as where u and v are the in-plane displacements at any point (x, y, z) in direction of x and y respectively; and u 0 and v 0 denote the in-plane displacements of point (x, y, 0) on the mid-plane in x and y direction respectively, and ( ) i f z is placed from Table 1.The kinematic relations can be obtained as follows: where

Constitutive equations
It is assumed that the laminate is manufactured from orthotropic layers of pre-impregnated unidirectional fibrous composite materials (see Figure 3).Neglecting z σ , the stress-strain relations for each layer in the ( , , ) x y z coordinate system may be written as(6) where ij Q are the plane stress-reduced stiffness values, which are known in terms of the engineering constants in the material axes of the layers:  ,  ,  ,  1 1 1 where and the compact form of Eq. ( 9) will be The components of Q     for each laminated plate has been discussed by Reddy (2004).Figure 5: Pre-buckling system of displacements in a strip.

Finite strip method
In this section, the rectangular plate is modeled by a number of finite strips, each of which has three equally spaced nodal lines (see Figure 5) (Cheung, 1976).For the th m harmonic, the displacement parameters of nodal line i are The unknown displacement field functions (Eq.( 5)) are assumed as follows: (11) where r is the number of harmonics and m S is the th m term of the basic function series (see Appendix) corresponding to particular end conditions, and X , Y , b R , s R are the interpolation matrices defined by Eq. ( 14).
In the above equations,

{ }
; ; ; ; Using Eqs.(12a) through (12d) and (4) the linear strain vector { } where im B     is the strain matrix.The total strain energy U stored during buckling may be written as where V is the volume of the strip.Hence, by substituting Eqs. ( 10) and ( 17) into Eq.( 18) the stiffness matrix is obtained from Latin American Journal of Solids and Structures 12 (2015) 561-582 is the stiffness matrix corresponding to nodal lines i and j , and it can be expressed as where m and n denote the related series terms.
The strip is subjected to in-plane stresses x σ and y σ shown in Figure 4.The potential energy reduction of these stresses ( p V ) during buckling is given by By appropriate substitution, the stability matrix in which Where and( 24) In the equations (24a-c), are the stability matrices.Once the stiffness matrix where λ is a scaling factor related to the critical load and { } ∆ is the eigenvector.

NUMERICAL RESULTS
The numerical program has been written in the MATLAB environment which can model various boundary conditions and three types of isotropic, orthotropic and laminated composite plates.
In this section, to verify the accuracy of the RPT in predicting the buckling behavior of orthotropic and asymmetric cross-ply laminates under different boundary conditions, various numerical examples are presented for laminates with the following properties, and the results of the RPT are compared with those of the classical plate theory (CPT), first-order shear deformation theory (FSDT) and higher-order shear deformation plate theory (HSDT).The explanations of various displacement models are given in Table 2. Material type (1) Reddy ( 2004) Material type (2) Reddy ( 2004) To more conveniently present the numerical results in graphical and tabular forms, they are dimensionless using the following relation: In obtaining the results, plate strips with 14 degrees of freedom have been used.Also in all the results, except the mentioned cases, one harmonic and 10 strips have been used.
In all the tables and figures, a, b and h are the plate width, length and thickness, respectively; and k is shear correction factor for the first-order shear deformation theory (FSDT).

Buckling analysis of simply-supported square orthotropic plate
The dimensionless buckling loads of the simply-supported square orthotropic plate (a b = ) have been presented in Tables 3 and 4 as well as Figures 6, 8 and 9. Material type (1), 10 strips and the first harmonic are used.The results obtained from the RPT numerical solution agree well with the Kim's Navier solutions and the FSDT results.Also, the difference between the results of the present theory, FSDT ( k = 5/6), and CPT have been illustrated in Figures 6 and 7 as an increase in the a h ratio and in Figures 8 and 9 as an increase in the elasticity modulus.As shown in Table 3, the differences between the results of the present study and FSDT ( k = 5/6), and between the results of the present study and FSDT ( k = 1) are 15.42% and 1.6%, respectively, for the same case of square orthotropic plate ( ).The buckling load of a square orthotropic plate subjected to in-plane biaxial pressure was presented in Table 4 and Figure 7, which for converging the results, we used the first two harmonics (m =1 and m =2) and 10 strips.The first two buckling mode shapes of a simply supported square orthotropic plate boundary conditions and a h = 5 and subjected to in-plane uniaxial compressive load is depicted in Figure 10.

b h
Theories Orthotropic

Buckling analysis of simply-supported square orthotropic plate with various shear deformation theories
Table 5 has listed the critical buckling loads obtained from various shear deformation theories for simply-supported orthotropic square plates subjected to uniaxial compression.Material type (1), 10 strips and the first harmonic are used to solve the problem.As shown in Table 5, the nondimensional buckling loads obtained by sinusoidal and exponential functions are greater than those obtained by hyperbolic and parabolic functions.

Buckling analysis of square orthotropic plate with different boundary conditions
The non-dimensional buckling loads of square orthotropic plates (a b = ) with different boundary conditions have been shown in Table 6 and Figure  ply supported, clamped and free.Material type (1), 10 strips and the first harmonic term is used to solve the problem.In Table 6, a comparison has been made between the critical buckling loads of thin plates (a h = 100) achieved by the present RPT numerical solution, the Levy-Thai solution (2011) and the CPT solution.The changes of the critical buckling load with thickness ratio and PSDT model are shown in Figure 11.In Table 6, 1 β and 2 β are the load parameters that indicate the loading conditions.Positive values for 1 β and 2 β indicate that the plate is subjected to biaxial compressive loads.Also, a zero value for 1 β or 2 β shows uniaxial loading in the x or y direction, respectively.The buckling mode shapes of a square orthotropic plate with various boundary conditions and a h = 5, subjected to in-plane uniaxial pressure are shown in Figure 12.

CONCLUSIONS
The finite strip numerical solution and the use of the refined plate theory for orthotropic and laminated composite plates at different boundary conditions have been investigated.Also in this Latin American Journal of Solids and Structures 12 (2015) 561-582 solution, the results of various transverse shear functions have been compared.The important findings of this analysis can be expressed as follows: 1-In this paper, we employed the four transverse shear functions of PSDT, HSDT, ESDT and SSDT (Table 1).In section 3.2 (Table 5), all four functions are used for the analysis of different plate samples and demonstrated that the non-dimensional buckling loads of the PSDT and HSDT functions are less than those obtained the ESDT and SSDT functions.Therefore in Section 3.1, we only used the PSDT function for analysis.
2-In Section 3.4 (Table 8), it is shown that the results obtained by the PSDT function are closer to the exact solution.3-The present theory yields more accurate buckling load values than the first-order shear deformation theory.4-The buckling loads of the hyperbolic transverse shear function has a good accuracy compared with those of the first-order shear deformation theory.5-The buckling loads of the exponential transverse shear function is usually higher than those of the first-order shear deformation theory.6-This paper provided many examples for the the analysis of orthotropic plates with different boundary conditions and subjected to uniaxial and biaxial loading situations.Examples of laminated composite plats with different layers and sizes are presented in section 3.4, in all cases good accuracy is observed.7-The most significant feature of this theory is that it may apply the transverse shear strains across the thickness as parabolic, sinusoidal, hyperbolic and exponential functions.Also, by having fewer unknowns in the equations, this theory enjoys a simpler form which is close to that of the CPT.

Figure 1 :
Figure 1: Illustrations of displacements and plate meshing arrangement.
b u and bv are assumed to be similar to the displacements given by the CPT.Therefore, the expressions for b u

Figure 2 :
Figure 2: Variation of functions ( ) i f z along the plate thickness.

Figure 3 :
Figure 3: Coordinate system and layer numbering used for a typical laminated plate.

Figure 4 :
Figure 4: Pre-buckling stresses in a strip.Figure5: Pre-buckling system of displacements in a strip.
2 s x b η = and s b is the strip width.It should be noted that the Hermitian cubic polynomials used in the interpolation functions of b w and s w in the x direction, guarantee the inter-element continuity of the transverse displacement w and of its first derivatives been derived, and combined for each com-Latin American Journal of Solids and Structures 12 (2015) 561-582posite strip, they can be assembled into the respective global matrices K dard procedures.The buckling problem can then be solved by eigenvalue equations strips and first two harmonics) Table 4: Nondimensional critical buckling loads of simply-supported (SSSS) square plates ( a b = ) subjected to biaxial compression.

Figure 6 :Figure 7 :
Figure 6: The effect of side-to-thickness ratio on the critical buckling load of square plates subjected to uniaxial compression; 1 2 25 E E = .

Figure 8 :
Figure 8: The effect of modulus ratio on the critical buckling load of square plates subjected to uniaxial compression; 10 a h = .

Figure 9 :Figure 10 :
Figure 9: The effect of modulus ratio on the critical buckling load of square plates subjected to uniaxial compression; 20 a h = .
11.In this section, the boundary conditions of two loaded ends are simply supported and side edges boundary conditions are considered as sim-Latin American Journal of Solids and Structures 12 (2015) 561-582

Figure 11 :
Figure 11: The effect of side-to-thickness ratio on the critical buckling load of square plates with different boundary conditions subjected to uniaxial compression along the y-axis; 1 2 E E = 10 and PSDT model.

Figure 12 :
Figure 12: The buckling mode shapes of a square orthotropic plate with various boundary conditions; A: SSSS, B: SSSC, C: SSCC, D: SSFF
One end simply-supported and the other end clamped

Table 2 :
Description of various displacement models.

Table 3 :
Nondimensional critical buckling loads of simply-supported (SSSS) square plates subjected to uniaxial compression.

Table 6 :
Comparison between nondimensional critical buckling loads of square orthotropic plates with different boundary conditions ( a h = 100).

Table 7 :
Nondimensional critical buckling load of simply-supported asymmetric cross-ply square plates ( a b = ).