Semi-Analytical Finite Strip Transfer Matrix Method for Buckling Analysis of Rectangular Thin Plates

Plates and shells are main components of modern engineering structures, whose buckling analysis has been focused by researchers. In this investigation, rectangular thin plates with loaded edges simply supported can be discretized by semi-analytical finite strip technology. Then the control equations of the strip elements of the buckling plate will be rewritten as the transfer equations by transfermatrixmethod. A new approach, namely semi-analytical Finite Strip TransferMatrixMethod, is developed for the buckling analysis of plates. This method requires no global stiffness matrix of the system, reduces the systemmatrix order, and improves the computational efficiency. Comparing with some theoretical results and FEM’s results of two illustrations (the plates and the ribbed plates) under six boundary conditions, the method is proved to be reliable and effective.


Introduction
Buckling analysis is one of the important steps in the design of thin-walled structures which can be applied in different branches of engineering, including shipbuilding, civil architecture, and mechanical construction [1].Thin plate, a main kind of thin-walled structure, is widely utilized to lighten engineering structures as well as save materials [2].The reliability of one single plate lies in its stability chiefly, which has been studied by experimental or mathematical means [3].
In the early works, the vibration and buckling performances of rectangular plates loaded by in-plane hydrostatic forces for a wide variety of aspect ratios, boundary conditions, and loading magnitudes have been analyzed by numerical technology [4].The solutions of the differential equations of the buckling Mindlin plate are obtained in discrete forms by applying numerical integrations [5].And extensive numerical results have been presented for the critical buckling loads of simply supported, rectangular composite plates subjected to five types of loading conditions: (1) uniaxial, (2) hydrostatic biaxial, (3) compression-tension biaxial, (4) positive shear, and (5) negative shear [6].By introducing an unified analytical solution technique for a multitude of combinations of boundary conditions, an analytical method is presented for the problem of elastic buckling of orthotropic rectangular plates [7].If the biharmonic operator in the buckling control equations of rectangular plates is reduced by performing the Laplace's operator and the finite difference method, the buckling load of the plate can be investigated [8].Up to now, many methods have been used to analyze the buckling problems of rectangular plates, such as the extended Kantorovich method [9], differential quadrature procedures [10], asymptotic finite strip method [11], block GMRES method [12], firstorder shear deformation meshless method [13], radial point interpolation method [14], untruncated infinite series technology [15], discrete singular convolution approach [16], and hierarchical Rayleigh-Ritz and finite element method [17].
Among these methods, finite element method is a powerful tool for engineering analysis, while the choices of the elements and the mesh sizes have significant influences on the results of buckling analysis [18].When calculating the buckling problems of regular geometry shape structures, finite strip method can be regarded as an efficient way.And the arbitrary shaped plate may be discretized as many strip elements by the subparametric mapping concept [19].To consider the transverse shear effect, the spline strip method has been proposed to analyze the buckling behaviors of rectangular Mindlin plates with linearly tapered thickness in one direction [20].The buckling stresses and natural frequencies of rectangular laminated plates with arbitrary layups and general boundary conditions can be predicted by the improved spline finite strip, which combines the superstrip concept [21].Then a higher-order shear deformable plate finite strip element is developed and employed to investigate the critical buckling loads of composite laminated plates [22].The spline finite strip method and multilevel substructuring procedures are combined for the buckling stress analysis of composite laminated, prismatic shell structures with general boundary conditions [23].Buckling mode localization in ribstiffened plates with randomly misplaced stiffeners is studied by finite strip method [24].Notably, a comprehensive review of the finite strip method for structural analysis is given [25].A layerwise B-spline finite strip method is developed for free vibration analysis of truly thick and thin composite laminated plates [26].And a finite strip Fourier p-element is developed to analyze the natural vibration characteristics of the thin plate [27].Based on the concept of the semienergy approach, the finite strip method can be proposed to analyze the buckling [28], postbuckling [29], dynamic buckling behaviors of the laminated plates [30], and stability phenomena of cylindrical shell structures [31].The harmonic series satisfying the boundary conditions in the loaded ends of thin-walled members are generally employed in Semi-Analytical Finite Strip Transfer Matrix Method (SAFSM) [32].And the buckling stress of the cold-formed sections can be analyzed by SAFSM [33], which has been widely used in computer softwares (such as THIN-WALL [34], CUFSM [35]) to develop the signature curves [36] of the buckling stress versus buckling half wavelength for thin-walled members.Furthermore, the constrained finite strip method innovated from SAFSM is developed and applied in cold-formed steel design [37].Generalized constrained finite strip method for thin-walled members with arbitrary cross-sections can be used to analyze secondary modes [38] and primary modes [39].The impacts of basis, orthogonalization, and normalization in constrained finite strip method are discussed for stability solution of open thin-walled members [40].
Transfer matrix method has been developed as an effective tool for vibration analysis of engineering structures, especially for chain connected system from topological perspective [41].Then a combined finite element transfer matrix method is developed to study the statics [42] and dynamics [43] of structures.Later a new transfer matrix method based on the boundary element and transfer matrix technology is proposed for the vibration analysis of two-dimensional plate acted by uniform [44] and concentrated [45] loads.By introducing the numerical integration, nonlinear dynamics of structures [46], multi-rigid-body system [47], and multiflexible-body system [48] can be dealt with by transfer matrix method.And a procedure which combines multiport transfer matrices and finite elements has been developed to resolve the acoustic phenomena of automotive hollow body networks [49].It should be noted that three references enlighten us to start this investigation.The buckling analysis of the plate with built-in rectangular delamination has been implemented by strip distributed transfer function method [50].And the transfer matrix method can be used to analyze x, u z, w the instability in unsymmetrical rotor-bearing systems [51] and tall unbraced frames [52].
Here by combining two powerful methods of the semianalytical finite strip and the transfer matrix, an efficient technology named semi-analytical Finite Strip Transfer Matrix Method (FSTMM) for the buckling analysis of plates is developed.The text is organized as follows: In Section 2, the general theorem of the semi-analytical finite strip for buckling analysis of plates is shown.Section 3 presents semianalytical Finite Strip Transfer Matrix Method for buckling analysis.In Section 4, some results calculated by FSTMM and other methods are given, which can validate the proposed method.The conclusions are included in Section 5.

The Semi-Analytical Finite Strip Analysis
2.1.Degree of Freedom and Shape Function.In classical Kirchhoff plate theory, the membrane or in-plane translations are neglected and only the -axis's translation  and the axis's rotation  are considered as local displacements [1].The global coordinate system (, , ) and local coordinate system (, , ) are actually the same in the plate model.In finite strip method (FSM), a thin plate, as shown in Figure 1, can be divided into many strip elements.
We introduce a numbering system of finite strip model shown in Figure 1.The total number of nodal lines is ; therefore, the total number of strips is  − 1.Generally, one internal nodal line is always connected with two strips, while the first and last nodal lines are only connected with one strip.This numbering system will be used to depict the state vector of the nodal line and the transfer matrix of the strip in Section 3.
Here the plate is assumed to have two opposite loaded edges with the simply supported boundary; analytical trigonometric functions that satisfy the simply-simply supported boundary condition of the loaded edges can be given to express the transverse deflection of the plate along the longitudinal direction [35,37]: where  is the half-wave number, which also stands for certain half wavelength along the longitudinal direction,  is the circumference ratio,  is the longitudinal coordinate in local system,  is the length of the plate.
Four cubic polynomials that satisfy the boundary condition can be selected as the shape function matrix to depict the out-of-plane displacement of the plate strip along the transverse direction, namely [39,40], where  is the width of the strip as shown in Figure 1 and  is the coordinate along the width direction in local system.Then the transverse deflection  of any points (, ) in the strip can be denoted by the so-called nodal line displacement vector   and shape function matrix N  (, ): where N  (, ) can be denoted as the production of the shape functions of ( 1) and ( 2), can be defined as the -axis's translation  and the -axis's rotation  of the two nodal lines in one strip, the subscripts  and  denote two edges of one strip, which are connected with other strips or are the boundaries of the plate respectively, and  is the maximum half-wave number as well as order of modal employed in the analysis, which is a finite positive integer.

Fundamental Stiffness Matrix.
So far, many plate theories have been developed by engineers and researchers [2].The Kirchhoff plate theory is a classical and simple one, which will be used in this investigation.It must be noted that other plate theories can be applied in the proposed method if the elastic deformation energy and the nodal line displacement vector are modified accordingly.The bending strain vector of middle plane in Kirchhoff plate can be defined as We substitute (3) into above equation, the bending strain vector can be expressed by the nodal line displacement vector and shape function matrix; namely, where 4) matrix that gives the relationship between the strain vector   and the nodal line displacement vector   .As to general linear elastic material, the elastic deformation energy can be expressed as where  and  denote strain and stress vectors, respectively,  is the volume of the material.According to Kirchhoff plate theory with constant thickness, the elastic bending deformation energy of ( 8) can be rewritten as where D  is an elastic constant matrix of the material with a dimension of (3 × 3), which is related to the thickness of the plate [1,2].We substitute (4) into ( 9), the new deformation energy formula can be obtained as follows: where the elastic stiffness matrix of the strip can be concisely expressed in a form Above elastic stiffness is a (4×4) matrix.When the parameters ,  vary, the elastic stiffness changes accordingly.However, to the case of  ̸ = , elastic stiffness matrix always equals zero, that can be found by integrating (11).The reason is that the shape function N    is orthogonal about the elastic constant matrix D  , which can be expressed as As shown in Figure 1, the strip is loaded with linearly varying edge tractions.The in-plane compressive loads can be expressed as where   ,   are the forces in two edges of the strip,  is the width of the strip as shown in Figure 1, and  is the transverse coordinate.Similar to the deduction of the elastic stiffness matrix, the (4 × 4) geometric stiffness matrix caused by inplane compressive loads can be obtained [34,35]: where N   = N()(  ()/) is a (1 × 4) matrix.It can be proofed that N    includes an orthogonal system of functions; namely, The solution of the buckling problems can be deduced by the orthotropic condition of ( 12) and ( 15) conveniently.For other boundary conditions, different analytical functions can be selected to describe the buckling shape of the strip [53].These will be discussed in the successive related papers.

Semi-Analytical Finite Strip Transfer Matrix
Method for Buckling Analysis

Control Equations of Strip Element.
In both finite element transfer matrix method and boundary element transfer matrix method, transfer equations of the given substructure can be deduced by the control equations of this substructure which consider the interaction forces between this substructure and other structures.As to the proposed Finite Strip Transfer Matrix Method, the strip element can be regarded as the substructure.
The control equations of the buckling strip can be obtained by virtual work principle [1,2]: where k   is the elastic stiffness matrix of (11), k   is the geometric stiffness matrix as shown in (14),   is the nodal line displacement vector can be defined by (5),  is the half-wave number of the buckling strip shape, and R  can be regarded as the generalized internal forces acted on the strip that can be denoted as where  Assuming  0 () as the initial axial force, the real axial forces in the geometric stiffness matrix can be expressed as where  is the buckling coefficient.So the geometric stiffness matrix k   can be rewritten as the function of the constant geometric stiffness matrix k   |  0 caused by initial axial force  0 (), We substitute ( 19) into ( 16): the control equations of the strip can be rewritten as follows: To simplify the equations, the coefficient matrix of the nodal line displacement vector   in above equation can be marked by where both coefficient matrices k   and k   |  0 are constant values when the loads determined by ( 18) vary.

State Vector, Transfer Equations, and Transfer Matrix.
During the analysis of transfer matrix method, the state vector of the strip is an important concept that includes two parts: one part describes the configuration of the strip, the other part gives the generalized internal forces acted on the strip by other members in the system.For example, the state vector of the nodal line () can be defined as where the first subscript  denotes the number of the nodal lines, the second subscript  denotes the number of the strips,   is the generalized internal force vector acted on the nodal line  correspondingly.
Using the block forms of ( 22), the control equations (20) can be rewritten as the form of the transfer equations of this strip, which are where the transfer matrix of the strip  is where the superscript  denotes the strip element, the subscript  denotes the number of the strips, the subscripts  and  denote the two nodal lines of the strip , and K  (,  = 1, 2) are (2 × 2) block submatrices that can be determined by (21).Actually, coefficient matrix of ( 21) can be denoted as Follow the condition of displacement continuum and the law of action and reaction, the state vectors of the same nodal line in two connected strips can be obtained: where the subscript  denotes the nodal line number and the subscripts  and  + 1 denote the order numbers of the two strips which are connected by the nodal line .So the transfer equations between particular nodal lines of conjunctional strips can be obtained by multiplying ( 23) and ( 26); namely, where the transfer matrix of this substructure is where the superscripts  and  denote the strip element and the nodal line.
Introducing the same procedure in the classical transfer matrix method, the overall system transfer equations and the overall transfer matrix U all which relate the state vectors at two edges of the plate can be assembled and calculated as Three classical boundary conditions are considered in general engineering problems: simply supported, clamped supported, and free edge.Different boundary conditions of the plate can give appropriate limits to specific variables in the state vectors, which are shown in Table 1.For the buckling analysis of plates with simply-simply (S-S) supported boundary condition of loaded edges in this dissertation, the unloaded edges may sustain six kinds of boundary conditions, which can be expressed by SSss, SScc, SSsc, SSfs, SSfc, and SSff (s, c, and f mean simply, clamped, and free boundary conditions, resp.).Here the capital letters and lowercase letters denote the loaded edges and unloaded edges correspondingly.
Taking the boundary condition SSss, for example, the total transfer equations can be deduced as follows: where subscripts  and  denote the first and last edges of the plate, U all is the overall transfer matrix of the plate.Then the nonzero variables in the state vector of the first edge of the plate have the relationship that can be deduced by (30): where  12 ,  13 ,  42 ,  43 are the elements of U all .To make the nonzero solution of (31) possible, it must satisfy the following condition: det Above equation gives the characteristic equation of the buckling plate by the proposed semi-analytical Finite Strip Transfer Matrix Method which can be used to calculate the buckling coefficients.We combine ( 31) and ( 27); the buckling mode can be obtained.Eigen-matrices under various boundary conditions are shown in Table 2.

Examples and Analysis
2305 N/mm.The length of the plate is a variable parameter which is selected for each boundary condition.
The section is divided into five segments along the loaded edge.The relationship schemas between buckling coefficient  and length-width ratio  of the plate are obtained by the proposed FSTMM, finite element method (FEM), and theoretical analysis.FSTMM's results are compared with theoretical and FEM's results under the boundary conditions of SSss and SSff, as shown in Figures 2(a) and 2(b).For the remaining four boundary conditions, comparisons with FSTMM's and FEM's results are shown in Figures 2(c)-2(f).The six comparisons confirm the reliability of FSTMM for the buckling analysis of the plate.

Illustrations of Ribbed Plates.
Plates with ribs are important engineering components for the load-carrying.Many methods are available to analyze the buckling of prismatic flat and stiffened shell structure [54].For further application of FSTMM, ribbed plates are discussed below.Notably, due to the application condition of the Kirchhoff plate theory, the thickness of the ribs must be relatively small.Besides, comparisons with FEM's results are used to testify the FSTMM.the thickness is 0.1 mm.The ribs are 0.4 mm in width and 0.2 mm in thickness.Three ribs are uniformly spaced along the transverse direction, the width of the interval space is 0.95 mm.The length of the plate is a variable parameter ranging from 0.5 mm to 25 mm.In shell models of FSTMM and FEM, the initial axial force of the plate is 7.2305 N/mm and the initial axial force of ribs is 14.461 N/mm.In solid model of FEM, the initial axial pressure in the cross section is 72.305MPa.Then the loading forces of the shell and solid models are equivalent.Figure 3(a) shows the meshed grid of the plate in FSTMM and the meshed plate in FEM using the shell elements and the solid elements can be given in Figures 3(b) and 3(c), respectively.As the plate simply supported in four edges, Figure 4 shows the total buckling deformation of the plate with three ribs when length-width ratios are 2 and 3.5.There are two (or three) half waves in the buckling mode of the length-width ratio  = 2 (or  = 3.5) shown in Figures 4(a   with the shell elements and solid elements, as shown in Figure 5. Three curves have similar variation tendency which proves FSTMM can also be credible in the buckling analysis   of the ribbed plate.And there is an obvious difference about the computational consumption that depends on the element number among these three methods shown in Figure 3.So the proposed method is highly efficient.Furthermore, the buckling coefficient of FEM with solid model is always smaller than that of FEM with shell model, and the buckling coefficient of FEM with shell model is always smaller than that of FSTMM; the reason is that the shear strain is generally included in the FEM analysis with solid elements and partly included in the analysis with the FEM's shell elements while neglected in the FSTMM analysis.It also indicates that the classical Kirchhoff assumption in the plate theory will make the critical buckling loads a bit larger than the true results.Comparing with Figures 2(a) and 5, the buckling coefficients converge to, 4 and 6 respectively, when  increases.We can obviously find that the plate can be stiffened by the ribs.

Plate with Four
Ribs.The material properties are the same as stated in the former chapter.The geometrical parameters are as follows.The width of the plate is 5 mm and the thickness is 0.1 mm.The ribs are 0.2 mm in width and 0.2 mm in thickness.Four ribs are uniformly spaced along the transverse direction; the width of the internal space is 0.84 mm.The length of the plate is a variable parameter ranged from 0.5 mm to 35 mm.In shell model of FEM and FSTMM, the initial axial force of the plate is 7.2305 N/mm and the initial axial force of ribs is 14.461 N/mm.In solid model of FEM, the initial axial pressure in the cross section is 72.305MPa.
Then the loading forces of the shell and solid models are equivalent.The plate is simply supported in loaded edges, while two unloaded edges are free and clapped separately.
Figure 6(a) shows the buckling deformation of a plate with four ribs and boundary conditions of SSfc when the length is 10 mm; in other words, the length-width ratio is 2. Figure 6(b) shows the buckling deformation of a plate with the length 17.5 mm (length-width ratio 3.5).As to the case of Figure 6(a) (or Figure 6(b)), this critical buckling configuration includes one (two) half-wave(s) model.Actually, the minimal critical buckling load can be obtained by setting the half-wave number  = 1 (or  = 2) in the FSTMM.The value of the half-wave number  can be used to distinguish the buckling models of the plate which is convenient for the suppression of the instable failure.
In Figure 7, three curves obtained by three different methods give the relationship between buckling coefficient and length-width ratio of the four-ribbed plate with boundary conditions of SSfc.The good agreement among the three results shown in Figures 5 and 7 indicates that the proposed FSTMM can be used in the buckling analysis of the plate with several ribs and different boundary conditions.

Comparative Analysis.
As a general rule, the computational precision can be improved by increasing the number of the elements.Here the influence of the element number to the computational precision can be analyzed if FEM's shell model and FSTMM are applied to calculate the buckling problems   Figure 9 compares the influence of the element number to the computational precision in FEM's shell model and FSTMM.In order to make the errors between the FEM's and theoretical results acceptable, the element number may increase unexpectedly from 100, 264 to 1600, as shown in Figure 9(a).In contrast with this performance of the FEM, as the number of strips  = 2 in FSTMM is selected, the computational results have good agreement with the theoretical results, as shown in Figure 9(b).So it may be confirmed that the proposed FSTMM has good efficiency for the buckling analysis of rectangular thin plates under the boundary condition of simply supported loaded edges.

Conclusions
In this investigation, the semi-analytical Finite Strip Transfer Matrix Method is proposed to analyze the buckling problems of rectangular thin plates under the boundary condition of simply supported loaded edges.In order to validate the method, the examples of the plate and the plate with ribs can be designed and analyzed by different methods.
It may be found that the method holds several highlights: (1) it demands no global stiffness matrix and reduces matrix order in the system analysis, (2) the orthogonality among the analytical shape functions of the loaded direction of the plate leads to the clear computational mode of the buckling analysis, (3) both the semi-analytical finite strip and the transfer matrix technologies ensure the proposed method efficient.

Figure 1 :
Figure 1: Coordinates, degree of freedom, and loads of a typical strip.
(and    ) is the generalized internal force associated with the transverse deflection    (and    ) of the nodal line  (and ) and    (and    ) is the generalized internal force associated with the -axis's rotation    (and    ) of the nodal line  (and ), respectively.
can be regarded as the generalized displacement vector of the nodal line , and R

4. 2 . 1 .
Plate with Three Ribs.The material properties are the same as the plate as stated in Section 4.1.The geometrical parameters are as follows.The width of the plate is 5 mm, Boundary conditions of SSfc

Figure 2 :
Figure 2: Relations between  and  of the plate.

Figure 3 :
Figure 3: Meshed grids of the plate with three ribs.

Figure 4 :
Figure 4: Buckling deformation mode of the plate with three ribs and boundary conditions of SSss.
) and 4(b) correspondingly.As a matter of fact, the buckling modes of Figures4(a) and 4(b) can be obtained by setting the computational parameters of the half-wave number  = 2 and  = 3 in the FSTMM with clearer engineering meaning.The relation curves between buckling coefficient and length-width ratio of the plate with three ribs and boundary conditions of SSss can be calculated by the FSTMM, the FEM

Figure 5 :
Figure 5: Relations between  and  of the plate with three ribs and boundary conditions of SSss.

Figure 6 :
Figure 6: Buckling deformation mode of the plate with four ribs and boundary bonditions of SSfc.

Figure 7 :
Figure 7: Relations between  and  of the plate with four ribs and boundary conditions of SSfc.

Figure 8 :
Figure 8: Three mesh sizes in FEM for the same plate.

Figure 9 :
Figure 9: Relations between  and  of the plate with boundary conditions of SSss by using three mesh sizes.

Table 1 :
State vectors of nodal lines under various boundary conditions.