Buckling analysis of thin-walled members via semi-analytical finite strip transfer matrix method

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1 Special Issue Article Bucklin analysis of thin-walled members via semi-analytical finite strip transfer matrix method Advances in Mechanical Enineerin 16, ol. 8() 1 11 Ó The Author(s) 16 DOI: 1.11/ aime.saepub.com Yu Zhan, Bin He, Li-Ke Yao and Jin Lon Abstract Slender thin-walled members are main components of modern enineerin structures, whose bucklin behavior has been studied widely. In this article, thin-walled members with simply supported loaded edes can be discretized in the cross-section by semi-analytical finite strip technoloy. Then, the control equations of the strip elements will be rewritten as the transfer equations by transfer matrix method. This new method, named as semi-analytical finite strip transfer matrix method, expands the advantaes of semi-analytical finite strip method and transfer matrix method. This method requires no lobal stiffness matrix, reduces the size of matrix, and improves the computational efficiency. Compared with finite element method s results of three different cross-sections under axial force, the method is proved to be reliable and effective. Keywords Bucklin, thin-walled members, finite strip method, transfer matrix method, finite element method Date received: 1 December 1; accepted: April 16 Academic Editor: Chuanzen Zhan Introduction Bucklin analysis is the most important step durin the desin of slender elements which can be applied in different branches of enineerin, includin mechanical construction, marine applications, and civil architecture. 1 Thin-walled structure, a main kind of slender structure, is widely utilized to lihten enineerin structures as well as save materials. The bucklin phenomenon is one of the chief failure models of thin-walled structure, which has been studied by experimental or mathematical means. In the early works, the stability and vibration of thin flat-walled structure, acted by compression forces, have been analyzed by a matrix method. It is based on enery rule that the elastic bucklin modes of I-section beams has been studied. Up to now, many methods have been used to analyze the bucklin problems of thin-walled structure, such as finite difference and finite element methods (FEMs), 6 nonlinear FEM, eneralized beam theory, 8,9 direct strenth method, 1 semi-analytical finite strip and spline finite strip methods, 11 and constrained finite strip method (cfsm). 1 In addition, by introducin a computer procedure, the calculation of the stresses and failure models in thin-walled structural members has been presented. 1 And an experimental proram investiatin the column behavior of four sizes of square hollow sections has been introduced. 1 Althouh the FEM has been widely applied in the analysis of bucklin behavior of thin-walled structures, the choices of the elements and the mesh sizes have Department of Mechanics, Nanjin Tech University, Nanjin, China Correspondin author: Bin He, Department of Mechanics, Nanjin Tech University, No., Puzhu Road(S), Nanjin 11816, China. hebin1@njtech.edu.cn Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution. License ( which permits any use, reproduction and distribution of the work without further permission provided the oriinal work is attributed as specified on the SAGE and Open Access paes ( open-access-at-sae).

2 Advances in Mechanical Enineerin sinificant influences on the results. 1 When calculatin the bucklin problems of structures which only have complex eometry shape in their cross-section, finite strip method (FSM) can be rearded as an efficient and powerful technoloy. And usin the sub-parametric mappin concept, the arbitrary-shaped member can be discretized as many strip elements. 16 By introducin the spline finite strip method, the bucklin stresses and natural frequencies of prismatic plate and shell structures have been predicted. 1 If a fictitious shear strain is adopted, a drillin rotation is introduced in the standard Mindlin Reissner finite strip for the analysis of thin-walled sections. 18 Based on the concept of the semi-enery approach, the FSM can be proposed to analyze the bucklin, 19 shear bucklin, and stability analysis of composite laminated plate and cylindrical shell structures. 1 The lonitudinal harmonic series satisfyin the boundary conditions at the lonitudinal ends are enerally employed in semi-analytical finite strip method (SAFSM). The SAFSM based on the shallow shell theory is developed to the bucklin analysis of prismatic structures which have curved corners. And the SAFSM has been used in computer software (such as THIN-WALL and CUFSM ) to develop the sinature curves 6 of the bucklin stress versus bucklin half-wavelenth for thin-walled members. Furthermore, the cfsm innovated from SAFSM is developed and applied in the determination and classification of bucklin modes. By extendin the applicability of the cfsm to the domain of eneral finite element analysis, the bucklin modal identification of the thin-walled member has been demonstrated. 8 The classical transfer matrix method (TMM) has been developed as an effective tool for structural analysis, especially for chain connected system from topoloical perspective. 9 By combinin the TMM and FEM, the finite element-transfer matrix method (FE-TMM) is developed to analyze the static and dynamic of structural problems. And then a structural analysis method, named as boundary element-transfer matrix method (BE-TMM), is proposed for the vibration analysis of two-dimensional plate acted by uniform 1 and concentrated loads. If the numerical interation is used, the nonlinear dynamics of structures, the dynamics of multi-riid-body system, and multi-riidflexible-body system can be simulated by TMM. And a new method, named as transfer matrix method of linear multibody system (MSTMM), is developed to study the hybrid multibody systems dynamics. 6 By combinin FEM and discrete time transfer matrix method of multibody system (MS-DT-TMM), the dynamics of eneral planar flexible multibody systems includin flexible bodies with irreular shape is studied. Nowadays, the bucklin analysis of the plate with built-in rectanular delamination has been implemented by strip distributed transfer function method. 8 And the TMM can be used to analyze the instability in unsymmetrical rotor-bearin systems 9 and tall unbraced frames. The bucklin analysis of rectanular thin plates via semi-analytical finite strip transfer matrix method (FSTMM), which is enlihtened by above three references, has been developed. 1 In this article, FSTMM can be extended to analyze the bucklin problems of thin-walled member with simply supported loaded edes. This article is oranized as follows: in section The Semi-analytical finite strip analysis, the eneral theorem of the semi-analytical finite strip for bucklin analysis of thin-walled member is shown. In section Semi-analytical FSTMM for bucklin analysis, the FSTMM for bucklin analysis is studied. In section Examples and analysis, some results calculated by FSTMM and FEM are iven to validate the method. The conclusions are presented in section Conclusion. The semi-analytical finite strip analysis Deree of freedom and shape function In the FSM, a thin-walled member as shown in Fiure 1(a) can be discretized into many strips in lonitudinal direction. Two left-handed coordinate systems are used: lobal and local. The lobal coordinate system is denoted as X-Y-Z, with the Y axis parallel to the lonitudinal axis of the member. The local system is denoted as x-y-z, which is always associated with a strip and z axis is perpendicular to the strip as shown in Fiure 1(b). We introduce a numberin system of finite strip model as shown in Fiure 1(a). The total number of strips is s; therefore, the total number of nodal lines is s + 1 for open cross-section. Each nodal line i has two membrane derees of freedom (DOFs) u i and v i and two bendin DOFs w i and u i. This numberin system will be used to depict the state vector of the nodal line and the transfer matrix of the strip in section Semi-analytical FSTMM for bucklin analysis. The analytical trionometric functions of the lonitudinal coordinate that satisfy the simply supported boundary condition of the loaded edes can be used to represent the strip s deformed confiuration, Y p (y)=sin ppy, p = 1,,,..., n ð1þ a where p is the half-wave number, which also stands for certain half-wavelenth alon the lonitudinal direction; y is the lonitudinal coordinate in local coordinate system; and a is the lenth of the member. The shape function for the membrane DOFs uses a linear function alon transverse direction. And four cubic polynomials can be selected as the shape functions to depict the bendin displacement of the strip alon transverse direction. Then, the explicit expressions of u, v, and w can be iven as follows

3 Zhan et al. Fiure 1. Coordinate systems and displacements: (a) discretization and numberin of a member and (b) Deree of Freedom and loads of a strip. w= Xm 8 >< >: w ip u ip w jp u jp p=1 9 >= Y p >; u = Xm v = Xm p = 1 p = 1 1 x b + x b h 1 x x i uip b b u jp h 1 x x i vip b b v jp Y p Y 9 p a pp x x b + x x x x b b b b ðþ ðþ b x ðþ where subscripts i and j denote two nodal lines of one strip, and m is the maximum half-wave number employed in the analysis, which is a finite positive inteer. Fundamental stiffness matrix The elastic stiffness matrix of FSM can be established similar to the deduction of FEM. If the plane stress assumptions and Kirchhoff plate theory may be employed, respectively, the total strain e of a strip, includin both the membrane strains e M and the bendin strains e B, is expressed as 8 >< e = e M + e B = >: e x e y xy 9 >= >; M 8 >< + >: e x e y xy 9 >= >; u z w x x >< v >= >< = + z w >= y y u >: y + v >; x z w >: >; M x y B B ðþ where u, v, and w are iven in section Deree of freedom and shape function. As to eneral linear elastic material, the elastic deformation enery can be expressed as 1 U = 1 ð e T sd = 1 ð e T Ded = 1 ð B T DBdAd = 1 X m X m p = 1 q = 1 d pt k pq e dq ð6þ where e and s denote strain and stress vectors, respectively; is the volume of the material; D is the elastic constant matrix of the material; B defines the relationship between the strain vector and the displacement vector; d = ½d 1T d T d mt Š T is the displacement vector; and d p = ½u ip v ip u jp v jp w ip u ip w jp u jp Š T. The elastic stiffness matrix of the strip can be concisely expressed as ð k e = B T DBd ðþ v The (8 8) block elements k pq e of the elastic stiffness matrix k e iven in above equation can be expressed as the combination of membrane and bendin terms, namely k pq e = kpq em k pq ð8þ where k pq em and kpq eb are the ( ) membrane and bendin elastic stiffness matrices, respectively. As shown in Fiure 1(b), the strip is loaded with linearly varyin ede tractions. The membrane compressive loads can be expressed as eb x t x = t i t i t j b ð9þ

4 Advances in Mechanical Enineerin where t i, t j are the forces in two nodes of the strip, b is the width of the strip as shown in Fiure 1(b), and x is the transverse coordinate in local coordinate system. Similar to the deduction of the elastic stiffness matrix, the potential enery induced by the membrane compressive loads can be expressed as ð " 1 W = t u x + v + w # d y y y 1 = 1 ð t x G T GdAd = 1 X m X m d pt k pq dq p = 1 q = 1 ð1þ where G defines the relationship between the secondorder strain components and the displacement vector. The eometric stiffness matrix of the strip element can be expressed as ð k = t x G T Gd ð11þ The (8 8) block elements k pq of the eometric stiffness matrix k iven in above equation can be expressed as the combination of membrane and bendin terms, namely k pq = kpq M k pq B ð1þ where k pq M and kpq B are the ( ) membrane and bendin eometric stiffness matrices, respectively. Semi-analytical FSTMM for bucklin analysis Control equations of strip element In both FE-TMM and BE-TMM, the transfer equations of the iven sub-structure can be deduced by the control equations of this sub-structure which consider the interaction forces between this sub-structure and other structures. As to the proposed FSTMM, the strip element can be rearded as the sub-structure. If the orthoonal conditions about k pq e and k pq iven by Yao et al. 1 can be used, the control equations of the bucklin strip can be obtained by virtual work principle, which are k pp e k pp d p = R p ð1þ where k pp e is the elastic stiffness matrix of equation (8), k pp is the eometric stiffness matrix as shown in equation (11), d p is the nodal line displacement vector, and R p is the eneralized internal forces actin on the strip, which can be expressed as R p = ½P p i Q p i P p j Q p j F p i M p i F p j M p j ŠT ð1þ where P p and Q p are the eneralized internal forces associated with the membrane deflection u and v, F p is the eneralized internal force associated with the transverse deflection w, M p is the eneralized internal force associated with the y-axis s rotation u, and the subscripts i and j denote two nodal lines of one strip. Assumin t (x) as the initial axial force, the real axial force in the eometric stiffness matrix can be expressed as t(x)=lt (x) ð1þ where l is the bucklin coefficient. Therefore, the eometric stiffness matrix k pp can be rewritten as the function of the initial eometric stiffness matrix k pp t caused by initial axial force t (x), namely = lkpp ð16þ k pp t By substitutin equation (16) into equation (1), the control equations of the bucklin strip can be rewritten as follows k pp e lk pp t d p = R p ð1þ To simplify the equation, the coefficient matrix of the nodal line displacement vector d p in above equation can be expressed as K = k pp e lk pp t ð18þ where both coefficient matrices k pp e and k pp are constant when the loads determined by equation (1) t vary. State vector, transfer equations, and transfer matrix Durin the deduction of transfer matrix of the system, the state vector of the nodal line is an important concept that includes two parts: one part describes the eneralized displacement of the nodal line, and the other part ives the eneralized internal forces actin on the nodal line by other members in the system. For example, the state vector of the nodal line l can be defined as Z l, n = 6 d B, l R B, l d M, l R M, l n = ½u p l v p l P p l Q p l w p l u p l F p l M p l ŠT n (l = i, j) ð19þ where the first subscript l denotes the number of the nodal line, the second subscript n denotes the number of the strip, d B, l = ½u p l v p l ŠT and d M, l = ½w p l u p l ŠT can be

5 Zhan et al. rearded as the eneralized displacement vectors of the nodal line l, and R B, l = ½P p Q pšt and R l l M, l = ½F p M p Š T l l are the eneralized internal force vectors actin on the nodal line l correspondinly. Usin the block forms of equation (19), the control equations (1) can be rewritten as the form of the transfer equations of this strip, namely Z j, n = U n Z i, n where the transfer matrix of the strip n is U n = 6 K 1 1 K 11 K 1 K 1 K K 1 1 K 11 K K K 1 K K 1 K K K 1 K K K 1 ðþ n ð1þ where the subscript n denotes the number of the strip, K ij (i, j = 1,,, ) are the ( ) block sub-matrices that can be determined by equation (18). Actually, coefficient matrix of equation (18) can be denoted as K = 6 K 11 K 1 K 1 K K K K K n ðþ Accordin to the condition of displacement continuum and the law of action and reaction, the transformation of the state vector from strip n to strip n + 1 at nodal line j n (i n + 1 ) which is shown in Fiure at an anle a is overned by the followin transformation 6 u i v i P i Q i w i u i F i M i n + 1 = 6 cos a sina 1 cosa sina 1 sina cosa 1 sina cosa 1 Fiure. Transformation at the nodal line. Usin the same procedure used in classical TMM, the overall system transfer equation and the overall transfer matrix U all, which relates the state vectors at two edes of the member, can be assembled and calculated. That is 6 n u j v j P j Q j w j u j F j M j n Z j, s = U all Z i, 1 U all = U s T s 1 U s 1 T s U T 1 U 1 where the subscript s is the total number of strips. ðþ ð6þ ðþ Examples and analysis Illustrations of open cross-section members For the bucklin analysis of open cross-section members with simply-simply (SS) supported boundary condition of loaded edes in this dissertation, the two unloaded edes are free, which can be expressed by SSff. Here, the capital letters and lowercase letters denote the loaded edes and unloaded edes correspondinly. This can be simplified as the followin form Z i, n + 1 = T n Z j, n ðþ where T n is the transformation matrix of strip n and strip n + 1 at nodal line j n (i n + 1 ). Therefore, the transfer equations between particular nodal lines of conjunctional strips can be obtained by multiplyin equations () and (), namely Z i, n + 1 = T n U n Z i, n ðþ Take the boundary condition SSff into analysis, the total transfer equations can be deduced as follows ½u v w u Š T L = U all ½u v w u Š T ð8þ F where subscripts F and L denote the first and last nodal lines of the member and U all is the overall transfer matrix of the member. The non-zero variables in the state vector of the first nodal line of the member have the relationship that can be deduced by equation (8)

6 6 Advances in Mechanical Enineerin Fiure. Cross-section: (a) C-section member and (b) its FSM mesh. 6 L U 1 U U U 6 U = 1 U U U U 1 U U U 6 U 81 U 8 U 8 U 86 u v w u F ð9þ Fiure. Classic sinature curves of FSTMM and FSM of C- section member. where U 1, U,..., U 8, U 86 are the elements of U all. To make the non-zero solutions of equation (9) possible, the followin condition must be satisfied 1 U 1 U U U 6 U det 1 U U U 6 B6 U 1 U U U 6 A = U 81 U 8 U 8 U 86 ðþ Above equation is the characteristic equation of the bucklin of open cross-section member by the FSTMM, which can be used to calculate the bucklin coefficients. If we combine equation (9) with equation (), the bucklin mode can be obtained. In order to demonstrate the method, two typical examples are considered: a C-section member and a Z-section member. Illustrations of C cross-section member. The dimensions of the C-section member are presented in Fiure (a). The section heiht is mm, the flane width is 8 mm, the flane lip lenth is mm, the plate thickness is mm, and the initial axial force t (x)= N=mm. The material properties throuh this article are as follows: Youn s modulus E = 1 N=mm, Poisson s ratio n = :, and shear modulus G = E=(1 + n). Alon the loaded ede, the member is divided into 11 strips, as shown in Fiure (b). Fiure shows the classic sinature curve, which can be used to determine and classify the bucklin modes, by both FSTMM and conventional FSM for the section in axial compression. 1 We notice that two curves have ood areements. The relationship schema between bucklin coefficient l and lenth a of the member is obtained by the proposed FSTMM and FEM. FSTMM s results are compared with FEM s results under the boundary Fiure. Bucklin curves of C-section member. conditions of SSff, as shown in Fiure. It should be noticed that the bucklin shape transfers from local to lobal modes at a = mm. And since the lenth of the thin-walled member increase, two kinds of lobal bucklin modes, torsion and bendin, can be found. When the member lenth is relatively small, a little difference between the two curves of both FSTMM and FEM can be observed as shown in Fiure. That is because the shear strain is included in the finite element analysis but nelected in the finite strip transfer matrix analysis. Fiure 6 ives the local, lobal torsion and lobal bendin bucklin shape of C-section member when the lenth is 1,, and 8 mm, respectively. Illustrations of Z cross-section member. Another example to validate the theory is a Z-section member. The

7 Zhan et al. Fiure 6. Bucklin shapes of C-section member: (a) a = 1 mm, local bucklin; (b) a = mm, lobal torsion and (c) a = 8 mm, lobal bendin. Fiure. Cross-section : (a) Z-section member and (b) its FSM mesh. dimensions are presented in Fiure (a), the section heiht is 18 mm, the flane width is 6 mm, the flane lip lenth is mm, and the plate thickness is mm. The member is divided into 1 strips alon the loaded ede, as shown in Fiure (b). In this section, the numerical results concernin the bucklin behavior of the Z-section member subjected to axial compression and axial bendin are presented. Fiure 8 shows the relation between the bucklin coefficient l and the lenth a by FSTMM and FEM of the Z-section member under the initial axial force t (x)= N=mm. The bucklin shape transfer from local to lobal bucklin modes at the lenth a = mm. Only one kind of lobal bucklin mode, bendin, occurs in the analysis. Fiure 9 shows the local and lobal bucklin shapes for Z-section member at a = mm and a = mm. For the member under Z-Z axial bendin moment M ZZ = 1 6 N mm shown in Fiure 1(a), the stress distributions in the cross-section can be calculated by thin-walled structure mechanics, which can be found in Fiure 1(b). It should be attended that the positive (neative) number denotes the compressional (tensional) stress here. The bucklin coefficient l versus lenth a by both FSTMM and FEM are plotted in Fiure 11. Note that the distortional reion exists in Fiure 8. Bucklin curves of Z-section member in axial compression. Fiure 9. Bucklin shapes of Z-section member in axial compression: (a) a = mm, local bucklin; (b) a = mm, lobal bucklin. this loadin condition. The bucklin shapes correspondin to local, distortional, and lobal modes with lenths,, and 1, mm, respectively, are shown in Fiure 1.

8 8 Advances in Mechanical Enineerin Fiure 1. Z-section member in Z-Z axial bendin: (a) direction of bendin moment and (b) stress distributions. section is studied, as shown in Fiure 1. Different from open cross-section members illustrated in section Illustrations of open cross-section members, there is no unloaded ede in a closed cross-section member. In other words, the first and last nodal lines are the same nodal line in the analysis. To satisfy the closed forms, equation (8) can be modified as follows ½u v P Q w u F M Š T F = T s U all ½u v P Q w u F M Š T F ð1þ Fiure 11. Bucklin curves of Z-section member in Z-Z axial bendin. Illustrations of closed cross-section member In order to demonstrate the efficiency of FSTMM to analyze closed cross-section, a rectanular hollow where the subscript F denotes the first nodal line of the member, U all is the transfer matrix of the member, and T s is the transformation matrix of strip s and strip 1 at nodal line j s (i 1 ). Equation (1) can be rewritten as follows ðt s U all EÞ½u v P Q w u F M Š T F = ðþ where E is a (8 8) unit matrix. To make the non-zero solutions of equation () possible, it must satisfy the followin condition Fiure 1. Bucklin shapes of Z-section member in Z-Z axial bendin: (a) a = mm, local bucklin; (b) a = mm, distortional bucklin; and (c) a = 1, mm, lobal bucklin.

9 Zhan et al. 9 detðt s U all EÞ= ðþ Fiure 1. Cross-section: (a) rectanular hollow section member and (b) its FSM mesh. The eometrical properties of the member (Fiure 1(a)) are as follows: the heiht is 1 mm, the width is 6 mm, the plate thickness is 1. mm, and the initial axial force t (x)= N=mm. The member is divided into 1 strips alon the loaded ede, as shown in Fiure 1(b). Fiure 1 ives the bucklin coefficient l versus lenth a of FSTMM and FEM. The bucklin shape transfer from local to lobal bucklin modes at a = mm. Fiure 1 shows the local and lobal bucklin shapes for rectanular hollow section member at a = 8 mm and a = mm, respectively. Fiure 1. Bucklin curves of rectanular hollow section member. Fiure 1. Bucklin shapes of rectanular hollow section member: (a) a = 8 mm, local bucklin; (b) a = mm, lobal bucklin. Precision analysis As a eneral rule, the computational precision can be improved by increasin the number of elements. By comparin the results calculated from FEM s shell model and FSTMM of bucklin problems of C-section member, the influence of the strip number to the computational precision can be analyzed. The bucklin behaviors can be obtained by the FSTMM with the strip numbers, 8, and 11, respectively, shown in Fiure 16(a) (c). Fiure 16(d) shows the FEM s shell model which is used for comparative analysis. Fiure 1 compares the influence of the strip number to the computational precision in FSTMM. When the number of strips n = 8 in FSTMM is selected, the computational results have ood areement with the FEM s results. It can be confirmed that the proposed FSTMM has ood efficiency for the bucklin analysis of thin-walled members under the boundary condition of simply supported loaded edes. Conclusion In this article, the semi-analytical FSTMM is proposed to analyze the bucklin problems of open and closed cross-section members under the boundary condition of simply supported loaded edes. In order to validate the method, the examples of the open and closed crosssection members can be desined and analyzed by the different methods in section Examples and analysis. It may be found that the method holds several hihlihts: (1) demands no lobal stiffness matrix and Fiure 16. FSTMM and FEM meshes: (a) five strips; (b) eiht strips; (c) eleven strips and (d) FEM mesh.

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