Explicit local buckling analysis and design of fiber–reinforced plastic composite structural shapes

Pultruded fiber–reinforced plastic (FRP) composite structural shapes (beams and columns) are thin-walled open or closed sections consisting of assemblies of flat plates and commonly made of E-glass fiber and either polyester or vinylester resins. Due to high strength-to-stiffness ratio of composites and thin-walled sectional geometry of FRP shapes, buckling is the most likely mode of failure before material failure. In this paper, explicit analyses of local buckling of rectangular orthotropic composite plates with various unloaded edge boundary conditions (i.e., (1) rotationally restrained along both unloaded edges (RR), and (2) one rotationally restrained and the other free along the unloaded edges (RF)) and subjected to uniform in-plane axial action at simply-supported loaded edges are first presented. A variational formulation of the Ritz method is used to establish an eigenvalue problem, and explicit solutions of plate local buckling coefficients in term of the rotational restraint stiffness (k) are obtained. The two cases of rotationally restrained plates (i.e., the RR and RF plates) are further treated as discrete plates of closed and open sections, and by considering the effect of elastic restraints at the joint connections of flanges and webs, the local buckling of different FRP shapes under uniform axial compression is studied. The approximate expressions of the rotational restraint stiffness (k) for various common FRP sections are provided, and their application to sectional local buckling predictions is illustrated. The explicit local buckling formulas of rotationally restrained plates are validated with the exact transcendental solutions. The analytical predictions for local buckling of various FRP profiles based on the present discrete plate analysis and considering the elastic restraints of the flange–web connections are in excellent agreements with available experimental results and finite element eigenvalue analyses. A design guideline for local buckling prediction and related performance improvement is proposed. The present explicit formulation can be applied effectively to determine the local buckling capacities of composite plates with elastic restraints along the unloaded edges and can be further used to predict the local buckling strength of FRP shapes.     

Effect of rotationally restrained and Pasternak foundation on buckling of an orthotropic rectangular Mindlin plate

This article, based on first-order shear deformation theory, presents the buckling analysis of a rotationally restrained orthotropic rectangular Mindlin plate resting on a Pasternak elastic foundation. Thus, the Mindlin–Reissner plate theory is employed for which the governing equations are solved by the Rayleigh–Ritz method. Uniformly distributed in-plane loads are applied to two simply supported opposite edges of the plate and the other two edges have rotationally restrained conditions without loading. Finally, the effects of plate parameters, such as foundation stiffness coefficients, aspect ratios, and ratio of elastic modulus in the x to y direction on the buckling loads are presented. The results show that the buckling load would increase when the ratio of the elastic modulus in the x to y direction increases and the plate is close to isotropic. The variation of buckling load versus changing ratio of elastic modulus in the x to y direction in the state of without elastic foundation and with clamp support is more than the rest of the state.     

Exact buckling solutions for composite laminates: proper free edge conditions under in-plane loadings

Summary This paper considers the elastic buckling of symmetric cross-ply laminated rectangular plates with two parallel edges simply supported, one edge free and the remaining edge free, simply supported or clamped. The first-order shear deformation plate theory is used in the analysis. An error apparently made by previous researchers for boundary conditions at free edges subjected to in-plane loads is corrected. Closed-form buckling factors are obtained using a generalised Levy-type solution method to solve the differential equations which govern the buckling behaviour of the laminates. Comparisons are made with previously published results, and the differences between buckling factors obtained with the appropriate and inappropriate free edge conditions are examined. The variation of buckling factors with plate aspect ratio, thickness ratio and the number of layers is investigated. Sets of first-known buckling solutions for cross-ply laminates are reported in design charts and tables.     

A state of stress and displacement of elastic plates using simple and mixed shear deformation theories

This paper presents accurate two-dimensional solutions for bending response of four types of single-layer orthotropic rectangular plates. The plates considered are of the type having two opposite sides simply-supported, and two other sides having combinations of simply-supported, clamped, and free-boundary conditions. Analytical solutions for deflections and stresses of rectangular plates are developed by means of the simple (SFPT) and mixed (MFPT) first-order shear deformable plate theories. The present MFPT not only shows improvement on predicting frequencies, critical buckling loads, deflections and in-plane stresses, but also accounts for variable transverse shear stress distributions through the thickness. This puts into evidence the important role played by MFPT in the modeling of homogeneous plate theories, which in contrast to SFPT does not require the incorporation of a shear-correction factor. For illustrative purposes, sample free vibration, stability, and bending problems for simply supported orthotropic plates are considered and comparisons of the obtained results are made with the exact and higher-order shear deformation theory results given in the literature.     

Levy-type solution for buckling analysis of orthotropic plates based on two variable refined plate theory

This paper presents closed-form solution for buckling analysis of orthotropic plates using two variable refined plate theory. The theory accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Governing equations are derived from the principle of minimum total potential energy. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, loading condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the critical buckling load of orthotropic plates are investigated and discussed in detail.     

Buckling of symmetric cross-ply square plates with various boundary conditions

The buckling analysis of cross-ply laminated square plates subjected to three types of in-plane forces and various edge boundary conditions is presented on the basis of a unified five-degree-of-freedom shear deformable plate theory. The employment of the appropriate “shear deformation shape functions” in the theory leads to certain shear deformable plate theories developed previously, also, fulfills the requirement of the continuity conditions among the layers. The governing equations of buckling behaviour of completely simply supported cross-ply laminated plates are solved analytically. For the plates with different combinations of free, clamped and simply supported boundary conditions at their edges, the Ritz method is applied by assuming the displacement components as the double series of simple algebraic polynomials. The numerical results obtained on the basis of various plate theories for uniaxial, biaxial compression and compression–tension types of loading and different length-to-thickness ratios are presented and compared with the ones available in the literature.     

Levy Solution for Buckling Analysis of Functionally Graded Rectangular Plates

In this article, an analytical method for buckling analysis of thin functionally graded (FG) rectangular plates is presented. It is assumed that the material properties of the plate vary through the thickness of the plate as a power function. Based on the classical plate theory (Kirchhoff theory), the governing equations are obtained for functionally graded rectangular plates using the principle of minimum total potential energy. The resulting equations are decoupled and solved for rectangular plate with different loading conditions. It is assumed that the plate is simply supported along two opposite edges and has arbitrary boundary conditions along the other edges. The critical buckling loads are presented for a rectangular plate with different boundary conditions, various powers of FGM and some aspect ratios.     

受压对称迭层矩形板的自由振动分析

根据受压的对称迭层矩形板自由振动的微分方程可以求得各种解析解来求解各种边值问题。迭层板有两种,一种是正交铺设,其方向与坐标轴平行,属正交异性板,当板的四边为简支时,可用双正弦级数来求解自由振动的各阶频率及其振型以及均匀受压的各阶临界载荷及其屈型。另一种是角铺设,属各向异性板,当两相邻边为自由,另两边为简支或固支时可用复数级数来求解其最低频率及其振型以及最低临界载荷及其屈型。此时其特征方程的根为两对复根,且可表成三角级数和双曲线级数,以满足边界条件。另外用代数多项式和双正弦级数组成的解来满足角点条件。在算例中计算了若干板受压或不受压的振动频率和临界载荷,并与其他文献进行了对比。     

On the use of Levy's method for symmetrically laminated composite plates

For rectangular plates with two opposite edges simply supported and no bending twisting coupling, Levy's method can yield exact solutions for static deflections, free vibrations and buckling loads. However, three distinct solutions are obtained, depending on the relative magnitude of the plate rigidities, which in turn are functions of material properties, laminate thickness and lay-up. In this communication a stiffness invariant formulation is used to identify those laminates which fall under the three cases defined.     

An analytical approach for local buckling analysis of initially delaminated composite thin-walled columns with open and closed sections

Local buckling of intact thin-walled columns is generally performed by modeling the wall segments as long plates and by assuming that edges common to two or more plates remain straight. Thus, the buckling load can be determined by considering the wall segments as individual plates rotationally restrained by the adjacent wall segments. This technique is combined with plate theories as a new analytical method to predict the buckling load of an initially delaminated column with any arbitrary sections (open or closed). First, moments at the rotationally restrained edges of delaminated segment (web or flange) are obtained from the curvature and stiffness of the adjacent laminates. Then, the strain energy of this delaminated segment with distributed moment at edges is calculated based on the first-order shear deformation theory. Using the principal of minimum potential energy, the governing equations are obtained and solved by the Rayleigh–Ritz approximation technique. Results of the present approach are compared with three-dimensional finite-element results obtained from eigenvalue buckling analysis in ANSYS software for both box- and channel-section columns with cross-ply and angle-ply stacking sequences. Finally, the effects of delamination size and location are investigated on the buckling loads.     

Exact solutions for free vibrations of orthotropic rectangular Mindlin plates

Exact closed-form solutions are obtained for free vibrations of orthotropic rectangular Mindlin plates by using the separation of variables method although it is difficult to solve them. The plates have two opposite edges simply supported and all possible combinations of classical boundary conditions at the other two edges. The exact solutions of orthotropic rectangular Mindlin plates are compared with those of isotropic ones and their differences are discussed. The exact solutions are validated through both mathematical proof and numerical comparisons with available p-Ritz solutions and the differential quadrature finite element method solutions calculated by the authors.     

Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory

Natural frequencies and buckling stresses of plates made of functionally graded materials (FGMs) are analyzed by taking into account the effects of transverse shear and normal deformations and rotatory inertia. The modulus of elasticity of the plates is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional (2-D) higher-order theory for rectangular functionally graded (FG) plates is derived through Hamilton’s principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of FG plates with simply supported edges. In order to assure the accuracy of the present theory, convergence properties of the fundamental natural frequency are examined in detail. Critical buckling stresses of FG plates subjected to in-plane stresses are also obtained and a relation between the buckling stress and natural frequency of simply supported FG plates without in-plane stresses is presented. The distributions of modal displacements and modal stresses in the thickness direction are obtained accurately by satisfying the surface boundary conditions of a plate. The modal transverse stresses have been obtained by integrating the three-dimensional equations of motion in the thickness direction starting from the top or bottom surface of a plate. The present numerical results are also verified by satisfying the energy balance of external and internal works are considered to be sufficient with respect to the accuracy of solutions. It is noticed that the present 2-D higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported FG plates.     

余弦分布压力下矩形薄板的屈曲

针对不同支承条件,两对边受半余弦非线性分布压力下弹性矩形薄板的屈曲问题,进行了分析研究。对于只产生对称变形的矩形薄板,基于辛弹性力学的平面矩形域理论,给出了精确的面内应力分布。运用Galerkin法分析计算了半余弦分布压力下矩形薄板的屈曲载荷。根据各种不同支承矩形薄板弯曲的位移边界条件,借助于符号运算软件Maple,编写了相应的用户计算程序。对九种不同支承组合下的弹性矩形薄板进行了计算,得到了不同长宽比矩形薄板的屈曲载荷系数。通过与已有文献结果的比较表明,该文求解方法是有效和精确的。基于所给出的结果,可望为解决矩形薄板在非线性分布载荷下的屈曲分析提供一种新的研究方法。     

基于弹性扭转约束边界的波形钢板整体剪切屈曲分析

波形钢板的整体剪切屈曲对波形钢腹板PC组合箱梁桥的竖向抗剪设计有重要意义。该文在正交各向异性薄板理论的基础上,利用势能驻值原理和瑞利-里兹法,推导了基于弹性扭转约束边界的波形钢板弹性整体剪切屈曲荷载的分析方法,并给出了几种特殊边界条件下的简化计算公式。计算结果表明:对于四边简支板和四边嵌固板,该方法与现有公式计算结果一致;波形钢板强抗弯刚度方向上的约束条件是影响其整体剪切屈曲的决定性因素,随着该方向上约束的增强,屈曲模式将从四边简支情况向四边嵌固情况过渡;弱抗弯刚度方向上约束条件的影响很小,可以忽略。最后给出了将波形钢板在实际梁结构中受到的边界约束等效为弹性扭转弹簧的方法。     

A new inverse trigonometric shear deformation theory for isotropic and functionally graded sandwich plates

A new inverse trigonometric shear deformation theory is proposed for the static, buckling and free vibration analyses of isotropic and functionally graded (FG) sandwich plates. It accounts for a inverse trigonometric distribution of transverse shear stress and satisfies the traction free boundary conditions. Equations of motion obtained here are solved for three types of FG plates: FG plates, sandwich plates with FG core and sandwich plates with FG faces. Closed-form solutions are obtained to predict the deflections, stresses, critical buckling loads and natural frequencies of simply supported plates. A good agreement between the obtained predictions and the available solutions of existing shear deformation theories is found to demonstrate the accuracy of the proposed theory.     

弹性力学Hamilton体系下的稳定问题

通过在Hellinger-Reissner广义势能中引入应变的非线性项,推导出了弹性力学Hamilton体系下的屈曲基本方程。并运用弹性力学方程组一般解的统一理论给出其一般解。最后作为例子,给出了两端简支的梁、组合梁和四边简支板、组合板的临界载荷,并与经典解做了比较。结果是严格弹性力学意义(没有引入任何几何变形假设)下的精确解。为衡量各种计入剪切变形的薄板、中厚板理论的准确性提供了一个标准。     

Exact solution for stability analysis of moderately thick functionally graded sector plates on elastic foundation

In this article, an exact analytical solution for buckling analysis of moderately thick functionally graded (FG) sector plates resting on Winkler elastic foundation is presented. The equilibrium equations are derived according to the first order shear deformation plate theory. Because of the coupling between the bending and stretching equilibrium equations of FG plates, these plates have deflection under in-plane loads lower than the critical buckling load acting on the mid-plane. The conditions under which FG plates remain flat in the pre-buckling configuration are investigated and the stability equations are obtained based on the flat plate assumption in the pre-buckling state. The stability equations are simplified into decoupled equations and solved analytically for plates having simply supported boundary condition on the straight edges. The critical buckling load is obtained and the effects of geometrical parameters and power law index on the stability of functionally graded sector plates are studded. The results for the critical buckling load of moderately thick functionally graded sector plates resting on elastic foundation are reported for the first time.     

正交各向异性板剪切屈曲载荷的近似表达式研究

利用Rayleigh-Ritz法和通用有限元程序ANSYS分析了纵向边被转动弹簧或扭转加劲肋约束的正交各向异性长板的临界屈曲问题, 引入无量纲正交各向异性参数、约束系数和临界屈曲系数, 利用曲线拟合技术, 得到了这两类约束板在均匀剪力作用下的临界屈曲载荷近似解公式;利用临界纵横比, 把求得的针对转动弹簧约束的临界屈曲解应用到扭转加劲肋约束板的临界屈曲解中, 同样得到了扭转加劲肋约束板的临界屈曲载荷。将求得的近似表达式与有限元ANSYS数值解以及文献中的结果进行对比, 结果吻合良好。     

Buckling and vibrations of two-directional FGM Mindlin circular plates under hydrostatic peripheral loading

An analysis is presented for the effect of hydrostatic peripheral loading on the free axisymmetric vibrations of functionally graded moderately thick circular plates on the basis of first-order shear deformation theory. The mechanical properties of the plate material are supposed to vary according to a power-law in both the radial and transverse directions. The numerical solution of the governing differential equation derived by using Hamilton's energy principle for such simply supported and clamped boundary conditions has been obtained employing the harmonic differential quadrature method choosing zeros of Chebyshev–Gauss–Lobatto as the grid points. The effect of different parameters has been analyzed on the frequency parameter for the first three modes of vibration. The critical buckling loads for both the plates have been computed by putting the frequencies to zero. Three-dimensional mode shapes for particular plate have been plotted. Obtained results have been compared.     

Thermal buckling analysis of moderately thick functionally graded annular sector plates

In this article, thermal buckling analysis of moderately thick functionally graded annular sector plate is studied. The equilibrium and stability equations are derived using first order shear deformation plate theory. These equations are five highly coupled partial differential equations. By using an analytical method, the coupled stability equations are replaced by four decoupled equations. Solving the decoupled equations and satisfying the boundary conditions, the critical buckling temperature is found analytically. To this end, it is assumed that the annular sector plate is simply supported in radial edges and it has arbitrary boundary conditions along the circular edges. Thermal buckling of functionally graded annular sector plate for two types of thermal loading, uniform temperature rise and gradient through the thickness, are investigated. Finally, the effects of boundary conditions, power law index, plate thickness, annularity and sector angle on the critical buckling temperature of functionally graded annular sector plates are discussed in details.