Optimal orientation of orthotropic materials using an energy based method

In this paper, an energy based method is proposed to determine the optimal orientation of orthotropic materials under static loading. Instead of assuming that the strain or stress fields are fixed with respect to orientational variables, the dependency of strain and stress fields on material orientation is explored by introducing an energy factor in the inclusion model. From the derivations, the strain based method and the stress based method can be recovered and their limitations are discussed. Numerical examples from these methods are presented and compared.     

Optimal locations of internal line supports for rectangular plates against buckling

This paper concerns the optimal locations of straight line internal supports for rectangular thin plates, under uniaxial compression, so as to maximize the elastic buckling loads. The optimization problem can be readily solved by using the newly developed pb-2 Rayleigh-Ritz method for the buckling analysis. Since the method avoids any discretization, the problems associated with the need to move the nodal points so as to coincide with the internal line support are eliminated. Moreover, the optimization exercise may be performed on a personal computer since the method requires a relatively smaller memory and computational effort as a result of a lower number of unknown variables involved when compared to other discretization methods such as the finite difference or finite element methods. Optimal results will be presented for plates with various combinations of edge conditions, aspects ratios and one or two internal line supports. The results indicate large sensitivity of buckling loads to supporting conditions.     

On optimal orientation of orthotropic materials

In order to use an anisotropic material effectively it should be oriented optimally with respect to the actual strain condition. Orientations with extreme energy density are obtained for orthotropic materials. It is found that the optimal orientation depends on one non-dimensional material parameter only, plus the ratio of the two principal strains. Complete analytical results are derived, including local as well as global maxima and minima.     

Vibration and buckling analysis of axisymmetric polar orthotropic circular plates

The vibration and stability analysis of polar orthotropic circular plates using the finite element method is discussed. In order to formulate the eigenvalue problems associated with the vibration and stability analyses, the clement stiffness, mass, and stability coefficient matrices are presented. By assuming the static displacement function, which is an exact solution of the polar orthotropic circular plate equation, approximates the vibration and buckling modes, the mass and stability coefficient matrices are readily derived from the given displacement function. Results showing the effects of orthotropy on natural frequencies and buckling loads are compared with their isotropic counterpart.     

On optimal orientation of nonlinear elastic orthotropic materials

The problem of minimization (or maximization) of the elastic energy density is solved in the two-dimensional case for a nonlinear elastic solid. The material behaviour is simulated on the basis of a power law stress-strain relation. Closed-form solutions which include corresponding solutions for a linear elastic solid are obtained. The latter may give local as well as global maxima and minima, respectively.     

Framework methods for orthotropic plates

Equivalent frameworks are presented to simulate orthotropic plates in extension or flexure. The structural properties of the framework models are obtained such that the actual and equivalent systems behave in the same manner. The real values of axial, flexural and torsional rigidities are considered in deriving formulas for these properties. There are no restrictions imposed on the values of Poisson's ratios. The stress distributions in orthotropic plates are calculated from the nodal displacements obtained.The models developed are applicable to a vast number of isotropic or orthotropic plate problems. The examples used to elaborate on these models are solved on a mini computer. Micro computers and programmable calculators could also be used. The finite element method is compared with the models presented and excellent agreements are found.     

On the stress-based and strain-based methods for predicting optimal orientation of orthotropic materials

In this paper, the strain-based and stress-based methods for solving optimal orientation problems of orthotropic materials are studied. Closed form solutions for both methods are derived and classified. The optimal orientation of both shear weak and some shear strong orthotropic materials may coincide with the major principal stress direction in the stress-based method. Similar results from the strain-based method are also derived. From the derivations, however, it can also be shown that both methods may fail under a special condition called repeated global minimum condition.     

Analysis of optimized plates for buckling

This paper is concerned with the analysis of optimized plates for buckling. The Rayleigh-Ritz technique is used to solve variable thickness rectangular plates. A double sine series is used to represent the lateral displacements and convergence is attained for a six term series. The result yields a buckling load which is 44% higher than that of a uniform thickness plate having the same volume. The plate itself was optimized using variational calculus to obtain the optimality condition which states that the thickness is proportional to the strain energy density and a truncated Fourier series solution (one term) was used to obtain an optimal shape having a critical load 112% higher than the uniform thickness plate.     

Pattern flattening for orthotropic materials

In many applications it is necessary to define good-fitting 2D flattened patterns for user-defined regions of a larger 3D surface. This paper describes the major stages involved in pattern flattening and illustrates the process with examples. In generating 2D patterns, some distortion is inevitably involved if the target 3D surface is not developable. For situations where distortion is required, it can be quantified in terms of the energy that must be imparted to the 2D flattening in localised areas so that it takes-up the original 3D region of the surface. An orthotropic strain model is adopted to convert the strain values to energy values. Starting with a bi-parametric definition of a large 3D surface, an arbitrary defined region is specified by the user in terms of a contiguous series of cubic curves lying on the bi-parametric plane. To extract the 3D region, a polygon list is generated to represent the surface. The triangulation process is based on a ‘marching front’ algorithm. A process is described which then flattens this polygon list and performs an energy minimisation analysis every time the process attempts to flatten an over-constrained triangle. Further consideration is made of seam insertion in the 3D surface definition and of adaptively modifying the triangulation process so that more triangles are used in areas of high-energy concentration. Examples are also presented to illustrate the sensitivity of the strain profiles to the fabric grain direction when the pattern is applied to the 3D surface.     

Dynamic buckling of orthotropic shallow spherical shells

The dynamic axisymmetric behaviour of clamped orthotropic shallow spherical shell subjected to instantaneously applied uniform step-pressure load of infinite duration, is investigated here. The available modal equations, based on an assumed two-term mode shape for the lateral displacement, for the free flexural vibrations of an orthotropic shallow spherical shell is extended now for the forced oscillations. The resulting modal equations, two in number, are numerically integrated using Runge-Kutta method, and hence the load-deflection curves are plotted. The pressure corresponding to a sudden jump in the maximum deflection (at the apex) is considered as the dynamic buckling pressure, and these values are found for various values of geometric parameters and one value of orthotropic parameter. The numerical results are also determined for the isotropic case and they agree very well with the previous available results. It is observed here that the dynamic buckling load increases with the increase in the orthotropic parameter value. The effect of damping on the dynamic buckling load is also studied and this effect is found to increase the dynamic buckling load. It is further observed that this effect is more pronounced with increase in the rise of the shell.     

Seminumerical solution for buckling of rectangular plates

A semianalytical, seminumerical method of solution is presented for the governing partial differential equation of rectangular plates subjected to in-plane loads. The basic functions in the y-direction are chosen as the eigenfunctions for straight prismatic beams. The classical method of separation of variables is employed to obtain an ordinary differential equation. The resulting equation is solved by a one-dimensional finite difference technique. The problem is then reduced to a typical eigenvalue problem which on solution yields the buckling coefficient of the plate. The method is applied on plates with different edge conditions and under various loading conditions. The results are compared with those of existing solutions. Results for the case when one loaded edge is fixed and the other simply supported were reported in the literature for the first time.     

Optimal design of shells against buckling subjected to combined loadings

In this paper, the problem of optimal design of shells against instability is considered. A thin-walled shell is loaded, in general, by overall bending moment, constant or varying along an axis of a shell, by the appropriate shearing force and by an axial force and a constant torsional moment. We look for the shape of middle surface as well as the thickness of a shell, which ensures the maximum critical value of the loading parameter. The volume of material and the capacity of a shell are considered as equality constraints. The concept of a shell of uniform stability is applied.     

Vibration and buckling of skew plates with edges elastically restrained against rotation

This paper deals with vibration and buckling analyses of skew plates with edges elastically restrained against rotation using the spline strip method. The effect of rotational stiffnesses, skew angles and aspect ratios on these problems is analysed, and its characteristic charts are also presented.     

FE-FD model for magneto-elastic buckling of ferromagnetic plates

A cantilever beam-plate, of magnetically soft material, inserted with its middle surface normal to a uniform magnetic field will buckle when the magnetic field reaches a critical value. The system of differential equations derived in this study are based on the Moon-Pao's model. The basic difference lies in the abandonment of the assumption that the magnetic couple acting on an element of the plate is proportional to the rotation of the middle surface of the plate. This assumption is employed in previous studies and leads to linear differential equations that is similar to the buckling problem of compressed rods or plates. Consequently, the complexity of the problem is reduced considerably. The derivation in this paper shows that the magnetoelastic buckling problem of the beam-plate is nonlinear. The solution to the system of differential equations is derived by coupling a finite element model for the magnetic field and a finite difference model for the cantilever beam-plate. The critical magnetic field, derived in this study is about 20% smaller than the experimental one, while theoretical results from previous works are about 180% larger than the experimental ones.     

Trefftz-unsymmetric finite element for bending analysis of orthotropic plates

This work develops a new four-node quadrilateral displacement-based Trefftz-type plate element for bending analysis of orthotropic plates within the framework of the unsymmetric finite element method (FEM). In the present formulation, the modified isoparametric interpolations are employed to formulate the element’s test functions in which the deflection is effectively enriched by the nodal rotation degrees of freedom (DOFs). Meanwhile, the element’s trial functions are determined based on the Trefftz functions that can a prior satisfy the governing equations of orthotropic Mindlin–Reissner plates. Numerical benchmark tests reveal that the new unsymmetric plate element is free of shear locking problem and can produce satisfactory results for both the displacement and stress resultant. In particular, it exhibits quite good tolerances to the gross mesh distortion.

     

正交各向异性叠层板的非线性主共振分析

研究了在四边简支的边界条件下,正交各向异性矩形叠层板在横向简谐激励作用下的非线性主共振及其稳定性问题．在给出了正交各向异性叠层板的振动微分方程的基础上,利用伽辽金法导出了相应的达芬型非线性强迫振动方程．应用平均法对主共振问题进行求解,得到了系统在稳态运动下的幅频响应方程．基于李雅普诺夫稳定性理论,得到了解的稳定性判定条件．作为算例,分别给出了不同条件下,系统运动的幅频响应曲线图、振幅-激励幅值响应曲线图和动相平面图,并对解的稳定性进行了分析,讨论了各参数对系统非线性振动特性的影响．     

Elastic buckling of tapered circular plates

This paper is concerned with the elastic buckling of tapered circular plates. The study is prompted by the fact that results hitherto available are restricted to a narrow range of taper parameters and are somewhat different from each other. For the buckling analysis, a simple and yet accurate numerical method is presented. It is based on the shooting method and the Rayleigh-Ritz approach. Comprehensive generic buckling results of circular plates with linearly and parabolically varying thicknesses are generated. Comparison studies of the buckling results showed that some of the existing results were erroneous. Optimal values of taper parameters for such tapered plates are also given.     

Nonlinear flexural vibrations of laminated orthotropic plates

Large amplitude free flexural vibrations of laminated orthotropic plates are studied using C⁰ shear flexible QUAD-8 plate element. The nonlinear governing equations are solved using the direct iteration technique. Numerical results are obtained for isotropic, orthotropic and cross-ply laminated plates with simply-supported boundary conditions on immovable edges. It is observed that hardening behaviour is increased for thick plates and orthotropic plates.     

Post-buckling behaviour of cylindrically orthotropic annular plates

A finite element formulation for the study of the post-buckling behaviour of cylindrically orthotropic annular plates is presented in detail. The results for radial load ratios are presented in the form of empirical formulae in terms of the deflection at the inner edge to the thickness ratios of the plates for the first time in the literature for various values of the orthotropic parameters and radii ratios.     

Post-buckling of cylindrically orthotropic circular plates on elastic foundation with edges elastically restrained against rotation

The post-buckling behaviour of cylindrically orthotropic circular plates is investigated through a finite element formulation, with the plates resting on an elastic foundation and their edges are elastically restrained against rotation. Results are presented in the form of linear buckling load parameters and empirical formulae for radial load ratios for various values of spring stiffness, foundation stiffness and orthotropy parameter.