BEAMS ON TENSIONLESS ELASTIC FOUNDATION: APPROXIMATE QUANTIFIER ELIMINATION WITH CHEBYSHEV SERIES

The well-known Sturm's theorem (based on Sturm's sequences) for the determination of the number of distinct real zeros of polynomials in a finite or infinite real interval has been already used in elementary quantifier elimination problems including applied mechanics and elasticity problems. Here it is further suggested that this theorem can also be used for quantifier elimination, but in more complicated problems where the functions involved are not simply polynomials, but they may contain arbitrary transcendental functions. In this case, it is suggested that the related transcendental equations/inequalities can be numerically approximated by polynomial equations/inequalities with the help of Chebyshev series expansions in numerical analysis. The classical problem of a straight isotropic elastic beam on a tensionless elastic foundation, where the deflection function (incorporating both the exponential function and trigonometric functions) should be continuously positive (this giving rise to a quantifier elimination problem along the length of the beam) is used as an appropriate vehicle for the illustration of the present mixed (symbolic–numerical) approach. Two such elementary beam problems are considered in some detail (with the help of the Maple V computer algebra system) and the related simple quantifier-free formulae are established and seen to coincide with those already available in the literature for the same beam problems. More complicated problems, probably necessitating the use of more advanced computer algebra techniques (together with Sturm's theorem), such as the Collins well-known and powerful cylindrical algebraic decomposition method for quantifier elimination, can also easily be employed in the present approximate (because of the use of Chebyshev series expansions) symbolic–numerical computational environment.     

Reliability-based design optimization of knuckle component using conservative method of moving least squares meta-models

This paper discusses reliability-based design optimization (RBDO) of an automotive knuckle component under bump and brake loading conditions. The probabilistic design problem is to minimize the weight of a knuckle component subject to stresses, deformations, and frequency constraints in order to meet the given target reliability. The initial design is generated based on an actual vehicle specification. The finite element analysis is conducted using ABAQUS, and the probabilistic optimal solutions are obtained via the moving least squares method (MLSM) in the context of approximate optimization. For the meta-modeling of inequality constraint functions, a constraint-feasible moving least squares method (CF-MLSM) is used in the present study. The method of CF-MLSM based RBDO has been shown to not only ensure constraint feasibility in a case where the meta-model-based RBDO process is employed, but also to require low expense, as compared with both conventional MLSM and non-approximate RBDO methods.     

Computation analysis of interlaminar fracture of laminated composites

The interlaminar fracture behavior of laminated composites has been investigated. Contact and friction along the crack surfaces is taken into account in the finite element modeling of the delamination crack growth. Mode I, mode II and mixed mode loading conditions at the crack tip have been analyzed. For the cracks with contact and friction along the crack surfaces the virtual crack closure integral method is used in order to calculate separated energy release rates. Computational modeling and analysis of cross-ply laminates in three-point bending has been performed. Contact elements were used in order to prevent the material interpenetration along the crack surfaces. Comparison of the results obtained with and without using contact elements has been carried out and significant differences between the correlated values of the energy release rates have been found. The influence of the coefficient of friction on the energy release rates was found to be significant for short delamination crack lengths but insignificant for long cracks. Numerical analyses of experimental data obtained for unidirectionally reinforced glass fiber composites by double cantilever beam tests and by notched flexure tests have been carried out. For the double cantilever beam test geometric linear and nonlinear finite element analyses have been performed and critical energy release rates were calculated. For the end notched flexure test the contact problem has been solved taking into account that adjacent to the support contact and friction will occur. For the double cantilever beam test the critical energy release rates obtained by linear and nonlinear finite element solution has been compared with those from four different analytical data reduction methods (the area method, the Berry method, the modified beam analysis, the compliance method). For the end notched flexure test the critical energy release rates, calculated by the finite element analysis and taking into account contact and friction along the crack surfaces, have been compared with those obtained by conventional beam analysis. This revised version was published online in July 2006 with corrections to the Cover Date.     

Flexural-Torsional Buckling and Vibration Analysis of Composite Beams

In this paper the general flexural-torsional buckling and vibration problems of composite Euler-Bernoulli beams of arbitrarily shaped cross section are solved using a boundary element method. The general character of the proposed method is verified from the formulation of all basic equations with respect to an arbitrary coordinate system, which is not restricted to the principal one. The composite beam consists of materials in contact each of which can surround a finite number of inclusions. It is subjected to a compressive centrally applied load together with arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting problems are (i) the flexural-torsional buckling problem, which is described by three coupled ordinary differential equations and (ii) the flexural-torsional vibration problem, which is described by three coupled partial differential equations. Both problems are solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the method can treat composite beams of both thin and thick walled cross sections taking into account the warping along the thickness of the walls. The proposed method overcomes the shortcoming of possible thin tube theory (TTT) solution, which its utilization has been proven to be prohibitive even in thin walled homogeneous sections. Example problems of composite beams are analysed, subjected to compressive or vibratory loading, to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. Moreover, useful conclusions are drawn from the buckling and dynamic response of the beam.     

Elastic behaviour of an orthotropic beam/one-dimensional plate of uniform and variable thickness

In this paper, the analysis of the title problem is based on mixed first-order thick-beam one-dimensional plate theory, and on using a small-parameter approach towards its numerical solution. The boundary conditions at the edges of the beam may be quite general, and between these two edges the beam may have varying thickness. Closed-form solutions have been developed for the static response of orthotropic beams with nonlinear thickness variation subjected to uniform loading. The accuracy of the present model is demonstrated by problems for which exact solutions and numerical results are available, and the results are also presented for a variety of problems whose solutions are not available in the literature.     

Separation of delamination modes in composite beams with symmetric delaminations

Delaminated composite beam under general edge loading conditions is studied. Based on a technical engineering theory an analytical procedure for calculation of strain energy release rate and its separation into modes I and II of delamination is presented. By choosing a suitable displacement field based on second-order shear-thickness deformation theory and using the principle of minimum total potential energy, the equations of equilibrium are obtained along with the appropriate boundary conditions. The J integral and its definition for different modes of fracture is used for calculation of strain energy release rate and its separation into different modes. Double cantilever beam (DCB) problem is a special case of this general problem in which loading is in mode I of fracture. The results of this method shows good agreement with FEM (i.e., finite element method) results and experimental data.     

功能梯度梁静力与动力分析的格林函数法

功能梯度梁静动态响应的数值分析方法一般局限于有限元法,存在有限元法的固有缺点,有必要发展新的数值求解方法。将功能梯度梁静力分析的控制微分方程转化为与匀质材料梁静力分析控制微分方程相一致的形式,并利用匀质材料梁静力问题的格林函数,开展功能梯度梁的静力分析。在此基础上,进一步推导获得功能梯度梁的柔度矩阵,据此建立功能梯度梁的运动方程,开展功能梯度梁的动力特性分析和动力响应分析。数值算例表明,采用格林函数法可以高效准确地分析功能梯度梁的静力响应与动力行为,验证了方法的计算精度与效率。     

SHAPE OPTIMIZATION AND MINIMUM WEIGHT LIMIT DESIGN OF ARCHES

The paper is concerned with the optimization of arches, using classical beam finite elements, for the minimum elastic displacements and the minimum weight designs under ultimate loading conditions.

The concept of separate but dependent design spaces for node coordinates and member plastic capacities is introduced. The shape optimization problem, whereby a norm of the elastic displacement vector is minimized, is formulated in the space of node coordinate variables. Then the minimum weight limit design problem in the space of member plastic capacities is considered using the static theorem of limit analysis. An iterative procedure alternating these two approaches is presented in the paper. The nonlinear unconstrained optimization and linear programming techniques are used to solve the corresponding numerical problems. The proposed method is illustrated by numerical examples.     

Application of the Laplace transform dual reciprocity boundary element method in the modelling of laser heat treatments

The Laplace Transform Dual Reciprocity Boundary Element Method (LTDRM or LT-DRBEM) provides with an alternative numerical technique to finite difference (FDM) or finite element methods (FEM) for solving transient diffusion problems. With this method, solutions are calculated directly at any specific time thus avoiding the use of time-stepping schemes. Besides, domain integrals are removed from the problem formulation.In this work we study the applicability of the LT-DRBEM method for laser heat treatment modelling purposes. A simple model was developed based on a two dimensional transient heat conduction equation, in which the laser beam is included as a heat flux boundary condition of gaussian shape. Results corresponding to a stationary and a moving beam are presented and discussed. Non-linear formulations of the problem as those given by temperature dependent material properties are also considered. Good accuracy results were obtained for the stationary beam approach, whereas severe limitations were found for the moving beam case.     

The boundary element method applied to the analysis of two-dimensional nonlinear sloshing problems

This paper deals with an application of the boundary element method to the analysis of nonlinear sloshing problems, namely nonlinear oscillations of a liquid in a container subjected to forced oscillations. First, the problem is formulated mathematically as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions, assuming the fluid to be inviscid and incompressible and the flow to be irrotational. Next, the governing equation (Laplace equation) and boundary conditions, except the dynamic boundary condition on the free surface, are transformed into an integral equation by employing the Galerkin method. Two dynamic boundary condition is reduced to a weighted residual equation by employing the Galerkin method. Two equations thus obtained are discretized by the use of the finite element method spacewise and the finite difference method timewise. Collocation method is employed for the discretization of the integral equation. Due to the nonlinearity of the problem, the incremental method is used for the numerical analysis. Numerical results obtained by the present boundary element method are compared with those obtained by the conventional finite element method and also with existing analytical solutions of the nonlinear theory. Good agreements are obtained, and this indicates the availability of the boundary element method as a numerical technique for nonlinear free surface fluid problems.     

On the analysis of a mixed mode bending sandwich specimen for debond fracture characterization

The mixed mode bending specimen originally developed for mixed mode delamination fracture characterization of unidirectional composites has been extended to the study of debond propagation in foam cored sandwich specimens. The compliance and strain energy release rate expressions for the mixed mode bending sandwich specimen are derived based on a superposition analysis of solutions for the double cantilever beam and cracked sandwich beam specimens by applying a proper kinematic relationship for the specimen deformation combined with the loading provided by the test rig. This analysis provides also expressions for the global mode mixities. An extensive parametric analysis to improve the understanding of the influence of loading conditions, specimen geometry and mechanical properties of the face and core materials has been performed using the derived expressions and finite element analysis. The mixed mode bending compliance and energy release rate predictions were in good agreement with finite element results. Furthermore, the numerical crack surface displacement extrapolation method implemented in finite element analysis was applied to determine the local mode mixity at the tip of the debond.     

Eigenfrequency optimization of stiffened plate using Kriging estimation

This paper describes the optimization of the beam stiffeners attached to plates in an eigenfrequency problem. The solution space is estimated using the Kriging method. A finite element analysis is carried out to evaluate the objective function at the sample points used for the estimation. The gradient method is used as a local optimizer. The Kriging estimation incurs relatively low cost, and is easy to combine with the gradient method. In this paper, we solve eigenfrequency optimization problems for a fully supported plate to maximize the minimum eigenfrequency and the difference between the 1st- and 2nd-order eigenfrequency. An optimization problem for an L-shaped plate with an eigenfrequency constraint is also solved. Good solutions are obtained for each example, and all the optimizations for these problems can be done at a lower computational cost. The results highlight the effectiveness of the method to solve the eigenfrequency optimization problems for the stiffened plate.     

Two general equations for deriving SIF expressions of some axisymmetric finite domain problems

Using the previous analytical method (Wang QZ, The crack-line (plane) stress field method for estimating SIFs—a review. Engineering Fracture Mechanics 1996; 55(4): 593–603.) and Green's function approach, two general equations are formulated for deriving approximate stress intensity factor (SIF) expressions for two categories of finite domain problems: (1) a finite-width strip with a center crack; (2) a circular cylinder with a concentric penny-shaped crack, both under various axisymmetric tensile loading at the crack faces. Examples with concentrated and distributed (up to quadratic variation) loading conditions are given to show the efficiency of these two general equations. As compared with the previous method, now the necessity of finding out the exact crack-line (plane) stress solution for the counterpart infinite problem is eventually waived. Another merit is that some SIF results for concentrated loading cases derived by using the general equations may have better accuracy than those given by the previous method. These two general equations are almost identical in form except for a small difference. Examples also show that the dimensionless SIF expressions for some problems in category (1) are identical with those in category (2), and there exists a regular correspondence between their loading conditions. Such identities in the dimensionless SIF expressions are useful in applications. Several example solutions given in this paper fill in the vacancy of missing solutions in present SIF handbooks, while other solutions are much simpler than the corresponding solutions in SIF handbooks.     

A decomposition procedure for large-scale optimum plastic design problems

A decomposition procedure is proposed in this paper for solving a class of large-scale optimum design problems for perfectly-plastic structures under several alternative loading conditions. The conventional finite element method is used to cast the problem into a finite dimensional constrained nonlinear programming problem. Structures of practically meaningful size and complexity tend to give rise to a large number of variables and constraints in the corresponding mathematical model. The difficulty is that the state-of-the-art mathematical programming theory does not provide reliable and efficient ways of solving large-scale constrained nonlinear programming problems. The natural idea to deal with the large-scale structural problem is somehow to decompose the problem into a collection of small-size problems each of which represents an analysis of the behaviour of each finite element under a single loading condition. This paper proposes one such way of decomposition based on duality theory and a recently developed iterative algorithm called the proximal point algorithm.     

On the feasibility of using Smoothed Particle Hydrodynamics for underwater explosion calculations

SPH (Smoothed Particle Hydrodynamics) is a gridless Lagrangian technique which is appealing as a possible alternative to numerical techniques currently used to analyze high deformation impulsive loading events. In the present study, the SPH algorithm has been subjected to detailed testing and analysis to determine the feasibility of using PRONTO/SPH for the analysis of various types of underwater explosion problems involving fluid-structure and shock-structure interactions. Of particular interest are effects of bubble formation and collapse and the permanent deformation of thin walled structures due to these loadings. These are exceptionally difficult problems to model. Past attempts with various types of codes have not been satisfactory. Coupling SPH into the finite element code PRONTO represents a new approach to the problem. Results show that the method is well-suited for transmission of loads from underwater explosions to nearby structures, but the calculation of late time effects due to acceleration of gravity and bubble buoyancy will require additional development, and possibly coupling with implicit or incompressible methods.This work was performed at Sandia National Laboratories, which is operated for the U.S. Department of Energy under Contract No. DE-AC04-94AL85000, and was partially funded by the Naval Surface Warfare Center under WFO proposal #15930816     

Meso-Scale Finite Element Simulations of 3D Braided Textile Composites: Effects of Force Loading Modes

Meso-scale finite element method (FEM) is considered as the most effective and economical numerical method to investigate the mechanical behavior of braided textile composites. Applying the periodic boundary conditions on the unit-cell model is a critical step for yielding accurate mechanical response. However, the force loading mode has not been employed in the available meso-scale finite element analysis (FEA) works. In the present work, a meso-scale FEA is conducted to predict the mechanical properties and simulate the progressive damage of 3D braided composites under external loadings. For the same unit-cell model with displacement and force loading modes, the stress distribution, predicted stiffness and strength properties and damage evolution process subjected to typical loading conditions are then analyzed and compared. The obtained numerical results show that the predicted elastic properties are exactly the same, and the strength and damage evolution process are very close under these two loading modes, which validates the feasibility and effectiveness of the force loading mode. This comparison study provides a suitable reference for selecting the loading modes in the unit-cell based mechanical behavior analysis of other textile composites.     

Formulation of reference surface element and its applications in dynamic analysis of delaminated composite beams

This paper presents a new finite element formulation, referred to as reference surface element (RSE) model, for numerical prediction of dynamic behaviour of delaminated composite beams and plates using the finite element method. The RSE formulation can be readily incorporated into all elements based on the Timoshenko beam theory and the Reissner–Mindlin plate theory taking into account the transverse shear deformations. The ‘free model' and ‘constrained model' for dynamic analysis of delaminated composite beams and/or plates have been unified in this RSE formulation. The RSE formulation has been applied to an existing 2-node Timoshenko beam element taking into account the transverse shear deformations and the bending–extension coupling. Frequencies and vibration mode shapes are determined through solving an eigenvalue problem. Numerical results show that the present RSE model is reliable and practical when used to predict frequencies and mode shapes of delaminated composite beams. The RSE formulation has also been used to investigate the effects of the number, size and interfacial loci of delaminations on frequencies and mode shapes of composite beams.     

Nonlinear analysis of liquid motion in a container subjected to forced pitching oscillation

A nonlinear analysis is carried out for the motion of the inviscid, incompressible fluid in a two-dimensional, rigid, open container which is subjected to forced sinusoidal pitching oscillation. Firstly, the problem is defined as a nonlinear initial-boundary value problem by the use of a governing differential equation and boundary conditions. Next, the problem is formulated in the form of a pseudo-variational principle, which provides a basis for our discretization. The finite element method and finite difference method are used spacewise and timewise, respectively. Due to the strong nonlinearity of the problem, an incremental method is used for the numerical analysis. Numerical results obtained by the present method are compared with solutions of the linear theory and experimental data. The difference between linear and nonlinear analysis has been clearly indicated.     

Buckling response of laminated composite stiffened plates subjected to partial in-plane edge loading

This article presents the buckling analysis of laminated composite stiffened plates subjected to partial in-plane edge loading. The finite element method is used to carry out the analysis. The eight-noded isoparametric degenerated shell element with C⁰ continuity and first-order shear deformation and a compatible three-noded curved beam element are used to model the plate skin and the stiffeners, respectively. The eigen value analysis is carried out to track the buckling load. The convergence study is performed for some specific problems and the results are compared with the available results in the literature. It is observed that the convergence of results is very fast for this finite element model. Effect of different parameters like orientation of fibers, number of layers, and loading types are considered in the present investigation. It is also observed that all these parameters have significant effect on the buckling response of the composite stiffened plate.     

Finite difference solution for circular plates on elastic foundations

In the present work the authors have developed a finite difference method of analysis for any circular plate with any kind of loading on semi-infinite elastic foundations. No assumption regarding the contact pressure distribution has been made. The equations have been developed in non-dimensional form and also the results have been obtained in non-dimensional form. These results have been compared with the available experimental results and the agreement between them is found to be much better than that of the previous works. The same method with slight modification can be applied for Winkler type foundations and problems of circular plates with varying thickness.