Stress analyses around holes in composite laminates using boundary element method

A three-dimensional (3D) boundary element method (BEM) is developed for the analysis of composite laminates with holes. Instead of using Kelvin-type Green's functions of anisotropic infinite space, 3D layered Green's functions with the materials of each layer being generally anisotropic, derived recently in the Fourier transform domain, are implemented into a 3D BEM formulation. A novel numerical algorithm is designed to calculate layered Green's functions efficiently. It should be noted that since layered Green's functions satisfy exactly the continuity conditions along the interfaces and top and bottom free surfaces a priori, the model becomes truly 2D and discretization is only needed along the hole surface and prescribed traction and/or displacement boundaries. To test the validity and accuracy of the proposed method, the present layered BEM formulation is applied to the problem of an infinite anisotropic plate with a circular hole where the analytical solution is available. It is found that even with a very coarse mesh, the present BEM can predict the hoop stress very accurately along the hole surface. The BEM formulation is then applied to analyze two composite laminates (90/0)_s and (−45/45)_s, under a remote in-plane strain, that have been studied previously with different approaches. For the (90/0)_s case, the hoop stresses along the hole surface predicted by the present layered BEM formulation are in very close agreement with the previous results. For the (−45/45)_s case, however, it is found that a nearly converged solution (less than 5% convergence by doubling the mesh) by the present method is at significant variance with the previous ones that are lack-of-convergence checks. It can be expected that for designing the bolted joints of composites with many layers, a computational tool developed based on the present techniques would be robust and offer a much better solution with regard to accuracy, versatility and design cycle time.     

Crack propagation analysis of welded thin-walled joints using boundary element method

 Tube-to-plate nodal joints under cyclic bending are widely used in the road transport and agricultural industry. The square hollow sections (SHS) used in these constructions are thin-walled and cold formed, and they have thicknesses of less than 4 mm. Some fatigue failures have been observed. The weld undercut may affect the fatigue life of welded tubular joints especially for thin-walled sections. The undercut dimensions were measured using the silicon imprint technique. Modelling of thin-walled cruciform joints, as a simplification of welded tubular joints, is described in this paper to determine the effect of weld undercut on fatigue propagation life. The Boundary Element Analysis System Software (BEASY) is used. The results of the effect of weld toe undercut from this analysis are compared with results from previous research to determine the comparative reduction in fatigue life between thin-walled joints (T=3 mm) and those made of thicker sections (T=20 mm). The loss in fatigue strength of the thin-walled joints is found to be relatively more than that for thicker walled joints. A 3D model of a tube to plate T-joint is also modelled using the boundary element software, BEASY. The nodal joint consists of a square hollow section, 50×50×3 SHS, fillet welded to a 10-mm thick plate, and subjected to cyclic bending stress. Fatigue analyses are carried out and the results are compared with the only available S–N design curve.     

Stress analysis of a front bumper fascia using the boundary element method

Stress concentration is one of the most common problems related to automotive components and numerical analysis can be of great interest to deal with such problems. The boundary element method (BEM) is a technique which can be used in stress analysis and it is specifically applied to the design of a car component here. This work presents the efficacy of the application of a procedure based on BEM sub-model for stress analysis in a proposed design change of a front bumper fascia. The results confirm the consistency of the proposed procedure compared to the finite element method (FEM), a consolidated method for stress analysis in the automotive industry.     

Dynamic analysis of cracks using boundary element method

This paper presents a procedure for dynamic stress intensity factor computations using traction singular quarter-point boundary elements. Cracks in a complete space, a half-space and a finite body loaded by steady state waves are studied. Curves for elastodynamic stress intensity factors vs frequency are presented. Transient stress intensity factors are computed by means of Fourier transform. The results are compared with other authors and shown to be accurate in all cases. The dynamic stress intensity factors are computed in a very direct and easy way to implement. This versatile procedure allows for the study of problems with complex geometry that include one or several cracks.     

Analysis of mechanically fastened composite joints by boundary element methods

This article focuses on the boundary element formulation development for analyzing a mechanically bolted composite. Boundary equations are formulated for all the member panels of the composite joints. These equations are solved together with the fastener equations to get the resultant contact forces for all the fasteners involved. The fasteners are then modeled as 1D springs that are governed by linear relationship between the fastener forces and the displacements of member panels at the respective fastener centers. After obtaining all the fastener forces from the global analysis, detailed stress analysis is performed for region around an individual fastener. The stress distributions around fastener holes are then used to evaluate the margin of safety of the composite panels. The numerical predictions on the fastener forces, failure modes and failure loads of two typical bolted composite joints using the proposed method agree well with that of the experimental results.     

Stress analysis with arbitrary body force by boundary element method

Linear stress analysis without body force can be easily carried out by means of the boundary element method. Some cases of linear stress analysis with body force can also be solved without the domain integral. However domain integrals are generally necessary to solve the linear stress problems with complicated body forces. This paper shows that the linear stress problems with complicated body forces can be solved approximately without the domain integral. In order to solve these problems, the domain is divided into small areas using contour lines of body force. In these areas, the distributions of body force are assumed approximately to satisfy the Laplace equation.     

Stress analysis of plastic IC package containing various interface delaminations using the boundary element method

The purpose of this paper is to carry out stress calculation of the integrated circuit (IC) plastic package under the remote loading or the pressure acting on the delamination surfaces by the boundary element method (BEM). Based on symmetrical property of the problem geometry and the loading, the corresponding fundamental solution can be obtained to further reduce the number of the boundary elements used in the discretization of the problems. Near the interface delamination tips, singular boundary elements are used to accurately obtain the stress intensity factors. For the pre-assumed delamination spaces, the corresponding stress distributions in each domain can be obtained so that the effect of the interface delamination on the stresses can be observed. In order to assure the rightness of the developed code, some special problems in which the analytical solutions are available are studied. The results show that the developed code can produce numerical results with high accuracy.     

基于渐进损伤分析的复合材料螺栓连接强度包线法研究

针对中国民机采用T800级复合材料这一新材料体系而基础数据匮乏的现状,采用渐进损伤分析(PDA)替代试验以显著降低研究周期和成本。综合渐进损伤方法和工程算法各自的优点,提出以渐进损伤分析替代应力集中减缓因子(SCRFs)测定试验,进而建立强度包线,并进行多钉连接强度预测的数值策略。为验证该数值策略的可行性,针对典型铺层应力集中减缓因子,测定试样,并开展渐进损伤分析,获得了试验件强度预测值来计算应力集中减缓因子,采用旁路载荷修正的强度包线法,绘制了典型铺层复合材料多钉连接旁路载荷修正强度包线,预测多钉连接的失效载荷,并与试验结果进行对比。结果表明:采用该数值策略预测的强度包线、多钉连接的失效载荷和失效模式均与试验结果吻合良好,证明了该数值策略的可行性。     

Modeling of concrete cracking—A hybrid technique of using displacement discontinuity element method and direct boundary element method

This paper presents a new approach by making use of a hybrid method of using the displacement discontinuity element method and direct boundary element method to model concrete cracking by incorporating fictitious crack model. Fracture mechanics approach is followed using the Hillerborg's fictitious crack model. A boundary element based substructure method and a hybrid technique of using displacement discontinuity element method and direct boundary element method are compared in this paper. In order to represent the process zone ahead of the crack, closing forces are assumed to act in such a way that they obey a linear normal stress-crack opening displacement law. Plain concrete beams with and without initial crack under three-point loading were analyzed by both the methods. The numerical results obtained were shown to agree well with the results from existing finite element method. The model is capable of reproducing the whole range of load–deflection response including strain-softening and snap-back behavior as illustrated in the numerical examples.     

Static and dynamic analysis of composite shear walls by the boundary element method

Summary A boundary element solution for the static and dynamic analysis of composite shear walls is developed. The composite shear wall consists of a matrix material surrounding a finite number of inclusions (in-fill walls) with different elastic constants firmly bonded to it. The problem is formulated in terms of the displacement components and all kinds of boundary conditions are encountered. Several numerical examples are worked out to demonstrate and attest the efficiency of the method. The results, whenever possible, are compared with those obtained from analytical or other numerical solutions. The case of homogeneous shear wall with or without holes is included as a special case.With 8 FiguresPresented at the 1st Nat. Cong. on Mechanics of the Hellinic Society for Theoretical and Applied Mechanics (HSTAM), Athens, June 1986.     

Modal analysis of anisotropic plates using the boundary element method

A numerical formulation for analysis of dynamic problems of thin anisotropic plates bending is presented. The bending behavior follows Kirchhoff's hypothesis. The formulation is based on the direct boundary element method. The problem is simplified by using the elastostatic fundamental solution of an infinite plate. Domain integrals arising from inertial terms are transformed into boundary integrals using the dual reciprocity technique. Boundary integrals are discretized and evaluated numerically. Natural frequencies for free vibration are obtained and the respective mode shapes are shown. The accuracy of numerical results obtained is assured by comparison with analytical or finite element results.     

The direct boundary element method in plate bending

The direct boundary element method based on the Rayleigh-Green identity is employed for the static analysis of Kirchhoff plates. The starting point is a slightly modified version of Stern's equations. The focus is on the implementation of the method for linear elements and a Hermitian interpolation for the deflection w. The concept of element matrices is developed and the Cauchy principal values of the singular integrals are given in detail. The treatment of domain integrals, the handling of internal supports, the properties of the solution and the effect of singularities are discused. Numerical examples illustrate the various techniques. In the appendix the influence functions for the second and third derivatives of the deflection w are given.     

Structural analysis of composite bolted joints using the complex potential method

The problem of a pin-loaded hole in a symmetric composite plate with finite dimensions is considered within the scope of the classical laminate theory. The analysis is performed by means of the Lekhnitskii complex potential method. For the given problem, an appropriate power series expansion of the complex potentials is stipulated, where the coefficients are determined from the underlying boundary conditions. The present approach provides an efficient method for the calculation of stresses and displacements in the neighbourhood of the hole where failure is likely to occur.     

A hybrid technique of modeling of cracks using displacement discontinuity and direct boundary element method

A hybrid technique to model two dimensional fracture problems which makes use of displacement discontinuity and direct boundary element method is presented. Direct boundary element method is used to model the finite domain of the body, while displacement discontinuity elements are utilized to represent the cracks. Thus the advantages of the component methods are effectively combined. This method has been implemented in a computer program and numerical results which show the accuracy of the present method are presented. The cases of bodies containing edge cracks as well as multiple cracks are considered. A direct method and an iterative technique are described. The present hybrid method is most suitable for modeling problems involving crack propagation.     

Three-dimensional magnetic field analysis method using scalarpotential formulated by boundary element method

A computational method for magnetic fields formulated by a BEM (boundary element method) has been developed. In the method, a reduced scalar potential is selected as an unknown variable to simplify the calculation of the boundary conditions. Its use requires a high numerical accuracy of the potential gradient. Conventional BEM does not provide this, because numerical element integration for the singular kernal causes a large error. To overcome this difficulty, a highly accurate numerical integration scheme is proposed based on the BEM, and it is applied to magnetic field problems. Calculation results for a spherical permeable material in a problem proposed by the Institute of Electrical Engineers of Japan (the problem of a magnetic field generated by a coil) agreed with the exact solution and the experimental data within 5%     

Modal analysis of plates using the dual reciprocity boundary element method

This paper presents a new method for determining the natural frequencies and mode shapes for the free vibration of thin elastic plates using the boundary element and dual reciprocity methods. The solution to the plate's equation of motion is assumed to be of separable form. The problem is further simplified by using the fundamental solution of an infinite plate in the reciprocity theorem. Except for the inertia term, all domain integrals are transformed into boundary integrals using the reciprocity theorem. However, the inertia domain integral is evaluated in terms of the boundary nodes by using the dual reciprocity method. In this method, a set of interior points is selected and the deflection at these points is assumed to be a series of approximating functions. The reciprocity theorem is applied to reduce the domain integrals to a boundary integral. To evaluate the boundary integrals, the displacements and rotations are assumed to vary linearly along the boundary. The boundary integrals are discretized and evaluated numerically. The resulting matrix equations are significantly smaller than the finite element formulation for an equivalent problem. Mode shapes for the free vibration of circular and rectangular plates are obtained and compared with analytical and finite element results.     

Transient analysis of three-dimensional wave propagation using the boundary element method

In the past the time domain solution of the wave equation has been limited to simplified problems. This was due to the limitations of analytical methods and the capacity of computers to manipulate and store ‘large’ blocks of spatial information. With the advent of ‘super computers’ the ability to solve such problems has significantly increased. This paper outlines a method for transient analysis of wave propagation in arbitrary domains using a boundary element method. The technique presented will allow the definition of a domain, the input of impedance conditions on the domain's surface, the specification of inputs on the surface, and the specification of initial conditions within the domain. It will produce a complete solution of the wave equation inside the domain. The techniques are demonstrated using a program with a boundary element formulation of Kirchhoff's equation. The elements used are triangular and compatible.     

Numerical analysis of damage progression and strength of countersunk composite joints

A numerical investigation is conducted into the damage progression and strength of bolted joints between fibre-reinforced composite laminates using countersunk fasteners. Experimental tests were previously conducted on a bearing test specimen and countersunk fastener single-lap joints. In this work, computational models are developed for Abaqus/Explicit, with continuum shells employed to model in-plane ply failure. The bolt-nut assembly is modelled with rigid elements, and the models account for bolt torque and frictional contact. The material properties required in the computational model are determined from standard tests, with the compression fracture toughness of composite plies calibrated against experimental data from the bearing test. The analysis approach captures the load-carrying capability of all configurations, and provides reasonable accuracy in predicting damage patterns. The effects of bolt torque, clearance and countersink height ratio are investigated, and the analysis results compare well with experimental findings. Furthermore, the analysis provides rich insight into the damage progression and joint behaviour at the ply level, with the in-plane and through-thickness damage patterns mapped for increasing applied load. Delamination is incorporated using a cohesive element layer at the start of the countersunk region, though has minimal influence on damage progression and load-carrying capability, which agrees with the experimental results.     

Stress and velocity in 2D transient elastodynamic analysis by the boundary element method

This paper is mainly concerned with the development of integral equations to compute stress and velocity components in transient elastodynamic analysis by the boundary element method. All expressions required are presented explicitly. The boundary is discretized by linear isoparametric elements whereas linear and constant time interpolation are assumed, respectively, for the displacement and traction components. Time integration is carried out analytically and the resulting expressions are presented. An assessment of the accuracy of the results provided by the present formulation can be seen at the end of the article, where two examples are presented.