Finite element analysis of static and dynamic stresses around suddenly punched holes in plates and shells

The magnitude and distribution of stresses around suddenly punched holes in initially stressed plates and shells is of interest to insure that cracks will not precipitate from stress concentration. This problem is of practical interest to pressure vessel designers to preclude catastrophic failure when holes are punched in vessels to release gas. This paper presents a finite element analysis of several problems investigating static and dynamic stress fields around suddenly punched circular holes.

The first problem deals with the investigation of the radial and tangential stress fields in the vicinity of a suddenly punched hole in a stretched, elastic, isotropic plate subjected to an initial hydrostatic stress field. The wave propagation from a punched hole in the plate under a hydrostatic state of stress was solved analytically, using transform techniques, by Miklowitz; the finite element analysis of this problem presented in this paper confirms the analytical solution. Two grid meshes were investigated and results are presented to show the effect of grid mesh on solution accuracy and the power of finite element techniques for solving stress unloading problems. A formula for determining integration step size is found to be a function of the minimum element length and the wave propagation velocity. A similar investigation into the stress effects around a suddenly punched hole in the plate subjected to an initial uniaxial state of stress was also carried out as a prerequisite for the final problem studied.

The last problem is an anisotropic composite shell of varying thickness under an initial stress field due to internal pressure. The static and dynamic stress fields are computed from an unloading wave that radiates outward from a reinforced circular hole that is cut in the shell in 20 μs. A finite-element model of the shell is developed using quadrilateral and triangular plate elements and both in-plane and bending stiffness is included in the analysis as is nonlinear differential stiffening incorporated into the analysis as a single step approximation. Both bending and in-plane waves radiate outward from the cut hole and the dynamic stresses around the hole edge are computed for both unloading waves. The effects of the unloading waves are temporally spaced due to different wave velocities.

The paper demonstrates that fast response stress problems are readily amenable to finite-element analysis. For holes other than circular, the power of finite-element methods is apparent since these shapes lead to mathematically intractable problems if closed form solutions are attempted.     

基于分层模型的复合材料耦合Lamb波频率—波数域特性研究

针对复合材料层合板中耦合Lamb波的传播问题,基于分层模型提出解析建模与有限元数值模拟相结合的方法对其进行预测和评估。利用Legendre正交多项式展开法推导多层各向异性复合材料层合板中耦合Lamb波的控制方程,并对频率-波数域频散特性曲线实现数值求解。基于平面壳单元构建复合材料层合板的有限元模型,采用波结构加载法生成单一Lamb波基本模态,设计复合材料层合板的不同纤维取向、边界和界面约束条件,并经二维傅里叶变换获得有限元模拟数据的频率-波数域频散特性曲线。通过对比验证,结果表明两种方法均有较好的吻合性。     

Accurate computation of critical local quantities in composite laminated plates under transverse loading

The design of laminated composite based components requires a detailed analysis of the response of the structure when subjected to external loads. For the analysis of laminated composite plates, several plate theories have been proposed in the literature. Generally, these plate theories are used to obtain certain global response quantities like the buckling load. However, the use of these theories to obtain local response quantities, i.e. point-wise stresses; interlaminar stresses and strains, can lead to significant errors.In this paper, a detailed study of the quality of the point-wise stresses obtained using higher-order shear deformable, hierarchic and layerwise theories is done for a plate under transverse loading. The effect of equilibrium based post-processing on the transverse stress quantities is also studied. From the detailed study it is observed that the layerwise theory is very accurate. However, for all the models proper mesh design is required to capture boundary layer effects, discretization error, etc. Using focussed adaptivity, and post-processed state of stress, accurate representation of the local state of stress can be obtained, even with the higher-order shear deformable theories. Using this approach, the first-ply failure load is obtained with the Tsai–Wu criterion. It is observed that use of an adaptive procedure leads to significantly lower failure loads as compared to those given in the literature.     

Finite element formulation including interlaminar stress calculations

A quadrilateral plate element is developed on the basis of utilizing the compatibility equations to obtain the in-plane stresses, and the equilibrium equations to obtain both transverse shear and normal stresses. A plate as opposed to shell or solid formulation serves to provide efficient solutions for thin to moderately thick laminated composite configurations. The element formulation involves relaxation of the Kirchhoff hypothesis via superposition of a shear rotation upon a midplane rotation. The displacement field is carefully selected to obtain the desired transverse stress variation. Results are compared to both closed form and numerical solutions.     

Analysis of two-dimensional mixed-mode crack problems by finite element alternating method

A finite element alternating method is presented and applied to analyze two-dimensional linear elastic mixed-mode fracture problems with single or multiple cracks. The method involves the iterative superposition of the finite element solution of a bounded uncracked plate and the analytical solution of an infinite two-dimensional plate with a crack subjected to arbitrary normal and shear loadings. The normal and shear residual stresses evaluated at the location of fictitious cracks are fitted by appropriate polynomials through the least-squares method. Based on those coefficients of the determined polynomials, the mixed-mode stress intensity factors can be calculated accurately. The interaction effects among cracks are also considered. This method provides a highly efficient way to deal with two-dimensional fracture problems.     

Estimation of low-velocity impact damage in laminated composite circular plates using nonlinear finite element analysis

Nonlinear finite element analysis is used for the estimation of damage due to low-velocity impact loading of laminated composite circular plates. The impact loading is treated as an equivalent static loading by assuming the impactor to be spherical and the contact to obey Hertzian law. The stresses in the laminate are calculated using a 48 d.o.f. laminated composite sector element. Subsequently, the Tsai-Wu criterion is used to detect the zones of failure and the maximum stress criterion is used to identify the mode of failure. Then the material properties of the laminate are degraded in the failed regions. The stress analysis is performed again using the degraded properties of the plies. The iterative process is repeated until no more failure is detected in the laminate. The problem of a typical T300/N5208 composite [45 °/0 °/ − 45 °/90 °]_s circular plate being impacted by a spherical impactor is solved and the results are compared with experimental and analytical results available in the literature. The method proposed and the computer code developed can handle symmetric, as well as unsymmetric, laminates. It can be easily extended to cover the impact of composite rectangular plates, shell panels and shells.     

Analysis of residual stresses in hot-rolled complex beams

This paper deals with the analysis of residual stresses in hot-rolled complex beams. After rolling, residual stresses appear during the cooling period when the temperature is not uniform in the cross-section. These temperatures are calculated by a transient nonlinear program. The thermal stresses are estimated by a two-dimensional thermoelastoplastic or thermoelasto-viscoplastic finite element idealization. Bending effects are introduced in a generalized plane-strain formulation assuming circular curvature along the beam. The method leads to a plane-strain calculation which is very interesting from the computational point of view; the temperature dependence of the physical properties can be taken into account. The technique has important industrial applications. It permits the optimization of the state of residual stress in hot-rolled complex beams by testing different cooling conditions.     

A spectrally formulated plate element for wave propagation analysis in anisotropic material

A new spectrally formulated plate element is developed to study wave propagation in composite structures. The element is based on the classical lamination plate theory. Recently developed method based on singular value decomposition (SVD) is used in the element formulation. Along with this, a new strategy based on the method of solving polynomial eigenvalue problem (PEP) is proposed in this paper, which significantly reduces human intervention (and thus human error), in the element formulation. The developed element has an exact dynamic stiffness matrix, as it uses the exact solution of the governing elastodynamic equation of plate in frequency–wavenumber domain as the interpolating functions. Due to this, the mass distribution is modeled exactly, and as a result, a single element captures the exact frequency response of a regular structure, and it suffices to model a plate of any dimension. Thus, the cost of computation is dramatically reduced compared to the cost of conventional finite element analysis. The fast Fourier transform (FFT) and Fourier series are used for inversion to time–space domain. This element is used to model plate with ply drops and to capture the propagation of Lamb waves.     

Transient analysis of composite plates by radial basis functions in a pseudospectral framework

This paper presents a study of the linear transient response of composite plates using radial basis functions and collocation method in a pseudospectral framework. The first-order shear deformation plate theory is used to define a set of algebraic equations from the equations of motion and boundary conditions. The transient analysis is performed by a Newmark algorithm. In order to assess the quality of the present numerical method, an analytical solution was also developed. Numerical tests on square and rectangular cross-ply laminated plates demonstrate that the present method produces highly accurate displacements and stresses when compared with the available results.     

二维多域弹性问题虚边界无网格伽辽金法分析

针对复杂的不同材料属性的多域组合问题（比如复合材料交界面上接触应力的计算）,虚边界无网格伽辽金法被进一步研究,提出了二维多域弹性问题虚边界无网格伽辽金法。简要介绍了多域组合思想、子域虚边界元法,详细推导了二维多域弹性问题分析的虚边界无网格伽辽金法,得到具体的离散格式,便于编程,推广研究。方程的加权系数为位移、面力、连续边界上的位移与面力关系式偏导,数值意义明确,公式具体。最后通过计算数值实例为复合材料交界面上接触应力的计算,给出了复合圆盘接触面上的法向、径向应力,分多种方案调整每个子域的虚边界半径值,所得结果与解析解、其他数值方法进行比较。结论是二维多域弹性问题虚边界无网格伽辽金法的方法计算可行、精确性与稳定性好。     

Elastic–plastic thermal stress analysis in a thermoplastic composite disc applied linear temperature loads via FEM

The aim of this study is to obtain thermal stresses in a thermoplastic composite disc unidirectionally reinforced by steel fibers. Finite element method was used to calculate the thermal elastic and elastic–plastic stress distributions within the composite disc. Therefore, the solution was carried out using the ANSYS software. The temperature loading was chosen so as to vary linearly from inner surface to outer surface along the radial sections of the disc. Linear thermal loads were selected as to differ from each other. They were also adjusted from 90 to 130 °C. Thermal stresses were formed within the disc by the linear temperature loads due to its having different thermal expansion coefficients in radial and tangential directions. In line with the thermal analysis results, the magnitudes of the tangential stress components for both elastic and elastic–plastic solutions were above the radial stress components. In addition, the residual stress components were also calculated using both elastic and elastic–plastic solution results. The results obtained pointed out that the magnitudes and distributions of the thermal stresses and residual stresses were greatly influenced by the increase in linear temperature loads.     

Development of special fastener elements

A special finite element (FASNEL) is developed for the analysis of a neat or misfit fastener in a two-dimensional metallic/composite (orthotropic) plate subjected to biaxial loading. The misfit fasteners could be of interference or clearance type. These fasteners, which are common in engineering structures, cause stress concentrations and are potential sources of failure. Such cases of stress concentration present considerable numerical problems for analysis with conventional finite elements. In FASNEL the shape functions for displacements are derived from series stress function solutions satisfying the governing difffferential equation of the plate and some of the boundary conditions on the hole boundary. The region of the plate outside FASNEL is filled with CST or quadrilateral elements. When a plate with a fastener is gradually loaded the fastener-plate interface exhibits a state of partial contact/separation above a certain load level. In misfit fastener, the extent of contact/separation changes with applied load, leading to a nonlinear moving boundary problem and this is handled by FASNEL using an inverse formulation. The analysis is developed at present for a filled hole in a finite elastic plate providing two axes of symmetry. Numerical studies are conducted on a smooth rigid fastener in a finite elastic plate subjected to uniaxial loading to demonstrate the capability of FASNEL.     

OPCM 动态传感性能有限元分析与实验研究􀀂

通过建立被测构件与压电正交异性复合材料(OPCM)传感元件的三维有限元模型研究OPCM的动态传感特性.采用对被测构件施加冲击载荷的方法获得传感元件输出的瞬态响应时域信号,进行傅立叶变换分析,计算和实验结果都表明OPCM传感元件对同平面内相互正交的应力波具有不同的频响特性,且两者之间有较好的吻合性.     

A high precision coupled bending–extension triangular finite element for laminated plates

It is well recognized that the estimation of interlaminar stresses and strain energy release rates is important in designing laminated composite panels. Generally coupled bending–extension finite elements are necessary to study laminates to include the effects of coupling and/or combined transverse and extensional loads. Such elements are normally formulated adapting the classical theory of bending and extension. While the classical laminated plate theory of bending has provision to obtain interlaminar stresses due to transverse loading, it is necessary to include certain higher order terms in the extensional theory in order to obtain the interlaminar stresses due to inplane loads. A high precision triangular element based on a theory which includes both the bending and extension with necessary higher order terms is presented in this paper. The performance of this element is validated with the aid of examples. Numerical results for displacements in symmetric and unsymmetric laminates under bending loads have been given. Numerical results for interlaminar stresses in symmetric and unsymmetric laminates have been given for the well-known benchmark problem of a coupon with free edges. Strain energy release rate components at the delamination tip in coupons with unsymmetric sublaminates have been given. The effects of delamination length and location on the components of the strain energy release rate have been studied. Results indicated that with the use of this element, the interlaminar stresses can be estimated reasonably accurately, over a major part of the laminate except in a small local region close to the free edge. Global–local analysis with three-dimensional elements in the local region, is suggested to obtain local stresses more accurately. Interlaminar stresses at the boundary of a hole in a perforated plate under extension have been obtained to illustrate the use of the present element in a global–local analysis strategy.     

Transient analysis of the dynamic stress intensity factors using SGBEM for frequency-domain elastodynamics

In this paper, a two-dimensional symmetric-Galerkin boundary integral formulation for elastodynamic fracture analysis in the frequency domain is described. The numerical implementation is carried out with quadratic elements, allowing the use of an improved quarter-point element for accurately determining frequency responses of the dynamic stress intensity factors (DSIFs). To deal with singular and hypersingular integrals, the formulation is decomposed into two parts: the first part is identical to that for elastostatics while the second part contains at most logarithmic singularities. The treatment of the elastostatic singular and hypersingular singular integrals employs an exterior limit to the boundary, while the weakly singular integrals in the second part are handled by Gauss quadrature. Time histories (transient responses) of the DSIFs can be obtained in a post-processing step by applying the standard fast Fourier transform (FFT) and algorithm to the frequency responses of these DSIFs. Several test examples are presented for the calculation of the DSIFs due to two types of impact loading: Heaviside step loading and blast loading. The results suggest that the combination of the symmetric-Galerkin boundary element method and standard FFT algorithms in determining transient responses of the DSIFs is a robust and effective technique.     

Thermal postbuckling of thin-walled composite stiffeners

A study is made of the thermal postbuckling response of composite stiffeners subjected to prescribed edge displacement and a temperature rise. The flanges and web of the stiffeners are modeled by using two-dimensional plate finite elements. A mixed formulation is used with the fundamental unknowns consisting of the generalized displacements and the stress resultants of the plate. A reduction method is used in conjunction with mixed finite element models for determining the postbuckling response of the stiffeners. Sensitivity derivatives are evaluated and used to study the effects of variations in the different lamination and material parameters of the stiffeners on their postbuckling response characteristics. Numerical studies are presented for anisotropic stiffeners with Zee and channel sections.     

Finite element elastic-plastic-creep analysis of two-dimensional continuum with temperature dependent material properties

A technique is presented for performing finite element elastic-plastic-creep analysis of two-dimensional continuum composed of material with temperature dependent elastic, plastic, and creep properties. The plastic analysis utilizes the Prandtl-Reuss flow equations assuming isotropic material properties and linear strain-hardening. A power creep flow law formulated by Odquist is used to determine the steady state creep strain rate. The plastic and creep flow laws are employed to derive a ‘softened’ plastic-creep stress-strain matrix. These modified stress-strain relations are then used to formulate the element stiffness matrix in the usual manner. The differences in the elastic, plastic, and creep properties of the material due to the temperature change during the increment result in the formation of pseudo stresses, which in turn lead to load terms that appear on the right hand side of the equilibrium equations. The load terms resulting from these pseudo stresses not only keep the solution on the temperature dependent stress-strain curve of the material, but also correct for the elastic ‘overshoot’ that occurs when an element changes from an elastic to a plastic state. The effect of large displacements is included by the formulation of the geometric stiffness matrix for each element being used in the computer code. With this procedure it becomes economically feasible to perform elastic-plastic-creep stress analysis of two-dimensional continuum subjected to transient thermal and mechanical loadings. Several examples of both elastic-plastic and creep analyses are presented, and the finite element solutions are compared to either other theoretical solutions or experiment.     

Numerical Simulation of Cylindrical Solitary Waves in Periodic Media

We study the behavior of nonlinear waves in a two-dimensional medium with density and stress relation that vary periodically in space. Efficient approximate Riemann solvers are developed for the corresponding variable-coefficient first-order hyperbolic system. We present direct numerical simulations of this multiscale problem, focused on the propagation of a single localized perturbation in media with strongly varying impedance. For the conditions studied, we find little evidence of shock formation. Instead, solutions consist primarily of solitary waves. These solitary waves are observed to be stable over long times and to interact in a manner approximately like solitons. The system considered has no dispersive terms; these solitary waves arise due to the material heterogeneity, which leads to strong reflections and effective dispersion.     

Analysis of Nonlinear Multibody Systems with Elastic Couplings

This paper is concerned with the dynamic analysis of nonlinear multibody systems involving elastic members made of laminated, anisotropic composite materials. The analysis methodology can be viewed as a three-step procedure. First, the sectional properties of beams made of composite materials are determined based on an asymptotic procedure that involves a two-dimensional finite element analysis of the cross-section. Second, the dynamic response of nonlinear, flexible multibody systems is simulated within the framework of energy-preserving and energy-decaying time-integration schemes that provide unconditional stability for nonlinear systems. Finally, local three-dimensional stresses in the beams are recovered, based on the stress resultants predicted in the previous step. Numerical examples are presented and focus on the behavior of multibody systems involving members with elastic couplings.