Analysis of Bernoulli beams with 3D stochastic heterogeneity

The behavior of stochastically heterogeneous beams, composed of isotropic sub-elements of randomly distributed stiffness is studied. Cross sectional as well as longitudinal heterogeneity are included. Average displacements, reaction forces and their statistical variance are found analytically by a functional perturbation method. Ratio of sub-element to beam characteristic size is not negligible and the use of an equivalent homogeneous structure with the classical effective material properties is not sufficient. The major aim is to study the relation between various microstructure properties (grain size, shape, modulus, statistical correlation lengths etc.) and the overall behavior of linear elastic Bernoulli beams. For the statically determinate case, only cross sectional 2D microstructure statistics is found to affect the elastic response, so that an equal average displacement can be achieved by an equivalent, non-isotropic homogeneous beam. For the indeterminate case, the average values of macro properties are affected by the 3D morphological features. Therefore, the proper equivalent homogeneous beam has to include non-local elastic properties. A simple reciprocal relation, connecting two separate loading systems is found, relating their external forces and displacement statistical variances. Morphological parameters, like two point probability moments, used in the final results are derived analytically, and their physical interpretations are discussed.     

A novel application of the FPM to the buckling differential equation of non-uniform beams

In this study the buckling load (P) of non-uniform, deterministic and stochastically heterogeneous beams, is found by applying the Functional Perturbation Method (FPM) directly to the Buckling (eigenvalue) Differential Equation (BDE). The FPM is based on considering the unknown P and the transverse deflection (W) as functionals of heterogeneity, i.e., the elastic bending stiffness “K” (or the compliance S=1/K). The BDE is expanded functionally, yielding a set of successive differential equations for each order of the (Frèchet) functional derivatives of P and W. The obtained differential equations differ only in their RHS, and therefore a single modified Green function is needed for solving all orders. Consequently, an approximated value for the buckling load is obtained for any given morphology. Four examples of simply supported columns are solved and discussed. In the first three, deterministic realizations of K are considered, whereas in the fourth, K is assumed to be the stochastic field. The results are compared with solutions found in the literature for validation.     

Periodic array of staggered planar cracks in an anisotropic elastic medium

The problem of determining the elastic displacements and stresses in an infinite anisotropic medium containing a periodic array of staggered planar cracks is considered. It is reduced to a system of Hadamard finite-part singular (hypersingular) integral equations with the crack-opening displacements as unknown functions. The integral equations may be solved numerically by a collocation technique. Numerical results for specific cases involving isotropic and transversely isotropic materials are obtained.     

Static deformation due to a long buried dip-slip fault in an isotropic half-space welded with an orthotropic half-space

Closed-form analytical expressions for the displacements and the stresses at any point of a two-phase medium consisting of a homogeneous, isotropic, perfectly elastic half-space in welded contact with a homogeneous, orthotropic, perfectly elastic half-space due to a dip-slip fault of finite width located at an arbitrary distance from the interface in the isotropic half-space are obtained. The Airy stress function approach is used to obtain the expressions for the stresses and the displacements. The case of a vertical dip-slip fault is considered in detail. The variations of the displacements with the distance from the fault and with depth have been shown graphically.     

Wave propagations in exponentially graded transversely isotropic half-space with potential function method

Time-harmonic response of a vertically graded transversely isotropic, linearly elastic half-space is analytically determined by introducing a new set of potential functions. The potential functions are set in such a way that the governing equations be simple and with physical meaning as well. In addition, the potential functions introduced in this paper are degenerated to a complete set of potential functions used frequently for wave propagations in homogeneous transversely isotropic media. Utilizing Fourier series and Hankel integral transforms, the governing equations for the potential functions are solved, after which the displacements and stresses are presented in the form of line integrals. Both the displacements and stresses determined here are collapsed on the solution previously reported for the constant profile transversely isotropic material. Because of complicated integrand functions, the integrals are evaluated numerically and presented graphically, where the effect of degree of change of material properties plays a major role, which may be recognized easily.     

双材料板条的界面应力──两端面承受一般载荷的情况

本文给出双材料板条的Ｓｕｈｉｒ界面应力微分方程一般解，利用此解求解了叠层板条结构两端面承受任意载荷时的界面应力问题。作为例子，求解了著名的Ｇｏｌａｎｄ－Ｒｅｉｓｓｕｅｒ胶合接头界面应力问题，并将极条厚度ｈ１＝ｈ２，材料弹性模量Ｅ１＝Ｅ２时的结果与张福范等的结果作了比较。     

System reliability analysis of spatial variance frames based on random field and stochastic elastic modulus reduction method

This paper presents the stochastic elastic modulus reduction method for system reliability analysis of spatial variance frames based on the perturbation stochastic finite element method (PSFEM) and the local average of a random field. The stochastic responses and reliability index of each element of a structural frame are characterized by the PSFEM and the first-order second-moment method, to properly handle the correlation structures and scale of fluctuation of random fields. A strategy of elastic modulus adjustment for the estimation of system reliability is developed to determine the range and magnitude of elastic modulus reduction, by taking the element reliability index as a governing parameter. The collapse mechanism and system reliability index of a stochastic framed structure are determined through iterative computations of the PSFEM. Compared with the failure mode approaches in traditional system reliability analysis, the proposed method avoids two major difficulties, namely the identification of significant failure modes and estimation of the joint probability of failure modes. The influences of the correlation structure and scale of fluctuation of the random field upon system reliability are investigated to demonstrate the accuracy and computational efficiency of the proposed methodology in system reliability analysis of spatial variance frames.     

Stochastic finite element analysis of a cable-stayed bridge system with varying material properties

Stochastic seismic finite element analysis of a cable-stayed bridge whose material properties are described by random fields is presented in this paper. The stochastic perturbation technique and Monte Carlo simulation (MCS) method are used in the analyses. A summary of MCS and perturbation based stochastic finite element dynamic analysis formulation of structural system is given. The Jindo Bridge, constructed in South Korea, is chosen as a numerical example. The Kocaeli earthquake in 1999 is considered as a ground motion. During the stochastic analysis, displacements and internal forces of the considered bridge are obtained from perturbation based stochastic finite element method (SFEM) and MCS method by changing elastic modulus and mass density as random variable. The efficiency and accuracy of the proposed SFEM algorithm are evaluated by comparison with results of MCS method. The results imply that perturbation based SFEM method gives close results to MCS method and it can be used instead of MCS method, especially, if computational cost is taken into consideration.     

Deformation due to mechanical and thermal sources in generalised orthorhombic thermoelastic material

A dynamical two-dimensional problem of thermoelasticity has been considered to investigate the disturbance due to mechanical (horizontal or vertical) and thermal source in a homogeneous, thermally conducting orthorhombic material. Laplace-Fourier transforms are applied to basic equations to form a vector matrix differential equation, which is then solved by eigenvalue approach. The displacements, stresses and temperature distribution so obtained in the physical domain are computed numerically and illustrated graphically. The numerical results of these quantities for zinc crystal-like material are illustrated to compare the results for different theories of generalised thermoelasticity for an insulated boundary and a temperature gradient boundary.     

The tangential differential operator applied to a stress boundary integral equation for plate bending including the shear deformation effect

Boundary integral equations (BIEs) for stresses are widely used in elastic and inelastic analyses, and those for tractions are essential in fracture mechanics problems. The existence of strong singularities in the fundamental solution kernels of BIEs for stresses at boundary points and for traction forces requires additional care in numerical implementations with respect to that employed for a displacement BIE. The use of the tangential differential operator (TDO) in conjunction with integration by parts is one way to reduce the order of strong singularities in these fundamental solution kernels when Kelvin-type fundamental solutions are used. Two formulations for stress and traction BIEs using the TDO are presented in this study. The TDO and integration by parts were employed in the first formulation only to reduce the strong singularity without changing other fundamental solution kernels. In the second formulation, the TDO was applied to all fundamental solution kernels involving the multiplication of generalized displacements to reduce the singularities, and the resulting kernels were combinations of those from the displacement BIE. Finally, plate problems were solved with both traction BIEs employing the TDO instead of the displacement BIEs to evaluate the accuracy of these formulations.     

Analysis of thermoelastic response in a functionally graded spherically isotropic hollow sphere based on Green–Lindsay theory

This paper is concerned with the investigation of thermoelastic displacements and stresses in a functionally graded spherically isotropic hollow sphere due to prescribed temperature in the context of the linear theory of generalized thermoelasticity with two relaxation time parameters (Green and Lindsay theory). Both the surfaces of the body are free from radial stresses, and the inner surface is subjected to a time-dependent thermal shock whereas the outer one is maintained at constant temperature. The basic equations have been written in the form of a vector–matrix differential equation in the Laplace transform domain which is then solved by an eigenvalue approach. The numerical inversion of the transforms is carried out using a method of Bellman et al. The displacements and stresses are computed and presented graphically. It is found that the variation of the thermophysical properties of a material as well as the thickness of the body strongly influence the response to loading. A comparative study with the corresponding homogeneous material has also been made. The solution of the problem of a spherically isotropic infinite medium containing a spherical cavity has been derived theoretically by tending the outer radius to infinity, as a particular case.     

DEM: A new computational approach to sheet metal forming problems

Many current approaches to finite element modelling of large deformation elastic—plastic forming problems use a rate form of the virtual work (equilibrium) equations, and a finite element representation of the displacement components. Called the incremental method, this approach produces a three-field formulation in which displacements, stresses and effective strain are dependent variables. Next, the formulation is converted to a one-field displacement formulation by an algebraic time discretization which uses a low order explicit time-stepping procedure to integrate the equations. This approach does not produce approximations which satisfy the discrete equilibrium equations at all times and, moreover, the advantage of the single-field algebraic formulation is realized at the expense of very small time steps needed to produce stability and accuracy in the numerical calculations. This paper describes a variant of the mixed method in which all three field variables (displacements, stresses and effective strain) are given finite element representations. The discrete equilibrium equations then generate a nonlinear system of algebraic equations whose solutions represent a manifold, while the constitutive equations form a system of ordinary differential equations. A commercially available, variable time step/variable order code is then used to integrate this differential/algebraic system. When applied to the problem of hydrostatic bulging of a membrane, the new approach requires far less computer time than the incremental method.     

Clamped nano-beams as adsorption induced sensors: Linear and non-linear effects

This work analyzes the deflection of clamped nano-beam due to stochastic surface stresses, induced by adsorption/desorption of surrounding particles. Both linear and non-linear effects (mid-plane stretching) are considered. A mechanical model for 1D nano-beam is first introduced and includes the surface effects via their effective cross sectional residual force and moment. The model considers local non-stationary surface residual stresses, governed by Langmuir’s interaction model. Local adsorption relations are described by non-deterministic model, from which the statistics of the surface residual stresses are extracted and their time-space correlations are calculated. A straightforward perturbation method is used to evaluate the non-linear effects. In each order of the approximated solution, the nano-beam deflections are governed by a stochastic differential operator with a non-deterministic load. Equations are solved analytically by the Functional Perturbation method (FPM) and validated by Monte-Carlo simulations. It is found that the non-deterministic nature of the nano-beam deflections can be used for in situ sensing applications, in cases of very fast or slow adsorption schemes, for which the microscopic sensors are not sufficient. Geometric non-linear effects can be used in order to achieve fine tuning of the sensitivity.     

The Boussinesq-Flamant problem for an elastic nonlocal half-infinite space

Summary Transmission of a concentrated force into a half-infinite elastic medium is examined assuming that the medium has nonlocal properties. The values of the nonlocal moduli are adopted from the known studies on the wave propagation in nonlocal media. Field equations are solved by using the Fourier transform technique. Inversion of the equations obtained for the stresses and displacements shows that the equations for stresses coincide with those predicted by the conventional theory. The equations for the displacements differ, however, from their classical counterparts, and can only be evaluated approximately. The first approximation leads to the classical equations. The fourth approximation derived for the Poisson material close to the line of loading displays deviations from the classical values amounting to 35%.     

Exact elasticity solution for the vibration of functionally graded anisotropic cylindrical shells

An exact elasticity solution is presented for the free and forced vibration of functionally graded cylindrical shells. The functionally graded shells have simply supported edges and arbitrary material gradation in the radial direction. The three-dimensional linear elastodynamics equations, simplified to the case of generalized plane strain deformation in the axial direction, are solved using suitable displacement functions that identically satisfy the boundary conditions. The resulting system of coupled ordinary differential equations with variable coefficients are solved analytically using the power series method. The analytical solution is applicable to shallow as well as deep shells of arbitrary thickness. The formulation assumes that the shell is made of a cylindrically orthotropic material but it is equally applicable to the special case of isotropic materials. Results are presented for two-constituent isotropic and fiber-reinforced composite materials. The homogenized elastic stiffnesses of isotropic materials are estimated using the self-consistent scheme. In the case of fiber-reinforced materials, the effective properties are obtained using either the Mori–Tanaka or asymptotic expansion homogenization (AEH) methods. The fiber-reinforced composite material studied in the present work consists of silicon-carbide fibers embedded in titanium matrix with the fiber volume fraction and fiber orientation graded in the radial direction. The natural frequencies, mode shapes, displacements and stresses are presented for different material gradations and shell geometries.     

弹性介质模糊有限元控制方程的快速解法

线弹性模糊有限元方法是分析弹性介质体模糊特性对结构响应产生不确定性影响的有效方法。即使对弹性介质体而言,模糊有限元控制方程的求解时间问题也是困扰其推广应用的主要障碍。为获得可靠可行的模糊有限元控制方程的快速求解方法,在深入研究弹性介质体的模糊源特点基础上,提出当引起结构模糊特性的力学参数为单源模糊数时,可以利用单源模糊数的运算特点来求解模糊有限元的控制方程,进而利用合成运算求解结构的模糊位移和模糊应力的分布。推导了基于单源模糊数运算的弹性介质模糊应力和模糊位移的计算表达式。应用模糊有限元求解的区间解法和快速解法对算例进行比较分析,结果表明了快速解法的正确性。     

Statics of thin-walled pretwisted beams

The displacement and strain fields of thin-walled pretwisted beams are prescribed in terms of generalized displacements for extension, bending, torsion and warping. Differential equations and boundary conditions are obtained from the elastic potential energy functional without assuming coincidence of the beam axis with any of the structural axes. This procedure gives a unique consistent definition of sectional moments and generalized forces. Some simple explicit formulae are derived for homogeneous tension–torsion. For the general case a computer code is developed on the basis of discretized generalized displacements and a modified energy functional, devised to obtain consistent lengthwise variation of the stresses and a unique decomposition of the torsional moment. Examples show agreement with analytical results for cylindrical beams and illustrate the various coupling effects for beams with pretwist. They also demonstrate the usefulness of the explicit formulae for homogeneous tension–torsion.     

Stress intensity factors around a crack in a nonhomogeneous interfacial layer between two dissimilar elastic half-planes

The stresses around a crack in an interfacial layer between two dissimilar elastic half-planes are obtained. The crack is parallel to the interfaces. The material constants of the layer vary continuously within a range from those of the upper half-plane to those of the lower half-plane. An internal gas pressure is applied to the surfaces of the crack. To derive the solution, the nonhomogeneous interfacial layer is divided into several homogeneous layers with different material properties. The boundary conditions are reduced to dual integral equations, which are solved by expanding the differences of the crack face displacements into a series. The unknown coefficients in the series are determined using the Schmidt method, and a stress intensity factor is calculated numerically for epoxy-aluminum composites.     

A Crack in the Homogeneous Half Plane Interacting with a Crack at the Interface Between the Nonhomogeneous Coating and the Homogeneous Half-Plane

A fundamental solution is established for a crack in a homogeneous half-plane interacting with a crack at the interface between the homogeneous elastic half-plane and the nonhomogeneous elastic coating in which the shear modulus varies exponentially with one coordinate. The problem is solved under plane strain or generalized plane stress conditions using the Fourier integral transform method. The stress field in the homogeneous half plane is evaluated by the superposition of two states of stresses, one of which is associated with a local coordinate system in the infinite fractured plate, while the other one in the infinite half plane defined in a structural coordinate system.     

Closed and approximate analytical solutions for rectangular Mindlin plates

Summary Basing on the Nádai-Lévy and the Vlasov-Kantorovich methods closed and approximate analytical solutions of Mindlin's plate equations in the case of rectangular plates are discussed. For elastic, homogeneous and isotropic plates three unknowns of the governing two-dimensional boundary value problem are formulated as series of products of functions depending on a single coordinate. Specifying the functions for one of the in-plane coordinate directions the governing partial differential equations for a special type of boundary conditions and the principle of virtual displacements for the general case yield a set of ordinary differential equations. The analytical solution of these equations provides expressions for functions depending on the other in-plane coordinate. For plates with simply supported edges for one of the coordinate directions and for arbitrary homogeneous boundary conditions for the other one the Nádai-Lévy method provides a closed or exact solution in the sense that the infinite series for displacements and stress resultants can be truncated to obtain any desired accuracy. In the general case of nonsimply supported edges the iterative Vlasov-Kantorovich method yields an approximate analytical solution. Both methods are nonsensitive to a reduction of the thickness with respect to accuracy and represent the boundary layer solutions in terms of exponential functions. Applications to rectangular plates with various types of boundary conditions are presented.