Stress concentration near a thin elastic inclusion under interaction with harmonic waves in the case of smooth contact

We solve the problem on the interaction of plane harmonic waves with a thin elastic plate-shaped inclusion. The ambient medium is assumed to be in plane strain. The smooth contact conditions are satisfied on both sides of the inclusion. The bending displacements of the inclusion are determined from the corresponding differential equation. In the statement of boundary conditions for this equation, one should take into account the transverse forces and bending moments applied to the lateral edges of the inclusion, while the boundary conditions are posed on the midplane of the inclusion. Using the discontinuous solution method, we reduce the problem to a system of two singular integral equations, which are solved numerically by the mechanical quadrature method. We obtain approximate formulas for the stress intensity coefficients near the ends of the inclusion and for the transverse forces and moments applied to the inclusion.     

On the theory of transverse bending of elastic plates

Departing from a self-contained two-dimensional formulation of the linear-theory problem of transverse bending of plates, three distinct topics are considered.The first of these concerns the integration problem for the case of orthotropy, specifically in regard to the factorization of a certain sixth-order master-equation.The second topic concerns the boundary layer aspects of contracted or reduced boundary conditions for the interior solution contribution for the case of isotropic plates. The analysis of this is based on a new form of the well-known general solution in terms of a deflection and a stress function variable, with this new form making it possible to distinguish between first- and second-order transverse shear deformation effects; the former being associated with the edge zone and the latter with the interior domain of the plate, with the shear correction terms for the interior being generalizations of the Timoshenko shear correction terms for beams.The third topic is a new system of contracted boundary conditions, both for the stress and for the displacement boundary value problem, in such a way that first-order transverse shear deformation effects are explicitly incorporated in the interior-domain solution contribution.     

薄板弯曲问题的虚边界元-最小二乘法

本文提出求解任意形状的薄板弯曲问题的虚边界元-最小二乘法。本法首先利用薄板弯曲平衡方程的格林函数和离开实际边界上分布的未知的横向荷载和法向弯矩函数建立满足实际边界条件的积分方程;然后采用最小二乘法和沿虚边界分段离散化的待定的分布横向荷载和法向弯矩函数得到求上述积分方程离散化数值解的线性代数方程组。导出了一系列的数值积分的公式,并求解了许多例题,数值结果说明本法完全避免了奇异积分及其复杂的处理方法和耗时的运算,而且在边界及其附近区域解的精度比普通边界元(以后简称边界元)法大大地提高了。     

Dynamical bending of rigid-plastic annular plates

Dynamical bending of circular rigid-plastic annular plates, fixed along the exterior boundary and free on the interior boundary, when subjected to instantaneously applied transverse uniformly distributed blast-type load[1], is investigated in this paper.It is shown that annular plates are preferable to plates without holes, since their load capacity increases while residual deflections decrease. A so-called boundary parameter is introduced to estimate the effect of boundary conditions on the radial bending moment.A procedure for determining the residual deflections at every point of a plate is developed for use on an electronic computer. Numerical examples are given. In the end of the paper, the particularities of solution of our problem for annular plates, corresponding to the setting of Wang[2], Wang and Hopkins[3] for plates without holes are discussed.     

Boundary layers of hierarchical beam and plate models

The boundary behavior of a family of hierarchical models of linearly elastic, isotropic plates is studied. The hierarchical models are obtained by spectral semidiscretization of the displacement fields in the transverse direction and strain energy projection. The well known Reissner-Mindlin model is contained in the hierarchy as a special case. A decomposition of the boundary layers of any model in the hierarchy into a bending and a torsion layer, both of which are model dependent, is given. It is shown further that the bending and torsion layers arise as Galerkin approximations of certain (nonlinear) eigenvalue problems in the plate cross section with the subspaces used to derive the hierarchical model. It is shown that the bending layers converge, as the order of the plate model tends to infinity, to the so-called Papkovich functions on an elastic strip.The regularity of the solution on polygonal plates is investigated for the whole hierarchy of plate models and shown to equal the regularity of the plane elasticity problem.Partially supported by AFOSR Grant No. F49620-92-J0100.     

Nonlinear dynamic analysis of circular plates with varying thickness

The BEM is developed for nonlinear free and forced vibrations of circular plates with variable thickness undergoing large deflections. General boundary conditions are considered, which may be also nonlinear. The problem is formulated in terms of displacements. The solution is based on the concept of the analog equation, according to which the two coupled nonlinear differential equations with variable coefficients pertaining to the in-plane radial and transverse deformation are converted to two uncoupled linear ones of a substitute beam with unit axial and unit bending stiffness, respectively, under fictitious quasi-static load distributions. Numerical examples are presented which illustrate the method and demonstrate its accuracy.     

Solving the problem of linear viscoelasticity for multiply connected anisotropic plates

This paper describes the method for solving the problems of linear viscoelasticity for thin plates under the influence of bending moments and transverse forces. The small parameter method was used to reduce the original problem to a sequence of boundary-value problems solved via complex potentials of the bending theory of multiply connected anisotropic plates. The general representations of complex potentials and boundary conditions for their determination are obtained. The method for determining the stress state of the plate at any time with respect to complex approximation potentials is developed by replacing the powers of the small parameter by the Rabotnov operators. The problem of a plate with elliptical holes is solved. The numerical calculation results in the case of a plate with one or two holes are given. The variation of bending moments in time until stationary condition is reached is studied, and the influence of geometric characteristics of the plate on these variable is described.     

Nonlinear analysis of shear deformable beam-columns partially supported on tensionless three-parameter foundation

In this paper, a boundary element method is developed for the nonlinear analysis of shear deformable beam-columns of arbitrary doubly symmetric simply or multiply connected constant cross-section, partially supported on tensionless three-parameter foundation, undergoing moderate large deflections under general boundary conditions. The beam-column is subjected to the combined action of arbitrarily distributed or concentrated transverse loading and bending moments in both directions as well as to axial loading. To account for shear deformations, the concept of shear deformation coefficients is used. Five boundary value problems are formulated with respect to the transverse displacements, to the axial displacement and to two stress functions and solved using the Analog Equation Method, a BEM-based method. Application of the boundary element technique yields a system of nonlinear equations from which the transverse and axial displacements are computed by an iterative process. The evaluation of the shear deformation coefficients is accomplished from the aforementioned stress functions using only boundary integration. The proposed model takes into account the coupling effects of bending and shear deformations along the member as well as the shear forces along the span induced by the applied axial loading. Numerical examples are worked out to illustrate the efficiency, wherever possible, the accuracy and the range of applications of the developed method.     

A Hamiltonian State Space Approach for 3D Analysis of Circular Cantilevers

A 3D exact analysis of extension, torsion and bending of a cantilever of a circular cross section is studied with emphasis on the fixed-end effect. Through Hamiltonian variational formulation, the basic equations of elasticity in cylindrical coordinates and the boundary conditions of the problem are formulated into the state space setting in which the state vector comprises the displacement vector and the conjugate stress vector as the dual variables. Upon delineating the Hamiltonian characteristics of the system, 3D solutions for transversely isotropic circular cantilevers subjected to an axial force, a torque, terminal couples and transverse forces are determined, thereby, the fixed-end effects and applicability of the solutions of generalized plane strains and the elementary theory of bending of beams are evaluated.     

On an inverse problem of bending of a physically nonlinear inhomogeneous plate

An elastic plate with a physically nonlinear inclusion of an arbitrary shape is considered. This plate is subjected to pure bending under the action of transverse forces and bending moments applied at the external boundary of the plate. There are no loads distributed over the surface. The problem of finding external actions that provide a necessary uniform moment state in the inclusion, i.e., prescribed constant moments and curvatures, is formulated and solved. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 104–107, September–October, 2007.     

Benchmark trigonometric and 3-D elasticity solutions for an exponentially graded thick rectangular plate

The bending problem of a transverse load acting on an isotropic inhomogeneous rectangular plate using both two-dimensional (2-D) trigonometric and three-dimensional (3-D) elasticity solutions is considered. In the present 2-D solution, trigonometric terms are used for the displacements in addition to the initial terms of a power series through the thickness. The effects due to transverse shear and normal deformations are both included. The form of the assumed 2-D displacements is simplified by enforcing traction-free boundary conditions at the faces of the plate. No transverse shear correction factors are needed because a correct representation of the transverse shearing strain is given. The plate material is exponentially graded, meaning that Lamé’s coefficients vary exponentially in a given fixed direction (the thickness direction). A wide variety of results for the displacements and stresses of an exponentially graded rectangular plate are presented. The validity of the present 2-D trigonometric solution is demonstrated by comparison with the 3-D elasticity solution. The influence of aspect ratio, side-to-thickness ratio and the exponentially graded parameter on the bending response are investigated.     

Vibrations of highly prestressed anisotropic plates via a numerical-perturbation technique

The method of matched asymptotic expansions is used to reduce the problem of the transverse vibrations of a highly prestressed anisotropic plate into the simpler problem of the vibration of an anisotropic membrane with modified boundary conditions that account for the bending effects. In the absence of an exact solution the membrane problem can be solved by any well-known numerical technique. The numerical-perturbation results for a clamped circular plate with rectangular orthotropy and a uniform tensile stress applied on its boundary show an excellent correlation with finite-element solutions for the original problem. Furthermore, the solutions obtained for annular plates form the basis for solutions to problems involving near-annular plates.     

On asymmetric bending of functionally graded solid circular plates

Based on the three-dimensional elasticity equations, this paper studies the elastic bending response of a transversely isotropic functionally graded solid circular plate subject to transverse biharmonic forces applied on its top surface. The material properties can continuously and arbitrarily vary along the thickness direction. By virtue of the generalized England’s method, the problem can be solved by determining the expressions of four analytic functions. Expanding the transverse load in Fourier series along the circumferential direction eases the theoretical construction of the four analytic functions for a series of important biharmonic loads. Certain boundary conditions are then used to determine the unknown constants that are involved in the four constructed analytic functions. Numerical examples are presented to validate the proposed method. Then, we scrutinize the asymmetric bending behavior of a transversely isotropic functionally graded solid circular plate with different geometric and material parameters.     

点支撑预应力中厚矩形板的弯曲

基于Reissner-Mindlin一阶剪切变形板理论，讨论在预加面内机械荷载或温度场作用下，点支撑中厚矩形板的弯曲问题。温度场假定沿板表面为均布，沿板厚方向为线性分布的。利用考虑剪切变形影响的Timoshenko梁函数，采用Rayleigh-Ritz法给出不同边界条件下点支撑中厚板在横向荷载作用下的挠度和弯矩分布。结果表明，均匀温度场与预加面内压力将使板的挠度和弯矩增加。支撑点位置的变化、边界约束条件和横向剪切变形效应都对板的内力大小和分布有显著影响。     

Creeping of a rotating beam

We determine the critical time in the problem of buckling of a beam rotating about its axis corresponding to the unstrained state. We draw an analogy between the transverse bending of a rotating beamand the problem of longitudinal bending [1], which is topical in applied geomechanics.     

Bending and vibration of composite laminated plates

Hybrid-stress finite element method is applied for analysis of bending and vibration of composite laminated plates in this paper. Firstly, based on the modified complementary principle, a rectangular hybrid-stress plate bending element is presented which applies to analysis of laminates. Inside the element, different stress parameters are assumed according to different layers. The boundary displacements are determined by means of the assumption of YNS theory on the boundary of elements. The element formed in this way not only can take effects of transverse shear deformation and local warping into account, but also has less degrees of freedom. Then, problems of bending and vibration of laminates are solved by using this element, and the numerical results are compared with the exact solutions. This shows that the results obtained in the paper are very close to the exact results.     

Flexural testing of an epoxy oversized strand model under traction

Experimental flexural tests are conducted on an epoxy oversized 1×6 strand; the objectives of these tests are to measure the bending stiffness and the associated strains. Three sets of boundary conditions are applied. The bending stiffness is deduced from the transverse midspan deflection and is found to be dependent on the axial force and the moment distribution. The deduced bending stiffness is compared to values computed from the theoretical bending stiffnesses.     

Dynamic analysis of 3-D beam elements including warping and shear deformation effects

In this paper the dynamic analysis of 3-D beam elements restrained at their edges by the most general linear torsional, transverse or longitudinal boundary conditions and subjected in arbitrarily distributed dynamic twisting, bending, transverse or longitudinal loading is presented. For the solution of the problem at hand, a boundary element method is developed for the construction of the 14 × 14 stiffness matrix and the corresponding nodal load vector of a member of an arbitrarily shaped simply or multiply connected cross section, taking into account both warping and shear deformation effects, which together with the respective mass and damping matrices lead to the formulation of the equation of motion. To account for shear deformations, the concept of shear deformation coefficients is used, defining these factors using a strain energy approach. Eight boundary value problems with respect to the variable along the bar angle of twist, to the primary warping function, to a fictitious function, to the beam transverse and longitudinal displacements and to two stress functions are formulated and solved employing a pure BEM approach that is only boundary discretization is used. Both free and forced transverse, longitudinal or torsional vibrations are considered, taking also into account effects of transverse, longitudinal, rotatory, torsional and warping inertia and damping resistance. Numerical examples are presented to illustrate the method and demonstrate its efficiency and accuracy. The influence of the warping effect especially in members of open form cross section is analyzed through examples demonstrating the importance of the inclusion of the warping degrees of freedom in the dynamic analysis of a space frame. Moreover, the discrepancy in the dynamic analysis of a member of a spatial structure arising from the ignorance of the shear deformation effect necessitates the inclusion of this additional effect, especially in thick walled cross section members.     

面内变刚度薄板弯曲问题的挠度?弯矩耦合神经网络方法

发展了一种求解面内变刚度功能梯度薄板弯曲问题的神经网络方法. 面内变刚度薄板弯曲问题的偏微分控制方程为一复杂的4阶偏微分方程, 传统的基于强形式的神经网络解法在求解该偏微分方程时可能会遇到难以收敛、边界条件难以处理的情况. 本文基于Kirchhoff薄板弯曲理论, 提出了一种直角坐标系下任意面内变刚度薄板弯曲问题的神经网络解法. 神经网络模型包含挠度网络与弯矩网络, 分别用于预测薄板的挠度与弯矩, 从而将求解4阶偏微分方程转换为求解一系列二阶偏微分方程组, 通过对挠度、弯矩试函数的构造可使得神经网络计算结果严格满足边界条件. 在误差的反向传播中, 根据本文提出的误差函数公式计算训练误差并结合Adam优化算法更新模型的内部参数. 求解了不同边界条件、形状的面内变刚度薄板弯曲问题, 并将所得计算结果与理论解、有限元解进行对比. 研究表明, 本文模型对于求解面内变刚度薄板弯曲问题具备适应性, 虽然模型中的弯矩网络收敛较挠度网络要慢, 但本文方法在试函数的构造上更为简单、适应性更强.      

On the solution singularity in the plane elasticity problem for a cantilever strip

The plane elasticity problem of bending of a cantilever strip whose material is assumed to be incompressible in the transverse direction is solved. It is shown that, in the classical statement of of the boundary condition for the fixed edge of the strip, the solution has a singularity at the corner points of the edge. Several cases of the strip fixation and loading characterized by the presence or absence of the solution singularity are considered. The strength of glass beams of three types, for which the theory of elasticity predicts whether the normal stress has a singularity, is studied experimentally. It is shown that the limit stresses for the beams of the types under study are practically the same, which testifies that the solution singularity does not have any physical nature.