Stability of orthotropic plates

The possibilities of the algorithm for applying the boundary element method to solving boundary value problems are discussed on the example of the two-dimensional Poisson differential equation. The algorithm does not change significantly when the type of boundary conditions changes: the Dirichlet problem, the Neumann problem, or a mixed boundary value problem. The idea of the algorithm is taken from the work of John T. Katsikadelis [1]. The algorithm is described in detail in the next sequence of actions. 1) The boundary- value problem for a two-dimensional finite domain is formulated. The desired function in the domain, its values, and its normal derivative on the boundary contour are connected by means of the second Green formula. 2) We pass from the boundary value problem for the Poisson equation to the boundary value problem for the Laplace equation. This simplifies the process of constructing an integral equation. We obtain the integral equation on the boundary contour using the boundary conditions. 3) In the integral equation, we divide the boundary contour into a finite number of boundary elements. The desired function and its normal derivative are considered constant values on each boundary element. We compose a system of linear algebraic equations considering these values. 4) We modify the system of linear algebraic equations taking into account the boundary conditions. After that, we solve it using the Gauss method. The computer program has been developed according to the developed algorithm. We used it in the learning process. The software implementation of the algorithm takes into account the capabilities of modern computer technology and modern needs of the educational process. The work of the program is shown in the test case. Further modification of the described algorithm is possible

The article deals with a laminated orthotropic plate. Between the layers there is rigid contact as well as contact with frictionless slippage. On the end faces at the level of the neutral surface of the slipping layers the plate is loaded by tangential loads. The model of loss of stability is based on the hypotheses on displacements of the plate that take transverse shear strain into account. Normal displacements of the plate do not depend on the transverse coordinate. The hypotheses derived by iteration make it possible to satisfy the conditions of interlaminar contact, both rigid and with frictionless slippage, with respect to displacements and stresses, and also the conditions on the outer surfaces of the plate. With the aid of Lagrange's variational principle we obtained differential equations under conditions of the precritical and the postcritical state, and also the corresponding boundary conditions. The equations of the postcritical state are written in terms of displacements. On the basis of the presented equations the problem of the stability of a homogeneous orthotropic rectangular plate with pin support along the periphery is solved. It is shown that the results obtained with the suggested model coincide with the known results. The article also examines a square three-layered orthotropic plate under conditions of frictionless slippage contact between the layers and pin support along the periphery. The central layer is loaded by tangential loads on the end faces at the level of its neutral surface. It is shown that when slippage is taken into account, the critical forces are substantially reduced.

Bending of rectangular orthotropic plates with rotationally restrained and free edges: Generalized integral transform solutions

Elastic-plastic analysis of a nozzle corner crack by finite element method

Elastic-plastic FEM-analysis of a nozzle corner crack and discussion of the results by some fracture mechanics concepts

In order to obtain information on the lateral buckling strength of stainless steel I-beams to propose the design formulae for the stability factors, this paper focuses on the overall buckling of laterally-unrestrained stainless steel I-beams. The design methods in current codes are discussed, and the results from these codes are compared with the finite element analysis (FEA) results. The influence from the material property on the overall stability of stainless steel beams is also investigated. Based on the FEA results, the design method on overall buckling strength of steel beams specified in the Chinese standard GB 50017-2003 is modified to calibrate the overall stability of stainless steel bending members with welded I-section. Through the comparison with EC3: 1993-1-4 and the FEA results, the proposed formulae in this paper are proved to be applicable in the design of overall stability of stainless steel bending members.

The plane problem of a loaded Euler-Bernoulli beam of finite length in frictionless bilateral contact with a microstructured half-plane modelled by the couple stress theory of elasticity is considered here. The study is aimed to investigate the size effects induced on the beam internal forces and moments by the contact pressure and couple stress tractions transmitted across the contact region. Use is made of the Green’s functions for point force and point couple applied at the surface of the couple stress elastic half-plane. The problem is formulated by imposing compatibility of strain between the beam and the half-plane along the contact region and three alternative types of microstructural contact conditions, namely vanishing of couple stress tractions, vanishing of microrotations and compatibility between rotations of the beam cross sections and microrotations of the half-plane surface. The first two types of boundary conditions are usually assumed in the technical literature on micropolar materials, without any sound motivation, although the third boundary condition seems the most correct one. The problem is thus reduced to one or two (singular) integral equations for the unknown distributions of contact pressure and couple stress tractions, which are expanded in series of Chebyshev orthogonal polynomials displaying square-root singularity at the beam ends. By using a collocation method, the integral equations are reduced to a linear algebraic system of equations for the unknown coefficients of the series. The contact pressure and couple stress along the contact region and the shear force and bending moment along the beam are then calculated under various loading conditions applied to the beam, varying the flexural stiffness of the beam and the characteristic length of the elastic half-plane. The size effects due to the characteristic length of the half-plane and the implications of the generalized contact conditions are illustrated and discussed.

The problem of modeling the spatial distribution of the magnetic field in a ferromagnetic medium with nonlinear characteristics is solved. For the calculation, it is proposed to use block elements in the form of polygons of arbitrary shape. Discretization of the considered boundary value problems is performed by partitioning the computational domain into block elements and linearizing the material equation within each element. The distribution of the field potential inside each block element is proposed to be specified using the fundamental solutions of the Laplace equations and their conjugate functions. The algorithm for solving the boundary condition inversion problem in each block element is given, the application of which allows to set the coupling equations for the potential and the normal derivative on the element boundary. It is proposed to replace block elements with multipoles described by equations with undefined conductance matrices. The conductance matrices that define the relations between magnetic fluxes and voltages are determined by inverting the boundary conditions in each block element by the results of solving boundary value problems formulated in a special way. The analysis procedure of the magnetic circuit formed by multipoles, the connection of which corresponds to the accepted partition of the computational domain, is described. The algorithm for numerical solving an overdetermined system of linear algebraic equations arising in the analysis of a magnetic circuit is constructed. The example of calculating the fluxes distribution is given and the error is estimated.

A three-dimensional finite element model has been developed to simulate the MIG P/S welding process of two aluminum plates. The finite element calculations are performed using ANSYS finite element code, which takes into account the thermal and mechanical non-linear material properties. The results of finite element analysis compared with those of experiment to show its validity in view of distortions. Parametric studies are carried out on the validated model to assess the effects of various factors on the final residual distortion. Large deformations, temperature dependent material properties are included in the model. Finally, the formulas of fitting curves of angular distortion transverse shrinkage, and longitudinal shrinkage have been proposed.

This paper presents a method for the analysis of two-dimensional crack problems using the boundary element method. In the present method, a domain to be analyzed is divided into two regions : the near-tip region, which is circular in shape and centered at the crack tip, and the outer region. The displacements and stresses in the near-tip region are represented by the truncated infinite series of the eigenfunction with unknown coefficients. The stress intensity factor is represented by one of the coefficients. The outer region is formulated by the boundary element method. The system of linear algebraic equations with respect to nodal displacements, nodal tractions and the coefficients of the series are constructed by discretized boundary integration in the outer region and the continuity conditions. To test the accuracy of the present method the center slant cracked rectangular plate under a uniform tension has been analyzed. The isoparametric linear elements are used. The calculated dimensionless stress intensity factors are in good accuracy.

In this work, we propose a new decomposed subspace reduction (DSR) method for reduced-order modeling of fracture mechanics based on the integrated singular basis function method (ISBFM) with reproducing kernel approximation enriched by crack-tip basis functions. It is shown that the standard MOR approach based on modal analysis (ISBFM-MA) with a direct employment of the crack-tip enrichment functions yields an inappropriate scaling effect to the stiffness matrix, and results in the loss of essential crack features and the erroneous representation of inhomogeneous Dirichlet boundary conditions in the reduced subspace. On the other hand, the solution of ISBFM-DSR is not affected by the arbitrary scaling of the enrichment functions, and it properly captures the singularity and discontinuity properties of fracture problems in its low-dimensional reduced-order approximation. It is also shown that the inhomogeneous boundary conditions can be accurately represented in the ISBFM-DSR solution. Validations are given in the numerical examples.

Seismic performance of a square HSS column to H-section beam bolted connection with double cover plate

Concrete is a complex heterogeneous material, and thus, it is important to develop numerical modeling methods to enhance the prediction accuracy of the fracture mechanism. In this study, a two-dimensional mesoscale model is developed using a non-ordinary state-based peridynamic (NOSBPD) method. Fracture in a concrete cube specimen subjected to pure tension is studied. The presence of heterogeneous materials consisting of coarse aggregates, interfacial transition zones, air voids and cementitious matrix is characterized as particle points in a two-dimensional mesoscale model. Coarse aggregates and voids are generated using uniform probability distributions, while a statistical study is provided to comprise the effect of random distributions of constituent materials. In obtaining the steady-state response, an incremental and iterative solver is adopted for the dynamic relaxation method. Load-displacement curves and damage patterns are compared with available experimental and finite element analysis (FEA) results. Although the proposed model uses much simpler material damage models and discretization schemes, the load-displacement curves show no difference from the FEA results. Furthermore, no mesh refinement is necessary, as fracture is inherently characterized by bond breakages. Finally, a sensitivity study is conducted to understand the effect of aggregate volume fraction and porosity on the load capacity of the proposed mesoscale model.

Asymptotic analysis of Timoshenko-like orthotropic beam with elliptical cross-section

Regression modeling and prediction of road sweeping brush load characteristics from finite element analysis and experimental results