Unilateral buckling of thin elastic plates by the boundary integral equation method

The unilateral buckling of thin elastic plates, according to Kirchhoff's theory, is studied by using a boundary integral method. A representation for the second member of the equation is given. In the matrix formulatiea, boundary unknowns are eliminated; therefore, the unilateral buckling problem reduces to compute the eigenvalues and the eigenvectors of a matrix depending on the contact zone with the rigid foundation. An iterative process allows this zone and the buckling load to be computed. The capacities of the proposed method are illustrated by four examples.     

Numerical convergence of simple and orthogonal polynomials for the unilateral plate buckling problem using the Rayleigh–Ritz method

Unilateral buckling is a contact problem whereby buckling is confined to take place in only one lateral direction. For plate structures, this can occur when a thin steel plate is juxtaposed with a rigid concrete medium and the steel may only buckle locally away from the concrete core. This paper investigates the use of simple and orthogonal polynomials in the Rayleigh–Ritz method for unilateral plate buckling. The orthogonal polynomials used are the classical Chebyshev types 1 and 2, Legrende, Hermite and Laguerre. The study presents a comparison between the efficiency of the polynomial‐based displacement functions with regard to elastic bilateral and unilateral plate buckling, where efficiency is measured as a function of their convergence characteristics. Some buckling solutions for plates with varying boundary conditions and in‐plane shear loads are also provided as an illustration. Copyright © 1999 John Wiley & Sons, Ltd.     

On the influence of the unilateral damage behaviour in the stability of cracked beam/columns

This paper deals with the buckling analysis of cracked beam-columns. The crack is modelled with a unilateral elastic bending-stiffness behaviour, represented by a unilateral rotational spring. This model takes into account the crack closure effect associated with the phenomenon of breathing crack. The rotational spring may model a one-sided crack (or a unilateral joint for other applications in biomechanics or civil engineering). The unilateral elastic constitutive law of the crack is introduced from a variational procedure based on energy arguments. It is shown that this constitutive law can be formulated as a function of a physically-based damage parameter. A one-crack and a two-crack hinged-hinged beam-column are theoretically investigated to illustrate the unilateral effect of the crack behaviour on the buckling load. It is shown that the open crack analogy can be adopted for the one-crack beam-column, whereas this simplified assumption cannot be retained for the two-crack beam-column. In this last case, the crack closure effect strongly affects the buckling load, with respect to the open crack analogy of the same problem. More specifically, the crack-closure phenomenon tends to increase the buckling load, with respect to the open crack assumption mainly retained in the literature.     

斜直井内钻柱的非线性螺旋屈曲

基于最小位能原理,研究了斜直井内钻柱的非线性螺旋屈曲问题。利用圆柱面坐标系,使接触非线性问题简化成了纯几何非线性问题。对于上端周向自由,下端铰支的钻柱,用Rayleigh-Ritz法得到了有一定横向载荷作用时,确定螺旋屈曲临界载荷无量纲形式的非线性积分方程。应用符号运算软件Maple10,对具体的工程算例进行了求解。研究发现:该文的计算结果与已有的实验结果吻合,说明钻柱的螺旋屈曲问题是非线性问题,同时也证实了该文的非线性算法的有效性。该方法可用于求解斜直井内的钻柱屈曲问题。     

Unilateral buckling of elastically restrained rectangular mild steel plates

 This paper considers the elastic unilateral buckling of rectangular mild steel plates that are restrained elastically and subjected to bending and axial actions. A variational formulation of the Ritz method using linear combinations of harmonic functions for the buckling deformations is used to establish an eigenproblem to determine the plate local buckling coefficients. The motivation for the study is the retrofit of reinforced concrete beams by gluing and then bolting steel plates to the sides of the beam. Such plates, when acting compositely with the concrete beam, are subjected to predominantly bending and axial actions which may cause unilateral local buckling. Whereas the bolts provide complete restraint against buckling at discrete points, the glue may also inhibit local buckling between these nodal points since it acts as a continuous elastic restraint. The influence of the glue stiffness, support conditions and plate proportions on the unilateral buckling of such plates are assessed.     

Existence of solutions of Signorini problems with friction

A linear elastic body in unilateral contact with a rigid support and subjected to conditions of Coulomb friction is considered. We prove that there exists a solution of this Signorini problem with friction [1]. We show also that if the coefficient of friction is “small”, then the problem has a unique solution.     

刚体撞击弹塑性直杆时冲击荷载之数值解

使用数值方法模拟了刚体撞击一端固支的弹塑性杆的全过程,得到不同质量比、冲击速度时的弹性和弹塑性冲击荷载,比较了冲击荷载的弹性解析解、弹性数值解和弹塑性数值解。结果表明:对于弹性冲击荷载,数值解和解析解非常相近且数值解有合理的升压段。在质量比不变的情况下,刚体速度和冲击荷载的峰值成线性关系,而对接触时间无影响。质量比的变化对冲击荷载的峰值和分布形式有影响。质量比越小,冲击荷载峰值越大,接触时间越长。对于弹塑性模型,杆中产生的塑性变形引起冲击荷载峰值减小和荷载曲线分布形式发生变化。塑性会延长接触时间,延长的接触时间就是塑性发展的时间。当速度越大,质量比越小,则杆中塑性发展时间越长,弹性和弹塑性杆冲击荷载的峰值和分布形式差别越大。     

A method of solution for the creep buckling of structural members of a linear visco-elastic material

Summary In this paper a method for solving creep buckling problems for structural members of a linear visco-elastic material is presented. The solution is found in the form of an expansion of the eigenfunctions of the corresponding elastic problem. The method can be used as either an analytic or a numerical technique.     

Buckling Analysis of Plates Stiffened by Parallel Beams

In this paper a general solution for the elastic buckling analysis of plates stiffened by arbitrarily placed parallel beams of arbitrary doubly symmetric cross section subjected to an arbitrary inplane loading is presented. According to the proposed model, the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. These tractions are integrated with respect to each half of the interface width resulting two interface lines, along which the loading of the beams as well as the additional loading of the plate is defined. The unknown distribution of the aforementioned integrated tractions is established by applying continuity conditions in all directions at the two interface lines, while the analysis of both the plate and the beams is accomplished on their deformed shape. The method of analysis is based on the capability to establish the elastic and the corresponding geometric stiffness matrices of the stiffened plate with respect to a set of nodal points. Thus, the original eigenvalue problem for the differential equation of buckling is converted into a typical linear eigenvalue problem, from which the buckling loads are established numerically. For the calculation of the elastic and geometric stiffness matrices six boundary value problems are formulated and solved using the Analog Equation Method (AEM), a BEM-based method. Numerical examples with practical interest are presented. The accuracy of the results of the proposed model compared with those obtained from a 3-D FEM solution is remarkable.     

Buckling of fibers in fiber-reinforced composites

Elastic stability of fibers in fiber-reinforced composite materials subject to compressive loading is studied. The transversal buckling mode is considered, and two limiting cases, the dilute and non-dilute composites are analyzed. In the case of a non-dilute composite, the cylindrical model and the lubrication approximation are applied. The original problem is reduced to a problem of stability of a rod on elastic foundation. Through the solution of this problem a simple formula for the buckling load is obtained. In the case of a dilute composite, the solution of a problem of stability of a compressed rod in elastic plane is used. On the basis of the obtained solutions in two limiting cases the interpolation formulae are derived. These formulae describe buckling of fiber in the fiber-reinforced composite for any value of the fiber volume fraction. Comparison with known numerical and experimental results is carried out, and the sufficient accuracy of the derived formulae is demonstrated.     

Buckling of triangular plates with elastic edge constraints

Summary This paper presents the first-known investigation on the buckling behavior of triangular plates with both translational and rotational elastic edge constraints. The energy functional of a general triangular plate with elastic edge supports is derived, and thep-Ritz method is utilized to derive the governing eigenvalue equation for the buckling problem. Convergence and comparison studies are carried out to verify the validity and accuracy of the solution method. Extensive buckling factors are presented for several selected isosceles and right-angled triangular plates of various edge support conditions and subjected to isotropic inplane compressive load. The influence of elastic edge supports on the buckling factors for triangular plates of various vertex angles (aspect ratios) and boundary conditions is examined.     

正弯矩区组合梁腹板局部稳定性研究

对正弯矩区组合梁腹板局部稳定性进行研究,首先要建立板件在拉、弯、剪共同作用下的相关屈曲方程。该文在前人研究的基础上,根据能量方法研究了四边简支板件在拉、弯、剪共同作用下的弹性相关屈曲方程,并利用通用有限元软件ANSYS建模进行计算,验证了理论推导结果,最终提出简化公式。随后在相关方程的基础上,推导出相应的弹性高厚比限值。并将该限值与未考虑弯拉作用的限值进行对比,结果有明显提高,多数情况下可放宽26%以上。该文成果为进一步研究正弯矩区组合梁腹板考虑残余应力和初始缺陷的弹塑性屈曲提供基 础。     

Localized direct boundary-domain integro-differential formulations for incremental elasto-plasticity of inhomogeneous body

A quasi-static mixed boundary value problem of incremental elasto-plasticity for a continuously inhomogeneous body is considered. Using the two-operator Green–Betti formula and the fundamental solution of a reference homogeneous linear elasticity problem, with frozen initial or tangent elastic coefficients, a boundary-domain integro-differential formulation of the elasto-plastic problem is presented, with respect to the displacement rates and their gradients. Using a cut-off function approach, the corresponding localized parametrix of the reference problem is constructed to reduce the elasto-plastic problem to a nonlinear localized boundary-domain integro-differential equation. Algorithms of mesh-based and mesh-less discretizations are presented resulting in sparsely populated systems of nonlinear algebraic equations for the displacement increments.     

Analysis of frictional contact problem using boundary element method and domain decomposition method

In this paper, we study the bilateral or unilateral contact with Coulomb friction between two elastic solids, using a domain decomposition method coupled with the boundary element method. The decomposition method we have selected is the Schur complement method, a non‐overlapping technique. It enables to reduce the solution of the global problem to the solution of a problem defined only on the contact surface. Moreover, its principal advantage is that computing is done separately on each solid. We have chosen to associate it with the boundary element method. Indeed, it only requires the discretization of the boundaries of solids. This technique of coupling reduces the number of unknowns and the time of computing. We have applied it to the study of indentation of an elastic foundation by an elastic flat punch and a sphere. In this last case, our results are in conformity with the Hertz theory and the analytical solution of Spence. Moreover, we have shown the influence of friction on the size of the contact radius and on the normal pressure at centre. Copyright © 1999 John Wiley & Sons, Ltd.     

Topology optimization of periodic microstructures with a penalization of highly localized buckling modes

The problem of determining highly localized buckling modes in perfectly periodic cellular microstructures of infinite extent is addressed. A double scale asymptotic technique is applied to the linearized stability problem for a periodic structure built from linearly elastic microstructures. The obtained stability condition for the microscale level is then used to establish a comparative analysis between different material distributions in the base cell subjected to the same strain field at the macroscale level. The idea is illustrated by some two‐dimensional finite element examples and used to design materials with optimal elastic properties that are less prone to localized instability in the form of local buckling modes at the scale of the microstructure. Copyright © 2002 John Wiley & Sons, Ltd.     

Indentation of an incompressible elastic film

The problem of impressing a rigid flat-ended cylindrical indenter onto an incompressible elastic film is considered. An integral transform solution is developed to reduce the solution of the problem to a Fredholm integral equation of the second kind with a symmetrical kernel depending on the boundary conditions (frictionless/bonded) in the areas of contact. The relationship between the applied load and the indentation depth is derived, which provides a guideline for measuring the elastic constants of thin films and determining the degree of adhesion between a thin film and a stiffer substrate.     

The normal aging viscoelastic contact problem

The problem of a mechanically smooth rigid indentor of arbitrary asymmetric curved profile being pressed against the surface of a linear viscoelastic half-space is reconsidered. A mechanical argument is used to remove restrictions and extend the range of validity of the solution given in [2] which is simplified and given for an aging material. Still, the shape of the contact area is always the same as that of some member of the one parameter family of contact areas associated with the corresponding elastic problem and the viscoelastic solution is expressed in terms of the allied elastic solutions.     

Elasto-plastic localised responses in one-dimensional structural models

This work complements recent developments concerning the buckling of beams lying on a nonlinear (non-convex) elastic foundation, and also reports on some investigations on the role of material nonlinearity. Two structural models are studied using a simple elasto-plastic constitutive relationship, and buckling problems are formulated as reversible fourth-order differential equations. It is demonstrated that modulated responses are possible under certain circumstances. Some numerical simulations are presented supporting the analytical findings.     

A linear model of elasto-plastic and adhesive contact deformation

Rigorous non-linear models of elasto-plastic contact deformation are time-consuming in numerical calculations for the distinct element method (DEM) and quite often unnecessary to represent the actual contact deformation of common particulate systems. In this work a simple linear elasto-plastic and adhesive contact model for spherical particles is proposed. Plastic deformation of contacts during loading and elastic unloading, accompanied by adhesion are considered, for which the pull-off force increases with plastic deformation. Considering the collision of a spherical cohesive body with a rigid flat target, the critical sticking velocity and coefficient of restitution in the proposed model are found to be very similar to those of Thornton and Ning’s model. Sensitivity analyses of the model parameters such as plastic, elastic, plastic-adhesive stiffnesses and pull-off force on work of compaction are carried out. It is found that by increasing the ratio of elastic to plastic stiffness, the plastic component of the total work increases and the elastic component decreases. By increasing the interface energy, the plastic work increases, but the elastic work does not change. The model can be used to efficiently represent the force-displacement of a wide range of particles, thus enabling fast numerical simulations of particle assemblies by the DEM.     

Local Buckling Delamination of a Rectangular Sandwich Plate Containing Interface Embedded Rectangular Cracks and Made From Elastic and Viscoelastic Materials

A three-dimensional buckling delamination problem for a rectangular sandwich plate made from elastic and viscoelastic materials is studied. It is supposed that the plate contains interface embedded rectangular cracks and that the edge-surfaces of these cracks have initial infinitesimal imperfections. The evolution of these initial imperfections with an external bi-axial compressed force (for the case where the materials of the layers of the plate are elastic) or with duration of time (for the case where the materials of the layers of the plate are viscoelastic) is investigated. The corresponding boundary value problem is formulated within the framework of the piecewise homogeneous body model with the use of three-dimensional geometrically nonlinear field equations of the theory of viscoelastic bodies. This problem is solved by employing boundary form perturbation techniques, Laplace transform and FEM. According to the initial imperfection criterion, the values of the critical parameters are determined. Numerical results on the critical force and critical time are presented and discussed. In particular, it is established that the values of the critical forces obtained for the buckling delamination around the rectangular embedded interface cracks are significantly greater than those obtained for the corresponding edge and band cracks.