On the behaviour of infinitely long rigid-plastic beams under transverse concentrated load

This paper is concerned with the behaviour of an infinitely long uniform beam, made of a non-hardening plastic-rigid material, under the action of a single transverse concentrated load. The effects of transverse shearing deformation and rotatory inertia are assumed negligible, and the yield condition is assumed to depend only upon the bending moment. This problem is solved subject to the condition that the ratio of the impact velocity of the concentrated load to the time is non-increasing in time. The impact velocity is allowed to vary with the time. The associated problem concerning the motion of the beam following the removal of the concentrated load is also solved. Hitherto the only problem of the present type whose solution has been published is that involving constant impact velocity, but even there the solution of the unloading problem was not found. The general analysis of this paper is applied to solve completely the constant impact velocity problem, and some detailed results for other cases are also given. The motion, subject to the above restriction upon the impact velocity, is such that there are always a central, fixed yield-hinge and two lateral, outwards-moving yield-hinges, and is also such that the beam does not come to rest in any finite time.     

Viscoplastic deformation of a ring under the action of a concentrated load

International Applied Mechanics -     

A fluid-saturated porous medium under the action of a moving concentrated load

The problem of motion of a concentrated load along the surface of a fluid-saturated porous medium is studied for a subsonic range of speeds. An analytical solution is found. It is shown that there exists a critical speed equal to the speed of the Rayleigh-type surface waves in a porous elastic medium. If this critical speed is exceeded, then the behavior of the solution and the free surface shape are changed. The free surface shape is analyzed at different speeds.     

Elasto-plastic behaviour of double-curved shells under a concentrated load

The paper presents the application of the so-called geometrical elements method to the solution of the elasto-plastic behaviour of spherical shells subjected to an axisymmetrical concentrated load. The approach is based on the observation that during large deformations, the shell structure deforms in a nearly isometrical manner. The shell is sub-divided into elements of two kinds: purely-isometrically deformed elements and quasi-isometrically deformed elements. Equilibrium of the structure is defined by the stationariness of the total potential energy. The total energy is compared with Pogorelov's result for the same strain energy. The solution obtained defines the large deformation behaviour and motion of the plastic zones on the surface of the shell.A simplified solution for similar problems of the shells with double positive Gaussian curvature is also presented.     

Dynamics of ideal fibre-reinforced rigid-plastic beams

A mechanism is proposed by which discontinuities in slope can propagate along an ideal fibr-ereinforced beam which is inextensible in the direction of its axis. The equations of motion of the beam are formulated, including the dynamical conditions which must be satisfied at the discontinuity. Constitutive equations for a rigid-plastic fibre-reinforced beam are established, and it is shown that slope discontinuities may propagate in a strain-hardening material, but are stationary in a perfectlyplastic beam. The theory is illustrated by its application to the problem of a beam moving in a direction normal to its axis brought to rest by striking a rigid stop at its mid-point. It is shown that in the subsequent motion slope discontinuities travel outwards from the centre of the beam. A complete explicit solution is obtained for the case of a beam with linear strain-hardening.     

Quasi-static response of linear viscoelastic cantilever beams subject to a concentrated harmonic end load

This study evaluates the response of a uniform cantilever beam with a symmetric cross-section fixed at one end, and submitted to a lateral concentrated sinusoidal load at the free extremity. The beam material is assumed to be homogeneous, isotropic and linear viscoelastic. Due to the nature of the loading and the beam slenderness, large displacements are developed but the strains are considered small. Consequently, the mathematical formulation only involves geometrical non-linearity. It is also assumed that the beam is inextensible (neutral axis length is constant) and that inertial forces are negligible, i.e., dynamic effects are insignificant and the system can thus be modeled quasi-statically. The beam is therefore subject to oscillations caused by the sinusoidal time-dependent load, leading to a transient response until the material stabilizes and the system exhibits a periodic response, which can be conveniently described in the frequency domain. The time domain solution of this problem is elaborated by considering the quasi-static response for each time interval. The mathematical equations are presented in dimensional and dimensionless forms, and for the latter case, a numerical solution is generated and several case studies are presented. The problem is governed by a set of non-linear ordinary differential equations encompassing functions of space and time that relate the curvature, rotation angle, bending moment and geometrical coordinates. In this study, an elegant solution is deduced using perturbation theory, yielding a precise steady-state solution in the frequency domain with considerable computational economy. The solutions for both time and frequency domain methods are developed and compared using a case study for a series of dimensionless parameters that influence the response of the system.     

Viscoelastic layer under the action of a moving load

Recently, problems concerning the dynamic behavior of imperfect continuous media under various types of actions have been widely investigated. The method of Laplace transformation is very convenient for describing physical processes concerning unsteady phenomena. In viscoelastic media two complications are added: the representation of the properties of a medium depending on time, and the inversion of the obtained solutions containing this additional complication. Certain approximate methods of inversion in the analysis of viscoelastic stresses are discussed in [1]. In [2, 3] a discussion is given for an effective method of constructing the solution of unsteady problems for finite and for infinite imperfect media using auxiliary functions, and a solution is presented for a half-space. Making use of the idea of the inversion of transforms, discussed in [4], in [5] a solution is obtained and a complete picture is presented for the dynamics of the variation of the stress field in a viscoelastic half-space. In the present study we consider the action of a normal moving load that is suddenly applied to the free surface of a viscoelastic layer. By Laplace and Fourier integral transformations we obtain a solution in the form of a uniformly converging series based on longitudinal and transverse waves reflected in the layer. By means of inverting the transforms by the method discussed in [4, 5], we obtain an exact solution for the stress field in the medium under investigation. We consider the special case of a viscoelastic medium of Boltzmann type, for which we obtain a numerical realization of the solution on a digital computer.     

Dynamic response of elastic layer on stiff foundation under time harmonic surface vertical concentrated load

In this paper, the line-load integral equation method proposed in reference [1] is first used for solving the elastodynamic problems. A set of one-dimensional regular integral equation is derived for calculating the dynamic response of elastic layer on stiff foundation under time harmonic surface vertical concentrated load. And the numerical solution of the integral equation is obtained.     

Bending of corner-supported rectangular plate under concentrated load

In this paper the solution for the bending of corner-supported rectangular plate under concentrated load at any point (α/2, η) of the middle line of the plate is given by means of a conception called modified simply supported edges and the method of superposition. Some numerical example is presented. The solution obtained by this method checks very nicely with what was obtained by G.T. Shih^[3] by means of spline finite element method when η=d/2. This shows that this method of solution is satisfactory.     

Large deflections of non-prismatic nonlinearly elastic cantilever beams subjected to non-uniform continuous load and a concentrated load at the free end

This work studies large deflections of slender,non-prismatic cantilever beams subjected to a combined loading which consists of a non-uniformly distributed continuous load and a concentrated load at the free end of the beam.The material of the cantilever is assumed to be nonlinearly elastic.Different nonlinear relations between stress and strain in tensile and compressive domain are considered.The accuracy of numerical solutions is evaluated by comparing them with results from previous studies and with a laboratory experiment.     

On the nonlinear stability of a truncated shallow spherical shell under a concentrated load

In this paper, the axisymmetric nonlinear stability of a clamped truncated shallow spherical shell with a nondeformable rigid body at the center under a concentrated load is investigated by use of the modified iteration method. The analytic formulas of second approximation for determining the upper and lower critical buckling loads are obtained. This paper was read at The Third East China Symposium on Solid Mechanics, Jiuhuashan, October, 1986.     

Axisymmetric bending for thick laminated circular plate under a concentrated load

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