Effects of elasticity, plasticity, and geometrical nonlinearity in problems of static and dynamic bending of plates

Abstract

There are many studies devoted to investigating static and dynamic processes of deformation and the limiting states in thin wall constructions (beams, plates, and shells) with using rigid plastic models [1][2][3][4][5][6]. However, the problems of applicability of rigid plastic models in the calculations of metal construc tions actually having elasto plastic properties are poorly studied. It is known that it is necessary for using the rigid plastic model [6] that the elastic deforma tion energy was an order of magnitude lower than the plastic deformation work and, consequently, the deformations should exceed the yield strength by an order of magnitude. For example, the plastic deforma tions should exceed 1% for steels and 3% for alumi num alloys. At such bending deformations of thin wall constructions, especially beams and plates, it is neces sary to take into account geometrically nonlinear effects of deformation, which were disregarded in pub lications [1-6] on the rigid plastic analysis known to authors.This work is devoted to the investigation of applica bility of the rigid plastic model by the example of the problems of the quasi static and dynamic bending of round plates for small and large deflections. The inves tigation is carried out on the basis of comparative anal ysis of the results of the numerical solution of prob lems on the bending of plates with a joint motionless and mobile support in various formulations: geometri cally linear and nonlinear with using the elasto plastic and rigid plastic models of deformation. The numeri cal method used underwent long term verification on a wide class of problems of statics and dynamics of plates and shells [7,8]. NUMERICAL SIMULATION OF PROCESSES OF DYNAMIC AND QUASI STATIC BENDING OF PLATESThe numerical solution of problems is carried out in the axisymmetrical formulation. The plate or shell is considered in the general cylindrical system of coordi nates roz. In addition, the system of coordinates s, ξ, connected to a deformable median surface is intro duced. The general and local system of orthogonal coordinates are related as , where are the direction cosines of the normal, and s is the arch length along the generatrix. The process of loading is divided into stages. Within each stage, the increment of displacements and defor mations is assumed as small. The current configura tion of the plate, which is transformed to a shell during the deformation, is represented as where r in = r in (s) and z in = z in (s) is the initial configu ration, u r = u r (s, t) and u z = u z (s, t) are the displace ments, and t is the time. The variation of the plate thickness in time is determined from the incompressibility condition. According to the theory of Timoshenko, the velocities of displacements are set in the form where is the angular velocity of rotation of the cross section, which is summed of the velocity of rota tion of the normal and the shear velocity .The deformation rates are calculated from the velocities of displacements in the current state met rics:r z z r s...

Buckling Analysis and Stability of Compressed Low‐Carbon Steel Rods in the Elastoplastic Region of Materials

“…e geometrical and material characteristics of the carbon steel (S235) columns with various grades are presented in Table 1. A diagram depicting the stress-strain relationship for carbon steel is constructed in Figure 4(a) according to Table 1, and a diagram depicting the slenderness ratio of critical stresses (equation (2)) for carbon steel is constructed in Figure 4(b), Curve 1 [9,15,21,28].…”