The theory of rolling given in the present paper is based upon the following assumptions:(1) There is no lateral spreading.(2) The material used is imconpressible,. isotropic, and viscous. It also obeys von Mises-Hencky's condition of yielding.(3) The elastic deformation of rolls is neglected.(4) The contact region between the material and rolls is divided into three parts: namely the forward slipping region, sticking region, and backward slipping region. The conditions of continuity of velocity and pressure are used.The results obtained from this theory agree very well with the experimental results given by Siebel and Lueg in 1933.
The theory of rolling given in the present paper is based upon the following assumptions:(1) There is no lateral spreading.(2) The material used is imconpressible,. isotropic, and viscous. It also obeys von Mises-Hencky's condition of yielding.(3) The elastic deformation of rolls is neglected.(4) The contact region between the material and rolls is divided into three parts: namely the forward slipping region, sticking region, and backward slipping region. The conditions of continuity of velocity and pressure are used.The results obtained from this theory agree very well with the experimental results given by Siebel and Lueg in 1933.
The equation for the large deflection of thin plates established by Th. von Karman has been well known for many years. But so far there are only a few problems been studied with numerical certainty. S. Levy was the first to apply this equation to solve the problem of a clamped plate under uniform pressure by the method of power series. After this, S. Levy got the solution of the simply supported rectangular plate also under uniform load by the method of double trigonometric series. These two methods used nearly the same procedure of determining the numerical value of the coefficents. But their numerical works are too cumbersome. Lately, W. Z. Chien treated Way's problem again by means of the perturbation method and obtained excellent results.In this paper, the problem of large deflection of a circular plate with a circular hole at the center is treated with the perturbation method.Recently, C. A. AлeKceeB worked out the same problem with the membrance theory, but his results differ greatly from the practical case of a thin plate. The reason of this is chiefly due to his neglecting the effect of bending. The results obtained in this paper are compared with those of AлeKceeв and discussed. We conclude that under concentrated load, the bending effect is momentous and therefore it cannot be neglected. The problem can be extended to other boundary conditions with the maximum deflection YM as parameter.
The equation for the large deflection of thin plates established by Th. von Karman has been well known for many years. But so far there are only a few problems been studied with numerical certainty. S. Levy was the first to apply this equation to solve the problem of a clamped plate under uniform pressure by the method of power series. After this, S. Levy got the solution of the simply supported rectangular plate also under uniform load by the method of double trigonometric series. These two methods used nearly the same procedure of determining the numerical value of the coefficents. But their numerical works are too cumbersome. Lately, W. Z. Chien treated Way's problem again by means of the perturbation method and obtained excellent results.In this paper, the problem of large deflection of a circular plate with a circular hole at the center is treated with the perturbation method.Recently, C. A. AлeKceeB worked out the same problem with the membrance theory, but his results differ greatly from the practical case of a thin plate. The reason of this is chiefly due to his neglecting the effect of bending. The results obtained in this paper are compared with those of AлeKceeв and discussed. We conclude that under concentrated load, the bending effect is momentous and therefore it cannot be neglected. The problem can be extended to other boundary conditions with the maximum deflection YM as parameter.