Elastic stress analysis of circular cylinders

Abstract

When two cylinders of different elastic properties are joined together and the compound cylinder thus formed is subjected to external loading, the stresses at and in the neighborhood of the joint or the interface get affected due to the mutual restraint of the two cylinders. The elastic stress analysis of these types of problems is of interest in thermal stress analysis, in the stress analysis of adhesive joints between materials of different elastic properties, and numerous other engineering applications in which the stress and displacement distributions at the interface of two cylinders are of concern.

The characteristic difficulty of the problem of stress distribution in composite (compound) media is that neither the stress nor the displacement is specified at the interface(s).

The method of approach given by Sundara Raja Iyengar for the stress analysis of composite infinite strips, and the method used for solving the problem of an infinite compound bar by the above author and Alwar, can be adopted for solving such problems. The stresses at the interface have been assumed as Fourier integrals, and the unknown functions involved have been determined using continuity conditions at the interface. This method has also been used for finding stresses in a laminated circular cylindrical shell by Sundara Raja Iyengar and Yogananda, as indicated earlier.

The method adopted for the solution of this type of problem is as follows: The unknown normal and shear stresses at the interface are assumed in the form of Fourier series or integrals with unknown functions. The composite body is virtually split into component bodies, and the stress functions are written for each of these bodies, satisfying the boundary conditions, including those normal and shear stresses assumed at the interface. The unknown functions in these assumed quantities are found using displacement continuity conditions at the interface.

Based on the above method, an elasticity solution is presented below for an infinite compound cylinder subjected to uniform tension at the ends (Figs. 4.1a and 4.1b). It is assumed that there is perfect adhesion between the two cylinders at the interface. Though the two cylinders have different elastic properties, each of the cylinders is assumed to be homogeneous and isotropic. The curved boundaries are assumed to be free of any loading. Two different values have been assumed for the elastic constants: Young抯 modulus and Poisson抯 ratio. In Chapter 3, the effect of Poisson抯 ratio on the stress and displacements in a semi-infinite cylinder subjected to end loads has been given. The assumption of assuming Poisson抯 ratio as zero for practical problems has been justified.

The application of the above problem to the anchorage zone stress problem in post-tensioned prestressed concrete beams has been discussed. It is observed that a spalling zone is developed if the radial stress is taken as the relevant transverse stress, corresponding to _r in the three-dimensional rectangular prism solution of Sundara Raja Iyengar and Prabhakara. Comparison has been made with some existing theoretical solutions. If the area of the loading is taken as corresponding to that of the solid cylinder, the results agree fairly well with the exact three-dimensional solution.

In Chapter 4, an elasticity solution has been presented for the analysis of an infinite composite cylinder subjected to uniform tension at the ends. Detailed numerical work has been done only for five typical Young抯 modulus ratios, and the convergence of the series, effect of Young抯 modulus ratio, and Poisson抯 ratio have been discussed.

It is noted here that the elasticity analysis presented for the infinite composite cylinder is applicable to the analysis of thermal stresses in composite cylinders.

A few elasticity problems which are not solved are mentioned here:

The study of the effect of Poisson抯 ratio on a finite solid cylinder would be useful.

The problem of a finite composite cylinder can be solved by using the same method as that used in Chapter 4.

The solution can also be extended to: a. infinite hollow composite cylinder b. finite hollow composite cylinder c. solid composite cylinder when cylinders of two different radii are joined.