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National Advisory Committee for Aeronautics, Technical Notes - A Biharmonic Relaxation Method for Calculating Thermal Stress in Cooled Irregular Cylinders

A numerital method was developed for calculating thermal stresses
in irregular cylinders cooled.by one or more internal passages. The
use of relaxation methods and elementary methods of finite differences
was found to give approximations to the correct values when compared_
with previously known solutions for concentric circular cylinders
possessing symmetrical and asymmetrical temperature distributions.

The evolution of aircraft propulsion systems has led to the fre-
quent employment of cooled structures. The concentric hollow cylinder
is a familiar example, although in some cases irregular cylinders such
as cooled turbine blades with several internal passages (reference 1) v
are under consideration. A method of calculating stresses in thin-
walled turbine blades.of the air-cooled type is presented in refer-
ence 2, but the general problem of calculating thermal stresses in
long, hollow, thick—walled irregular cylinders has not been solved.

Several methods of calculating thermal stresses for various special
distributions of temperature in long hollow cylinders of particular
shape have been developed. Some theoretical aspects of the general -
problem have also been discussed. In reference 3 the problem is mm
regarded as an ordinary stress problem with given body and surface
forces replacing the effects due to temperature distribution, whereas
in reference 4 the equilibrium and boundary conditions of the theory
of elasticity are used without modifications to exhibit the temperature
effects as body and surface forces. The method of reference 4 was
applied to several special problems that had already been solved as
well as to problems of composite bodies and eccentric circular
cylinders. No application of analytical or numerical methods of
calculating thermal stresses in cooled irregular multiply connected
cylinders has been published.

An investigation was conducted at the NADA Lewis laboratory to
calculate thermal stresses in cooled irregular multiply connected
cylinders. The problem of thermal stresses in irregular cylinders
is formulated in a manner that permits solution by the use of finite—
difference methods. The contour integrals of reference 5 expressing
the single-valued character of the displacements for arbitrary circuits
around the internal boundaries are written in forms suitable for
numerical methods of differentiation and integration. The boundary
conditions based on the assumption of force-free boundaries and single-
valued_displacements are formulated in terms of derivatives of the
stress function as suggested.for uniform—temperature problems in
reference 5, and stress functions are set up in a manner that is an
extension to the thermal—stress problem of the work of reference 6
on doubly connected domains at uniform temperatures.