The Determination of Temperature, Stresses, and Deflections in Two-Dimensional Thermoelastic Problems

Abstract

An analytical successive-approximation method for the solution of linear partial differential equations is presented first in general terms, and then applied to the solution of two-dimensional heat and thermal stress problems. The method is applicable when solutions are desired for bars or plates—i.e., for bodies with one dimension small compared to the others. The final expressions given by this procedure for example for the stress aconsist of a number of terms (a= So-^), where the term at is proportional to the quantity [/3 d r/(d:x;)~']; (3 is the ratio of height to length of the bar, and x measures the distance along the span. The first term of the series in that corresponding to the assumption that sections plane before heating remain plane after heating; the solution obtained thus shows that this term provides a good approximation for thermal loadings varying smoothly along the span, and for thin bars. Similar results are obtained for the temperature and the deflections. Explicit formulas for the calculation of stresses and deflections are given. The validity of the Bernoulli-Euler hypothesis of beam-theory is examined. Illustrative examples are presented for all the above developments. The use of the method in problems in which the material properties are functions of the temperature is outlined.