Solution of the Dynamic Problem of Thermoelasticity in Stresses for a Rectangular Beam

A methodology for constructing a solution to the two-dimensional dynamic problem of thermoelasticity in stresses for a rectangular beam is proposed.  The system of stress equations describing the plane-deformed state of the beam under nonstationary thermal and mechanical loadings is chosen as the initial one.  The methodology is based on the approximation of the distributions of all components of the dynamic stress tensor by cubic polynomials along the thickness coordinate of the beam.  As a result, the system of initial two-dimensional nonstationary equations for these components is reduced to a system of one-dimensional nonstationary equations for the thickness-integral characteristics (analogous to forces and moments) of these components.  The re-approximation of their distributions along the transverse coordinate of the beam by cubic polynomials is used.  As a result, a system of ordinary differential equations with respect to the time variable was obtained for the integral characteristics of the analogs of forces and moments.  Taking into account the initial conditions for these integral characteristics, the general solutions of the Cauchy problems for the integral characteristics of forces and moments are found using the Laplace transform in the time variable.  The expressions of the components of the dynamic stress tensor and the stress intensity are derived from the expressions obtained in this way.  A criterion for evaluating the bearing capacity of a beam is proposed.  The resonant frequencies of four types for a beam are identified, which make it possible to study the resonant effects in a beam under nonstationary thermal and mechanical loadings.