Nonlinear Rapid Heating of Shallow Arches

The analysis of large amplitude thermally induced vibrations of a shallow curved beam is considered in this article. The beam is made from an isotropic homogeneous material. One surface of the arch is subjected to rapid surface heating while the other surface is kept at reference temperature. The temperature profile is evaluated by means of the 1D heat conduction equation across the thickness of the beam. The thermally induced moments and forces are evaluated from the temperature profile. These resultants are inserted into the equations of motion of the arch. To establish the equations of motion, kinematic assumptions of a shallow arch with considerations of the von Karman type of geometrical nonlinearity, Euler-Bernoulli beam assumptions, and linear stress-strain-temperature law are employed. The equations of motion are discreted using the conventional polynomial Ritz formulation. The resulting equations are nonlinear and at each time step the Newton-Raphson technique is used to obtain the solution. The equations of motion of the arch are traced in time domain using the Newmark time marching scheme, which results in the temporal evolution of lateral displacement at arbitrary point of the arch. Numerical results are provided to explore the effects of different parameters. It is shown that thermally induced vibrations indeed exist especially for thin arches. As the beam becomes thicker or shorter, the thermally induced vibrations fade up and for sufficiently thick arches, quasi-static and dynamic responses of the arch become identical.