Saint-Venant and Karman equations for orthotropic prestretched plate when exposed to temperature

Abstract

In space technology, thin plates are used, which are preliminarily stretched with the help of forces in its plane and attached to rigid ribs. In fire rescue technology, plate designs are being developed that represent a tension fabric supported by drones to extinguish the energy of a person falling from a height, during his evacuation both from a high-rise object and in other exceptional cases. The plates are thin and usually consist of a composite material. Shear forces predominate as loads; to reduce deflection, the fabric is prestretched onto a rigid contour. In this work, the equations of B. Saint-Venant and T. Karman for an orthotropic plate are obtained, taking into account the temperature increment. The former are the equations of equilibrium in displacements with initial forces, and the latter are a system of non-linear equations of the continuity of deformations and non-linear equations of equilibrium. The form of representation of models is differential. Examples of calculation of a plate for the action of a concentrated force and preliminary tension are considered. The plate continuum is replaced by a discrete region; differential relations are replaced by finite-difference analogs. Nonlinear equations were solved by iterations. The calculation of a thin plate for the action of a concentrated force showed that the resulting longitudinal forces are so large that the stresses are two to three orders of magnitude higher than the stresses allowed for the considered orthotropic material. To reduce this effect, the plate is pre-stretched. The bending surface becomes more monotonous, the deflection decreases, which leads to a decrease in the stress level. Comparison of calculations obtained from the action of a concentrated force and a change in temperature showed that in this flexible plate of small thickness, the effect of temperature exposure is insignificant. The apparatus of the Karman theory is relatively difficult to implement numerically. The mixed form of the model in stresses and displacements requires additional studies of the convergence of solutions. The Saint-Venant deformation model, as a model of a flexible plate with a small deflection, makes it possible to solve the problems of ensuring the rigidity and strength of a complex longitudinal-transverse bending of an orthotropic plate.