Saint-Venant and Karman equations for an orthotropic pre-stretched plate under tem-perature

Thin plates that are preliminarily stretched with the help of forces in their plane and attached to rigid ribs are used in space technology. In fire rescue technology, plate designs are being developed that repre-sent a life net supported by drones to cancel the energy of a person falling from a height during his evacua-tion 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 life net is pre-stretched onto a rigid contour. In this work the equations of B. Saint-Venant and T. Karman for an orthotropic plate are obtained, tak-ing 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 deformations continuity and non-linear equations of equilibrium. The form of models’ representation is differential. Examples of plate calculation for the action of a concentrated force and preliminary stretching are con-sidered. The plate continuum is replaced by a discrete region; differential ratios 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 stresses, the plate is pre-stretched. The bending surface be-comes 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 temperature changes showed that in this flexible plate of small thickness, the effect of temperature exposure is insignificant. The apparatus of the Karman theory is relatively complex in numerical implementation. 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 orthotropic plates.