Nauka i Obrazovanie (Jan 2015)
Calculation of Thermal Strains and Stresses in Composite Conical Fairing
Abstract
Fairings in the form of thin-walled circular conical shells made from composite materials are a commonly used structural element of the cutting-edge rocketry. In the course of operation they are subjected to intense thermal and force loading. To calculate them the unnecessarily time consuming (laborious) methods using complicated finite element software systems are often applied. The article proposes an analytical method to calculate thermal strains and stresses in composite conically shaped fairing under uneven heating. The proposed method of calculation can be useful at the preliminary stage of design process. The article describes a design model. For its construction the theory of thin orthotropic conical shells under axisymmetric loading, based on the Kirchhoff-Love hypotheses, is used. The article considers uneven heating of the fairing, when the temperature varies nonlinearly along the slant height and across the thickness. The fairing is made from composite material with woven filler. Duhamel-Neumann relations describe thermo-elastic properties of the material. Boundary conditions are formulated. The system of resolving ordinary nonhomogeneous differential equations of the second order with respect to the rotation angle of the normal in the meridian direction and shearing forces is obtained. The system of equations is transformed to the Bessel equation via auxiliary complex function. Its order is determined by the ratio of the elasticity modulus along the weft to the modulus of elasticity along the warp. For the practically important case when this ratio is equal approximately to 0.5, the analytical solution of the homogeneous system of equations is obtained. This solution is expressed by elementary functions. The variant of a particular solution for the nonhomogeneous system of differential equations is proposed. For the "long" conical shell a simplified solution is given. Using the obtained relations for the displacements and internal forces, stress-strain state of composite fairing under uneven heating can be analyzed. It is noted that the numerical solution obtained by S.G. Godunov’s method of orthogonalization coincides with the analytical solution. The article considers an example of calculating displacement and stress in the composite fairing. It shows that for a given temperature field, compressive stresses on the outer surface of the composite fairing are of essential value. For the fairing made from composite material with woven filler they can cause failure. Axial and radial displacements are small as compared to the wall thickness of the fairing.
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