Study of stress concentrations in a shell under local forces by the transition matrix method

Spacecraft structures like shells connected by frames are inevitably loaded by local forces that may cause dangerous stress concentrations. Universal numerical techniques like finite difference and finite element methods provide results with unpredictable errors in these areas. In this paper, an analytical method for studying stress concentrations in shells is first proposed. Differential equations of the mechanics of shells are solved analytically in terms of Cauchy-Krylov special functions. The property of these functions to satisfy arbitrary initial conditions is used to construct an algorithm that reduces a boundary value problem to an initial problem. Another property of Cauchy-Krylov functions, that is, to represent a solution multiplicatively also forms the basis of this algorithm. By way of example, the critical parameters of a cylindrical shell, which determine the upper bound for stable calculations without orthogonalization and normalization of the solution, are computed. The results of studies will be useful when designing spacecrafts to decrease their weight.

References

[1] Karmishin A.V., Liaskovets V.A.,Miachenkov V.I., Frolov A.N. Statika i dinamika tonkostennykh obolochechnykh konstruktsii [Statics and dynamics of thin-shell structures]. Moscow, Mashinostroenie publ., 1975. 375 p.

[2] Vinogradov Yu.I. A multiplicative method of solving boundary value problems of shell theory. Journal of Applied Mathematics and Mechanics, 2013, vol. 77, issue 4, pp. 445–451.

[3] Vinogradov Yu.I. A method of solving linear ordinary differential equations. Dokl Ross Akad Nauk, 2006, no. 409(1), pp. 15–18.

[4] Vinogradov A.Yu, Vinogradov Yu.I. A method of transferring the boundary conditions of Cauchy-Krylov functions for stiff linear ordinary differential equations. Dokl Ross Akad Nauk, 2000, no. 373(4), pp. 474–476.

[5] Vinogradov A.Yu, Vinogradov Yu.I. Cauchy-Krylov functions and algorithms for solving boundary value problems in mechanics of shells. Doklady Physics, 2000, vol. 45, issue 11, pp. 620–622.

[6] Vinogradov A.Iu., Vinogradov Iu.I.,Gusev Iu.A., Kliuev Iu.I. Metod resheniia dvukhtochechnoi kraevoi zadachi s ispol’zovaniem funktsii Koshi-Krylova [Method for solving two-point boundary problem using Cauchy function-Krylov]. Izvestiia rossiiskoi akademii nauk. Mekhanika tverdogo tela [Mechanics of Solids]. 2003, no. 2, pp. 150–157.

[7] Beliaev A.V., Vinogradov Iu.I. Modifikatsiia metoda Godunova resheni ia kraevykh zadach teor i i obolochek [Modification of Godunov’s method for solving boundary value problems in the theory of shells]. Inzhenernyi vestnik [Engineering Gazette]. 2013, no. 7. Available at: http://engbul.bmstu.ru/doc/597785.html (accessed 20 October 2013).