tailieunhanh - A numerical method for shallow shell vibration and stability problems

On the base of the integral representation of displacement functions through Green's functions the author has proposed a numerical method for solving the differential equations of the problem. These equations were solved approximately after producing them into linear algebraic equations by finite difference technique. | Jou;nal of Mechanics, NCNST of Vietnam T. XVI, 1994, No 2 {7 - 12) A NUMERICAL METHOD FOR SHALLOW SHELL VIBRATION AND STABILITY PROBLEMS TRAN DUC CHINH Hanoi University of Civil Engineering §0. INTRODUCTION The stability and vibration problems of shallow shells have been studied by many scientists [1, 2]. The usual approaches for those Problems were based on the partial differential equations of high order with unknown functions being displacement w and stress ip functions. Integrating these equations by analytical method usually are too difficult because of the high order of the differential equations even if for bending problems [3]. On the base of the integral representation of displacement functions through Green's functions the author has proposed a numerical method for solving the differential equations of the problem. These equations were solved approximately after producing them into linear algebraic equations by finite difference technique. §1. GOVERNING EQUATIONS Vlasov's governing differential equations for thin shallow shell with variable curvatures in the form of the. three displacements (U, V, W) have been employed [4, 5) I I I where Ln, £12, . ,L33 ~linear differential operators of the shell, h- thickness of the shell; X 0 , YO, Zo - harmonic surface loads situated ·on the shell, m- density of the mass for an unit- area, E - Young's modulus, r; - Poisson's coefficient. For convenience in integration and computation, the dimensionless cartesian coordinates are used. In the case of free vibration ¥o = Yo = Zo = 0. The three displacements in the governing equations are assumed in the form u(X, Y, t) = u(X, Y) sinwt, ii(X, Y, t) = v(X, Y) sinwt, w(X, Y, t) = w(X, Y) sinwt. 7 () Substituting the aboves into the governing equations for free vibration of the shells gives + L,z(v) + LB(w) = >.u; Lz,(u) + Lzz(v) + Lz3(w) = >.v; L31(u) + L3z(v) + L33(w) = >.w. Lu(u) () In the Case of elastic stability the governing equations of the .