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THEORY OF ELASTICITY
B. N. FRADLIN and S. M. SHAKHNOVSKII
CONSTRUCTION OF THE GREEN TENSOR FOR THE PROBLEM OF EQUILIBRIUM OF A SHALLOW SHELL OF DOUBLE CURVATURE
(Presented by Academician Yu. N. Rabotnov, 9 VII 1959)
- As is known \((^{1,3})\), by means of N. A. Kil’chevskii’s method the system of functional equations for a shallow shell rectangular in plan, in displacements, under arbitrary boundary conditions, is represented in the form*
\[ u_{(i)\beta}(M,N)=v_{(i)\beta}(M,N)- \]
\[ -\int_0^a\int_0^b K^j_{(i\beta)}(Q,M)\,u_{(i)j}(Q,N)\,dx_Qdy_Q +A_{(i)\beta}(M,N). \tag{1} \]
Using the arbitrariness of the regular part of the auxiliary displacements, we choose it so that the operators \(A_{(i)\beta}(M,N)\) vanish. For this purpose the auxiliary displacements \(v^*_{(i)\beta}\) must satisfy the conditions of fastening of the shell contour.
It may be verified that, when the shell contour is rigidly fastened, the desired displacements \(v^*_{(i)\beta}\) may be given the form**
\[ v^*_{(i)\beta}(M,N)=v_{(i)\beta}(M,N)+\sum_{m,n} z^\beta_{mn}(M)\,Z^i_{mn}(N); \tag{2} \]
\[ v_{(i)\beta}(M,N)=\sum_{m,n} A^{(i)\beta}_{mn}\,Z^\beta_{mn}(M)\,Z^i_{mn}(N), \tag{3} \]
where
\[ Z^1_{mn}(M)=\cos\frac{m\pi x_M}{a}\sin\frac{n\pi y_M}{b},\qquad Z^2_{mn}(M)=\sin\frac{m\pi x_M}{a}\cos\frac{n\pi y_M}{b}, \]
\[ Z^3_{mn}(M)=\sin\frac{m\pi x_M}{a}\sin\frac{n\pi y_M}{b}; \]
\[ A^{(1)1}_{mn}=\frac{4\varepsilon}{\pi^2Eh}\frac{\gamma_{mn}}{\omega^2_{mn}},\qquad A^{(2)2}_{mn}=\frac{4\varepsilon}{\pi^2Eh}\frac{\delta_{mn}}{\omega^2_{mn}},\qquad A^{(3)3}_{mn}=\frac{48(1-\nu^2)\varepsilon a^2}{\pi^4Eh^3}\frac{1}{\omega^2_{mn}}, \]
\[ A^{(1)2}_{mn}=A^{(2)1}_{mn}=-\frac{4(1+\nu)^2\varepsilon^2}{\pi^2Eh}\frac{mn}{\omega^2_{mn}},\qquad A^{(\alpha)3}_{mn}=A^{(3)\alpha}_{mn}=0; \]
\[ \gamma_{mn}=(1-\nu^2)m^2+2(1+\nu)\varepsilon^2 n^2,\qquad \omega_{mn}=m^2+\varepsilon^2 n^2, \]
\[ \delta_{mn}=2(1+\nu)m^2+(1-\nu^2)\varepsilon^2 n^2,\qquad \varepsilon=\frac{a}{b}; \]
* Here and below \(i,j,\beta,\gamma=1,2,3;\ \alpha=1,2;\ m,n,k,l=1,2,\ldots,\infty\).
** It can be shown that the operations performed subsequently on the series (2) and (3) are valid in accordance with the theory of generalized functions.
\[ z_{mn}^{1}(M)=\frac{\beta_m x_M-a}{a}\sin\frac{n\pi y_M}{a},\qquad z_{mn}^{2}(M)=\frac{\beta_n y_M-b}{b}\sin\frac{m\pi x_M}{a}, \]
\[ z_{mn}^{3}(M)=\psi_m(x_M)\left[\psi_n-(y_M)-\sin\frac{n\pi y_M}{b}\right] -\psi_n(y_M)\sin\frac{m\pi x_M}{a}; \]
\[ \psi_m(x_M)=\frac{m\pi}{a^3}x_M(x_M-a)(\alpha_m x_M-a), \]
\[ \psi_n(y_M)=\frac{n\pi}{b^3}y_M(y_M-b)(\alpha_n y_M-b), \]
\[ \alpha_k=1+(-1)^k,\qquad \beta_k=1-(-1)^k. \]
On the basis of what has been said, system (1) is transformed into the system of integral equations
\[ u_{(i)\beta}(M,N)=v_{(i)\beta}^{*}(M,N) -\int_0^a\int_0^b K_{(\beta)}^{*j}(Q,M)u_{(i)j}(Q,N)\,dx_Qdy_Q, \tag{4} \]
whose kernels can be represented in the form
\[ K_{(\beta)}^{*j}(Q,M)=\sum_{m,n} f_{mn}^{(\beta)j}(Q)Z_{mn}^{\beta}(M). \tag{5} \]
We do not write out the expressions for the functions \(f_{mn}^{(\beta)j}(Q)\) because of their unwieldiness.
- We seek the solution of system (4) in the form
\[ u_{(i)\beta}(M,N)=\sum_{m,n}\left[E_{mn}^{(i)\beta}(N)Z_{mn}^{\beta}(M) +A_{mn}^{(i)\beta}Z_{mn}^{i}(N)z_{mn}^{\beta}(M)\right], \tag{6} \]
where the functions \(E_{mn}^{(i)\beta}(N)\) are to be determined.
Substituting (6) into (4) and introducing the notation
\[ R_{mnkl}^{(\beta)j}=\int_0^a\int_0^b f_{mn}^{(\beta)j}(Q)Z_{kl}^{j}(Q)\,dx_Qdy_Q, \qquad T_{mnkl}^{(\beta)j}=\int_0^a\int_0^b f_{mn}^{(\beta)j}z_{kl}^{j}(Q)\,dx_Qdy_Q, \]
we obtain the following infinite system of equations for the unknown functions:
\[ E_{mn}^{(i)\beta}(N)+\sum_j\sum_{k,l}R_{mnkl}^{(\beta)j}E_{mn}^{(i)\beta}(N) = A_{mn}^{(i)\beta}Z_{mn}^{i}(N) -\sum_j\sum_{k,l}A_{kl}^{(i)j}T_{mnkl}^{(\beta)j}Z_{mn}^{i}(N). \tag{7} \]
- In the case of hinged fastening of the shell contour, one should take
\[ v_{(i)\beta}^{*}=v_{(i)\beta},\qquad K_{(\beta)}^{*j}(Q,M)=K_{(\beta)}^{j}(Q,M) =\sum_{m,n}B_{mn}^{(\beta)j}Z_{mn}^{j}(Q)Z_{mn}^{\beta}(M), \tag{8} \]
where
\[ B_{mn}^{(\alpha)1}=B_{mn}^{(\alpha)2}=0,\qquad B_{mn}^{(1)3}=-\frac{4\varepsilon k_1}{\pi a}\frac{m\alpha_{mn}}{\omega_{mn}^{2}}, \qquad B_{mn}^{(2)3}=-\frac{4\varepsilon^2 k_1}{\pi a}\frac{n\beta_{mn}}{\omega_{mn}^{2}}, \]
\[ B_{mn}^{(3)1}=-\frac{48\varepsilon\rho_1 k_1a}{\pi^3h^2}\frac{m}{\omega_{mn}^{2}}, \qquad B_{mn}^{(3)2}=-\frac{48\varepsilon^2\rho_1 k_1a}{\pi^3h^2}\frac{n}{\omega_{mn}^{2}}, \qquad B_{mn}^{(3)3}=\frac{48\varepsilon\rho_3 k_1a}{\pi^4h^2}\frac{1}{\omega_{mn}^{2}}, \]
\[ \alpha_{mn}=\rho_1m^2-(\chi-\nu-2)\varepsilon^2n^2,\qquad \beta_{mn}=[1-(\nu+2)\chi]m^2-\rho_2\varepsilon^2n^2, \]
\[ \rho_1=1+\nu\chi,\qquad \rho_2=\nu+\chi,\qquad \rho_3=1+2\nu\chi+\chi^2,\qquad \chi=\frac{k_2}{k_1}. \]
In this case the solution of the problem is considerably simplified, and the components of the Green tensor are represented in the form
\[ u_{(i)\beta}(M,N)=\sum_{m,n}D_{mn}^{(i)\beta}Z_{mn}^{(\beta)}(N)Z_{mn}^{i}(M), \tag{9} \]
where the coefficients \(D_{mn}^{(i)\beta}\) are determined from the system of equations
\[ D_{mn}^{(i)\beta}+\frac{a^{2}}{4\varepsilon}\sum_j B_{mn}^{(\beta)j}D_{mn}^{(i)j}=A_{mn}^{(i)\beta}. \tag{10} \]
As a result of the calculations we obtain
\[ D_{mn}^{(1)1}=\frac{4\varepsilon}{\pi^{2}Eh}\frac{1}{\omega_{mn}^{2}} \left(\gamma_{mn}+\frac{Cm^{2}\alpha_{mn}^{2}}{\Omega_{mn}}\right), \qquad D_{mn}^{(2)2}=\frac{4\varepsilon}{\pi^{2}Eh}\frac{1}{\omega_{mn}^{2}} \left(\delta_{mn}+\frac{C\varepsilon^{2}n^{2}\beta_{mn}^{2}}{\Omega_{mn}}\right), \]
\[ D_{mn}^{(3)3}=\frac{4\varepsilon}{\pi^{2}Eh}\frac{C}{k_{1}^{2}a^{2}}\frac{1}{\omega_{mn}^{2}} \left(1-\frac{C\theta_{mn}^{2}}{\Omega_{mn}}\right), \]
\[ D_{mn}^{(1)2}=D_{mn}^{(2)1} =-\frac{4\varepsilon^{2}}{\pi^{2}Eh}\frac{mn}{\omega_{mn}^{2}} \left[(1+\nu)^{2}+\frac{C\alpha_{mn}\beta_{mn}}{\Omega_{mn}}\right], \]
\[ D_{mn}^{(1)3}=D_{mn}^{(3)1} =\frac{4\varepsilon C}{\pi Ehk_{1}a}\frac{m\alpha_{mn}}{\Omega_{mn}}, \qquad D_{mn}^{(2)3}=D_{mn}^{(3)2} =-\frac{4\varepsilon^{2}C}{\pi Ehk_{1}a}\frac{n\beta_{mn}}{\Omega_{mn}}, \]
where
\[ \Omega_{mn}=\omega_{mn}^{4}+C\theta_{mn}^{2}, \qquad \theta_{mn}=\varkappa m^{2}+\varepsilon^{2}n^{2}, \qquad C=\frac{12(1-\nu^{2})a^{4}k_{1}^{2}}{\pi^{4}h^{2}}. \]
The expression for the deflection of a shell under the action of a normal concentrated unit force, found by V. Z. Vlasov \((^{2})\), and the expressions for the components of displacement caused by the action of a tangential concentrated unit force, found by M. Mishonov \((^{4})\), coincide respectively with the components of the Green tensor \(u_{(3)3}\) and \(u_{(1)1}\), \(u_{(1)2}\), \(u_{(1)3}\), determined by formulas (9).
The results obtained indicate, in particular, the equivalence of the systems of integral and differential equations of shell equilibrium.
Kyiv Polytechnic Institute
Received
18 VI 1959
CITED LITERATURE
\(^{1}\) M. O. Kilchevsky, Collection of Works of the Institute of Mathematics, Academy of Sciences of the Ukrainian SSR, 8, 97 (1946).
\(^{2}\) V. Z. Vlasov, General Theory of Shells, 1949.
\(^{3}\) B. N. Fradlin, S. M. Shakhovsky, Izv. Acad. Sci. USSR, Mechanics and Machine Engineering, No. 1, 144 (1959).
\(^{4}\) M. Mishonov, Applied Mathematics and Mechanics, 22, issue 5, 691 (1958).