Full Text
Reports of the Academy of Sciences of the USSR
1964. Volume 158, No. 6
THEORY OF ELASTICITY
A. N. GUZ
ON THE SOLUTION OF PROBLEMS FOR A SHALLOW SPHERICAL SHELL IN THE CASE OF MULTIPLY CONNECTED DOMAINS
(Presented by Academician A. Yu. Ishlinskii on 6 V 1964)
In the work \((^1)\), for the study of the stress state in a shallow spherical shell in the case of multiply connected domains, a method of successive approximations was proposed; moreover, the authors restricted themselves only to the first approximation, which made it possible to determine at what distance the holes do not affect one another. Here a procedure is proposed for reducing problems on the stress state of shallow spherical shells in the case of multiply connected domains to infinite systems of algebraic equations.
§ 1. Consider the stress state of a shallow spherical shell which, in the plane, occupies an \((m+1)\)-connected domain \(S\), bounded by the circle \(L_0\) (Fig. 1). \(L_k\) are circles with centers to which the coordinate systems \((x_k, y_k)\) are attached; the coordinate system \((x,y)\) is attached to \(L_0\).
The study of the stress state reduces to the solution of the equation
\[ \nabla^2(\nabla^2-i\varkappa^2)\Phi=\frac{q r_0^4}{D} \tag{1} \]
under the boundary conditions
\[ \mathscr{L}_k^{(t)}\Phi\big|_{L_k}=f_{kz}(\theta_k); \tag{2} \]
\[ t=1,2,3,4;\quad z_k=x_k+iy_k;\quad z_k=r_k e^{i\theta_k};\quad k=0,1,\ldots,m;\quad z=z_0;\quad z=re^{i\theta}, \]
where \(\Phi=w+i g\varphi\); \(D\) is the cylindrical stiffness; \(q\) is the intensity of the normal load; all coordinates are dimensionless, referred to \(r_0\), and are related by the relations \(z=z_k+l_k\); \(g=\sqrt{12(1-\nu^2)/E^2h^4}\); \(\varkappa=r_0\sqrt[4]{12(1-\nu^2)/R^2h^2}\); \(\mathscr{L}_k^{(t)}\) are the differential operators of the boundary conditions.
The solution of equation (1) will be written in the form of the sum
\[ \Phi=\Phi_1+\Phi_2+\Phi_3; \tag{3} \]
\(\Phi_1\) is a solution of the Laplace equation, \(\Phi_2\) is a solution of the Helmholtz equation, and \(\Phi_3\) is a particular solution.
\[ \Phi_1=i\sum_{k=1}^{m}B_k\ln (z-l_k)(\bar z-\bar l_k)+\varphi(z)+\overline{\psi(z)};\quad \operatorname{Im}B_k=0, \tag{4} \]
where \(\varphi(z)\) and \(\psi(z)\) are functions holomorphic in \(S\), which, following \((^2)\), we represent in the form
\[ \varphi(z)=\sum_{k=1}^{m}\sum_{p=1}^{\infty}\frac{\alpha_{kp}}{(z-l_k)^p}+\sum_{p=0}^{\infty}\beta_p z^p;\quad \psi(z)=\sum_{k=1}^{m}\sum_{p=1}^{\infty}\frac{\alpha^*_{kp}}{(z-l_k)^p}+\sum_{p=0}^{\infty}\beta^*_p z^p. \tag{5} \]
Fig. 1
To determine the components of the stress and deformation states corresponding to \(\Phi_1\), in the coordinate system \((r_\mu,\theta_\mu)\), the relations obtained are:
\[ S^1_{r_\mu\theta_\mu}+iT^1_{r_\mu} = -\frac{1}{gr_0^2}z_\mu \frac{\varphi''(z)-\psi''(z)}{\bar z_\mu} + \frac{2i}{gr_0^2} \sum_{k=1}^{m} \frac{B_k z_\mu}{(z-l_k)^2\bar z_\mu}; \qquad T^1_{\theta_\mu}=-T^1_{r_\mu}; \]
\[ G^1_{r_\mu} = -D\frac{1-\nu}{r_0^2}\operatorname{Re} z_\mu \frac{\varphi''(z)+\psi''(z)}{\bar z_\mu}; \qquad G^1_{\theta_\mu}=-G^1_{r_\mu}; \]
\[ \widetilde Q^1_{r_\mu} = -\frac{2}{r_0 r_\mu}G^1_{r_\mu} + D\frac{1-\nu}{r_0^3 r_\mu}\operatorname{Re} z_\mu^2 \frac{\varphi'''(z)+\psi'''(z)}{\bar z_\mu}; \tag{6} \]
\[ u^1_\mu+iv^1_\mu = -\frac{1+\nu}{Ehgr_0} \left[ \overline{\varphi'(z)}-\overline{\psi'(z)} -2i\sum_{k=1}^{m}\frac{B_k}{z-l_k} \right] \frac{\bar z_\mu}{r_\mu} - \frac{r_0}{R}\frac{\bar z_\mu}{r_\mu} \int[\varphi(z)+\psi(z)]\,dz + i\frac{r_0}{R}\frac{\bar z_\mu}{r_\mu} z\,\operatorname{Im}(\beta_0+\beta_0^*). \]
We represent the solution of the Helmholtz equation for the domain \(S\) in the form
\[ \Phi_2 = \sum_{k=1}^{m}\sum_{p=0}^{\infty} \begin{pmatrix} a_{kp}+ib_{kp}\\ a_{kp}^*+ib_{kp}^* \end{pmatrix} H_p^{(1)}(r_k\chi\sqrt{-i}) \begin{matrix} \cos p\theta_k\\ \sin p\theta_k \end{matrix} + \sum_{p=0}^{\infty} \begin{pmatrix} a_p+ib_p\\ a_p^*+ib_p^* \end{pmatrix} J_p(r\chi\sqrt{-i}) \begin{matrix} \cos p\theta\\ \sin p\theta \end{matrix}; \tag{7} \]
\[ H_p^{(1)}(r_k\chi\sqrt{-i}) = \operatorname{her}_p r_k\chi+i\,\operatorname{hei}_p r_k\chi; \qquad J_p(r\chi\sqrt{-i}) = \operatorname{ber}_p r\chi+i\,\operatorname{bei}_p r\chi. \]
The components of the stress and deformation states corresponding to \(\Phi_2\) and \(\Phi_3\), in the coordinate system \((r_\mu,\theta_\mu)\), are calculated by the usual formulas for the polar coordinate system (3). The displacements corresponding to \(\Phi_2\) satisfy the single-valuedness conditions for displacements; assuming that these conditions are also satisfied by the displacements corresponding to \(\Phi_3\), from the single-valuedness conditions for displacements, taking (6) into account, we obtain \(\alpha_{k1}+\alpha_{k1}^*=0\).
Fig. 2
Thus the components of the stress and deformation state are determined in any coordinate system \((r_\mu,\theta_\mu)\). Having calculated, from these components, the quantities entering into the boundary conditions (2), we expand them in Fourier series on the \(\mu\)-th contour. When computing integrals containing the function \(\Phi_1\), one should pass to the domain of the complex variable (4), where they are computed rather easily by the residue theorem; and when computing integrals containing the function \(\Phi_2\), one must use the addition theorem for cylindrical functions (5). Equating to zero the coefficients of the harmonics on each of the contours, we obtain an infinite system of algebraic equations containing \(4(m+1)\) rows of undetermined constants.
§ 2. As an example, let us write the infinite system of algebraic equations for the problem of the stress state in a spherical shell around two equal circular holes that are free in the sense of (6), of radius \(r_0\), the distance between whose centers is equal to \(lr_0\), under uniform internal pressure of intensity \(q\). We shall assume the basic stress state to be momentless (6). The solution of this problem reduces to the solution in the infinite domain \(S\) (Fig. 2) of the homogeneous equation (1) (\(\Phi_3=0\)) under deter-
conditions “at infinity” (⁶) and the following boundary conditions:
\[ T_{r_\mu}\big|_{r_\mu=1}=-p_0h;\qquad S_{r_\mu\theta_\mu}\big|_{r_\mu=1}=0;\qquad G_{r_\mu}\big|_{r_\mu=1}=0;\qquad \widetilde Q_{r_\mu}\big|_{r_\mu=1}=-\frac{qr_0}{2}; \]
\[ p_0=\frac{qR}{2h};\qquad \mu=1,2. \tag{8} \]
By virtue of the force and geometric symmetry, \(\Phi_1\) and \(\Phi_2\) have the form
\[ \Phi_1(z)=\varphi(z)+\overline{\psi(z)};\qquad \psi(z)=\overline{\varphi(z)}; \]
\[ \varphi(z)=\sum_{p=1}^{\infty}\alpha_p \left[\left(z+\frac{l}{2}\right)^{-p} +(-1)^{-p}\left(z-\frac{l}{2}\right)^{-p}\right], \tag{9} \]
\[ \Phi_2=\sum_{p=0}^{\infty}(a_p+ib_p) \left[H_p^{(1)}(r_1\chi\sqrt{-i})\cos p\theta_1 +(-1)^pH_p^{(1)}(r_2\chi\sqrt{-i})\cos p\theta_2\right]. \]
From the conditions of single-valuedness of the displacements, \(\operatorname{Re}\alpha_1=0\). We note that \(\Phi_1\) and \(\Phi_2\) in (9) satisfy the conditions “at infinity” (⁶). Let us write out the infinite system for determining \(a_p,b_p,c_p\), and \(d_p\) \((\alpha_p=c_p+id_p)\):
\[ c_0=0;\qquad d_0=0;\qquad c_1=0;\qquad C_n^1=0;\qquad C_n^2=0;\qquad C_{np}^1=0;\qquad C_{np}^2=0; \]
\[ D_n^3=0;\qquad D_n^4=0;\qquad D_{np}^3=0;\qquad D_{np}^4=0; \]
\[ A_n^t a_n+B_n^t b_n+C_n^t c_n+D_n^t d_n +\sum_{p=0}^{\infty}\left(A_{np}^t a_p+B_{np}^t b_p+C_{np}^t c_p+D_{np}^t d_p\right) \]
\[ =-p_0\frac{R}{E}\delta_n^0\left(\delta_t^1+\chi\delta_t^4\right); \tag{10} \]
\[ t=1,2,3,4;\qquad n=0,1,\ldots,\infty;\qquad \delta_n^k= \begin{cases} 1, & n=k,\\ 0, & n\ne k; \end{cases} \qquad \varepsilon_n= \begin{cases} 1/2, & n=0,\\ 1, & n\ne 0. \end{cases} \]
Here the following notation has been introduced:
\[ A_n^1=\operatorname{her}_n\chi-\operatorname{hei}'_n\chi;\qquad A_n^2=n(\chi\,\operatorname{hei}'_n\chi-\operatorname{hei}_n\chi); \]
\[ A_n^3=(1-\nu)\operatorname{her}''_n\chi-\nu\,\operatorname{hei}_n\chi;\qquad A_n^4=\operatorname{hei}'_n\chi+n^2\frac{1-\nu}{\chi^2}\left(\chi\,\operatorname{her}'_n\chi-\operatorname{her}_n\chi\right); \]
\[ B_n^1=-\operatorname{hei}_n\chi-\operatorname{her}'_n\chi;\qquad B_n^2=n(\chi\,\operatorname{her}'_n\chi-\operatorname{her}_n\chi); \]
\[ B_n^3=-(1-\nu)\operatorname{hei}''_n\chi-\nu\,\operatorname{her}_n\chi;\qquad B_n^4=\operatorname{her}'_n\chi-n^2\frac{1-\nu}{\chi^3}\left(\chi\,\operatorname{hei}'_n\chi-\operatorname{hei}_n\chi\right); \]
\[ C_n^3=2\frac{1-\nu}{\chi^2}\,n(n+1);\qquad C_n^4=-2\frac{1-\nu}{\chi^3}\,n^2(n+1); \]
\[ D_n^1=-2\frac{n(n+1)}{\chi^2};\qquad D_n^2=-2n(n+1); \]
\[ A_{np}^1=\varepsilon_n\{[\operatorname{her}_{p+n}l\chi+(-1)^n\operatorname{her}_{p-n}l\chi] (\operatorname{ber}_n\chi-\operatorname{bei}'_n\chi) \]
\[ -[\operatorname{hei}_{p+n}l\chi+(-1)^n\operatorname{hei}_{p-n}l\chi] (\operatorname{bei}_n\chi+\operatorname{ber}''_n\chi)\}; \tag{11} \]
\[ A_{np}^2=n\varepsilon_n\{[\operatorname{her}_{p+n}l\chi+(-1)^n\operatorname{her}_{p-n}l\chi] (\chi\,\operatorname{bei}'_n\chi-\operatorname{bei}_n\chi) \]
\[ +[\operatorname{hei}_{p+n}l\chi+(-1)^n\operatorname{hei}_{p-n}l\chi] (\chi\,\operatorname{ber}'_n\chi-\operatorname{ber}_n\chi)\}; \]
\[ A_{np}^3=\varepsilon_n\{[\operatorname{her}_{p+n}l\chi+(-1)^n\operatorname{her}_{p-n}l\chi] [(1-\nu)\operatorname{ber}''_n\chi-\nu\,\operatorname{bei}_n\chi] \]
\[ -[\operatorname{hei}_{p+n}l\chi+(-1)^n\operatorname{hei}_{p-n}l\chi] [(1-\nu)\operatorname{bei}''_n\chi+\nu\,\operatorname{ber}_n\chi]\}; \]
\[ \begin{aligned} A_{np}^{4}={}&\varepsilon_n \Bigl\{[\operatorname{her}_{p+n} l\chi+(-1)^n\operatorname{her}_{p-n}l\chi] \Bigl[\operatorname{bei}'_n\chi+n^2\frac{1-\nu}{\chi^3} (\chi\operatorname{ber}'_n\chi-\operatorname{ber}_n\chi)\Bigr]+\\ &\quad+[\operatorname{hei}_{p+n}l\chi+(-1)^n\operatorname{hei}_{p-n}l\chi] \Bigl[\operatorname{ber}'_n\chi-n^2\frac{1-\nu}{\chi^3} (\chi\operatorname{bei}'_n\chi-\operatorname{bei}_n\chi)\Bigr]\Bigr\};\\ B_{np}^{1}={}&-\varepsilon_n\Bigl\{[\operatorname{her}_{p+n}l\chi+(-1)^n\operatorname{her}_{p-n}l\chi] (\operatorname{bei}_n\chi+\operatorname{ber}''_n\chi)+\\ &\quad+[\operatorname{hei}_{p+n}l\chi+(-1)^n\operatorname{hei}_{p-n}l\chi] (\operatorname{ber}_n\chi-\operatorname{bei}''_n\chi)\Bigr\};\\ B_{np}^{2}={}&n\varepsilon_n\Bigl\{[\operatorname{her}_{p+n}l\chi+(-1)^n\operatorname{her}_{p-n}l\chi] (\chi\operatorname{ber}'_n\chi-\operatorname{ber}_n\chi)-\\ &\quad-[\operatorname{hei}_{p+n}l\chi+(-1)^n\operatorname{hei}_{p-n}l\chi] (\chi\operatorname{bei}'_n\chi-\operatorname{bei}_n\chi)\Bigr\};\\ B_{np}^{3}={}&-\varepsilon_n\Bigl\{[\operatorname{her}_{p+n}l\chi+(-1)^n\operatorname{her}_{p-n}l\chi] [(1-\nu)\operatorname{bei}''_n\chi+\nu\operatorname{ber}_n\chi]+\\ &\quad+[\operatorname{hei}_{p+n}l\chi+(-1)^n\operatorname{hei}_{p-n}l\chi] [(1-\nu)\operatorname{ber}''_n\chi-\nu\operatorname{bei}_n\chi]\Bigr\};\\ B_{np}^{4}={}&\varepsilon_n\Bigl\{[\operatorname{her}_{p+n}l\chi+(-1)^n\operatorname{her}_{p-n}l\chi] \Bigl[\operatorname{ber}'_n\chi-n^2\frac{1-\nu}{\chi^3} (\chi\operatorname{bei}'_n\chi-\operatorname{bei}_n\chi)\Bigr]-\\ &\quad-[\operatorname{hei}_{p+n}l\chi+(-1)^n\operatorname{hei}_{p-n}l\chi] \Bigl[\operatorname{bei}'_n\chi+n^2\frac{1-\nu}{\chi^3} (\chi\operatorname{ber}'_n\chi-\operatorname{ber}_n\chi)\Bigr]\Bigr\};\\ C_{np}^{3}={}&2\frac{1-\nu}{\chi^2}l^{-p-n}\frac{(n+p-1)!}{(n-2)!(p-1)!};\qquad C_{np}^{4}=2\frac{1-\nu}{\chi^3}nl^{-p-n}\frac{(n+p-1)!}{(n-2)!(p-1)!};\\ D_{np}^{1}={}&-2\frac{l^{-p-n}}{\chi^2}\frac{(n+p-1)!}{(n-2)!(p-1)!};\qquad D_{np}^{2}=2l^{-p-n}\frac{(n+p-1)!}{(n-2)!(p-1)!}. \end{aligned} \]
In system (10), for \(n=0\) the second equation (\(t=2\)) drops out, while the first and fourth (\(t=1,4\)) coincide; for \(n=1\) the first and second equations (\(t=1,2\)) also coincide. Under these conditions, system (10) gives the solution of the stated problem.
Restricting ourselves to a finite number of equations in system (10), we obtain an approximate solution in any approximation.
In the case of the plane problem, systems of equations for several approximations were obtained by a similar method in (7). It turned out that the discrepancy in the first approximation, as compared with the practically exact solution, is small even when the distance between the holes is equal to half the radius of a hole.
Institute of Mechanics
Academy of Sciences of the Ukrainian SSR
Received
25 IV 1964
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- A. S. Kosmodamianskii, Some Problems of the Theory of Elasticity on Stress Concentration, abstract of doctoral dissertation, Kiev, 1963.