Full Text
Reports of the Academy of Sciences of the USSR
1968. Volume 182, No. 3
UDC 539.3
THEORY OF ELASTICITY
L. S. SRUBSHCHIK
ASYMPTOTICS OF REISSNER’S EQUATIONS
IN THE NONLINEAR THEORY OF SYMMETRICALLY LOADED
SHELLS OF REVOLUTION
(Presented by Academician A. Yu. Ishlinskii on 31 I 1968)
- E. Reissner \((^{1})\) obtained more exact equations for the finite symmetric deformation of thin shallow shells of revolution, abandoning assumptions concerning any smallness of the angles of rotation of a shell element as a result of deformation:
\[ (\Phi-\Phi_0)''-(\Phi-\Phi_0)' \left(\frac{r_0}{\alpha_0}\right)' \bigg/ \frac{r_0}{\alpha_0} -\frac{\alpha_0^2\cos\Phi}{r_0^2}(\sin\Phi-\sin\Phi_0) + \]
\[ +\nu\,\frac{\alpha_0}{r_0}\Phi_0'(\cos\Phi-\cos\Phi_0) = \frac{\alpha_0^2}{r_0D}(\Psi_H\sin\Phi-\Psi_V\cos\Phi), \]
\[ \Psi_H''+\Psi_H' \left(\frac{r_0}{\alpha_0}\right)' \bigg/ \frac{r_0}{\alpha_0} - \left[ \frac{\alpha_0^2}{r_0^2}\cos^2\Phi -\nu\,\frac{\alpha_0}{r_0}\Phi'\sin\Phi \right]\Psi_H = \]
\[ = \frac{\alpha_0^2}{r_0}C(\cos\Phi-\cos\Phi_0) +\nu\,\frac{\alpha_0\sin\Phi}{r}\Psi_H' + \]
\[ + \left[ \frac{\alpha_0^2}{r_0^2}\cos\Phi\sin\Phi + \frac{\alpha_0}{r_0}\nu\Phi'\cos\Phi \right]\Psi_V - \frac{\alpha_0}{r_0}(r_0^2p_H)' -\nu\alpha_0^2p_H\cos\Phi, \tag{1} \]
\[ \Psi_V=-\int r_0\alpha_0p_V\,d\xi, \qquad C=Eh, \qquad D=\frac{Eh^3}{12(1-\nu^2)}, \qquad 0<\nu<0.5 . \]
Here the surface of the shell is specified by the parametric equations \(r_0=r_0(\xi)\), \(z_0=z_0(\xi)\); \(\Phi_0(\xi)\) is the angle which the shell element makes with the axis of abscissas before deformation at the corresponding point \(\xi\), \(\Phi(\xi)\) is the angle after deformation; \(\Psi_H\) and \(\Psi_V\equiv (r_0V)\) are, respectively, the horizontal and vertical components of stress; \(E\) is Young’s modulus; \(\nu\) is Poisson’s ratio; \(h=\mathrm{const}\) is the shell thickness; \(p_H\) and \(p_V\) are, respectively, the horizontal and vertical components of the load, which depend on the load intensity \(\rho(\xi)\) and the angle \(\Phi(\xi)\).
Introduce the geometric parameter \(\varepsilon\), characterizing the relative thinness of the shell: \(\varepsilon=h/d\gamma\), where \(\gamma^2=12(1-\nu^2)\) and \(d\) is the radius of the largest section perpendicular to the axis of revolution. We pass to dimensionless quantities. To this end we put
\[ r_0=rd,\qquad z_0=zd,\qquad \alpha_0=\alpha d, \]
\[ \Psi=\Psi_H/d^2\gamma E\varepsilon^k, \qquad p=p_H/E\gamma\varepsilon^k, \qquad q=p_V/E\gamma\varepsilon^k, \tag{2} \]
\[ T=\frac{\Psi_V}{d^2\gamma E\varepsilon^k} = -\int r\alpha q\,d\xi . \]
Substituting (2) into (1), we obtain Reissner’s system of equations written in dimensionless form
\[ \varepsilon^{3-k} \left\{ \frac{r}{\alpha}(\Phi-\Phi_0)'' + \left(\frac{r}{\alpha}\right)'(\Phi-\Phi_0)' - \frac{\alpha}{r}\cos\Phi(\sin\Phi-\sin\Phi_0) + \right. \]
\[ \left. +\nu\Phi_0'(\cos\Phi-\cos\Phi_0) \right\} = \alpha(\Psi\sin\Phi-T\cos\Phi), \]
\[
\varepsilon^{k-1}\left\{\frac{r}{\alpha}\Psi''+\left(\frac{r}{\alpha}\right)'\Psi'-
\left[\frac{\alpha}{r}\cos^2\Phi-\nu\Phi'\sin\Phi\right]\Psi\right\}=
\]
\[
=\alpha(\cos\Phi-\cos\Phi_0)+\varepsilon^{k-1}\nu\sin\Phi T'
-\varepsilon^{k-1}(r^2p)'
+\varepsilon^{k-1}\left[\frac{\alpha}{r}\cos\Phi\sin\Phi+\nu\Phi'\cos\Phi\right]T
-\nu\varepsilon^{k-1}r\alpha p\cos\Phi .
\tag{3}
\]
For example, let us prescribe boundary conditions (reduced to dimensionless form) corresponding to a clamped shell along the edge:
\[ \Phi=\Phi_0,\quad u\equiv d\varepsilon^{k-1}\left(\frac{r}{\alpha}\Psi'+r^2p-\nu\Psi\cos\Phi-\nu T\sin\Phi\right)=0 \quad \text{for } \xi=b, \tag{4} \]
and boundary conditions corresponding to an immovable hinged support:
\[ M_\xi\equiv Eh^2\varepsilon\left[\frac{\Phi'-\Phi_0'}{\alpha} +\nu\frac{\sin\Phi-\sin\Phi_0}{r}\right]=0;\qquad u=0\quad \text{for } \xi=b. \tag{5} \]
- For an approximate solution of equations (3), one can apply the Galerkin method in the form of P. F. Papkovich, analogously to how this is done within the framework of the semimembrane theory \((^{2,3})\). To this end, from the first equation (3) we find \(T\) and, substituting into the second equation, obtain:
\[
\varepsilon^{k-1}\left\{\frac{r}{\alpha}\Psi''+\left(\frac{r}{\alpha}\right)'\Psi'-\frac{\alpha}{r}\Psi\right\}
=\alpha(\cos\Phi-\cos\Phi_0)+\varepsilon^{k-1}\nu\sin\Phi\cdot T'
\]
\[
-\varepsilon^{k-1}(r^2p)'-\nu\varepsilon^{k-1}r\alpha p\cos\Phi
+\frac{\varepsilon^2}{\alpha}\left[\frac{r}{\alpha}(\Phi-\Phi_0)''
+\left(\frac{r}{\alpha}\right)'(\Phi-\Phi_0)'\right.
\tag{6}
\]
\[
\left.
-\frac{\alpha}{r}\cos\Phi(\sin\Phi-\sin\Phi_0)
+\nu\Phi_0'(\cos\Phi-\cos\Phi_0)\right]
\left[\frac{\alpha}{r}\sin\Phi-\nu\Phi'\right]\equiv f(\Phi).
\]
Let, further, \(\{\varphi_k\}\) be an orthonormal basis in the corresponding Hilbert space. We shall seek the solution in the form
\[ \Phi_n=\sum_{k=1}^{n}c_{nk}\varphi_k \qquad (n=1,2,\ldots). \tag{7} \]
Substituting (7) into the right-hand side of (6), from the resulting linear differential equation with the prescribed boundary conditions we find \(\Psi_n\). The expression found for \(\Psi_n\), as well as \(\Phi_n\), is then substituted into the first equation (3), and we require that the resulting expression be orthogonal to all \(\varphi_k\) \((k=1,2,\ldots,n)\); then we obtain a system for \(c_{nk}\), for the solution of which one can apply the method \((^3)\).
- Equations (3) contain the natural small parameter \(\varepsilon\) multiplying the highest derivatives. This leads to the conclusion that asymptotic methods should be applied for the analysis of sufficiently thin shells. Moreover, the application of the Galerkin method in this case is difficult, since it requires the use of high approximations. The parameter \(\varepsilon\) may enter (3) in different ways depending on the manner of transition to dimensionless form. It seems natural to choose this method according to formulas (2), depending on the order of magnitude of the loads acting on the shell. Therefore, for loads of different orders of magnitude, different asymptotic expansions of the solutions will be constructed.
In what follows, for definiteness, a spherical shell under the action of hydrostatic pressure will be considered. For this, in (3) one must set \((0\le \xi\le b<\pi)\)
\[ d=R,\quad r=\sin\xi,\quad z=-\cos\xi,\quad \alpha=1,\quad p=-\rho\sin\Phi,\quad q=\rho\cos\Phi . \tag{8} \]
It should be noted, however, that everything that follows below is easily carried over to convex shells \(\bigl(\Phi_0(\xi)\) is an increasing function, with \(m\xi\le \Phi_0(\xi)<\pi,\ m>0\bigr)\).
A. Let the intensity of the normal load \(\rho\) be less than or of order \(\varepsilon^2\). Then in (2), (3) one must take \(k=2\), i.e., in both equations (3), for the higher derivatives, there will be \(\varepsilon^1\). In this case the asymptotic expansions are constructed in powers of \((h/\gamma R)^{1/2}\).
Indeed, denote \(\varepsilon=\mu^2\) and seek solutions in the form of series in powers of \(\mu\) (the first iteration process \((^4,^5)\))
\[ \Phi=\sum_{s=0}^{N}\mu^s u_s,\qquad \Psi=\sum_{s=0}^{N}\mu^s v_s. \tag{9} \]
Substituting (9) into (3) for \(k=2\), \(\varepsilon=\mu^2\), and equating to zero the coefficients of \(\mu^0,\mu^1,\mu^2,\ldots\), we obtain systems of equations for determining \(v_s,u_s\). Thus, to determine \(v_0,u_0\) we have
\[ v_0\sin u_0+\rho\cos u_0\int_{0}^{\xi}\sin \xi \cos u_0\,d\xi=0,\qquad \cos u_0=\cos \xi. \tag{10} \]
Equations (10) have two solutions
\[ 1)\quad u_0=\xi,\ v_0=-\frac14\rho\sin 2\xi;\qquad 2)\quad u_0=-\xi,\ v_0=\frac14\rho\sin 2\xi. \tag{11} \]
The first of them describes a shell shape close to the initial one and corresponding to a momentless stressed state; the second is close to the mirror-reflected shape and corresponds to the equilibrium state of the shell in the postcritical stage. Accordingly, the asymptotics is constructed in a neighborhood of each solution separately. The solutions (11), as well as \(v_s,u_s\) \((s\geq 1)\), do not satisfy the boundary conditions (4) or (5). To satisfy them, the second iteration process \((^4,^5)\) is used, for which the solutions are sought in the form
\[ \Phi=\sum_{s=0}^{N}\mu^s(u_s+g_s),\qquad \Psi=\sum_{s=0}^{N}\mu^s(v_s+h_s). \tag{12} \]
We substitute (12) into (3), use the equation for \(v_s,u_s\), expand the known functions in Taylor series in a neighborhood of the point \(\xi=b\), make the substitution \(\xi=b-\mu t\), and equate to zero the coefficients of \(\mu^0,\mu^1,\mu^2,\ldots\). To determine \(g_0,h_0\) we obtain the system of equations
\[ \sin b\,g_0''-\frac{v_0(b)}{\cos u_0(b)}\sin g_0-h_0\sin [u_0(b)+g_0]=0, \]
\[ \sin b\,h_0''-\cos [u_0(b)+g_0]+\cos u_0(b)=0 \tag{13} \]
with boundary conditions (in the case of problem (3), (4)) for the first solution
\[ g_0(0)=0,\qquad h_0'(0)=0,\qquad g_0(\infty)=h_0(\infty)=0 \tag{14} \]
and for the second solution
\[ g_0(0)=-2b,\qquad h_0'(0)=0,\qquad g_0(\infty)=h_0(\infty)=0. \tag{15} \]
We note that, for an approximate solution of problem (13), (15), the method \((^6)\) (pp. 34–38) works well. The equations for \(g_1\) and \(h_i\) are not written out here because of their cumbersomeness. In the case of problem (3), (5) we obtain \(h_0'(0)=g_0'(0)=0\), and, consequently, from (13) we have \(g_0=h_0=0\). (An analogous result is obtained for the first solution of problem (3), (4).) Then for \(g_1,h_1\) we obtain
\[ g_1''-\frac12\rho g_1-h_1=0,\qquad h_1''+g_1=0. \tag{16} \]
with boundary conditions for the first solution
\[ g'_1(0)=0,\qquad h'_0(0)=-\frac{1}{2\rho}(1-\nu),\qquad g_1(\infty)=h_1(\infty)=0 \tag{17} \]
and for the second solution in (11)
\[ g'_1(0)=-2(1+\nu),\qquad h'_1(0)=\frac{1}{2\rho}(1-\nu),\qquad g_1(\infty)=h_1(\infty)=0. \tag{18} \]
B. \(\rho\sim O(\varepsilon)\), \(k=1\), and the small parameter \(\varepsilon^2\) occurs only with derivatives \(\Phi\). Here the asymptotic expansions are constructed in powers of \(h/R\gamma\). To determine \(v_0,u_0\) we obtain the membrane equations:
\[ \sin\xi\,v''_0+\cos\xi\,v'_0+v_0/\sin\xi = \cos u_0-\cos\xi-\rho(\sin^2\xi\sin u_0)', \tag{19} \]
\[ v_0\sin u_0+\rho\int_0^\xi \sin\xi\cos u_0\,d\xi\cos u_0=0. \]
For solution (18) one can apply a method combining S. A. Chaplygin’s method and the method of power series \((^5)\). Let us write the first equations for the boundary layer:
\[ h''_0=0,\qquad g''_0-k_1\sin g_0=0, \]
\[ k_1=v_0(b)/\cos u_0(b)\sin(b)>0, \tag{20} \]
\[ h_0(\infty)=0,\qquad g_0(\infty)=0, \]
whence \(h_0\equiv0\), \(g_0=-4\arctan C\exp[-\sqrt{k}\,(b-\xi)/\varepsilon]\), where the constant \(C\) is determined from the boundary condition for \(g_0\) at \(t=0\).
C. \(\rho\sim O(1/\varepsilon)\), \(k=-1\). Here the asymptotic expansions are constructed in powers of \((h/R\gamma)^2\). The membrane equations have the form (18), but in the first equation the difference \(\cos u_0-\cos\xi\) is absent. To determine \(g_0,h_0\) we obtain (20).
- In the case of a hinged immovable fastening of the shell along the contour, the existence of solutions is proved for which asymptotic expansions are valid, and error estimates are given. Along the way, in case A the nonuniqueness of the number of solutions in the problem of equilibrium of a loaded spherical shell is established. To prove these facts, methods developed in works \((^5,^7)\) are used.
The author expresses gratitude to I. I. Vorovich and V. I. Yudovich for their attention to the work.
Rostov State University
Received
27 I 1968
REFERENCES
\(^1\) E. Reissner, Proc. Symposia in Appl. Math., 3, 27 (1950).
\(^2\) V. I. Feodos’ev, Elastic Elements of Precision Instrument Design, Moscow, 1949.
\(^3\) I. I. Vorovich, V. F. Zhilova, PMM, 29, no. 5 (1965).
\(^4\) M. I. Vishik, L. A. Lyusternik, UMN, 12, issue 5, 3 (1957).
\(^5\) L. S. Srubshchik, V. I. Yudovich, PMM, 26, no. 5 (1962).
\(^6\) A. V. Pogorelov, Geometrical Theory of the Stability of Shells, “Nauka,” 1966.
\(^7\) L. S. Srubshchik, PMM, 31, issue 4 (1967).