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Investigation of static and dynamic behavior of anisotropic inhomogeneous shallow shells by spline approximation method/Lekstuju anizotropiniu nehomogeniniu kevalu statines ir dinamines elgsenos tyrimai, taikant splainu aproksimacijos metoda.

1. Introduction

Shallow shells made of orthotropic materials are widely used for construction of structure elements in modern engineering (Fig. 1). To estimate their strength under possible conditions of service operation, it is necessary to have the information about the stress-strain state (Cowper et al. 1970; Gould 1988; [TEXT NOT REPRODUCIBLE IN ASCII]. 1987) and dynamic characteristics (Graff 1991; Lee et al. 1984; Liew et al. 1997) of the mechanical objects being considered.

Currently the problems of computational mathematics, mathematical physics, and mechanics, spline-functions are widely solved (Fan and Cheung 1983; [TEXT NOT REPRODUCIBLE IN ASCII]. 1980). It is due to advantages of the spline-approximation techniques in comparison with others. As basic advantages, the following can be referred: stability of splines in respect to local disturbances, i.e. behaviour of the spline near a point does not affect the behaviour of the spline as a whole as, for instance, this holds in the case of the polynomial approximation; fast convergence of the spline-interpolation in contrast to polynomial one; simplicity and convenience in realization of algorithms for constructing and calculating splines by personal computers. Use of spline unctions in various variational, projective, and other discrete-continual methods makes it possible to obtain appreciable results in comparison with those the classical apparatus of polynomials would yield, to simplify essentially their numerical realization, and to obtain the desired solution with a highdegree accuracy (Grigorenko and Zakhariichenko 2004; Grigorenko and Yaremchenko 2004).


2. Basic relations and constitutive equations

2.1. Free vibrations of shallow shells in classic formulation

According to the Mushtari-Donnell-Vlasov's theory of shallow shells, the natural transverse vibrations of these shells are described by the equations


where x and y are the Cartesian coordinates of a point on the mid-surface (0 [less than or equal to] x [less than or equal to] a, 0 [less than or equal to] y [less than or equal to] b), t is time, w--the shell deflection, and [rho]--the density of the material (rotary and in-plane inertia are not included there).

The normal and shear forces [N.sub.x], [N.sub.y], and S and the bending and twisting moments [M.sub.x], [M.sub.y], and H satisfy the following relations:




[E.sub.1], [E.sub.2], [G.sub.12], [v.sub.1], [v.sub.2] are the elastic and shear moduli and Poisson's ratios; [k.sub.1] and [k.sub.12]--the curvatures of midsurface, u, v, w--components of displacements vector.

The system of equations (1-2) yields 3 equivalent differential equations for the 3 displacements u, v, and w of the mid-surface:


It is assumed that all points of the plate vibrate harmonically with a frequency [omega], i.e. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (the symbol "~" is omitted hereafter).

Finally we obtain



[a.sub.m] = [a.sub.m] (x, y), [b.sub.m] = [b.sub.m] (x, y),

m = 1, ..., 8; [c.sub.n] = [c.sub.n](x,y), n=1, ..., 9,11,12,

[c.sub.10] = [c.sub.10](x, y, [omega]).

Boundary conditions for displacements are specified on the boundaries x = 0, a and y = 0, b. Clamped boundary at y = coast.

u = v = w = [partial derivative]w/[partial derivative]y = 0 at y = 0, y = b; (6)

hinged boundary:

u = [partial derivative]v/[partial derivative]y = w = [[partial derivative].sup.2]w/ [partial derivative][y.sup.2] = 0 at y = 0, y = b; (7)

one boundary hinged and the other clamped:

u = [partial derivative]v/[partial derivative]y = w = [[partial derivative].sup.2]w/[partial derivative][y.sup.2] = 0 at y = 0,

u = v = w = [partial derivative]w/[partial derivative] = 0 at y = 0 at y = b. (8)

Similar conditions can also be prescribed on the boundaries x = const (replacing y by x and v by a in Eqs (6-8)).

2.2. Stress-strain state of shallow shells in refined formulation

The equilibrium equations of refined Timoshenko-Mindlin type shell theory (Fpnropexxo n up. 1987) are


where [N.sub.x], [N.sub.y], [N.sub.xy], and [N.sub.yx] are the tangential forces; [Q.sub.x] and [Q.sub.y] are the shearing forces; [M.sub.x], [M.sub.y], [M.sub.xy], and [M.sub.yx] are the bending and twisting moments.

The elastic relations for orthotropic shells symmetric across the thickness about the chosen coordinate surface are




where [[epsilon].sub.x], [[epsilon].sub.y], [[epsilon].sub.xy], are the tangential strains of the coordinate surface; [[kappa].sub.x], [[kappa].sub.x], [[kappa].sub.x]--the flexural strains of the coordinate surface; [[??].sub.x], [[??].sub.y]--the angles of rotation of the normal regardless of transverse shear; [[gamma].sub.x], [[gamma].sub.y] are--angles of rotation of the normal due to transverse shear; [[psi].sub.x], [[psi].sub.x]--the complete angles of rotation of the rectilinear element.

From (9)-(11) we obtain


Clamped boundary at y= coast.

u = v = w = 0, [[psi].sub.x] = [[gamma].sub.y] = 0; at y = 0, y = b; (13)

hinged boundary:

u = w = 0, [partial derivative]v/[partial derivative]y = 0, [[psi].sub.x] = 0, [partial derivative] [[psi].sub.y]/[partial derivative]y = 0;

at y = 0, y = b. (14)

3. Method of solution

3.1. Free vibrations of shells

The solution of the system of equations (5) is sought in the form


where [u.sub.i](x), [v.sub.i](x), and [w.sub.i](x) (i=0, ..., N) are the unknown functions; [[phi].sub.i](y), [[chi].sub.i](y)--functions constructed using cubic B-splines and [[psi].sub.i](y)--functions constructed using quintic B-splines ([TEXT NOT REPRODUCIBLE IN ASCII] 1980) and they are selected so as to satisfy the boundary conditions at y = const using linear combinations of cubic and quintic B-splines (Grigorenko and Kryukov 1995).

Substituting (15) into Eqs (5), we require that they be satisfied at prescribed collocation points [[xi].sub.k] [member of] [0, b], k = 0, ..., N. If the mesh has an even number of nodes (N=2n+1) and the collocation points are such that [[xi].sub.2i] [member of] [[y.sub.2i], [y.sub.2i+1]], [[xi].sub.2i+1] [member of] [[y.sub.2i], [y.sub.2i+1]], (i= [bar.0..n]), then the interval [[y.sub.2i], [y.sub.2i+1]] has 2 collocation points, and the adjacent intervals [[y.sub.2i+1], [y.sub.2i+2]] do not have such points. Within each of the intervals [[y.sub.2i], [y.sub.2i+1]], collocation points are selected as follows:


where [z.sub.1] = 1/2 - [square root of 3]/6; [z.sub.2] = z/2 = 1/2 + [square root of 3]/6 are the roots of a quadratic Legendre polynomial on the interval [0, 1]. Such collocation points are optimal and substantially increase the accuracy of approximation. As a result, we obtain a system of 3(N + 1) linear differential equations for [u.sub.i], [v.sub.i], and [w.sub.i] With the notation


This system can be normalized:

and the notation [bar.c] x A of the matrix [[c.sub.j][a.sub.ij]], where

A = [[a.sub.ij], (i, j = 0, ..., N) is a matrix and [[bar.c].sup.T] = {[c.sub.0], ..., [c.sub.N]} is a vector, the system of differential equations becomes


This system can be normalized:


A(x,[omega])) is a square matrix of order 8(N+1)x8(N+1). The boundary conditions for this system can be expressed as

[B.sub.1][bar.Y](0) = [bar.0], [B.sub.2][bar.Y](a) = [bar.0]. (18)

To solve the eigenvalue problem for the system of ordinary differential equations (17) with the boundary conditions (18), we will combine discrete orthogonalization with incremental search ([TEXT NOT REPRODUCIBLE IN ASCII] 1986).

3.2. Stress-strain state of shells

The solution of boundary-value problem (12)-(14) can be represented as


where [u.sub.i], [v.sub.i], [w.sub.i], [[psi].sub.xi], [[psi].sub.yi] are the searched functions of the variable x [[phi].sub.ji](y) (j = [bar.1, 5]; i = 0,1, ..., N) are the linear combinations of B-splines third power.

If a resolving function is equal to zero, then


If the derivative of a resolving function with respect to s is equal to zero, then


The functions [[phi].sub.j,N-1](y) and [[phi].sub.j,N] (y) can be represented similarly.

Substituting (19) into Eq. (12) and boundary conditions (14), we require that they be satisfied at prescribed collocation points. We obtain a dimensional boundary problem that can be solved by the discrete orthogonalization method. The full solving technique is described in 3.1.

The results of calculation are presented for square in plane isotropic shell displacements wE/q at all hinged boundaries in cross-section y = a/2 (Table). The parameters of shell are a = 10, h = 0.4, [k.sub.1] = 0.05, [k.sub.2] = 0, v = 0.3, q = const.

The results were obtained by spline-approximation and Fourier series methods. As follows from Table, the solution approximate to the exact one with increase in quantity of collocation points. It can be reliability criterion of the technique proposed.

4. Numerical results

4.1. Studying the natural vibrations of shells basing on the Mushtari-Donnell-Vlasov's theory

We will use the proposed approach to study the spectrum of natural vibrations of a square shallow shell with varying thickness and different boundary conditions. The thickness of the plate varies by the formula

h(x) = [h.sub.0] [[alpha](6 [x.sup.2]/[a.sub.2] - 6 [x/a] + 1) + 1]. (20)

The material of the shell is orthotropic ([TEXT NOT REPRODUCIBLE IN ASCII] 1957) with Young's moduli [E.sub.1] = 4.76 x [10.sup.4] MPa, [E.sub.2] = 2.07 x [10.sup.4] MPa, shear moduli [G.sub.12] = 0.531 x [10.sup.4] MPa, [G.sub.13] = 0.501 x [10.sup.4] MPa, [G.sub.23] = 0.434 x [10.sup.4] MPa and Poisson's ratios [v.sub.1] = 0.149 , [v.sub.2] = 0.0647 , 1/[k.sub.1] = 1/[k.sub.2] = 12.5; 3.125; 1.5625 (there 1/[k.sub.1] and 1/[k.sub.2] are dimensionless radiuses of curvatures)

The following boundary conditions were used:

--the entire boundary is clamped (A);

--two adjacent sides are clamped and the other sides are hinged (B).

Figs 2-4 show the dimensionless natural frequencies of the shell [bar.[omega].sub.i] = [[omega].sub.i][a.sup.2] [square root of [rho][h.sub.0]/[D.sub.11]] as a function of the parameter [alpha] for A (solid line) and B (dashed line) boundary conditions.

From Figs 2-3 follows, that the first frequencies of orthotropic shells of a variable thickness at the big radiuses of curvature increase, and the second frequencies decrease practically linearly at increasing [alpha]. Under boundary conditions B, first two frequencies increase with increasing [alpha]. At the further reduction of the main radiuses of curvature the first frequencies decrease, and for the second both increasing and decreasing under certain boundary conditions is possible with increasing [alpha] (Fig. 4). The higher frequencies, basically, increase nonlinearly, though their decreasing is possible also since some value [alpha]. Such behaviour of frequencies is caused by simultaneous influence both of variable thickness and the orthotropy of material.




4.2. Stressed state of non-thin shallow shells basing on the Timoshenko-Mindlin's theory

Let us analyze, as an example, the stress-strain state a doubly curved isotropic shallow shell with square plan-form and varying thickness under uniform normal pressure q = [q.sub.0] = const. The thickness of the shell (Fig. 1) varies by (20). The input data: a = b = 10, [k.sub.1] = 1/10, [k.sub.2] = 0, [h.sub.0] = 1, [alpha] = -0.4, -0.2, 0, 0.2, 0.4, Figs 5-7 show the thickness dependence of the displacements and stresses in the section y = a/ 2 on the lateral surfaces of the shell clamped at 3 edges and hinged at one edge. It can be seen that w,[[sigma].sup.+.sub.x], and [[sigma].sup.-.sub.x] are distributed asymmetrically.

Fig. 5 demonstrates that the maximum displacement is slightly shifted from the point of the rise toward the hinged edge, the maximum increasing with [alpha]. As the thickness increases in this zone, the deflection decreases insignificantly. Fig. 6 shows how the stress on the outside surface depends on the thickness. It can be seen that the maximum of [[sigma].sup.+.sub.x], is shifted from the point of therise toward the hinged edge and increases with [alpha].

Fig. 7 shows the stress distribution on the inside surface. The stress patterns on the inside and outside surfaces of the shell are qualitatively close and differ by sign. Quantitatively, the maximum stresses [[sigma].sup.-.sub.x] are almost twice as great as [[sigma].sup.+.sub.x].




5. Conclusions

1. The paper proposes a numerical-analytical approach to investigation of the stress-stain state and natural vibrations of orthotropic varying thickness plates and shells. The approach includes 2 stages. At the first stage an initial eigenvalue or boundary problem for the systems of partial differential equations is reduced to the eigen-value (boundary) problem for the system of high-order ordinary differential equations by representing the desired solution in the form of segment of series in spline-collocations and choosing collocation points in the domain under consideration. The obtained one-dimensional eigen-value (boundary) problems are solved by the stable numerical method of discrete-ortogonalization in combination with the step-by-step search method what provides highly accurate solution.

2. The applied problems for natural vibrations (Mushtari Donnell-Vlasov's theory) and stress-strain state (Timoshenko-Mindlin's theory.) of shallow shells with varying thickness under different boundary conditions are solved.

Received 15 May 2008, accepted 26 Dec 200


Cowper, G. R.; Lindberg, G. M.; Olson, M. D. 1970. Shallow shell finite element of triangular shape, Int. J. Solids Struct 6(8): 1133-1156.

Fan, S. C.; Cheung, Y. K. 1983. Analysis of shallow shells by spline finite strip method, Eng. Struct. 5 (4) : 255-263.

Gould, P. L. 1988. Analysis of shells and plates. New York: Springer-Verlag. 483 p.

Graff, K. F. 1991. Wave motion in elastic solids. New York: Dover. 650 p.

Grigorenko, Ya. M.; Kryukov, N. N. 1995. Solution of problems of the theory of plates and shells with spline functions (survey), International Applied Mechanics 31(6): 413-434.

Grigorenko, Ya. M.; Yaremchenko, S. N. 2004. Stress analysis of orthotropic noncircular cylindrical shells of variable thickness in a refined formulation, International Applied Mechanics 40(3): 266-274.

Grigorenko, Ya. M.; Zakhariichenko, L. 1. 2004. Stress-strain analysis of elliptic cylindrical shells under local loads, International Applied Mechanics 40(10): 1157-1163.

Lee, J. K.; Leissa, A. V.; Wang, A. J. 1984. Vibrations of blades with variable thickness and curvature by shell theory, Trans. ASMEJ Eng. Gas Turbines Power 106: 11-16.

Liew, K. M.; Lim, C. W. 1997. Vibratory characteristics of cantilevered rectangular shallow shells of variable thickness, Inst. Aerounaut. Astronaut. J. 32(2): 387-396.

Liew, K. M.; Lim, C. W.; Kitipomchai, S. 1997. Vibration of shallow shells: A review with bibliography, Appl. Mech. Rev. 50(8): 431-444.

[TEXT NOT REPRODUCIBLE IN ASCII] [Grigorenko, J.M.; Bespalova, E. I.; Kitajgorodskij, A. B.; Schinkar, A. I. Natural vibrations of shells structural elements]. Knell: [TEXT NOT REPRODUCIBLE IN ASCII] 172 c.

[TEXT NOT REPRODUCIBLE IN ASCII] [Grigorenko, J. M.; Vasilenko, A. T.; Golub, G. P. Statics of anisotropic shells of finite shear stiffness]. [TEXT NOT REPRODUCIBLE IN ASCII] 216 c.

[TEXT NOT REPRODUCIBLE IN ASCII] [Zavjalov, J. C.; Kvasov, J. I.; Miroschnichenko, V. L. Methods of spline functions]. MOCKBA: HAYKA. 352 c.

[TEXT NOT REPRODUCIBLE IN ASCII] [Lechnickij, S. G. Anisotropic Pplates]. [TEXT NOT REPRODUCIBLE IN ASCII]. 463 c.

Alexander Grigorenko, Sergiy Yaremchenko

S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Nesterov str. 3, Kyiv 03057, Ukraine


Alexander GRIGORENKO. Professor, Head of Department of Numerical Methods, S. P. Timoshenko Institute of Mechanics, National Academy of Sciences (NAS) of Ukraine. Member of National committee of theoretical and applied mechanics. His research interests are dynamic behaviour of shells and 3D objects.

Sergiy YAREMCHENKO. PhD, Senior staff scientist of Department of Numerical Methods, S. P. Timoshenko Institute of Mechanics of NAS of Ukraine. His research interests are stress-strain state problems of shells.
x Spline-approximation method Fourier
a N = 9 N = 17 N = 21 solution

0.1 1098.8 1117.3 1119.3 1121.0
0.2 2016.7 2052.9 2056.6 2060.0
0.3 2679 2730 2735.4 2740.0
0.4 3070.9 3132.3 3138.6 3144.3
0.5 3199.8 3264.9 3271.7 3277.7
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Author:Grigorenko, Alexander; Yaremchenko, Sergiy
Publication:Journal of Civil Engineering and Management
Article Type:Report
Date:Mar 1, 2009
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