30 Bulletin of TICMI. x i

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1 30 Bulletin of TICMI ON CONSTUCTION OF APPOXIMATE SOLUTIONS OF EQUATIONS OF THE NON-LINEA AND NON-SHALLOW CYLINDICAL SHELLS Gulua B. I. Veua Institute of Applied Mathematics of Iv. Javahishvili Tbilisi State University Sohumi State University Abstract. In the present paper we consider the geometrically non-linear and non-shallow cylindrical shells when components of the deformation tensor have non-linear terms. By means of I. Veua method two dimensional problem is obtained. Approximate solutions of I. Veua s equations for approximations N = 1 are constructed. Concrete problem is solved when the components of the external force are constants. Key words: Non-shallow cylindrical shells small parameter. MSC 000 : 74K5. The refined theory of shells is constructed by reduced the three- dimensional problems of the theory of elasticity to the two-dimensional problems. I.Veua had obtained the equations of shallow shells [1][]. It means that the interior geometry of the shell does not vary in thicness. This method for non-shallow shells in case of geometrical and physical non-linear theory was generalized by T.Meunargia [3][4]. A complete system of equations of the three-dimensional nonlinear theory of elasticity can be written as: 1 g i gσ i + Φ = 0 i = x i σ i = λ j j U + 1 j U j U i + i U + i j U + j i U + i U j U j + j U g is the discriminant of the metric tensor of the space σ i are contravariant stress vectors Φ is an external force λ and are Lame s constants i and i are covariant and contravariant base vectors of the space and U is the displacement vector. In the present paper we consider the system of equilibrium equations of the two-dimensional geometrically non-linear and non-shallow cylindrical shells which is obtained from the three-dimensional problems of the theory of elasticity for isotropic and homogeneous shell by the method of I. Veua. The displacement vector Ux 1 x x 3 is expressed by the following formula [1] Ux 1 x x 3 = ux 1 x + x3 h vx1 x.

2 Volume Here ux 1 x and vx 1 x are the vector fields on the middle surface x 3 = 0 h is the thicness of the shell x 3 is a thicness coordinate h x 3 h x 1 and x are isometric coordinates on the cylindrical surface. The system of equilibrium equations of the two-dimensional geometrically non-linear and non-shallow cylindrical shells may be written in the following form approximation N = 1: σ 11 + σ 1 + ε 0 σ F1 = σ 1 + σ + 0 F = σ 13 + σ 3 ε 0 σ F3 = σ 11 + σ h σ 31 + ε 1 σ F1 = σ 1 + σ 3 0 σ h F = σ σ σ 33 ε 1 σ F3 = 0 m F = m Φ + m + 1 [ ] 1 + ε + σ 3 1 m 1 ε σ 3 h σ ij m Φ m = m + 1 h h h 1 + x 3 x3 σ ij ΦP m dx 3. h 1 σ 3 x 1 x ±h = ± σ 3. Here P m are Legendre polynomials of order m ε = h 0 0 is the radius of the middle surface of the cylinder. Let us construct the solutions of the form [5] is a small parameter u i = u i ε v i = =1 v i ε i = =1 u i and v i are the components of the vectors u and v respectively. Formal substitution of 3 into and 1 shows that series 3 will satisfy equations 1 if the following equations are fulfilled: u 1 + λ + 1 θ +λ 1 v3 = X1 u + λ + θ +λ v3 = X 4 ] v 3 3 [λ θ +λ + v 3 = X3

3 3 Bulletin of TICMI X p v 1 + λ + 1 Θ 3 1 u3 + v 1 = X4 v + λ + Θ 3 u3 + v = X5 5 u 3 + Θ = X6 = 1... p = are the components of external force and well-nown quantity defined by functions u 0 i... 1 u i v 0 j... 1 v j. The general solutions of systems 1 end are written in the following form u + = 5λ + 6 3λ + v 3 = λ 3λ + v + = u 3 = 1 ϕz z ϕ z ψz ϕ z + ϕ z + χz z + v3 λ χz z + u+ 6λ + z 4λ + f z + zf z + fz g z + i wz z + v+ 3 z z fz + z fz + gz + gz + u3 u + = u 1 + i u v + = v 1 + i v z = x 1 + ix z = 1 x + i 1 x z = 1 x i 1 x ϕz ψz fz and gz are any analytic functions of z χz z and wz z are the general solutions of the following Helmholtz s equations respectively: χ η χ = 0 w γ w = 0 η = γ = 3. 1λ + λ + Here v3 u+ and v+ u 3 are particular solutions of the non-homogeneous equations 1 and respectively. We solve the problem when the middle surface of the body after developing on the plane is the circle with the radius. Let s consider the concrete problem when the components of the external force are constant X 1 = X = 0 X 3 = q. Boundary conditions are u r + iu ϑ = 0 z = v 3 = 0 z = 6 v r + iv ϑ = 0 z = u 3 = 0 z = 7

4 Volume This problem for the approximation = 1 is a well nown case in the theory of elasticity for which we have u 1 λ + + = 3λ + a λ 1 + 1λ + q λη z 1λ + α 0I 1 ηre iθ 1 v 3 = α 0 I 0 ηr λ + 6λ + q 4λ 3λ + a 1 v 1 + = 3 3 qz + 8λ + 8λ + qz z u 1 3 = q 1 16λ qz z 8λ + 3 3λ + qz z a 1 = α 0 = λ + λλ+ηi 1η 1λ+ 7λ+ I 0 η λ+ 3λ+ λ ηi 1 η 3λ+3λ+I 0 η λ + 6λ + λ q + λ λ+ηi 1 η 3λ+ 18λ+ I 0 η λ + λ ηi 1 η 3λ+ q I 0 η. The system of equilibrium equations for the approximation = are: v + + λ + z Θ 3 z u 3 + v + = A 1 + A z z + A 3 z z 8 +A 4 z + z + A 5 I 1 ηre iϑ + I 1 ηre iϑ u 3 + Θ = B 1 + B z z + B 3 z z + B 4 z + z + B 5 z 3 z + z 3 z 9 A 1 = 3λ q 16λ + A = 3λ q 4 8λ + 9λq A 3 = 64λ + A 4 = 3 q 8λ + A 5 = 3q 8λ + λ + α 0 9 q B 1 = 18λ q 8λ + + 3λ q 4 16λ + B = λq 33λ + 10q 8λ + 8λ + 7 q 18λ + + 3q 4λ + B 3 = 9 q 7λ + q 16λ + 18λ + 9λq 64λ +

5 34 Bulletin of TICMI B 4 = 9q q 8λ + 9q B 5 = 18λ +. 9 q 18λ + The general solutions of systems 8 end 9 are written in the following form v + = u 3 = 1 4λ + 3 f z + zf z + fz g z + i wz z + N 0 + N 1 z + N z + N 3 z + N 4 z + N 5 z z + N 6 z z + + N 7 z 3 z + N 8 I 0 ηr + N 9 I 1 ηre iϑ + N 10 I 3 ηre 3iϑ z fz + z fz z + gz + gz + M 0 z z + z z M 1 z 3 z + z 3 z + M z z + M 3 z 3 z 3 + M 4 z 4 z 4 + M 5 I 0 ηr + + M 6 I ηre iϑ + I ηre iϑ M 0 = A 1 16λ + λ + M 1 = B 4 A 4 4λ + λ + M = B + 3B 1 16λ + A 4 4λ + M 3 = B 3 B 7λ + B 3 M 4 = 384λ + M 5 = A 5 1λ + M 6 = A 5 4λ + N 0 = A 1 3 N 1 = B 1 N = A 4 3 N 3 = A 6 8A 3 9 M 0 N 4 = 4B 5 9 4λ + B B 5 N 5 = A 3 16A 3 4M 0 N 6 = B 9 4M 4 N 7 = B 3 3 6M 6 N 8 = λ + ηa 5 63λ + N 9 = λ + ηa 5 63λ + M 6 N 10 = λ + ηa 5 6λ + Boundary conditions are M 6. v r + i v ϑ = 0 u 3 = 0 z =. 10

6 Volume Let us introduce the functions fz gzand wz z by the series fz = c n z n n=1 gz = d n z n n=0 wz z = β n I n γre inθ. 11 I n ηr are Bessel s modifications functions. By substituting 11 into 10 we obtain c 1 = N 1 N 6 N 7 4 c = N 0 + N 5 + N 8 I 0 η + I 0γ N I γ 3 + M 0 I0 γ + 1 I γ + 8λ+ c 3 = N + N 9 I 1 η + I 1γ N I 3 γ 10I 3 η + 4M 1 3 I1 γ + 1 I 3 γ 3 + 8λ+ c 4 = I γ I 4 γ + 1 N 4 + 8λ+ d 0 = 1 4 N1 + N N 7 6 M 4 M 3 5 M 4 8 M 5 I 0 η d 1 = c M 0 d = c 3 1 M1 4 + M 6 I 3 η d 3 = c 4 β 1 = β = i N3 c γi 3 γ i N10 I 3 η 3 c 3. γi 3 γ The system of equilibrium equations for u + and v 3 are: u + + λ + z ϑ +λ z v3 = C 1 z + C z + C 3 z z + C 4 zz 1 u 3 + Θ = D 1 + D z z + D 3 z z + D 4 z + z 5λ + 9 C 1 = 8λ + q 3 q 16λ + C = λ q λ 8λ + λ + q 3q C 3 = 16λ + C 3λ + q 4 = 16λ +

7 36 Bulletin of TICMI D 1 = 3λ λ + q + 3 q q 4 16λ + 8λ + D = 3λ λ + q + 4 8λ + 8λ + 9λq D 3 = 64λ + D 3λ 6q 4 = 3λ + q. The general solutions of systems 1 are written in the following form: u + = 5λ + 6 3λ + ϕz z ϕ z ψz λ χz z + K 0 z 13 6λ + z +K 1 z 3 + K z z + K 3 z z + K 4 z 3 z + K 5 z z + K 6 z z 3 + K 7 z 4 z + K 8 z 3 z v 3 = λ ϕ z + ϕ z + χz z + L 0 + L 1 z + z 14 3λ + +L z z + L 3 z z + z z + L 4 z 3 z + z 3 z + L 5 z z L 0 = λ + λ + [ λ6c4 C 3 36λ + + D 18 ] λ + λλ + + 7λ + D 3 4λ + C 4 L 1 = λc 1 8λ + λ + D 4 6λ + L = λc 4λ + λ + D 6λ + L 3 = λλ + λ + C 3 L 4 = λc 3 λ + L 5 = λc 4 48λ + λ + D 3 6λ + K 0 = λl 0 λ + K 1 = λ + C 1 + 4λL 1 4λ + K = λ + C 1 + 4λL 1 8λ + K 3 = C 4 λ + C + 4λL K 4 = K 5 = 4λL 3 16λ + 4λ + K 6 = λ + 3C 3 6λL 4 4λ + K 8 = C 4 1 λ + C 4 + 8λL 5. 48λ + Boundary conditions are u r + i u ϑ = 0 K 7 = λ + C 3 + 6λL 4 48λ + v 3 = 0 z =. 15 Let us introduce the functions ϕz ψzand χz z by the series ϕz = ρ n z n ψz = ϱ n z n χz z = δ n I n ηre inθ. 16 n=1 n=0

8 Volume By substituting 16 into 15 we obtain: ρ 1 = K 0 + K ληi 1ηL 0+L +L 54 1λ+I 0 η λ+ + 3λ+ λ ηi 1 η 3λ+3λ+I 0 η ρ = N 0 + N 5 + N 8 I 0 η + I 0γ N I γ 3 + M 0 I0 γ + 1 I γ + 8λ+ ρ 3 = N + N 9 I 1 η + I 1γ N I 3 γ 10I 3 η + 4M 1 3 I1 γ + 1 I 3 γ 3 + 8λ+ ρ 4 = I γ I 4 γ + 1 N 4 + 8λ+ ϱ 0 = 1 4 N1 + N N 7 6 M 4 M 3 5 M 4 8 M 5 I 0 η ϱ 1 = c M 0 ϱ = c 3 1 M1 4 + M 6 I 3 η d 3 = c 4 i δ 1 = N3 c γi 3 γ i δ = N10 I 3 η 3 c 3. γi 3 γ The problems when the middle surface of the body after developing on the plane are the circular ring with the radiuses radiuses 1 and will be solved. e f e r e n c e s [1] I.N. Veua. Theory on Thin and Shallow Shells with Variable Thicness. Tbilisi Metsniereba 1965 in ussian. [] I.N. Veua. Some General Methods of Construction of Different Variants of the Theory of Shells. Moscow Naua 198 in ussian. [3] T. Meunargia. On one method of construction of geometrically and physically nonlinear theory of non-shallow shells. Proc. of A. azmadze Math. Institute [4] T. Meunargia. On the application of the method of a small parameter in the theory of non-shallow I.N. Veua s shells. Proc. of A. azmadze Math. Institute [5] P.G. Ciarlet. Mathematical Elasticity. I. North-Holland Amsterdam New-Yor Toyo [6] B. Gulua. On construction of approximate solutions of equations of the non-shallow cylindrical shell. Proc. of I.Veua Inst. of Appl. Math eceived ; revised ; accepted