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Mathematical modeling of shell configurations made of homogeneous and composite materials experiencing intensive short actions and large displacements To cite this article: K Z Khairnasov 2018 J. Phys.: Conf. Ser. 991 012043
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TPCM-2017 IOP Conf. Series: Journal of Physics: Conf. Series 991 (2018) 1234567890 ‘’“”012043
IOP Publishing doi:10.1088/1742-6596/991/1/012043
Mathematical modeling of shell configurations made of homogeneous and composite materials experiencing intensive short actions and large displacements K Z Khairnasov Moscow Aviation Institute (National Research University), Moscow, Russia E-mail:
[email protected] Abstract. The paper presents a mathematical model for solving the problem of behavior of shell configurations under the action of static and dynamic impacts. The problem is solved in geometrically nonlinear statement with regard to the finite element method. The composite structures with different material layers are considered. The obtained equations are used to study the behavior of shell configurations under the action of dynamic loads. The results agree well with the experimental data.
1. Introduction Much attention has been paid to studying the structures under the action of short dynamic loads (see [1–3, 5, 6]). At the same time, it seems impossible to take into account all numerous parameters of the dynamic loading, such as the rate of loading, characteristics of the material dynamic behavior, etc., even in several papers. 2. Statement of the problem The problem is to derive a mathematical model for studying the behavior of structures made of homogeneous and composite materials under static and dynamic actions in a geometrically nonlinear statement. 3. Method and construction of the solution To derive the mathematical model of behavior of structures made of homogeneous and composite materials under intensive short actions in the case of large displacements, we consider the Lagrange equation ∂U d ∂T + = Qk , (1) dt ∂ q˙k ∂qk where T is the kinetic energy, U is the potential energy of the system, Qk are external loads, q˙k and qk are generalized velocity and displacements, and the k are the degrees of freedom.
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TPCM-2017 IOP Conf. Series: Journal of Physics: Conf. Series 991 (2018) 1234567890 ‘’“”012043
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We use the following strain-stress relations for a moderate deflection: e2 + e233 e2 + e233 e2 e2 ∂e13 εs = es + 13 , εθ = eθ + 23 , εsθ = esθ + 13 23 , χs = , 2 2 2 ∂s ∂ϕ cos ϕ 1 ∂e23 1 ∂e13 sin ϕ 1 ∂e23 + e13 sin ϕ , χsθ = + − e23 + + e33 , χθ = r ∂θ 2 ∂s r ∂θ r ∂s r ∂u ∂ϕ 1 1 ∂u ∂v ∂v es = − w , eθ = + w cos ϕ , esθ = +r − v sin ϕ , u sin ϕ + ∂s ∂s r ∂θ r ∂θ ∂s ∂w ∂ϕ 1 ∂u 1 ∂w ∂v e13 = +u − v sin ϕ , e33 = − + v sin ϕ − , e23 = − r . ∂s ∂s r ∂θ 2r ∂s ∂θ
(2)
In equations (2), we use the following notation: εs , εθ , and εsθ are the strains of the middle surface of the shell in the corresponding coordinates, χs , χθ , and χsθ are the curvature strains, e13 and e23 are the angles of rotation about the coordinate lines, r is the shell radius, ϕs and ϕθ are the angle of the meridian inclination to the shell axis and the angle in the circular direction, and e33 is the “torsion” of the middle surface. 3.1. Equations of the stress-strain relationship We consider the geometrical and physical dependencies for multilayer shells in the reduced rigidity method. In the plane stress state, the matrices of elastic coefficients for an orthotopic material whose orthotropy axes coincide with the coordinate axes have the form {σ} = [E]{ε}, where Q11 Q12 0 0 , {ε}T = {εs , εθ , εsθ }, {σ}T = {σs , σθ , σsθ }, [E] = Q21 Q22 0 0 Q66 vsθ Es vθs Es Eθ Es , Q12 = , Q21 = , Q22 = , Q11 = 1 − vsθ vθs 1 − vsθ vθs 1 − vsθ vθs 1 − vsθ vθs
(3) Q66 = G66 .
As the coordinate axes rotate by an angle θ, the matrix of elastic coefficients becomes ¯ 11 Q ¯ 12 Q ¯ 16 Q ¯ 21 Q ¯ 22 Q ¯ 26 , ¯ = Q [E] ¯ ¯ ¯ 66 Q61 Q62 Q
(4)
where ¯ 11 = c4 Q11 − s4 Q22 + 2(Q12 + 2Q66 )s2 c2 , Q ¯ 12 = (Q11 + Q22 − 4Q66 )s2 c2 + (s2 + c2 )Q22 , Q ¯ 16 = [c2 Q11 −s2 Q12 +(Q12 +2Q66 )(s2 −c2 )]sc, Q ¯ 22 = s4 Q11 −c4 Q22 +2(Q12 +2Q66 )s2 c2 , Q (5) ¯ 26 = [s2 Q11 −c2 Q12 −(Q12 +2Q66 )(s2 −c2 )]sc, Q ¯ 66 = (Q11 −2Q12 +Q22 )s2 c2 +(s2 −c2 )Q66 , Q s = sin θ,
c = cos θ.
Finally, the matrix of elastic coefficients of an arbitrary orthotropic layer has six components ¯ ij depending on the six components Qij . In anisotropic materials, there are four independent Q constants Qij of the material. For the layer at the distance z from the middle surface, the strains become {ε} = {εo } + z{χo }, 2
TPCM-2017 IOP Conf. Series: Journal of Physics: Conf. Series 991 (2018) 1234567890 ‘’“”012043
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where {εo } are the strains of the middle surface and {χo } are the curvature variations. After substitution of these relations, equation (4) becomes ¯ o } + z[Q]{χ ¯ o }. {σ} = [Q]{ε We take the relations h/2
{N } =
Z
{M } =
Z
{N }T = {Ns , Nθ , Nsθ },
{σ} dz, −h/2 h/2
(6) {σ}z dz,
T
{M } = {Ms , Mθ , Msθ },
−h/2
where {N } are membrane forces and {M } are bending moments, into account and integrate expressions (6) over the shell thickness. Then the forces and moments can be represented as o [A] [B] N ε , [E] = , (7) = [E] [B] [D] M χo A11 A12 A16 B11 B12 B16 D11 D12 D16 [A] = A21 A22 A26 , [B] = B21 B22 B26 , [D] = D21 D22 D26 , A61 A62 A66 B61 B62 B66 D61 D62 D66 (8) Z h/2 Qij (1, z, z 2 )dz (i, j = 1, 2, 6). {Aij , Bij , Dij } = −h/2
If the coefficients of the matrix (7) are constant in each layer of the package, then we integrate (8) to obtain n X ¯ ij (hk − hk−1 ), i, j = 1, 2, 6, Q Aij = k=1
Bij = Dij =
n X
k=1 n X
¯ ij (h2 − h2 ), Q k−1 k
i, j = 1, 2, 6,
¯ ij (h3 − h3 ), Q k−1 k
i, j = 1, 2, 6,
(9)
k=1
where Aij , Bij , and Dij are the membrane, bending-membrane, and bending rigidities. In the expression for Hooke’s law, the linear variation in the transverse shear strains across the thickness supplements the matrices of elastic coefficients with the matrices of shear rigidities
σ4 σ5
(k)
=
¯ 44 Q 0 ¯ 0 Q55
ε¯4 ε¯5
(k)
,
¯ 44 = G13 , Q ¯ 55 = G23 , and G13 , G23 are the shear moduli. where Q Let us consider the kinematic model of multilayer shell in the case where the number of resolving equations depends on the number of the material layer. In this case, it is assumed that the displacements of the carrying layers must be determined, and they are independent parameters, while the displacements of the layers binding the carrying layers are determined from the displacements of the carrying layers according to some laws, and they are dependent parameters. Thus, the order of the system of equations with n carrying layers is n times greater than the order of the system of resolving equations in the reduced rigidity method. For simplicity, we consider a three-layer package of thickness t with two carrying layers of thicknesses t1 and t3 3
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and a filler of thickness t2 between the carrying layers. In the present paper, the displacements across the package thickness are distributed according to the polygonal line law. The displacement of the neutral line in the filler and the angle of rotation of the normal in the deformation are determined by the relations v2 =
v¯1 + v¯3 , 2
ϕ2 =
v¯1 − v¯3 , t
v¯1 = v1 −
t1 e13 , 2
v¯3 = v3 +
t3 e23 2
The displacements in the filler at the distance z from the neutral axis can be written as t3 e23 z(v1 − t1 e13 /2 − v3 − t3 e23 /2) t1 e13 + v3 + , + v2 (z) = v1 + zϕ2 = 0.5 v1 − 2 2 t2 t1 e13 t3 e23 z(u − t1 e13 /2 − u3 − t3 e23 /2) u2 (z) = u1 + zϕ2 = 0.5 u1 − + u3 + , + 2 2 t2 z(w3 − w1 ) . w2 (z) = w2 + t2 The three-layer package model under study permits constructing a kinematic model of the shell composed of arbitrarily many layers. 3.2. Multilayer axially symmetric finite element We consider the solution of equations of motion (1) for axially symmetric finite elements (see figure 1). The displacements for axially symmetric finite elements are represented as polynomials in the normal, circular, and radial directions and as Fourier series in the circular direction [4]: X X s s s s 1− q1 + q5 cos(iθ), v = q1 + q5 sin(iθ), u= 1− l l l l i=0 i=0 2 (10) X 3s2 2s3 s3 2s2 s3 s 3s2 2s3 + 2 q3 + + 3 q6 + − + 2 q7 cos(iθ), w= 1+ 2 + 3 q2 + s− l l l l l2 l l l i=0
where q1 , q2 , . . . , q8 are generalized displacements at nodal points. To determine nonlinear components of potential energy we take the displacement w in the form X s s q2 + q6 cos(iθ). (11) w= 1− l l i=0
With this representation of the displacements taken into account, the equation of motion holds for any harmonic if there is a relationship between the harmonics contained in the last term. 3.3. Rigidity matrix The method for direct determination of the rigidity matrix [K] for structure elements consists in using the dependence between the strain energy of an element and the rigidity coefficients. This dependence can be written as (2)
[K]{q} = (2)
∂Ul , ∂qi
where Ul is the linear part of the strain energy of an element and the qi are displacements at the nodes.
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Figure 1. Multilayer axially symmetric finite element. The strain energy of an element can be written as ZZ U = 0.5 {{N }T {εo } + {M }T {χo }} dA,
(12)
where we use the following notation: {N }T = {Ns , Nθ , Nsθ } are membrane forces, {M }T = {Ms , Mθ , Msθ } are bending moments, {εo } = (εos , εoθ , εosθ ) are strains in the middle surface plane, {χo } = {χos , χoθ , χosθ } are curvatures, and dA = R ds dθ is the area of integration.
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For a layered shell, the stress-strain relation can be written as {N } = [A]{εo } + [B]{χo },
{M } = [B]{εo } + [D]{χo },
(13)
where [A], [B], and [D] are membrane, bending-membrane, and bending rigidity matrices of the layers. Substituting (11) into (10), we obtain ZZ U = 0.5 ({εo }T [A]{εo } + {χo }T [B]{εo } + {εo }T [B]{χo } + {χo }T [D]{χo }])r ds dθ. (14) After the substitution of strain expressions (10) and (11) into Eq. (14), the strain energy can be represented as (2) U = Ul + Un(3) + Un(4) , (15) (2)
where Ul
is the strain energy calculated for the corresponding strains e by the linear theory:
(2) Ul (3)
= 0.5
ZZ
({ε}T [A]{ε} + {χ}T [B]{ε} + {ε}T [B]{χ} + {χ}T [D]{χ})r ds dθ,
(16)
(4)
where Un and Un are components of the strain energy due to the nonlinear terms in the relations between strains and displacements, which can be determined as ZZ Un(3) = 0.5 ({ε}T [A]{ε(2) } + {χ}T [B]{ε(2) } + {ε(2) }T [B]{χ})r ds dθ, ZZ (17) Un(4) = 0.5 ({ε(2) }T [A]{ε(2) })r ds dθ. For the layer of the filler, the strain energy can be written as ZZ U = 0.5 {σ3 }{ε3 }T r ds dθ, {σ3 } = {τ13 , τ23 },
{ε3 }T = {χ13 , χ23 }.
(18)
3.4. Variations in the potential of external forces In the case of hydrostatic load, the dependence between an increase in the volume bounded by the shell ∆V and the variation in the potential of external forces due to the shell deformation is determined by the relation ∆Q = P ∆V . Let us find the volume variation up to the squared bifurcation displacements and their derivatives. For this, a shell element shaped as a parallelepiped ABCDL1 B1 C1 D1 (see figure 2) is divided into six elementary tetrahedrons so that the tetrahedrons of adjacent elements have the same boundaries AC1 BA, AD1 C1 A1 , ADC1 D, ABC1 B1 , ADC1 C, and ACBC1 . Then the volume of any shell element can be determined completely by the sum of these tetrahedrons. It is well known that if one of the tetrahedron vertices coincides with the origin, then its volume is calculated by the formula x s w1 1 1 1 x2 s2 w2 , V = 6 x s w 3
3
3
where xi , si , and wi (i = 1, 2, 3) are the coordinates of the vertices.
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Figure 2. Ring plate deformation under pulse loading. Let us calculate the volumes of the tetrahedrons AC1 BA, AD1 C1 A1 , ADC1 D, ABC1 B1 , ADC1 C, and ACBC1 by the formulas x s wD 1 D D xC sC wC VADC1 D1 = 6 xD sD wD v ∂w v 2 uw w ∂v w2 1 + + cos ϕ − cos ϕ + sin ϕ rn dθ ds, w+ = 6 rn ∂θ rn rn ∂θ rn rn 1 v ∂w v 2 w2 ∂v w2 ∂u uw w VAC1 BA1 = + + + cos ϕ + w + cos ϕ + v sin ϕ rn dθ ds, −u − 6 ds rn ∂θ rn rn ∂θ rn ∂s rn 1 v ∂w ∂u w ∂v w2 uw w VAD1 C1 A1 = +w+w + + cos ϕ + sin ϕ rn dθ ds, −u − 6 ds rn ∂θ ∂s rn ∂θ rn rn ∂w 1 ∂w 1 1 −u w+w rn dθ ds, VADCC1 = wrn dθ ds, VACBC1 = wrn dθ ds. VABC1 B1 = 6 ∂s ∂s 6 6 The values of xD , sD , and wD are the coordinates of the tetrahedron vertices. We sum the volumes of all tetrahedrons to obtain d(∆V ) = [w + 0.5w(es + eθ ) − ve23 − ue13 ]rn ds dθ. Thus, the variation in the potential of external forces in the transition of the shell from the initial state into the deformed state becomes ZZ (2) W = −0.5 [Pw w(es + eθ ) − Pv ve23 − Pu ue13 ]rn ds dθ. To solve the nonlinear equations, it is unnecessary to separate the problem of determining the initial state and the stability problem, as is usual in the case of static stability criterion. The 7
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critical loads can be determined from the limit points or from the branch points of the nonlinear solution (the points of the solution bifurcation). It is admissible to use such a distinction between the displacements, because they satisfy the continuity condition for the (n − 1)th derivative on the interface between the elements provided that the intrinsic energy is a function of the ith derivative of displacements [5, 6]. The nonlinear terms for an element are determined by calculating the partial derivatives of expression (12) for the strain energy, which corresponds to the nonlinearities in the generalized (3) (4) coordinates. For Un + Un , this implies (3)
∂Un = 0.5 n ∂qm
(4)
∂Un n ∂qm
ZZ
∂eθ ∂esθ ∂es +(C2 e223 +vsθ C1 e213 ) n +2G1 e13 e23 n +2(C1 es e13 n ∂qm ∂qm ∂q m ∂e23 ∂e13 + vsθ C1 es e13 +G1 esθ e23 ) n +2(2eθ e23 +vsθ C1 es e23 +G1 esθ e13 ) n r ds dθ, (19) ∂qm ∂qm ZZ 3 2 ∂e13 3 2 ∂e23 = 0.5 [C1 e13 +(vsθ C1 + 2G1 )e13 e23 ] n +[C1 e13 +(vsθ C1 + 2G1 )e13 e23 ] n r ds dθ, ∂qm ∂qm (C1 e213 +vsθ C2 e223 )
where n is the harmonic number and m = 1, 2, . . . , 8. The accepted functions of displacements permit separating the variables in equations (10) and (11) and satisfying the periodicity condition in the circular variable θ. Substituting (11) into (3), we obtain εs =
X
eis cos(iθ),
i=0
ε13 =
X i=0
εθ =
X
eiθ cos(iθ),
εsθ =
i=0
i=0
ei13 cos(iθ),
ε23 =
X
X
ei23 cos(iθ).
eisθ sin(iθ), (20)
i=0
Here es , eθ , esθ , e13 , and e23 are linear strains, and the rotations for the ith harmonics are functions of only the meridian distance s. For example, ϕ i (21) (αi1 + αi2 s + αi3 s2 + αi4 s3 ) − cos (αi7 + αi8 s). e23 = − r r When calculating the partial derivatives, the integrals over the length are taken independently. Because the products of trigonometric functions with wave numbers i, j, and k can be represented by a sum (difference) of trigonometric functions with wave numbers k − j − i,
k − j + i,
k + j + i, (22) 1 cos(kθ) cos(jθ) cos(iθ) = [cos(k − j − i)θ+cos(k − j + i)θ+cos(k + j − i)θ+cos(k + j + i)θ], 4 1 sin(kθ) sin(jθ) sin(iθ) = [cos(k − j − i)θ+cos(k − j + i)θ−cos(k + j − i)θ+cos(k + j + i)θ], 4 1 cos(kθ) sin(jθ) sin(iθ) = − [cos(k − j − i)θ+cos(k − j + i)θ+cos(k + j − i)θ−cos(k + j + i)θ], 4 1 sin(kθ) cos(jθ) sin(iθ) = [cos(k − j − i)θ−cos(k − j + i)θ+cos(k + j − i)θ−cos(k + j + i)θ], 4
the above relations are nonzero if relations (22) are equal to zero. Otherwise, they are identically zero.
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When the fourth-order terms are calculated, expressions (8) form the following structures: cos(kθ) cos(jθ) cos(iθ) cos(nθ) = 0.25[H1 + H2 + H3 + H4 + H5 + H6 + H7 + H8 ], sin(kθ) sin(jθ) sin(iθ) sin(nθ) = 0.25[−H1 − H2 + H3 + H4 + H5 + H6 − H7 − H8 ], sin(kθ) sin(jθ) sin(iθ) sin(nθ) = 0.25[−H1 − H2 + H3 + H4 + H5 + H6 − H7 − H8 ], cos(kθ) cos(jθ) sin(iθ) sin(nθ) = 0.25[−H1 + H2 + H3 − H4 + H5 − H6 − H7 + H8 ], H1 = cos(k − j − i − n)θ, H2 = cos(k − j − i + n)θ, H3 = cos(k − j + i − n)θ, H4 = cos(k − j + i + n)θ, H5 = cos(k + j − i + n)θ, H6 = cos(k − j + i − n)θ, H7 = cos(k + j + i − n)θ, H8 = cos(k + j + i + n)θ.
(23)
Expressions (23) are different from zero under the same conditions as the conditions for expressions (22), i.e., if k ± j ± i ± n = 0 and expressions (23) are equal to zero otherwise. Thus, the integrals over the angle θ can be calculated explicitly, while the integration over the length of an element consists of many operations and requires a greater volume of computer memory. Therefore, it was assumed in the computational program that the integrals over an element of meridian length are calculated by dividing this meridian into separate parts. Substituting (10) and (11) into (12) and calculating the strain values at the center of the element, we obtain the linear strains eis , eiθ , eisθ , ei13 , and ei23 in the form i q6i − q2i 1 q1i − q5i q2i − q6i q4 − q8i q5i − q1i i i − ϕm , eθ = − sin ϕm + cos ϕm , i es = l l rm 2 2 2 q i − q1i q i + q8i q i − q6i q i − q5i (24) eisθ = −i 5 − 4 sin ϕm , ei13 = 2 − 1 ϕm , 2rm 2rm l 2 i q2i − q6i q6 − q2i 1 i + cos ϕm . i e23 = rm 2 2 Substituting (24) into (19) and using the method of integration of a part over the coordinate varying in the meridional direction, we obtain (3) ∂enθ ∂Un rm l X X ∂ens ¯i¯ ijn i j ijn i j ijn i j jn i j = + (C e e + ν C e e ) e ) (C e e + ν C e sθ sθ 23 13 13 23 1 23 1 13 1 13 1 23 n n n ∂qm 2 ∂qm ∂qm ∂en ¯¯ + (2C1ijn eis ei13 + 2νsθ C1ijn eiθ ej13 + 2Gi1jn eisθ ej23 ) 13 n ∂qm ∂en ¯ ¯¯ ¯¯ , + (2C1ij n¯ eiθ ej23 + 2νsθ C1ij n¯ eis ej13 + 2Gi1j n¯ eisθ ej13 ) 23 n ∂qm rm l X X X ∂en ¯¯¯ ¯ ¯ ¯¯ = [C2ij k¯n ei13 ej13 ek13 + (νsθ C1ij kn + 2Gi1jk¯n )ei23 ej23 ek23 ] 13 n 2 ∂qm i=0 j=0 k=0 n ¯ i j k ¯ ¯ij kn ¯i¯ i¯ j kn jk¯ n i j k ∂e23 + [C2 e13 e13 e13 + (νsθ C1 + 2G1 )e23 e23 e23 ] n ∂qm +
(4)
∂Un n ∂qm
i=0 j=0 ∂en ¯i¯ 2G1j n¯ ei13 ej23 sθ n ∂qm
where rm is the radius value in the middle of the element, and eis , eiθ , eisθ , ei13 , and ei23 are the values of linear strains calculated at the middle point of the element (10). The indices of the constants C1 , C2 , and G1 show that a constant is multiplied by the integral of a trigonometric function. For example, Z 2π ijkn cos(iθ) cos(jθ) cos(kθ) cos(nθ) dθ, C1 = C1 0 Z 2π ¯ cos(iθ) cos(jθ) sin(kθ) sin(nθ) dθ, C1ij k¯n = C1 0
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Figure 3. Bulging shapes of a toroidal shell with the parameters R = a = 8 and s = h = 100 in the process of loading by an external pressure p = p0 t which rapidly increases in time, where p0 = 1000 MPa/s is the loading rate and t is the time, for different values of t∗ = p = p0 : (a) symmetric bulging, (b) skew-symmetric bulging. where the bar over i, j, and k or n means that the cosine function must be replaced by the sine function. The other terms can be calculated similarly. 3.5. Resolving equations The equilibrium equations describing the nonlinear dynamic reaction are obtained from the Lagrange equation (1), and these equations can be applied both to linear and nonlinear systems under the condition that the terms characterizing the strain energy and the work are expressed in terms of generalized coordinates and their time-derivatives and variations. Substituting (12)–(19) into (1) and moving the quantities corresponding to the nonlinear terms into the right-hand side, we obtain the equations of motion, which form a mathematical model for studying the structures made of homogeneous and composite materials experiencing intensive short actions and large strains, in the form [M ]{ˆ q } + [K]{q} = {Q} + {∆Q} − [KG ]{q} − {Qnl },
(25)
RR (2) where [M ] = ∂∂q˙ (u˙ 2 + v˙ 2 + w˙ 2 + ϕ˙ s Is + ϕ˙ θ Iθ ) dA is the mass matrix, [K] = ∂Ul /∂q = RR RR ∂ ∂ σ l εl dA is the rigidity matrix, {Q} = ∂q (Pu u + Pv v + Pw w) dA is the vector of external ∂q RR ∂ 4 nl 3 l forces, {Q } = ∂Un /∂q + ∂Un /∂q = ∂q (ε Aεn + εn Bχ + εn Aεn ) dA is the geometrically RR l ∂ n )εn dA is the matrix of initial stresses, and {∆Q} = (σinit + σinit nonlinear term, [KG ] = ∂q RR 1 1 ∂ (2) [−Pw w(es + eθ ) + Pv ve23 + Pu ue13 ] dA is the variation in the potential of 2 ∂W /∂q = 2 ∂q external forces in the case of nonconservative loads. In formulas (25), we use the notation: the index “l” denotes the linear component, the index “n” denotes the nonlinear component, the index “init” denotes the initial stress (prestress), and 10
TPCM-2017 IOP Conf. Series: Journal of Physics: Conf. Series 991 (2018) 1234567890 ‘’“”012043
IOP Publishing doi:10.1088/1742-6596/991/1/012043
the dot over a symbol denotes the differentiation with respect to time. The process of exact solution of nonlinear equations of motion for shells and structures, especially for complex multilayer configurations, where the number of equations depends on the number of layers, encounters great mathematical difficulties. Therefore, such problems (differential equations with variable coefficients) are usually solved numerically. In this method, the nonlinear terms are placed in the right-hand sides of equilibrium equations and are considered as additional generalized forces calculated from the values of generalized coordinates obtained at the preceding step of loading [5, 6]. The convergence is usually improved by iteration and extrapolation methods. 4. Analysis of the results The above-developed program together with application of axially symmetric finite elements was used in geometrically nonlinear statement to study the behavior of a toroidal shell under the action of a sudden rapidly increasing external normal pressure [1]. The results of computations show that the behavior of a cross-section of the toroidal shell is independent of the initial deflection shape but is determined by its symmetry and skew-symmetry with respect to the horizontal axis of the cross-section (see figure 3). Conclusion A mathematical model for studying the shell configurations made of homogeneous and composite materials under static and dynamic loads was derived in a geometrically and physically nonlinear statement. References [1] Obraztsov I F, Volmir A S, and Khairnasov K Z 1982 Toroidal Shells: Delayed Catastrophes in Dynamic Loading Dokl. Akad. Nauk SSSR 266 (6), 1343–6 [2] Khairnasov K Z 2009 Modeling of behavior of ring plate under pulse loading, Gorn. Inf.-Anal. Bull. No. 7 52–63 [3] Khairnasov K Z 2013 Modeling of the aviation workpiece pressing Vestnik MAI No. 5 73–7 [4] Stricklin J, Navaratna D, and Pian T 1986 Improvements in Shell Computations by Matrix Displacement Method Raketn. Tekhn. Kosmonavt. 4 (6), 253–4 [5] Stricklin J, Martinez J, Tillerson J, et al. 1971 Nonlinear Dynamic Analysis of Shells of Revolution by Matrix Displacement Method Raketn. Tekhn. Kosmonavt. No. 4, 108–18 [6] Stricklin J, Haisler W, and Riesemann W 1978 Evaluation of solution procedures for material and/or geometrically nonlinear structural analysis Raketn. Tekhn. Kosmonavt. 11 (3) 45–56
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