CC BY
2Magazine of Civil Engineering. 2023. 120(4). Article No. 12003
Magazine of Civil Engineering issn
2712-8172
journal homepage: http://engstroy.spbstu.ru/
Research article UDC 539.3
DOI: 10.34910/MCE.120.3
Stress-strain state of elastic shell based on mixed finite element
Yu.V. Klochkov1 , V.A. Pshenichkina2 , A.P. Nikolaev1 , O.V. Vakhnina1 B , M.Yu. Klochkov2
1 Volgograd State Agrarian University, Volgograd, Russian Federation
2 Volgograd State Technical University, Volgograd, Russian Federation M ovahnina@bk.ru
Keywords: thin-walled shell-type structure, modified mixed functional, compliance matrix, four-node sampling element, force and kinematic unknowns
Abstract. In a mixed formulation, a four-node finite element was developed, which is a fragment of the middle surface of the elastic shell. Longitudinal forces and bending moments, as well as displacements and their first derivatives with respect to curvilinear coordinates, were taken as nodal unknowns. To obtain the compliance matrix, the Reissner functional was used, in which the stresses, when using the direct normal hypothesis, are represented by dependences on the forces and bending moments of the middle surface, the approximation of which was carried out by bilinear functions. In the interpolating expressions for the kinematic sought quantities, Hermite polynomials of the third degree were used. As a result of minimizing the transformed functional with respect to the force and kinematic nodal unknowns, the compliance matrix of the accepted discrete element was formed. Verification of the developed discrete element in a mixed formulation was carried out on the examples of calculations of cylindrical shells with circular and elliptical cross sections. The values of the force parameters found using the developed algorithm adequately satisfied the conditions of static equilibrium (the calculation error was less than 0.5 %). An analysis of the obtained finite element solutions showed the effectiveness of the developed algorithm and made it possible to note the possibility of its use in calculations of thin-walled structures made of incompressible materials.
Citation: Klochkov, Yu.V., Pshenichkina, V.A., Nikolaev, A.P., Vakhnina, O.V., Klochkov, M.Yu. Stressstrain state of elastic shell based on mixed finite element. Magazine of Civil Engineering. 2023. Article no. 12003. DOI: 10.34910/MCE.120.3
1. Introduction
Definition of the object of the study. The current widespread use of thin-walled shell-type structures (pipelines, tanks, hangars, domed roofs, wide-span ceilings, and others) puts forward a rather urgent task of creating domestic computational algorithms for analyzing the stress-strain state of such technospheric systems and objects.
Literature review. At present, when choosing the optimal shapes and sizes of thin-walled shell-type structures, numerical methods for analyzing their SSS [1-6] come to the fore, with FEM taking the priority position. It is widely used in calculations of plates and shells both under elastic [7-12] and elastoplastic [13, 14] deformation. FEM is essential in the analysis of SSS structures made of composite materials [15-17], as well as in matters of shell stability [18]. Three-dimensional finite elements are used both in the analysis of the stress-strain state of bulk structures and thin-walled structures [19-21].
The relevance of the research. Most of the currently created finite element computing systems are based on FEM in the formulation of the displacement method, which inevitably leads to the need to calculate second-order partial derivatives of the normal component of the displacement vector when using the theory of thin shells [22] based on the Kirchhoff-Love hypotheses. At the same time, finite element algorithms for
© Klochkov, Yu.V, Pshenichkina, V.A., Nikolaev, A.P., Vakhnina, O.V., Klochkov, M.Yu., 2023. Published by Peter the Great St. Petersburg Polytechnic University.
determining the stress-strain state of shell structures in a mixed formulation [23, 24] make it possible to obtain the desired internal force quantities (longitudinal forces and bending moments) directly in the process of solving the system of equations formed as a result of minimizing the mixed Reissner functional. This can also achieved without organizing additional computational procedures that greatly complicate the finite element algorithm for calculating thin-walled shell-type structures.
The purpose and objectives of the study. This paper presents the derivation of the modified Reissner functional, in which the total specific work of stresses is expressed in terms of the specific work of longitudinal forces and bending moments at a point of the middle surface on deformations and curvatures of the middle surface at this point. By minimizing the modified mixed functional with respect to force (longitudinal forces and bending moments) and kinematic (displacement vector components and their firstorder partial derivatives) nodal unknowns, the compliance matrix and the column of nodal forces of a quadrangular discretization element, which is a fragment of the middle surface of a thin-walled shell-type structure, are assembled.
The verification of the developed algorithm was carried out on the example of determining the SSS of cylindrical shells with circular and elliptical cross sections. An analysis of the results of the obtained finite element solutions made it possible to conclude that the developed algorithm is effective and that the calculation accuracy of the required force and kinematic nodal unknowns is acceptable.
2. Materials and Methods
The median surface of a thin-walled shell-type structure can be given by the radius vector
R0 = xi + y(x, t) j + z(x, t)k, (1)
where t is a parameter counted from the vertical axis in a plane perpendicular to the axis Ox, which is at a distance of x from the origin.
Basis vectors of a point M0 are determined by derivatives
a o = R0. -0 = R0. a 0 = -0 v -ol (2)
a- = Rx. a.2 = Rt. a = a- v aw wa0 , (2)
where a0 = (a° • a-0) (a° • a°) - (a-0 • a°) .
The derivatives of the basis vectors of point M0 are determined by the components in the same basis [25]
a0 =r0pa0 + b0 a0. a0 = b0pa0 (3) aa,p = 1 apap + bapa ; a,p = -bp ap , (3)
where indices a, p, p take values 1, 2; ^p are Christoffel symbols of the second kind; ¿p°p are mixed components of the curvature tensor.
The position of the point of the shell at a distance of Z from the point of the middle surface M0, as well as its position after the application of a given load, are determined by the radius vectors
R0Z = R0 +za0; RZ = R0Z+F. (4)
The displacement vector V of point
m 0Z according to the direct normal hypothesis [22] can be
represented by the following expression
V = v + z(a-a0), (5)
where v = vpa.p + va0 is the displacement vector of point
M 0Z; a = a- v a2/-Ja is unit vector of the
normal at point M; ap = a0 + vp are covariant vectors of the local basis of point M of the deformed state.
Here and below, the comma means the operation of differentiation with respect to global coordinates x and t.
The basis vectors of points M 0Z and M z are determined by the corresponding differentiation (4) with respect to x and t
gP = ROZ = ap0 -zb0Ya0; g, = R,p = gp + vp +z(a>p -bp0Yay0). (6)
Deformations at point M z of a thin-walled shell-type structure are determined by the difference between the components of the metric tensors at the point of the initial and deformed states [26]
SPY (g PY g Py)/2'
(7)
Four-node element. The finite element is represented by a quadrangular part of the middle surface with nodes i, j, k, l. Taking into account that when implementing the mixed formulation of the FEM, there is no need to include the desired unknown higher-order derivatives in the structure, the column of nodal variable parameters of the used quadrangular sampling element in the local -1 n<1 and global x, t coordinate systems was chosen in the form
{UL i
1x60
{u
1x60
0 f =
{Nf {M}T {v1L f {v2L f {vL f
1x12 1x12 1x12 1x12 1x12
{N}7 {M}7 {v10 }7 {v20 f {v0 }}
1x12 1x12 1x12 1x12 1x12
(8)
(9)
where
{N}7 = {N 11iN 11JN11kN111N22i...N221N 12\..N121};
1x12
{M}T = {m 11iM11JMnkM111M22i...M221M12i...M121} are columns of power parameters;
( L^ ( i j k l i l i M ( G)J ( i j k l i l i l\ {q } =\qqJqqqA ... q,^ ... q,n); {q ) ={qqJqqqx ... qxqt ... qt}
1x12 1x12
kinematic parameters in local -1 n<1 and global x, t coordinate systems, respectively. Here, q means the values vp, v.
Bilinear functions of local coordinates n [27] were used as shape functions for the force unknowns
\T
are columns of
Naß = {^}T {Naß}; Maß={^}T {Maß},
1x4 4x1 1x4 4x1
(10)
and for the kinematic required unknowns, the products of Hermite polynomials of the third order were applied [27]
q = MT {qL }•
1x12 12x1
(11)
Compliance matrix of a four-node bin. To obtain the compliance matrix of a four-node discretization element, one can use the Reissner functional written in the following form
ns ={{°PYf {4}dV-0.5J{aPY}T [C]{aPY}dV-0.5 J{U}T {P}dF, (12)
V V F
where {aPYf ={ayW2}; {sPYf ^efc^}; }T ={v1v2v}; {P}7 ={p1p2p} is column of the external surface load vector.
In accordance with [26], the elasticity matrix [C] included in (12) determines the relationship between columns {spy} and {^PY}
{spy| = [C ]{*"}. (13)
Column {^pY}, on the basis of the theory of thin shells [22], can be expressed in terms of the required force unknowns, which are the longitudinal forces NaP and bending moments Map
Ky} = [ NM }, (14)
3x1
1/h 0 0 Z/I 0 0 " 0 1/h 0 0 Z/I 0 0 0 1/h 0 0 Z/I
the shell thickness; I = h3/12 is moment of inertia.
3x6 6x1
where [DCT ] =
3x6
{NM}T ={N11N22N12M11M22M12 h is
The column of covariant components of the strain tensor at point M Z, taking into account the direct normal hypothesis [22], Cauchy relations (7), and interpolation dependence (11), can be represented by the matrix relation
{sPr} = [De]{Bpy} = [D8][5]{,i} = [D8][5] [T] p}
3x6 6x1 3x6 3x36 36x1 3x6 3x36 36x36 36x1
(15)
where [ Ds ] =
s,
3x6
10 0 z oo" 0 10 0 z 0
T
{sPY} ={s11s222s12K11K222K12} is a column of
1x6
00100 z
deformations and curvatures at point M of the middle surface; {uL} = <{v1L} {v2L} {vL}
1x36
1x12 1x12 1x12
P }'=|P }>2G }
1v36 L 1v12 1v12 1v12 from the local coordinate system n to the global one x, t.
; [ T] is transformation matrix of the column of kinematic quantities
The column of power variable parameters {NM} is interpolated through its nodal values using relations (10)
{NM } = [ H ]{Gaß},
6x1 6x24
24x1
where
{gaß }T = J {N11 }T {N22 }T {N 12 }T {M 11 }T {M22 }T {M 12 }T
1v4 1v4 1v4 1v4 1v4 1v4 Functional (12), taking into account (14) and (16), can be represented as
T T
n s = {gaß} J[H ]T [ DCTf [De][B] dV [T ] {uG }-
1x24 V 24x6 6x3 3x6 6x36 36x36 3^1
- 0.5{Gaß}T J[H]T [DCT]T [C][D0][H]dV{Gaß} -
(17)
1x24 V 24x6 6x3 3x3 3x6 6x24
24x1
- 0.5 {uG }T [T] T J[A]T {P} dF,
1x36
36x36 f 36x3 3x1
W 1x12 0 0
[ A] = 3x36 0 w 1x12 0
0 0 M 1x12
Applying to (17) the procedure of minimization with respect to the required unknowns {GaP}7 , we can obtain the following matrix expression
dns d{GaP}7 = [S] {uG }- [Z] {GaP} = 0,
24v36 36v1 24v24 24v1
where [ S ] = J[ H f [Daf [De][B] dV [V]; [Z ]=J[H f [Daf [C][Da][H ] dV.
24v36 V 24v6 6v3 3v6 6v36 36v36 24v24 v 24v6 6v3 3v3 3v6 6v24
The first integral in functional (12) can be represented in the following form
\T
(18)
JM^}dV = {{sy {o^}dV =
V 1x3 3x1 V 1x3 3x1
= {uG}T [T] T J [B]T [D8]T [Do]T [H]dV{Gaß}.
1x36 36x36 V36x6 6x3 3x6 6x24 24x1
(19)
By minimizing the functional (17) taking into account (19) with respect to the kinematic unknown unknowns {uG } , we can write the following matrix relation
dns/d{uG }T = [S ] T {GaP}-{R} = 0, (20)
where {R} = [T]T J[a] {P} dF.
36v24 24v1 36v1
TT F
The system of equations obtained as a result of minimizing the functional ns with respect to
{g ap}T and {uG } can be represented in the matrix form
-[ * ] [ S ]
24x24 24x36
[ S ]T [0]
36x24 36x36
{Gaß}
24x1
{uG}
36x1
{0}
24x1 {*}
36x1
or in a more compact form
[ K ]{U0 } = { f }, (22)
where [ K ] =
60x60
-[ * ] [ * ]
24x24 24x36
[ S ]T [0]
.36x24 36x36,
60x60 60x1 60x1
is the flexibility matrix of a four-node sampling element;
T I T T I
{f) = < {o) {R} > is column of nodal forces.
1x60 [1x24 1x36 J
An analysis of the structures of matrices [Z] and [S] in the compliance matrix [K] shows that matrix [K] is also a determinable value in the case of an incompressible material at a transverse strain coefficient of v = 0.5.
Analyzing the resulting compliance matrix [K], it can be noted that it contains a significant zero
block [0] , which can significantly reduce the conditionality of the global compliance matrix of the entire
36x36
shell-type structure. To eliminate this problem, this paper proposes to carry out the following transformations.
Let us express from equation (18) the column of force nodal unknowns
{g °P)= [ Z ]-1 [ S ]{uG ) (23)
24x1 24x24 24x36 36x1
and substitute relation (23) into equation (20)
[S] T [*]-1 [S]{u0 }-{R}= 0. (24)
36x24 24x24 24x36 36x1 36x1
Transforming (24), we can obtain the following matrix expression
[L] T ={uG ) = {R), (25)
36x36 36x1 36x1
T —1
where [L] = [S] [Z] [S] is the modified compliance matrix of the four-node bin.
36x36 36x24 24x24 24x36
Analyzing (25), it can be noted that [L] does not contain a zero block and differs from [K] in
36x36 60x60
a significantly smaller dimension, which reduces the requirements for the amount of RAM used by computer equipment when studying the stress-strain state of a thin-walled shell-type structure.
Based on the obtained modified compliance matrix [L] , with the help of the index matrix [28], the
36x36
global compliance matrix of the entire calculated thin-walled shell-type structure is assembled and the solution of the global system of algebraic equations is performed, the unknowns of which are only the
{uG }.
kinematic nodal unknowns {u
G
After calculating the kinematic nodal unknowns {uG} using (23), without any difficulty, one can
obtain the values of the desired force unknowns at any point of interest to the designer in the considered thin-walled shell-type structure.
Verification of the developed computational algorithm based on the use of the modified compliance matrix of the four-node discretization element [L] was performed on specific calculation examples.
36x36
3. Results and Discussion
Calculation example 1. As a test example, a circular cylinder was calculated, rigidly clamped on the right end and having a free edge on the left end. The radius vector (1) in this case will look like
R0 = xi + R sin t j + R cos tk.
(26)
The cylinder was loaded with an internal pressure of intensity qw and a uniformly distributed axial load qu applied along the free left end. The design scheme of the shell is shown in Fig. 1.
Figure 1. Calculation scheme of a circular cylinder with a uniformly distributed axial
load qu and internal pressure qw .
The following initial values are accepted: R = 0.9 m; h = 0.02 m; L = 0.8 m; E = 2105 MPa; v = 0.3; qw = 5 MPa; qu = 500 kN/m.
The values of stresses in the edge sections of the shell are presented in Table 1 for various variants of discretization of the shell fragment, considered according to the symmetry conditions.
Table 1. Stress values in sections of a cylindrical shell.
Characteristic section
Stress, MPa
Grid of Discretization Nodes
21x21
41x41
51x51
61x61
Analytical Solution
in 410.5 417.5 418.3 418.8 —
out -360.5 -367.5 -368.3 -368.8 -
G11
midl 25.00 25.00 25.00 25.00 25.00
Rigid °11
termination in a22 117.5 122.3 123.0 123.3 -
out -113.7 -113.1 -113.0 -112.9 -
a22
,^midl 4.48 7.24 7.62 7.84 -
CT22
in 24.98 24.99 25.00 25.00 -
G11
out 25.02 25.01 25.00 25.00 -
G11
midl 25.00 25.00 25.00 25.00 25.00
G11
Free end
in a22 226.8 226.8 226.8 226.8
out 222.0 222.0 222.0 222.0 -
a22
midl 224.5 224.5 224.5 224.5 225.0
a22
An analysis of the data presented in Table 1 allows us to state the fact of a fairly fast convergence of the computational process as the grid of discretization nodes thickens. In addition, it should be noted that the numerical values of normal stresses correspond to the physical meaning of the problem being
solved. Meridional stresses an on the middle surface in the outer and inner fibers of the edge sections
of the cylindrical shell correspond to a given axial external load
a™d/ = qu/h = 500 kN/m/0.02 m = 25.0 MPa.
Ring stresses of the middle surface at the free end of the cylinder g^1 = 224.5 MPa correspond to the specified internal pressure qw with an acceptable level of error 5 = 0.22 %.
_midl qw 'R
°22 =—.-
h
5 MPa • 0.9m 0.02 m
= 225.0 MPa.
The developed algorithm for determining the stress-strain state of thin shells, which implements a mixed version of the FEM, makes it possible to immediately obtain internal force factors (longitudinal forces and bending moments) at any point of the shell structure of interest to the researcher without excessive labor-intensive calculations. "Physical" values of forces and moments in the edge sections of the cylindrical shell, referred to the middle surface, are presented in Table 2, the structure of which is similar to Table 1.
Table 2. Values of forces and moments in a circular cylinder.
Characteristic section
Efforts, N; Grid of discretization nodes
moments, N • m 21x21 41x41 51x51 61x61 Analytical So
Nu 500.0 500.0 500.0 500.0 500.0
N22 89.7 144.7 152.5 156.8 -
Mu -2570.0 -2616.6 -2622.3 -2625.4 -
M22 -771.2 -785.8 -787.6 -788.6 -
Nu 500.0 500.0 500.0 500.0 500.0
N22 4490.5 4490.3 4490.3 4490.3 4500.0
M11 0.131 0.035 0.023 0.016 0.000
M22 -49.3 -49.3 -49.3 -49.3 -
Rigid termination
Free end
The data in Table 2 testify to the stable convergence of the computational process in terms of forces and moments. The values of the axial longitudinal forces Nu in the edge sections of the cylindrical shell
correspond to a given axial load of qu = 500 kN/m. The value of the longitudinal ring force also
corresponds to a given internal pressure qw with a minimum error ô = 0.2 %.
The bending moment Mu tends monotonically to zero in the end section.
On the basis of the foregoing, it can be concluded that the developed algorithm is correct and that the accuracy of calculating the controlled strength parameters of the SSS of shell structures is sufficient for engineering practice.
Calculation example 2. The stress-strain state of a cylindrical shell with an elliptical cross section, rigidly fixed at the ends, loaded with an internal pressure of q = 5 MPa is determined. Due to the presence of symmetry, 1/8 of the shell was considered. The design scheme is shown in Fig. 2.
Figure 2. Calculation scheme of a cylindrical shell with elliptical cross section.
The radius vector expression (1) for an elliptical cylinder will look like this:
R0 = xi + b sin t j + c cos tk.
(27)
In the problem under consideration, the following initial data are accepted: b = 1.0 m; c = 0.8 m; l = 1.0 m; h = 0.02 m; E = 2105 MPa; v = 0.3.
The values of normal stresses and bending moments in the shell sections for various variants of discretization of the calculated fragment of the shell are presented in Table 3.
Table 3. Values of normal stresses and bending moments in an elliptical cylinder.
Grid of discretization nodes
Stress, mom N^m
Characteristic .
.. MPa; moments,
section 41x41 51x51 61x61 81x81 101x101
Solution in the formulation
of the displacement method, 61x61
Rigid termination,
X = 0.0;
t = 0.0 rad.
(point A)
Rigid termination,
X = 0.0;
t = n / 2 rad.
(point B)
in G11
_out a11
in a22
out a22
Mu M22
in G11
out a11
in a22
a
out 22
Mi
11
M.
Mid-span, X = L / 2; t = 0.0 rad. (point C)
22
in a11
out a11
in a22
out a22
642.1 -544.6 183.5 -172.5 -395.6 -118.7 264.8 -118.3 85.21 -29.70 -127.7 -38.90 71.35 95.77
312.4
338.5
642.4 -544.9
185.1 -171.1 -395.8 -118.8
265.2 -118.7 84.00 -31.12 -128.0 -38.95 71.34 95.79
312.3
338.5
642.5 -545.0 186.1 -170.1 -395.8 -118.9
265.4 -118.9
83.16 -32.10 -128.1 -39.00 71.33 95.80 312.3
338.5
642.7 -545.2
187.4 -169.0 -396.0 -118.9 265.6 -119.1
82.06 -33.32 -128.2 -39.00 71.33 95.80 312.3
338.5
642.8 -545.2 188.1 -168.3 -396.0 -118.9 265.7 -119.2 81.38 -34.06 -128.3 -39.00 71.32 95.81 312.3 338.5
642.4 -544.8 192.7 -163.4
265.5 -119.0 79.65 -35.70
71.27 95.67
312.0
338.1
Grid of discretization nodes Solution in the
Characteristic section Stress, MPa; moments, N^m 41x41 51x51 61x61 81x81 101x101 formulation of the displacement method, 61x61
in G11 37.72 37.72 37.73 37.73 37.73 37.84
Mid-span, X = L / 2; out a11 9.81 9.81 9.81 9.81 9.81 9.83
t = n / 2 rad. in a22 out a22 173.9 173.9 174.0 174.0 174.0 174.7
(point D) 144.2 144.2 144.2 144.3 144.3 144.8
As follows from the analysis of tabular data, the convergence of the computational process in terms of both stresses and moments is very stable. In order to verify the developed algorithm, the rightmost column contains the stress values found based on the use of a quadrangular finite element, the stiffness matrix of which was composed on the basis of a finite element procedure in the formulation of the displacement method [27]. As shown by a comparative analysis of the finite element solutions obtained on the basis of the developed algorithm, and the FEM in the formulation of the displacement method, the numerical values of the normal stresses practically coincide at all characteristic points with an acceptable
minimum discrepancy in the values of a 22 at points A and B. When performing this comparative analysis
one should take into account the fact that when implementing the developed algorithm in a mixed formulation, it is possible to directly obtain the numerical values of force factors (forces and moments) and stresses at any of the nodal points of the calculated shell. When using FEM in the formulation of the displacement method, to obtain numerical values of stresses, it is required to perform several stages of computational procedures, namely: after obtaining the displacement values and their first derivatives, it is necessary to calculate the values of the second derivatives of the normal displacement using an interpolation procedure. Then, using the Cauchy relation [22], it is necessary to calculate the deformations of the midsurface point. Next, it is necessary to proceed, on the basis of the Kirchhoff-Love hypotheses, to deformations at a point of an arbitrary layer of the shell, and only after that, using the relations of Hooke's law, it is possible to obtain the stress values. All the above computational procedures complicate the calculation algorithm and increase the calculation error. The use of a mixed formulation of the FEM implemented in the developed algorithm makes it possible to avoid additional cumbersome computational procedures and makes it possible to directly obtain the desired strength parameters of the calculated shell structure, which ultimately makes the developed algorithm the most preferable in the analysis of SSS of shell structures of various configurations.
Calculation example 3. The quadrangular discretization element developed in this work in a mixed formulation can be effectively used to study the SSS of shells made of an incompressible material. The problem was solved to determine the strength parameters of an elliptical cylinder, the design scheme, the geometric and physical characteristics of which coincide with the data of calculation example 2. The difference was that Poisson's ratio was taken equal to v = 0.5, i.e. it was assumed that the shell is made of an incompressible material. The results of the numerical experiment are presented in tabular and graphical forms. Table No. 4, the structure of which is similar to the structure of Table No. 3, presents the numerical values of normal stresses and bending moments in the support and span sections of an elliptical cylinder, depending on the degree of refinement of the grid of discretization nodes of the calculated shell fragment. Analyzing the tabular data, one can state the stable convergence of the computational process as the grid of discretization nodes becomes denser.
Table 4. Values of normal stresses and bending moments in an elliptical cylinder made of incompressible material.
Characteristic Stress, MPa; moments, N^m Grid of discretization nodes
section 41x41 51x51 61x61 81x81 101x101
in G11 689.8 690.0 690.1 690.2 690.3
Rigid termination, X = 0.0; out a11 in a22 -493.3 334.4 -493.5 336.0 -493.6 337.1 -493.7 338.4 -493.8 339.1
t = 0.0 rad. (point A) out a22 Mu -257.0 -394.3 -255.6 -394.5 -254.7 -394.6 -253.6 -394.6 -252.9 -394.7
M22 -197.6 -197.7 -197.8 -197.8 -197.8
in G11 247.8 248.1 248.3 248.5 248.6
Rigid termination, X = 0.0; out a11 in a22 -41.27 129.5 -41.60 128.3 -41.77 127.4 -41.95 126.3 -42.03 125.7
t = n / 2 rad. (point B) out a22 M11 -14.97 -96.36 -16.52 -96.57 -17.55 -96.69 -18.83 -96.81 -19.60 -96.86
M22 -49.39 -49.47 -49.51 -49.55 -49.56
Mid-span, X = L / 2; in G11 out a11 108.7 156.3 108.7 156.3 108.7 156.3 108.7 156.3 108.7 156.3
t = 0.0 rad. (point C) in a22 out a22 305.4 347.3 305.4 347.2 305.4 347.2 305.4 347.2 305.4 347.2
Mid-span, X = L / 2; in G11 out a11 73.92 34.0 73.92 34.0 73.92 34.0 73.91 34.0 73.91 34.0
t = n / 2 rad. (point D) in a22 out a22 179.4 139.2 179.4 139.2 179.4 139.3 179.5 139.3 179.5 139.3
in out
Fig. 3 shows the graphs of changes in normal stresses on the inner a and outer a surfaces of the shell, as well as bending moments Mn, on the middle surface along the generatrix of the
elliptical cylinder.
The analysis of the graphical material shows that the maximum values of the edge effect appear directly in the rigid embedment, gradually fading towards the zone located at a distance of 0.1 L from the reference section, which corresponds to the physical meaning of the problem being solved.
er, MPa
1
3 V
S(t* 1 . -— ■ -
0 r ' 10 iy / 15 20 25 30 35 40 45 50
sy 6 x/ 4 J ¿s Generating length
// 5 4 2
er0"
"II
5 ■ Mii
6 — - -Vf-n
Figure 3. Diagrams of normal stresses and bending moments along the generatrix at t = 0.0 rad.
m out
Fig. 4 shows the changes in the normal stresses a and a , as well as the bending momei Mn, M22 along the arc of the shell cross section in a rigid enclosure (x = 0.0 m).
Analyzing the graphs presented in Figure 4, it can be noted that the controlled strength characteristics (normal stresses and bending moments) reach a maximum at the value of parameter t equal to zero. Then the values of normal stresses and bending moments gradually decrease (by about two times) to their minimum values in the reference section at the value of parameter t equal to n/2.
Figure 4. Diagrams of normal stresses and bending moments in rigid termination at x = 0.0 m.
4. Conclusions
Taking into account the results of the numerical studies, we can draw the following conclusions.
1. The convergence of the computational process using the developed finite element in a mixed formulation is stable in terms of both force and kinematic factors.
2. The obtained numerical values of the stresses at the controlled points are in adequate agreement with the stress values found from the conditions of static equilibrium (the calculation error does not exceed
0.5.%).
3. The use of the developed mixed finite element leads to the possibility of determining the power parameters directly as a result of solving the system of resolving equations.
4. The developed finite element in a mixed formulation is suitable for determining the SSS of thin-walled structures made of incompressible materials (v = 0.5 ).
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Information about authors:
Yuriy Klochkov, Doctor of Technical Sciences ORCID: https://orcid.org/0000-0002-1027-1811 E-mail: klotchkov@bk.ru
Valeria Pshenichkina, Doctor of Technical Sciences ORCID: https://orcid.org/0000-0001-9148-2815 E-mail: vap hm@list.ru
Anatoly Nikolaev, Doctor of Technical Sciences ORCID: https://orcid.org/0000-0002-7098-5998 E-mail: anpetr40@ yandex. ru
Olga Vakhnina, PhD in Technical Sciences ORCID: https://orcid.org/0000-0001-9234-7287 E-mail: ovahnina@bk.ru
Mikhail Klochkov,
ORCID: https://orcid.org/0000-0001-6751-4629 E-mail: m. klo4koff@yandex.ru
Received 01.03.2023. Approved after reviewing 26.04.2023. Accepted 27.04.2023.
