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4
Sabirov Nizambay Khayitbaevich, Head of the Department "Information Technologies", Tashkent Institute of Textile and Light Industry E-mail: sobir1.n@mail.ru Yuldashev Tojimat, Doctor of technical science Department of Structural Mechanics Tashkent Institute of Railway Transport Engineering E-mail: str.mech@gmail.com Abdusattarov Abdusamat, Professor, Doctor of technical science Department of Structural Mechanics Tashkent Institute of Railway Engineering E-mail: str.mech@gmail.com.
EQUATIONS OF MOTION AND THE FORMATION OF A DIFFERENCE SCHEME FOR CALCULATING THE CONICAL PART OF SHELL STRUCTURES
Abstract: Based on the variation principle of the Hamilton-Ostrogradsky principles, mathematical models are deformed and finite-difference schemes for calculating the conical part of shell structures are formulated.
Keywords: conical shell, design scheme, equations of motion, equilibria, stresses, deformation, sweep method.
As is known, composite thin-walled shell structures are widely used in modern technology and have a significant specificity of structural forms, manufacturing technology, operating conditions and material properties. Simulation of deformation processes, the development of numerical methods and the creation of software tools for calculating composite shell structures (tank type) consisting of cylindrical, spherical or conical shell elements under static and dynamic loads have acquired particular urgency [1-4]. In this connection, the paper considers the equations of motion (equilibrium) and the formation of a finite-difference scheme for calculating the conical part of shell structures. In this case, the Hamilton-Ostrogradsky variational principle and the Bubnov-Galerkin's method are used [3].
Consider a truncated conical shell of thickness h and length l = S - S0 along the generator. The position of the shell point will be determined in the conic coordinates
(a, p,y} .Concerning the middle surface and the length along the generatrix, we have the following relations.
-2 <y < 2, 0 <p<p,, S0 " S" S. ( * 0)
The coefficients of the first quadratic form are equal to A = 1; B = S ■ sin a , and the radii of curvature of the middle surface R1 =<*>, R2 = ctgd / S. For this problem, the Lamé coefficients have the form:
H1 = 1; H2 = S sinO + ycosO; H3 = 1 (1) Denote the displacements by U1 = Ua,U2 = Up, U3 = Us. Using Kirchhoff-Love, according to which the deformation of the shell occurs without deformation of shear l12 and l13 in the plane of normal sections and without deformation of elongation l11 over the thickness of the shell:
I12 = I13 = I11 = 0. (2)
For U 1,U2,U3 we have:
U = W (a, ß, U2 =| 1 + Y ctg0\V—l V s ) s ■ si
■ sine dp
According to formula (3), we represent deformations in the following form:
1
y dW TT TT dW
, U3 = U-r —
ds
(3)
_du- dW. 1 __
133 _ Y "Z ~.; '22 _ . ^ ^
da da a sine + y cose
1 + Y ctg0]dV + W cos0 + (u-y^^ I sine;
a J dß a sine dß2 ^ da 1
I.-.?
1
dU
asin0 + 7cos0 dß , /i 1
1
1
d2W
1 + ctgd I--y,
a J da ^asinö asinö + Ycosö Jdadß
1
+
, dWUi+Lctge)v
a^asinö asinö + Ycosa) dß a^ a )
The stress components are determined by the formulas:
^ = ( + 2G)) + M№; aßß = Xlaa + ( + 2G)), aaß = Glaß
Now, on the basis of the Hamilton-Ostrogradski's principle, we make the variational equation [1]:
J(öT-ön + öÄ )dt = 0
(4)
(5)
(6)
In determining the variation of the kinetic, potential energy, and the work of external forces, the following relationships are used [3; 4]:
Jsr = JJp| dU sdUSU+U sdU
dt dt dt dt dt dt
jöTT = | ¡(CT22ÖI22 + O33ÖI33 + a 238X23 yivti
t t v
J S A = J J [Xl5Ul + X2ÖU2 + X 35U3 ] (a ■ sin G + y ■ cosd)dadßdydt
j(asinö + ycosO)dadßdydt (7)
01+5U11 + |WsU2 f+|'] + 03+5U3 f+3
01-^U 11 -2Wuf-2WU[-2
asinö + —cosd 2
h
asinO- —cosO \ \dadßdt +
(8)
+| J J [U1 + P25U2 + P3SU 3 ] dadydt + J J J [q[ + q2SU2 + q 3SU3 ]] (a sin 0 + y cos 0 )dfidydt
t y a t y fi
Following [3], we form the variational equation taking into account (4) - (5) and (7) - (8), then we apply the Bubnov-Galerkin procedure with respect to the coordinate:
wi wi (9)
u = XU (x,t)sinP; y = (x,t)cosn ; W = YWn (x,t)sin—P
n Pi n Pi n Pi
As a result, we obtain the following system of differential equations of motion of the conical shell with the corresponding boundary and initial conditions:
-a
(1)
-a
(2)
d2Un dt2
d 2Vn
dt2
■Z1)
d3W„
-- a
(1)
d3W„
d2U„
■Z1)
dU„
dxdt2 3 dx3 dx2 dx -ai8)Un + afV - a1(0)Wn + Xn = 0 ;
-a
(1)
,(2)
d2Wn dt2
,(2)
d2Vr dx2
+a(2)U - a(2)V + a(2W + Y = 0
-a
,(2>
(2)
d2Wn dx2
(2h,
,(2)
dUn dx
,(2)
dVn dx
dVn dx
,(2)
dWn dx
dWn dx
- +
(10)
-a
(3)
d2Wn dt2
,(3)
d2V dt2
-a
(3)
d3Un dx dt2
,(3)
diWn dx 2dt2
-- a,
(3)
d4Wn dx4
,(3)
d3Ur dx3
-a
(3)
d3W„ dx3
2
2
7
2
4
6
-a.
(3)
d2V
dx
2 9
«dx - a (3)
2 10
dx2
dUn dx
,(3)
dVn dx
-a
(3)
dWn dx
- a(3)Un + a(X - a1(3)Vn + Zn = 0
i(3h
i(3h
Border conditions:
b« ^X - bi U - b (u + blV - bW + T *
dx2
dx
SU.
= 0 ;
-b^+b22)dW- - b32Un+b?V - b5(2)Wn+s;n
dx
dx
5V
= 0 ;
(11)
-b (3)
d2U.
b(3)
d 2W„
1 dt2 2 ôxôt2
■b(3)
d3Wn dx3
- b(3)
d2Un dx2
-b(3)
d2Wn dx2
-b (3)
dx
,(3)
dWn dx
-bfUn -b9(3Vn -b1(3)Wn + R'ln 1\5Wn = 0;
-I x
-b.W^K + b(Ä-b(4Ä-b(4)V + b(4)W -M*
dx2 2 dx
dx
5
dW„
dx
= 0.
Initial conditions:
(()
,(2)
-c
dUn dt
dWn ~dt
-c
(()
dVn
HT
-c
(1)
d 3W„
- + c
(()
d2U„
3 dx2dt 4 dxdt
-- c
(2)
d2Wn dx dt
SU.
= 0 ;
,(3)
ÔV»
--c
(3)
öWn
dWn ' dt
= 0 ;
(12)
SV.
= 0.
To form a difference scheme, first the system of result of approximation we have the following systems differential equations (10) is represented in vector form. of algebraic equations:
We introduce the following vectors
Un = (wnUVn)T ; Fn =(ZnXnYn)T (i 3)
According to (13), the equation of motion of the conical shells (10) takes the form:
A(Un + Aßl + Aß" + AU + A5Ün" +
+ A6U n + AU + AsUn + Fn = 0 ( 14) The initial conditions will also be represented in vector form:
[3(Un + 32U'n + 3U'1 ]öUn\ = 0 (15)
A.U.,., + 3.U., , + C.U.,. + DU_ +
EU
)
i+2
3
F = 0
(18)
where An = A)4 -A5—; 3n = -4A(N4 + A5N3 + A6N2 - A7 N ; Cn = 6A4N4 - 2A6N2 + A8 ; ; Dn = -4A(N4 - A5N3 + A6N2 + A7 N ;
Here the matrices u are ofthe third order, respectively, with the elements a . (x, n) and b. (x, n).
4 N
En = A(N4 + A5 —;
(19)
form:
We consider the difference boundary conditions. We assume that the conical shell is clamped when x = 0 and We rewrite the boundary conditions (11) in the x = 1. It follows from the conditions: (16)
ZmSU\ = 0 ; Zln8V\ = 0 ;
(Z3n - b(]Un + bfWn )swn = 0; Zin5W'n\ = 0 .(16)
V } x 'x
Equation (15) is written without taking into account inertial terms:
AU: + a5U': + A6i7: + AU + AU + F = 0 (17) Substituting (21) into the system of difference
Nowwe use central difference schemes approximating equations (19), we obtain a system of linear algebraic the derivatives with second-order accuracy [5]. As a equations in the form:
Wn,0 = 0 ,Un,0 = 0 ,Vn,0 = 0 ,Wn,N = 0 ,Un,N = 0 ,Vn,N = 0 Wn,-( = Wn,( ,Wn,N+( = Wn,N-( ; (20)
These relations can be written in vector form:
Un,0 = 0, A-Un,_( = A-Un,( ,Un,0 = 0,
A(+(Un N+( = A(+(Un ,N-( (21)
x
x
7
x
AU = b (b = Fn)
(22)
where
A =
Q A E,
B2 C2 D2 E2
A3 B3 C 3 D3 E3 —
— An—3 BN—3 CN—3 Dn—3 E N—3
AN —2 BN—2 CN—2 Dn—2
V AN—1 Bn—1 CN—1 y
here C i = AnAl_i + Q ; cn-i = CN-1 + EN+1
To the solution of the system of linear algebraic equations (22) we apply Godunov's method of sweeping. As a result, we obtain the following recurrence relations on the forward run [5]:
Un= f - HkUnk+1 - FkUnM2 (23)
where
f = Gt \bk - Akfk-2-( - AHk-2 ))-i]; Hk = Gk [Dk-(Bk - AkHk-2 )Fk-i] Fk = GkEk ; Gk = [Ck - AF -2-(Bk - AHk -2 )Hk-i
a) b)
Figurel. Epure of deflections and bending moment
The direct run of the sweep ends at k = N -1 and determine the value of the last unknown vector Un,N or Un N _1. Then, using the formula (23), perform the reverse move and find the value of the grid functions Un ,k for an arbitrary term of the series (9).
As an example of the use of software tools, the VAT of the conical shell, pinched by both contours, is subject-
f nB^
ed to the action of distributed load q = q0 1 + cos—
Pi
V
/
where q0 is the external pressure directed perpendicular to the axis of the shell.
The calculation is performed with grid spacing N = 40 with the following geometric and mechanical charac-
teristics: a = (a1 -a0) 1a0 = 0,93 ; S = h(a1 -a0) = = 0,0547 ; q0 = 1kg / cm2 ; h = 1cm ; 9=n/3; E = 2,1 ■ 106 kg/sm 2; The projections of external forces have the form: = q ■ cos d, 02+ = q ■ sin d.
The numerical convergence of the calculated values at various grid steps (N = 10.20.30.40) and for various terms of the series (6) (N '= 1.5.9) is analyzed.
The solution of the problem is presented in the form of graphs and tables. For illustration, Figures 1 (a, b) show the deflection (W) and bending moment (R1) curves of different values (3. It can be seen from the figure that these values on the meridian P = 0 reach their maximum values, while for P = 0,7 they have mini-
mal values. A similar picture is observed for the displace- the conical part of shell structures with the use of the
ment (U), and shearing forces (Qj). sweep method and also the diagrams of the calculated
Conclusion. Equations of motion and finite-differ- quantities are given. ence schemes for solving boundary-value problems of
References:
1. Vlasov V. Z. General theory ofshells and its applications in engineering. - Moscow.: Gostekhizdat, - 1949. - 761 p.
2. Kovalenko A. D., Grigorenko Ya. M., Ilyin L. A. The theory of thin conical shells. Kiev, publishing house ANUK, -1963. - 287 p.
3. Abdusattarov A., Yuldashev T., Sobirov N.Kh. Modeling of calculation of shell structures of tank type. Tashkent, Uzbekistan, - 2016. - 108 p.
4. Abdusattarov A., Yuldashev T., Sobirov N.Kh., To the formation of the difference boundary-value problem of the spherical part of shell structures // Bulletin of Tashit, - 2017. - No. 1. - P. 35-44.
5. Godunov S. K., Ryabenky V. S. Difference schemes. - M:. Nauka, - 1977. - 440p.
