CC BY
0ALGORITHM FOR CALCULATING A THREE-LAYER ROD WITH BOTH ENDS RIGIDLY FIXED
1Anarova Sh., 2Ismoilov Sh., 3Shokirov D.
1Professor of TUIT, Tashkent, Uzbekistan 2PhD, NamECI, Namangan, Uzbekistan 3PhD student, NamECI, Namangan, Uzbekistan https://doi.org/10.5281/zenodo.13120745
Abstract. This article details a mathematical framework and computational method for examining the stress-strain conditions of a three-layer beam, fixed rigidly at both ends and subjected to spatial forces. The development of this model involves applying the Ostrogradsky -Hamilton principle, Cauchy's equations, and Hooke's law. A mathematical model for a three-layer rod is developed with appropriate equations, generalized initial and natural boundary conditions. The computational algorithm for the given problem is developed using the centralfinite difference method, and the implicit scheme of this method is employed in the solution process. The results obtained using the matrix driving method for second-order differential equations in the computational algorithm are presented through graphs.
Keywords: three-layer rods, mathematical model, Hooke's law, Cauchy relations, stress, deformation, transverse bending, displacement, Ostrogradskiy-Hamilton variational principle, finite differences, driving method, kinetic energy, potential energy, external forces.
In today's world, where digital technologies are advanced, three-layer rods and plates are an integral part of modern architecture and engineering. They are widely used in the construction of bridges, sports facilities, large illuminated areas, and other structures with large spans. The construction of these rods consists of three layers: the upper and lower layers and the filler layer. The main advantages of using three-layer rods include high strength, lightness, aesthetic appearance, and cost-effectiveness. These benefits make the use of three-layer rods an integral part of modern architecture and engineering, playing a significant role in meeting the construction needs of society.
In our country, many scientists have conducted research to improve the theoretical foundations and develop computational methods for multi-layer rod structures. Among them, academician V.Q. Qobulov developed a refined theory of the linear deformation processes of structural elements and proposed algorithmic approaches for solving practical problems [1]. In Uzbekistan, the creation of algorithmic systems for the algorithmization and automation of various types of mechanical problems was initially proposed by academician V.Q. Qobulov, who also developed the theory of algorithmization. This work has been further developed by academician T. Buriyev, K.Sh. Bobomurodov, F.B. Badalov, B. Kurmanbayev, T. Yuldashev, Sh.A. Nazirov, X. Eshmatov, B. Mardonov, M. Usarov, B. Babajanov, and their students.
Statement of the Problem. We will develop mathematical models for the displacement problems of points in three-layer rods with both ends rigidly fixed, based on the Ostrogradsky-Hamilton variational principle. For the load-bearing layers of the three-layer rod, Bernoulli's hypothesis is applicable, and for the filler layer, the exact relations of elasticity theory with a linear approximation of the displacements of points along the height are valid.
In developing the mathematical model for this problem, the Ostrogradsky-Hamilton variational principle is utilized, with the fundamental formula of this principle given as follows [1, 6-8]:
K -n + A)dt = 0. (1)
t
Here, K, n and A represent the kinetic energy, potential energy of the beam, and the work done by external forces, respectively.
The displacement of points in a three-layer rod will be as follows [2, 3]:
i ;„ A
u(1) = u -
(2)
u — u2 -
K
z - c —1 2 y
h
am, w(1) = w (c < z < c + K );
z + c + —
a
(2)
w(2) = w2 (-c - h2 < z < -c);
(3)
u( ) =
\+z V1
(3)
wy ' = — 2
V c y 1
k
\
—u, + —a
/1)
V 2
y
1 - z
2
1
f 1+ z 1 w+ Í1 - z ]
V c y V c y
V c y W, (-c < z < c).
h2 (2)^ —u —a '
V 2 2 4
(2)
y
Here, longitudinal and transverse displacements are denoted as u(k\x, z) and w(k\x, z).
Where k is the layer number. The thicknesses of the layers are h2, h = 2c.
Main part. Determining the variation of kinetic energy for a three-layered bar. According to the boundary conditions and formula (2), the deformation components under spatial loads are determined as follows [1, 13]:
1
£V 2
du duf du, du.
dx,- dx.
v j 1
dx dx
(3)
j y
The kinetic energy of the three-layer rods is calculated using the following expression: \SKdt = \\p^ ^^u^ dvdt + (k) dvdt + JJpJ ^^ u (k)dvdt.
t t v k=1 & t v k=1 &
dt
(4)
t v k=
By substituting (2) into (4), the resulting formula gives rise to these notations. pF = JJpdzdy, pSy = JJpzdzdy, pIy = JJ pz2dzdy,
y z y z y z
F - cross - sectional area of the rod, S - static moment, I - moment of inertia, p - density.
After performing several operations such as differentiation, variation, integration, and similar procedures on the above, we obtain the equations for calculating the kinetic energy of the three-layered bar and the corresponding natural initial conditions [6, 7].
According to Hooke's Law, the relationship between stress and strain is as follows [4]:
&u (-) „(du(k) &u(k) ^
T(k) _ ;11
- E s(k) - E
- Ekbu - Ei
dx
<13 - G3
1 -+- 3
dz
dx
(5)
Here, E - modulus of elasticity, G3- shear modulus, <ri/ - stress.
in=xxib jfr = ЯК(1) + 0ц(2)&п(2) "
t t=1 ]=1 v t v t
+0 ™8eu (3) + a13 (1813 (1) +a13 т081Ъ (2) + a13 (3813 (3) ) dtdv;
(6)
'13 1 ^13 1 ^13
Based on (5), when calculating the potential energies, substituting (2) into (6) gives us the following equality:
\smt = JJk(1)
8
du(1) „ dam , „ dam , h да(1)Л
dx
-8z-
dx
- + 8c-
dx
■ + 8-
2 dx
+0
(2)
8
du (2) h
—--8za(2) -8ca(2)-8^a(2)
dx 2
+ -o
(3)
8
du(1) dx
(i)
+ 08
h da 2 dx
- + 8—
du(1) ~z h da - + 8- n
(i)
c dx
c 2 dx
8
du(2) h da(2) z du(2) czh da
-8 л 1л 1
(2)
(7)
dx
+ -o
(3)
2
13
2 dx . dw,
-8—
c dx
+ 8—
. z dw,
dw
c 2 dx . z dw-,
8^ + 8-^ + 8^-8— dx c dx dx c dx
+
■ dvdt.
+8 - u(1) + 8^- a(1)-8 - u(2) + 8 a( c 2 c c 2c
After several mathematical calculations (differentiation, variation, integration, notations) in (7), we obtain the following:
Here, N, M, Q - forces and moments.
N¡1 = J J dzdy; M^ = J J^(1)zdzdy; N^ = J JV2)dzdy; M^ = JJ^(2)zdzdy;
j z
j z
у z
j z
N<1 = J|oii(3)dydz; M® = JJan(3)zdydz; ß® = JJa®dzdy; M® = JJza®dzdy.
У z
y z
y z
y z
The work done by external forces is determined by the following expression:
J8Atdt=JJ
J F(k )8ul (k )dvdt + J F3 (k )8u3 (k )dvdt
k=1
^JJ
k=1
+
JJ
t I
J ^(k )8m1 (k )d^dt + J ^(k )8мз (k )d^dt
k=1
k=1
+
J f(k )8u (k )d^dt + J f(k )8u (k ) dsxdt
k=1
k=1
(8)
Here, P- body forces, q - surface forces, f - boundary forces.
The work done by external forces is calculated. To do this, substitute u1 and u3 from (2) into (8), and we obtain the following:
' ) - (z - c - |)a(1) V F({2)S (u2) - (z + c + j +
J8Adt =JJ
t t v
+F (3)8
41)
F (1)8
(1 + z)(!u(1) + h- a(1)) + (1 - -)(!u(2)- ^a(2)) V c 2 4 c 2 4 J
+ F3 [i)8w1 + F3 {2)8w2 + F3 (3)8
43);
1(1 + z )w1 +1(1 - z )w2 V 2 c 1 2 c \
dvdt +
t v
1
2
1
t v
+jj
t l
q(1)öfu(1) - (z - c - —)a()) ] + q(2)öf u(2) - (z + c + ^)a(2)'
v 2 ) V 2 )
A _ 1 U _ 1 7, A
V
(1 + z )() u<" + — a<") + (1 - z )() u'J>-a">) c 2 4 c 2 4
(9)
(3);
1
1
+ q3{1)Swi + q^Sw, + 1(1 + -)wi +1(1 - -)W2
V 2 c 2 c )
)
dldt +
-J[f:
())Ä ,(i)
+ 1 \ f, öu()) -Mf,{»öa{) + cf (1)Sa(1) + ^ f (1)Sa(1) + /r'öu^ -
h 7 ())
,(2) .
(2)
-Mt(2W2) - c/ öa(2) - /1
—-f fa*+1 / (3)su«+—-f :3)
öa(1) +
+
—Mt(3)öu(1) +—Mfl(3)öa(1) +1 f (3)Su(2) - — /(3)Sa(2)
2c
4c
M^W 2)+—■ Mt1(3)öa(2) + f3))öW) + f 32)öW2 +—f 33)öw—
+
+ Mf3(3)öW) + ) f 33)öW2Mf3(3)öW2 2c 2 2c
dt
+— 2
By expanding the parentheses and integrating the expression for body forces acting on the solid with respect to the y and z axes, and integrating the surface forces with respect to the x axis (the length of the solid), we obtain the following:
J Adt = JJ J J F (1W:) dydz - JJ F(1) z8a(X) dydz + cJJ F (1W1} dydz +
t t x y z y z y z
h J J F(1 W1}dydz + J J F{2)Su(2)dydz - J J F(2)zSa™dydz -
y z y z y z
c JJ F (2W2) dydz - ^ JJ f (2)Samdydz + - JJ F (3W {%dz +
y z 4 y z 2 y z
h- JJ F {3)Samdydz + — JJ F(3) zSu^dydz + A- JJ F(3) zSamdydz +
4 y z 2c y z 4c y z
- JJ F (3)^u(2)dydz - ^ JJ F ^a2 dydzJJ F(3) z^u {2)dydz +
+— 4
+— 2
j z
y z
y z
+-
—L 4 c
ii F(3) zöa(2) dydz + ü F {l)öWldydz + J J F1 {2)öW2dydz + ) J J F3 (3)öWldydz +
+ — J JF3zöWjdydz + ) J JF3i-3^öW2dydz - — J JF3(3)zöW2dydz 2c 11 2 J J 2c ^
(10)
dxdt +
+J J qil)8uil)dl - J q(1)z8amdl + c J q(l)8a(r>dl +hJ qil)8amdl
' +
+J qi2)8ui2)dl - J q(2)z8a(2)dl - c J q[2]8a{2)dl - ^ J q{2'^8a{2)dl l l l 4 l
+1J q (3)8u >l)dl + ^ J q (3)8a(1) dl + — J q(3) z8u (1)dl + A- J q (3) z8a(1)dl + 2l 4 l 2c l 4c l
+1 f q(3)8u(2)dl-h2 Jq(3)8a(2)dl-A J q(3)z8u(2)dl + A Jq(3)z8a(2)dl■ 2 i 4 z 2cJl 4cJl
+J q(18wxdl + J q(28w2dl+1J qi~i^8wldl+—J q(3)z8w^dl +
7 7 2 7 2c 7
+1J q^8w2dl - — J q3^3)z8w2dl dt + J
2 • 2cJ,
J f (1)8u- J f(1)z8amdsl +
+cJ + h Jf(1)8a(1)+ J f(2)8u(2Ц - J f(2)z8a(2)ds ■
z
2c
cJf^W2^ - A Jf^W2^ +1Jf(3)8u(1Ц + hL Jf (3)a(1)ds +
s1 s1 s1
J f(3)z8u+ A J f(3)z8a(r>dsl +1J f <3)8u(2Ц - A J f (3)8a(2)ds
s1 s1 s1 s1
-A J f(3)z8u(2Ц + A J f(3)z8a(2)dsl + J f {l)8wlds + J f {2)8w2dsx + s1 s1 s1 s1
+1Jf3(3)8w1ds1 + A.Jf3(3)z8w1ds1 +1Jf^dS, -AJf3(3)z8w2ds1
Here,
F1 = J J F(1)dydz, Fi = J J F(2)dydz, Fi = J J F(3)dydz, MF,(1) = J J F(1)zdydz,
y z y z y z y z
MF(2) = J J F(2) zdydz, MF(3) = J J F(3)zdydz, F31) = J J F(1)dydz, F32) = J J F(2)dydz,
y z y z y z y z
F33) = JJF(3)dydz, MF3(1) = J JF3(1)zdydz, MF3(2) = J JF3(2)zdydz, MF3(3) = J JF3(3)zdydz;
y z y z y z y z
(11)
q(1) =JJq(1)dydz, q(2) =JJq(2)dydz, q(3) =JJq(3)dydz, Mq(1) =JJq(1)zdydz,
y z y z y z y z
Mq(2) =JJq(2)zdydz, Mq(3) =JJq(3)zdydz, q(1) =JJq(1)dydz, q(2) =JJq(2)dydz,
yz yz yz y z
q(3) = J J qf'dydz, Mq(V) = J J q(1) zdydz, Mq3(2) = J J q(2) zdydz, Mq(3) = J J q(3) zdydz.
13 J J ^3 -"-"Î3 J J ^3 ""S™", -"-"Î3
y z y z y z y z
By substituting the developed expressions (4), (7), and (9) into the fundamental formula (1), we obtain the equations with both ends rigidly fixed, along with the initial and boundary conditions [5-7].
The generalized equations of the three-layered rod under spatial loads are:
s
x
2«fl)
PF
d 2u
dt2
+ pcF
d2 a
— da
dt2
- + p —F
dt2
- + -
d 2u()) —
PF
dt2
+ — pF
d2a
dt2
1 T d2u(1) h , d2a(1) ^d2u(2) — ^d2a(2) 1 d2u(2)
+ —:pIv-T" + —VpIv-+ pF------pF------pIv-— +
c2 y dt2 2c y dt2 dt2 2 dt2 c2 y dt2
+
—
2 (2)
2c
pIy
d2a
2 ^ y
dt2
r dN1" ) dN3 ) dM1(3) ) ^
))
f (1)+2 f (3)+)+r q(—+2 q™+¿MT)
Q(3)
v dx 2 dx 2c dx 2c )
öu(1) = 0
f
2„ (2)
pF
d2u
2 (2)
dt2
- pcF
d2a
— ^ d a
(2) i f
dt
-pJL F 2
dt2
■ + — 4
d 2u()) —
pF
dt2
+ — pF 2
d2a
dt2 d2u(2)
1 T d2u(1) — , d2a(1) ^d2u(2) — ^d2a(2) )
--7Ply-^---T ply-r" + pF-ö---- pF-+ — pIv
c1 y dt2 2c y dt2 dt2 2 dt2 c2 y dt
—
2c
2 pIy
d2a(2)^ fdN^ ) dM3" ! dM3
a
"dt5
Q(3)
v dx 2 dx 2c dx 2c )
+V f (2)+2 F » - ¿Mf u qf+2 ü" - ¿m»
öu(2) = 0
f ^d 2u(1) Ä d 2u(1) rd 2a(1) _ d 2a(1) ^ d 2a(1) pcF-— + p—F-— + pIy-— + pc F-— + pc—F-— +
dt2
— d 2a(1) —2 d 2a
dt2
2 p2 ri)
+p — cF , 2 dt2
+ p—F- _ 4 dt2
)
■ + — 4
y dt2
dt2
— d2u(1) —
dt2
2^0)
- pF- , 2 dt2
d2a
+ — pF- , 4 dt2
+
—
2c2
pI
d2u
2" y dt2 2
+
— , d2a(1) — ^d2u(2) — — ^d2a(2) — , d2u(2) h— T d2a
4c
dt2
dt2
dt2
2c
2 ^ y ^2
dt2
4c
2y
dt2
))
+
M)+c — — dNSi+— ML —L dQ)3) +
dx dx 2 dx 4 dx 4c dx 4c dx V
— -(—) (3) — (3)^ r ()) -o) — -(l) —1-(3) — (3)N
+— F1 +— F1 + —MFjl ) l + l-Mq}"1 + cq + —q + —q + —Mq(3)
2
4
4c
4c
öa(— = 0
rd 2u(2) — rd 2u(2) rd 2a( 2) 2p
-pcF----p— F-— + pIy-+ pc F ,
dt2 2 dt2 y dt2 dt2
— ^d2a(2) —2 d a(2) 1
+pc—F-— + F-— + -
2 dt2 4 dt2 4
5V2) dt:
— ^ d2u(1) h—2 J 2a(1)
— d 2a(2)
+ pc—F-— +
2 dt2
-—pF , ^ dt2
4
pF-
dt2
+
—
d2u(1) hfc r d2a(1) — ^d2u(2) —2 d2a
2 (2) i2
2 (2)
+pIy dt2
pI
- "ir pF + -f pF"
4c y dt2 2 dt2 ^ dt2
h
2„(2)
2c
Py
d2u
hl
2 y
dt2
4c
py
~fl (2) f da*
2 ^ y
dt
JJ
SM1(12) d#1(12) h d-Nf — c ■
dr
dr
2 dr
h ^ + h, M) - A dQ£Vf-MF (2) - cFf> - h F(2) - hiFf) +
4 dr 4c dr 4c dr
2
4
(12)
h
+ ^MF(3) l + l -Mq(2)öa(2) -cq(2)q\ 'q, + ^Mq[
■(2) ^ -(2) h _(3) h
,(3)
4c
4c
öa(2) = 0
^ 77 d2 w 1 ^ pF—r1 + -
dt2
4
, d2 w
pF^1 + p— I c
1 d2 Wj
, d 2w,
dt2
Tv -, -p~r Iv n
2 y dt2 dt2 c2 y dt2
1 T d2
J
+
f 1 dö1(33) 1 dM1(33) ^ f-(1) 1 —(3) 1
2 dr 2c dr
-(1) 1 (3) 1
+ , F, + 2Fr + -MF?- 1 + 1 qr + 2q33) + ^q?
öw, = 0
f „ d2w 1 f r- d2W 1 r d2w
pF^n2 + ~ pF—1 -p- 1
4 1 dt2 c
dt2
, d 2w,
1 T d2
pF^1 -p^-F^1 + pF—r2 + p^r Iv ,
dt2 c2 y dt2 dt2 c2 y dt2
+
f 1 dQ(3 1 dM1(33) ^
2 dr 2c dr
+ f F 32) +1F 33) -A MF. o 1 +
2c
-(2) 1 -(3) 1 q3 +- q3 - — Mq:
(3)
Natural generalized initial conditions
Z7 du(1) j-. ■ pF~~ + pcF~
dt
öw, = 0
da(1) \ da(1) 1 f du(1) h ^ da(1) h da(1)
■ + p — F-+ -
dt 2 dt 4
pF
+ — pF- , dt 2 dt 2c 2 y dt
+ -
pIy
■ +
+ T pIv
du^ dt
+pF
du(2) K ^da(2) 1 T du(2) h
pF
dt 2 dt
- — pIv
dt 2c
pIv
2 ^ y
da(2) dt
öu
(1)
= 0
pF
du(2) dt
- pcF -
da(2) ^ da(2) 1 f du(1) ^ da(1) 1
-p—F dt 2 dt 4
- + -
pF
dt 2
+ — pF-
dt c
- — pIv
ßu^ dt
h T da(1) dw(2) h .da(2) 1 , dw(2) h , da(2)' -V pF-+ pF---2 pF-+ — pF---\ pl.-
2 ^ y ^ ^ ^ O ^ ^ y ~ 2 r v
2c
dt
dt
2
dt c
dt 2c
2 y
dt
öu
(2)
= 0
pcF
du(1) ^ du(1) . da(1) 2 da(1) \ da(1) h i7da(1)
- + p — F dt 2
'(1) , da(1)
dt -+pIv a+pcF-
+ pc — F-dt 2 dt
■ + p—cF -2 dt
+
h2 da( ) 1 f h r du(1) h ^ da( ) h , da(1) h T du(1)
+p — F-+--1 pF-+ — pF-+ -V pIv-+ —V pIv-+
4 dt 4,2 dt 4 dt 4c y dt 2c y dt
h ^ du(2) hh .da(2) h t du(2) hh , da(2) ^
pF---pF----V pIv-+ -2r pIv-
^ dt 4 dt 2c y dt 4c y dt
öa
(1)
= 0
-pcF
du
( 2)
h ^ du
(2)
-p — F—— + pIv dt 2 dt }
da
(2)
da
(2)
h ^ da
dt
+ pc2F—--+ pc—F -
dt 2 dt
(2)
A u da
(2)
+ pc—F -2 dt
■ +
h2 da(2) 1 f h ^ du(1) hh ^ da(1) h , du(1) hh , da(1)
--1----2 nP---L^ nP--1--nT--1--L-L nT -
+p^ F-4 dt
pF
4, 2
dt
4
pF
+ -dt 2c
2 pIy
dt 4c
2 pIy
dt
h ^du(2) h!da(2) h , dw(2) h r da(2) ^ —2 pF-+ — pF---V pIv-+ —h pIv-
2 dt 4 dt 2c y dt 4c^ y dt J
öa
(2)
= 0
c
v
T7dW1 , 1
pF—1 + — dt 4
öw, 1 dw, dw 2 1 dw2A dt c y dt dt c y dt
öw1
= 0
Pdw2 1 pF—2 + — dt 4
( dw, 1 dw, ^dw, 1 dw2 I V dt c y dt dt c y dt j
öw-,
= 0
(13)
Natural generalized boundary conditions:
öu(1)
r (() -L. I (3) V ff1 /3) , 1^(3)A
^n +-^3) +—+ f1 +-f1 +—Mf1 2 2c ) v 2 2c
^ +1 ^(3) -_! Mu
11 2 11 2c
(3)
—(2) 1 —(3) 1 ,
f() +1 f() Mf1
2 2c
(3)
öu
(2)
= 0
= 0
M + + h + h <3) + A. M(3) I +
+
4c
+cf 11'+h f +h f r+f m/1")
2 4 4c
öa
(1)
^-M1(,2) - cM2) - h M2) - A ^ + A
4c
+(-m/i2) - cf (2)-hf((2) - £ f r+£ m/1(3)
= 0
+
öa(2)
= 0
-( 2 ^+¿M!? )+( f 31'+2 f 33)+¿m/™ I
-( 1 a? m;» )+( f S»+1 f 33' - ± M/j31
öw.
öw9
= 0
= 0
(14)
Calculation algorithm: To develop the algorithm based on the equations (12), (13), and (14) of the three-layered rod, these equations are expressed in vector and matrix form [7-10]. Vibration equation of the three-layered rod:
A/fd2U .d2U BdU n7} - .
M—¿- + A—T + B-+ CU + DF = 0
dt dx dx
Natural initial and boundary conditions:
(15)
[M ]
dUk dt
= 0,
(16)
(17)
The vector and matrix form of the equations are given in formulas (15), (16), and (17),
where M, A, B, C, D, M, B are matrices [11-13].
In developing the calculation algorithm for the three-layered rod, we use the explicit
scheme of the finite difference method and transform the equation into the following form [9-10]:
ij+i m ./+1
J+1
7+1. j+\
= K .
(18)
The system of three-diagonal equations (18) can be expressed in matrix form as follows using the matrix elimination method:
Here, âi+l and the unknown coefficients are of the following form
Д
a.
¿+i
A -C
A+1 =
Fr~Aßr
2-с
these are determined using the formulas.
To verify the deformation of a three-layered rod with fixed ends subjected to a 10 kN force, you need to use the given geometric parameters. Length l=2000 mm, width b=100 mm and layer Heights h1=10mm, h2=10mm, h3=120mm. Mechanical parameters Ei,2=2* 10HPa, E3=71*109 Pa, pi.2=7850 kg/m3, p3=2770 kg/m3.
Figure 1. The general displacements relative to the principal axis of the cross-section are considered along the X-axis.
Figure 2. The general displacements relative to the centroidal axis of the cross-section are considered along the Z-axis.
Conclusion. The mathematical model of the three-layered bar problem under spatial loads, based on the Ostrogradskiy-Hamilton principle, demonstrates the effectiveness of the numerical methods employed above. These methods allow for a more accurate analysis of the stress and strain conditions of three-layered bars under various conditions and provide more precise conclusions.
REFERENCES
1. Кабулов В.К. Алгоритмизация в теории упругости и деформационной теории пластичности. - Ташкент: Фан, 1966. - 391 с.
2. Плескачевский Ю. Деформирование металлополимерних систем / Минск:Бел. навука., 2004. 342 с.
3. Старовойтов Э. Трехслойние стержни в терморадиационних полях. / Минск.: Беларуская наука, 2017. 275 с.
4. С.П.Тимошенко, Дж. Гудьер — "Теория упругости"Перевод с английского М.Н.Рейтмана. М:. — "Наука" 1979.
5. Nuraliyev F M. Algorithmization in Magneto-elasticity of Thin Plates and Shells of the Complex Configurations // Comput. Sci. Inf. Technol. -2015. Vol. 3, -№ 3. -P. 66-69.
6. Анарова Ш.А., Шокиров Д.А., Исмоилов Ш.М. Современное состояние и постановка задачи исследования трёхслойных стержней // Проблемы вычислительной и прикладной математики. - 2022. - № 4(42). - С. 54-78.
7. Анарова Ш.А., Шокиров Д.А., Исмоилов Ш.М. Современное состояние и постановка задачи исследования трёхслойных стержней // Проблемы вычислительной и прикладной математики. - 2023. - № 5(52). - С. 56-82.
8. S. Anarova, S. Ismoilov and D. Shokirov, "Nonlinear Mathematical Model of Oscillation Processes of Spatially Loaded Rods with Account for Temperature," 2021 International Conference on Information Science and Communications Technologies (ICISCT), Tashkent, Uzbekistan, 2021, pp. 1-5, doi: 10.1109/ICISCT52966.2021.9670072.
9. Анарова Ш.А., Шокиров ДА. Вычислительный алгоритм расчета трёхслойных стержней при пространственных нагрузках // Проблемы вычислительной и прикладной математики. - 2024. - № 3(57). - С. 57-76.
10. Anarova, S.A., Shokirov, D.A., Amanov, O.T., Amonova, O.A. Mathematical Support for The
11. Study of Three-Layer Rods Under Spatial Loads // ACM International Conference Proceeding Series, 2024, -pp 25-31. DOI: 10.1145/3660853.3660858
