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IJE Transactions B: Applications Vol. 24, No. 3, October 2011 - 227
THE ANALYSIS OF A BEAM UNDER MOVING LOADS
Department of Civil Engineering, Urmia University, Urmia, Irane.mardani@urmia.ac.ir
*Corresponding Author
(Received: March 15, 2010 – Accepted in Revised Form: September 15, 2011)
doi: 10.5829/idosi.ije.2011.24.03b.03
Abstract It is assumed that a beam made of material has a physical nonlinear behavior. This beam is analyzed under the moving concentrated and distributed continuous loads. The vibration equations of motion are derived from the Hamilton's Principle and Euler–Lagrange Equation. In this study, the amplitude of vibration, circular frequency, bending moment, stress and deflection of the beam has been calculated. At the state of concentrated moving load, the obtained analytic solution has been exemplified. The results of this study indicate that when the material of the beam is considered physically nonlinear, there is no critical velocity and the resonance phenomenon doesn’t happen.
Keywords: Moving Load, Hamilton Principle, Euler- Lagrange Equation, Duffing Equation, Resonance
1. INTRODUCTION
The study of dynamical effect of moving loads at highway and railroad bridges has a history of more than one and a half century. The collapse of Jester Bridge in England in 1847 encouraged both the theoretical and experimental studies. The catastrophe caused tremendous human losses and created a lot of excitement in civil engineering.
Presently, there are many structures made from materials which are not subject to the Hook’s law. Therefore, there is a great tendency to study stress and strain in elements of structures made of physical nonlinear material under various static and dynamic loads. In the linear theory, the property of material is not taken into consideration; however, all of relevant parameters are taken into consideration in the theory of nonlinear. Thus, the physical nonlinear theorydemonstrates an exact calculation method for the analysis of stress, strain, and other internal forces
in structural elements.In order to represent all the possible states of
the material by one mathematical law, the following algebraic function may be introduced:
32
This equation is not different from the general and.
f
Manzhaalovsky employed Empeher’s method and proposed the following two-term equations of the parabolic type on the basis of the tests carried out by Glushkov [1].
111
2222
چکيده فرض ميشود كه يك تير ساخته شده از مصالح با رفتار غيرخطي فيزيكي تحت بار متمرکز متحرك و گسترده يكنواخت متحرک آناليز مي شود. معادلات ديفرانسيل حركت ارتعاشي بااستفاده از اصل هاميلتون و معادلات اويلر- لاگرانژ بدست مي آيد. در اين يررسي, دامنه حركت ارتعاشي, فركانس طبيعي دوراني, ممان خمشي و خيز تير محاسبه مي شود. در حالت بار متمركز متحرك حل تحليلي بصورت مثال عددي ارائه شده و در نتيجه مشخص شده است كه وقتي مصالح تير
داراي رفتار غيرخطي فيزيكي است سرعت بحراني وجود ندارد و پديده رزونانس اتفاق نمي افتد.
E. Mardani*
(1)
(2)
(3)at small deformations
relationship between
1 2 3
2
for tension (4)
for compression
228 - Vol. 24, No. 3, October 2011 IJE Transactions B: Applications
where, 1, 2, 1 and 2 are empire elasticity. Trinomial equations of the parabolic type were suggested for cast iron by Glushkov who proposed the use of a similar relationship for metal wire [1]:
32 cb (5)
In addition to exponential and power laws, hyperbolic relationships were suggested for brittle materials. Accordingly, as introduced the following function for cast iron:
(6)
Glushkov [1] has developed a theory for bars and discs made of brittle materials, employed the following hyperbolic laws:
1
Finally, the relationship between stress and strain, in the case of physical nonlinear, is presented by Kauderer [2]. As the equationproposed by Kauderer is comprehensive and expresses the relationship between the stress and strain in three dimensional states, we preferred to use the equation for the analysis of the physical nonlinear stress and strain
20
00
where, ij 0
stress:
(10)
K and G at small deformation are respectively volume contraction and shear elastic module. The relationship among K, E, and G is indicated through the following equation:
E (11)
K () is average stress function and 20tl is
shear stress function; it can be indicated through the following equation:
0
202
404
202
20
00
202010
...1)(
...1)(
n
nn
n
nn
tltltltl
KKKK (12)
Researches have demonstrated that K () in physical nonlinear material on andeformation is close to the straight line (i.e., K () = 1). Also, the two first sentences of the shear stress function are enough.
202
20
2l is the physical nonlinear coefficient.
As the Eq. (9) indicates, stress components of Z, y, and xy are created. The simplification of the equation suggests that the stress components are too small in comparison with the Z. These new components of the stress have less impact on the frequency of vibration and on the other parameters during the vibration of the beam. As a result, the following equation is obtained from the Eq. (9) at a two dimensional surface:
)33
3
2 (14)
The purpose of this paper was the analysis of a beam made of physical nonlinear material under the moving concentrated and distributed continuous loads, which was discussed through examples analytically.
2. THEORY
2.1. Analytical solution for moving concentrated load To study the effect of moving load on the prismatic beam, first, we are discussing moving concentrated load. Thus, it is assumed that the load P moves along the beam (Fig. 1).
(9)
(8)
(7)
(13)
average relative
E
ij ij 0 ij
K ( ) l(t )
3K 2Gi , j = 1 , 2 , 3
3
9KG
0
b
is Croneker symbols, and is average
( δ )
1
A B
x y z
5K G
27 GE( lZ Z Z
2 E
l (t ) 1 l t
In the above equation,
IJE Transactions B: Applications Vol. 24, No. 3, October 2011 - 229
Figure 1. Schematic view of a prismatic beam under moving load
To consider the effect of external moving load, it is assumed an equivalent distributed load which depends on z and t as follows [3]:
1sinsin
2),(
k L
zk
L
ak
L
Ptzq
(15)
where,L
zkzp
sin)( , (k = 1,2,3,…) is the head
vibration mode of the beam.
Therefore, the potential and kinetic energy of the system will be as follows [4]:
L
dzz
wJ
G
El
z
wEJ
0
4
2
2
13
4
2
2
2
2
0 54
1
2
1 (16)
L
dzt
wFKi
0
2
2
1 (17)
Also, the work of the external moving load will be as follows [5]:
L
L
dztzwL
zk
L
ak
L
PA
dztzwtzqA
0
0
,sinsin2
,.,
(18)
The Hamilton's Principle for this beam will be
dtdzt
wF
tzwL
ak
L
ak
L
P
z
wJ
G
El
z
wEJdtKiAH
k
t
t
t
t
L
.2
1
,sinsin2
54
1
2
1
2
1
4
2
2
13
4
2
2
1
2
1 0
2
2
2
0
(19)
where, dxdyyJdxdyyJ 41
20 ,
Considering:
tL
z0, , which , varies respectively
from 0 to and 0 to 2. where,
0 is circular frequency vibration of the
system in linear case and expressing in Hamilton principle (19), the following equations are obtained:
.
ddt
wFwk
kL
Pw
LJ
G
E
LEJ
LH
k
w
.2
1,.sin
sin2
54
1
2
1
220
1 0
4
2
2
8
8
13
4
2
2
2
4
4
00
(20)
We assumed that the deformation of the beam would be found from the following expression [7]:
qpw , (21)
where, p and q are coordinate and generalized functions respectively.
By substitution of expression (21) in expression (20), we obtain the following expressions:
ddqpF
qpkL
vtk
L
P
qpL
JG
El
qpL
EJL
H
k
.2
1
sinsin2
54
1
2
1
2220
1
448
8
13
4
2
2
22
0 0 4
4
00
(22)
The following expression is derived from the calculations indicated above in Eq. (22) [8]:
2
00
2
0
220
42
0
NdL
dqdqcbqaqL
H (23)
P
a=vt
Z
L
P
[6]:expressed as below
230 - Vol. 24, No. 3, October 2011 IJE Transactions B: Applications
L
vtk
L
Pd
dpFc
dpL
JG
Elb
dpL
EJa
sin
2
154
1
2
1
0
2
0
48
8
13
4
2
0
24
4
0
(24)
For Integral (23), Euler equation gives [9]:
0422
42
22
0
320
3
20
20
dbqaqqc
dbqaqq
N
qcqc
q
N
q
N
(25)
L
vtk
L
Pdq
a
baqqc
sin2122 220
(26)
By substitution of 0
, t
L
kinto Eq. (26)
we’ll have:
sin21 220 dq
a
bq
c
aq
(27)
where,cL
Pd
2,
0
By substitution of qd
X20
into Eq. (27), X is
dynamic coefficient. We will have:
sin1 2 eXXX (28)
where,40
2
2d
a
be (29)
To solve the Duffing Eq. (28), we follow the following procedure [10]:
5sin3sin
sinsin
53
,5,3,11
XX
XXXn
n (30)
If we substitute the expression (30) into 28, and compare the similar coefficients of sin n , we will get a lot of cubic nonlinear algebraic equations. To our knowledge, there is no exact solution for these equations. Thus, we employed an approximated method. For this purpose, we applied three constraints Eq. (30) and we assumed that Xn
>> Xn+1: therefore, in this case, we will have the following system of nonlinear equations:
02
3
2
3
4
3
4
3
4
3251
04
3
2
3
2
3
2
3
4
3
11
191
12
3
4
3
2
3
4
3
2
3
4
31
52
1523
2313
21
355
2
52
12533
21
53133
313
2
2515
23
231
32
15313
112
XXXX
XXXXXeX
XXXXXX
XXXXXeX
XXXXXX
XXXXXXeX
(31)
With solving of Eq. (31) by the method of Zeidel, we will have [11]:
14
31 3
112 eXX
1254
3
194
23131
5
2
31
3
XXXex
X
xeX
(32)
When the load is out of the beam, Eq. (27) will be as below in which the system will have free vibration.
021 2
22
2
q
a
bq
c
a
d
dq
(33)
Finally, by solving Eq. (33), we found the period of vibration:
k
Qa
b
c
aT
21
4 (34)
Consequently, the circular frequency is as follows:
KQ
a
b
c
a 11
22
(35)
where:
IJE Transactions B: Applications Vol. 24, No. 3, October 2011 - 231
where,
642 sin
256
25sin
64
9sin
4
11
2K
is the circular frequency of vibration, Q is amplitude of vibration, and k ( ) is a second order elliptic integral. As the above equation indicates,
2Qa
b has a minus sign, which decreases the in
comparison with 0 (as it was indicated in 35).
Based on the equations expressed above, the explanation of the analysis of deflection, bending stress, and bending moment are as follows.
Deflection: Deflection is obtained from the Eq.(21).
dpEJ
PldandtVa
l
aX
l
aX
l
aX
l
anX
l
anX
l
zdtzW
nn
nn
230
3
2
5315,3,1
5,3,12
",
5sin
3sinsinsin
sinsin),( (36)
As it will be indicated in the tables below, the coefficients X3 and X5 are too small and neglected.
l
a
l
zX
dtzW
sinsin),( 12 (37)
Bending Stress: Bending Stress is calculated by
using Eq. (14) and 2
2
z
wyZ
, which is called
KIRHOF principle.
3
2
2
3
3
22
2
27
2
z
wy
G
El
z
wyEZ (38)
Bending Moment: Bending Moment in every section of the beam is designed by the following equations:
2
2
0
13
3
22
2
0
2
2
3
3
22
2
.27
21
27
2
z
w
J
J
G
El
z
wEjM
dydxz
wy
G
El
z
wyEdydxyM Z (39)
The expression 2
2
z
w
equals the following equation:
l
a
l
z
l
d
z
w
sinsin.2
2
20
2
2
Now the obtained analytic solutions are being applied to the following example. It has been assumed that the material of the beam is copper (Fig. 2).
KNPmL
l
cmkgE
cmkgG
cmJ
cmJ
20,2
1018.0
/10241.1
/1046.0
1044.58
10236.1
62
26
26
641
440
R = 6.615cm
Figure 2. Section of beam
Then vibration amplitudes are determined and are shown in Table 1.
TABLE 1. Dimensionless vibration amplitudes
2 X1
X3 X511
12
13
0.0 -40.2 -39.83 1.00 -16.11 -22.04-3
-4
-6 -11
Based on Table (1) the diagram of resonance
Figure 3. Resonance Curve
10 cm
Y
2
is indicated in Fig. 3.
X X X
0.84 -21.07 10.54 0.1902 -1.31 10
1 -11.73 0.041 -1.01 10
2 1.00 3.69 10 -1.63 10
|X|
dddddddd
232 - Vol. 24, No. 3, October 2011 IJE Transactions B: Applications
Deflection, stress, and bending moment in the section of the beam are calculated by using equations (36), (37), and (38) for azand 12 , which are indicated in Tables 2 and 3.
moment at the point of loada
8
l
4
l8
3l
2
l
85
l4
3l
87
l
W (cm)- 0.054 - 0.1856 - 0.3468 - 0.3712 - 0.316 - 0.1856 - 0.054
M (K.N.m)
2.057 6.842 11.922 14.05 11.922 6.842 2.057
Y (cm) 6.61 7.9676 8.9816 10
)(mpaz 49.49 58.37 66.91 70.25
Bending stress in the middle section of the beam is indicated in Fig. 4.
70.25
70.25
Figure 4. The diagram of stress in the middle section of the beam
The Critical velocity for given example is obtained 135.95 m/s Dynamical Coefficients according to velocity is obtained and are written in Table 4.
Dynamic Coefficients at linear state are derived by Kesiliv [12]:
l
a
l
a
sinsin1
1
12
where, l
vvta
,,
The Dynamical Coefficient when the load is in the middle of the Span is estimated and shown at Fig. 5.
Figure 5. Dynamical coefficient-velocity
As shown at the Fig. 5, when 12
the
critical V equals to 135.95 m/s and resonance happens at linear state. Whenever, at nonlinear state, X has definite value, equals 11.73 in Table 4.
Based on v=50 m/s, (v=180 km/h), deflection, bending moment, and bending stress in the middle of the span are found by the obtained equations for nonlinear state, and for linear state they are found by equations which are given by Kesiliv [12]:
l
a
l
al
alp
M
l
a
l
apy
.sin.sin1
.sin2
1sin.sin
1
22
2
Y
X
21a
TABLE 4. Dynamical CoefficientsV (m/s) 0 25 50 75 100 124.57 135.95
22
0 0.0338 0.1353 0.3036 0.541 0.83895 1
|X| 1 1.036 1.158 1.438 2.193 9.27 11.73
V
00 20 40 60 !80 100 120 140
TABLE 3. Stress in the middle section of the beam
V(m/s)
5
10
15
20
25
TABLE 2. Calculation of deflection and bending
IJE Transactions B: Applications Vol. 24, No. 3, October 2011 - 233
The numerical resultants have written at Table 5.
TABLE 5. A comparison of deflection, stress, and bending moment at linear and nonlinear state
State W (cm) mpa M (KN.M)
Nonlinear 0.025 7.73 9.5
Linear 0.034 10.29 12.48
2.2. Analytical solution for moving distributed load To consider the effect of moving distributed continuous load, it is assumed that the moving distributed continuous load moves along the prismatic beam as it is shown in Fig. 6.
Figure 6. Schematic view of a prismatic beam under moving load
The potential and kinetic energy of this system can be written as follows:
dzz
wI
G
El
z
wEI
l
0
4
2
2
13
4
2
2
2
2
0 54
1
2
1
(40)
and:
dzdt
dwmdz
t
wFki
ll 2
00
2
2
1
2
1
(41)
where, E, G, l2, ρ, m, and F denote modulus of elasticity, modulus of elasticity in shear, nonlinearity coefficient and density, mass of load per unit length and cross sectional area,
beam is as follow:
dtdzdt
dwm
t
wF
Z
w
IG
El
z
wEIdtkiH
t
t
lt
t
224
2
2
13
4
22
2
2
0
0
)(2
1)(
2
1)(
54
1)(
2
1)(
2
1
2
1
By considering, dw
dt
w
tv
w
z
and Euler
equation:
2
20
z
L
u t
L
k z
L
i( ) ( ) ( )
where, uw
zk
w
ti
w
z
2
2, ,
and Eq.differential motion equation can be written as:
4
4
4
4
2 2
20
2
2
2
0
2
2
2
3
31
0
8
8
2
2
3
32
2
2
4
4
2
92
l
w mv
l EI
w F m
EI
w
lE
G
I
I l
w w w w
( )
( )
where, ., tl
z
CosSinkQw .00 (45)
where, Q0 , is amplitude of vibration of beam at linear state.
If Eq.(44)derived:
)3.33.39
3..3(72
1
)(
308
88
0
13
3
2
2
2
0
2
2
2
02
22
4
4
4
4
CoskSinCoskSin
CosSinkCosSinkQl
k
I
I
G
El
w
EI
mFw
EIl
mvw
l
(46)
Private solving of Eq. (11) is defined as follow:
3.3.3
3..),(
43
21
CoskSinaCoskSina
CosSinkaCosSinkaw
(47)
where, a1 ,a2 , a3 , a4 are constant coefficients, by substitution Eq.comparing the same coefficients Sin kξ, Cos τ, a1, a2, a3, a4 and are obtained, which are as follows [13-15]:
l
v)/( mkgm
, by considering Eq. (43)
(45) is substituted at right side of Eq. and is simplified, the following equation is
(47) into Eq. (46) and
respectively. The principle of Hamilton for this
(42) and further simplification the
(43)
(44)
(42)
As solving of the Eq. (44)
is difficult the consecutive approximated method is used. Furthermore, to make the solving procedure easier, it is assumed at linear state:
234 - Vol. 24, No. 3, October 2011 IJE Transactions B: Applications
308
88
0
13
3
2
2
12
222
14
44
24
1Q
k
I
I
C
E
EI
mFa
EI
mvka
k
(48)
308
88
0
13
3
2
2
2
22
222
24
44
72
1
9
Qk
I
I
C
E
aEI
mFa
EI
mvka
k
(49)
308
88
0
13
3
2
3
2
32
222
34
44
8
1
981
Qk
I
I
C
E
aEI
mFa
EI
mvka
k
(50)
308
88
0
13
3
2
4
2
42
222
44
44
24
1
9981
Qk
I
I
C
E
aEI
mFa
EI
mvka
k
Thus, circular frequency of system ω is obtained:
2 02 2
2
2 2
2
2
0
2
4
3 1
6 6
4 02
2 20
2 211
24
1
EI
F m
k
l
k
l
mv
EI
lE
GI
k
lQ
k EI mv l
( ). ( )
( )
The critical velocity is derived by considering the criteria ω = 0.0
2
04
44
03
3
20
24
11 Q
l
k
I
I
G
El
m
EI
l
kVcr
If l2=0.0 the critical velocity is obtained at
linear state, m
EI
l
kVcr
0
If at Eq. (52), m= 0.0 that is the load is out of beam:
)24
11( 2
04
44
0
1
3
3
20
2
22
Ql
k
I
I
G
El
F
EI
l
k
ω is the circular frequency at free vibration at physical nonlinear . It is seen from Eq.circular frequency of system depends on nonlinearity of material and the velocity of load.
To understand the analysis obtained here an example is presented in this section.assuming beam which is shown in Fig. (1), the parameters used in studied equation are as follows:
E =2.1 105Mpa G = 0.87105Mpal2 = .085106
q = mg =20 KN/m (unit weight of load)P = ρfg = 0.66 KN/m (unit weight of beam)I0=1.5110-3m4
I1 = 1.310-4m6
L=12m
Circular frequency is obtained from Eq. (52) and plotted in Fig. 8.
Figure 8. Frequency – Velocity Curve
Vcr =103.044m/sec. The dynamic coefficient (Yd/Ys) at the middle point of beam is derived from Eq.in Fig. 9.
Figure 9. Dynamic Coefecient- Velocity Curve
V (m/s)
Dyn
amic
Coe
ffic
ient
Freq
uenc
y
Figure 7. Section of beam
(51)
(52)
(53)
(54)
(52) the
(52) and plotted
Velocity(m/s)
For the
The critical velocity is obtained from Eq. (53):
IJE Transactions B: Applications Vol. 24, No. 3, October 2011 - 235
Diagram of motion at interval of a period is
Figure 10. The time history of the oscillation at the middle section of the beam during one cycle
The bending moment at the middle point (a = L/2)
Figure 11. Bending moment – Velocity Curve
3. CONCLUSION
The effect of material nonlinearity on the response parameters of bridge under concentrated and distributed moving loads are investigated analytically. The Hamiltonian principles and Euler’s equations employed to found the nonlinear vibration equation of the system. The Fourier series is used to decompose the deflection as a multiplication of functions in time and space. The resulting equation in time is the well known Duffing’s equation. Solving the Duffing equation by perturbation method the response parameters of the system is evaluated. In the case of concentrated moving load and linear material, theoretically with increasing the speed of the moving load resonance might happen. However considering the material
nonlinearity, resonance doesn't happen, and the internal forces will have definite values. Taking into account the material nonlinearity the internal forces for velocities blew critical velocity reduces as much as 10-15 percent in comparison with the linear case. Using the results dynamic amplification factors is calculated for the system. Increasing the material nonlinearity, results in decreasing in the value of vibration amplitude. In the case of distributed continuous moving loads using analytical solution, vibration frequency and the dynamic amplification factors and bending moment are evaluated for different velocities, blew critical velocity. Analysis shows that the more is the speed of the moving load, the more is the amplitude of the vibration.
NotationA = external work of the moving load = the distance of the concentrated load from the supportE = module of elasticity F = an area of the sectionG = sheer elastic moduleH = Hamilton PrincipleJ0 = moment of inertiaJ1 = dydxx4
K = volume contraction 0K = average stress function
Ki = kinetic energyL = the span of the beam 2
0tl = sheer stress function
2l = nonlinear coefficient
p = coordinate functionP = concentrated loadp(z) = the head vibration made of the beam q() = generalized functionq(z , t) = equation distributed loadw (z , t) = deformation of the beam X = dimensionless amplitude (dynamic coefficient)Z = a distance of any point from the support = stress0 = average stressX , Y , Z = original stress = strain ij = three – dimensional strain = density
Velocity (m/se)
Ben
ding
Mom
ent
(t.m
)
shown in Fig. 10 (for v = 30m/,
is shown in Fig. 11.
Freq
uenc
y
Time (s)
50m/s)
236 - Vol. 24, No. 3, October 2011 IJE Transactions B: Applications
= potential energy0 = circular frequency = circular frequency of loadV = velocity
4. REFERENCES
1. Glushkov, G. Engineering Methods for Analysing Strength and Strength and Regidity, Moscow, 356 p, (1974).
2. Kauderer, H., Nichtlinnear Mechanic, Springer Verlag, Berlin. PII: S0020 -7683, (1961) (99) 00029 - 3.
3. Qaribov, R. and Hasanov, N. “Analysis of deformation of prismatic beams made of physical nonlinear material under impulse loads”, Scientific works in Structural Mechanic
4. Meirovitch. L. 1997. Principles and Techniques of Vibrations. ISBN - 9780023801419.
5. Nayfeh, A.H. and Mook, D.T., “Nonlinear oscillations”, Willy Classics Library Edition Published, (1997), 56-58.
6. Clough, R.W. and. Pennzien. J., (1993), “Dynamics of Structures”, McGraw - Hill, New York.
7. Mardani, E., “The Nonlinear Behavior of Materials under Moving Loads. In the Tenth Est”, Asia – Pacific Conference on structural Engineering and Construction (EASEC– 10), 3 -5 August, (2006), Bangkok, Thailand.
8. Wang, P.H., TaLin, H. and Tang, T.Y., “Study on nonlinear analysis of a highly redundant cable - stayed bridge”, Journal of Computers & structures, Vol. 80,
No. 2, (2002), 165-182. Doi: 10. 1016/s0045 -7949(01)00166 - 3.
9. Mardani, E., Garibov, R. and Isayev, A., “Physical nonlinear analysis of a beam under moving load”, the second international conference on Bridges, (1996),Tehran,
10. Ghorashi, M. and Nitzche, F., “Nonlinear Dynamic Response of an Accerating Composite Rotor Blade using perturbations”, Journal of Mechanics of Materials and structures, Vol. 4, (2009), 693-718.
11. Bicin, B., “Vibration of Beams with Multiple open cracks subjected to Axial Forces”, Journal of sound and vibration, Vol. 222, No. 3, (1999), 409-423.
12. Kesiliv, V.A. Structural mechanic. Publish soviet scientific and technical literature, Moscow, 550 P,(1972).
13. Sadhakar R. M., “Advances in nonlinear Vibration analysis of structures”, Sadhana Academy proceedings
14. Duan H. J. and Song, Z. S., “Nonlinear free vibration analysis of thin – walled curved beam with no-symmetric cross section”,
Vol. 13, No. 2, (2008), 150-154, Doi: 10, 1007/s12204-4008-150-4.
15. Mardani, E. and Garibov, R., “Physical Nonlinear of a Prismatic Beam under Moving Continuous Distributed Load”, Scientific works in Structural Mechanic, No7.ch, 2, (1997), Baku, ELM, 75-80.
, No 11, (2001), Baku, ELM, 178-180.
Iran. 20 – 25.
in Engineering Sciences, Vol. 26, No. 3, (2001), 243- 249,Doi:10.1007/BF02703386.
Journal of Shanghai Jiaotong University (Sciences),
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