THE ANALYSIS OF A BEAM UNDER MOVING LOADS

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, [email protected]

*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,


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


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:


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


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


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:


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):


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)


= 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.

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THE ANALYSIS OF A BEAM UNDER MOVING LOADS