analytical solution for large deflections of a cantilever beam under nonconservative load based on...

1You have full text access to this content

Analytical solution for large deflections of acantilever beam under nonconservative load basedon homotopy analysis method

1. A. Kimiaeifar1,

2. G. Domairry2,*,

3. S. R. Mohebpour3,

4. A. R. Sohouli2,

5. A. G. Davodi4

Article first published online: 6 NOV 2009

DOI: 10.1002/num.20538

2009 Wiley Periodicals, Inc.

Issue

Numerical Methods for Partial Differential Equations

Volume 27, Issue 3, (/doi/10.1002/num.v27.3/issuetoc) pages 541553, May 2011

Additional Information

How to Cite

Kimiaeifar, A., Domairry, G., Mohebpour, S. R., Sohouli, A. R. and Davodi, A. G. (2011), Analytical solution for

large deflections of a cantilever beam under nonconservative load based on homotopy analysis method. Numer.Methods Partial Differential Eq., 27: 541553. doi: 10.1002/num.20538

Author Information

Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

23

4

Keywords:

homotopy analysis method; nonconservative load; large deformation; nonlinear differential equation

Department of Mechanical and Civil Engineering, Babol University of Technology, Babol, Iran

Department of Mechanical Engineering, Persian Gulf University, Bushehr, Iran

Department of Civil Engineering, Shahrood University of Technology, Shahrood, Iran

Email: G. Domairry ([email protected])

*Department of Mechanical and Civil Engineering, Babol University of Technology, Babol, Iran

Publication History

1. Issue published online: 6 NOV 20092. Article first published online: 6 NOV 20093. Manuscript Accepted: 18 MAY 20094. Manuscript Received: 16 FEB 2009

Abstract (/doi/10.1002/num.20538/abstract)Article

References (/doi/10.1002/num.20538/references)Cited By (/doi/10.1002/num.20538/citedby)

Enhanced Article (HTML) (http://onlinelibrary.wiley.com/enhanced/doi/10.1002/num.20538) Get PDF (267K)(/doi/10.1002/num.20538/pdf)

Abstract

1. Top of page

2. Abstract

3. I. INTRODUCTION

4. II. GOVERNING EQUATION

5. III. APPLICATION OF HOMOTOPY ANALYSIS METHOD

6. IV. CONVERGENCE OF HAM SOLUTION

7. V. RESULT AND DISCUSSION

8. VI. CONCLUSIONS

9. NOMENCLATURE

10. References

In this article, large deflection and rotation of a nonlinear beam subjected to a coplanar follower static loading is

studied. It is assumed that the angle of inclination of the force with respect to the deformed axis of the beam

remains unchanged during deformation. The governing equation of this problem is solved analytically for the first

time using a new kind of analytic technique for nonlinear problems, namely, the homotopy analysis method

(HAM). The present solution can be used in wide range of load and length for beams under large deformations.

The results obtained from HAM are compared with those results obtained by fourth order Range Kutta method.

Finally, the load-displacement characteristics of a uniform cantilever under a follower force normal to the

deformed beam axis are presented. 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq

27:541553, 2011

I. INTRODUCTION

1. Top of page

2. Abstract

3. I. INTRODUCTION





8. VI. CONCLUSIONS

9. NOMENCLATURE

10. References

Most scientific problems in solid mechanics are inherently nonlinearity. Except a limited number of these problems,

most of them do not have analytical solution. Nonconservative forces are associated with theoretically interesting

and from the practical point of view, are very important problems in aerospace engineering, also in other fields of

modern applied mechanics and engineering such as fluid, and aerodynamics, aeroelasticity, and electrical engineering

[115]. This type of loading cannot, in general, be associated with a stationary single-valued function dependent

only on the generalized displacements, as standard gravitational loads. Therefore, for nonconservative systems, the

nature of instability requires a special investigation for each problem [1, 4, 6, 7]. To understand the behavior of such

flexible structures and to evaluate their elastic limits many different formulations and numerical procedures for the

displacement and rotation analysis of large deformations have been proposed [1, 8].

Therefore, these nonlinear equations should be solved using other methods. Some of them are solved using

numerical techniques and some are solved using the analytical method of perturbation [9]. In the numerical method,

stability and convergence should be considered so as to avoid divergence or inappropriate results. In the analytical

perturbation method, we should exert the small parameter in the equation. Therefore, finding the small parameter

and exerting it into the equation are deficiencies of this method.

Figure 1. The geometry and boundary conditions of a beam

under nonconservative load.

Download figure to PowerPoint

(/doi/10.1002/num.20538/figure.pptx?

figureAssetHref=image_n/nfig001.jpg)

One of the semiexact methods which does not need small/large parameters is homotopy analysis method (HAM),

first proposed by Liao in 1992 [9, 10]. This method has already been applied successfully to solve many problems

in solid mechanics such as large deformation of a cantilever beam, KdV equations, solitary wave, etc. [1113] and

in fluid mechanics [1426]. In this method, we can adjust and control the convergent region and it is the most

important feature of this technique in comparison with other techniques [3133].

In this study, HAM has been applied to find the analytical solutions of nonlinear ordinary differential equation

governing large deflections of a cantilever beam. The numerical solution based on shooting method and fourth order

Runge Kutta method developed by these authors. The results obtained from HAM are compared with the results

obtained from numerical methods. Finally, the load-displacement characteristics of a uniform cantilever under a

follower force normal to the deformed beam axis are presented.

II. GOVERNING EQUATION

1. Top of page

2. Abstract

3. I. INTRODUCTION





8. VI. CONCLUSIONS

9. NOMENCLATURE

10. References

The cantilever OA is subjected to a coplanar terminal loading consisting of a bending moment MA, of an axial

compressive force PA and of a transverse force QA (Fig. 1).

PA and QA are follower forces, i.e., after the deformation rotate with the end section, A, of the bar and at all times,

remains, respectively, tangential and perpendicular to its deformed axis. Therefore, at any point of coordinates x(s),

y(s) the external moment M is [27]:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

where xA, yA, and A denote the coordinates and the normal slope at the end section. The classical Euler-Bernoulli

hypothesis assumes that the bending moment at any point of the beam is proportional to the corresponding

curvature,

Substituting Eq. (1) into Eq. (2), differentiating with respect to the arc length and using the relations:

Nonlinear differential equation governing the problem has been obtained as follow:

where

The boundary conditions associated to (4) are

where m = MA/EA.

After introducing the following position,

as a consequence (4) becomes:

And the boundary conditions:

For , Eq. (8) together with the boundary conditions of Eq. (9) describes the Beck cantilever problem [2,

28, 29]. In the corresponding conservative case (a straight cantilever subjected to a vertical compressive load ;

) Eqs. (4, 6) yield:

As is well/known [28, 29] (Kirchhoff kinetic analogy) previous relations represent also the steady oscillations of a

rigid pendulum as solved by Liao using HAM [30].

(11)

(12)

(13)

(14)

III. APPLICATION OF HOMOTOPY ANALYSISMETHOD

1. Top of page

2. Abstract

3. I. INTRODUCTION





8. VI. CONCLUSIONS

9. NOMENCLATURE

10. References

According to Eq. (8), a nonlinear operator has been defined as follows:

Using the Taylor's series expansion for and , yields:

Substituting Eq. (12) and Eq. (13) into Eq. (11), result in

where q [0,1] is the embedding parameter, is a nonzero auxiliary parameter. As the embedding parameter

increases from 0 to 1, (s;q) varies from the initial guess 0(s) to the exact solution (s);

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

Expanding (s;q) in Taylor's series with respect to q yields:

where

Homotopy analysis method can be expressed by many different base functions10, according to the governing

equation; it is straightforward to use a set of base function:

In the form

That bn is a coefficient to be determined. Besides determining a set of base function, the auxiliary function H(s),

initial approximation 0(s), and the auxiliary linear operator must be chosen in such a way that all solutions of the

corresponding high-order deformation equations exist and can be express by this set of the base function and the

other expressions such as sn sin(ms) must be avoided.

This provides us with the so-called rule of solution expression [10].

We choose linear operator, as:

With the property

where c1, c2 are integral constant. Under the rule of solution expression and initial condition it is straightforward to

choose

The zero order deformation equation is:

According to the rule of solution expression denoted by Eq. (19) and from Eq. (23), the auxiliary function H(s) can

be chosen as follows:

(24)

(25)

(26)

(27)

Differentiating Eq. (23), m times with respect to the embedding parameter q and then setting q = 0 and finally

dividing them by m! and From Eq. (14) and (19) the so-called mth-order deformation equation for m 1 is

obtained:

where

and

It's time to choose H(s) uniquely under the rule of solution expression and rule of coefficient ergodicity. It can be

shown that when (k = 2n|n = 1, 2, 3, ), the terms s5 up to s2n + 1 always disappear in the solution expression of

m(s), so that the coefficients of the terms s5 up to s2n + 1 are always zero and cannot be modified even if the order

of approximation tends to infinity. This disobeys the so-called rule of coefficient ergodicity, which was expressed by

Liao [10], if we choose (k = 2n + 1|n = 1, 2, 3, ), bases like s2m + 1 will disappear which disobey rule of solution

expression and rule of coefficient ergodicity, on the other hand when k = 1, 2, 3, the terms slns, lns would

appear that clearly disobey rule of solution expression, and finally when we take (k = 1 n|n = 5, 6, 7, . there

will be terms like in the solution expression of m(s) which disobey the rule of solution expression. So we have to

choose k = 0. Consequently the corresponding auxiliary function was determined uniquely H(s) = 1. Now the

answer is obtained successively as follows:

(28)

The solution has been developed up to 14th order of approximation of (s) (see Figs. 2, 3 and Tables I and II).

Table I. Comparison between present HAM solution, numerical solution for (0.5)

by 12th/order of approximation of (s) with = 0.3 and 0.8 for various value of .

HAM solution HAM solution Numerical solution

0.3 0.4955 0.8 0.4959 0.4959

0.3 0.9678 0.8 0.9682 0.9682

0.3 1.3983 0.8 1.3984 1.3987

0.3 1.7727 0.8 1.7731 1.7764

0.3 2.1028 0.8 2.0981 2.0982

Table II. Comparison between HAM solution and numerical solution for (s) with

12th-order approximation, L = 1 and = 0.8. (Relative error define as |(HAM

numeric)/numeric|).

, m = 0,

s h HAM Numeric Relative error

0.0 0.8 2.99832 2.99272 0.00187

0.1 0.8 2.77151 2.77169 0.00006

0.2 0.8 2.45800 2.45826 0.00010

0.3 0.8 2.06828 2.06849 0.00010

0.4 0.8 1.63178 1.63200 0.00013

0.5 0.8 1.18918 1.18955 0.00031

0.6 0.8 0.78296 0.78346 0.00064

0.7 0.8 0.44669 0.44699 0.00069

0.8 0.8 0.19947 0.19973 0.00133

0.9 0.8 0.04989 0.05000 0.00202

Figure 2. Comparison of convergence for different orders, (a)

(0.5), (b) , and (c) .




Figure 3. The curve, (a) , m = 0, , (b) , m =

5, , (c) , m = 5, . The 12th-order

approximation.




0.9 0.8 0.04989 0.05000 0.00202

1.0 0.8 0.00000 0.00000 0.00000

IV. CONVERGENCE OF HAM SOLUTION

1. Top of page

2. Abstract

3. I. INTRODUCTION





8. VI. CONCLUSIONS

9. NOMENCLATURE

10. References

The series in Eq. (14) is the solutions for the position parameter, , if one guarantees the convergence of these

series. The convergence region and rate of solution series can be adjusted and controlled by means of the auxiliary

parameter . In general, by means of the so-called -curve, it is straightforward to choose an appropriate range for

which ensure the convergence of the solution series, as pointed out by Liao [10]. To influence of on the

convergence of solution, the so-called -curve of (0.5) and by 12th-order approximation of solution are

plotted, as shown in Figures 2 and 3. it is easy to find out the valid region of , for example from Figure (3b) it is

clear that the acceptable region of is 1.5 0.5 and from Figure (3c), the range for the acceptable values of

is 1.4 0.6. Moreover, increasing the order of approximation, the range for the acceptable values of

increases (see Fig. 2 and I).

V. RESULT AND DISCUSSION

1. Top of page

2. Abstract

3. I. INTRODUCTION





8. VI. CONCLUSIONS

9. NOMENCLATURE

10. References

In Tables IIIV we make a comparison for the rotation of beam obtained using HAM and numerical solutions under

different loads. It is quite clear that the rotation of the free end of the beam increases with an increase in the shearing

force. The results obtained from the HAM method are quite consistent with this result, as shown in the Tables 1 and

2.

Table III. Comparison between HAM solution and numerical solution for (s) with

12th-order approximation, L = 1 and = 1.3. (Relative error define as |(HAM

numeric)/numeric|).

, m = 5,


0.0 1.3 1.81189 1.81126 0.00035

0.1 1.3 1.77015 1.77123 0.00061

0.2 1.3 1.63888 1.64060 0.00105

0.3 1.3 1.43753 1.43929 0.00122

0.4 1.3 1.19023 1.19087 0.00053

0.5 1.3 0.91863 0.91876 0.00014

0.6 1.3 0.64564 0.64560 0.00007

0.7 1.3 0.39405 0.39399 0.00015

0.8 1.3 0.18733 0.18726 0.00037

0.9 1.3 0.04913 0.04908 0.00092

1.0 1.3 0.00000 0.00000 0.00000

Table IV. Comparison between HAM solution and numerical solution for (s) with

12th-order approximation, L = 1 and = 1. (Relative error define as |(HAM

numeric)/numeric|).

, m = 1,


0.0 1.0 1.26003 1.26044 0.00032

0.1 1.0 1.09738 1.09807 0.00063

0.2 1.0 0.93876 0.93946 0.00075

0.3 1.0 0.78085 0.78133 0.00061

0.4 1.0 0.62395 0.62350 0.00073

0.5 1.0 0.46853 0.46888 0.00074

0.6 1.0 0.32380 0.32321 0.00182

0.7 1.0 0.19480 0.19440 0.00206

0.8 1.0 0.09177 0.09160 0.00188

0.9 1.0 0.02411 0.02405 0.00262

1.0 1.0 0.00000 0.00000 0.00000

Figure 4. Effects of on the slope, L = 1.




Figure 5. Effects of on the slope, L = 1 and .




Figure 6. Effects of on the position of beam, L = 1.




Figure 7. Effects of s on the position of beam, L = 1 and .




It should be noticed that as the load increases, i.e., means , and take larger numbers, the angle of the tip of

beam approaches and 2. For example, if takes the number 10, angle of tip of beam will be 2.99. It should also

be mentioned that in this article the term cos( + m(s l)) has been approximated by Taylor's series which causes

some small negligible errors. If loads take large numbers, the nonlinear term in Eq. (13) will increase, but even in this

case HAM will give a very accurate solution. As shown in Tables IIIV the obtained solution, as compared with the

numerical ones, represents a remarkable accuracy even for large values of the loads.

HAM formulations can be successfully employed to provide analytical solutions for large deflections static problems

for beams loaded by circulatory forces. By using of this method and confidence of results(see Tables IIIV) effects

of load-displacement and load-slope are investigated. The effect of shear force on the slope is presented in Fig. 4

and effects of compressive, tensile, and shear forces on the slope are shown in Fig. 5. Effects of external force and

moment are shown in Figs. 6 and 7.

VI. CONCLUSIONS

1. Top of page

2. Abstract

3. I. INTRODUCTION





8. VI. CONCLUSIONS

9. NOMENCLATURE

10. References

In this article the HAM is given for analysis of a cantilever beam under nonconservative load. Converging, accurate

results with few iterations are obtained, thus reducing the computational cost. An excellent rate of convergence has

been demonstrated and the results are in good agreement with the solution of numerical results. It can be concluded

that the presented method is a convenient and efficient method for the nonlinear analysis of beam problems and can

be used in a wide range of load and length for beams under large deformations.

NOMENCLATURE

1. Top of page

2. Abstract

3. I. INTRODUCTION





8. VI. CONCLUSIONS

9. NOMENCLATURE

10. References

Table .

Eb Young's modulus

H(s) Auxiliary function

Auxiliary parameter

12

I Moment of inertia

L Undeformed length

xA Cartesian coordinate

Reminder term

MA External moment

PA Compressive force

QA Transverse force

s Independent dimensionless parameter

A Normal slope at the end section

yA Cartesian coordinate

References

1. Top of page

2. Abstract

3. I. INTRODUCTION





8. VI. CONCLUSIONS

9. NOMENCLATURE

10. References

Argyris, J. H., Symeonidis, S. P., Nonlinear finite element analysis of elastic systems under

nonconservative loading-natural formulation. I. Quasistatic problems, Comput Methods

Appl Mech Eng 26, ( 1981), 75124

CrossRef (/resolve/reference/XREF?id=10.1016/0045-7825(81)90131-6), Web of Science

Times Cited: 85 (/resolve/reference/ISI?id=A1981LT34000004), ADS (/resolve/reference/ADS?

id=1981CMAME..26...75A)

Bolotin, V. V., Nonconservative problems of the theory of elastic stability, Pergamon, New

York, 1963

34

5

6

7

8

9

10

11

12

13

14

Dowell, E. H., Curtiss, H. C., Scanlan, R. H., Sisto, F., A modern course in aeroelasticity,

Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1978

Dowel, E. H., Aeroelasticity of Plates and Shells, Noordhoff, Leyden, The Netherlands, 1975

Web of Science Times Cited: 47 (/resolve/reference/ISI?id=A1975V442800001)

Nemat-Nasser, S., On elastic stability under nonconservative loads, H. H. E.Leipholz, Solid

Mechanics Division, University of Waterloo, Waterloo, Ontario, 1972 194

Leipholz, H. H. E., Six lectures on stability of elastic systems, 3rd ed., Solid Mechanics

Division, University of Waterloo, Waterloo, Ontario, 1975

Leipholz, H. H. E., Aspects of dynamic stability of structures, ASCE, J Eng Mech Div, 101,

EM2, ( 1975), 109124

Web of Science Times Cited: 7 (/resolve/reference/ISI?id=A1975W017200001)

Argyris, J. H., Straub, K., Symeonidis, S. P., Nonlinear finite element analiysis of elastic

system under nonconservative loading natural formulation. II. Dynamic, Comput Meths

Appl Mech Eng, 26, ( 1981), 75124


Times Cited: 85 (/resolve/reference/ISI?id=A1981LT34000004), ADS (/resolve/reference/ADS?

id=1981CMAME..26...75A)

Liao, S. J., The proposed homotopy analysis technique for the solution of nonlinear

problems, Ph.D. Thesis, Shanghai Jiao Tong University, China, 1992

Liao, S. J., Beyond perturbation: introduction to homotopy analysis method, Chapman and

Hall/CRC Press, Boca Raton, 2003

CrossRef (/resolve/reference/XREF?id=10.1201/9780203491164)

Wanga, J., Chena, J. K., Liao S. J., An explicit solution of the large deformation of a

cantilever beam under point load at the free tip, J Computat Appl Math, 212, ( 2008),

320330

CrossRef (/resolve/reference/XREF?id=10.1016/j.cam.2006.12.009), Web of Science Times

Cited: 14 (/resolve/reference/ISI?id=000253149100014), ADS (/resolve/reference/ADS?

id=2008JCoAm.212..320W)

Abbasbandy, S., The application of homotopy analysis method to solve a generalized Hirota-

Satsuma coupled KdV equation, Phys Lett A, 361, ( 2007), 478483

CrossRef (/resolve/reference/XREF?id=10.1016/j.physleta.2006.09.105), CAS

(/resolve/reference/CAS?id=1:CAS:528:DC%2BD2sXit1yntg%3D%3D), Web of Science

Times Cited: 93 (/resolve/reference/ISI?id=000244508400007), ADS (/resolve/reference/ADS?

id=2007PhLA..361..478A)

Sami Bataineh, A., Noorani, M. S. M., Hashim, I., On a new reliable modification of

homotopy analysis method, Commun Nonlinear Sci Numer Simulat, 14, 2009), 409423

CrossRef (/resolve/reference/XREF?id=10.1016/j.cnsns.2007.10.007), Web of Science Times


id=2009CNSNS..14..409S)

Domairry, G., Mohsenzadeh, A., Famouri, M., The application of homotopy analysis method

to solve nonlinear differential equation governing Jeffery--Hamel flow, Commun Nonlinear

Sci Numer Simulat, 14, ( 2009), 8595

15

16

17

18

19

20

21

22



id=2009CNSNS..14...85D)

Domairry, G., Nadim, N., Assessment of homotopy analysis method and homotopy

perturbation method in non-linear heat transfer equation, Int Commun Heat Mass Transfer,

35, ( 2008), 93102

CrossRef (/resolve/reference/XREF?id=10.1016/j.icheatmasstransfer.2007.06.007), CAS

(/resolve/reference/CAS?id=1:CAS:528:DC%2BD2sXhsVCksbnI), Web of Science Times

Cited: 20 (/resolve/reference/ISI?id=000252913500011)

Ziabakhsh, Z., Domairry, G., Analytic solution of natural convection flow of a non-

Newtonian fluid between two vertical flat plates using homotopy analysis method, Commun

Nonlinear Sci Numer Simulat, 14, ( 2009), 18681880



id=2009CNSNS..14.1868Z)

Liao, S. J., On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids

over a stretching sheet, J Fluid Mechanics, 488, ( 2003), 189212

CrossRef (/resolve/reference/XREF?id=10.1017/S0022112003004865), Web of Science

Times Cited: 193 (/resolve/reference/ISI?id=000185258000006), ADS

(/resolve/reference/ADS?id=2003JFM...488..189L)

Liao, S. J., Cheung, K. F., Homotopy analysis of nonlinear progressive waves in deep water,

J Eng Math, 45, ( 2003), 105116

CrossRef (/resolve/reference/XREF?id=10.1023/A:1022189509293), Web of Science Times


Liao, S. J., Pop, I., Explicit analytic solution for similarity boundary layer equations, Int J

Heat and Mass Transfer, 47, ( 2004), 7585

CrossRef (/resolve/reference/XREF?id=10.1016/S0017-9310(03)00405-8), Web of Science

Times Cited: 94 (/resolve/reference/ISI?id=000185820900008)

Abbasbandy, S., The application of homotopy analysis method to nonlinear equations

arising in heat transfer, Phys Lett A, 360, ( 2006), 109113

CrossRef (/resolve/reference/XREF?id=10.1016/j.physleta.2006.07.065), CAS

(/resolve/reference/CAS?id=1:CAS:528:DC%2BD28XhtFekur%2FE), Web of Science Times


id=2006PhLA..360..109A)

Hayat, T., Khan, M., Homotopy solutions for a generalized second grade fluid past a porous

plate, Non-linear Dynamics, 42, ( 2005), 395405

CrossRef (/resolve/reference/XREF?id=10.1007/s11071-005-7346-z), Web of Science Times


Hayat, T., Khan, M., Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant

fluid, Int J Eng Sci, 42, ( 2004), 123135

CrossRef (/resolve/reference/XREF?id=10.1016/S0020-7225(03)00281-7), Web of Science

Times Cited: 119 (/resolve/reference/ISI?id=000186480200002)

23

24

25

26

27

28

29

30

31

32

33

Hayat, T., Khan, M., Asghar, S., Homotopy analysis of MHD flows of an Oldroyd 8-constant

fluid, Acta Mechanica, 168, ( 2004), 213232

CrossRef (/resolve/reference/XREF?id=10.1007/s00707-004-0085-2), Web of Science Times


Abbas, Z., Sajid, M., Hayat, T., MHD boundary layer flow of an upper-convected Maxwell

fluid in porous channel, Theor Comput Fluid Dyn, 20, ( 2006), 229238

CrossRef (/resolve/reference/XREF?id=10.1007/s00162-006-0025-y), Web of Science Times


id=2006ThCFD..20..229A)

Hayat, T., Abbas, Z., Sajid, M., Asghar, S., The influence of thermal radiation on MHD flow

of a second grade fluid, Int J Heat Mass Transfer, 50, ( 2007), 931941

CrossRef (/resolve/reference/XREF?id=10.1016/j.ijheatmasstransfer.2006.08.014), Web of

Science Times Cited: 63 (/resolve/reference/ISI?id=000244575000014)

Sajid, M., Hayat, T., Asghar, S., Non-similar solution for the axisymmetric flow of a third

grade fluid over a radially stretching sheet, Acta Mechanica, 189, ( 2007), 193205

CrossRef (/resolve/reference/XREF?id=10.1007/s00707-006-0430-8), Web of Science Times


Alliney, S., Tralli, A., Extend variational formulation and F. E. models for nonlinear beams

under nonconservative loading, Computer Methods Appl Mechanics Eng, 46, ( 1984),

177194


Times Cited: 7 (/resolve/reference/ISI?id=A1984TS39000005), ADS (/resolve/reference/ADS?

id=1984CMAME..46..177A)

Love, A.E., A treatise on the mathematical theory of elasticity, Dover, New York, 1944

Timoshenko, S.P., Gere, J. M., Theory of elastic stability, McGraw-Hill, New York, 1961

Wang, J., Chena, J. K., Liao, S., An explicit solution of the large deformation of a cantilever

beam under point load at the free tip, J Computat Appl Math, 212, ( 2008), 320330

CrossRef (/resolve/reference/XREF?id=10.1016/j.cam.2006.12.009), Web of Science Times


id=2008JCoAm.212..320W)

Ganji, D. D., Rafei, M., Sadighi, A., Ganji, Z. Z., A comparative comparison of He's method

with perturbation and numerical methods for nonlinear vibrations equations, Int J

Nonlinear Dynamics Eng Sci, 1, ( 2009), 120

Abbasbandy, S., Shirzadi, A., The variational iteration method for a family of fifth-order

boundary value differential equations, Int J Nonlinear Dynamics Eng Sci, 1, ( 2009), 3946

Golbabai, A., Ahmadian, D., Homotopy Pade method for solving linear and nonlinear

integral equations, Int J Nonlinear Dynamics Eng Sci, 1, ( 2009), 5966

Enhanced Article (HTML) (http://onlinelibrary.wiley.com/enhanced/doi/10.1002/num.20538) Get PDF (267K)(/doi/10.1002/num.20538/pdf)

More content like this

Find more content:

like this article (/advanced/search/results?articleDoi=10.1002/num.20538&scope=allContent&start=1&resultsPerPage=20)

Find more content written by:

A. Kimiaeifar (/advanced/search/results?searchRowCriteria[0].queryString="A.Kimiaeifar"&searchRowCriteria[0].fieldName=author&start=1&resultsPerPage=20)G. Domairry (/advanced/search/results?searchRowCriteria[0].queryString="G.Domairry"&searchRowCriteria[0].fieldName=author&start=1&resultsPerPage=20)S. R. Mohebpour (/advanced/search/results?searchRowCriteria[0].queryString="S. R.Mohebpour"&searchRowCriteria[0].fieldName=author&start=1&resultsPerPage=20)A. R. Sohouli (/advanced/search/results?searchRowCriteria[0].queryString="A. R.

Sohouli"&searchRowCriteria[0].fieldName=author&start=1&resultsPerPage=20)A. G. Davodi (/advanced/search/results?searchRowCriteria[0].queryString="A. G.Davodi"&searchRowCriteria[0].fieldName=author&start=1&resultsPerPage=20)All Authors (/advanced/search/results?searchRowCriteria[0].queryString="A. Kimiaeifar" "G. Domairry" "S.R. Mohebpour" "A. R. Sohouli" "A. G.Davodi"&searchRowCriteria[0].fieldName=author&start=1&resultsPerPage=20)

analytical solution for large deflections of a cantilever beam under nonconservative load based on...

Documents