Post-buckling analysis of variable cross-section cantilever beams under combined load via differential quadrature method

KSCE Journal of Civil Engineering (2010) 14(2):207-214DOI 10.1007/s12205-010-0207-4

− 207 −

www.springer.com/12205

Structural Engineering

Post-Buckling Analysis of Variable Cross-Section Cantilever Beams underCombined Load via Differential Quadrature Method

Omid Sepahi*, Mohammad Reza Forouzan**, and Parviz Malekzadeh***

Received June 21, 2009/Revised August 21, 2009/Accepted September 9, 2009

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Abstract

Based on the geometrically nonlinear theory of extensible elastic rods, the governing equations for post-buckling of variable cross-section cantilever beams subjected to a concentrated axial load at its free end and a non-uniformly distributed axial load areformulated. The Differential Quadrature Method (DQM) as a simple and computationally efficient numerical method is used toobtain the critical buckling load. The accuracy of the method is verified by comparing the results with those of the analytical resultfrom the elliptical integral, and other methods such as shooting method or multiple scale method. The post buckling configuration ofthe beam is obtained for different loading conditions.Keywords: differential quadrature, non-linear geometry, cantilever beam, post-buckling

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1. Introduction

Cantilever beams are common elements of many aeronautical,civil and mechanical engineering systems. Thin cantileverbeams, being flexible, exhibit large deflections and slopes, notonly when they are subjected to external loads but also due totheir own weight. Because of the large deflections, geometricalnon-linearity occurs. When the axial extensibility of the rod isconsidered, the differential line element of the deformed centralaxis is not equal to that of the pre-deformed axial axis line,namely, ds≠ dx, so that the deformed arc length of the rodbecomes one of the unknown functions of the problem (Wu,2003). Then it greatly adds the difficulties of solving theproblems.

Lee (2001) has analyzed the post-buckling of uniformcantilever column under a combined load consisting of auniformly distributed axial load and concentrated load at the freeend. Wu (2003) studied the post-buckling of a non-uniformcantilever beam under combined load with shooting method. Vazand Mascaro (2005) presented the solution of the behavior ofslender elastic rods subjected to axial terminal forces and self-weight via perturbation method. Lacarbonara (2008) constructedthe post-buckling solutions of non-uniform linearly and non-linearly elastic rods via a higher-order perturbation approach.Mazzilli (2009) dealt with post-buckling of all five classical kindof Euler buckling with multiple scale solution and pursued thecritical load and post-buckling configuration. Shavartsman

(2009) presented a direct method for analysis of flexiblecantilever beam subjected to two follower forces. Thermal post-buckling analysis of a heated elastic rod is studied by Li (2002).

The application of DQM in large deflection analysis of beamand plate is studied in the works of Malekzadeh et al.(Malekzadeh and Farid, 2007a, 2007b; Malekzadeh andSetoodeh, 2007; Karami and Malekzadeh, 2002). Based on theprevious studies of different engineering problems, it can beconcluded that DQM procedures offer comparable accuracy withless computational effort in comparison with those of theGalerkin method (Fazelzadeh et al., 2007), the finite differencemethod (Malekzadeh and Rahideh, 2009) and the finite elementmethod (FEM) (Malekzadeh and Vosoughi, 2009). In addition, itis mathematically simpler than FEM and also in spite of FEM itdiscretizes the strong form of the governing equations and therelated boundary conditions, which increases the accuracy ofDQM.

In order to overcome the drawback of the conventional DQMfor the higher order differential equation (order≥ 3), differentmethodology were introduced by the researchers (Karami andMalekzadeh, 2002; Liu and Wu, 2001a, 2001b). In this regards,Liu and Wu introduced the generalized differential quadraturerule.

In this paper, based on the theory of extensible elastic rods, thebuckling and post-buckling of extensible variable cross-sectioncantilever column subjected to a combined load, consisting of anaxial concentrated load at its free end and the non-uniform

*Ph.D. Student, Dept. of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran (E-mail: [email protected])**Assistant Professor, Dept. of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran (E-mail: [email protected])

***Assistant Professor, Dept. of Mechanical Engineering, Persian Gulf University, Bushehr 75168, Iran (Corresponding Author, E-mail: [email protected], [email protected])

Omid Sepahi, Mohammad Reza Forouzan, and Parviz Malekzadeh

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distributed axial load was analyzed using the differentialquadrature method. After demonstrating the convergence andaccuracy of the method, some parametric studies are performedto exhibit the post-buckling behavior of such columns.

2. Mathematical Formulation

Consider a variable cross-section cantilever column of un-deformed length L subjected to a tip load P at its free end anddistributed axial load q, as shown in Fig. 1. Denote a point of thepre-deformed axial line of the column as C(x,0), with x ∈ [0, L],where x and y are the Cartesian coordinate variables. When therod is in a buckled state, the material point C moves to point C' (x+ u, w), in which u and w are the displacement of the point C inthe x and y-directions, respectively. Here, it is presumed that thedeformed central axis is still in Oxy plane as shown in Fig. 1.

Based on the geometrically nonlinear theory of axially ex-tensible elastic rods the following basic equations can be writtenas Kinematic relations (Vaz and Silva, 2003, Madhusudan et al,2003):

, , (1)

, (2)

Equilibrium equations:

(3)

(4)

Constitutive relations:

, (5)

where s(x) is the arc length of the deflected axial line, R(x) is thestretch ratio of the axial line, θ(x) is the rotation angle of thecross-section, ε(x) is the strain of the axial line, κ(x) is thecurvature of the deflection curve, N(x) is the axial internal force,M(x) is the internal bending moment, P is the constrained forceat the ends along x direction, E is the Young' s modulus, A is thecross-section area and I is the moment of inertia of the cross-

section. Substituting Eq. (5) into Eqs. (3) and (4), one obtains:

(6)

(7)

Differentiating Eq. (6) with respect to x, yields:

(8)

The related boundary conditions become:

, , at

, , at (9)

The following dimensionless quantities are introduced:

, , , , , ,

(10)

where A0, I0 are the area and moment of inertia of the cross-section of the column at the x=0, respectively. Then Eqs. (1), (7)and (8) can be written in terms of the non-dimensional forms as:

, , (11)

(12)

(13)

where . A value of λ0=120 will be used in the nextcomputations.

Also, the boundary conditions can be expressed in thedimensionless form as:

, , at

, , at (14)

3. Solution Procedure

Because of the strong nonlinearity of Eqs. (11), (12) and (13),it is difficult to find an analytical solution of them. Herein, thedifferential quadrature method is used to study the criticalbuckling load and the post buckling behavior of the beam forsome different types of geometry and loading conditions.

3.1 Buckling Analysis In the first step, the critical buckling load will be found. Each

type of loads, concentrated or distributed load, can effect the

dsdx----- R= du

dx------ Rcosθ 1–= dw

dx------- Rsinθ=

ε ds dx–dx

---------------- R 1–= = κ 1R---dθ

dx------=

N x( ) P q η( ) ηd0x∫+( )cosθ– Vsinθ–=

M x( ) P w0 w x( )–( ) q µ( ) ηd0x∫ w x( ) w η( )–( )⋅+=

N AE R 1–( )= M EIdθdx------=

EIdθdx------ P w0 w x( )–( ) q η( ) ηd0

x∫ w x( ) w η( )–( )⋅+=

R 1 cosθ P q η( ) ηd0x∫+( ) AE⁄–=

EId2θdx2-------- EdI

dx-----dθ

dx------+ P q η( ) ηd0

x∫+( )dw

dx-------–=

s 0= dθdx------ 0= x 0=

u 0= θ 0= w 0= x L=

x xL---= s s

L---= u u

L---= w w

L----= I I

I0---= P PL2

EI0---------= q qL3

EI0--------=

dsdx----- R= du

dx------ Rcosθ 1–= dw

dx------- Rsinθ=

I d2θdx2-------- dI

dx-----dθ

dx------+ P q0

x∫ η( )dη+( )dw

dx-------–=

R 1 cosθ P q0x∫ η( )dη+( ) Iλ0

2( )⁄⋅–=

λ02 A0L

2 I0⁄=

s 0= dθdx------ 0= x 0=

u 0= w 0= θ 0= x 1=

Fig. 1. Vertical Cantilever Column under a Combined Load

Post-Buckling Analysis of Variable Cross-Section Cantilever Beams under Combined Load via Differential Quadrature Method

Vol. 14, No. 2 / March 2010 − 209 −

buckled state, so it must be specified the weight of each load inbuckling occurrence. A column with circular cross-section and adiameter changing linearly along the x axis is considered; also itis supposed that the distributed load q is the weight of thecolumn, thus:

(15)

, , (16)

where r0 and α are the radius at x=0 and the beam taper angle,respectively.

Now, by applying the DQM, Eq. (15) can be discretized. Afterthe discretization and applying the boundary conditions, thebuckling load can be obtained.

The basic idea of the DQM is that the derivative of a functionwith respect to a space variable at a given sampling point isapproximated as a weighted linear sum of the function value atsampling points in the domain of that variable. In order toillustrate the DQ approximation, consider a function f(x) havingits field on a domain 0≤ x≤ L. Let, in the given domain, thefunction values be known or desired on a grid of samplingpoints. According to DQ method, the rth derivative of a functionf(x) can be approximated as:

for i=1, 2, ..., n and r=1, 2, ..., n-1 (17)

where is called the weighting coefficients of the rth-orderderivative, n and fj are the total number of grid points and thesolution values at grid point j, respectively. From this equation,one can deduce that the important components of DQ approxi-mations are the weighting coefficients and the choice ofsampling points. In order to determine the weighting coeffici-ents, a set of test functions should be used in Eq. (17). Forpolynomial basis functions DQ, a set of Lagrange polynomialsare employed as the test functions in Eq. (17). Then, the weight-ing coefficients for the first-order derivatives in x-direction arethus determined as (Shu, 2000):

; i, j=1, 2…, n (18)

where

The weighting coefficients of the second order derivative canbe obtained as (Shu, 2000):

(19)

Grid points with uniform or non-uniform spacing can be used.However, it was found that using the roots of orthogonalpolynomials as the grid points yields more accurate results thanusing the uniform grid spacing (Karami and Malekzadeh, 2002,Shu, 2000). Hence, here in numerical computations, the roots ofChebyshev polynomials are used, that is (Shu, 2000):

for i=1, 2, ..., n (20)

Now, by applying the DQ method in inner grid points, n-2equations can be obtained which with two boundary conditionsperform a set of complete system of equations (n by n). Theresults can be expressed as:

(21)

(i=2, 3, ..., n-1)

And the boundary conditions become:

, (22)

The problem includes the combined load, thus, three kinds ofthe critical load can be obtained, which are denoted as whenq=0, when P=0 and , when P≠ 0, q0≠ 0. In thematrix form, the final set of equations can be written as:

(23)

where . The smallest value of λ yields the criticalvalue of load.

3.2 Post-Buckling AnalysisBy using the resulting solutions, the post-buckling analysis is

pursued. Finally, the nonlinear set of equations for the post-buckling analysis can be expressed as:

2≤ i≤n -1

, (24)

The Newton-Raphson iteration method is used to solving theabove set of nonlinear equations and the post-buckling results areplotted as shown as in the next section.

3.3 Numerical ResultsBased on the presented formulations, a FORTRAN code is

developed to calculate the critical buckling load. The resulting

I d2θdx2-------- dI

dx-----dθ

dx------+ P q0

x∫ η( )dη+( )Rsinθ–=

q0x∫ η( )dη ρg0

x∫ πr2 x( )( ) L3

EI0------- dx⋅ ⋅ ⋅ q0Ini= =

q0ρg πr2⋅

EI0-----------------= Ini

r x( )r0

2---------0

xi∫ dx= r x( ) r0 x L tan α( )⋅ ⋅+=

drf x( )dxr

-------------x xi=

cijr( )f xj( )

j 1=

n

∑ cijr( )fj

j 1=

n

∑= =

cijr( )

cij1( )

1L--- M xi( )

xi xj–( )M xj( )------------------------------ for i j≠

cij1( )

j 1=i j≠

Nx

∑– for i j=

⎩⎪⎪⎪⎨⎪⎪⎪⎧

=

M xi( ) xi xj–( )j 1= i j≠,

n

∏=

cij2( )[ ] cij

1( )[ ] cij1( )[ ] cij

1( )[ ]2= =

xi

L--- 1

2--- 1 cos i 1–( )π

n 1–( )-----------------–

⎩ ⎭⎨ ⎬⎧ ⎫

=

Ii cij2( )θj

j 1=

n

∑dIi

dx------ cij

1( )θjj 1=

n

∑+ P q0xi∫ η( )dη+[ ]Risinθi–=

dθdx------ x 0=( ) cij

1( )θjj 1=

n

∑ 0= = θ x 1=( ) θn 0= =

Pcr*

q0cr* Pcr q0cr

A{ } n n×( ) θ{ } n( ) λ B[ ] n n×( ) θ{ } n( )=

λ P q0+=

Ii cij2( )θj

j 1=

n

∑dIi

dx------ cij

1( )θjj 1=

n

∑ P q0xi∫ η( )dη+[ ]Risinθi+ + 0=

c1 ij,1( ) θj

j 1=

n

∑ 0= θn 0=

Omid Sepahi, Mohammad Reza Forouzan, and Parviz Malekzadeh

− 210 − KSCE Journal of Civil Engineering

solution is examined with the available references. Table 1shows the values of and for the different values of α. InTable 2 the values of for some prescribed values of α and Pare presented. It can be shown that the value of α has a greatinfluence on the buckling load magnitude, however, the effect ofα reduces with the increasing the value of α (Tables 1 and 2). Itcan be seen that the close agreement between the results of thispaper and other reference (Wu, 2003; Lee, 2001) exist.

Here, the post-buckling behavior of the column is studied.

Table 3 presents the examination results for the developed codeand shows the post-buckling solution for a uniform cantileverwhen q=0. It can be seen that the results of this paper are in goodagreement with those of the other references.

In the next figures the following symbols are used: #1: β =20o,#2: β =40o, #3: β =60o, #4: β =80o, #5: β =100o, #6: β =120o,#7: β =140o, #8: β =160o.

q0cr* Pcr

*

q0cr

Table 1. The Values of q*0cr and P*

cr Corresponding to Different Values of α

Method α (deg) 0 0.5 1 1.5 2 2.5

Present 7.8473 23.882 51.183 91.099 144.62 212.48

Wu et al. (2003) 7.8357 23.884 51.348 91.158 144.77 212.81

Present 2.4674 7.9193 17.515 31.777 51.036 75.493

Wu et al. (2003) 2.4678 7.9191 17.517 31.791 51.068 75.557

Table 2. The Values of q*0cr for Some Prescribed Values of α and P

P

0 0.05 0.1 0.2 1 2

q0cr

α = 0Present 7.8473 7.6873 7.5368 7.2349 4.7685 1.5549

α = 0Wu et al. (2003) 7.8357 7.5389 7.2402 7.2309 4.7688 1.5578

α = 1Present 51.183 51.061 50.939 50.696 48.740 46.246

α = 1Wu et al. (2003) 51.348 51.079 50.949 50.711 48.749 46.351

α = 2Present 144.62 144.52 144.42 144.21 142.60 140.56

α = 2Wu et al. (2003) 144.77 144.67 144.56 144.37 143.14 140.70

q0cr*

q0cr*

Pcr*

Pcr*

Table 3. The Result of Post-Buckling Analysis of a Uniform Col-umn When q0 =0 (Continued)

Method β P W(0) 1-U(0)

Lee (2001) 80 3.192 0.7196 0.5594

Elliptical integral method 80 3.1903 0.719 0.56

Present 100 3.7465 0.7914 03489

Lee (2001) 100 3.7458 0.7916 0.349

Elliptical integral method 100 3.7455 0.792 0.349

Present 120 4.650 0.8031 0.1230

Lee (2001) 120 4.6498 0.8032 0.1231

Elliptical integral method 120 4.6486 0.803 0.1213

Present 140 6.2716 0.7504 -0.107

Lee (2001) 140 6.2719 0.7505 -0.107

Elliptical integral method 140 6.2697 0.75 -0.107

Present 160 9.9396 0.6247 -0.3406

Lee (2001) 160 9.9428 0.6247 -0.3404

Elliptical integral method 160 9.9411 0.625 -0.34

Table 3. The Result of Post-Buckling Analysis of a Uniform Col-umn When q0 =0

Method β P W(0) 1-U(0)

Present 0 2.4674 0 1

Lee (2001) 0 2.4671 0 1

Elliptical integral method 0 2.4674 0 1

Present 20 2.5058 0.2194 0.9696

Lee (2001) 20 2.5049 0.2194 0.9698

Elliptical integral method 20 2.5044 0.22 0.97

Present 40 2.6247 0.4221 0.8810

Lee (2001) 40 2.624 0.4223 0.8812

Elliptical integral method 40 2.6228 0.422 0.881

Present 60 2.8420 0.5931 0.741

Lee (2001) 60 2.8412 0.5933 0.741

Elliptical integral method 60 2.8424 0.593 0.741

Present 80 3.192 0.7194 0.5593

Post-Buckling Analysis of Variable Cross-Section Cantilever Beams under Combined Load via Differential Quadrature Method

Vol. 14, No. 2 / March 2010 − 211 −

Fig. 2(a) shows the post-buckling configuration for thedifferent values of P when q0 =0 and α=0. Fig. 3(a) presents thesame results for α=0 and q0=5. The effect of gravitational fieldon the post-buckling configuration can be seen in Figs. 2(a) and3(a) also Figs. 3(a) and 6(a) present the effect of α on the post-buckling configuration. It is observed that α has a greet influence

on the post-buckling capacity of column. Also Fig. 8(a) presentsthe effect of gravitational field on the post-buckling shapes whenthe column is under self weight. The variation of bending momentand axial force parameters for the corresponding post-bucklingconfigurations are plotted in Figs. (4), (7) and (9).

Fig. 2. Post-Buckling of a Column with α=0 and q=0: (a) Post-Buckling Configuration, (b) Variation of β across P

Fig. 3. Post-Buckling of a Column with α=0 and q0 =5: (a) Post-Buckling Configuration, (b) Variation of β across P

Fig. 4. Post-Buckling of a Column with α=0 and q0=5: (a) Variation of Bending Moment Parameter, (b) Variation of Axial Force Parameter

Omid Sepahi, Mohammad Reza Forouzan, and Parviz Malekzadeh

− 212 − KSCE Journal of Civil Engineering

4. Conclusions

As a simple but powerful numerical method, the differentialquadrature method is used for the solution of buckling and post-buckling of an extensible elastic cantilever beam under com-

bined load. The effect of gravitational field and greet influence ofthe taper angle of the column on the post-buckling capacity ispresented. Solutions show the good agreement with availablereferences. Some numerical results are prepared which may beuseful for other researchers.

Fig. 5. Post-Buckling of a Column with α=2 and q0 =0: (a) Post-Buckling Configuration, (b) Variation of β across P

Fig. 6. Post-Buckling of a Column with α=2 and q0 =5: (a) Post-Buckling Configuration, (b) Variation of β across P

Fig. 7. Post-Buckling of a Column with α=2 and q0=5: (a) Variation of Bending Moment Parameter, (b) Variation of Axial Force Parameter

Post-Buckling Analysis of Variable Cross-Section Cantilever Beams under Combined Load via Differential Quadrature Method

Vol. 14, No. 2 / March 2010 − 213 −

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Omid Sepahi, Mohammad Reza Forouzan, and Parviz Malekzadeh

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