Acomprehensive study on the linear viscoelastic buckling of Euler-Bernoulli (E-B)and Timoshenko beams is presented. The mathematical formulation for visco elas-tic E-B beam under combined time variant general axial and transverse loading

and for the viscoelastic Timoshenko beam only subjected to time variant general axialloading is developed. Various boundary conditions are defined then clamped edge con-straints are applied. The E-B and Timoshenko cantilever beams are investigated undergeneral axial loading. Both differential and integral forms of the beam governing equa-tions are derived, but only the differential equations are solved, using the finite differencenumerical technique assuming Kelvin material model. A computer programme is imple-mented for computing the beam deflections and a technique is described for obtainingthe critical buckling loads. Where possible the finite difference numerical results are com-pared with other available results. The correlations of the results are extremely well. Theimportant contributing factors are the general nature of the loading and the technique forcalculating the critical buckling loads.

Manouchehr Salehi* and Faroukh Ansari

Mechanical Engineering Department, Amir Kabir University of TechnologyP.O. Box: 15875-4413, Tehran, I.R. Iran

Received 6 February 2005; accepted 31 January 2006

Viscoelastic Buckling of Euler-Bernoulli andTimoshenko Beams under Time Variant

General Loading Conditions

Euler-Bernoulli beam;

Timoshenko beam;

viscoelastic beam;

critical buckling load;

finite-difference;

Kelvin model.

A B S T R A C T

Key Words:

INTRODUCTION

(*) To whom correspondence to be addressed. E-mail: msalehi@aut.ac.ir

Iranian Polymer Journal

15 (3), 2006, 183-193

Many different types of analyses forelastic and plastic behaviour forisotropic [1], and anisotropic E-Band Timoshenko beams have beencarried out [2]. An analytical modelto predict the linear viscoelasticbehaviour of thin-walled laminated

fibre-reinforced plastic (FRP) com-posite beams is presented [2]. Corre-spondence principle is used and themodel integrates micro/macro-mechanics of composites andmechanics of thin-walled laminatedbeams to perform beam analysis in

Available online at: http://journal.ippi.ac.ir

Laplace or Carson domain. Results for a box-beam in ten-sion and under bending are obtained. A Maxwell solidand three parameter solid type models are used to studythe bending behaviour of viscous material in higher-orderelastic beams by Johnson and Tessler [3]. A viscoelsticinternal variable constitutive theory is applied to both thebeam theory and the finite element formulation. Finally,numerical examples for the problems of relaxation, creepand cyclic creep are presented for an orthotropic Maxwelltype solid beam [3]. However, viscoelastic behaviour ofbeams, in particular Timoshenko beams, has received aminor attention in compression. Recently, the creep buck-ling of linear viscoelastic E-B beam under axial andtransverse loads is carried out [4]. A Maxwell modelmaterial is used for investigating the behaviour of nuclearfuel rods at elevated temperatures. The quasi-static anddynamic analysis of viscoelastic Timoshenko beam usingthe hybrid Laplace transform/finite element method ispresented by Chen [5]. The limited study observed in thisfield encouraged the first author to undertake a compre-hensive study on the linear viscoelastic buckling behav-iour of E-B and Timoshenko beams subjected to timevariant general loading conditions. In very special cases,analytical solutions are presented for solving Maxwelltype fluid material model [4].

With increasing the application of polymer basedmaterials in several fields of engineering like mechani-cal, aerospace, electrical poles etc. [6,7], and theincreasingly common use of such material in structuralelements like beams, columns, plates and shells, itseems reasonable to start investigating the mechanicalbehaviour of these types of elements. Polymer basedresins are used in composite structures, in which vis-coelastic behaviour ought to be considered. For metalstructures at elevated temperatures viscoelastic type ofanalysis is essential.

In the following sections of the paper, beam govern-ing equations, loading functions, boundary conditionsand the mechanical properties of the Kelvin model vis-coelastic material are given. The finite difference for-mulation and the numerical solutions of the beam gov-erning equations together with comparison results andseven examples are presented in the following sections.

BEAM COORDINATE SYSTEM AND LOADING

The coordinate system and the loading are shown in

Figure 1 for combined axial and transverse loading andin Figure 2 for axial loading only. The deformed shapeof the beam is highly exaggerated. In Figure 1 the beamcan be loaded with point loads at different locationsalong the length of the beam as axial and transverseloads as well as non-uniformly distributed loads in bothaxial and transverse directions. This type of loading isused for E-B beam.

For Timoshenko beam a similar type of loading asabove is used but only in axial direction, as is shown inFigure 2.

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Iranian Polymer Journal / Volume 15 Number 3 (2006)

Figure 1. Loading on Euler-Bernoulli beam.

Figure 2. Loading on Timoshenko beam.

BEAM DIFFERENTIAL EQUATIONS

The deflection variation of every point of the beam isdue to bending only in E-B beam and due to combinedbending and shear in Timoshenko beam.

Euler-Bernoulli Beam Differential EquationsThe bending moment due to axial load and transverseload is given as follows:

M = MF+Mq (1)

where MF and Mq are bending moments due to axialand distributed transverse loads, respectively. The bend-ing moments are in terms of time and position along thebeam length. The bending moment equations due togeneral axial and transverse loadings are given as:

where υ(x,t) is the deflection function, F(x,t) is theaxial load profile and q(x,t) is the transverse load pro-file. The loading function includes point, distributedand couple type of loading. Consequently, eqn (1) canbe written as:

The relation between the beam deflection and bendingmoment [8] given below:

PE(M(x,t)) = QE (I.υ”(x,t)) (5)

where the differential operators are stated as follows:

Now substituting eqn (4) into eqn (5) results the fol-

lowing general equation:

where I is the second moment of area of the beamcross-section. This is the most general equation forEuler-Bernoulli beam at uniform temperature.

In the especial case of constant cross-section withrespect to time and neglecting the Poisson effect [8],eqn (6) becomes:

In the case of pure transverse loading, eqn (7) changes to:

and in the case of pure axial loading, eqn (7) becomes:

Timoshenko Beam Differential EquationsThe deflection of every point of the beam is due tobending and shear. In homogeneous and isotropic linearviscoelastic material at uniform temperature the defor-mations due to bending and shear are independent [8].Consequently the process of deformation can be shownto be as follows:

The indices M and S refer respectively to bending and

Viscoelastic Buckling of Euler-Bernoulli and ...Salehi M. et al.

Iranian Polymer Journal / Volume 15 Number 3 (2006) 185

[ ]∫ ′⋅′⋅′υ−υ=x

o

F xd)t,x(F)t,x()t,x(M (2)

∫ ′⋅′⋅′−=x

o

q xd)t,x(q)xx(M (3)

( ) [ ]

∫

∫

′⋅′⋅′−

+′⋅′⋅′υ−υ=

x

o

x

ot,x

xd)t,x(q)xx(

xd)t,x(F)t,x()t,x(M (4)

( ) ( )[ ] ( ) ( ) ( )

⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂

υ∂⋅

=⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅′⋅′−+′⋅′⋅′υ−υ ∫∫

2

)t,x(

2

)t,x(

E

x

o

x

o

E

xIQ

xdt,xqxxxdt,xFt,xt,xP (6)

( ) ( )[ ] ( ) ( ) ( )

⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂

υ∂⋅

=⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅′⋅′−+′⋅′⋅′υ−υ ∫∫

2

)t,x(

2

E

)x(

x

o

x

o

E

xQI

xdt,xqxxxdt,xFt,xt,xP (7)

( ) ( ) ( )⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂

υ∂⋅=⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅′⋅′−∫ 2

t,x

2

E

)x(

x

o

E

xQIxdt,xqxxP (8)

S

ji

)t,x(

ji

M

ji

)t,x(

ji

ji

)t,x(

ji

xtxtxt ∂∂

υ∂∂+

∂∂

υ∂∂=

∂∂

υ∂∂ (10)

( ) ( )[ ] ( )

( ) ⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂

υ∂⋅

=⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅′⋅′υ−υ∫

2

)t,x(

2

E

x

x

o

E

xQI

xdt,xFt,xt,xP (9)

∑∑== ∂

∂⋅=∂∂⋅=

n

0ii

iE

i

En

0ii

iE

i

E

tqQ,

tpP

shear effects. Each term in eqn (10) can be separatelyevaluated.

a: Bending Effect

The relationship between the bending moment and theaxial load can be shown to be as follows:

To make eqn (11) suitable for the application of bound-ary conditions it is differentiated twice with respect toas follows:

b: Shear Effect

The rate of change of the slope (or curvature) due toshear for the linear elastic beam in terms of internalaxial force is [9]:

where n is the shear correction factor, υs is the deflec-tion due to shear, N is the internal axial force, A is thecross sectional area and G is the shear modulus of elas-ticity. A similar equation can be obtained for a linearviscoelastic material. This equation in Laplace space isfound to be as follows:

where QG and P G are differential operators due toshear strain and shear stress, respectively and:

Eqn (14) can be written as:

or in inverse Laplace transform as:

Finally, the equation relating the deflection due to shearand the axial force is found to be as follows:

Consequently, eqns (10), (12) and (17) constitute thebeam differential equations.

Mechanical Properties for Kelvin Model Material

In Kelvin model the operators PE and QE which arerestated as follows:

are assumed to have values as given below:

the superscripts K refer to Kelvin. Neglecting the effectof Poisson ratio, we obtain:

Pνk = 1, Qνk = 0 (19)

The ratio of the operators due to shear are then asfollows:

thus we conclude that:

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Iranian Polymer Journal / Volume 15 Number 3 (2006)

2

2

2

S2

xAGnN

x ∂υ∂⋅=

∂υ∂

(13)

SGG)s(Q

ANn)s(P υ′′⋅=υ′′⋅⋅ (15)

)(Q)A

nN(P SGG υ′′=υ′′⋅ (16)

υ′′⋅

⎟⎟⎠

⎞⎜⎜⎝

⎛=υ′′

)s(P

)s(QA

Nn

G

G

S

(14)

( ) ( )[ ] ( )

⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂

υ∂⋅

=⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅′⋅′υ−υ

∂∂ ∫

4

M

)t,x(

4

)t,x(

E

x

o

E2

2

xIQ

xdt,xFt,xt,xPx

(12)

( ) ( )[ ] ( )

⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂

υ∂⋅

=⎟⎟⎠

⎞⎜⎜⎝

⎛′⋅′⋅′υ−υ∫

2

M

)t,x(

2

)t,x(

E

x

o

E

xIQ

xdt,xFt,xt,xP (11)

2

S2S

2

2

x,

x ∂υ∂=υ′′

∂υ∂=υ′′

⎟⎠⎞⎜⎝

⎛∂

υ∂=⎟⎠⎞⎜⎝

⎛∂

υ∂⋅⎟⎠⎞⎜⎝

⎛ ′⋅⋅ ∫ ′ 4

S4G

4

4x

o )t,x(

G

xQ

xxdF

AnP (17)

K

K

EK

EK

GK

GK

PQ1

PQ

PQ

ν

ν

+= (20)

tEQ

1P

GK

GK

∂∂η+=

=(21)

)0i(,0q,q,Eq

)0i(,0P,1P

EK

1

EK

1

EK

EK

i

EK

o

o

>=η==

>==(18)

∑∑== ∂

∂⋅=∂

∂⋅=n

0ii

iE

i

En

0ii

iE

i

E

tqQ,

tpP

Viscoelastic Buckling of Euler-Bernoulli and ...Salehi M. et al.

Iranian Polymer Journal / Volume 15 Number 3 (2006) 187

where E is the modulus of elasticity and η is the viscos-ity coefficient.

Now substituting the properties obtained in eqns(18), (19) and (21) in eqns (12) and (17) the differentialforms of the equations become:

the integral form of the equations transform to:

Boundary ConditionsUsing the following equations, the boundary conditionsare defined:

(a) Clampedυ = 0

The total deflection curve has a zero slope at theclamped edge. The deflection can include shear effector it may be solely due to bending. (b) Internal Supportυ = 0

A similar explanation as that given above for thedeflection is valid here too. The second of eqn (29)shows that at the support the slope is continuous.

NUMERICAL SOLUTION

Beam Governing Differential EquationsThe closed form solution of the E-B beam governingequations for Maxwell model is presented by Yu [4].However, this solution is only obtained for a specialloading and boundary conditions. Here, on the otherhand, a very general loading for E-B and Timoshenkobeams with variable time is assumed for a Kelvin mate-rial model.

The governing differential equations for E-B andTimoshenko beams, i.e. eqns (7), (10), (12) and (17) aresolved utilizing the finite difference numerical solutiontechnique for obtaining the numerical results. In orderto minimize the computational effort, the E-B beamgoverning equation is not discretized instead theTimoshenko beam governing equations are written infinite difference form, where E-B beam results arerequired the shear term in the Timoshenko beam equa-tions is ignored. Consequently, only eqns (10), (12) and(17) are stated in finite difference form (Figure 3) asfollows:

[ ]

))(t

E(I

xd)t,x(F)t,x()t,x(

M

)t,x()x(

x

o

υ′′∂∂η+

=′⋅′⋅′υ−υ∫ (22)

∫∫

′⋅′∂

⎟⎠⎞⎜⎝

⎛ υ′′⋅⎟⎠⎞⎜⎝

⎛ ′⋅∂⋅′−=υ′′

′′t

o

x

o )t,x(GKS td

t

xdF)tt(J

An (25)

⎟⎟

⎠

⎞⎜⎜

⎝

⎛

∂

υ∂=⎟⎠

⎞⎜⎝⎛

∂∂

4

M

)t,x(

4

E

)x(2

F2E

xQI

xMP (26)

⎟⎠⎞⎜⎝

⎛∂

υ∂=⎟⎠⎞⎜⎝

⎛∂

υ∂⋅⎟⎠⎞⎜⎝

⎛ ′⋅∂∂ ∫ ′ 4

S4G

4

4x

o )t,x(2

2G

xQ

xxdF

xP

An

(27)

+− ∂υ∂=∂

υ∂oo

xx xx(29)

S

)t,x(

M

)t,x()t,x(υ′′∆+υ′′∆=υ′′∆ (30)

0x

=∂υ∂ (28)

))(t

E(n

AxdF S

)t,x(

)x(

)t,x(

x

o )t,x(υ′′

∂∂η+=υ′′⋅⎟⎠

⎞⎜⎝⎛ ′⋅∫ ′

(23)

( )∫ ′⋅′∂∂

⋅′−=υ′′ ′t

o

F

t,xEK

)x(

M

)t,x(td

t

M)tt(J

I1 (24)

tI

IExF)( M

)tt,x(

xxi

oi)t,xi()tt,xi()tt,x(

M

)t,x(

∆⋅η

υ′′−⎥⎥

⎦

⎤

⎢⎢

⎣

⎡∆⋅⋅υ ′′−υ′′

=υ′′∆

∆−

∆=

=∆⋅∆−∆⋅∆−∑

(31)

t

nA

nAExF S

)tt,x()t,x(

xxi

oi)t,xi(

S

)t,x(∆⋅η

υ′′⋅−υ′′⋅⎥⎥

⎦

⎤

⎢⎢

⎣

⎡∆⋅

=υ′′∆∆−

∆=

=∆∑

(32)

BOUNDARY CONDITIONS IN FINITEDIFFERENCE FORMSThe boundary conditions were stated above, i.e. theclamped end and the internal supports, are used in thepresent analysis. Consequently, the boundary condi-tions stated in eqn (28), in finite difference form aregiven bellow:

CONVERGENCE STUDY

In order to establish the degree of accuracy of theresults, a convergence study is carried out. The numberof nods which is characterized by the ratio (L/∆x)varies from 3 to 1000 and the percentage of error isshown in Table 1. The convergence curve for the criti-cal load is shown in Figure 4. In Figure 4 as the num-ber of nodes increases, the critical load converges to a

fixed value. As shown in Table 1, with increasing thenumber of nodes the percentage of error induced reach-es to a practically negligible value. This is obviouslyexpected to happen. The relative error is calculatedfrom the following relation:

NUMERICAL RESULTS AND DISCUSSION

The numerical results obtained from the solution of thegoverning equations include the Euler-Bernoulli andTimoshenko beam results. Initially the results are com-pared with existing results in the literature and, follow-ing the comparison of the results, several new resultsparticularly for Timoshenko beams are presented. It isworthwhile mentioning that, if desired, in order toobtain Euler-Bernoulli beam result one may eliminatethe shear term in the Timoshenko beam theory.

Comparison of ResultsThe results for viscoelastic buckling of E-B andTimoshenko beams are presented in Table 2 for com-parison purposes. The properties used in computing theresults are given bellow:

I = 160 � 10-6 m4, L = 10 m, n = 1.2 (used only forTimoshenko beam), E = 200 GPa , A = 0.001 m2 , η =0.001 (kg/m.s), Pν = 1, Qν = 0

Here, in fact, the Poisson’s ratio is set to zero which is

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Iranian Polymer Journal / Volume 15 Number 3 (2006)

Figure 3. Beam represented by finite difference model.

∑∆

=∆⋅

∆⋅υ′=υx

x

oi)t,xi()t,x(

x (33)

∑∆

∆=

∆⋅∆⋅υ′′−=υ′

xL

xxi

)t,xi()t,x(x (34)

cr

cr3

F

FFError

−= (35)

Figure 4. Convergence of critical load with mesh size.

Table 1. Percentage relative error.

Number of nodes xL

∆3 4 5 10 100 1000

Percentage relative error 4.7 2.6 1.6 0.4 0.003 0.00003

a legitimate assumption in viscoelastic beams. The typeof examples solved is shown in Figure 5.

In Figure 5c, a cantilever subjected to a concentrat-ed axial load in its third buckling mode is shown. Forsolving this type of problems internal boundary condi-tions are required. This type of internal supports is stat-ed in eqn (29). The results are given in Table 3.

As seen from the results presented in Tables 2 and3 the results obtained from the finite difference numer-ical solution correlate very well with the exact theoret-ical results.

New ResultsIn this section seven new examples of the type of

Timoshenko beam are solved. The properties used arethe same as those used above.

Example OneThe buckling of a cantilever beam with Kelvin typesolid model is considered under the loading shown inFigure 6.The loading profile is defined as:

when (Fcrd ) is the magnitude of the uniformly distrib-uted critical load which is obtained to be(1063033.301(N/m)) with (∆t = 5�10-11s) and the erroris within (10-5). Employing the implemented theoryand the techniques the critical load profile is found tobe as follows:

Examples Two to SixIn examples two to six a similar critical load profile isfound as follows:For example two: F(x) = 344912.262 × (x/L)(N/m)For example three: F(x) = 280228.604 (N/m)

Viscoelastic Buckling of Euler-Bernoulli and ...Salehi M. et al.

Iranian Polymer Journal / Volume 15 Number 3 (2006) 189

Figure 5. Examples for E-B and Timoshenko beams, (a) E-B

and Timoshenko, (b) and (c) Timoshenko.

Table 2. Critical load comparison for examples in Figures 5a and 5b.

Examples Present results Reference [10]

Figure 5 (a) E-B beam

∆t = 7.332 × 10-10s, Error 10-5

Figure 5 (a) Timoshenko

∆t = 7.332 × 10-10s, Error 10-5

Figure 5 (a) Timoshenko (n=1000)

∆t = 7.332 × 10-10s, Error = 3 × 10-5

Figure 5 (b) Timoshenko

∆t = 6.259 × 10-10s, Error 10-5≤20010

nLx ==∆

100010

nLx ==∆

≤20010

nLx ==∆

≤20010

nLx ==∆

789568.352 N

808901.531 N

808893.842 N

258651.0 N/m

789575.529 N

808893.442 N

808893.442 N

258649.278 N/m

)m/N)(L/x(FFcrd)x(

⋅=

Table 3. Critical load comparison for example Figure 6c for

∆t = L/n = 10/500, ∆t = 0.421 × 10-10s, Error 10-5.≤

Example Present Reference [10]

Figure 5 (c) Timoshenko 20222538.28 N 20222336.05 N

)m/N)(L/x(301.1063033F)x(

×=

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Iranian Polymer Journal / Volume 15 Number 3 (2006)

For example four: F(x) = 453313.259 × (L-2x)/L(N/m)For example five: F(x) = 720270.436 × (2xL)/L(N/m)For example six: F(x) = 346782.311 × (L-x)/L(N/m)

Consequently the critical load can be calculated at adesired position along the beam. The complete resultsare summarized in Table 4.

Example SevenIn this problem a cantilever beam under a general axialloading as shown in Figure 7 is considered. The gener-al loading is defined as follows:

(F1) is applied at x1 = 0 m

( F2) is applied at x2 = 0.2 m( F3) is applied at x3 = 0.4 m( F4) is applied at x4 = 0.7 m

and (F5) is a non-uniformly distributed load. Themechanical and geometrical properties are given as follows:

I = 160 × 10-6, E = 200 GPa, η = 0.001 (kg/m.s), L= 1 m, A = 0.001 m2, Pv = 1 , Qv= 0, n = 1.2, and spec-ifying ρ = 0 shows that the problem is a quasi-staticone.(a) (Fcr) is assumed to be the critical axial load appliedat the free end, then let: (F1= 0.5 Fcr ), (F3= 0.2 Fcr), (F4= 0.3 Fcr), and F5 = (1/meter3) × x(x-0.5)Fcr (0mx 0.6 m). (F2cr) should be determined for the equilib-rium of the beam.(b) If at time (t = 0) the initial deflection of the beam is:

and

and (F2 = 10F2cr) and the other loads are similar inmagnitude and position to those defined in (a) deter-mine the time that it takes for maximum slope of thebeam to reach a value of (0.06) from its initial value.

≤≤

Figure 7. Beam and its loading for example seven. Figure 8. Convergence analyses for part (b).

Figure 6. Distributed loading in the form of increasing ramp

along the beam.

)meter()2L

x(sin10 3

)0,x(π⋅=υ ×−

...),3,2,1i(,0t

0t

i

)t,x(

i

==∂

υ∂

=

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Iranian Polymer Journal / Volume 15 Number 3 (2006) 191

Table 4. Critical load profile results for various loading conditions for cantilever Timoshenko beams.

New examples Critical load profile Loading condition

Example one:( ) )m/N()L/x(301.1063033Fx

×=

Example two:( ) )m/N()L/x(262.344912Fx

×=

Example three:( ) )m/N(604.280228Fx

=

Example four: ( ) )m/N(L)/x2L(259.453313Fx

−×=

Example five:( ) )m/N()L/x2(436.720270Fx

×=

Example six: ( ) )m/N(L)/xL(311.346782Fx

−×=

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Iranian Polymer Journal / Volume 15 Number 3 (2006)

(c) If the initial deflection at time (t = 0) is given asfollows:

and

and that

F2 = 10F2cr + F2cr/10.t.(1/s)

and all the other forces are the same as above, again,determine the time that it takes for the maximum slopeto reach a value of (0.06) from its initial value.

Solution for (a)Initially (Fcr) is found to be (584.52�106 N) for (∆x =0.01 m) and (∆t = 10-15 s ). In part (a) (F2 ) is set to (F2 = 0)and check for the stability of the beam and that it is foundto be stable. Therefore, a critical value for (F2) can beobtained. Using the procedure described earlier the crit-ical value for (F2cr) is found to be (F2cr = 0.61235Fcr ).

Solution for (b)Having found (Fcr) and (F2cr), the loading profile isknown. This loading and the boundary conditions fort = 0 are applied and the required time is found to be (ta= 1.4442�10-12s) and that the maximum slope is at (x= 0). The accuracy of this result is specified by the con-

vergence study as shown in Figure 8. For the aboveresults (∆t) is taken to be (∆t = 0.3×10-15s).

Solution for (c)Similarly (Fcr) and (F2cr) are calculated. Applying theloading and the boundary conditions at (t = 0), therequired time is computed to be (tb) for ta = 1.2415×10-12s.The convergence of (tb) is shown in Figure 9.

CONCLUSION

The differential and integral forms of the Euler-Bernoulli and Timoshenko beams governing equationswith time-dependent axial and transverse loadings aredeveloped. The loading on the Euler-Bernoulli beamcan be axial and simultaneously transverse of a generaltype. However, the loading on the Timoshenko beamcan be pure axial but still of a general nature. Themechanical properties for a Kelvin type material modelare defined. The boundary conditions are successfullyimplemented. The equations developed here determinethe deflection profile under the prescribed loading.Consequently, a method for calculating the criticalbuckling load is used. The finite difference numericalmethod is utilized to solve the differential forms of thebeam equations and a computer program is developedto generate the numerical results. The comparisons ofthe present results with the existing available results arevery satisfactory. Seven examples are presented withloading profiles to be used for predicting the bucklingloads. Consequently, the outcome of the present paperis two fold:(a) the nature of the general type of loading and (b) thespecial mathematical technique used to calculate thecritical buckling loads.

ACKNOWLEDGEMENTS

The authors would like to express their sincere thanksto the Mechanical Engineering Department, Amir KabirUniversity of Technology for supporting this research.

NOMENCLATURE

A : Beam cross sectional area

Figure 9. Convergence analyses for part (c).

)meter)(2L

xsin(10 3

)0,x(

π⋅×=υ −

...),3,2,1i(,0t

0t

i

)t,x(

i

==∂

υ∂

=

Viscoelastic Buckling of Euler-Bernoulli and ...Salehi M. et al.

Iranian Polymer Journal / Volume 15 Number 3 (2006) 193

E : Modulus of elasticityFcr : Critical buckling loadFcrd : Uniformly distributed critical loadF(x,t) : Axial loading profileG : Shear modulus of elasticityI : Second moment of areaI(x,t) : Second moment of area as function of x and tJE

(t): Creep compliance function for normal defor-mationK,K’: Mechanical properties of the elastic supportsL : Beam lengthM: Total bending momentMF: Bending moment due to axial loadMq: Bending moment due to transverse loadM(x,t) : Total bending moment as function of x and tn : Shear correction factorN : Internal axial forcePE,QE : Differential operatorsPG,QG : Differential operators in Laplace spaceq(x,t) : Transverse loading profilet : Timetf : Final timet’ : Dummy variable of integration

υ(x,t): Deflection as function of x and tυ’(x,t) : First derivative of deflection w.r.t.x as func-tion of x and tν”(x,t) : Second derivative of deflection w.r.t.x asfunction of x and tυM

(x,t) : Deflection due to bendingυS

(x,t) : Deflection due to shearx : Position along the beamx’ : Dummy variable of integrationεxx : Normal strainτxx : Normal stress

η : Viscosity coefficient∆( ) : Incremental value of the variable ( )

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