Flexural Buckling Analysis of Composite Beams of Variable Cross-Section by BEM

8/14/2019 Flexural Buckling Analysis of Composite Beams of Variable Cross-Section by BEM

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Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering

COUPLED PROBLEMS 2005

M. Papadrakakis, E. Oate and B. Schrefler (Eds)

CIMNE, Barcelona, 2005

FLEXURAL BUCKLING ANALYSIS OF COMPOSITE BEAMS OFVARIABLE CROSS-SECTION BY BEM

E.J. Sapountzakis*and G.C. Tsiatas

*

*School of Civil Engineering, National Technical University, Zografou Campus, GR-157 80 Athens,

Greece

e-mail: [email protected]

Key words: Flexural buckling, composite, variable cross-section, beam, boundary integral

equation, analog equation method

Abstract. In this paper a boundary element method is developed for the flexural buckling

analysis of composite Euler-Bernoulli beams of arbitrary variable cross section. The

composite beam consists of materials in contact each of which can surround a finite number

of inclusions. Since the cross-sectional properties of the beam vary along its axis, the

coefficients of the governing differential equation are variable. The beam is subjected to a

compressive centrally applied load together with arbitrarily axial and transverse distributed

loading, while its edges are restrained by the most general linear boundary conditions. The

resulting boundary value problems are solved employing a boundary integral equation

approach. Besides the effectiveness and accuracy of the developed method, a significant

advantage is that the displacements as well as the stress resultants are computed at anycross-section of the beam using the respective integral representations as mathematical

formulae. Several beams are analysed to illustrate the method and demonstrate its efficiency

and wherever possible its accuracy. The influence of the boundary conditions on the buckling

load is demonstrated through examples with great practical interest. The flexural buckling

analysis of a homogeneous beam is treated as a special case.

1 INTRODUCTION

Elastic stability of beams is one of the most important criteria in the design of structuressubjected to compressive loads. The flexural buckling coupled analysis is much more

complicated in the general case of a composite beam of variable cross-section. Namely, a

beam consisting of a relatively weak matrix material reinforced by stronger inclusions or of

materials in contact. The extensive use of the aforementioned structural elements necessitates

a reliable and accurate analysis of the flexural buckling problem.

Although there is an extensive research on the coupled flexural buckling analysis of

homogeneous beams of variable cross-section using analytical1-5, semi analytical6 or


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E.J. Sapountzakis and G.C. Tsiatas

numerical methods7-8, to the authors knowledge, publications on the solution to the general

composite problem do not exist.

In this investigation, an integral equation technique is developed for the flexural buckling

analysis of composite beams of arbitrary variable cross section. The composite beam consists

of materials in contact, each of which can surround a finite number of inclusions. The beam is

subjected to a compressive centrally applied load together with arbitrarily axial and transverse

distributed loading, while its edges are restrained by the most general linear boundary

conditions. The solution method is based on the concept of the analog equation9. According to

this method, the fourth order ordinary differential equation with variable coefficients is

replaced by an equivalent one pertaining to the transverse deformation of a substitute beam

with unit bending stiffness subjected to a fictitious transverse load distribution under the same

boundary conditions. Several beams are analysed to illustrate the method and demonstrate its

efficiency and wherever possible its accuracy. The influence of the boundary conditions onthe buckling load is demonstrated through examples with great practical interest. The flexural

buckling analysis of a homogeneous beam is treated as a special case.

2 STATEMENT OF THE PROBLEM

Let us consider an initially straight Euler-Bernoulli composite beam of length lhaving a

doubly symmetric arbitrary variable cross-section. The cross-section variation is assumed to

be smooth, so that the Euler beam theory is valid10. The beam is subjected to a compressive

centrally applied load P and to the combined action of distributed loading ( )x xp p x= ,

( )y yp p x= and ( )z zp p x= in thex , y and zdirections, respectively, where the x axis

coincides with the beam centroidal axis andy , z are the cross sections axes of symmetry

(Fig. 1a). The composite cross-section consists of materials in contact, each of which can

surround a finite number of inclusions, with modulus of elasticity jE , occupying the regions

j ( )j 1,2,...,K= of the y,z plane. The materials of these regions are assumedhomogeneous, isotropic and linearly elastic. Let also the boundaries of the nonintersecting

regions j be denoted by j ( )j 1,2,...,K= . These boundary curves are piecewise smooth,i.e. they may have a finite number of corners (Fig. 1b).

Taking into account the two planes of symmetry of the beam cross-section, the flexural

buckling problem can be uncoupled into two independent problems considering bending in

both zand yzplanes. Thus, the evaluation of the buckling load will be accomplished byregarding the beam subjected separately to bending in xzand in yzplanes, while the smaller

buckling load of the obtained ones will be of importance in engineering design.

Thus, considering the beam subjected to the compressive load P and to the combined

action of distributed loading ( )x xp p x= , ( )z zp p x= , the angle of rotation of the cross-section according to the linear theory of beams satisfies the following relations

cos 1 (1a)


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dwsin

dx

(1b)

x

z 0KE =

zp

xp

1E2E

y

l

yp

P

P

(a)

y

z

2( ) 2

C

3

3( )

K( )K

n

t

s

q

P r q P= s

1

Kjj=

= n

s

1( ) 1

(b)

Figure 1: Beam subjected to axial and bending loading (a) with variable composite cross section of arbitraryshape, occupying the two dimensional region (b).

while the curvature of the beam is given as

2

2

d d w

dx dx

= (2)


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zV

P

z

N dN+

zp

H dH+

yM

y yM dM+N

x

xp

dx

HQ

Q dQ+

z zV dV+

d+

P

Figure 2: Forces and moments acting on the deformed element in the xzplane.

Referring to Fig. 2, the stress resultants H, zV acting in the , zdirections, respectively,

are related to the axial Nand the shear Q forces as

H N cos Q sin = (3a)

zV N sin Q cos = + (3b)

which by virtue of eqns. (1) become

dwH N Q

dx= (4a)

z

dwV N Q

dx= + (4b)

The second term in the right hand side of eqn. (4a), expresses the influence of the shear force

Q on the horizontal stress resultant H. However, this term can be neglected since Q is much

smaller than Nand thus eqn. (4a) can be written as

H N (5)

The governing equation will be derived by considering the equilibrium of the deformed

element. Thus, referring to Fig. 2 we obtain

x

dHp

dx= (6a)


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zz

dVp

dx

= (6b)

ydMQ

dx= (6c)

Substituting eqns. (5), (4b) into eqns. (6a,b) and using eqn. (6c) to eliminate Q , we obtain

x

dNp

dx= (7a)

yz

dMd dwN p

dx dx dx

+ =

(7b)

Eqn. (7a) can be solved independently to evaluate the axial force N. Thus, integrating theaforementioned equation we obtain

( ) xN x p x c= + (8)

which by virtue of the boundary condition at the beam end l=

( )N l P= (9)

yields for the unknown constant c

xc P p l = + (10)

Substituting eqn. (10) into eqn. (8) yields

( ) ( )xN x p x l P= (11)

Using the above equation to eliminate the axial force Nfrom eqn. (7b) we obtain

( )2 2

yx z2 2

d M d dw d wp x l P p

dx dxdx dx

=

(12)

The expression for the bending moment of a composite beam can be written as

2

y 2d wM D( x ) D( x ) dx= = (13)

where

K

j jj 1

D( x ) E I ( x )=

= (14)

is its variable bending stiffness. Substituting eqn. (13) into eqn. (12) we obtain the governing

differential equation of the problem as


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( ) ( )

2 2 2

x z2 2 2

d d w d dw d w

D x p x l P pdx dxdx dx dx

+ + = inside the beam (15)

Moreover, the pertinent boundary conditions of the problem are given as

( )1 2 z 3w x V ( x ) + = (16a)

( )1 2 y 3

dw xM ( x )

dx + = at the beam ends 0,l= (16b)

where i i, ( i 1,2,3 )= are given constants. Eqns. (16) describe the most general boundary

conditions associated with the problem at hand and can include elastic support or restrain. It is

apparent that all types of the conventional boundary conditions (clamped, simply supported,free or guided edge) can be derived form these equations by specifying appropriately the

functions i and i (e.g. for a clamped edge it is 1 1 1 = = , 2 3 2 3 0 = = = = ).

Similarly, considering the beam subjected to the compressive load Pand to the combined

action of distributed loading ( )x xp p x= , ( )y yp p x= we obtain the boundary value problem

of the flexural buckling problem concerning bending in xy plane as

( ) ( )

+ + =

2 2 2

x y2 2 2

d d v d dv d vD x p x l P p

dx dxdx dx dx inside the beam (17)

( ) + =1 2 y 3

v x V ( x ) (18a)

( ) + =1 2 z 3

dv xM ( x )

dx at the beam ends x 0,l= (18b)

where i i, ( i 1,2,3 )= are given constants.

3 INTEGRAL REPRESENTATIONS - NUMERICAL SOLUTION

According to the precedent analysis, the flexural buckling problem of a beam reduces in

establishing the displacement components ( )w x , ( )v x having continuous derivatives up to

the fourth order satisfying the governing equations (15), (17) inside the beam and the

boundary conditions (16), (18) at the beam ends 0,l= , respectively. The numerical solutionof the boundary value problems described by eqns (15), (16) and (17), (18) is similar. For this

reason, in the following we analyze the solution of the problem of eqns (15), (16) noting any

alteration or addition for the problem of eqns (17), (18).

Eqn (15) is solved using the Analog Equation Method as it is developed for ordinary

differential equations in Katsikadelis9. This method is applied for the problem at hand as

follows. Let ( )w x be the sought solution of the boundary value problem described byeqns (15) and (16a,b). Differentiating this function four times yields


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( )=4

4

d wb x

dx

(19)

Eqn (19) indicates that the solution of the original problem can be obtained as the transverse

deflection of a beam with unit stiffness subjected to distributed fictitious loading ( )b x underthe same boundary conditions. The fictitious loading is unknown. However, it can be

established using BEM as follows.

The solution of eqn (19) is given in integral form as11

( )

l3 * 2 2 * 3 *

l * *

3 2 2 30

0

d w dw d w d w dw d ww x bw dx w w

dx dxdx dx dx dx

= +

(20)

where *w is the fundamental solution given as

( )3 2* 31w l 2 312

= + (21)

with r / l= , r x = , ,points of the beam, which is a particular singular solution of

the equation

4 *

4

d w( x, )

dx = (22)

Employing eqn (21) the integral representation (20) can be written asl

3 2l

4 4 3 2 13 200

d w d w dww( x ) b ( r )dx ( r ) ( r ) ( r ) ( r )w

dxdx dx

= + + +

(23)

where the kernels ,i( r ) ( i 1,2,3,4 ) = are given as

1

1( r ) sgn

2 = (24a)

2

1( r ) l( 1 )

2

= (24b)

23

1( r ) l ( 2 )sgn

4 = (24c)

( )3 234 1( r ) l 2 312

= + (24d)

Notice that in eqn (23) for the line integral it is r x = , , points inside the beam,

whereas for the rest terms it is r x q= , inside the beam, q at the beam ends 0, l.


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Differentiating eqn (23) results in the integral representations of the derivatives of the

transverse deflection w as

l3 2

l3 3 2 13 20

0

dw( x ) d w d w dwb ( r )dx ( r ) ( r ) ( r )

dx dxdx dx

= + +

(25a)

l2 3 2

l2 2 12 3 20

0

d w( x ) d w d wb ( r )dx ( r ) ( r )

dx dx dx

= +

(25b)

l3 3

l1 13 30

0

d w( x ) d wb ( r )dx ( r )

dx dx

=

(25c)

The integral representations (23) and (25a), when applied for the beam ends ( 0,l), together

with the boundary conditions (16a,b) are employed to express the eight unknown boundary

quantities ( )w q , ( )xw, q , ( )xxw, q and ( )xxxw, q ( q 0,l = ) in terms of b . This isaccomplished numerically as follows.

The interval ( )0,l is divided into Nequal elements (Fig. 3), on which ( )b x is assumed to

vary according to certain law (constant, linear, parabolic etc). The constant element

assumption is employed here as the numerical implementation becomes very simple and the

obtained results are very good.

Nodal points

2

x

1 N

l

Figure: 3. Discretization of the beam interval and distribution of the nodal points.

Employing the aforementioned procedure, the following set of linear equations is obtained

=

311 12 14

22 23 x 3

331 32 33 34 xx

42 43 44 xxx 4

,

,

,

+

0E E 0 E w

00 E E 0 w b

FE E E E w 0

0 E E E w F0

(26)

where 22E , 23E , 1iE , ( i 1,2,4= ) are 2 2 square matrices including the values of the

functions 1 2 1 2a , a , , of eqns (16a,b); ijE , ( i 3,4= , j 1,2,3,4= ) are square 2 2 known

coefficient matrices resulting from the values of the kernels i at the beam ends; 3 , 3 are


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2 1 known column matrices including the boundary values of the functions 3 3a , of

eqns (16a,b) and iF ( i 3,4 )= are 2 N rectangular known matrices originating from theintegration of the kernels on the axis of the beam. Moreover,

( ) ( ){ }T

w 0 w l =w (27a)

( ) ( )T

x

dw 0 dw l ,

dx dx

=

w (27b)

( ) ( )T

2 2

xx 2 2

d w 0 d w l ,

dx dx

=

w (27c)

( ) ( )T3 3

xxx 3 3

d w 0 d w l ,

dx dx

=

w (27d)

are vectors including the two unknown boundary values of the respective boundary quantities

and { }T

1 2 Nb b ... b=b is the vector including the Nunknown nodal values of the fictitious

load.

Discretization of eqns (23), (25) and application to the Ncollocation points yields

( )4 1 2 x 3 xx 4 xxx , , ,= + + +w C b E w E w E w E w (28a)

( )x 3 1 x 2 xx 3 xxx , , , ,= + +w C b E w E w E w (28b)( )xx 2 1 xx 2 xxx , , ,= +w C b E w E w (28c)

xxx 1 1 xxx, ,= w C b E w (28d)

where iC ( i 1,2,3,4 )= are N N known matrices; iE ( )i 1,2,3,4= are N 2 also known

matrices and w , x,w , xx,w , xxx,w , xxxx,w are vectors including the values of the transverse

deflection ( )w x and its derivatives at the Nnodal points.The above equations, after eliminating the boundary quantities employing eqns (26), can

be written as

= +w Tb t (29a)

x x x, = +w T b t (29b)

xx xx xx, = +w T b t (29c)

xxx xxx xxx, = +w T b t (29d)

where T , xT , xxT , xxxT are known N N matrices and t , xt , xxt , xxxt are known N 1

matrices. It is worth here noting that for homogeneous boundary conditions ( 3 3 0 = = ) it is


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x xx xxx= = = =t t t t 0 .

In the conventional BEM, the load vector b is known and eqns (29) are used to evaluatethe transverse deflection w and its derivatives at the Nnodal points. This, however, can not

be done here since b is unknown. For this purpose, N additional equations are derived,

which permit the establishment of b . These equations result by applying eqn (15) to the N

collocation points, leading to the formulation of the following set of N simultaneous

equations

( )xxP+ =A T b c (30)

In the above equation the matrices A and c are evaluated from the expressions

= + + + +x xxx xx xx x1 xx x2 x2A D D T D T P T P T (31a)

( )= + + + +z x xxx xx xx xx x1 xx x2 x2 Pc p D t D t t P t P t (31b)

where D , xD , xxD are N N diagonal matrices including the values of the stiffness and its

derivatives at the nodal points. The values of the derivatives of the bending stiffness are

approximated by appropriate central, forward or backward finite differences. Moreover, x1P ,

x2P are also N N diagonal matrices including the values of ( )xp x l , ( ) +x x xp , x l p at

the nodal points and zp is the vector containing the values of the external transverse

distributed loading at the nodal points.

Solving the linear system of eqns (30) for the fictitious load b , the deflection and its

derivatives in the interior of the beam are computed using eqns (29).

Buckling equation

In this case it is 3 3 0 = = (homogeneous boundary conditions) and z=p 0 . Thus, eqn (30)

becomes

( )xxP+ =A T b 0 (32)

The condition that eqn (32) has a non-trivial solution yields the buckling equation

( )xxdet P 0+ =A T (33)

4 NUMERICAL EXAMPLES

On the basis of the analytical and numerical procedures presented in the previous sections,

a computer program has been written and representative examples have been studied to

demonstrate the efficiency, wherever possible the accuracy and the range of applications of

the developed method.


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Example 1

For comparison reasons, a beam of length lwith three different types of boundary conditionsat the beam ends and corresponding varying stiffnesses has been studied. The first one of

these types has hinged-hinged boundary conditions and stiffness variation according to the

relation

( )20D( x ) D 1 = + (34)

The second one of the aforementioned types has fixed-hinged boundary conditions and

stiffness variation given from

o

(0)

(L)y

z l(a)

y

z

tf=1cm

ohwh0=18cm

bf

b0=8cm(I)

tw=1cm

tf=1cm

EA

ES

(tf=tw=t)

C

(b)

Fig.4. Longitudinal (a) and transverse (b) sections of the variable I - section beam of Example 2.


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20

9 3D( x ) D

16 4

= +

(35)

while the final one has fixed-fixed boundary conditions and stiffness variation according to

the relation

20

1D( x ) D

6

= +

(36)

where / l= . In Table 1 the computed buckling load ( )20P P / D / l= for theaforementioned cases is presented as compared with those obtained from an analytical

solution5 (in all three cases the exact buckling load is 2

0

12D / l ). The accuracy of the

obtained results using the proposed procedure is remarkable.

hinged-hinged fixed-hinged fixed-fixed

BEM Exact5 BEM Exact5 BEM Exact5

12.000 12.000 12.000 12.000 12.000 12.000

Table 1. Buckling load Pof the beam of example 1.

Example 2

The elastic stability of the composite I-section beam (flange material = 8 2fE 2.1 10 kN / m ,

web material = 7 2wE 7.0 10 kN / m ) of Fig. 4, with length =l 1.0m , has been studied. The

flange width ( )f 0 1b x b r x= and the web height ( )w 0 2h x h r x= are both varying linearly,

while all other dimensions remain constant; 1r , 2r are constants stating the rate of change of

the cross-section. Since Boley10has shown that the discrepancy between 2-D elasticity and

engineering beam theory for a beam with unit constant width and a rate of change of the

cross-section r 0.35 leads to an error of 7.5% , while for r 0.17 the error is 1.8% , the

maximum value of the aforementioned constant 2r has been set equal to 0.16, for the studiedexample. Three types of boundary conditions are considered. In Figs. 5-7 the variation of the

buckling load Pwith the rate of change 2r for various values of 1r for the cases of hinged-

hinged, fixed-hinged and fixed-fixed boundary conditions, respectively are presented. The

influence of the boundary conditions on the buckling load is remarkable.


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0 0.04 0.08 0.12 0.16Rate of change r

2

0

1

2

3

4

Buckling

load

(x104k

N)

r1=0.00

r1=0.02

r1=0.04

Figure 5: Variation of the buckling load with the rate of change of the web height for various values of the rate

of change of the flange width, for the hinged-hinged beam of Example 2.

0 0.04 0.08 0.12 0.16Rate of change r

2

0

2

4

6

8

Buckling

load

(x104k

N)

r1=0.00

r1=0.02

r1=0.04


of change of the flange width, for the fixed-hinged beam of Example 2.


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0 0.04 0.08 0.12 0.16Rate of change r

2

0

4

8

12

16

Buckling

load

(x104k

N)

r1=0.00

r1=0.02

r1=0.04


of change of the flange width, for the fixed-fixed beam of Example 2.

Example 3

The elastic stability of the composite beam of Fig. 8 with length l 5.0m= , consisting of asteel I-beam ( 8 2sE 2.1 10 kN / m= ) of variable flange width filled with concrete

( 7 2cE 3.0 10 kN / m= ) has been studied. The flange width varies according to the law

( ) ( )f Ib x b 2a sin x / l = while all other dimensions remain constant; Ib 0.60m= and a is

a shape parameter of the cross-section. In Table 2 the computed values of the buckling load

P for various values of the shape parameter a and for the cases of hinged-hinged, fixed-

hinged and fixed-fixed boundary conditions are presented. The influence of the boundary

conditions on the buckling load is once more verified.

5 CONCLUDING REMARKS

In this paper a boundary element method is developed for the flexural buckling analysis of

composite Euler-Bernoulli beams of arbitrary variable cross section. The composite beam

consists of materials in contact each of which can surround a finite number of inclusions. The

beam is subjected to a compressive centrally applied load together with arbitrarily axial and

transverse distributed loading, while its edges are restrained by the most general linear

boundary conditions. The main conclusions that can be drawn from this investigation are


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x

y

l=5.00

(I) (II)

a

0.60ma

(I)

( ) ( )2 sin /f Ib x b a x l = (a)Steel:

ES

Concrete:

EC

0.02m

0.02m

0.60m

bIIa a

bI= 0.60m

z

y

(b)

Figure 8: Plan view (a) and cross section (b) of the composite beam of Example 3.

hinged-hinged fixed-hinged fixed-fixeda

xzbending yzbending xzbending yzbending xzbending yzbending

0.00 29.2208 17.9108 59.7809 36.6425 116.8972 71.6518

0.05 25.3557 11.2531 53.4717 25.4703 105.9140 51.9743

0.10 21.4278 6.3634 46.9566 16.4903 94.3827 35.4015

0.15 17.4116 3.0482 40.1585 9.5451 82.1033 21.7737

Table 2. Buckling load P ( 410 kN ) for various values of the shape parameter a and for various cases ofboundary conditions of the composite beam of example 3.


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a. The numerical technique presented in this investigation is well suited for computer aided

analysis for beams of arbitrary variable cross section, subjected to any linear boundaryconditions and to an arbitrarily distributed or concentrated loading.

b. Accurate results are obtained using a relatively small number of beam elements.c. The displacements as well as the stress resultants are computed at any cross-section of the

beam using the respective integral representations as mathematical formulae.

d. The significant influence of the boundary conditions on the buckling load is remarkable.e. The flexural buckling analysis of a homogeneous beam can be treated as a special case.f. The developed procedure retains the advantages of a BEM solution over a pure domain

discretization method since it requires only boundary discretization.

ACKNOWLEDGMENTS

Financial support for this work provided by the Pithagoras: Support of Research Groups inUniversities, an EU funded project in the special managing authority of the Operational

Program in Education and Initial Vocational Training.

REFERENCES

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forces,International Journal of Solids and Structures, 27, 135-143 (1991).

[3] D.E. Panayotounakos, Classes of solutions in the problem of stability analysis in bars

with varying cross-section and axial distributed loading,International Journal of Solids

and Structures, 21, 3229-3236 (1995).

[4] I. Elishakoff and O. Rollot, New closed-form solutions for buckling of a variable

stiffness column by Mathematica, Journal of Sound and Vibration, 224, 172-182

(1999).

[5] I. Elishakoff, Inverse buckling problem for inhomogeneous columns, International

Journal of Solids and Structures, 38, 457-464 (2001).

[6] F. Arbabi, and F. Li, Buckling of variable cross-section column: Integral-equation

approach,Journal of Structural Engineering(ASCE), 117, 2426-2441 (1991).

[7] N.M. Newmark, Numerical procedure for computing deflections, moments and

buckling loads, Trans. ASCE108, 1161-1188 (1943).

[8] D.L. Karabalis and D.E. Beskos, Static, dynamic and stability analysis of structurescomposed of tapered beams, Computers and Structures, 16, 731-748 (1983).

[9] J. T., Katsikadelis, The Analog Equation Method-a Powerful BEM-Based Solution

Technique for Solving Linear and Non-linear Engineering ProblemsBoundary Element

Method XVI, ,167-182 1994.

[10] A.B Boley, On the accuracy of the Bernoulli-Euler theory for beams of variable

section,Journal of Applied Mechanics, 30, 373-378 (1963).

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Documents

Flexural Buckling Analysis of Composite Beams of Variable Cross-Section by BEM