Approximative formula for post-buckling analysis of non-linearly elastic columns with...

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Approximative Formula for Post-Buckling Analysis of Nonlinearly

Elastic Columns with Superellipsoidal Cross-Sections

Mihael Brojan*, Franc Kosel

* Laboratory for Nonlinear Mechanics, Faculty of Mechanical Engineering, University of Ljubljana, Askerceva

6, SI-1000 Ljubljana, Slovenia. E-mail: [email protected]

Abstract

Approximative formulas for post-buckling analysis of nonlinearly elastic columns made of

Ludwick material are developed for free-clamp, hinge-hinge, and clamp-clamp supports. The

columns have a superellipsoidal cross-section. Comparison between analytically obtained and

numerical solutions showed good agreement. Additionally, post-buckling configurations for

all three types of columns and materials are given in diagrams, from where the influence of

material constants on the shape of the deflection curve can be examined.

Keywords

Nonlinearly Elastic Column, Superellipsoidal Cross-Section, Post-buckling, Approximative

Solution, Analytical Formula.

Introduction

Slender members under sufficient axial compressive loadings may exhibit large lateral

displacements which usually lead to sudden failure of structures long before the maximum

stresses exceed the limit stress. In designing structures and machines this phenomenon called

buckling is thus of major concern. For example, when a straight uniform column is subjected

to an axial compression force P , Figure 1a), it remains straight when P is small and

(usually) deflects laterally when P exceeds a certain critical value. This critical value crP is

often named the Euler buckling force.

In the present article, buckling and post-buckling of uniform columns which are made of

nonlinearly elastic material is studied as a continuation of the subject covered by the authors

in [1]. In the last few years similar investigations of geometrically and materially nonlinear

problems of beam bending have been reported by number of authors. Contributions that are

most closely related to the problem addressed here can be found in [2-4] where the

mechanical behavior of a cantilever beam made of non-linear Ludwick’s bimodulus material

is analyzed when subjected to pure bending; in [5] where the moment–curvature relation was

derived for superellipsoidal cross-section fibers made of nonlinear material for the case of

pure bending; and in [6,7] where the beam made of functionally graded non-linearly elastic

material of Ludwick type is considered.

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M. Brojan, F. KoselJournal of Reinforced Plastics and Composites, 2011, Volume 30, Number 5, Pages 409-415

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Figure 1. Trivial and post-critical shape modes of I, II and IV Euler column.

Boundary conditions and critical loads of the columns investigated in this contribution are

listed in Table 1. Critical loads are the solutions of the transcendental equations, cos 0ω = ,

sin 0ω = , and sin 2cos 2 0ω ω ω+ − = ,

which can be obtained via linearization of the problem.

Table 1. Boundary conditions and critical loads.

Euler case support boundary conditions 2

cr / ( )P L EI⋅

I clamp-free (0) 0ϑ = , '( ) 0Lϑ = 2 / 4π

II hinge-hinge '(0) 0ϑ = , '( ) 0Lϑ = 2π

IV clamp-clamp (0) 0ϑ = , ( ) 0Lϑ = 24π

Formulation of the problem

Let us consider a slender, initially straight column of length L subjected to an axial

compression force P , as shown in Figure 1. The column has a cross-section of constant width

2a and constant height 2b in the shape of the superellipse (cf. §2.1). The mathematical

model of the discussed problem is based on the elastica theory. The material of which the

column is made is assumed to be homogenous, incompressible, and isotropic. The nonlinear

stress–strain relation is given by the Ludwick’s formula

1/

( ) sign( )n

Eσ ε ε ε= , (1)

where E and n are material constants.

From static equilibrium of internal forces and inner bending moments which act on an

infinitesimal element of the deformed beam, cf. Figure 2, geometrical relations

/ cosdx ds ϑ= , / sindy ds ϑ= , from the expression for inner bending moment as a function

of normal stress A

M ydAσ= −∫ , Equation (1), and the normal strain-curvature expression

1yε ρ −= − and 1 /d dsρ ϑ− = , we can deduce an equation

(1 )/ 2

1 2

1sin sin 0

n n

n

d dEI P H

n ds ds

ϑ ϑϑ ϑ

−

+ + + =

, (2)

which together with the accompanying boundary conditions (cf. Table 1) describes the post-

buckling behavior of columns subjected to an axial force. Variable s , 0 s L≤ ≤ , denotes a

curvilinear coordinate along the centroidal axis measured from the fixed end of the column

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and ( )sϑ represents the angle between the positive direction of the x-axis and the tangent to

the centroidal axis at point s .

Remark 1. It should be noted that H and 0M are the reactive force and moment acting in the

upper support of the column, Figure 1b)-d). Furthermore, 0, 0H M = for I Euler’s case,

0H ≠ , 0 0M = for II Euler’s case and 0, 0H M ≠ for IV Euler’s case.

Figure 2. Free body diagram and infinitesimal element of the deflected column.

By introducing quantities

1 n

nPp

EI +

:= , 1 n

nHh

EI +

:= , /(1 )n nt p s+:= , (3)

where (1 )[0 ]n nt p L/ +∈ , , and introducing parameter η

h

pη := , (4)

dividing Equation (2) by p , we obtain

(1 ) 2

2sin cos 0

n nd d

dt dt

ϑ ϑϑ η ϑ

− / + + =

. (5)

Superellipse

A generalized ellipse or superellipse is a closed curve defined by the following implicit

equation

1z y

a b Ra b

α β

α β +| | | |+ = , , , , ∈ . (6)

where a and b are semi-axes. They are special cases of curves which are known in analytical

geometry as Lame curves. The name superellipse was proposed by Piet Hein, a Danish poet

and scientist who popularized these curves for design purposes, [8].

Remark 2. It can be noted that an ordinary ellipse is obtained if 2α β= = , and further if

1a b= = the unit circle is obtained. In the limit case α β, → ∞ equation (6) yields a

superellipse which resembles a rectangle whereas in the limit case 0α β, → it resembles a

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cross. Some more special cases are depicted in Figure 3, where values of a , b , α , β are

given in brackets, ( )a b α β, , , .

Figure 3. Cross-sections defined by superellipse.

Superellipse may also be represented parametrically by

2

2

sign(cos ) cos

sign(sin ) sin

z a

y b

α

β

ϕ ϕ

ϕ ϕ

/

/

=

= ,

where [0 2 ]ϕ π∈ , . The 1 n+ -th moment of area can therefore be determined as shown below

1

1

†2

22

2 1 2(2 ) 1

0

1† Green's theorem

2

8(cos ) (sin )

(2 )

n

nA

n

A

nn

I y dA

y dzn

abd

n

π α βϕ ϕ ϕα

++

+

∂

+ / / − + / +

:=

=−+

= .+

∫

∫

∫

The integral above can be expressed by the Beta function,

2 2 1 2 1

0( ) 2 (cos ) (sin )p q

B p q dπ

ϕ ϕ ϕ− −, := ∫ , (7)

which leads to

2

1

4 1 21

(2 )

n

n

ab nI B

nα α β

+

+

+= , + +

. (8)

By using the identities, cf. [9]

( ) ( ) ( 1) ( )q

B p q B q p B p q B p qp q

, = , , , + = ,+

(9)

one can get

2

1

4 1 2

(2 )

n

n

ab nI B

nα β α β

+

+

+= , . + +

(10)

Remark 3. In the limit case, when α β, → ∞ , it follows

2

1

4

2

n

n

abI

n

+

+ =+

, (11)

and further for 1n = a well known formula 3

2 4 / 3I ab= is obtained.

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Remark 4. The area of a superellipse A

dA∫ can be obtained by setting 1n = − in Equation

(10),

4 1 1ab

A Bα β α β

= , +

. (12)

Determination of the critical force

In immediate post-buckling ( )tϑ is expected to be small for all (1 )[0 ]n nt p / +∈ , . Therefore

approximating sinϑ ϑ� is reasonable. In the I Euler case 0H ≡ . Then Equation (5) can be

reduced to

(1 ) 2

20

n nd d

dt dt

ϑ ϑϑ

− / + =

. (13)

Accompanying boundary conditions are

( )(1 )(0) 0 0n np Lϑ ϑ / +′= , = . (14)

Introducing function u , such that

'( ) : ( ( ))t u tϑ ϑ= . (15)

and differentiating with respect to variable t gives

''( ) '( ( )) ( ( ))t u t u tϑ ϑ ϑ= . (16)

The Equation (13) can now be rewritten as

1' 0nu u ϑ/ + = . (17)

Integrating and considering Equation (15) yields

2

(1 )( )1 2

n nnc

n

ϑϑ + /′ + =

+. (18)

It follows from the boundary condition (1 )( ) 0n np Lϑ / +′ = that

2

2

ecϑ

= , (19)

where (1 )( )n n

ep Lϑ ϑ / +:= . From Equations (18) and (19) one can obtain

(1 ) (1 )(1 )

2 21 1

2

n n n nn n

e

d n

dt n

ϑϑ ϑ

/ + / +/ +

+ = −

. (20)

Rewriting the equation above and integrating over the domains of ϑ and t on each side of

equation lead to

(1 ) (1 )

(1 ) (1 )( )

(1 )(0) 02 2

1 1

2

n n n nn n n n

p L p L

n n

e

d ndt

n

ϑ

ϑ

ϑ

ϑ ϑ

/ + / + / + / +

/ +

+ = −

∫ ∫ . (21)

The integral on the RHS is equal to

(1 ) (1 )

(1 )1 1

2

n n n n

n n

RHS

nS p L

n

/ + / +/ ++ =

. (22)

As already mentioned, ( )tϑ is expected to be small at immediate post-buckling and it is

therefore reasonable to linearize function ϑ , so that

(1 ) 1( ) n n

et p L tϑ ϑ − / + −� . (23)

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Taking this into account, the integral on LHS can be written as

(1 )(1 ) (1 )

(1 )(1 ) 0 2 (1 ) 2 21

n nn np L

eLHS n nn n

n n

dtS

p L p L t

ϑ / +− / +

/ +/ + − / + −

=−

∫ . (24)

Introducing variable 2 (1 ) 2 2n nw p L t− / + −= ,

(1 ) (1 )

11 2 (1 )

0(1 )

2

n nn ne

LHSS w w dw

ϑ − / +− / − / += −∫ . (25)

Beta function B , given by Equation (7), can also be written in the following form, cf. [9]

1

1 1

0( ) (1 )p q

B p q x x dx− −, = −∫ . (26)

Hence

(1 ) (1 )

1 1

2 2 1

n n

eLHS

S Bn

ϑ − / + = , + . (27)

Equating (27) and (22) results in

(1 )(1 )

1 (1 )

1 1( )

2 (1 ) 2 1

n nn n

e

n n n

np n B

n L n

ϑ + /− /

/ + /

= , + + (28)

and

(1 )(1 )

1I 1 (1 )

1 1( )

2 (1 ) 2 1

n nn n

e n

n n n

EIP n B

n L n

ϑ + /− /+

/ + /

= , + + . (29)

which follows from 2(3) additionally.

Approximative formula for post-buckling behavior analysis of nonlinearly elastic columns

with superellipsoidal cross-sections is therefore

(1 )(2 1) (1 ) 2

I (1 )

2 1 2 1 1( )

(1 )( (2 ) ) 2 1

n nn n n n n

e

n n

E ab nP n B B

n n L n

ϑα β α β

+ /− / − / +

+ /

+ = , , + + + + . (30)

The critical force crP for the linearly elastic column can be found by setting 1n =

2

I,cr 2 24P EI

L

π= . (31)

Expression for II ( )P n can be derived in a similar way. Namely, considering obvious

natural symmetry in the deflection curve of II Euler case, the first boundary condition in (14)

can be replaced by ( )(1 ) 2 0n np Lϑ / + / = . Thus an equation, equivalent to Equation (21), can be

written (1 ) (1 )

(1 ) (1 )

( 2) 2

( )2 2

n n n n

n n n n

p L p L

p L p L… …

ϑ

ϑ

/ + / +

/ + / +

/ /=∫ ∫ Parameter (1 )( )n n

ep Lϑ ϑ / +:= and approximation

(1 )(2 ( ) 1)n n

ep L tϑ ϑ / +/ −� are used to get the following expressions

(1 )(1 )

1II (1 )

2 1 1( )

(1 ) 2 1

n nn n

e n

n n

EIP n B

n L n

ϑ + /− /+

+ /

= , + + , (32)

(1 )3 (1 ) 2

II (1 )

2 1 2 1 1( )

(1 )( (2 ) ) 2 1

n nn n n

e

n n

E ab nP n B B

n n L n

ϑα β α β

+ /− / +

+ /

+ = , , + + + + , (33)

from where

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2

II,cr 2 2P EI

L

π= . (34)

arises in the case of 1n = .

From double natural symmetry in the deflection curve of IV Euler case an expression for

VI ( )P n can be derived. In this case the second boundary condition is replaced by

( )(1 ) 4 0n np Lϑ / +′ / = . An equation, equivalent to Equation (21), can now be written

(1 ) (1 )( 4) 4

0 04 4

n n n np L p L

… …ϑ / + / +/ /

=∫ ∫ Parameter e

ϑ and approximation for ϑ are now

(1 )( 4)n n

ep Lϑ ϑ / +:= / and (4 )

ex tϑ ϑ /� . Thus

(1 )(1 2 ) (1 )

1IV (1 )

2 1 1( )

(1 ) 2 1

n nn n n n

e n

n n

EIP n B

n L n

ϑ + /+ / − /+

+ /

= , + + , (35)

(1 )(1 4 ) (1 ) 2

IV (1 )

2 1 2 1 1( )

(1 )( (2 ) ) 2 1

n nn n n n n

e

n n

E ab nP n B B

n n L n

ϑα β α β

+ /+ / − / +

+ /

+ = , , + + + + , (36)

and for 1n =

2

IV,cr 2 2

4P EI

L

π= . (37)

Remark 5. The results represented by Equations (31), (34), and (37) are identical to well

known formulas found in the literature, e.g. cf. [10].

Examples

In this section, we show a comparison between the results obtained from the formulas (30),

(33), (36) we constituted and numerical solutions which were obtained applying the Runge-

Kutta-Fehlberg integration and shooting method. Additionally, post-buckling configurations

for all three types of columns and materials ( 1n < , 1n = , 1n > ) are given for illustration.

I Euler case

Post-buckling force P as a function of angle of rotation ( )e

Lϑ ϑ= is shown in Figure 4 for a

free-clamp supported column. The results of numerical and analytical calculations are in good

agreement even at relatively large angles of rotation. Namely, at 0 285e

ϑ = . , 0 518e

ϑ = . ,

0 701e

ϑ = . , and 1 030e

ϑ = . , the difference between numerically and analytically calculated

post-buckling force in the 2n = case are 1 1 %. , 3 5 %. , 6 4 %. , and 13 6 %. , respectively. In

the case of 0 6n = . there is a 1 1 %. , 3 2 %. , 6 4 %. , and 12 1 %. difference at 0 296e

ϑ = . ,

0 517e

ϑ = . 0 735e

ϑ = . , and 1 021e

ϑ = . , respectively.

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Figure 4. Comparison between analytical and numerical solution.

Figure 5 illustrates the results calculated from the analytical formula (30). It shows the post-

buckling load as a function of angle e

ϑ for 0 6n = . for several different shapes of the cross-

section. It should be emphasized that these results arise from one formula only, cf. (30).

Figure 5. Post-buckling load as a function of the angle of rotation e

ϑ .

The influence of material constant n on post-buckling configurations for a free-clamp

supported column is depicted in Figure 6. The diagrams are displayed at constant values of

0 5e

ϑ = . , 1 0e

ϑ = . , 2 0e

ϑ = . , and 3 0e

ϑ = . .

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Figure 6. Post-buckling configurations for free-clamp supported column.

II Euler case

Results comparable to those in the previous case are obtained for a hinge-hinge supported

column, Figure 7. The difference between post-buckling loads calculated via numerical and

analytical approaches are 1 3 %. , 3 2 %. , 6 4 %. , and 13 6 %. at 0 308e

ϑ = . , 0 495e

ϑ = . ,

0 701e

ϑ = . , and 1 030e

ϑ = . , respectively for 2n = . For 0 6n = . there is a 1 1 %. , 3 2 %. ,

5 8 %. , and 12 1 %. difference at 0 296e

ϑ = . , 0 517e

ϑ = . , 0 699e

ϑ = . , and 1 021e

ϑ = . ,

respectively.


A similar influence of material constant n on post-buckling configurations is also noticeable

in the case of hinge-hinge supported column, Figure 8.

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Figure 8. Post-buckling configurations for hinge-hinge supported column.

IV Euler case

As expected, the results obtained for a clamp-clamp supported column are comparable to

those in both previous cases, Figure 9. The difference between post-buckling loads calculated

via numerical and analytical approaches are in this case 1 2 %. , 3 7 %. , 6 4 %. , and 13 6 %. at

0 296e

ϑ = . , 0 530e

ϑ = . , 0 701e

ϑ = . , and 1 030e

ϑ = . , respectively for 2n = . For 0.6n = there

is a 1 3 %. , 3 1 %. , 6 4 %. , and 11 7 %. difference at 0 342e

ϑ = . , 0 507e

ϑ = . , 0 735e

ϑ = . , and

1.0e

ϑ = respectively. In this case ( / 4)e

Lϑ ϑ= .


Post-buckling configurations in correlation with the influence of material constant n on

deformation for a clamp-clamp supported column can be found in Figure 10.

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Figure 10. Post-buckling configurations for clamp-clamp supported column.

Conclusion

The problems which involve geometrically exact mechanics as e.g. post-buckling analysis of

columns are usually difficult to solve. Since analytical solutions are quite rare one has to rely

on finding the solution numerically which can be very time consuming. The results of this

article are useful in engineering practice, when analyzing post-buckling behavior of non-

linearly elastic columns, e.g. made from rubber. A relatively simple analytical formulas are

constituted for free-clamp, hinge-hinge, and clamp-clamp supports of columns which have

superellipsoidal cross-section and can therefore be of an arbitrary shape between an ellipse

and a rectangle. The accuracy of the analytical formula has been validated numerically,

applying the Runge-Kutta-Fehlberg integration and shooting method. A good agreement

between the numerical and analytical approach has been confirmed for up to relatively large

angles of rotation for all three types of support discussed. Additionally, post-buckling

configurations for all three types of columns and materials ( 1n < , 1n = , 1n > ) are given for

illustration. The diagrams from which the influence of material constant n on the shape of the

deflection curve can be examined are displayed at four constant values of angle e

ϑ .

References

[1] M. Brojan, A. Puksic, F. Kosel, Buckling and post-buckling of a nonlinearly elastic

column, Z. Angew. Math. Mech., p. 518–527, Vol. 87, No. 7, 2007.

[2] C. Baykara, U. G uven, I. Bayer, Large Deflections of a Cantilever Beam of Nonlinear

Bimodulus Material Subjected to an End Moment, J. Reinf. Plast. Comp., p. 1321–1326, Vol.

24, No. 12, 2005.

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[3] M. Brojan, T. Videnic, F. Kosel, Non-prismatic nonlinearly elastic cantilever beams

subjected to an end moment, J. Reinf. Plast. Comp., p. 1071–1082, Vol. 26, No. 11, 2007.

[4] A.S. Shatnawi, S. Al-Sadder, Exact Large Deflection Analysis of Non-prismatic

Cantilever Beams of Nonlinear Bimodulus Material Subjected to Tip Moment, J. Reinf. Plast.

Comp., p. 1253–1268, Vol. 26, No. 12, 2007.

[5] K.W. Lee, Bending Analysis of Nonlinear Material Fibers with a Generalized Elliptical

Cross-section, Textile Res. J., p. 710-714, Vol. 75, No. 10, 2005.

[6] Y.A. Kang, X.F. Li, Bending of functionally graded cantilever beam with power-law non-

linearity subjected to an end force, Int. J. Nonlin. Mech., p. 696-703, Vol. 44, No. 6, 2009.

[7] Y.A. Kang, X.F. Li, Large Deflections of a Non-linear Cantilever Functionally Graded

Beam, J. Reinf. Plast. Comp., p. 1761-1774, Vol. 29, No. 12, 2010.

[8] A. Jaklic, A. Leonardis, F. Solina, Segmentation and Recovery of Superquadrics, Ch. 2 -

Superquadrics and their geometric properties, in series Computational imaging and vision,

Kluwer Academic Publishers, Dordrecth, 2000.

[9] W.W. Bell, Special Functions for Scientists and Engineers, Dover Publications, 2004.

[10] C.M. Wang, C.Y. Wang, J. N. Reddy, Exact Solutions for Buckling of Structural

Members, CRC Press, Boca Raton, FL, 2005.

List of Figures

Figure 1. Trivial and post-critical shape modes of I, II and IV Euler column.

Figure 2. Free body diagram and infinitesimal element of the deflected column.

Figure 3. Cross-sections defined by superellipse.


Figure 5. Post-buckling load as a function of the angle of rotation e

ϑ .

Figure 6. Post-buckling configurations for free-clamp supported column.


Figure 8. Post-buckling configurations for hinge-hinge supported column.


Figure 10. Post-buckling configurations for clamp-clamp supported column.

List of Tables

Table 1. Boundary conditions and critical loads.

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