View of Critical Length of a Column in View of Bifurcation Theory

1 Introduction

Linear formulation of the eigenvalue problem for a verti- cal rod loaded with its own weight has apparent drawbacks, e.g.,

a) all the critical lengths of the rod (i.e., the eigenvalues) compose a discrete set, outside of which all the lengths are noncritical, i.e., nonzero deflections are impossible, and

b) all the critical deflections (i.e., the eigenfunctions) are of an arbitrary magnitude.

To a certain extent, they may be eliminated using the precise (nonlinear) form of the curvature. The nonlinear problem was solved long ago by L. Euler (1744) and by J. L. Lagrange (1773), who gave the solution of the simplest problem for a homogeneous rod loaded with a longitudinal force (neglecting its own weight) in the form of elliptic fun- ctions. Detailed analysis of the problem was published e.g., in [1], [2] and it is mentioned in the first part of the paper as an introduction to the bifurcation problem.

The second part of this paper contains the formulation of nonlinear eigenvalue problem for a vertical homogeneous rod loaded with its own weight only, and the corresponding linear problem is solved. The last part is devoted to applica- tions of bifurcation theory (see e.g., [3], [4]) to the nonlinear problem and the approximate solution of the problem is found.

All the computations and pictures were performed with use of the Maple V software.

2 Critical load of a column in view of nonlinear theory

The problem of finding the critical vertical load, which is applied to the free end of a vertical homogeneous rod of uniform cross section fixed at the bottom is a well-known eigenvalue problem that has been explicitly solved in many mechanics textbooks. The example is mentioned here to illustrate the differences between the results of linear and nonlinear theories.

If we denote the horizontal deflection of the column byw (only such displacements are taken into account), its curvature byr, the variable length measured from the free end bys, the length of the whole rod byl, the angle between thex-axis and the tangent line at the pointsof the rod byq(s), the equa- tion of the moments is ^{EI r =P w s}

( ( )

^-w

( )

⁰

)

^,whereEis

Young’s modulus,Iis the moment of inertia and Pis the above mentioned load(in the directionxof the column). Dif- ferentiating (bys) the equation and substituting the relations

( )

1

r

q q

=d = -

d d

d s

w s

, s sin , (1)

into it, we get d d

2q q

s₂ + p²sin =0, (2)

wherep² =P EI. In the case of small deflections, the equa- tion may be approximated by the linear equation

d d

2q q

s₂ + p² =0,

having (together with the appropriate boundary conditions) the discrete set of eigenvalues

( )

p k

l k

k2 2

2 1

2 1 2

=æ -

èç ö

ø÷ = ¼

p , , ,

and corresponding eigenfunctions with an arbitrary magni- tude. But these results do not correspond to reality (see the introduction). For this reason many engineers and mathema- ticians have tried to solve the nonlinear equation (2).

Using the identity d d

d d

d d

d d

d d

2q

r q r

q

q r

s² s s ²

1 1 1

= æ 2 èçç ö

ø÷÷ = æ èçç ö

ø÷÷ = æ èçç ö

ø÷÷

on the left hand side of (2), integrating byqandtaking into account the boundary condition q(0)=q0, 1/r(q0)=0, we get

( )

¹

( )

2 ²

2

r q = p cosq -cosq₀ . (3) Computing

r =dq d

s

from (3), we must choose the sign in front of the root. As the most important eigenvalue is the smallest one, we choose the minus sign on the whole interval (0,q0), and then we will con- sider the smallest critical force and its right neighborhood.

Integrating the root of (3) and taking into account the boun- dary conditions(0)=l, we get the implicit form of the fun- ctionq=q(s):

Critical Length of a Column in View of Bifurcation Theory

M. Kopáčková

The paper investigates nonlinear eigenvalue problem for a vertical homogeneous rod loaded with its own weight only. The critical length of the rod, for which the rod loses its stability, is found by use of bifurcation theory. Dependence of maximal deflections of the rods on their lengths is given.

Keywords: critical length of column, eigenvalue problem, bifurcation theory, approximation of solution, Bessel function.

( )

l s

p

- ^q = ¹²

ò

^{0q cosq -}d^{qcosq0. (4)} The valueq₀is obtained by use of the substitution

sinq sin sinq

2 2

= F 0

in (4) and limiting the result forq®q0,s®0 : lp=

- æèç ö ø÷

ò

₁ ^dF _F

2

0 2

0 2

sinq sin

p

, (5)

which is the implicit form of the functionq0(p). It is seen, that the smallestpsatisfying (5) is the smallest eigenvaluep₁of the linear problem with correspondingq0=0. The dependence of the maximum deflectionsw₀ºw(0) on the forceP, resp.

pis given by the formula w

( )

p

p

0 2 0

= sinq 2 ,

which is obtained by integrating the second equality of (1) with the use of (3) and the fact

d d

d d

d d

d d w

s w

s

= × = w×

q q

q r 1

and limiting the result forq ® q0,s®0 (see the Fig. 1).

The dependence of the ratio

( ) ^{( )}

r p l x l

º -l

onpis given by the following formula

( )

r p = -lp - æ

èç ö

ø÷ -

- æèç ö

ø÷ æ

1 1

2 1 2

1

1 2

0 2

0 2

sin sin

sin sin q

F q è F

çç çç ç

ö

ø

÷÷

÷÷

ò

÷^dq

p

0 2

,

which is calculated (similarly asw₀) integrating the equality

( )

d d x s

s =cosq,

and limiting the result forq ® q₀,s®0 (see Fig. 2).

Two improvements comparing with the linear theory are evident:

1. If ptends to the first eigenvalue p₁= p /2lof the linear problem from the right hand side of the interval, the maxi- mum deflectionw₀tends to zero, whereas no deflection of the rod exists forp£p₁.

2. For every p³p₁ there exists the unique solution w(q), q Î(0,q0] with maximum deflectionw0(p).

The same considerations hold forp_k,k= 2, 3, …

3 Critical lengths of a column bent with its own weight – nonlinear formulation and some results of linear theory

In the further discussion we answer the question which length of a homogeneous column of a constant cross section is bent with its own weight only, i.e., finding the smallestl, for which there exists nonzero deflectionwof the column.

Let us consider the equality of the moments in the form

( ) ( )

( ) ( )

EI fS w s w s , l

s

r =

ò

⁰ - s ^{d ,}s Î ^{0 ,} wherefis the density of the column andSis its cross section (the other notation coincides with that of part 1). Differen- tiating the equation and substituting the relations

1

( )

r

q q

=d = -

d d

d s

w

s s

, sin

into it, we get d

( )

d

2q q

s fS

EIs s

2 + sin =0.

To eliminate the unknown length l of the column, we transformsto the new variables/l(denoting it agains) in the last equation and we denote the unknown functionq in the new variable byu(s). Thus, we get the equation

( ) ( ) ( )

d d

2

2 0 0 1

u

s s +lssinu s = , s Î , , (6)

where

l= fSl EI

3

(7) is an unknown eigenvalue of the problem to Equation (6) and to the boundary conditions

( ) ( )

¢ = =

u 0 0, u1 0. (8)

The first condition follows from the assumption that up- per end of the column is free of tension, i.e., 1/r= 0, and the second condition expresses the fixed end at the bottom.

In the case of small deflections, Equation (6) may be linearized using the following approximations

( )

sinq q= = ¢. . w s .

Thus, we get the linear eigenvalue problem

( ) ( ) ( ) ( ) ( )

d d

d d

2

2 0 0 1 0 0 1 0

v

s s s v s s v

s v

+l = , Î , , = , = , (9)

here we denotedv(s) =w¢(s).

The solutions of (9) are the discrete set of the eigenvalues ln zn

=9 n= ¼

4 0 1

2

, , ,

and the corresponding eigenfunctions

( ) ( )

v_n s =C s J_-1 3 z s_n 3 2 n= ¼ 0 1

, , ,

wherez_n(n= 0, 1, …) are all the roots of the Bessel function

( )

J_{-1 3} z andCis an arbitrary constant.

The first three eigenfunctionsv₀,v₁, v₂of the problem (whereC is chosen in such a way thatv_i(0) = 1,i= 0, 1, 2) are given in Fig. 3, whereas the corresponding deflections w₀, w₁, w₂ are drawn in Fig. 4. The critical lengths of the column are determined by the eigenvalueslnand the rela- tion (7), i.e.,

l EIz

n3 9 f S2n

= 4 .

The length for which the homogeneous column loses its stability (i.e., nonzero deflection of the column exists without any force apart from its own weight),

l EIz

0 9 f S02 1 3

=æ 4 èçç ö

ø÷÷

corresponds to the smallest eigenvaluel₀of the above prob- lem, where

l0 02

9

4 7 83734744

= z =

. (10)

and the corresponding eigenfunction is of the form

( ) ( ) ^{( )}

v s0 s J1 3 s3 2 s

186635086 0 1

= _- . , Î , . (11)

4 The solution of the bifurcation problem

The assumption of small deflections, and then the validity of linear theory, implies the same insufficiency as that men- tioned in the introduction. To remove it, we have to take into account the nonlinear equation (6) representing together with the boundary conditions (8) the problem of a bifurcation of the nonlinear operator L(u) – F(u,t) (given below) at the point [0, 0], (whereL(0) –F(0,t) = 0), i.e., there exists a nonzero solution of the equation (14) (see below) in every neighborhood of [0, 0].

Every eigenvalue ln gives a bifurcation point [0, ln] and the solution of the problem is similar to each other f o r n= 0, 1, … Then, we choose the smallest eigenvaluel0, because this number determines the loss of stability of the column.

Equation (1) may be rewritten into the form

( ) ( ) ( ( ) ( ) ) ( )

¢¢ + = - Î

u s l0s u s s l0u s lsinu s , s 0 1, . (12) Let us denote by L(u) the linear operator as the closure (inH¹(0, 1)) of the operator given by the values

( ) ( ) ( )

L u s º ¢¢u s +l0s u s

for the functions satisfying boundary conditions (8) and

( ) ( ( ( ) ( ) ) ( ) )

F u t, =s l0 u s -sinu s - sinu s , (13) where =l - l0. Now, we may reduce the problem of finding nonzero solutions to (6), (8) for l near to l0 to solving the equation

( )

Lu=F u, (14)

(inL²(0, 1) ) for sufficiently small .

All the solutions of the homogeneous equation Lu=0

form a one-dimensional subspace N of the eigenfunctions v₀(s) given by the formula (11) corresponding to the eigen- valuel0. Any solutionu(s) to the equation

Fig. 3: Three eigenfunctionsv₀,v₁,v₂of the linear program

Fig. 4: Three deflections of the column

( ) ( ) ( )

Lu s = f s , sÎ 0 1, (15)

exists and is of the form

( ) ( ) ( ) ( ) ( ( ) ( ) ) ( )

u s b f v u s v s u Cv s

s

=

ò

^{0 s 0 s 0} - ^{0 0 s ds}+ ^{0 (16)} if and only if the functionf is orthogonal to v₀ inL²(0, 1), i.e.,

( ) ( )

f s v s₀ s

0 1

0 d =

ò

^{, (17)}

whereCis an arbitrary constant,

( ) ( )

b= 2 u s = s J z s

3 3 ^{0 1 3 0}

p , 3 2 .

u₀is a solution of

( ) ( )

¢¢ + =

u s l0s u s 0,

representing together withv₀a fundamental system of solu- tions to the above equation. Formula (16) is easily obtained by solving the linear problem (15), (8) with the use of variation of parameters. As operatorLis not invertible, we split equa- tion (14) into two equations:

( ) ( ( ) )

Q Lu =Q F u t, , (18)

(

^{I -Q}

)

^×

(

^{Lu-F u t}

( )

^,

)

^{=0 , (19)}

whereQis a projector operator ofL²(0, 1) on

( ) ( ) ( )

N^{^} =ìf ÎL f s v s s=

íï îï

üý ï

ò

þï

2 0

0 1

0 1, : d 0 .

Due to the uniqueness of the solution to (15) in N^{^}, the operatorQ(Lu) is invertible inN^{^}. Then the solutionu(s) of (14) may be rewritten in the form

( )

u=U U1 2 +U2

whereU₁(U₂) is the solution of (18) for fixed but arbitraryU₂, andU₂ is afterwards obtained as a solution of (19) where U₁=U₁(U₂) was substituted.

Now, we will computeU₁,U₂solving problems (18), (19), or, more precisely, their approximations. Let us introduce a sufficiently small parametertand use the following series

( ) ( ) ( ) ( )

l l- 0= + + + ¼

= + + + ¼

= -

1 22 33

0 2 2

3 3

t t t

u s v s t u s t u s t

u u

,

,

sin u³ u⁵

3! + 5! - ¼

(20)

in (12). Comparing the coefficients corresponding to the same power tⁿ, n= 1, 2, 3, …, we get an infinite system of equations, three of which are written below

( ) ( )

¢¢ + =

v s₀ l₀s v s₀ 0, (21)

( ) ( ) ( )

¢¢ + = -

u s₂ l₀s u s₂ 1v s₀ , (22)

( ) ( )

¢¢ + = - -æ + + -

èç ö

ø÷ u s_{3 0}s u s₃ s ⁰ v²₀ v₀ u₃ v₀³

2 6

l l

2 1 1 . (23)

Other equations may easily be obtained continuing the computations. (21) is automatically satisfied due to (11).

Any solution of (21) is of the form (16) forf(s)=- 1v₀(s) satisfying (17), which implies ₁= 0. In this case, the only solution u₂ÎN^{^} is u₂(s)º0. Substituting these values into (23), we get the equation foru₃:

( ) ( ) ( )

¢¢ + = æ -

èç ö

ø÷ Î u s_{3 0}s u s₃ s ⁰v₀² v₀ s

2 0 1

l l

2 , , . (24)

And again, any solution of (24) satisfying (8) is of the form (16) under the assumption (17) on the right hand side of (24), which determines the unique ₂. The constantCfrom (16) is determined by the conditionu₃ÎN^{^}.

The computation of _n, u_n(s) (n= 2, 3, …) gives appro- ximations of the solutionu(s) of (14), whereas _n,u_n+1(s) is a zero for odd indicesn. The approximations

( ) ( ) ( )

u s .=v s t0 +u s t3 3 (25) for the values of parametert= ±0.5, ±1.0, ±1.5 are given in Fig. 5.

The dependence of two approximations ofl(denoted by L₁,L₂on the picture 6):

l l= 0+ 2

2t , l l= 0 + 2+ 4

2t 4t (26)

is shown in Fig. 6.

Excluding parameter t from (25) and (26), we get the maximum anglej0(denoted byU₀in Fig. 7) as a function of the lengthlof the column (see Fig. 7).

Fig. 5: The approximate solutionsu(s) fort= ±0.5, ±1.0, ±1.5

Fig. 6: Two approximations ofl:L₁,L₂

5 Conclusions

The last figure illustrates the fact that the nonlinear theory gives reasonable results:

a) the deflections of the column loaded with its own weight are zero forl £ l₀(the first eigenvalue of the linear prob- lem),

b) the maximum deflection increases continuously with in- creasing length of the column.

Formula (25) represents the solution of the problem as the sum of the first eigenfunctionv₀(s) of the linear problem multiplied by parametert:t®0 forl®l0and “a perturba- tion”, which is orthogonal tov₀inL². The above mentioned

method is applicable to other stability problems described by nonlinear ordinary and partial differential equations, e.g., lateral buckling.

Acknowledgement

The author is grateful to prof. J. Šejnoha for his stimu- lating discussions and valuable advice.

The research of the author was supported by the founda- tion of the Ministry of Education in the project “Výzkumný záměr No. J04/98:210000001”.

References

[1] Rzhanicyn, A. R.:Ustoichivost ravnoviesija uprugich sistem.

Moskva 1955

[2] Collatz, L.: Eigenwertaufgaben mit technischen Anwendun- gen. Leipzig 1963

[3] NIRENBERG, L.:Topics in Nonlinear Functional Analysis.

New York 1974

[4] Krasnoselskij, M. A., Zabrejko, P. P.:Geometricheskije me- tody nelinejnogo analiza. Moskva 1975

[5] Char, B. W., Geddes, K. O., Gonet, G. H., Leong, B. L., Monagan, M. B., Watt, S. M.: First Leaves: A Tutorial Introduction to Maple V. Springer-Verlag 1992

RNDr. Marie Kopáčková, CSc.

Department of mathematics phone:+420 2 2435 4385

e-mail: marie.kopackova@fsv.cvut.cz Czech Technical University in Prague Faculty of Civil Engineering

Thákurova 7, 166 29 Praha 6, Czech Republic Fig. 7: Graph of the maximum anglej0(l)ºu(l)