1.39 Stability problems

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In practice there are no stability problems, because even in “perfect” structures

we find eccentricities. But then also in stress problems, failure occurs if the

critical load level is reached, as in the example of the Euler beam I in Fig.

1.139. If the horizontal force H is absent, it is a stability problem, and with

the force H it becomes a stress problem, but the critical load

Pcrit = π2

4

EI

l2 (1.546)

also dominates the stress problem, because when the load reaches Pcrit, which

corresponds to ε = π/2, the bending moment at the base of the column

becomes infinite

M = −H l

ε

tan ε , ε2 = l2

|P|

EI

, (1.547)

because tan ε = ∞ for ε = π/2.

In a true stability problem there are no lateral loads p. The only external

load, the compressive force P, enters the problem via the differential equation.

Formally it does not count as an external load.

In stability problems the potential energy Π consists only of the internal

energy Π(w) = 1/2 a(w,w), and Π is zero when the structure buckles (!)

Π(wcrit) =

1

2 a(wcrit, wcrit) =

1

2

_ l

0

[M2

crit

EI

− P (w

_

crit)2] dx = 0 (1.548)

so that wcrit cannot be found by minimizing the potential energy. It also

makes no sense to search for a work-equivalent load case ph, because in stability

problems p = 0. Instead Galerkin’s method (weighted residual method)

194 1 What are finite elements?

is applied. The buckled shape wcrit of the beam must satisfy the differential

equation

EIwIV (x) +P w

__(x) = 0 (1.549)

and homogeneous boundary conditions such as w(0) = 0 and/or w_(0) = 0,

etc. All elastic curves w which satisfy the geometric boundary conditions form

the space V .

The beam is subdivided into m finite elements and allowed to assume

under compression only those shapes that can be expressed by the n nodal unit

displacements of the free nodes,

_

i uiϕi, where the ϕi are usually the nodal

unit displacements of the first-order beam theory (!). These shape functions

form the basis of the subspace Vh ⊂ V .

Because of (1.549), the right-hand side of the exact deflection curve w =

wcrit is orthogonal to all shape functions ϕi ∈ Vh:

_ l

0

[EIwIV (x) +P w

__(x)] · ϕi dx = 0. (1.550)

After integration by parts, it follows—because the shape functions ϕi ∈ Vh

satisfy the boundary conditions—that the strain energy product between w

and the shape functions must also be zero:

a(w,ϕi) =

_ l

0

[EIw

__

ϕ

__

i

−P w

_

ϕ

_

i] dx = 0 i = 1, 2, . . . , n . (1.551)

The FE solution wh tries to imitate this property of the true solution. That

is, the nodal displacements ui must satisfy the system

(K − P ×KG)u = 0 (1.552)

where

k ij =

_ l

0

EI ϕ

__

i ϕ

__

j dx kG

ij =

_ l

0

ϕ

_

i ϕ

_

j dx . (1.553)

The trivial solution would be u = 0, which is the neutral position of the

beam. Because the right-hand side is zero, a solution u             = 0 can only exist if

the determinant of the matrix is zero:

det (K − P ×KG) = 0. (1.554)

The smallest positive number P >0, for which this holds is the approximate

buckling load Ph

crit.

We know that the pitch of a guitar string will increase with the tension

in the string. The opposite tendency we observe in a column. The frequency

will decrease if the compression increases and if the column finally buckles

1.39 Stability problems 195

Fig. 1.140. The buckling load and the first eigenmode

the return frequency has reached its lowest possible value, namely zero, which

means that it takes the column infinitely long to perform one full swing.

Not all stability problems possess a distinctive lowest eigenvalue. In some

cases a geometric nonlinear analysis with proper imperfections is not only

more concise but sometimes also the only possible way to predict the limit

state of a structure.

Rayleigh quotient

In FE analysis the buckling load Ph

crit is an overestimate. This follows from

the fact that the buckled shape wcrit minimizes the Rayleigh quotient on V ,

and that the minimum value is just Pcrit:

Pcrit =

_ l

0

EI(w

__

crit)2 dx

_ l

0

(w

_

crit)2 dx

. (1.555)

But because the minimum on a subspace Vh is always greater than the minimum

on the whole space V , it follows that Ph

crit

≥ Pcrit.

Usually the eigenvector u that belongs to the eigenvalue Ph

crit is normalized

in the sense that |ui| ≤ 1. If the associated shape

wh =

_

i

ui ϕi (1.556)

is substituted element-wise into the differential equation EIwIV (x)+Pw__(x)=

0 and the associated nodal forces and moments are studied, the FE load case

ph is recovered. The latter is an expansion in terms of the unit load cases p i

ph =

_

i

ui pi . (1.557)

Because the nodal unit displacements ϕi of first-order beam theory as for

example

ϕ1(x) = 1 − 3x2

l2 +

2x3

l3 , (1.558)

196 1 What are finite elements?

Fig. 1.141. The load case ph solved by the first eigenmode

are not homogeneous solutions of the differential equation of second-order

beam theory

(EI

d4

dx4 + P

d2

dx2 ) ϕ1(x) = P (

12 x

l3

− 6

l2 ) (1.559)

lateral loads hold the “buckled” beam in place. That is, the FE solution is

the solution of a stress problem, even though it shares with the exact curve

the property that it is orthogonal to all ϕi ∈ Vh. In normal FE analysis such

functions would be called spurious modes, because they do not interact with

the other shape functions ϕi.

In FE analysis to the “buckled” shape of a beam or a plate belongs a load

case ph            = 0 which is orthogonal to all nodal unit displacements.

If the homogeneous solution of second-order beam theory were used as

shape functions, the FE program would be an implementation of the secondorder

slope-deflection method, and the buckled shape would be exact because

then wcrit would lie in Vh.

Example. An FE analysis of the continuous beam in Fig. 1.140 with two

elements yielded for the buckling load the value

Ph

crit =

16.48 EI

l2 >

12.7 EI

l2 = Pcrit (1.560)

and the buckled shape

_

u4

u6

_

=

_

−0.707

1

_

. (1.561)

If the FE solution wh is substituted into the differential equation and the

jumps in the shear force V and the bending moment M are measured at the

nodes, then this gives an impression of the load case ph (see Fig. 1.141). But

note that this arrangement is only a snapshot because the load case ph can

be scaled in an arbitrary way, since any multiple of the “buckling mode”wh

is also a possible solution.

As can be seen in Fig. 1.141 forces are necessary to hold the buckled

beam in place. This is equivalent to saying that the opposing forces prevent

1.40 Interpolation 197

object than a perfect

circle

the beam from buckling. This explains why the approximate buckling load

is greater than the exact load. The same phenomenon is experienced by two

acrobats in Fig. 1.142. The acrobat on the perfect circle finds himself in an

unstable position, while his colleague profits from the fact that the vertices of

the polygon hamper rotation, and are marginally stabilizing his position.

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