4 Strut & Column Buckling

Consider a simply-supported strut with length L and flexural rigidity EI loaded axi-ally with a force P (Fig. 22). At buckling it assumes a profile (mode) y(x), where at x

�

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�

Figure 22: Buckling strut.

we have a bending moment M introduced by the mode shape eccentricity,

M = −Py. (28)

Bending theory states: M = EI × curvature, which from linear bending theory can bewritten:

M = EId2y

dx2. (29)

Combining eqs. (28) and (29) we obtain the differential equation:

EId2y

dx2+ Py = 0 (30)

Rewriting this in the form:d2y

dx2+

P

EIy = 0

gives the form of the equation for a free undamped oscillator, or the equation for simpleharmonic motion. It is useful to differentiate this equation twice with respect to x so thattwo boundary conditions can be applied at each support.

d4y

dx4+

P

EI

d2y

dx2= 0 (31)

The solution of this equation has the form:

y(x) = A sin ωx + B cos ωx + Cx + D (32)

where A, B, C and D are constants dependent on the boundary conditions of the strutand ω is defined:

ω =

√

P

EI(33)

In the case shown in Fig. 22

y(0) = y′′(0) = 0 ⇒ B = D = 0

y(L) = y′′(L) = 0 ⇒ C = 0, A sin ωL = 0(34)

23

where primes denote differentiation with respect to x, a nontrivial solution (A 6= 0), isgiven by:

sin ωL = 0 ⇒ ωL = 0, π, 2π, . . . , nπ where: n ∈ Z. (35)

Therefore the condition for a nontrivial y(x) is

P =n2π2EI

L2. (36)

Unless the strut is restrained somewhere the lowest value of P (n = 1) is the practicalvalue for buckling—the Euler load (PE)

PE =π2EI

L2. (37)

In reality the equilibrium paths are not flat, they exhibit a stable-symmetric profile.However, this stable nature is not significant until deflections get very large,3 so theyare assumed to be flat and the critical load is a fair estimate of the failure load whenstruts buckle.

However it is important to note that linearized deflections, which we have assumed inthe struts, can only give information on critical loads and say nothing for post-bucklingresponses. This means that analysing systems which are asymmetric or unstable withlinearized deflections would always overestimate the real structural strength because oftheir inherent sensitivity to imperfections. For systems that have a significant post-buckling stiffness (e.g. rectangular plates supported on all edges) assuming linearizeddeflections can seriously underestimate the real structural strength.

4.1 Potential Energy Formulation of Euler Buckling

The following potential energy functional contains large displacement (small-strain)expressions for strain energy and work done by load P of the elastica—the inextensiblestrut:

V =

∫ L

0

1

2EI

(

d2y

dx2

)2[

1 −

(

dy

dx

)2]

−1

− P

1 −

√

1 −

(

dy

dx

)2

dx . (38)

This can be expanded as a power series (with primes denoting d/dx):

V =

∫ L

0

[

1

2EIy′′2

(

1 + y′2 + y′4 + . . .)

− P

(

1

2y′2 +

1

8y′4 + . . .

)]

dx (39)

Small deflection assumptions allow us to truncate the series to first order (lineariza-tion):

V =

∫ L

0

(

1

2EIy′′2 −

1

2Py′2

)

dx (40)

3In this case, eq. (12) becomes valid for curvature and the problem becomes that of the elastica (seeSection 1.3 of Thompson & Hunt 1973 or Section 2.7 of Timoshenko & Gere 1961).

24

To obtain the equilibrium equation of the structure, the stationary point of the potentialenergy needs to be found. As this is in a form of an integral, the condition for anintegral to be stationary is that the first variation of the integral must be zero. If V isexpressed thus:

V =

∫ L

0

L(y′′, y′) dx (41)

where L is the integrand and is like a Lagrangian function from dynamics theory, wecan express the first variation of V (denoted as δV ) thus:

δV =

∫ L

0

[(

∂L

∂y′′

)

δy′′ +

(

∂L

∂y′

)

δy′

]

dx. (42)

whereL =

1

2EIy′′2 −

1

2Py′2, (43)

and∂L

∂y′′= EIy′′,

∂L

∂y′= −Py′. (44)

If we also say that:

δy′′ =d

dxδy′ and δy′ =

d

dxδy (45)

and then integrate by parts twice we get the following:

δV = [EIy′′δy′ − (EIy′′′ + Py′)δy]L

0+

∫ L

0

(EIy′′′′ + Py′′) δy dx. (46)

So if δV = 0 at equilibrium, the terms in the square brackets must vanish when appro-priate boundary conditions are applied:

• Simple supports (zero deflection and bending moment): y = y ′′ = 0

• Fixed supports (zero deflection and slope): y = y ′ = 0

• Free (zero bending moment and shear force): y ′′ = 0, EIy′′′ + Py′ = 0.

More importantly the integral vanishes when the integrand is equal to zero, namelywhen:

EIy′′′′ + Py′′ = 0 , (47)

which gives the equation of the Euler column. This analysis has used a technique calledthe calculus of variations which provides the link between Newton’s laws and potentialenergy.

4.2 Effective Length Concept

With the solution of the governing ODE we could consider any boundary conditioncombination by obtaining a solution of y(x). However, there is a more intuitive way

25

of analysing different cases using the concept of Effective length (Le). Common casesfollow, the adjusted Euler loads take account of the effective lengths thus:

PE =π2EI

L2e

. (48)

1. Simple supports both ends:

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�

2. Fixed both ends:

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�

3. Cantilever (free → fixed):

����������

4. “Sway” column (fixed both ends but can also sway across):

�������

�

5. Simple support → fixed:

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)

There are many cases where end fixity lies between the above cases. This is especiallyprevalent in frames. Fig. 23 shows two examples. When in doubt, take the largest effec-tive length Le that gives the lowest P C, and always choose the correct mode carefullyto avoid the example pitfall in Fig. 24.

26

�

��

(a) Sway

�

� �

(b) Non-sway

Figure 23: Sway and non-sway frames. For the effective lengths of the vertical mem-bers: (a) L < Le < 2L (between cases 1 & 3); (b) 0.5L < Le < 0.7L (between cases 2 &5).

�

�

(a) System

��� �

���� �

(b) Correct mode

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������� �!��

(c) INCORRECT

Figure 24: Choosing buckling effective lengths carefully. The central restraint is a pin.

27

4.3 Strut and Column Analysis

Sometimes the combination of boundary conditions and non-uniform cross-sectionproperties of columns are such that exact evaluation of critical loads is not possibleusing the differential equation (31). However, good approximations can be made ifthe system is formulated by total potential energy. The strain energy U is given by theexpression in eq. (14), but the work done by the load is thus:

P∆ = P

∫ L

0

1

2

(

dy

dx

)2

dx, (49)

which makes the expression for V:

V =

∫ L

0

[

1

2EI

(

d2y

dx2

)2

−P

2

(

dy

dx

)2]

dx (50)

For the standard column, substituting:

y =∑

n=1

Qn sinnπx

L(51)

gives correct results since the chosen function in this case reproduces the correct buck-ling mode. Approximate results for P C can be obtained for approximate expressionsfor y(x).

4.3.1 Rayleigh’s Method

1. Assume a one DOF form for y(x).

2. Substitute y into V given in eq. (50).

3. For critical equilibrium, set:d2V

dQ2= 0

Example

For a simply-supported strut of length L, assume:

y = Qx(L − x) (52)

Substituting the assumed y into V gives:

U =

∫ L

0

1

2EI

(

d2y

dx2

)2

dx =

∫ L

0

1

2EI(−2Q)2 dx = 2EILQ2 (53)

P∆ =

∫ L

0

P

2

(

dy

dx

)2

dx =

∫ L

0

P

2(QL − 2Qx)2 dx =

PL3Q2

6(54)

V = 2EILQ2 −PL3Q2

6(55)

28

Applying d2

V

dQ2= 0 gives:

PC =12EI

L2(56)

which is around 22% too high, this is a pretty poor estimate as the assumed form for ydoes not satisfy all the boundary conditions.

4.3.2 Timoshenko’s Method

Note from eqs. (28) and (29) that when a strut buckles assuming small deflections:

d2y

dx2= −

Py

EI(57)

hence for U we could write

U =1

2EI

∫ L

0

(

d2y

dx2

)2

dx =P 2

2EI

∫ L

0

y2 dx. (58)

Example

Assuming the same form as in eq. (52) this time gives:

U =

∫ L

0

P 2

2EIw2 dx =

∫ L

0

P 2

2EIQ2x2(L − x)2 dx =

P 2L5Q2

60EI(59)

P∆ =

∫ L

0

P

2

(

dy

dx

)2

dx =

∫ L

0

P

2(QL − 2Qx)2 dx =

PL3Q2

6(60)

V =P 2L5Q2

60EI−

PL3Q2

6(61)

Applying d2

V

dQ2= 0 gives:

PC =10EI

L2(62)

which is only 1.3% too high, this is a very good estimate even though the assumedform for y does not satisfy all the boundary conditions.

4.3.3 Notes on the approximate methods

• The approximation for y should always satisfy the geometric boundary conditionsto be useful. In the example, the geometric conditions at the boundaries are thezero displacements. The zero bending moments at the boundaries are not satis-fied by y—this is an example of a static boundary condition.

• The origin of integration should be a position of zero bending moment.

29

• Timoshenko’s method relies on the expression for relating curvature to deflec-tion. Timoshenko’s method is a special case of Rayleigh’s method only reallyuseful for columns.

• As y is generally modelled more precisely than its 2nd derivative, Timoshenko’smethod turns out to be more accurate.

• Rayleigh’s method can be used in a wide variety of applications including plates,shells and even structural dynamics.

• Extending the Rayleigh technique to more than one DOF makes the approxima-tion better. Generally, this is called the Rayleigh–Ritz method.

• It can be shown that an approximate energy method always arrives at upperbound for PC.

30

5 Real Columns

For columns that are designed and used in practice, the Euler buckling load is onlypart of the story. We also have to consider the effect of material failure, i.e. plasticityor yielding. A idealized view is given in the graph in Fig. 25 where λ is the nondimen-

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3�465#7%89;:=<0>-:@?A4!96B,C

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Figure 25: Idealized graph of real column failure.

sional slenderness ratio of the section defined from the second moment of area I , thecross-sectional area A and the sectional radius of gyration r. Thus:

I = Ar2, (63)λ = Le/r, (64)

σE =π2EI

AL2e

=π2E

λ2. (65)

The expression in eq. (65) is the mean axial stress in the section during Euler buckling.The slenderness is defined to incorporate the section property and the effective lengthinto one parameter: high values imply buckling, low values imply yielding failure.

The codes, however, also take account of initial imperfections from experimental data.

5.1 Column Formulae

5.1.1 Some preliminaries

We can plot σE against λ to obtain a 2nd order hyperbola, the so-called Euler hyperbola.The Euler curve is clearly an upper bound on column strength. We can normalize both

31

stress and slenderness with respect to the material yield stress σy thus:

Stress : σ/σy Slenderness : λ/λr (66)

where λr = π√

E/σy.

For a stocky column, we do not expect buckling but rather squashing (yielding) of thecolumn. For low values of λ, this occurs when the mean axial stress reaches σy. Thissecond upper bound on column strength is thus defined when σ/σy = 1.0 for all λ/λr.

5.1.2 The Real Column: Perry–Robertson Formula

Object: to determine the value of P that causes the largest stress in the column to reachσy.

At any section we have a combination of (i) a mean axial stress (σ = P/A) and (ii)bending stresses corresponding to a moment (M = Py). The latter is compressive insidethe curve, and tensile outside. Since P/A is compressive, the most stressed location is atmidspan on the inside of the column (i.e. T in the Fig. 26), and this is where first yield

yiyi + y

T

No Load Load P

Figure 26: Imperfect column under axial load P .

will occur as P is increased. At T the stress σT is given by

σT =Md

I+

P

A(67)

where d is the distance from extreme fibre to the neutral axis. As M = P (y + y i) and

32

letting σ = P/A, we get

σT =P (y + yi)d

I+ σ

=P

A

A

Id

(

yi

σ

σE − σ+ yi

)

+ σ

= σ

(

dyi

r2

)(

σE

σE − σ

)

+ σ. (68)

This is after defining the magnification factor σ/ (σE − σ) which relates the deflectionowing to the load, y, to the imperfection yi. Putting η = dyi/r

2 finally gives

σT = σ

(

ησE

σE − σ+ 1

)

. (69)

If we put σT = σy and solve for the value of σ which causes yield at T we obtain:

σ =σy + (1 + η)σE

2−

√

[

σy + (1 + η)σE

2

]2

− σyσE (70)

This is the Perry–Robertson Formula for the mean axial stress to cause failure in acolumn. This leads to a graph (Fig. 27) that rounds the corners of the curve in Fig. 25.BS 449 used the Perry–Robertson curve and factored it downwards using an allowablestress σa criterion:

σa = 0.6σfailure. (71)

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Figure 27: Perry–Robertson curve in comparison with the ideal column failure curve.

BS 5950 improved on 449 by combining experimental data with the Perry–Robertsoncurve with factors accounting for:

33

• section shape, whether UC, UB etc;

• residual stresses in rolled and welded sections;

• the stocky column effect where σfailure & σyA;

• real support conditions.

It has been plotted for specific values of η which reflect the influence of geometricimperfections and residual stresses.

Values of η from BS5950:η = 0.001d′(λ − λ0) ≮ 0

where λ = L/r and the limiting slenderness λ0 = 0.2π√

E/σy.

Values of d′:

Curve a d′ = 2.0Curve b d′ = 3.5Curve c d′ = 5.5

Curve a represents rolled hollow sections with relative strengths decreasing as we gofrom curves a to c.

5.1.3 Designing columns to BS 5950

Section 4.7 in BS 5950 Part 1 (2000 revision) is the relevant section:

• Table 22 defines the effective length;

• Table 23 defines which of the curves a, b or c to use—depending on section shape,manufacture and so on;

• Table 24 has the curves a, b and c tabulated against slenderness to obtain thedesign strength.

34