elastic instability modes

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STEEL CONSTRUCTION: APPLIED STABILITY

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STEEL CONSTRUCTION:

APPLIED STABILITY

Lecture 6.3: Elastic Instability Modes

OBJECTIVE/SCOPE

To describe the elementary elastic instability modes and to derive the principal critical

loads for columns, beams and plates.

PREREQUISITES

Lecture 6.1: Concepts of Stable and Unstable Elastic Equilibrium

RELATED LECTURES

Lectures 6.6: Buckling of Real Structural ElementsLecture 7.7: Buckling Lengths

RELATED WORKED EXAMPLES

Worked Example 6.1: Energy Methods I

Worked Example 6.2: Energy Methods II

SUMMARY

This lecture explains how critical buckling loads are determined by solution of thedifferential equilibrium equations for the structure. The critical loads, assuming simple

loading and boundary conditions, are then calculated for the principal cases, namely:

flexural buckling of columns. lateral buckling of beams. buckling of plates.

1. INTRODUCTION

Instability can occur in all systems or members where compression stresses exist. Thesimplest type of buckling is that of an initially straight strut compressed by equal and

opposite axial forces (Figure 1).


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Other buckling modes also of great practical interest in steel constructions, are:

lateral buckling of beams (Figure 2). plate buckling (Figure 3).

shell buckling (Figure 4).


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The fundamental differences in behaviour of columns, plates and shells are shown in

Figure 5. For behaviour in the elastic range the critical load and the maximum load carried

by an actual (imperfect) column are in reasonable agreement. For the plate, if the

postcritical strength is achieved with acceptably small lateral deflections, a greater loadthan the critical load might be acceptable. For thin-walled cylinders, however, the


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maximum load in the real (imperfect) situation is much less than the theoretical critical

load.

For compressed struts, the flexural buckling illustrated in Figure 1 is not the only possible

buckling mode. In some cases, for example, a torsional buckling (Figure 6) or a

combination of torsional and flexural buckling can be seen; if a member is thin-walled,one can also observe a plate buckling of the elements of the cross-section (Figure 7) which

can interact with the overall buckling of the member.


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Determination of the critical load using bifurcation theory takes advantage of the fact that

the critical situation is associated with a neutral equilibrium condition; equilibrium in a

slightly deflected shape can, therefore, be established, leading to differential equations

which are simple to manage, at least for certain classes of structures. The critical load

gives information on the level of stability of a system, or member; it is also used as a basic

value (bound) for the calculation of the ultimate load for structures in danger of instability,

as shown in later lectures. In this lecture, the critical loads are calculated by solving the

differential equilibrium equations describing the phenomenon. These solutions are

available only for the simplest cases of loading and boundary conditions. A generalmethod for assessing critical loads, based on an energy approach is given in Lecture 6.4.

2. FLEXURAL BUCKLING OF COLUMNS

At the critical load, the stable equilibrium of the straight column is at its limit and there

exists a slightly deflected configuration of the column which can also satisfy equilibrium

(Figure 1). For this configuration, the bending moment at any cross-section is given, for a

pin-ended strut, by:

M = N.y (1)


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Using the differential equation given by simple bending theory, and considering large

deflections:

(2)

or the approximation

(3)

which is reasonably accurate for loads approaching critical load and for small deflections;by introducing Equation (1) this becomes:

(4)

where EIzis the bending rigidity of the column in the plane of buckling.

The general solution of this equation is:

y = A sin kx + B cos kx (5)

where

k2= (6)

(only positive solutions, i.e. compression forces, are of interest).

A and B are constants of integration which must be adjusted to satisfy the

boundary conditions:

y = 0 for x = 0 (7a)

and

y = 0 for x = l (7b)

The first boundary condition gives B = 0; the second one gives:

A sin kl = 0 (8)

which requires either A = 0 (in this case there is no deflection), or sin kl=0, i.e.


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kl = n (9)

where n- is any integer.

Finally, the critical load is obtained from the following:

Ncr,n= (10)

Figure 8 shows the first three buckling modes (n = 1, 2 and 3 respectively).

Normally, the smallest value of kl, and therefore of the critical load Ncr, which satisfies

Equation (9) is obtained by taking n = 1; this critical load is called the Euler load; in the

case where bracing is used, higher buckling modes may be decisive.

The critical load for a pin-ended column was calculated by Leonhard Euler in 1744.Historically speaking, it is the first solution given to a stability problem. The same

procedure may be used for cases with other boundary conditions.

The critical load given above does not take into account the effect of shear forces; this can

be done by adding the shear deformation:

= (11)

where V, the shear force, is given by:


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V = N (12)

and Av- is the shear area of the cross-section.

By adding the change in slope of the deflection curve produced by the shear force, the

differential equation of the buckling phenomena becomes:

(13)

which gives the critical load:

Ncr*= (14)

Thus, owing to the action of the shear forces, the critical load is reduced when compared

to Euler's load. In the case of solid columns, the influence of shear can generally be

neglected; however, in the case of laced or battened compression members, this effect may

become of practical importance and should be considered.

3. LATERAL BUCKLINGWhen a beam is bent about its strong axis, it normally deflects only in that plane.

However, if the beam does not have sufficient lateral stiffness or lateral supports to ensure

that this occurs, then it may buckle out of the plane of loading, as shown in Figure 2.

For a straight elastic beam, there is no out-of-plane displacements until the applied

moment reaches its critical value, when the beam buckles by deflecting laterally and

twisting (Figure 2); lateral buckling, therefore, involves lateral bending and torsion. For

the simplest case, of a doubly symmetric simply supported beam, loaded in its stiffer

principal plane by equal moments (Figure 2), the differential equilibrium equations of the

beam are as follows:

minor axis bending:

(15)

torsion:

E.It. (16)


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where EIz- is the weak axis bending rigidity; Mis the lateral bending moment inducedby the twisting, , of the beam; GItis the Saint-Venant torsional rigidity; EIwis

the warping rigidity and M is the torque induced by the lateral deflection v.

When these equations are both satisfied at all points of the beam, then the deflected and

twisted position is one of equilibrium which can be found by differentiating Equation (16)

and substituting Equation (15); then, the differential equation of lateral buckling is given

by:

E.Iw (17a)

or

Cw (17b)

This expression was established, for the first time in 1899, by Prandtl. The general

solution of this equation is:

= A1sinh k1x + B1cosh k1x + A2sin k2x + B2cos k2x (18)

where

= (19)

= - (20)

in which A1, A2, B1 and B2 are constants of integration which must satisfy the

boundary conditions:

= 0 for x = 0 (21a)

and

= 0 for x = l (21b)

= 0 for x = 0 (22a)

and


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= 0 for x = l (22b)

Equations (21) and (22) show that, in the case of a so-called simply supported beam, the

supports must prevent both lateral deflection and twist but the section is free to warp at the

ends.

The four boundary conditions give:

A1= B1= B2= 0 (23)

and

A2sin k2l = 0 (24)

which requires either A2= 0 (in this case there is no twist), or

sin k2l = 0, i.e: k2l = n (25)

where n- is any integer.

Substituting Equation (25) into Equation (20) and rearranging, using the smallest value of

k2l gives the critical moment for the beam:

(26)

4. BUCKLING OF PLATES

The simplest example of this phenomenon is that of a rectangular plate with four edges

simply supported (prevented from displacing out-of-plane but free to rotate) loaded in

compression as shown in Figure 3. As for compressed struts, the plate remains flat until

the applied load reaches its critical value, at which time it buckles with lateral deflections.

The differential equation for plate buckling, established by Bryan in 1891, gives for the

case shown in Figure 3:

(27)

where D is the bending rigidity of the plate:

D = Et3/{12(1-2)} (28)

The general solution of this equation is:


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w = A sin mx/a . sin n/b (29)

which satisfies the boundary conditions:

w = = 0 for x = 0 (30a)

and

w = = 0 for x = a (30b)

w = = 0 for y = 0 (31a)

and

w = = 0 for y = b (31b)

Substituting Equation (29) in Equation (27), gives:

N = (32)

where mand n- are the number of half-waves in the directions xandyrespectively.

The smallest value of N, and therefore the critical load Ncr, will be obtained by taking n

equal to 1. This shows that the plate buckles in such a way that there can be several half-

waves in the direction of compression but only one half-wave in the perpendicular

direction. Therefore, the expression for the critical load becomes:

Ncr= k (33)

where:

k = (34)

If the plate buckles in one half-wave, then m = 1 and k acquires its minimum value

(equal to 4), when a = b, i.e. for a square plate.


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Similarly, if the plate buckles into two half-waves, then m = 2 and k reaches its minimum

value (also equal to 4), when a = 2b.

Similarly assuming m = 3, 4,..., one obtains the series of curves given in Figure 9. It isinteresting to note that, at the values 2, 6,... of the ratio a/b, there is a coincidence oftwo buckling modes.

5. CONCLUDING SUMMARY

For compression members, such as struts, beams, plates and shells, the critical loadis the upper bound value for the ultimate load of an actual (imperfect) member.

The critical load is associated with the neutral equilibrium condition of themember.

For simple cases, the critical loads may be calculated by solving the differentialequilibrium equations describing the phenomena.

6. ADDITIONAL READING

1. Timoshenko, S.P. and Gere, J.M., "Theory of Elastic Stability", McGraw-Hill, 2nd

edition, New York, 1961.

2. Allen, H.G. and Bulson, P.S., "Background to Buckling", McGraw-Hill, London,

1980.

3. Shanley, F.R., "Strength of Materials", McGraw-Hill, New York, 1957.

4. Murray, N.W., "Introduction to the Theory of Thin-Walled Structures", Clarendon

Press, Oxford, 1984.

elastic instability modes

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