Barber J.R. Intermediate Mechanics of Materials

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12.8 Quick estimates for the buckling force 545

12.8 Quick estimates for the buckling force

We have already seen in §§12.2, 12.3 that initial imperfections or lateral loads can

cause substantial deﬂections of axially loaded members below the theoretical criti-

cal force.

It is therefore important to design compression members to operate well

below the critical force. A good working rule is to stay below half the critical force

wherever possible and if circumstances make it advantageous to try to exceed this

limit, take great care over the accuracy both of manufacture and loading of the beam.

One incidental advantage of this necessity to design well below the critical force

is that it is not necessary to know the theoretical critical force to a very high degree

of accuracy. If we are going to apply a safety factor of 2, anything better than 20%

accuracy in a calculation is so much wasted effort. The Rayleigh-Ritz method will

usually give this kind of accuracy with an elementary function with a single degree

of freedom, but care must be taken to ensure that the assumed deformation is similar

in shape to that which is likely to occur. If in doubt, perform two or more calculations

with different trial functions and use the one which gives the lowest critical force.

In beam problems, an even quicker estimate can be made if the deformed shape

can be realistically estimated. The shape of the buckled section is dominated by the

sinusoidal functions of equation (12.4) and the buckling force is uniquely related to

the wavelength of the resulting sinusoid. Most buckled conﬁgurations involve less

than a complete wave, so it is convenient to deﬁne the length of a quarter wave as

1/4

. It then follows that the buckling force is

∗

1/4

. (12.71)

Thus, if the expected deformed shape includes N quarter waves in a beam of length,

L, we have L

1/4

=L/N and

∗

. (12.72)

This formula is clearly exact for the modes of the simply-supported beam of Fig-

ure 12.2 (a), but it also gives a reasonable estimate for more complicated problems.

For example, Figure 12.26 (a) shows a beam that is built in at one end and simply

supported at the other, loaded by an axial force P. The anticipated deformed shape

is shown in Figure 12.26 (b) and this contains approximately three quarter waves,

as indicated by the division of the beam into segments by the dotted lines. Equation

(12.72) then predicts that the buckling force will be

∗

≈

22EI

For this reason, it is quite difﬁcult to get an accurate measurement of the critical force

experimentally.

546 12 Elastic Stability

Figure 12.26: (a) A beam built in at one end and simply supported at the other; (b)

the expected deformed shape

This problem can be solved by the equilibrium method. After a fairly extensive

calculation, we obtain the exact value

20.1EI

so the one line estimate of equation (12.72) is really quite good. The reader is advised

to check this prediction even in cases where a more accurate result is required, since

it provides a useful check against casual calculation errors.

12.9 Summary

A perfectly straight beam will suffer no lateral deﬂection under purely axial loading

until a critical compressive force is reached, at which the system is unstable and the

beam buckles. A related instability known as whirling occurs in rotating shafts when

the rotational speed reaches a critical value.

If the beam is slightly curved or if there are also lateral loads, the lateral deﬂec-

tions are increased by a compressive force and increase without limit as the critical

condition is approached. In these cases, signiﬁcantly enhanced displacements and

stresses can be produced well below the critical force, so it is necessary to use a

good factor of safety against buckling failure.

Buckling failures are particularly critical in thin-walled members, which in other

respects are extremely efﬁcient load bearing structures. Buckling is therefore often a

limiting consideration in optimal design.

Two basic methods are available for determining the critical force. In the neu-

tral equilibrium method, differential equations are derived deﬁning the conditions

which must be satisﬁed if the system is to be capable of remaining in equilibrium in

(a)

(b)

Problems 547

a ‘non-trivial’ state of lateral deformation. Solution of these equations and the impo-

sition of boundary conditions leads to an algebraic equation for the critical condition.

Beam problems can generally be solved in this way, but more complex two and three

dimensional problems are seldom analytically tractable.

An alternative approach is to use the Rayleigh-Ritz method in which a functional

form is assumed for the deformed shape and the stability criterion is based on the

requirement that the total potential energy should be a local minimum. This method

is very versatile, but it depends crucially on our ability to predict and represent the

probable buckled shape of the structure.

Further reading

J.M.T. Thompson and G.W. Hunt (1973), A General Theory of Elastic Stability, John

Wiley, London.

S.P. Timoshenko and J.M. Gere (1961), Theory of Elastic Stability, McGraw-Hill,

New York, 2nd edn.

Problems

Section 12.1

12.1*. Figure P12.1 shows a beam of ﬂexural rigidity EI and length L which is simply

supported at the points A (z=L/4) and B (z=3L/4). It is compressed by two forces P

whose lines of action are always parallel to AB. Find the critical force P

and sketch

the shapes of the ﬁrst two buckling modes.

12.2. The composite beam AB of Figure P12.2 consists of three segments. The seg-

ments AC and DB are both rigid and of length a, whilst the segment CD is of ﬂexural

rigidity EI and length (L−2a). The beam is simply supported at A,B and compressed

by a force P. Find the equation determining the critical value of P for instability and

show that your result reduces to equation (12.10) when a=0. Solve the equation for

the case where a= L/4.

Figure P12.1

548 12 Elastic Stability

Figure P12.2

12.3. Find the critical force for the beam of Figure P12.2 for the case where the

segment CD is rigid and the segments AC, DB have ﬂexural rigidity EI.

12.4. Figure P12.4 (a) shows a rigid bar AB of length L, which is pinned at B and

compressed by a horizontal force P at A. The side of the bar is bonded to a block

of rubber which can be modelled as an elastic foundation of modulus k, as shown

in Figure P12.4 (b). In other words, the rubber exerts a vertical force w(z) per unit

length given by

w(z) = ku(z) ,

where k is a constant and u(z) is the local vertical displacement.

Find the value of P at which the system becomes unstable.

(a)

(b)

Figure P12.4

Sections 12.2, 1 2.3

12.5. The cantilever beam of Figure P12.5 of length L and ﬂexural rigidity EI has an

initial deﬂection deﬁned by the equation

A B

rubber

A B

(L - 2a)

Problems 549

(z) =

where

≪L. Find the maximum bending moment in the beam when it is subjected

to an axial force

P =

u (z)

Figure P12.5

12.6. A uniform simply supported beam of ﬂexural rigidity EI and length L is sub-

jected to a lateral uniformly distributed load w

per unit length and an axial force

P as shown in Figure P12.6. Find the maximum bending moment if P is equal to

70% of the buckling force. What is the ratio between this moment and the value that

would be obtained if P were zero?

12.7. The uniform beam in Figure P12.7 has a circular cross section of diameter d

and length L. It is subjected to an axial compressive force P which is lower than the

ﬁrst critical force for the beam, but whose line of action is a distance d/4 from the

centreline of the beam.

Find the maximum bending moment in the beam.

Figure P12.7

w per unit length

Figure P12.6

550 12 Elastic Stability

12.8.* Figure P12.8 shows a cantilever beam of length L loaded by a lateral force F

and a tensile axial force P. Show that the bending moment at the support is reduced

by the application of the axial force and ﬁnd this moment as a function of F and P.

If the beam is a circular cylinder of diameter, d, are there any circumstances in

which the maximum tensile stress in the beam is reduced as a result of the application

of P.

Note: The homogeneous solution for this problem is different from (12.4).

Figure P12.8

Section 12.4

12.9. Find the critical force for the indeterminate problem of Figure 12.11.

12.10. The free end of a cantilever beam of length L and ﬂexural rigidity EI is re-

strained from lateral displacement by a frictionless support, which offers no resis-

tance to rotation (see Figure P12.10).

Find the critical force P

at which the beam will buckle.

Figure P12.10

Section 12.5

12.11. Figure P12.11 shows a beam of length L and ﬂexural rigidity EI, which is

supported on two springs, each of stiffness k, and loaded by a compressive force P.

Treating the beam as rigid, determine the value of P at which the system is un-

stable. Hence ﬁnd the critical value k

, beyond which increasing k does not increase

the buckling force.

Problems 551

Figure P12.11

12.12. A beam of ﬂexural rigidity EI and length L is compressed by an axial force

P. The two ends of the beam are restrained from lateral motion (u = 0 at z = 0, L),

but rotation (end slope) is resisted by torsion springs which exert moments M

respectively, given by

= k

; i = 1,2 .

Find the ﬁrst buckling force P

for the system. Show that it is a monotonically

increasing function of the spring stiffness k and ﬁnd the value of k for which

— i.e. twice the value that would be obtained if k were zero.

Section 12.6

12.13. The mounting assembly of Problem 7.7 is subjected to an axial force P whose

line of action passes through the centroid of the steel strip. Find the critical force at

which buckling occurs and the wavelength of the resulting deformed conﬁguration.

12.14. A long steel beam of second moment of area I = 5×10

is supported on

an elastic foundation of modulus k = 10 MPa and loaded by an axial force P. Find

the buckling force P

and the wavelength of the resulting deformed conﬁguration

steel

=210 GPa).

12.15. An inﬁnite beam of ﬂexural rigidity EI is supported on an elastic foundation

of modulus k and loaded by a concentrated moment M

at z = 0. The beam also

transmits an axial compressive force P. Find the rotation

(0) due to the moment as

a function of P and hence ﬁnd the axial force needed to increase this rotation by a

factor of two.

12.16. A beverage can of diameter 2.5 inch and wall thickness 2×10

−3

inch is made

of aluminium alloy for which Young’s modulus is E =11 ×10

psi and

=0.25. Find

the axial compressive force at which it will buckle in an axisymmetric mode and the

wavelength of the resulting deformed conﬁguration. What is the mean compressive

stress in the can just before buckling and how does it compare with typical yield

strengths for aluminium alloys?

552 12 Elastic Stability

12.17.* Figure P12.17 shows an idealization of a ﬁbre-reinforced composite material

in which uniaxial ﬁbres of diameter 1 mm and Young’s modulus E = 180 GPa are

embedded in a matrix of modulus E = 1.4 MPa. The resistance of the matrix to

the lateral displacement of a single ﬁbre is estimated to be equivalent to an elastic

foundation of modulus 0.5 MPa. If there are 25×10

ﬁbres per m

of cross-sectional

area, estimate the uniaxial compressive stress at which the composite will fail by

ﬁbre buckling when loaded along the ﬁbre axis.

fibres

1 mm

matrix

Figure P12.17

12.18. A solid cylindrical shaft of diameter 10 mm and length 1 m is built in at one

end, the other end being unsupported. Find the ﬁrst critical speed of the shaft, if the

material is steel for which E =210 GPa,

=7700 kg/m

12.19. The transmission shaft for a rear wheel drive car consists of a hollow steel

tube of mean diameter 30 mm, wall thickness 2 mm and length 1.8 m. Determine

the critical rotational speed of the shaft, treating it as simply supported at the ends.

steel

=210 GPa,

steel

=7700 kg/m

12.20. A turbo-alternator for a power generation plant can be idealized as a solid steel

cylinder of diameter 300 mm and length 14 m. It is supported in three bearings, one

at each end and one at the mid-point, all of which can be treated as simple supports.

Find the ﬁrst critical rotational speed (E

steel

=210 GPa,

steel

=7700 kg/m

12.21. A solid wheel of diameter 200 mm and thickness 30 mm is supported at the

mid-point of a shaft of diameter 20 mm and length 600 mm. The shaft is simply

supported at its ends. Find the critical rotational speed if all the components are

made of steel (E =210 GPa,

=7700 kg/m

12.22. A student swings a 2 lb weight (mass 1/16 slug) around her head at the end

of a spring of stiffness 3 lb/in and length 10 in. What is the theoretical maximum

rotational speed she could achieve?

Problems 553

12.23.* A thin-walled elastic tube of Young’s modulus E, wall thickness t, radius

a and length L is simply supported at its ends. An inviscid ﬂuid of density

ﬂows

through the tube at velocity V . Find the value of V for which the tube will buckle.

Neglect gravity.

Hint: Treat the tube as a beam of appropriate ﬂexural rigidity. Lateral forces and

hence bending moments are generated by the acceleration of the ﬂuid when the tube

is in a deformed conﬁguration.

12.24.* Find the ﬁrst whirling speed for the shaft of Figure 12.20, taking into ac-

count the mass of both the shaft and the wheel and using the boundary conditions

(12.49). Plot a graph of

Ω

as a function of the dimensionless ratio M/mL (the ratio

of the mass of the wheel to that of the shaft) and compare it with the approximate

expression (12.48). How good is the approximation?

Section 12.7

12.25. Figure P12.25 shows a rigid bar supported by a torsion spring at one end and

loaded by an axial force P. Find the critical force P

, if the spring exerts a moment

M = k

, where

is the rotation at the support.

Figure P12.25

12.26. The uniform cantilever shown in Figure P12.26, of ﬂexural rigidity EI and

length L, is loaded by an axial force P and its free end is restrained by a spring of

stiffness k.

Use the Rayleigh-Ritz method to estimate the critical force, assuming the beam

deforms into a parabolic shape.

Figure P12.26

12.27. A uniform beam of ﬂexural rigidity EI and length L is simply supported at its

ends and also at a point a distance b from one end. Use the Rayleigh-Ritz method

and the trial function u=Cx(x−b)(x−L) to estimate the buckling force for the beam.

Use your results to sketch a graph of P as a function of b in the range 0< b < L.

554 12 Elastic Stability

12.28. Figure P12.28 shows a solid cylindrical tower of height h and diameter

d, made from an elastic material of density

and Young’s modulus E. Use the

Rayleigh-Ritz method to estimate the maximum value of h if the tower is not to

buckle under its own weight.

Use a one degree of freedom approximation to the buckled shape. Be particularly

careful in determining the expression for the potential energy of the distributed force

due to self-weight.

What is the critical height for a steel tower of diameter 10mm? (E

steel

=210 GPa,

steel

=7700 kg/m

Figure P12.28

12.29. In an attempt to make the tower of Problem 12.28 higher without using addi-

tional material, it is proposed to make the diameter d vary with distance z from the

top according to the equation

d(z) = Cz

where C,n are constants.

The total volume of material available is constant and equal to V. What value of

n should be used to achieve the maximum height h without buckling?

12.30. A beam of length L is built in at z = 0 and loaded by an axial force P at z = L

as shown in Figure P12.30. The beam is made of a material with Young’s modulus

E and has a rectangular cross section of sides a,b, where

a = c ; b = c

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