Beams under transverse and axial loads 163

Structural instability

6.4 Beams under transverse and axial loads 163

where A and B are unknown constants and A' = P / E I . Substituting the boundary conditions v = 0 at z = 0 and 1 gives

W W

A = - , B = ( 1 - cos X I ) X2P X2P sin X I

so that the deflection is determinate for any value of w and P and is given by v = - X2 P

[

^cosXz+

('~i-~~)sinXz]

+ & ( z 2 - l z - $ - ) (6.30) In beam-columns, as in beams, we are primarily interested in maximum values of stress and deflection. For this particular case the maximum deflection occurs at the centre of the beam and is, after some transformation of Eq. (6.30)

v m a = ~ ( s e c 2 - i ) A1 - 8 p w12 The corresponding maximum bending moment is

M,,, = -Puma - - w12 8 or, from Eq. (6.31)

(6.31)

(6.32) We may rewrite Eq. (6.32) in terms of the Euler buckling load PCR = d E I / 1 2 for a pin-ended column. Hence

(6.33) As P approaches PCR the bending moment (and deflection) becomes infinite.

However, the above theory is based on the assumption of small deflections (otherwise d2v/d2 would not be a close approximation for curvature) so that such a deduction is invalid. The indication is, though, that large deflections will be produced by the presence of a compressive axial load no matter how small the transverse load might be.

Let us consider now the beam-column of Fig. 6.10 with hinged ends carrying a concentrated load W at a distance u from the right-hand support. For

and for

d2v W

d.9 -

I

z 2 1 - a, ^EI - - - M = -Pv - - ( I - u> (1 - z)

(6.34)

(6.35)

Writing

X2 = P / E I

164 Structural instability

Y +

Fig. 6.10 Beam-column supporting a point load.

Eq. (6.34) becomes

d2v Wa

- + A v = --

d z 2 EII

the general solution of which is

(6.36) Wa

w = A cos Xz

+

^{Bsin X Z - - z} PI Similarly, the general solution of Eq. (6.35) is

(6.37) v =

C

cos Xz

+

^{D sin Az - - ( I - a)(, - z)}

where A , By C and D are constants which are found from the boundary conditions as follows.

When z = 0, v = 0, therefore from Eq. (6.36) A ⁼ 0. At z = I , w = 0 giving, from Eq. (6.37),

C

= -DtanXI. At the point of application of the load the deflection and slope of the beam given by Eqs (6.36) and (6.37) must be the same. Hence, equating deflections

W PI

Wa Wa

PI PI

B sin X(1 - a) - - ( I - a) = D[sin X(1- a) - tan XI cos X(1 - a)] - - ( I - a) and equating slopes

Wa W

PI PI

BXcosX(1-a) --= DX[cosX(I-a)+tanXIsinX(I-a)] +-(I-a)

Solving the above equations for B and D and substituting for A , By C and D in Eqs (6.36) and (6.37) we have

W sin Xa Wa PI

PA sin XI sink--z for z

<

I - a

V =

W sin X(1 - a) W

PA sin XI PI

V = sinX(I-2)--(I-a)(/-z) for z 2 l - a (6.39) (6.38)

These equations for the beam-column deflection enable the bending moment and resulting bending stresses to be found at all sections.

A particular case arises when the load is applied at the centre of the span.

The deflection curve is then symmetrical with a maximum deflection under the

6.5 Energy method 165

Y +

Fig. 6.1 1 Beam-column supporting end moments.

load of

w

^XI Wl v,,, =-tan---

2PX 2 4P

Finally, we consider a beam-column subjected to end moments M A and MB in addition to an axial load P (Fig. 6.11). The deflected form of the beam-column may be found by using the principle of superposition and the results of the previous case. First, we imagine that MB acts alone with the axial load P. If we assume that the point load W moves towards B and simultaneously increases so that the product Wu = constant = MB then, in the limit as a tends to zero, we have the moment M B applied at B. The deflection curve is then obtained from Eq. (6.38) by substituting Xu for sin Xa (since Xu is now very small) and MB for Wa. Thus

V

=%

^{(7sin Xz &)} _(6.40)

In a similar way, we find the deflection curve corresponding to M A acting alone. Sup- pose that W moves towards A such that the product W(I - a ) ⁼ constant = MA.

Then as ( I - a) tends to zero we have sin X ( 1 - a ) = X ( I - a ) and Eq. (6.39) becomes MA sinX(1- z) ( I - z)

v=-[ P sin XI

--I

^{I (6.41)}

The effect of the two moments acting simultaneously is obtained by superposition of the results of Eqs (6.40) and (6.41). Hence for the beam-column of Fig. 6.11

(6.42) Equation (6.42) is also the deflected form of a beam-column supporting eccentrically applied end loads at A and B. For example, if eA and eB are the eccentricities of P at the ends A and B respectively, then M A = PeA, MB = PeB, giving a deflected form of

MB (sinXz z ) I

7 ^[

sinX(1 -z) ( I - z ) P sinX1

I

sin XI

--I

¹

v=-

(6.43) Other beam-column configurations featuring a variety of end conditions and

sin Xz sin X ( 1 - z)

(I

- z)

v = e B ( , X r - 5 ) sin XI

--I

¹

loading regimes may be analysed by a similar procedure.

The fact that the total potential energy of an elastic body possesses a stationary value in an equilibrium state may be used to investigate the neutral equilibrium of a buckled

166 Structural instability

Y t

Fig. 6.12 Shortening of a column due to buckling.

column. In particular, the energy method is extremely useful when the deflected form of the buckled column is unknown and has to be ‘guessed’.

First, we shall consider the pin-ended column shown in its buckled position in Fig. 6.12. The internal or strain energy

U

of the column is assumed to be produced by bending action alone and is given by the well known expression

or alternatively, since EZ d2v/& =

-M

(6.44)

(6.45) The potential energy

V

of the buckling load PCR, referred to the straight position of the column as the datum, is then

v

= -PCR6

where 6 is the axial movement of PcR caused by the bending of the column from its initially straight position. By reference to Fig. 5.15(b) and Eq. (5.41) we see that

giving

(6.46) The total potential energy of the column in the neutral equilibrium of its buckled state is therefore

or, using the alternative form of U from Eq. (6.45)

(6.47)

(6.48) We have seen in Chapter 5 that exact solutions of plate bending problems are obtainable by energy methods when the deflected shape of the plate is known. An identical situation exists in the determination of critical loads for column and thin