Chapter 8 Columns 255 Chapter 9 Thin Plates 294
8.4 Stability of beams under transverse and axial loads
Stresses and deflections in a linearly elastic beam subjected to transverse loads as predicted by simple beam theory, are directly proportional to the applied loads. This relationship is valid if the deflections are small such that the slight change in geom- etry produced in the loaded beam has an insignificant effect on the loads themselves.
This situation changes drastically when axial loads act simultaneously with the trans- verse loads. The internal moments, shear forces, stresses and deflections then become dependent upon the magnitude of the deflections as well as the magnitude of the exter- nal loads. They are also sensitive, as we observed in the previous section, to beam imperfections such as initial curvature and eccentricity of axial load. Beams supporting both axial and transverse loads are sometimes known as beam-columns or simply as transversely loaded columns.
We consider first the case of a pin-ended beam carrying a uniformly distributed load of intensity w per unit length and an axial load P as shown in Fig. 8.11. The bending moment at any section of the beam is
M=Pv+wlz 2 − wz2
2 = −EId2v dz2 giving
d2v dz2 + P
EIv= w
2EI(z2−lz) (8.29)
The standard solution of Eq. (8.29) is
v=A cosλz+B sinλz+ w 2P
z2−lz− 2 λ2
where A and B are unknown constants and λ2=P/EI. Substituting the boundary conditionsv=0 at z=0 and l gives
A= w
λ2P B= w
λ2P sinλl(l−cosλl)
Fig. 8.11Bending of a uniformly loaded beam-column.
8.4 Stability of beams under transverse and axial loads 269
so that the deflection is determinate for any value of w and P and is given by v= w
λ2P
cosλz+
1−cosλl sinλl
sinλz
+ w
2P
z2−lz− 2 λ2
(8.30) 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. (8.30)
vmax= w λ2P
secλl
2 −1
−wl2
8P (8.31)
The corresponding maximum bending moment is Mmax= −Pvmax−wl2
8 or, from Eq. (8.31)
Mmax= w λ2
1−secλl 2
(8.32) We may rewrite Eq. (8.32) in terms of the Euler buckling load PCR=π2EI/l2 for a pin-ended column. Hence
Mmax= wl2 π2
PCR P
1−secπ 2
P PCR
(8.33) As P approaches PCRthe bending moment (and deflection) becomes infinite. However, the above theory is based on the assumption of small deflections (otherwise d2v/dz2 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. 8.12 with hinged ends carrying a concentrated load W at a distance a from the right-hand support. For
z≤l−a EId2v
dz2 = −M = −Pv−Waz
l (8.34)
and for
z≥l−a EId2v
dz2 = −M = −Pv−W
l (l−a)(l−z) (8.35) Writing
λ2= P EI Eq. (8.34) becomes
d2v
dz2 +λ2v= −Wa EIlz
Fig. 8.12Beam-column supporting a point load.
the general solution of which is
v=A cosλz+B sinλz−Wa
Plz (8.36)
Similarly, the general solution of Eq. (8.35) is v=C cosλz+D sinλz− W
Pl(l−a)(l−z) (8.37) where A, B, C and D are constants which are found from the boundary conditions as follows.
When z=0, v=0, therefore from Eq. (8.36) A=0. At z=l, v=0 giving, from Eq. (8.37), C= −D tanλl. At the point of application of the load the deflection and slope of the beam given by Eqs (8.36) and (8.37) must be the same. Hence, equating deflections
B sinλ(l−a)− Wa
Pl(l−a)=D[ sinλ(l−a)−tanλl cosλ(l−a)]−Wa Pl(l−a) and equating slopes
Bλcosλ(l−a)− Wa
Pl =Dλ[ cosλ(l−a)−tanλl sinλ(l−a)]+W Pl(l−a) Solving the above equations for B and D and substituting for A, B, C and D in Eqs (8.36) and (8.37) we have
v= W sinλa
Pλsinλlsinλz−Wa
Plz for z≤l−a (8.38)
v= W sinλ(l−a)
Pλsinλl sinλ(l−z)− W
Pl(l−a)(l−z) for z≥l−a (8.39) 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 load of
vmax= W 2Pλtanλl
2 − Wl 4p
8.5 Energy method for the calculation of buckling loads in columns 271
Fig. 8.13Beam-column supporting end moments.
Finally, we consider a beam-column subjected to end moments MAand MBin addi- tion to an axial load P (Fig. 8.13). 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 MBacts alone with the axial load P. If we assume that the point load W moves towards B and simultaneously increases so that the product Wa=constant=MB then, in the limit as a tends to zero, we have the moment MBapplied at B. The deflection curve is then obtained from Eq. (8.38) by substitutingλa for sinλa (sinceλa is now very small) and MBfor Wa. Thus
v= MB P
sinλz sinλl − z
l
(8.40) In a similar way, we find the deflection curve corresponding to MAacting alone. Suppose that W moves towards A such that the product W (l−a)=constant=MA. Then as (l−a) tends to zero we have sinλ(l−a)=λ(l−a) and Eq. (8.39) becomes
v= MA P
sinλ(l−z)
sinλl −(l−z) l
(8.41) The effect of the two moments acting simultaneously is obtained by superposition of the results of Eqs (8.40) and (8.41). Hence for the beam-column of Fig. 8.13
v= MB P
sinλz sinλl −z
l
+ MA P
sinλ(l−z)
sinλl − (l−z) l
(8.42) Equation (8.42) is also the deflected form of a beam-column supporting eccentrically applied end loads at A and B. For example, if eAand eBare the eccentricities of P at the ends A and B, respectively, then MA=PeA, MB=PeB, giving a deflected form of
v=eB
sinλz sinλl −z
l
+eA
sinλ(l−z)
sinλl −(l−z) l
(8.43) Other beam-column configurations featuring a variety of end conditions and loading regimes may be analysed by a similar procedure.