. way, gradually increasing the load until it is found that 
_ further, may be called the critical load for the 
strut. 
_ slender columr. may be demonstrated as follows. 
_ the deflection being u,. Let Z be the modulus 
RR ee ee re 
COLUMNS AND STRUTS 163 
at (a). A load insufficient to bend the strut is applied, and the strut is 
then slightly bent by pressing it sideways at the middle of its length. 
When this side pressure is removed the strut straightens 
itself. A small addition is made to the load and the side 
pressure is again applied and removed, the strut bends and 
straightens again. The experiment is continued in this 
P 
when the side pressure is applied and then removed the 
strut remains bent. When this point is reached it will be (a) 
found that, whatever amount of deflection be given to the 
strut by the applied lateral force, the strut will retain 
that amount of deflection when the lateral force is removed, , G 
provided that the elastic limit is not exceeded. But if yg, 930, 
the load on the strut be further increased and the strut 
be slightly bent as before, the load will increase the amount of bend- 
ing until the strut takes a permanent set or collapses. This load, which 
will keep the strut bent but will not bend it 
That there is a critical load for a long 
Let the strut shown at (a), Fig. 231, be in equi- 
librium under the load P, and lateral force Q,, 
of the cross section, and let 7, be the maximum 
stress due to bending. ; 
The total bending moment is Py + Qe, Fia. 231. 
_ and the moment of resistance to bending is f,Z, therefore Pw, + Qe =f,Z. 
Tf Q,; be now diminished to zero and P, be increased to P, the deflec- 
tion u, remaining the same, it follows that Pu, =,Z or P=i7, 
1 
If the same strut be in equilibrium under the load P, and lateral 
force Q,, as shown at ()), Fig. 231, the deflection being u,, then decreasing 
Q, to zero, and increasing P, to P’, the deflection wu, remaining the same, 
it follows that py 7, where f, is the maximum stress due to bending. 
Us 
But Aah, therefore P’ =P. 
1 
From the foregoing it follows that the strut will be in equilibrium 
under the load P for any deflection within the elastic limit. 
For a load greater than P the force Q, acting as shown at (c), Fig. 
231, would be necessary to prevent further deflection of the strut, because 
the bending moment due to P+W is Pu,+Ww,, and the moment of 
resistance of the strut to bending is 7,Z, but Pu,=/,Z, therefore the 
bending moment Wz, must be balanced by the moment’ of Q, hence 
QU=Wu,. Within the clastic limit, /Z, the moment of resistance of the 
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