ABSTRACT
Consider a pin-ended, uniform and initially straight strut of length I carrying an axial compressive load P (with sign compression positive rather than negative). If the strut deflects laterally such that its curve is v(x) as shown in Fig. 7.1, the horizontal end reactions must remain zero and the bending moment at lengthwise coordinate x can be determined from equilibrium as
M = —Pv. Substituting this into the elastic bending formula, Eq. (6.4), gives
d2v E J _ + P t , = 0 (7.1)
Whereas this differential equation always has a trivial solution v = 0, there are specific values of P for which non-zero solutions exist. With the known end conditions v = 0 at x = 0 and x = I, the smallest such load is
P = n 2 ¥ (7.2)
in which case v = a $>m(7Tx/l) where a is an indeterminate constant. This load is known as the Euler buckling load and designated ?E.
