44 Mechanics of Materials 2 $2.9 i.e. =ll+意c/0】 (2.26) This formula is known as the Smith-Southwell formula. Unfortunately,since o =P/A,the above equation represents a function of P(the required unknown)which can only be solved by trial and error or graphically.A good approximation however,is obtained as shown below: Webb's approximation From above Omax 0 + 2 (2.25)bis) nL Let 2 Then (司)=V(保)-) Now for 0 between 0 and x/2, 1+0.26 26 1+0.2 P P。+0.26P secθa Pe P 1一p P。-P 、元 Therefore substituting in egn.(2.25) eh 1+ Pe+0.26P Pe-P + P「 h Pe+0.26P\ Pe-P (2.27) where omax is the maximum allowable stress in the strut material,Pe is the Euler buckling load for axial loading,and P is the maximum allowable value of the eccentric load. The above equation can be re-written into a more readily observed quadratic equation in P,thus: rl-02s筒-r.(+)+aA个 +OmaxAPe =0 (2.28) For any given eccentric load condition P is the only unknown and the equation can be readily solved. (b)One end fixed,the other free Consider the strut shown in Fig.2.11. BM.=El dx2 =P(eo-y)44 Mechanics of Materials 2 52.9 i.e. (2.26) This formula is known as the Smith-Southwell formula. Unfortunately, since a = P/A, the above equation represents a function of P (the required unknown) which can only be solved by trial and error or graphically. A good approximation however, is obtained as shown below: Webb’s approximation From above a, = e [ 1 + $ sec $1 nL 2 Let - =e Then Now for 8 between 0 and nI2, = ‘.d(k) =&E) L2 P = $5) 2 P 1 + 0.26- 1-- P, + 0.26P 1 + 0.26 ($) - Pe - P- P, - P sece 2 - 1 - (;)2 pe Therefore substituting in eqn. (2.25) P, + 0.26P =‘[1.$( p,-p )] A (2.25)(bis) (2.27) where a,, is the maximum allowable stress in the strut material, P, is the Euler buckling load for axial loading, and P is the maximum allowable value of the eccentric load. The above equation can be re-written into a more readily observed quadratic equation in P, thus: For any given eccentric load condition P is the only unknown and the equation can be readily solved. (6) One endjixed, the other free Consider the strut shown in Fig. 2.1 1 d2 Y BM,. = EI- = P(eo - y) dx2