$2.2 Struts 35 The particular solution is F y-, 化-0=p化- 2EI The full solution is therefore F y=Acosnx+Bsinnx+(L-x) FL Whenx=0,y=0, ∴，A=一 P F When x=0,dy/dx=0,.'.B= nP cosnx+ FL F npsinnx+pL-划 F = np-nL cosnx+sinnx+n亿-x划 But when x=L,y=0 nL cos nL sin nL tan nL nL The lowest value of nL(neglecting zero)which satisfies this condition and which therefore produces the fundamental buckling condition is nL =4.5 radians. =45 20.25EI or P.= (2.5) L2 or,approximately 2r2E1 P.= (2.6) 2.2.Equivalent strut length Having derived the result for the buckling load of a strut with pinned ends the Euler loads for other end conditions may all be written in the same form, π2E1 i.e. P= (2.7) 12 where I is the equivalent length of the strut and can be related to the actual length of the strut depending on the end conditions.The equivalent length is found to be the length of a simple bow (half sine-wave)in each of the strut deflection curves shown in Fig.2.6.The buckling load for each end condition shown is then readily obtained. The use of the equivalent length is not restricted to the Euler theory and it will be used in other derivations later.92.2 Struts 35 The uarticular solution is F F n2EI P y=- (L-x)=-(L-x) The full solution is therefore F P y = Acosnx + Bsinnx + -(L - x) FL P When x = 0, y = 0, .A=-- .. F When x = 0, dyldx = 0, :. B = - nP F. F osnx + - sinnx + -(L - x) FL y=--c P nP P F nP = -[-nL cos nx + sin nx + n (L - x>I But when x = L, y = 0 .. nL cos nL = sin nL tannL = nL The lowest value of nL (neglecting zero) which satisfies this condition and which therefore produces the fundamental buckling condition is nL = 4.5 radians. or 20.25EI P, = ~ L* or, approximately 2.2. Equivalent strut length Having derived the result for the buckling load of a strut with pinned ends the Euler loads for other end conditions may all be written in the same form, i.e. X~EI P, = - 12 (2.7) where 1 is the equivalent length of the strut and can be related to the actual length of the strut depending on the end conditions. The equivalent length is found to be the length of a simple bow (half sine-wave) in each of the strut deflection curves shown in Fig. 2.6. The buckling load for each end condition shown is then readily obtained. The use of the equivalent length is not restricted to the Euler theory and it will be used in other derivations later