§2.6 Struts 39 Table 2.3. Material Oy or o a (MN/m2) Pinned ends Fixed ends Low carbon steel 315 17500 1/30000 Cast iron 540 1/1600 1/64000 Timber 35 1/3000 1/12000 N.B.a for pinned ends =4 x (a for fixed ends) 2.6.Perry-Robertson formula The Perry-Robertson proof is based on the assumption that any imperfections in the strut, through faulty workmanship or material or eccentricity of loading,can be allowed for by giving the strut an initial curvature.For ease of calculation this is assumed to be a cosine curve,although the actual shape assumed has very little effect on the result. Consider,therefore,the strut AB of Fig.2.8,of length L and pin-jointed at the ends.The initial curvature yo at any distance x from the centre is then given by πx yo=Co cos L -P -L2 Fig.2.8.Strut with initial curvature. If a load P is now applied at the ends,this deflection will be increased to y+yo. 8u:=狼=-P6+om) 2+后(+Gs咒)=0 the solution of which is y=Asin ()+a()+[(会)/（信-】 where A and B are the constants of integration. Now when x=±L/2,y=0 A=B=0 y=[(会o)/(信-司】-[ecw)/(提-P刃$2.6 Material Low carbon steel Cast iron Timber Struts t~? or a,. a (MN/m2) Pinned ends Fixed ends 315 117500 1/3OOOO 540 1/1600 1 /64 000 35 1/3000 1/12OOO 39 2.6. Perry-Robertson formula The Perry-Robertson proof is based on the assumption that any imperfections in the strut, through faulty workmanship or material or eccentricity of loading, can be allowed for by giving the strut an initial curvature. For ease of calculation this is assumed to be a cosine curve, although the actual shape assumed has very little effect on the result. Consider, therefore, the strut AB of Fig. 2.8, of length L and pin-jointed at the ends. The initial curvature yo at any distance x from the centre is then given by TX yo = cocos - L Fig. 2.8. Strut with initial curvature. If a load P is now applied at the ends, this deflection will be increased to y + yo. .. "1 BMc=EI- d2 Y = -P(y+cocos- - + - (y + cocos "") L = 0 dx2 L d2y P dx2 EI the solution of which is y=Asin,/(&)x+Bcos,/(&)x+ [(%cos?) / (5 - k)] where A and B are the constants of integration. Now when x = fL/2, y = 0 .. A=B=O .. y= [($cos7)/($-;)] = [(Pcocos~)/(F-P)] T~EI