$2.1 Struts 31 The above comments and the contents of this chapter refer to the elastic stability of struts only.It must also be remembered that struts can also fail plastically,and in this case the failure is irreversible. 2.1.Euler's theory (a)Strut with pinned ends Consider the axially loaded strut shown in Fig.2.1 subjected to the crippling load Pe producing a deflection y at a distance x from one end.Assume that the ends are either pin-jointed or rounded so that there is no moment at either end. y Fig.2.1.Strut with axial load and pinned ends. B.M.at C=E/ d2y dx2 =-Pey d2y xz+Pey=0 EI d2y Pe Eiy÷0 i.e.in operator form,with D=d/dx, (D2 +n2)y=0.where n2=Pe/EI This is a second-order differential equation which has a solution of the form y A cos nx +B sinnx i.e. Now atx=0,y=0 .A=0 and atx=L,y=0 .B sin L√/(Pe/E)=0 either B=0 or sin =0 If B=0 then y =0 and the strut has not yet buckled.Thus the solution required is ny()-0/()= π2E1 P= (2.1)52.1 Struts 31 The above comments and the contents of this chapter refer to the elastic stability of struts only. It must also be remembered that struts can also fail plastically, and in this case the failure is irreversible. 2.1. Euler’s theory (a) Strut with pinned ends Consider the axially loaded strut shown in Fig. 2.1 subjected to the crippling load P, producing a deflection y at a distance x from one end. Assume that the ends are either pin-jointed or rounded so that there is no moment at either end. Fig. 2.1. Strut with axial load and pinned ends. .. d2 Y B.M. at C = EI- = -Pey dx2 EI-+Pey=O d2 Y dx2 .. Le. in operator form, with D d/dx, (D2 + n2)y = 0, where n2 = PJEI This is a second-order differential equation which has a solution of the form y = A cosnx + B sin nx i.e. Now at x = 0, y = 0 and at x = L, y = 0 Y = A cos ,/( $)x + B sin ,/( $)x :. A = 0 :. B sin L,/(Pe/EZ) = 0 .. either B = 0 or sinL If B = 0 then y = 0 and the strut has not yet buckled. Thus the solution required is 2EZ P, = - L2