§21 Struts 33 N.B.-It is always convenient to arrange the diagram and origin such that the differential equation is achieved in the above form since the solution will then always be of the form y A cos nx+B sin nx +(particular solution) The particular solution is a particular value of y which satisfies eqn.(2.2),and in this case can be shown to be y =a. y A cos nx B sin nx+a Now whenx=0,y=0 A=-a when x =0,dy/dx =0 B=0 y =-acosnx +a But whenx=L,y=a a=-acosnL+a 0=cosnL The fundamental mode of buckling in this case therefore is given when nL = xEI or Pe= 4L2 (2.3) (c)Fixed ends Consider the strut of Fig.2.4 with the origin at the centre. Fig.2.4.Strut with fixed ends. In this case the B.M.at C is given by dx2=M-Py d2y P M dx2+Ely=Ei (D2+n2)y=M/EI$2.1 Struts 33 N.B.-It is always convenient to arrange the diagram and origin such that the differential equation is achieved in the above form since the solution will then always be of the form y = A cos nx + B sin nx + (particular solution) The particular solution is a particular value of y which satisfies eqn. (2.2), and in this case can be shown to be y = a. .. Now when x = 0, y = 0 .. A = -a when x = 0, dyldx = 0 .. B=O .. y=-acosnx+a But when x = L, y = a .. a = -acosnL +a y = A cosnx + B sinrzx + a 0 = cosnL The fundamental mode of buckling in this case therefore is given when nL = in. or ~EI P, = - 4LZ (c) Fixed ends Consider the strut of Fig. 2.4 with the origin at the centre. Fig. 2.4. Strut with fixed ends. In this case the B.M. at C is given by (2.3) d2y P A4 dx2 El (D2 + n2)y = M/EI