A Method for Determining the Ultimate Strength of a Column Swaged at Ends

Abstract

Let ABC, Fig. 2, be a pin ended strut of length La deflected under its Euler load . Let DBE be a pin ended strut with a smaller moment of inertia, a shorter length Lb, but with the same ultimate column strength, deflected under its Euler load . It is assumed that the ends of the two struts lie on the straight line which is the line of action of the force P and that they are so deflected by this force that their elastic axes are tangent at B. On page 564, Vol. II , of Strength of Materials by Timoshenko, it is shown that at any point the deflection of a strut fixed at one end and with a sufficiently large axial load applied at the other end is given by the equation y=5(l —cos Kx). The origin for and x is at the fixed end and 6 is the maximum deflection which occurs a t the free end. For convenience the deflection at any point may be measured from the line of action of the force which passes through the free end and is parallel to the x axis: d — y = 5cosKx. It is evident that the fixed ended strut may be considered to be half of a symmetrical pin ended strut and that the value, d cos Kx, yields the ordinates to the elastic curve of a pin ended strut when the origin is at the intersection of a line drawn through the ends and the plane of symmetry. If the origin is a t the end of the strut, the value of the ordinate to the elastic curve at any point measured from the line of action of the force P through the ends becomes 5 sin Kx. AtB, Fig. 2, we have the deflection equal to