Figure 5(a) shows a pin-ended slender compression member which is slightly curved initially, due to an imperfection e’. At any load level P, the total lateral displacements, measured from the vertical axis is S. (Allen and Bulson, 1980) have shown that the shape of the deflected compression member can be represented by a Fourier series, which can be reduced to the following expression that relates the lateral displacement Sat a distance z along the member’s length to the applied load P:
where L is the length of member and Pcr is the Euler buckling load and is given by:
where EI is the flexural rigidity of a prismatic member, function of Young’s modulus E and moment of inertia I of the member’s cross section. The effective length factor k accounts for boundary conditions and is taken as unity for pin-ended members.
It should be noted that Eq. 5 is valid for lateral displacements of value up to 10% of the length (j < L/10). It is also important to note that Equations 5 and 6 assume linear elastic behaviour of the material and that the residual stresses are not taken into consideration. In the following sections, methods are proposed to account for residual stresses, material non-linearity due to yielding, and the contribution of FRP.