Force Equilibrium and Moments

Figure 4(b) shows a cross-section at mid-height of a concentrically loaded slender compression member. Due to overall buckling, the axial force is eccentric with respect to the mid-height section. For a given strain gradient induced by the external eccentric axial load P, which is based on a strain level e at the extreme compression side and neutral axis depth c, the strain Ei in each element i located at a distance yi from the centroid can be determined as follows:

where h is the depth of the section. The strain Ei is then added to the residual strain Ers to obtain the total strain Ei + rs for the steel elements and check whether the element has yielded or not:

Ъ+rs =e, +є„ (2)

Possible stress distributions at various stages of loading are shown in Fig. 4(b). The total axial load at this loading level, for a given e and c, can be obtained by numerical integration of stresses over the cross section, for both the yielded and elastic elements as well as the FRP elements, as follows:

P = ‘L{f. E.A:)+ iMJ+ZMfAi) (3)

elastic steel plastic steel FRP

and the corresponding moment M is:

M = YfeiE*A*tyt)+ YXFzAslZi )+Zfe ЕЛу) (4)

elastic steel plastic steel FRP

where As and Af are the areas of steel and FRP elements, respectively, and Esi and Ef are Young’s moduli of steel and FRP elements, respectively.