The strain values in the steel cross section are incrementally increased, until the section reaches its full plastic capacity, as shown in Fig. 4(b). Elements with compressive residual stresses typically reach yielding before elements with tensile residual stresses. Eventually, all elements reach the yield strength. At this point, the effective moment of inertia of the steel cross section and the buckling load become zero, based on Equations 9 and 10. Consequently, a value for the lateral displacement Scan no longer be obtained using Eq. 5.
For FRP material in the compression side, the failure strains are limited by debonding and crushing at a strain value of 0.0013 mm/mm. After excessive overall buckling, the FRP on the outer surface may be subjected to some tensile strains, which are well below the ultimate value.
Generation of Full Responses
In order to obtain the load-lateral displacement (P-8) response in compression, the procedure can be summarized as follows:
1. Assume a value of the extreme compressive strain £and a neutral axis depth c (Fig. 4(b)).
2. For each element of the steel cross section, calculate its strain value £i, using Eq. 1, add the residual strain Ers as given by Eq. 2, and compare it to the yield strain value Ey to check whether the element has yielded or not.
3. Calculate the corresponding stress of each element, using the steel stress-strain curve. For ei+rs > Ey, the stress is limited to Fy.
4. Calculate the strain Et, using Eq. 1, and corresponding stress for each FRP element, based on linear stress-strain response. Compare the strain to ultimate values to check for failure of FRP.
5. Calculate the axial load P and bending moment M for the entire cross section, by using Equations 3 and 4, which are essentially a summary of steps 3 to 5 above.
6. Calculate the eccentricity e = M/P, which is induced by the non uniform stress distribution.
7. Calculate the cross sectional transformed effective inertia It, excluding both the yielded steel elements and failed FRP elements, using Eq. 12.
8. Calculate the critical load Pcr using the conservative approach described by Eq. 11 (or alternatively using the more accurate approach by using Eq. 10). Pcr is used to calculate the lateral displacement at mid-length 8 for a prescribed imperfection e’ using Eq. 5.
9. Compare the eccentricity e obtained in step 6 with the lateral displacement 8, calculated in step 8. If the two values are different, assume a new value of the neutral axis depth c and repeat steps 2 to 8 until the two values are equal. This will provide one point on the load-lateral displacement curve.
10. Enter a higher value of strain e in step 1 and repeat the process from steps 2 to 9 until the complete load-lateral displacement response is established.
In order to generate an approximate load-axial displacement P-А response of compression members, for an axial load P and corresponding lateral deflection 8, obtained earlier, the following procedure can be followed:
1. The axial shortening term Aa of the displacement is calculated using Eq. 14.
2. For a given lateral displacement 8, establish the deformed sine curve of a buckled member with mid-length amplitude of 8 and an arc length of L.
3. Calculate the chord length of the sine curve S.
4. Calculate the ‘curvature’ component of the axial displacement A b using Eq. 16.
5. The total axial displacement A is calculated using Eq. 13.
6. Repeat the previous steps for each load level P and its corresponding lateral displacement 8, until the complete P-А response is established.