Verification of Model and Parametric Study

The model described in the previous section has included several features, namely the P-8 effect (non­linear geometry), plasticity (yielding) of the steel section, the through-thickness residual stresses in the HSS section, the initial imperfection, and failure criteria of FRP in compression. In order to illustrate the significance of these features, the load-lateral displacement response of one specimen retrofitted with three CFRP layers, on two opposite sides, (3L-2S) has been predicted for five different cases. In case 1, Equations 5 and 6 have been used in their original form, assuming linear elastic materials (ignoring steel yielding and residual stresses). In case 2, plasticity of steel is considered, however, residual stresses are ignored. In case 3, both steel plasticity and residual stresses are accounted for but the failure criteria of FRP in compression is not applied (FRP is assumed fully intact in the compression side throughout the full response). In case 4, all the features of the model are applied, including the effect of variable inertia along the length, which is accounted for in calculating the buckling load, using the finite difference approach as shown in Fig. 6(b). Case 5 is similar to case 4, except that the simplified approach for calculating the buckling load is used, as shown in Fig. 6(c).

Figure 7 shows the experimental and analytical responses for the five cases. All predictions were made assuming initial imperfection (e’ = L/500), which is the limit permitted by Canadian standards (CAN/CSA-S16-01). The effect of initial imperfection itself will be discussed later. Figure 7 clearly shows that ignoring the steel plasticity (case 1) would highly over estimate the axial strength. Ignoring the residual stresses (case 2) would also overestimate the load at which transition occurs from the elastic to plastic response. Also by assuming that FRP is fully effective in compression (case 3), the ultimate load is somewhat overestimated. It is, therefore, clear that cases 4 and 5 represent the most accurate predictions, using the full capabilities of the model. It is also clear that the simplified conservative approach used in case 5 is quite reasonable and would, therefore, be used in this paper for the predictions of all responses and the parametric study, next.


Specimen 3L-2S (e’ – U500) Euler Load = 691 kN


____ о—————— о-*-

, Case 1


a. § ____________ ^

£ 1

•в в Jr Case 2 Case 3

z 400 X>

з s Case 4 _ 3 « o _ у Case 5

g 300 _1

f Case 1 Linear elastic HSS section


9 Case 2 Elasto-plastic HSS section + No residual stresses

f Case 3 Same as case 2 ♦ residual stresses


Я Case 4 Same as case 3 + FRP fails in compression ♦ variable

и (El) along length

0 (

У Case 5 Same as case 4 but simplified const. (El) along length

0 5 10 15 20 25 30 35 40 45

Lateral displacement (mm)

Fig. 7 Significance of various features of analytical model

The load versus lateral and axial displacements of the five specimens have been predicted and are presented in Figures 8(a) and 8(b), respectively. A summary of the predicted and experimental values is also presented in Table 1. The predictions are made for three different levels of initial imperfections, namely L/300, L/500, and L/1000. It is expected that most steel sections will practically have imperfection values less than L/500 (CAN/CSA-S16-01). The model shows reasonable agreement with test results for this common level of imperfections. However, for specimen 5L-2S, the behaviour was not accurately predicted since the maximum lateral displacement, measured experimentally, occurred near the quarter length point, rather than at mid-length. Also, for the same reason, this specimen did not show higher gain of strength compared to other specimens, with less number of CFRP layers. Figures 8(a) and 8(b) clearly emphasize the important effect of initial imperfection. The load-axial strain behaviour at two opposite sides of each specimen has also been predicted in Fig. 8(c), using the imperfection value that showed the best results for each respective specimen.

It has been shown that the model is capable of predicting the various responses of axially loaded slender HSS members retrofitted with CFRP sheets. It was also shown earlier, in the experimental results, that no clear correlation was established between the number of CFRP layers and strength gains, due to the variability of imperfections among the test specimens. In the following section, the model is used to study the sole effect of the number of CFRP layers, by fixing the level of imperfection. This study builds on the same specimens used to verify the model. The responses of control, 1L-2S, 3L-2S, and 5L-2S specimens are estimated twice, using two fixed values of imperfections, namely L/300 and L/1000, as shown in Fig. 9.