The typical failure modes of specimens were local buckling of steel plates and global outward bulge of steel tube at the concave corners near the mid-height of the column, as shown in Fig. (4). The maximum width-thickness ratios of steel plates of the specimens C1, C4 and C5 ( =27, 22 and 39, respectively) are relatively large among the specimens. The failure modes of them were similar. When reaching the maximum load, global outward bulge of steel tube at the concave corners were found. At the load descending stage, local buckling appeared randomly on the steel plates with relative larger width-thickness ratio, such as the side-α₁ steel plate and the side-b₁ steel plate. Global outward bulge of steel tube at the concave corners were more obvious at the late loading stage, and each leg of the cross-shaped steel tube deformed similarly as a single column which was subjected to eccentric compression. The failure modes of specimens C2 and C3, whose are relatively small among the specimens, were similar with those of the specimens C1, C4 and C5, but the globe outwart bulge at the concave corners was smaller, and no local buckling of steel plate was found till the end.

Fig. (4). Typical failure modes.

The experimental phenomena indicate that the width-thickness ratio of steel plate is a key factor affecting the failure modes of a cross-shaped CFT stub column. On the other hand, as distinguished from the CFT columns without concave corners, the global outward bulge of the steel tube at the concave corners of the cross-shaped CFT columns has significant influences on the mechanical behavior. Besides, increasing the yield strength of steel has little effect on the failure modes.

The curves of load (N) versus average longitudinal strain (ε) of all specimens are shown in Fig. (5). The loads corresponding to the initiation of local buckling (N_b) are marked on the curves, designated as B1 to B5, where the points B2 and B3 are absent because local buckling was not found for the specimens C2 and C3. Table 1 lists the key results on the curves, including the experimental maximum loads and corresponding average longitudinal strains (N_u and ε_u), the loads and corresponding average longitudinal strains at the points B (N_b and ε_b), the ratios N_u/N_un and N_b/N_un, and the ductility indexs DI=0.75×(ε₈₅/ε₇₅) [18]. As shown in Fig. (5) and Table 1, the characteristics of load bearing and deformability of the specimens are as follows.

1. The maximum loads (N_u) of the specimens C3 and C4 are larger than their nominal strengths (N_un), N_u/N_un≥1, respectively. The so-called nominal strength, N_un=f_ckA_c+f_ysA_s, is the theoretical maximum strength of a CFT column in a case that the average stresses of the concrete and the steel tube are f_ck and f_ys, respectively. When the parameters velevant to the longitudinal strength of specimens, such as the thickness and strength of steel tube, are changed, it is more meaningful to investigate the load bearing characteristics by the ratio N/N_un than by the practical load (N) directly. Essentially, when other parameters are the same, there are three factors affecting the ratio of a practical load (N) and the nominal strength (N_un), such as the N_u/N_un and the N₈₅/N_un, where N₈₅ refers to the load when it falls to 85% of the maximum load. Besides, they also affect the value of deformation in these conditions, such as the ε_u and ε₈₅. Take the ratio N_u/N_un for example, firstly, by increasing the confinement effects of the steel tube on the core concrete, the mean strength and the elastoplastic deformation capacity of core concrete are larger than the plain concrete, then the ratio N/N_un is increased, and the average longitudinal strain (ε_u) is increased too. Second, if the stresses of steel tube distribute more uniformly due to a smaller extent of outward bulge, more area of steel tube can reach yield at the peak load, then the ratio N_u/N_un is increased. It also contributes to a larger deformation (ε_u), because the plastic deformation capacity of the integral steel tube section is increased. Third, when the total strength of steel tube which contributes to the bearing capacity of the column is smaller, the ratio N_u/N_un and the relevant deformation (ε_u) are increased. It is because that the strength loss of steel tube caused by a same extent of outward bulge or local buckling is smaller. However, if the extent of outward bulge is effectively reduced and meanwhile the total strength of steel tube is increased, for instance, by increasing the thickness of steel tube, the ratio N_u/N_un is also enhanced. For other loading stage, these three factors also influence the ratio N/N_un and the relevant deformation (ε), only the extent of impact of each factor changes.

   Fig. (5). Load (N) - average longitudinal strain (ε) curves.

   
   For the specimen C3, the width-thickness ratio of the side-α₁ steel plate is the smallest of all specimens, and from the experiment, the outward bulge of steel tube is also the smallest. It gives rise to the largest confinement effects on the core concrete, and the longitudinal stresses of almost the whole steel tube can reach yield at the peak load, as discussed later. As a result, the ratio N_u/N_un reaches 1.17. For the specimen C4, the yield strength of steel is only 239MPa and its thickness of steel tube is relatively small (t=3.72mm), so the total strength of steel tube which contributes to the bearing capacity of the column is the smallest of all specimens. Besides, part of the core concrete near the convex corners is constrained. Both the cases above are beneficial to the increase of ratio N_u/N_un, and the beneficial effects are larger than the adverse effects due to a relatively small t. As a result, for the specimen C4, the ratio N_u/N_un reaches 1.1. For the specimens C1, C2 and C5, the steel tubes have larger contribution to the cross section bearing capacity than the case of specimen C4, and the outward bulge of steel tubes is larger than the case of specimen C3 because the width-thickness ratios of steel tubes are larger. As a result, the ratios N_u/N_un are all smaller than 1.0. Especially, for the specimen C5, the width-thickness ratio of the side-α₁ steel plate is the largest. The ratio N_u/N_un of it is only 0.75 on account of large outward bulge of steel tube.
2. The parameters of specimens C1 and C2 are nearly the same except for the thickness of steel tube, t_C2/t_C1=1.54. The situation is similar between the specimens C3 and C4, t_C3/t_C4=2.08. As shown in Fig. (5), with the increase of t, namely, the decrease of width-thickness ratio of each steel plate, the initial axial rigidity is larger, and the strength-descending rate after the peak load is generally smaller. There is an increase of the ratio N_u/N_un by 8% and an increase of the relevant ε_u by 59%, compared the specimen C2 with the specimen C1, as shown in Table 1. And, they are 6% and 56%, respectively, compared the specimen C3 with the specimen C4. In fact, by increasing t, the flexural stiffness for resisting outward bulge of steel tube is increased, so that the outward bulge of steel tube, especially the areas near the mid-width of each steel plate and the areas near the concave corners, is smaller. As a result, the confinement effects of steel tube on the core concrete are improved. Besides, the stresses on the steel tube distribute more uniformly, so that the relevant total axial strength of the steel tube is increased. These two effects lead to the increases of ratio N_u/N_un and the relevant ε_u.Furthermore, although the difference of thickness between the specimens C3 and C4 (t_C3/t_C4) is larger than that of the specimens C2 and C1 (t_C2/t_C1), the increasing rates of N_u/N_un and ε_u between the specimens C3 and C4 are merely approximative to those between the specimens C2 and C1. The reason is, as discussed before, the total strength of steel tube of the specimen C4 which contributes to the bearing capacity of the column is the smallest of all specimens, so that the ratio N_u/N_un and ε_u of the specimen C4 are both relatively large and more closer to those of the specimen C3. In addition, the ε₈₅ of the specimen C2 is 1.45 times that of the specimen C1, and it is 7.09 times compared the specimen C3 with the specimen C4. It indicates that the load resistant capacity at the post-peak stage is increased by increasing t. Furthermore, as distinguished from the ε_u, the increasing rates of ε₈₅ are considerably different for the specimens with varied yield strength of steel and varied increasing rate of t. As it can be seen, the increasing rate of ε₈₅ is smaller than that of ε_u, compared the specimen C2 with the specimen C1, t_C2/t_C1=1.54. But, it is significantly larger than that of ε_u, compared the specimen C3 with the specimen C4, t_C3/t_C4=2.08. It is because the trend of outward extrusion of core concrete and the trend of outward bulge of steel tube in the post-peak stage are extremely larger than those in the pre-peak stage. In the post-peak stage, it has a larger demand on the flexural stiffness of steel tube for resisting outward bulge and preventing large strength loss. Besides, with larger yield strength of steel, the strength loss of the specimen C2 caused by outward bulge of steel tube in the post-peak stage is larger, so that the increasing rate of ε₈₅ compared with the specimens C2 and C1 should be smaller. Similarly, although the indexes ε_u and ε₈₅ that reflected the deformability are both increased by increasing t, another index, the ductility index DI, does not change in the same way. The DI, which is essentially the ratio of ε₈₅ and ε₇₅, is increased by 4.57 times compared the specimen C3 with the specimen C4, but it is 0.91 times compared the specimen C2 with the specimen C1. It means that the ductility can only be increased in case that the increasing rate of t is large enough. It is because that the increasing rates of ε₇₅ and ε_u by increasing t are approximative according to the experimental data, but the increasing rate of ε₈₅ only becomes larger than that of ε_u by increasing a large amount of t, such as the case between the specimens C3 and C4.
3. The difference between the specimens C2 and C5 is the sectional dimension (k=α₁/b₂=b₁/a₂). As shown in Table 1, for the specimen C5 (k=2.25), the ratio N_u/N_un that is related to the confinement effects of concrete and axial strength of steel tube, and the values of ε_u, ε₈₅ and DI that are related to the deformability, are all smaller than those of the specimen C2 (k=1). It is because that the width-thickness ratios of the steel plates on the side-α₁ and side-b₁ increase by increasing k. Besides, the strain that local buckling of the specimen C5 appeared (ε_b) is the smallest of all specimens.
4. The parameters between the specimens C1 and C4 are similar except that the yield strength of steel (f_ys) of the specimen C1 is 1.46 times that of the specimen C4. As shown in Fig. (5), no influence on the initial axial rigidity of specimen can be found by varing the f_ys. As shown in Table 1, the values of N_u, N_u/N_un, ε_u, ε₈₅, and DI of the specimen C1 are 0.96, 0.81, 0.94, 1.15 and 1.22 times those of the specimen C4, respectively. By increasing f_ys, the transverse stresses of steel tube become larger at the peak load. It is beneficial to increase the confinement effects on core concrete. As a result, logically, it seems that the values of N_u, N_u/N_un, ε_u, ε₈₅, and DI should all become larger. However, other situations should be considered. First, increasing f_ys has no influence on the flexural stiffness of steel plate. It means that, for the specimens C1 and C4, the extent of outward bulge of steel tube and the relevant stress distribution pattern should be similar. Second, by increasing f_ys, the strength loss of steel tube caused by a same extent of outward bulge should be larger, because the axial strength of steel tube that contributes to the cross section bearing capacity becomes larger. Synthesize the adverse effect on the strength loss of steel tube and the beneficial effect on constraining the concrete, by increasing f_ys in a certain amount, such as that in this experiment, the values of N_u, N_u/N_un and ε_u may inversely become smaller. In addition, the outward extrusive defomation of concrete in the post-peak stage is larger. So, in the post-peak stage, the steel tube with a larger f_ys can provide larger transverse stresses to constrain the concrete than those in the pre-peak stage. Then, by increasing f_ys, the indexes ε₈₅ and DI related to the load resistant capacity in the post-peak stage may become larger.

The width-thickness ratio of the side-α₁ steel plate in the specimen C5 is the largest of all the specimens. As a typical example, the strain distribution characteristic on the side-α₁ steel plate in the middle height of the specimen C5 are investigated. The positions of wire strain gauges are shown in Fig. (6). The curves of normalized loads (N/N_u) versus longitudinal strains (ε_L) and transverse strains (ε_H) on the side-α₁ steel plate of the specimen C5 are shown in Fig. (7) and Fig. (8).

Fig. (6). Positions of strain gauges of the specimen C5.

Fig. (7). N/Nu-εL relationship curves of the specimen C5.

Fig. (8). N/Nu-εH relationship curves of the specimen C5.

The average longitudinal strains (ε) of all specimens, achieved from the readings on LVTDs, have reached yield at the peak load, as shown in Fig. (5). However, not all the longitudinal strains on the steel plate, achieved from the readings on strain gauges, can reach yield, as shown in Fig. (7). Besides, the compressive strains increase from the convex corner to the concave corner given by 6#<8#<10#. It is because that the areas near the concave corners tend to bulge outward integrally under the outward extrusion of core concrete. It makes each leg of the cross-shaped steel tube deform similarly as a single column subjected to eccentric compression, and the global outward bulge near the convex corner of each leg is the largest, as shown in Fig. (4). As a result, the longitudinal strains of steel tube cannot distribute uniformly. Furthermore, from the test results of other specimens, the longitudinal strains distribute more unevenly along with the increase of global outward bulge, and more area of steel plate cannot reach yield.

As shown in Fig. (8), when reaching the maximum load, all the transverse strains have not reached yield. It reflects that the confinement effects on the core concrete at the peak load are small. However, it can be seen, in the post-peak stage, the transverse strains develop faster due to larger extrusion deformation of concrete. On the other hand, the global outward bulge of steel tube near the concave corner is so obvious that ε_H of 7# and 9# develop faster than that of 5#, especially in the post-peak stage.

Table 2 gives the comparison between the maximum strengths obtained by the methods in the codes and the experimental maximum loads (N_uc/N_u). As shown in Table 2, the experimental results of the specimens C1, C2 and C5 are obviously overestimated by the methods in the codes GJB and DBJ to at least 10%, while the evaluated strengths of the specimens C3 and C4 are relatively closer to the experimental results, especially for the specimen C3. The methods in the codes GJB and DBJ take into account the confinement effects of square or rectangle steel tube on the core concrete by applying magnification factors to the axial strength of the whole composite section. They require smaller limits of the width-thickness ratio of steel plate than those in other codes. For example, the should be smaller than 40 according to the code GJB, while it is equal to 52 in the code EC4. The purpose of stipulating smaller width-thickness ratio limit is to prevent large outward bulge or local buckling on the mid-span region of each steel plate in the square or rectangular CFT columns, so that more area of the steel tube can reach yield and the confinement effects on the core concrete are larger. Furthermore, the magnification factors considered in the code DBJ are larger than those in the code GJB.

Theoretically, for the cross-shaped CFT columns, if the confinement effects on the concrete and the stress distribution condition of steel tube are similar to those of square or rectangular CFT columns, the maximum strength can be evaluated by the codes GJB and DBJ. However, although the maximum width-thickness ratios of steel plates of all specimens ( = 11~39) do not exceed the limits that stipulated in the codes GJB and DBJ, but only the experimental results of the specimen C3 and C4, with the width-thickness ratios only 27% and 54% of the limit, respectively, are approximate to the predicted results. In fact, at the peak load, the global outward bulge of cross-shaped steel tube near the concave corners has key and significant effects on the stress distribution pattern of the steel tube and the confinement effects on the concrete, because in the test, the outward bulge or local buckling that occurred near the mid-span region of each steel plate was not obvious in the pre-peak stage. Under the transverse tensile stresses of the side-α₁ and side-b₁ steel plates, the concave corner of steel tube is subjected to an outward resultant force. So the edge intersected by the side-α₁ and side-b₁ steel plates can rotate and deform outward, and cannot be seen as a strict restrained boundary of the steel plates. The boundary conditions of the side-α₁ or side-b₁ steel plate are more like being restrained elastically at the edge near the convex corner and being freedom at the edge near the concave corner. It is different from the boundary conditions of steel plates in a square or rectangular CFT column, which are restrained elastically at the both edges [24]. As a result, to restrict the global outward bulge at the concave corners and the local buckling in the mid-span of steel plates, the width-thickness ratio limit of the cross-shaped CFT columns should be smaller than that of the square or rectangular CFT columns, such as the case of the specimen C3.

Additionally, the adverse effects of outward bulge augment when the contribution of axial strength of steel tube to the bearing capacity is larger. For example, although the of the specimen C2 is smaller than that of the specimen C4, but its yield strength of steel is larger. Then the predicted results from the codes GJB and DBJ are unsafe for the specimen C2. Actually, for the specimen C4, although the differences between the N_uc and the N_u are less than 10%, but the maximum load in the experiment is somewhat overestimated, such as that the values of N_uc/N_u from the codes GJB and DBJ are 1.905 and 1.101, respectively. It is because that the thickness of steel plate of the specimen C4 is somewhat small. On the whole, for the cross-shaped CFT columns, conservatively, the methods in codes GJB and DBJ are suitable to evaluate the maximum strength of the columns with obvious confinement effects and with small loss of strength of steel tube caused by outward bulge, namely, with small width-thickness ratio of steel plate and small yield strength of steel, f_ys≤261MPa and ≤11, like the specimen C3.

The methods in the codes ACI and EC4 adopt the superposion method by simply combining the total yield strength of steel tube with the total axial strength of plain concrete. In fact, the theoretical value predited by the method in code EC4 is nearly the same as the nominal strength (N_un), except that the cylindar axial compressive strength of concrete (f_c^') is adopted in the fomula of code EC4, f_c^'=0.79f_cu,k. For the code ACI, an additional reduction coefficient 0.85 is adopted to decrease the strength of core concrete, so the calculated values are the smallest of the four codes. However, the strength reduction coefficient about core concrete cannot actually reflect the mechanical behaviour of cross-shaped CFT column. As shown in Table 2, both the standard deviation values of the N_uc/N_u, achieved from the codes ACI and EC4, are large. The method in code EC4 is suitable to evaluate the maximum strength of the columns without obvious confinement effects and with small loss of strength of steel tube, namely, with relative larger width-thickness ratio of steel plate than that of the specimen C3 and with small yield strength of steel, f_ys≤239MPa and ≤22, like the specimen C4. The method in code ACI, numerically, is suitable to evaluate the maximum strength of the columns without confinement effects and with large loss of strength of steel tube, namely, with relative large width-thickness ratio of steel plate and with large yield strength of steel, f_ys≤348MPa and ≤27, like the specimens C1 and C2. For the specimen C5, the yield strength of steel is large, f_ys=346MPa, and the width-thickness ratio of side-α₁ steel plate is the largest, =39, but it is still smaller than the limits required in the codes. However, the values of N_uc/N_u are 1.3~1.6, namely, its strength is significantly overestimated by all the methods in design codes.

By considering the confinement effects of the cross-shaped steel tube on the concrete, together with the loss of longitudinal strength of steel tube caused by outward bulge, a method for predicting the maximum strength of cross-shaped CFT columns with symmetrical section is proposed.

(1)

where A_si and A_c are the area of each steel plate and the area of core concrete, respectively; ϕ_si is the strength reduction coefficient of each steel plate; i is the label of each steel plate, i=1 denotes the typical steel plates in a symmetrical cross-shaped section which are composed of the side-α₁ and side-b₁ steel plates, and i=2 denotes the other typical steel plates, such as the side-α₂ steel plate; ϕ_c is the strength magnification coefficient of core concrete used to consider the confinement effects.

The method raised by Ge [4] to calculate the value of ϕ_si of the square or rectangular CFT columns is used for reference. It is suitable for the side-a₂ steel plate directly. For the side-α₁ and side-b₁ steel plates, as discussed above, their boundary conditions are more like being restrained elastically at the edge near the convex corner and being freedom at the edge near the concave corner. So, it can be simply assumed that these two steel plates deform together as one steel plate, even there is an angle between them. According to Ge [4], when R_i≤0.85, local buckling will not occur before reaching the maximum load, then, where R_i is the coefficient about width-thickness ratio of each steel plate, calculated by

(2)

where v is the Poisson’s ratio of steel; B_i is the width of each steel plate, especially, the side-α₁ and side-b₁ steel plates are assumed as one steel plate. When R_i>0.85, the loss of longitudinal strength of steel tube due to outward bulge or local buckling should be considered. ϕ_si can be calculated by

(3)

ϕ_c is obtained by regression analysis method from the experimental data in this paper, and it is expressed by

(4)

where , it is called the confinement coefficient adopted in the code GJB.

As shown in Table 3, most of the experimental results are well predicted by the proposed method, except that of the specimen C5. The N_uc/N_u of the specimen C5 is equal to 1.153. It is somewhat overestimated, but it is already far more close to the experimential results than those predited by the design codes. Conservatively, the proposed method can be adopted to accurately predict the maximum strength of the columns with ≤30 and f_ys≤345MPa.