Experimental hysteretic behavior of high strength concrete columns confined by butt-welded closed composite stirrups

3.1. Determination of skeleton curves

In this research to ascertain the restoring force model of high strength concrete columns confined with butt-welded closed composite stirrups, takes into account the effect of the hysteretic features of members, along the lines of the principles of the maximum degree simplification, and according to the relative test data and $P$/$P_y$-$\mathrm{\Delta }$/${\mathrm{\Delta }}_y$ skeleton curves as shown in Fig. 3, the skeleton curves of high strength concrete columns confined by butt-welded closed composite stirrups can be identified to be tri-linear curves as shown in Fig. 4, and the following are the detailed contents.

Fig. 3Non-dimensional skeleton curves

Fig. 4Suggested restoring force model

(1) Elastic stage: The test results show $P$-$\mathrm{\Delta }$ skeleton curves of high strength concrete columns confined by butt-welded closed composite stirrups have no obvious variation point when concrete samples begin to crack. This is contributed by the arrangement of the butt-welded closed composite stirrups having superior reserve action to concrete crack development. Hence, the concrete initial crack has little effect on the rigidity of the columns. Meanwhile, acknowledging the main purpose of elasto-plastic seismic response analysis is to research properties of the members after having entered into the plastic stage, the skeleton curves of high strength concrete columns confined by butt-welded closed composite stirrups before yielding is simplified by drawing a straight line connecting the origin point and yield point together, to obtain the elastic stage rigidity equation: $k_e=P_y/{\mathrm{\Delta }}_y$ ($0A$ in Fig. 4).

(2) Strengthening stage: The skeleton curves after specimen yield have an obvious strengthening stage. The specimen CCSHRC-1 with axial compression ratio $n=$0.28 has the post-yield strengthen factor $P/P_y$ of 1.25, while the specimen CCSHRC-4 with axial compression ratio $n=$0.47 has the post-yield strengthen factor $P/P_y$ of 1.08. Fig. 3 shows that the strengthen factor $P/P_y$ decreased as the axial compression ratio increased. However, the volume-stirrup ratio has great influence on the length of plastic hinge. Strengthening stage of skeleton curves can be simplified to a straight line segment connecting yield point ($P_y$, ${\mathrm{\Delta }}_y$) (point $A$ in Fig. 4) and maximum bearing capacity point ($P_{max}$, ${\mathrm{\Delta }}_p$) (point G in Fig. 4). Theoretically, to determine the strengthening stage the characteristics of the two point’s parameters ($P_y$, ${\mathrm{\Delta }}_y$) and ($P_{max}$, ${\mathrm{\Delta }}_p$) must be considered. However, to calculate ${\mathrm{\Delta }}_p$ we must consider many kinds of deformations such as the bending deformation, shearing deformation, and sliding deformation, the second-order effect, the length of the plastic phase hinges and other factors. Theoretical calculation is relatively discrete and complicated for ${\mathrm{\Delta }}_p$. Therefore, this paper suggests ${\mathrm{\Delta }}_p$ should be determined by empirical method. The polynomial fit method was considered with our test results, the following relationships between length of horizontal projection of the strengthening stages of the non-dimensional skeleton curves ($AG$) divided by the yield displacement ((${\mathrm{\Delta }}_p-{\mathrm{\Delta }}_y$) /${\mathrm{\Delta }}_y$) and it has a one power relationship with the axial compression ratio $n$ and the volume-stirrup ratio ${\rho }_v$_. After the fitting, the formula was obtained:

The slope of the strengthening stage is:

(3) Strength degradation stage: Fig. 4 shows the strength degradation stage ($GM$ line segment) and its rigidity relationships $k_d=\alpha k_e$. From Fig. 3, we concluded that the strength degradation stage of the specimens can be simplified to a straight line segment connecting maximum bearing capacity point ($P_{max}$, ${\mathrm{\Delta }}_p$) (point $G$ in Fig. 4) to the maximum displacement amplitudes point, hence the degradation rate of descending stage increased as the axial compression ratio increases. By analyzing the test results, the rigidity degradation rate of descending stage was determined by Eq. (3):

3.2. Determination of unloading rigidity

Fig. 5 shows the changing law of unloading stiffness degradation rate of the specimens with displacement ductility factor under various axial compression ratio and volume-stirrup ratio. According to the test results, when the displacement ductility factor was four, the stiffness of the specimen CCSHRC-1 with axial compression ratio $n=$0.28 decreased by 17.81 %. Similarly, the stiffness of the specimen CCSHRC-4 with maximum axial compression ratio $n=$0.47 decreased by 26.28 %. This decrease cause amplitude relatively increased by 47.6 %. However the stiffness of the specimen CCSHRC-6 with volume-stirrup ratio ${\rho }_v=$2.22 % decreases by 36.10 %. Likewise, the stiffness of the specimen CCSHRC-4 with the maximum volume-stirrup ratio ${\rho }_v=$3.63 % decreases by 26.28 %, and this decrease causes amplitude relatively decrease by 37.4 %. Nevertheless, based on the regression analysis, in this research, the unloading stiffness of restoring force model of high strength concrete columns confined by butt-welded closed composite stirrups was evaluated using Eq. (4), which was deduced from experimental data. After the fitting, the formula was obtained:

where $K_e$ is the equivalent elastic rigidity $K_e=P_y/{\mathrm{\Delta }}_y$, and ${\mathrm{\Delta }}_y$ is the yield displacement; $n$ is the axial compression ratio of the columns ranged between 0.28$\le n\le$0.46; $\mathrm{\Delta }$ is the corresponding displacement amplitudes of unloading point in the skeleton curves; and ${\rho }_v$ is the volume-stirrup ratio of the columns ranged between 2.22 %$\le {\rho }_v\le$ 3.63 %.

3.3. Paths and reduction in strength under cyclic loading conditions

The reloading strength degradation rate ($\gamma =\mathrm{\Delta }P/P$) in each cyclic loading under the same displacement amplitudes is directly related to the axial compression ratio of the specimens and the existing value of the displacement amplitudes. The strength degradation rate is usually expressed by the ratio of the maximum horizontal loads at the third and the first cycles under certain level displacement amplitudes. The strength degradation rate of every specimen with various axial compression ratios under various ductility factor are shown in Table 4. According to the test results, when displacement ductility factor was four, the strength of the specimen CCSHRC-1 with axial compression ratio $n=$0.28 decreased by 2.70 %, and for the specimen CCSHRC-4 when axial compression ratio increased to 0.47 its strength reduced by 4.66 %. This decrease cause amplitude relatively increased by 72.6 %; and the strength of the specimen CCSHRC-3 with volume-stirrup ratio ${\rho }_v=$2.22 % decreased by 3.36 %, while the strength of the specimen CCSHRC-1 with maximum volume-stirrup ratio ${\rho }_v=$3.63 % decreased by 2.70 %. This decrease cause amplitude relatively decreased by 24.4 %. Therefore, in the restoring force model, the influence of the axial compression ratio and volume-stirrup ratio to the strength degradation rule must be considered. In this research, by means of the statistical analysis, the strength degradation rate under cyclic loading conditions was evaluated using Eq. (5), which was deduced from experimental data. After the fitting, the formula was obtained:

where $\mu =\mathrm{\Delta }/{\mathrm{\Delta }}_y$ is the displacement ductility factor under existing value of the displacement amplitudes ranged between 1 ≤ $\mu$ ≤ 7; and other symbols were the same as above.

Fig. 5Changing law of unloading rigidity degradation rate of the specimens

3.4. Hysteretic loop rule

The following hysteretic loop rule of the restoring force model of the columns was concluded:

(1) The initial sheer force of the members has not exceeded its yield strength, the loading and unloading paths were all along the elastic phase of the skeleton curves ($0A$ in Fig. 4).

(2) The shear force of the members once exceeded its yield strength caused the loading paths to extend going forward along the skeleton curves ($AB$ in Fig. 4). According to the various displacement amplitudes, and the axial compression ratio as well as the volume-stirrup ratio of the columns, the stiffness from the skeleton curves of unloading patterns can be determined by using Equation (4) ($BC$ in Fig. 4).

(3) In consideration to the reverse loading and reloading paths, the initial shear force of the members has not exceeded its yield strength, and the reverse loading paths directly point to the direction of the yield point on the skeleton curves from the corresponding points on the displacement axis after unloading ($CD$ in Fig. 4); however, under reverse loading conditions, when the shear force of the members has exceeded yield point of the skeleton curves, the strength degradation amplitude during reverse loading and reloading can be calculated according to the Eq. (5) from the Fig. 4 (points $G^{\text{'}}$ and $J^{\text{'}}$). For example, by looking at Fig. 4, the maximum point $B\left({\mathrm{\Delta }}_i,P_i\right)$ of the loading phase and the reloading point $G\left({\mathrm{\Delta }}_i,P_i-\mathrm{\Delta }{P}_i\right)$ were determined, and the value for the strength degradation $\mathrm{\Delta }{P}_i$ was calculated.

(4) In certain loading condition, if the displacement amplitudes distributions have not exceeded the maximum displacement amplitudes in the same direction, the unloading stiffness value remained the same. However, under the reverse loading and reloading conditions in comparison to the original loading distribution without any consideration to the strength degradation distribution ($KL$ and $NO$ in Fig. 4) the direction remained the same.