Improvement of seismic performance of self-centering mid-rise RC frames by adding semi-rigid rocking columns

Abstract

In recent years, increased emphasis has been placed on the self-centering performance of buildings due to the elimination of plastic hinges and a significant reduction in residual deformations, which lowers the cost of post-earthquake repairs and thus allows the building to operate continuously. Previous studies primarily focused on self-centering low-rise reinforced concrete (RC) frames because the effects of higher modes and flexural deformations are more pronounced in medium- and high-rise frames. These effects result in non-uniform height deformation of the frame; in other words, the deformation pattern in buildings' height due to higher modes effect is curved rather than linear. This phenomenon causes problems with self-centering performance and compromises the structure's stability. On the other hand, the reduction of residual drift in mid-rise and high-rise structures is more important than in low-rise structures due to construction’s difficulties in rehabilitation after an earthquake. With these interpretations, the need for self-centering action in medium- and high-rise structures is high. This study introduced columns with a high flexural stiffness as semi-rigid elements in numerical models of self-centering RC frames with 5 and 8 stories. Steel angles are used in the beam-column joints as an energy-dissipating device in self-centering frames, and the column-base connection and post-tensioning tendons are also passed through the cross-section of the beam and column for self-centering performance. For the first time, the behavior of self-centering RC frames with and without semi-rigid columns was investigated with incremental dynamic analysis (IDA). According to IDA results of numerical models in OpenSees software, using semi-rigid columns in self-centering RC frames without increasing the frame's overall rigidity reduces the negative effects of higher modes and flexural deformations. In addition, this study shows that using semi-rigid columns in the self-centering RC frames leads to a more uniform story drift ratio. This effect is more significant in the 8-story frame than in the 5-story frame.

Appendix

Based on the previous study by Lu et al. (2019), this part explains the design method in the second phase of the design. After the preliminary design and the determination of the details of the beams and columns, as well as obtaining the elastic moment and elastic shear of the joints, the following procedure is applied:

Using Eq. (2), the value of $x$, which is the height of the tension block in the concrete section, is calculated.

$$M_{y} = \alpha f_{c} bx\left( {h - x} \right)/2 + A_{s} f_{s,y} \left( {h/2 - a_{s} } \right)$$

(2)

where $M_{y}$ = the largest elastic moment available in the beam-column connections recorded from preliminary design, α = parameter of the relationship between concrete axial strength and stress of equivalent stress block, $f_{c}$ = compressive strength of the concrete, $x$ = height of the rectangular stress block, $h$ = height of the cross section, $A_{s}$ = area of longitudinal bars subjected to compressive force only, $f_{s,y}$ = yield strength of the longitudinal bars, $a_{s}$ = distance between force resultant of longitudinal bars and compressive edge.

Using Eq. (3) and the value of $x$, the initial force of the tendons in the beams is calculated as,

$${\upalpha }f_{c} bx + A_{s} f_{s,y} - P_{0} = 0$$

(3)

where $P_{0}$ = initial PT force, $b$ = width of section.

The cross-sectional of the tendons is obtained using the yield stress of steel ($f_{y}$) and the calculated initial force ($P_{0}$). It should be noted that the yield stress of the post-tension steel is considered to be 0.6 $f_{y}$ in the calculations.

Shear resistance of self-centering connection is provided by friction force. To control whether the initial force of the tendon provides this shear resistance or not, Eq. (4) must be satisfied.

$$N + P_{0} \ge \left( {V_{G}^{d} + V_{e}^{d} } \right)/\mu_{f}$$

(4)

where N = axial load due to gravity, which is usually zero in the beam-column joint, $V_{G}^{d}$ = elastic shear due to gravity load, $V_{e}^{d}$ = elastic shear due to earthquake load, $\mu_{f}$ = the friction coefficient between contact interfaces.

All the previous steps are repeated to calculate the initial force of the tendons of the columns.

To determine the base shear demand of the structure in the target displacement, the same method as the displacement-based design is used. For this purpose, by an equivalent single degree of freedom, the base shear equivalent to this target deformation is calculated.

In the deformation of prefabricated members, the rotation of the rigid body is the dominant mechanism. This is a reason to assume that the displacement pattern of the self-centering frame should be linear along the height. By using the target drift of 2.5%, and Eq. (5), the displacement of each floor can be calculated.

$$\Delta_{i} = h_{i} \theta_{d}$$

(5)

where $h_{i}$ = elevation of each floor, $\theta_{d}$ = design drift ratio, $\Delta_{i}$ = displacement of the ith floor.

Based on Eq. (6), the design displacement of the equivalent single degree of freedom is calculated as,

$$\Delta_{d} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {m_{i} \Delta_{i}^{2} } \right)}}{{\mathop \sum \nolimits_{i = 1}^{n} \left( {m_{i} \Delta_{i} } \right)}}$$

(6)

where, $m_{i}$ = mass of each floor, $\Delta_{d}$ = the design displacement of the equivalent single degree of freedom.

In the ASCE 7–16 code, the acceleration spectrum with a damping ratio of 5% will be converted into a displacement spectrum ($\Delta \left( {T,\xi } \right) = S\left( {T,\xi } \right)T^{2} /\left( {4\pi^{2} } \right)$), and with an equivalent displacement ($\Delta_{d}$), the equivalent period ($T_{eq}$) can be obtained.

The mass of the equivalent single degree of freedom ($m_{eq} )$ is calculated by Eq. (7).

$$m_{eq} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {m_{i} \Delta_{i} } \right)}}{{\Delta_{d} }}$$

(7)

The stiffness of the equivalent single degree of freedom ($K_{eq}$) is calculated by Eq. (8).

$$K_{eq} = \frac{{4\pi^{2} m_{eq} }}{{T_{eq}^{2} }}$$

(8)

And finally, the required base shear (demand base shear = $V_{B}$) is calculated using Eq. (9).

$$V_{B} = K_{eq} \Delta_{d}$$

(9)

Modeling of the frame is done using the initial design information as well as the post-tension force of the tendons. With the initial assumption, steel angles are also modeled. The cyclic analysis is performed as a single cycle with a target drift of 2.5%. The following items are controlled based on the analysis results.

• 1. The performance of the self-centering of the frame is controlled. If it is not established, steel angles with less strength should be considered so as not to disturb the self-centering performance, or the diameter of the tendons is increased.

• 2. The base shear of analysis should be larger than the demand base shear ($V_{B}$). If it is not satisfied, the diameter of the tendons can be increased and at the same time steel angles with higher strength can be replaced.

The mentioned method is a trial-and-error procedure, and it is worth noting that the above two conditions must be met simultaneously.