This article was submitted to Earthquake Engineering, a section of the journal Frontiers in Built Environment
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Over the past two decades, precast concrete members have been utilized in seismically resilient structures. In developing these structures, different techniques have been used for connecting the precast members to the foundation. In building construction, unbonded post-tensioning (PT) tendons can anchor a precast wall to the foundation, resulting in the so-called rocking wall system. The rocking wall system develops a dry connection with the foundation and provides moment resistance by means of the PT tendons. The PT tendons remain elastic when the wall is subjected to design-level ground motions to preserve the re-centering capability of the wall. Moreover, the structural damage is concentrated near the wall toes and can be minimized with proper detailing of the toes. Rocking wall systems can consist of a Single precast Rocking Wall (SRW), which uses no supplemental damping, or walls with supplemental damping in the form of viscous or hysteretic energy dissipating devices. In addition to the supplemental damping, rocking walls dissipate the seismic energy through their impacts on the foundation base, their inherent viscous damping, and the hysteresis of concrete near the wall base. While the investigation of rocking walls continues to gain interest, there is no widely accepted means of modeling their dynamic behavior. This paper investigates two popular approaches for modeling rocking walls with and without supplemental damping: the finite element method and analytical modeling. The ability of the two approaches to capture the local and global responses of the walls is evaluated against shake table tests of walls with multiple-level intensity base motions. Next, the behavior characteristics of the two modeling approaches and their ability to simulate impact damping are discussed.
Single rocking wall (SRW) (
During the rocking motion, the PT tendons remain elastic and damage of the wall due to concrete nonlinearity is limited to the bottom toes of the wall, as shown in
To improve the hysteretic damping of SRWs, different wall systems have been developed utilizing supplemental damping devices. For example,
PreWEC (PREcast wall with end columns) system:
In the above-referenced studies, the lateral behavior of the walls was characterized through quasi-static testing, which inhibited the energy dissipation produced by viscous damping and wall impacts on the foundation. Though dynamic tests were also performed (e.g.,
More recently, research studies used free vibration and shake table tests of walls to investigate the dynamic sources of energy dissipation. In
However,
Following the experimental observations, researchers attempted to reproduce the behaviors of the different wall systems using modeling approaches. One approach was the use of existing finite element platforms with frame element models in the form of distributed or lumped plasticity models or with three-dimensional models (e.g.,
To improve modeling of impact damping in the finite element method, researchers utilized the numerical damping generated by the Hilber-Hughes-Taylor (HHT) time-integration scheme, which was shown to dissipate seismic energy in short-time intervals, thus closely simulating the impact damping (
This paper is an effort to understand and compare the accuracy of the finite element and analytical models for simulating unbonded post-tensioned precast concrete walls with and without supplemental damping. For this purpose, the finite element model (FEM) developed by
A total of eight unbonded post-tensioned precast concrete walls were tested using the National Earthquake Engineering Simulation (NEES) shake table facility at the University of Nevada, Reno by
From the above-referenced database, the shake table test results of one SRW, namely SRW2, and one PreWEC, namely PreWEC-2, were used in the present research study to compare with the FEM and the GDAA. The SRW2 and PreWEC-2 had an average shear resistance near 175 kN. The key design variables of the SRW2 and the PreWEC-2 are noted in
Summary of design variables for SRW2 and PreWEC-2.
Wall ID | Post-tensioning parameters | Total number of O- connectors, damping ratio at design drift (%) | Shear resistance at 2% drift (kN) |
||
---|---|---|---|---|---|
No., dia.(mm) of PT tendon | Initial PT stress (MPa), force (kN) | ||||
Design | Measured | ||||
SRW2 | 6, 12.7 | 0.64 |
0.5 |
N/A | 206/187 |
PreWEC-2 | 3, 15.2 | 0.56 |
0.53 |
8, 15.3% | 182/183 |
^{a}Following the SA method (
Fabrication and design details of the wall panels.
Shake table test setups at UNR:
The seismic performance of the SRWs and PreWECs was evaluated by exciting the shake table into a series of ground motions with different levels of intensities. The present study employed the following two shake table excitations:
A short duration motion, namely
A long duration motion, which corresponds to the earthquake record at the 1995 Kobe-Takatori station, with the PGA of 0.62 g.
The amplitudes of these records were scaled by a factor of 2.88 and 3.6, respectively, to meet the design basis (EQ-III) and maximum considered (EQ-IV) earthquake events for the scaled walls. The time step of the ground motion records was decreased by a factor of 5/18 to meet the scale. More information about the scaling procedure, the acceleration response spectra of the scaled ground motions, and the recorded shake table motions are presented in
Two approaches were employed for modeling the experimental responses of the SRW2 and PreWEC-2. The first approach is a Finite Element Model (FEM) that was developed using OpenSees (
The second approach is an analytical model, named in this research study as the Generalized Dynamic Analysis Approach (GDAA). This model was developed by
The Finite Element Model (FEM) was developed to simulate the lateral seismic responses of SRWs and PreWECs for the purpose of performance-based analysis up to lateral drift ratios of 4–5%, within which the walls experience little damage at their bottom toes and the material nonlinearity in the PT tendons is limited. A schematic view of the FEM as developed in OpenSees (OS) is presented in
Finite element model (FEM):
In addition to the hysteretic damping in the moment-rotation curve of the wall base and the O-connectors, a tangent stiffness proportional Rayleigh damping was included to simulate the impact energy dissipation. The assigned damping values were 3 and 2%, respectively for the SRW2 and the PreWEC-2, which were derived based on the experimental data as follows: average equivalent damping ratios of 1.5 and 1.0% were calculated experimentally for the SRW2 (
The GDAA was developed to simulate the seismic behavior of SRWs. Three damping components are included in the GDAA: 1) inherent viscous damping of the SRWs; 2) hysteretic damping due to damage of the wall toes and yielding of the unbonded post-tensioning steel; and 3) impact damping, which is implemented with an event-based approach. The GDAA simulates the rocking response of SRWs with respect to the foundation base and first-mode flexural response of the wall body. For the purpose of this paper, the flexural response is eliminated from the formulation of GDAA, because it is small in unbonded post-tensioned precast concrete walls (
The dynamic motion of the SRW is calculated with respect to the inertia frame
To capture the compressive deformations of the wall base, the base section is discretized into
The equation of motion of the GDAA for SRWs was developed by
Excluding flexure of the wall body, the equation of motion of the GDAA becomes
In
Moreover,
The GDAA formula of
The kinetic energy of the seismic mass,
The seismic mass contributes to the potential energy due to the horizontal ground excitation,
The variation of the work produced by the O-connectors,
Using
Using
This section describes the constitutive models used for the concrete fibers of the wall base section, the PT tendons, and the O-connectors.
The compressive stress-strain loading and reloading curves for the concrete fibers are computed with the constitutive model described by
The tensile loading and reloading stress-strain behaviors of the PT tendons are modeled with the constitutive model described by
A phenomenological hysteretic law is proposed in this research study for the O-connectors. The law was calibrated with data of the O-connector tests by
Force-deformation curves for the O-connectors;
The hysteretic behavior of the O-connectors is described in
If loading toward the negative direction continues beyond the deformation
The proposed hysteretic law is compared with the O-connector test data in
Rocking walls dissipate part of their kinetic energy during impact on the foundation base. The GDAA models the impact with an event-based approach, which computes the post-impact response of the wall with impulse-momentum equations. It is assumed that the impacts occur when either of the points of the wall base with coordinates
The shake table responses of the SRW2 and PreWEC-2 are compared with those computed using the FEM and the GDAA. As discussed previously, the FEM is a lumped-plasticity model which can compute the global responses of the SRW2 and PreWEC-2, including the maximum and residual lateral drifts, the absolute acceleration, the base shear and moment resistance due to the shake table excitations. Hence local wall responses, such as the wall contact length at the rocking interface and the variation of PT stress are not computed. If computing these local responses is of interest, the use of GDAA, is recommended. This point is discussed below.
Global responses of FEM and GDAA in comparison with the experimental data:
As noted previously, one advantage of using the GDAA as opposed to the FEM described in this paper is the capability to compute the local wall responses. As an example,
Local responses of GDAA in comparison with the experimental data during Sylmar:
Overall, the FEM satisfactorily reproduced the global behavior of the SRW2 and the PreWEC-2 for ground motions of different intensities. This model can be used for performance-based analysis of rocking wall systems, when estimating the global behavior of the walls is of interest, up to the lateral drift ratios of 4–5%. On the other hand, when estimation of the local wall responses is of interest, the GDAA may be used, because of its ability to compute these responses with good accuracy.
As described in
When subjected to large lateral drifts, the lateral stiffness and strength of the unbonded post-tensioning precast concrete walls may degrade due to yielding of the unbonded tendons and concrete damage at the compression toes. To examine the behavior of the FEM and GDAA in cases where wall degradation may occur, the horizontal component of the 1994 Northridge earthquake as recorded in the Sylmar-Converter Station (PEER 2020 NGA record sequence number or RSN—1084), was used to excite the FEM and the GDAA for the case of SRW2.
Responses of SRW2 to the horizontal component of Sylmar as modeled by the GDAA and the FEM:
On the other hand, the base shear vs. drift response of the FEM, shown in
The effect of vertical components of ground motions on the rocking behavior has been considered in previous analytical research studies of free-standing rigid blocks (e.g.,
The responses of the FEM and the GDAA were calculated considering 1) the horizontal component of the 1994 Northridge earthquake alone; and 2) the horizontal and vertical components of the 1994 Northridge earthquake combined. The lateral drift time histories for the two cases are plotted in
Lateral drift time histories of the FEM and the GDAA for
The impact damping influences the seismic behavior of SRWs, as observed experimentally by
The seismic response of the SRW2 to the Sylmar excitation (i.e., namely
Lateral drift time histories by the GDAA and FEM, respectively for various values of
Modeling of the impact damping can also influence the dynamic behavior of walls with supplemental damping devices, such as the PreWEC. This is discussed in
Based on the above discussion, accurate modeling of impact damping is important for capturing the experimental behavior of rocking walls. While the use of a generalized coefficient of restitution,
To investigate modeling of impact damping in the FEM for walls with different aspect ratios, five additional case study SRWs were designed by varying the wall aspect ratio from 2.0 to 4.0, as shown in
Design parameters of case study walls.
case study | Length of the wall panel (m), wall aspect ratio (i.e., height |
No., dia.(mm) of PT tendon, initial PT stress (MPa) | Moment, kN-m/Shear, kN, resistance at 2% drift |
---|---|---|---|
A (SRW2) | 1.91, 2.8 | 6, 12.7 strand, 923.9 (0.5 |
769.4, 180.3 |
B | 1.33, 4.0 | 6, 12.7 strand, 1,270.7 (0.68 |
771.1, 180.7 |
C | 1.52, 3.5 | 6, 12.7 strand, 937 (0.5 |
768.1, 180 |
D | 1.78, 3.0 | 5, 12.7 strand, 845.3 (0.45 |
769.2, 180.2 |
E | 2.13, 2.5 | 4, 12.7 strand, 754.3 (0.41 |
772.8, 181.0 |
F | 2.67, 2.0 | 3, 12.7 strand, 635.7 (0.34 |
767.7, 179.9 |
Height of the wall panels remained unchanged, i.e., 5.33 m.
Following the SA method (
Next, the seismic responses of the five SRWs were computed with the FEM by calibrating the tangent stiffness proportional Rayleigh damping to achieve an adequate comparison with the corresponding responses by the GDAA. The selected seismic excitation in this case was the Sylmar excitation (i.e., namely
Lateral drift time histories of case study SRWs during Sylmar–the FEM with selected damping ratios vs the GDAA:
Modeling of unbonded post-tensioned precast concrete walls with rocking connections was discussed in this research study. Two modeling approaches were investigated, namely 1) a Finite Element Model (FEM), which was developed based on lumped plasticity to capture the global wall responses up to lateral drift ratios of 4–5%; and 2) a Generalized Dynamic Analysis Approach (GDAA), which is an analytical model developed to capture both the global and local wall responses up to lateral drift ratios of 10%. The accuracy of the two models was evaluated with shake table tests of unbonded post-tensioned precast concrete walls without (SRW2) and with (PreWEC-2) supplemental damping devices. The following conclusions were made:
Both the FEM and the GDAA satisfactorily reproduced the experimental global behaviors of the SRW2 and the PreWEC-2.
The GDAA was capable of computing the experimental local responses of the SRW2 and the PreWEC-2, including the variation of the post-tensioning force and the contact length of the wall to the foundation base. Because the FEM is a lumped plasticity model, it cannot compute these local responses. The use of a fiber-based modeling approach can be investigated for improving the FEM in future research studies.
While the FEM satisfactorily reproduced the global behavior of the walls up to the drifts of 4–5%, the GDAA may be preferred for computing wall responses to larger drifts, because of its ability to simulate concrete damage and yielding of the PT tendons.
It was shown that the use of a fiber-based sectional analysis in the GDAA provided the capability to capture the effect of vertical ground excitations to the wall responses and that vertical excitations may alter the seismic behavior of the walls.
The FEM and the GDAA corroborated that small variations of the impact damping can influence the seismic behavior of unbonded post-tensioned precast concrete walls significantly. The GDAA captured impact damping with an event-based approach, while the FEM used a tangent stiffness-based equivalent impact damping ratio of 3% for SRWs, which was derived based on shake table test data.
Results of a case study analysis on five SRWs with different aspect ratios confirmed the accuracy of the FEM’s 3% impact damping for walls with aspect ratios of 2.5 and higher. It was found that an equivalent damping ratio of 5% is more suitable for modeling SRWs with lower aspect ratios.
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
DK: Modeling and Analysis (GDAA), Results and Discussions, Writing. MN: Modeling and Analysis (FEM), Comparison with the Shake Table Data, Results and Discussions, Writing.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.