- Open Access
Analytical Study of Force–Displacement Behavior and Ductility of Self-centering Segmental Concrete Columns
- Reza Hassanli^{1}Email authorView ORCID ID profile,
- Osama Youssf^{1, 2},
- Julie Mills^{1} and
- Mostafa Fakharifar^{3}
https://doi.org/10.1007/s40069-017-0209-4
© The Author(s) 2017
- Received: 2 November 2016
- Accepted: 16 June 2017
- Published: 18 September 2017
Abstract
In this study the behavior of unbonded post-tensioned segmental columns (UPTSCs) was investigated and expressions were proposed to estimate their ductility and neutral axis (NA) depth at ultimate strength. An analytical method was first employed to predict the lateral force–displacement, and its accuracy was verified against experimental results of eight columns. Two stages of parametric study were then performed to investigate the effect of different parameters on the behavior of such columns, including concrete compressive strength, axial stress ratio, diameter and height of the column, axial stress level, duct size, stress ratio of the PT bars, and thickness and ultimate tensile strain of fiber reinforced polymer wraps. It was found that the column’s aspect ratio and axial stress ratio were the most influential factors contributing to the ductility, and axial stress ratio and column diameter were the main factors contributing to the NA depth of self-centering columns. While at aspect ratios of less than ten, as the axial stress ratio increased, the ductility increased; at aspect ratios higher than ten, the ductility tended to decrease when the axial stress ratio increased. Using the results of parametric study, nonlinear multivariate regression analyses were performed and new expressions were developed to predict the ductility and NA depth of UPTSCs.
Keywords
- post-tensioned
- self-centering
- concrete columns
- ductility
- seismic response
- force–displacement behavior
1 Introduction
In the recent past considerable attention has been given to the use of self-centering precast concrete members and connections (Priestley et al. 1999; Kurama et al. 2002; Perez et al. 2007; Kim and Choi 2015) and bridge construction (Dawood et al. 2011; Kwan and Billington 2003; Youssf et al. 2015a, 2016; Kim et al. 2012; Kim 2013). Self-centering behavior can be induced to the system by employing unbonded post-tensioning (PT) steel. In these systems the residual drifts and the damage to the structure in seismic events are limited, resulting in reduced cost associated with structural repair and business downtime (Hassanli et al. 2016a). Self-centering concrete systems can display a ductile response and carry high levels of lateral loads (Henry 2011).
The conventional methods used to determine the lateral strength of concrete columns cannot be applied to self-centering columns due to the increase in the PT force that results in variable axial load. Moreover, strain compatibility is not a correct assumption for these columns (ElGawady et al. 2010; ElGawady 2011). As a result of the rotation of the columns at their base, deformations of self-centering columns are mainly due to the rocking mechanism (Wight et al. 2007). This rocking mechanism leads to variation in the level of axial stress at different drift ratios and hence the lateral load behavior of the columns. The lateral displacement of the self-centering columns also consists of elastic flexural deformation, shear deformation and relative movement of segments due to sliding. Deformation due to shear and sliding are usually ignored, however they might be significant in columns with low aspect ratios (Priestley et al. 1996a).
Recently, segmental precast concrete members have become a subject of interest. In segmental systems with PT bars/tendons, the PT force works as the clamping force which keeps the segments together. Unbonded post-tensioned segmental columns (UPTSCs) can be used in bridge structures to reduce construction times and improve structural quality. Monotonic, cyclic and dynamic behavior of these columns have been studied experimentally by many researchers (Chang et al. 2002; Chou and Chen 2006; Marriott et al. 2009; Yamashita and Sanders 2009; Wang et al. 2008; Shim et al. 2008; Bu et al. 2015). While UPTSCs are capable of withstanding large nonlinear displacement, due to their low level of damage under seismic excitation, they experience large displacements. To tackle this drawback, different approaches have been developed to increase the energy dissipation capacity of the rocking columns, including the use of metallic yielding components (ElGawady 2011; Ou et al. 2009, 2010), shape memory alloy bars (Roh and Reinhorn 2010; Roh et al. 2012), external energy dissipating dampers (Palermo et al. 2007) and columns with cast-in-place (emulative) base (Kim et al. 2015; Ou et al. 2013). To improve the behavior self-centering precast members the effect of using confinement jackets (Motaref et al. 2013; Hewes and Priestley 2002), fiber-reinforced concrete (Shajil et al. 2016), and rubber or elastomeric bearing pads (ElGawady 2011; Motaref et al. 2010) have also been investigated.
Research undertaken on the analytical approach to predict the behavior of UPTSCs is limited. Hewes and Priestley (2002) proposed a simplified analytical model to predict the lateral force- displacement relationship. Ou et al. (2007) extended the approach to hybrid UPTSCs (with longitudinal mild steel across the segments’ joints), and using a modified curvature to account for the effect of mild steel, they showed the accuracy of the method for hybrid UPTSCs. Palermo et al. (2007) showed that a similar design procedure considering lumped plasticity models can be used to design and model hybrid bridge structures. Chou et al. (2013) considered two plastic hinges to account for the joint opening between the column and its base as well as between the first two column segments and demonstrated that their model is capable of predicting pushover responses of UPTSCs.
Three dimensional finite element (FE) modeling has been performed to predict the force–displacement behavior of UPTSCs (Dawood et al. 2011; Ou et al. 2007; Chou et al. 2013; Sideris 2015; Youssf et al. 2015b). Ou et al. (2007) demonstrated that three dimensional FE modeling can be used to effectively predict the lateral response of UPTSCs. Chou et al. (2013) and Kim et al. (2010) showed that two dimensional FE modeling can also predict the force–displacement behavior response of UPTSCs.
Using experimental results of eight columns, this paper evaluates the accuracy of an analytical approach to predict the behavior of UPTSCs. The concept of the analytical method was similar to previous studies (Hewes and Priestley 2002; Ou et al. 2007; Pampanin et al. 2001); however, the displacement at the column top was considered as the independent variable rather than the curvature at the base of column. Moreover, the plastic hinge length expression proposed by Hassanli et al. (2017) was considered. Stress–strain relationships for un-confined and FRP-confined concrete, proposed respectively by Kent and Park (1971) and Lam and Teng (2003), were adopted. The analytical approach was developed based on the mechanics of rocking of columns about their base and geometric compatibility conditions. In this approach the column rotation was assumed to occur only at the column base.
Using the validated analytical approach, Stage I of a parametric study consisting of 43 columns was performed to investigate the effect of different parameters on the behavior of self-centering columns, including concrete compressive strength, axial stress ratio, diameter and height of the column, axial stress level, duct size, stress ratio of the PT bars, and the thickness and ultimate tensile strain of fiber reinforced polymer (FRP) wraps. The results of Stage I of the parametric study were then used to generate the configuration of the columns for Stage II of the parametric study. The ductility of the columns was then calculated and nonlinear multivariate regression analysis was performed to develop an expression to estimate the ductility of UPTSCs.
2 Analytical Approach
The conventional strain compatibility method to determine the lateral strength of concrete columns is not appropriate for unbonded PT specimens (Hewes and Priestley 2002; Bu and Ou 2013), as explained in the introduction section. Hassanli et al. (2017) developed an expression to predict the plastic hinge length of masonry walls and incorporated it with a step-by-step analytical procedure (Hassanli 2015). The analytical method they developed could effectively predict the lateral behavior of the masonry walls. Hassanli et al. (2016a) applied the procedure to concrete walls and showed that it could accurately predict their lateral force behavior. In this study the same procedure was considered, however, it was modified to be appropriate for column members. The steps of the analytical approach were as follows:
2.1 Force–Displacement Response at Decompression Point
Using Eqs. 1 and 6 a point of the force displacement response corresponding to the decompression point, \( \Delta_{0} \),\( V_{0} \), can be determined.
2.2 Force–Displacement Response Beyond Decompression Point
- 1.
Assume a top displacement of Δ (due to rocking) and calculate the corresponding column rotation, θ (Fig. 2) (θ = Δ/h _{ c }).
- 2.
Assume a value of neutral axis (NA) depth, c (Fig. 2).
- 3.Calculate the strain in the PT steel, \( \varepsilon_{ps} \), using Eq. 8.where \( l_{ps} \) is the unbonded length of the PT steel and \( \varepsilon_{se} \) is the effective strain in the PT steel after immediate stress losses. Equation 8 implies that the strain in the PT bar is the summation of the effective strain due to initial post-tensioning, \( \varepsilon_{se} \), and the strain developed due to the rocking mechanism.$$ \varepsilon_{ps} = \theta (r - c)/l_{ps} + \varepsilon_{se} $$(8)
- 4.Calculate the concrete strain at the extreme compressive fiber, \( \varepsilon_{c } , \) using Eq. 9,where \( L_{pl} \) is the plastic hinge length. No comprehensive study was found in the literature evaluating the plastic hinge length of UPTSCs, hence, the following expression, which was originally developed for wall member (Hassanli et al. 2015, 2016a) was adopted in this study,$$ \varepsilon_{c } = \theta c/L_{pl} + \varepsilon_{0} $$(9)where \( f_{c}^{'} \) is the axial compressive strength of unconfined concrete (MPa), and \( f_{c} \) is the axial stress level which is defined as$$ L_{pl} = 0.11(2r) + 3475\frac{{f_{c} }}{{f_{c}^{'} }} ({\text{mm}})$$(10)where \( A_{c} \) is the cross sectional area of concrete column.$$ f_{c} = \frac{{f_{se} A_{ps} + N}}{{A_{c} }} $$(11)
- 5.Calculate the stress developed in the PT steel using an appropriate constitutive model. In this study, the following elasto-plastic material model with linear kinematic hardening was considered,$$ \sigma_{ps} = \left\{ {\begin{array}{*{20}l} {\varepsilon_{ps} E_{ps} } \hfill & {\varepsilon_{ps} \le \varepsilon_{py} } \hfill \\ {\varepsilon_{py} E_{ps} + (\varepsilon_{ps i} - \varepsilon_{py} )(f_{pu} - f_{py} )/(\varepsilon_{pu} - \varepsilon_{py} )} \hfill & {\varepsilon_{py} < \varepsilon_{ps} \le \varepsilon_{pu} } \hfill \\ 0 \hfill & {\varepsilon_{pu} < \varepsilon_{ps} } \hfill \\ \end{array} } \right.. $$(12)
- 6.Calculate the corresponding concrete stress. An appropriate stress–strain relationship must be utilized. In this study the following models were adopted for unconfined and FRP-confined concrete,
- (a)Unconfined concrete In this study Kent–Park stress–strain relationships was used (Eq. 13) (Kent and Park 1971) for unconfined concrete.where,$$ f_{cj} \left( {\varepsilon_{cj} } \right) = \left\{ {\begin{array}{*{20}l} {f_{c}^{'} \left[ {\left( {\frac{{2\varepsilon_{cj} }}{0.002}} \right) - \left( {\frac{{\varepsilon_{cj} }}{0.002}} \right)^{2} } \right]} \hfill & {\varepsilon_{c j} < 0.002} \hfill \\ {f_{c}^{'} \left[ {1 - Z_{c} \left( {\varepsilon_{cj} - 0.002} \right)} \right]} \hfill & { 0.002 \le \varepsilon_{cj} \le \varepsilon_{cp} } \hfill \\ {0.2f_{c}^{'} } \hfill & { \varepsilon_{cj} > \varepsilon_{cp} } \hfill \\ \end{array} } \right. $$(13)where \( f_{cj} \left( {\varepsilon_{cj} } \right) \) is the concrete stress (MPa), and \( \varepsilon_{cj} \) is the concrete strain at distance \( x_{j} \) from the neutral axis (Fig. 3),$$ \varepsilon_{cp} = 1.6\left[ {\frac{{3 + 0.29f_{c}^{'} }}{{145f_{c}^{'} - 1000}}} \right] + 0.0015 $$(14)$$ \varepsilon_{cj} = (x_{j} /c) \varepsilon_{c } $$(15)
- (b)FRP-confined concrete Numerous models have been proposed to predict the stress–strain response of FRP-confined concrete. A comprehensive review of 88 models developed to predict the axial stress–strain behavior of FRP-confined concrete in circular sections was performed by Ozbakkaloglu et al. (2013). They considered both design-oriented and analysis-oriented models and to provide a comprehensive assessment of the models, a large test database of 730 FRP-confined concrete cylinders tested under monotonic axial compression was collected. Comparing all models, they concluded that the one developed by Lam and Teng (2003) was the most accurate model to predict the ultimate strength. Therefore, this model (provided below) was used in this study for FRP-confined specimens,where,$$ f_{cj} \left( {\varepsilon_{cj} } \right) = \left\{ {\begin{array}{*{20}l} {E_{c} \varepsilon_{cj} - \frac{{(E_{c} - E_{c2} )^{2} }}{{4f_{0} }}\varepsilon_{cj}^{2} } & {\varepsilon_{cj} < \varepsilon_{c1} } \\ {f_{c}^{'} + E_{c2} \varepsilon_{cj} } & {\varepsilon_{c1} \le \varepsilon_{cj} \le \varepsilon_{cu} } \\ \end{array} } \right. $$(16)$$ \varepsilon_{c1} = \frac{{2f_{c}^{'} }}{{E_{c} - E_{c2} }} $$(17)$$ E_{c2} = \frac{{f_{cc}^{'} - f_{c}^{'} }}{{\varepsilon_{cu} }} $$(18)$$ f_{0} = 0.872f_{c}^{'} + 0.371f_{l} + 6.258 $$(19)$$ f_{l} = \frac{{f_{FRP} t_{FRP} }}{r} $$(20)$$ f_{cc}^{'} = \left\{ {\begin{array}{*{20}c} {f_{c}^{'} (1 + 3.3\frac{{f_{lu,a} }}{{f_{c}^{'} }})} & {f_{l} \ge 0.07f_{c}^{'} } \\ {f_{c}^{'} } & {f_{l} < 0.07f_{c}^{'} } \\ \end{array} } \right. $$(21)where \( f_{cc}^{'} \) = axial compressive strength of FRP-confined concrete (MPa); \( f_{l} \) = confining pressure provided by the FRP jacket (MPa); \( t_{FRP} \) = thickness of the FRP jacket; \( f_{FRP} \) = ultimate tensile strength of FRP material (MPa); \( f_{lu,a} \) = actual lateral confining pressure at ultimate (MPa) (= 0.586 \( f_{FRP} \) for CFRP (Lam and Teng 2003)); \( \varepsilon_{FRP} \) = ultimate tensile strain of FRP material; and \( \varepsilon_{co} \) = axial strain of unconfined concrete at \( f_{c}^{'} \) and \( \varepsilon_{cu} \) _{=} ultimate axial strain of FRP-confined concrete.$$ \varepsilon_{cu} = 1.75 + 5.53(\frac{{f_{lu,a} }}{{f_{c}^{'} }})(\frac{{\varepsilon_{FRP} }}{{\varepsilon_{co} }})^{0.45} $$(22)
- (a)
- 7.Calculate the total compression force, c_{c}, and the total tension force, T, using Eqs. 23 and 24.$$ c_{c} = \mathop \smallint \limits_{0}^{c} f_{cj} dA = \mathop \smallint \limits_{0}^{c} f_{cj} b_{w} dx $$(23)where \( \sigma_{ps } \) can be determined using Eq. 12, \( f_{c j} \) can be determined using Eq. 13 or 16 for unconfined and FRP confined concrete, respectively, and \( b_{w} \) can be calculated form Eq. 32 or 33 for rectangular or circular cross sections, respectively.$$ T = \sigma_{ps} A_{ps} $$(24)
- 8.
If \( c_{c} = T - N \) go to next step otherwise return to step 2.
- 9.
Calculate \( \Delta_{f} = \Delta_{0} + \Delta \)
- 10.Take the moment about the neutral axis to calculate the total moment capacity, M, using Eq. 25.$$ M = \sigma_{ps} A_{ps} \left( {d - c} \right) + N \left( {r - c} \right) - \int \nolimits_{0}^{c} f_{cj} xb_{w} dx $$(25)
- 11.The point with the coordinate of (∆ _{ f }, V = \( M/h_{c} \)) corresponds to a point in the force–displacement curve. To obtain another point, return to step 1. The flowchart shown in Fig. 4 presents the proposed procedure of the design approach.
It is worth noting that the presented force–displacement procedure considers only elastic flexural deformation (before decompression) and rocking response (beyond decompression). The deformation due to shear and sliding is ignored. In rocking columns with large aspect ratios (usually of higher than 3) (Priestley et al. 1996a), shear and sliding deformations typically have small magnitudes (Wight et al. 2007) and hence can be neglected, otherwise the corresponding displacement, \( \Delta_{s} \), needs to be considered to determine the total displacement (\( \Delta_{f} = \Delta_{0} + \Delta + \Delta_{s} \)).
3 Verification of Analytical Approach
Properties of the test specimens.
Specimencode | \( f_{c}^{'} \) (MPa) | Bott. segment confinement | Axial PT load^{a} (kN) | Material^{b} | |
---|---|---|---|---|---|
Bott. seg. | Top three seg. | ||||
C50 | 55 | Unconfined | 50 | CC | CRC |
C100 | 55 | Unconfined | 100 | CC | CRC |
CF50 | 55 | Confined | 50 | CC | CRC |
CF100 | 55 | Confined | 100 | CC | CRC |
CR50 | 38 | Unconfined | 50 | CRC | CRC |
CR100 | 38 | Unconfined | 100 | CRC | CRC |
CRF50 | 38 | Confined | 50 | CRC | CRC |
CRF100 | 38 | Confined | 100 | CRC | CRC |
In the analytical approach, for both CC and CRC, the concrete material model presented in Eq. 12 was adopted. Due to rocking of columns about their bases and damage in only the bottom segments during the experiments, the material properties of CC and CRC were used for columns, C50, C100, CF50 and CF100 and columns CR50, CR100, CRF50 and CRF100, respectively.
Prediction of the lateral strength using the analytical approach.
Specimen code | Peak Strength (kN) | V _{ EXP } /V _{ Analysis } | PT force (kN) | T _{ EXP }/T _{ Analysis } | ||
---|---|---|---|---|---|---|
V _{ EXP } | V _{ Analysis } | T _{ EXP } | T _{ Analysis } | |||
C50 | 3.2 | 3.2 | 0.99 | 85.8 | 78.2 | 1.10 |
C100 | 4.3 | 4.6 | 0.93 | 123.2 | 121.9 | 1.01 |
CF50 | 4.3 | 4.3 | 1.00 | 116.9 | 117.4 | 1.00 |
CF100 | 5.6 | 5.3 | 1.05 | 162.7 | 164.8 | 0.99 |
CR50 | 2.6 | 3 | 0.88 | 83.0 | 74.2 | 1.12 |
CR100 | 4 | 4.1 | 0.97 | 113.7 | 123.7 | 0.92 |
CRF50 | 4.1 | 3.8 | 1.09 | 115.5 | 117.6 | 0.98 |
CRF100 | 5.1 | 4.6 | 1.1 | 157.75 | 164.3 | 0.96 |
The presented analytical procedure was developed for self-centering columns without energy dissipators; however, if internal mild steel or external dissipators or other similar devices are used, the approach can potentially be used if modified to account for the contribution of the dissipators on the moment capacity and stiffness of the column, as well as the possibility of joint opening between segments. Previous studies have shown that considering geometric compatibly concepts, a similar analytical approach can be used to predict the response of precast concrete hybrid frames and piers with external dissipators (Pampanin et al. 2001; Cao et al. 2015).
4 Parametric Study
As described, the presented analytical method was able to accurately predict the force–displacement behavior of UPTSCs and the force developed in the PT bars. The validated analytical approach was then used to conduct a parametric study to better understand the behavior of UPTSCs and also to develop expressions to predict the ductility of UPTSCs. The parametric study was performed in two stages by considering two different sets of column models. The first set was developed to examine the effect of different parameters on the behavior of UPTSCs, including concrete compressive strength, axial stress ratio, diameter and height of the column, axial stress level, duct size, stress ratio of the PT bars, and the thickness and ultimate tensile strain of the FRP wraps. Based on the analysis results obtained from the first set, design recommendations were provided for UPTSCs. Subsequently, matrices of column models were generated for the second stage (Stage II) of the parametric study. Stage II of parametric study was performed on a set of columns considering two parameters, axial stress ratio and aspect ratio as main variables to investigate the ductility of UPTSCs.
4.1 Parametric Study—Stage I
Column matrix of Stage I—parametric study.
Column code | Varying parameter | Lateral capacity (kN) | Parameters obtained from bi-linear idealized response | ||||
---|---|---|---|---|---|---|---|
Δ_{ y } (mm) | Δ_{ u } (mm) | V _{ y } (kN) | K _{ e } (kN/mm) | μ | |||
C1-1 | \( f_{c}^{'} \) = 25 MPa | 2.65 | 11.8 | 70.0 | 2.49 | 0.21 | 5.9 |
C1-2 | \( f_{c}^{'} \) = 35 MPa | 2.87 | 8.4 | 60.5 | 2.68 | 0.32 | 7.2 |
C1-3 | \( f_{c}^{'} \) = 45 MPa | 3.11 | 7.2 | 55.7 | 2.83 | 0.39 | 7.7 |
C1-4 | \( f_{c}^{'} \) = 55 MPa | 3.18 | 4.7 | 59.8 | 2.82 | 0.60 | 12.7 |
C1-5 | \( f_{c}^{'} \) = 65 MPa | 3.25 | 4.6 | 68.0 | 2.90 | 0.63 | 14.8 |
C1-6 | f \( f_{c}^{'} \) = 75 MPa | 3.30 | 4.4 | 107.0 | 2.89 | 0.66 | 24.3 |
C2-1 | f _{ c } / \( f_{c}^{'} \) = 0.032 | 2.24 | 7.1 | 129.4 | 1.98 | 0.28 | 18.3 |
C2-2 | f _{ c } / \( f_{c}^{'} \) = 0.042 | 2.5 | 6.0 | 93.3 | 2.15 | 0.36 | 15.6 |
C2-3 | f _{ c } / \( f_{c}^{'} \) = 0.057 | 3.18 | 4.7 | 59.8 | 2.82 | 0.60 | 12.7 |
C2-4 | f _{ c } / \( f_{c}^{'} \) = 0.109 | 4.61 | 4.6 | 56.8 | 4.19 | 0.92 | 12.5 |
C2-5 | f _{ c } / \( f_{c}^{'} \) = 0.160 | 5.68 | 4.5 | 57.9 | 5.22 | 1.15 | 12.8 |
C2-6 | f _{ c } / \( f_{c}^{'} \) = 0.212 | 6.55 | 4.6 | 57.0 | 6.03 | 1.31 | 12.4 |
C2-7 | f _{ c } / \( f_{c}^{'} \) = 0.263 | 7.23 | 4.9 | 54.4 | 6.67 | 1.37 | 11.2 |
C3-1 | f _{ c } = 1.4 MPa | 1.54 | 17.8 | 128.3 | 1.45 | 0.08 | 7.2 |
C3-2 | f _{ c } = 2.8 MPa | 2.47 | 12.2 | 80.1 | 2.33 | 0.19 | 6.5 |
C3-3 | f _{ c } = 5.7 MPa | 4.18 | 8.3 | 64.4 | 3.88 | 0.47 | 7.8 |
C3-4 | f _{ c } = 8.5 MPa | 5.84 | 7.2 | 56.2 | 5.46 | 0.76 | 7.8 |
C3-5 | f _{ c } = 11.3 MPa | 7.57 | 7.2 | 50.8 | 7.01 | 0.98 | 7.1 |
C3-6 | f _{ c } = 14.2 MPa | 9.14 | 6.9 | 47.5 | 8.53 | 1.23 | 6.8 |
C4-1 | D _{ f } = 0 | 3.52 | 5.0 | 63.0 | 3.12 | 0.63 | 12.7 |
C4-2 | D _{ f } = 0.03 | 3.18 | 4.7 | 59.8 | 2.82 | 0.60 | 12.7 |
C4-3 | D _{ f } = 0.07 | 2.85 | 4.6 | 56.9 | 2.54 | 0.55 | 12.3 |
C4-4 | D _{ f } = 0.10 | 2.54 | 4.6 | 55.1 | 2.26 | 0.49 | 12.0 |
C4-5 | D _{ f } = 0.13 | 2.24 | 4.6 | 53.4 | 1.99 | 0.43 | 11.5 |
C5-1 | D = 150 mm | 3.18 | 4.7 | 59.8 | 2.82 | 0.60 | 12.7 |
C5-2 | D = 250 mm | 7.60 | 9.4 | 113.9 | 7.21 | 0.77 | 12.2 |
C5-3 | D = 350 mm | 16.01 | 21.4 | 86.7 | 15.20 | 0.71 | 4.1 |
C5-4 | D = 450 mm | 28.14 | 37.9 | 79.0 | 28.84 | 0.76 | 2.1 |
C6-1 | h _{ c } = 0.5 m | 10.05 | 2.9 | 17.3 | 9.16 | 3.20 | 6.0 |
C6-2 | h _{ c } = 1.0 m | 4.72 | 4.2 | 38.5 | 4.22 | 1.00 | 9.1 |
C6-3 | h _{ c } = 1.5 m | 3.00 | 4.9 | 63.8 | 2.66 | 0.54 | 12.9 |
C6-4 | h _{ c } = 2.0 m | 2.17 | 6.3 | 91.6 | 1.94 | 0.31 | 14.4 |
C6-5 | h _{ c } = 2.5 m | 1.69 | 7.9 | 120.7 | 1.51 | 0.19 | 15.3 |
C6-6 | h _{ c } = 3.0 m | 1.38 | 9.4 | 150.8 | 1.23 | 0.13 | 16.1 |
C7-1 | t _{ FRP } = 0 mm | 3.18 | 4.7 | 59.8 | 2.82 | 0.60 | 12.7 |
C7-2 | t _{ FRP } = 0.1 mm | 4.21 | 36.1 | 78.6 | 4.30 | 0.12 | 2.2 |
C7-3 | t _{ FRP } = 0.2 mm | 4.74 | 51.7 | 108.0 | 4.91 | 0.10 | 2.1 |
C7-4 | t _{ FRP } = 0.3 mm | 5.05 | 61.3 | 137.4 | 5.27 | 0.09 | 2.2 |
C7-5 | t _{ FRP } = 0.4 mm | 5.72 | 71.7 | 156.6 | 5.73 | 0.08 | 2.2 |
C8-1 | ε _{ FRP } = 0.003 | 4.01 | 32.8 | 72.0 | 3.94 | 0.12 | 2.2 |
C8-2 | ε _{ FRP } = 0.01 | 4.26 | 35.5 | 89.4 | 4.26 | 0.12 | 2.5 |
C8-3 | ε _{ FRP } = 0.02 | 4.25 | 35.4 | 114.0 | 4.25 | 0.12 | 3.2 |
C8-4 | ε _{ FRP } = 0.03 | 4.63 | 36.5 | 106.8 | 4.38 | 0.12 | 2.9 |
The effect of duct size is particularly significant for hollow sections. For example, when concrete masonry blocks or precast concrete hollow sections are used, the movement of the PT steel within the duct must be considered.
4.1.1 Effect of Compressive Strength, \( f_{c}^{'} \)
To investigate the effect of \( f_{c}^{'} \), columns C1-1 to C1-6 with compressive strength of 25–75 MPa were generated. The responses are plotted in Fig. 7. As shown in the figure, as the compressive strength increased, the lateral strength and the normalized PT force increased and the NA depth decreased. As shown in Fig. 7b, columns with lower \( f_{c}^{ '} \) exhibited a decreasing trend of force developed in the PT steel at large drift ratios. This is attributed to the earlier damage of concrete in compression zone in columns with lower \( f_{c}^{ '} \). However, at higher \( f_{c}^{ '} \), the columns were intact for longer time at higher drift ratios as observed in specimens C1-5 and C1-6. The moment capacity of the columns increased from 3.8 to 4.7 kN m (24% increase) when the compressive strength increased from 25 to 75 MPa, see Fig. 7d.
4.1.2 The Effect of Axial Stress Ratio, \( f_{c} /f_{c}^{'} \)
The axial stress ratio is defined as the initial stress on the column, \( f_{c} \), divided by \( f_{c}^{'} \). The initial stress on the column was calculated as the summation of the initial PT force and the self-weight of the column and the loading beam, divided by the column cross sectional area, A _{ g }. In this parametric study, the axial stress ratio, \( f_{c} /f_{c}^{'} \), was varied from 0.032 to 0.263 in columns C2-1 to C2-7, by keeping the value of \( f_{c}^{'} \) constant and changing the PT force. As shown in Fig. 8 the level of \( f_{c} /f_{c}^{'} \), has significant effect on the lateral capacity, PT force and NA depth. Although in columns with higher \( f_{c} /f_{c}^{'} \), the lateral force capacity was higher, they exhibited a higher rate of post-peak strength degradation. This is attributed to the increased probability of concrete crushing at high stress levels that results in faster degradation of the lateral strength. The normalized PT force in the columns with lower \( f_{c} /f_{c}^{'} \) was significantly higher than that in columns with high levels of \( f_{c} /f_{c}^{'} \). While the NA depth comprised about 0.2D in the column C2-1 with \( f_{c} /f_{c}^{'} = 0.032 \) it increased to 0.45D in the column C2-7 with \( f_{c} /f_{c}^{'} = 0.263 \), see Fig. 8c. As shown in Fig. 8d, the moment capacity of column C2-1 was 3.2 kN m, however; it increased to 10.3 kN m in column C2-7. From all these comparisons it can be concluded that the axial stress ratio has significant effect on the capacity and behavior of UPTSCs.
4.1.3 The Effect of Axial Stress Level, \( f_{c} \)
Five different values of axial force were applied to columns C3-1 to C3-6, providing an axial stress ranging from 1.4 kN in column C3-1 to 14.2 kN in column C3-6. To maintain similar axial stress ratio (of 0.15) in all columns of this category, the value of \( f_{c}^{'} \) was varied accordingly. As shown in Fig. 9, as the axial stress level increased the lateral force capacity increased and the normalized PT force decreased. According to Fig. 9c, the NA depths of the columns of this category were similar and were about 0.35D at peak load for all columns. Consequently, as long as the axial stress ratio was constant, the NA depth was independent of the axial stress level. The moment capacity of the columns increased approximately linearly from 1.4 to 14.1 kN m as the axial stress level increased from 1.4 to 14.2 kN, see Fig. 9d.
4.1.4 The Effect of Duct Size, D _{ f }
If an oversized duct is used to accommodate the PT bar, then due to the rocking mechanism the PT bar moves towards the compression zone until it touches the duct’s inner wall. To investigate the effect of oversized ducts, five duct size factors, D _{ f } (see Eq. 26) were used ranging from 0 in column C4-1 to 0.13 in column C4-5. As shown in Fig. 10a, as the duct size factor increased, the lateral force capacity and the force developed in the PT steel decreased. This is attributed to the reduced lever arm. At peak strength, the NA depth was approximately similar in all columns of this category. Increasing the duct size factor from 0 to 0.13 caused the moment capacity to reduce from 5 to 3.2 kN m (about 36% reduction). Thus it is concluded that if an oversized duct or concrete hollow section is used, its negative effect on the effective PT depth, and hence the lateral strength of the columns, must be taken into account.
4.1.5 The Effect of Column Diameter, D
To investigate the effect of the diameter on the behavior of UPTSCs, columns C5-1 to C5-4 having diameters ranging from 150 to 450 mm were considered. As expected increasing the column diameter had significant effect on its behavior. The lateral strength, the PT force and the moment capacity increased as the diameter increased, as shown in Fig. 11. By increasing the column diameter, the lever arm of the concrete cross-section increases. This increases the moment capacity as well as the lateral strength of the column section and delays the column deterioration under cyclic loading, and hence results in a higher force in the PT bars (Fig. 11b).
4.1.6 Effect of Column HEIGHT, h _{ c }
Columns C6-1 to C6-6 (Table 3) having heights ranging from 0.5 to 3.0 m were generated to investigate the effect of the column’s height on the behavior of UPTSCs. As the height increased the lateral load capacity and the PT force decreased due to the decrease in the column stiffness by increasing its height. However, the moment capacity decreased slightly from 5.0 in column C6-1 to 4.1 in column C6-6. This can be attributed to the longer unbonded length of PT steel in longer columns compared with that of shorter columns which results in a lower PT strain, and hence lower PT forces compared with shorter columns. The decreased PT force results in reduced moment capacity in longer columns. Figure 12c shows that the NA depth of columns C6-1 to C6-6 remained relatively similar. At peak load the NA depth was about 0.20–0.25D for all columns in this category.
4.1.7 Effect of FRP Thickness, t _{ FRP }
As presented in Table 3, columns C7-1 to C7-5 were generated to investigate the influence of the FRP thickness, t _{ FRP }, on the behavior of FRP-confined specimens. As mentioned before for the FRP-confined concrete, the model propose by Lam and Teng (2003) was used. In this model the effect of FRP thickness (Eq. 20) is reflected in the stress–strain behavior of FRP-confined concrete (Eq. 16).
As shown in Fig. 13a, the stiffness of the columns remained relatively unchanged; however, as the FRP-wrap thickness increased the lateral force capacity and the level of PT force increased. By increasing the FRP thickness, the stiffness of the FRP jacket increases and its hoop strain decreases, and hence the micro cracks in the concrete core decrease. This keeps the column cross-section intact for a longer time at higher drifts which increases the column lateral strength and PT force development. The NA depth at the peak load was 0.2D for all columns of this category (Fig. 13c). As shown in Fig. 13d while the moment capacity of unconfined column, C7-1 was 4.5 kN m, it increased to 8.2 kN m in the FRP-confined column C7-5 with t _{ FRP } of 0.4 mm.
4.1.8 Effect of FRP Ultimate Tensile Strain, \( \varepsilon_{FRP} \)
A wide ultimate tensile strain range of 0.003–0.029 was reported for different types of FRP material (Lam and Teng 2001). Hence, FRP-confined columns C8-1, C8-2, C8-3, and C8-4 with \( \varepsilon_{FRP} = \) 0.003, 0.01, 0.02, and 0.03, respectively, were considered to investigate the effect of \( \varepsilon_{FRP} \) on the behaviour of UPTSCs. Note that the effect of FRP ultimate tensile strain is reflected in \( \varepsilon_{cu} \) in Eq. 22 which influence the stress–strain response of FRP-confined concrete (Eq. 16).
As shown in Fig. 14, increasing the FRP ultimate tensile strain resulted in an increase in the lateral force and PT force due to the increased deformation capacity of FRP with higher tensile strain. However, the NA depth at peak strength was about 0.2D in all columns. The moment capacity increased from 5.7 kNm in C8-1 to 6.6 kN m in column C8-4 (an increase of about 15%).
4.1.9 Ductility
To determine the displacement ductility of the UPTSCs, the capacity curves were first developed using the analytical method presented previously. The idealized bilinear curves of the capacity curves were then obtained using the following procedure, which has been widely used and adopted in previous studies (Priestley and Park 1987; Park and Paulay 1975; Ho and Pam 2002). The ultimate displacement, \( \Delta_{u} \), was taken as the displacement when the lateral strength of the column dropped by 20%. The yield displacement, \( \Delta_{y} \), yield strength,\( V_{y} \), and the effective yield stiffness, \( k_{e} \), was obtained using bilinear approximation of the force–displacement response of the columns, and the post-yield stiffness had a zero value (perfectly plastic). An iterative procedure was used to determine the idealized bilinear curves adopting the equal energy concept. The yield displacement and yield strength were determined such that the area under the idealized and capacity curves were equal and the two lines intersected at a strength of 0.75V _{ y }.
As shown in Fig. 15h and i adding FRP wrap resulted in decreased ductility. Also, the variation of the FRP thickness and ultimate tensile strain had negligible effect on ductility, and hence can be ignored. Consequently, for FRP-confined specimens, regardless of the thickness and type of FRP material, a minimum ductility of two can be considered. In the experimental study conducted by Hadi (Hadi 2009) the same adverse effect of FRP wrapping on the ductility of columns, beams and beam-column members was observed. This can be attributed to the method of bi-linearization of the capacity curve as well as the sudden rupture of the FRP layer at high drifts, which results in rapid strength degradation after the peak strength. Note that although the FRP-confined columns presented a low level of ductility they exhibited a large displacement capacity (Table 3).
4.2 Parametric Study—Stage II
In the previous section it was concluded that the ductility values of the specimens without FRP wrap depended mainly on two factors, the level of axial stress ratio and the column’s aspect ratio. The configuration of the columns in Stage II of the parametric study, which included 36 columns, was determined according to the conclusions obtained from Stage I of the parametric study. As mentioned, stage II of parametric study was performed on a set of columns considering two parameters, axial stress ratio and aspect ratio as main variables to investigate the ductility of UPTSCs. Axial stress ratios of 0.032, 0.042, 0.057, 0.109, 0.212 and 0.263 were considered by applying different PT forces. The aspect ratio was varied from 3.3 to 20, by considering six different column heights ranging from 0.5 to 3.0 m. The other parameters were fixed in all columns and were similar to the control specimen described in Stage I. Similar to Stage I, the force–displacement responses of the columns were first determined using the analytical procedure previously developed, and these responses where then used to calculate the ductility of the columns according to the procedure presented.
Figure 16b indicates the effect of axial stress ratio and aspect ratio on the moment capacity of the columns of Stage II of the parametric study. As shown in the figure, increasing the axial stress ratio resulted in an increase in the moment capacity of the column. This behavior was nearly independent of the column’s aspect ratio, especially for the columns with higher axial stress ratios. At low axial stress ratios, the moment capacity of columns with lower h _{ c } /D was slightly higher than those with higher h _{ c } /D. This can be attributed to the higher level of force in the PT steel in columns with lower h _{ c } /D. Between two identical columns with different heights, the unbonded length of the PT steel is usually smaller in the shorter one, resulting in a higher level of force developed in the PT steel. Of two identical columns with different diameters, at the same drift ratio, the elongation of the PT steel in the column with greater diameter is higher (due to greater lever arm), leading to a higher level of force in the PT steel.
The moment capacity of the columns with low axial stress ratio is comparatively small, as shown in Fig. 16b. In practice to reach the required strength and to provide an economical design the axial stress ratio is usually higher than 5% (Hassanli et al. 2016b). Moreover, a minimum level of axial stress ratio is required to prevent sliding of the segments and provide enough shear capacity transferred between the segments and at the column-footing interface.
4.3 Neutral Axis (NA) Depth
If the depth of neutral axis is known, the elongation of the PT steel and the concrete deformation at the compression zone at different column rotations can be determined accordingly. As shown in Figs. 7, 8, and 9 at peak strength, the concrete compressive strength and axial stress ratio influence the NA depth however, the level of axial stress seems to have no effect on the NA depth. The column height, FRP thickness and FRP ultimate tensile strains have negligible effect on the NA depth (see Figs. 12, 13, 14), hence, can be ignored, however as shown in Fig. 11, column diameter does have considerable an effect on the NA depth. Using the column matrix of Stage I (Table 3), multivariate regression analysis was performed and the following equation was obtained to determine the NA depth of UPTSCs,
5 Conclusion
- 1.
The analytical approach considering single joint rotation, could effectively predict the response of UPTSCs. It was shown that the moment capacity of UPTSCs was highly sensitive to the level of axial stress and axial stress ratio, and less sensitive to the concrete compressive strength.
- 2.
Oversized duct has negative effect on the effective depth of the PT bars, and hence the lateral strength of the columns. Hence, the strength reduction due to oversized ducts must be taken into account.
- 3.
The ductility of unbonded post-tensioned columns is strongly affected by two factors, the level of axial stress ratio and the aspect ratio. However, as long as the axial stress ratio, \( f_{c}^{'} /f_{c} \), is constant, the level of axial stress, \( f_{c} \), and compressive strength of concrete, \( f_{c}^{'} \), has nearly no effect on the ductility.
- 4.
For unconfined UPTSCs, for high levels of aspect ratio (h _{ c } /D > 10), the ductility decreased as the axial stress ratio increased; however, at low levels of aspect ratio (h _{ c } /D < 10) the ductility increased as the axial stress ratio increased. Equation 30 was proposed to estimate the displacement ductility of those columns.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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