Ductility Design of Reinforced Very-High Strength Concrete Columns (100–150 MPa) Using Curvature and Energy-Based Ductility Indices

Previous studies shows that unconfined VHSC behave different to Normal Strength Concrete (NSC) (Kristombu Baduge et al. 2018; Lee et al. 2018). Even though researchers have proposed various types of stress–strain models for unconfined HSC based on extensive experimental programs, only a few stress–strain models, such as those proposed by (Attard and Setunge 1996; Candappa et al. 1999; Légeron and Paultre 2003; Razvi and Saatcioglu 1999; Samani and Attard 2012) can predict the constitutive behaviour of concrete with strengths greater than 100 MPa. The stress–strain model developed by Légeron and Paultre (2003) is valid up to 130 MPa and Kristombu Baduge (2016) used this model for unconfined VHSC up to 150 MPa with some modifications. Thus, the model proposed by Légeron and Paultre (2003) is selected to model the unconfined behaviour of VHSC with a modified peak strain and modulus of elasticity.

The stress–strain models for confined VHSC proposed by researchers such as (Attard and Setunge 1996; Cusson and Paultre 1995; Razvi and Saatcioglu 1999) are valid up to compressive strengths of 125–130 MPa, and are developed assuming that the behaviour is governed by the confining effects from lateral reinforcement only. Previous experimental studies (Kristombu Baduge et al. 2018b, c) showed that the confined behaviour of VHSC is dependent on confinement from the lateral reinforcement, dowelling action of the lateral and longitudinal reinforcement, and the concrete shear force at the failure plane, as shown in Fig. 2. However, existing models are derived considering confinement effects only and the experimental program by Kristombu Baduge (2016) showed that incorporating the doweling action and shear force at the failure plane (Fig. 2) due to cohesion and friction can predict the results more accurately.

The model is developed for compressive strengths in the range of 120–160 MPa and is validated using experimental results (Kristombu Baduge 2016). The stress–strain model proposed by Kristombu Baduge (2016) was used in the analytical program (see Fig. 3b). Further details of the new stress–strain model can be found in elsewhere (Kristombu Baduge et al. 2018a). The stress–strain model for confined concrete is valid for circular sections so that the method proposed in the study can be used to predict the moment–curvature relationship of circular columns. The parameters for the material constitute models can be found in Kristombu Baduge et al. (2018).

The stress–strain model proposed by Samra (1990) is used for the analysis of the steel reinforcement. The experimental program by Kristombu Baduge (2016) on confined VHSC samples showed that the stress–strain behaviour of the confined core is related to the buckling of longitudinal steel and hoop fracture. Similarly, Mendis et al. (2000) showed that it is important to consider hoop fracture and buckling of longitudinal reinforcement in the prediction of accurate moment–curvature behaviour.

First, hoop fracture can be associated with the ultimate strain of confined concrete due to an abrupt reduction in the capacity and subsequent failure of hoops due to sudden load transfer. Equation (7) proposed by Wei and Wu (2014) is used to calculate the ultimate strain.where ε_cc and ε_co are the peal strain of confined and unconfined concrete, respectively, f_r is effective confinement pressure, f_co is compressive strength of concrete.

Buckling of longitudinal steel bars can occur due to higher axial compression after the spalling of cover concrete (Bae 2006, Chapter 6; Bayrak and Sheikh 2001; Kristombu Baduge 2016; Mander et al. 1984). The tensile and compressive behaviour of a steel bar is different when the steel is subjected to buckling. Therefore, accurate modelling of the buckling and stress–strain behaviour of steel is important to predict the post-peak behaviour of the moment–curvature relationship. Researchers such as (Bae et al. 2005; Bayrak and Sheikh 2001; Bresler and Gilbert 1961; Mau 1990; Mau and El-Mabsout 1989; Papia and Russo 1989; Scribner 1986) developed the stress–strain behaviour of steel under compression and incorporated inelastic buckling. The method proposed by Bae et al. (2005) is used in the analytical program.

Figure 3c presents the nonlinear full-range moment–curvature behaviour predicted with the MATLAB analytical program for a square column under axial load ratios corresponding to 0.1 to 0.5. For this column, the following parameters were used: ρ_l = 0.17, f_r = 4 MPa, f_c = 130 MPa: ρ_l is longitudinal steel ratio, f_r is confinement pressure and f_c is compressive strength of concrete.

Using the full-range moment–curvature diagram, the curvature ductility can be evaluated. Figure 4 illustrates the curvature ductility curve of VHSC columns with a similar geometry to that shown in Fig. 3a. The columns have a compressive strength of 130 MPa, different axial load levels (P/f_cA_c = 0.1, 0.2, 0.3, 0.4 and 0.5), and different confinement levels (f_r = 1, 2, 3 and 4 MPa).

Kristombu Baduge (2016) showed that the curvature ductility cannot represent the increase in the plastic energy of VHSC columns along with an increase in the peak moment (Fig. 5) due to the confinement effect, higher strength and higher axial load levels. Plastic energy is related to the energy dissipation characteristics during cyclic deformation in an earthquake event. Incorporating the plastic strain energy into ductility indices is vital for VHSC members because they dissipate relatively large energy compared to a normal strength concrete member of the same ductility (Elmenshawi 2008) as shown in Fig. 5, and can thereby have higher energy dissipation characteristics.

Considering the importance of incorporating plastic energy into the ductility index, the importance of flexural deformation in an earthquake event and the load sustaining capacity of columns, (Kristombu Baduge 2016) proposed the flexural rotation-based energy ductility index (E_θ) to evaluate the ductility of VHSC columns. The expression given by Eq. (8), based on the rotational energy of flexural deformation, is proposed for the flexural rotation-based energy ductility index, E_θ:where M and θ are the moment and rotation of the moment–rotation graph, respectively; θ_y is the yield rotation; and θ_u is the rotation at ϕ_u. The moment–rotation behaviour can be predicted using the moment–curvature relationship once the plastic hinge length and height of the column are known. The moment–rotation graph can be generated using the moment–curvature graph for a double-curvature column. The total rotation due to flexure deformation can be calculated by adding elastic rotation and plastic rotation as given in Eq. (9) and Eq. (10), respectively which proposed by Elmenshawi (2008): for θ_y ≥ θwhere θ is rotation of a prismatic element with double curvature, ϕ is curvature, and L is length of the column.

Assuming the plastic curvature is constant and uniformly distributed over the plastic hinge length, L_p: for θ_y < θwhere L_p is the plastic hinge length and ϕ_y is yield curvature. The method proposed by Park et al. (1982) for plastic hinge lengths of HSC columns is used to calculate the plastic hinge length of VHSC columns. The yield rotation is defined as the rotation related to the yield curvature (ϕ_y). The ultimate rotation for the index is taken as the rotation at 0.8 times the peak moment at the post-peak branch. Thus, the index has a provision for sustaining the load carrying capacity beyond the peak moment. Therefore, the new index gives better insight for ductility by considering energy dissipation characteristics, as well as the ability of VHSC columns to deform while sustaining its load carrying capacity. In order to calculate the rotational energy due to flexural rotation, rotation due  to bond slip and shear deformation were  not included in the calculation.

Figure 6 indicates the flexural rotation-based energy ductility (E_θ) for the column illustrated in Fig. 3a, with a compressive strength of 130 MPa, an axial load ratio varying from 0.1 to 0.5, a height of 3.3 m and confinement pressures in the range of 1 MPa to 4 MPa. The energy ductility values are higher than the curvature ductility for columns. Similarly, the curvature and energy ductility of columns with different geometries, compressive strengths, steel ratios and axial load levels can be evaluated using the analytical program. Subsequently, using a regression analysis on different parameters, the relationship between ductility and governing parameters can be established.

Axial load level ratio is the ratio of the applied axial load (P) to the axial compression capacity of the column (A_gf_c). The full axial compression capacity is calculated by taking the product of the gross cross-sectional area (A_g) and the compressive strength of concrete (f_c). The contribution to the load carrying capacity from the longitudinal steel is neglected. This factor represents the effect of axial load level on the ductility of columns.

Confinement pressure represents the amount of confinement given to the core. In addition, it represents the transverse steel requirement (ρ_s) to obtain a particular curvature ductility. In this dimensionless parameter, the volumetric ratio and effective transverse steel strength is included, which improve the ductility due to confinement effects and the dowelling action of transverse steel against relative sliding of the failure planes. The confinement pressure is calculated using the method proposed by Mander et al. (1988).

This ratio represents the effect of early spalling of cover concrete. The section area ratio is given by the ratio of the gross area (A_g) to the core area (A_c).

Even though many researchers have not considered the effects of longitudinal steel, the experimental results showed that longitudinal steel improves the descending branch of core concrete due to dowelling action and constrains the movement of concrete. Thus, the material ductility of the confined core is enhanced, which results in improved column ductility. The dowelling action is related to the longitudinal steel ratio (ρ_s), compressive strength of concrete (f_c) and yield strength of longitudinal steel (f_y). In addition, the longitudinal steel ratio relates to the confinement effectiveness coefficient so that it is related to confinement pressure.

Based on studies by Foster (1999), the limited ductile design in AS 3600 (2009) suggests that the ductility level of a 50 MPa column detailed according to the minimum requirements of AS 3600 (2009) is adequate for ordinary moment resisting structures. This is due to the fact that an equivalent 50 MPa column has adequate ductility and the code does not specify any additional confinement to achieve ductility. In this study, a similar approach is suggested for nominal ductility design with low-seismic demands. The basis for curvature-based nominal ductility design is to provide similar curvature ductility to a 50 MPa column with equivalent geometry and axial load level, detailed according to the minimum requirements specified in AS 3600 (2009). Thus, the required nominal ductility levels of Eq. (16) shall be similar to the curvature ductility level of an equivalent 50 MPa column. Using the analytical program, the curvature ductility of 50 MPa rectangular columns with different geometries and detailed according to the minimum requirements of AS 3600 (2009) is evaluated at different axial load levels; for a 50 MPa concrete column, the stress–strain behaviour of confined and unconfined concrete proposed by Mendis et al. (2000) is used in the analytical study. The columns with compressive strengths of 50 MPa used in the present study have different ductility levels which depend on the axial load level. Thus, the required nominal ductility levels in Eq. (16) for VHSC columns must be modified according to the different axial load levels of a 50 MPa column. The expression given by Eq. (17) is derived using regression analysis for the ductility of a 50 MPa column with different axial load levels. Thus, Eq. (17) is proposed for determining the nominal curvature ductility level of VHSC columns with different axial load levels.

Therefore, combining Eqs. (16) and (17), the following equation can be derived for the minimum ductile design of VHSC columns:

For lower axial load levels and lower longitudinal steel ratios, steel yields prior to the crushing of concrete and can sustain large deformation. Therefore, columns subject to the above criteria have higher curvature ductility levels. Thus, the required confinement level to achieve nominal ductility is less, and the equation may give negative or very low confinements for nominal ductility. As such, the transverse requirement for anti-buckling or spalling must be set as the minimum transverse reinforcement.

As discussed previously, considering the drawbacks of curvature ductility in capturing the enhanced plastic energy of VHSC columns at higher axial load levels and higher strengths, this study suggests that the flexural rotation-based energy ductility (E_θ−) as the ductility index, and the ductility level of an equivalent 50 MPa column with similar geometry and axial load levels as the nominal ductility level of VHSC columns, considering the fact that a 50 MPa column has adequate ductility for limited ductility demands; This is somewhat similar to the approach of AS (2009) which proposed by Foster (1999).

A parametric study showed that axial load ratio, confinement pressure, compressive strength, cover and longitudinal reinforcement ratio have effects on flexural rotation-based ductility index, E_θ. An equation format that is similar to Eq. (11) with similar dimensionless parameters is used in the regression analysis, considering the similarity of parameters for the curvature ductility index and energy ductility index. The energy ductility index is calculated for the range of parameters given in Table 4. Nonlinear regression analysis, similar to the previous analysis, is carried out for the prescribed equation and analytical results. Excluding the outliers, regression analysis is carried out for a better statistical fit of the analytical data. The regression analysis gives Eq. (19), with the following coefficients and indices, and the equation has a very good agreement with the analytical results. Once the outliers are removed, the coefficient of correlation of the proposed equation is 0.90, which is acceptable to predict the flexural rotation-based energy ductility (E_θ) of VHSC columns.

The equation can be further simplified into the following equation to find the required confinement level:

Figure 12 shows a comparison between the design equation and analytical predictions. It can be concluded that the new expression given by Eq. (19) has a very good correlation with the analytical results. Thus, Eq. (20) can be used to find the required confinement level for a particular energy ductility level. Since the equation is developed to target the confinement pressure instead of the volumetric ratio of transverse steel, the equation can be used for both circular and rectangular columns.

The following equation, which satisfies the 95th percentile, can be proposed for the confinement pressure versus the energy ductility index. Thus, Eq. (21) can be used to determine the required confinement level for energy-based nominal ductility design.

The energy ductility of 50 MPa columns detailed according to the minimum requirements of AS 3600 (2009) is suggested for nominal ductility design. Using regression analysis, the expression given by Eq. (22) is proposed for the energy ductility levels of 50 MPa columns, with the minimum detailing requirements specified in AS 3600 (2009). Thus, the nominal energy ductility level of VHSC columns is given by Eq. (22) for different axial load levels.

Combining and simplifying Eqs. (21) and (22), the required transverse steel quantity can be found for energy-based nominal ductile design, which is represented by the following expression:

The curvature-based method and energy-based method for nominal ductility design are compared with the existing expression proposed by CSA (2004), ACI 318 (2011) and EC 8 (2004a). However, it must be noted that these methods are not applicable or developed for strengths up to 150 MPa, and the results have been extended/extrapolated for strengths up to 150 MPa though it is not appropriate-The strength limits of guidelines and codes are given in Tables 1 and 2. The results are extended over the material strengths limits of codes and guidelines just to give an indication on confinement requirement of the new methods if strength limits can be extrapolated. For nominal ductility levels, NZS 3101.1 (2006) specified a 70% transverse reinforcement requirement of ductile design (with µ_ϕ = 19) due to anticipated higher seismic demands in the region, and regions with low-seismic demands may not require much ductility. Thus, NZS 3101.1 (2006) gives a large amount of transverse steel for nominal ductility design. Unrealistic volumetric ratios that cannot practically be provided for columns are given by NZS 3101.1 (2006). Therefore, NZS 3101.1 (2006) is excluded in the following comparison.

Figures 13, 14, and 15 compare the transverse steel requirements for a typical nominal ductile VHSC column with a square cross-section of 1500 × 1500 mm, compressive strengths of 100 to 150 MPa, axial load levels of 0.1, 0.3 and 0.5, and longitudinal ratios of 1.0%, 2.5% and 4.0%, with 500 MPa transverse and longitudinal steel. The first and second columns of graphs show the variations in the required volumetric ratio and spacing of transverse steel with the compressive strength of concrete, respectively. As mentioned previously, for lower axial load ratios, the required confinement level to gain nominal ductility is low and can be negative in some instances. Therefore, a minimum transverse steel amount is set to satisfy the requirement for anti-buckling as specified in AS 3600 (2009).

CSA 23.3 (2004) and EC 8 (2004a) standards do not give any guidelines for nominal ductility, and transverse steel requirements are governed by the anti-buckling condition. Thus, Constant volumetric ratios are given by these codes for nominal ductility design, regardless of different axial load levels, compressive strengths and longitudinal steel ratios: In order to calculate transverse steel requirement for anti-buckling, the diameter of longitudinal steel is considered as greater than or equal to 25 mm. AS 3600 (2009), which is based on an energy-based method (I₁₀), gives a volumetric ratio of transverse steel, considering the effects of axial load ratio and the compressive strength of concrete. The code has not included the effects from the dowelling action of longitudinal and transverse steel, although it considers the effects of steel configuration on the confinement pressure.

According to Figs. 13, 14, and 15, transverse steel requirements for curvature ductility-based design and energy-based design are increasing with the increase of strength and axial load levels. The material model used for confined VHSC considered the contribution of dowelling effects of longitudinal and transverse steel on the plastic behaviour of confined concrete. Thus, the longitudinal steel ratio has a contribution to the ductility of the material, and consequently, to the ductility of the column. The comparison between Figs. 13 and 15 showed that increasing the longitudinal steel ratio decreases the transverse steel quantity. The gap between curvature-based design and energy-based design is increasing with the increase of strength and axial load level. It can be concluded that the increment of transverse steel required for an increase of strength and axial load are not similar for curvature- and energy-based ductility design. The transverse steel requirement for curvature-based design is much higher than that for energy-based design when the strength of concrete and axial load levels increase. For higher compressive strengths and axial load levels, VHSC columns have higher peak moments relative to the yield moments. Therefore, it results in higher plastic energy in the moment–curvature graph. Thus, energy ductility is relatively high, requiring a lower amount of transverse steel relative to curvature-based design. However, the enhancement in energy is not captured by the curvature ductility index and gives higher confinement relative to the energy-based method. Therefore, the energy-based approach provides an economical design for VHSC columns. The comparison shows that VHSC columns can obtain nominal energy and curvature ductility levels using practical stirrup spacing and can be used in structures located in low-seismic regions with nominal ductility demands.