Seismic performance comparison between direct displacement-based and force-based design of a multi-span continuous reinforced concrete bridge with irregular column heights

The CHBDC 2010 considers response spectra corresponding 10% probability of exceedance in 50 years for bridge design, however, the draft CHBDC 2014 suggests response spectra corresponding 2% probability of exceedance in 50 years for FBD of bridges. Therefore in this study, the base shear demand has been computed through elastic response spectrum analysis, which has been conducted considering the acceleration response spectrum corresponding 2% probability of exceedance in 50 years for Vancouver (NBCC 2010). Elastic response spectrum analysis is performed by calculating the elastic base shear demand for the natural period of the bridge. The response spectrum for Vancouver is shown in Fig. 3. The effective stiffness and natural period of the bridge in longitudinal direction have been determined as 576 276 kN/m and 0.84 s, respectively. The effective stiffness for cracked section has been calculated by multiplying stiffness reduction factor (0.7) to the initial bending stiffness of the columns considering uncracked moment of inertia. The elastic base shear demand has been calculated as 30 102 kN by multiplying the mass of the bridge with the spectral acceleration corresponding to the natural period of the structure. According to CHBDC 2010, the response modification factor R = 3 for ductile reinforced concrete. Therefore, the inelastic base shear demand is one third of the elastic base shear demand (i.e., 10 034 kN). The total base shear has been distributed to each of the three columns according to their initial stiffness, which is given in Table 2. The design reinforcement for the columns is given in Table 3. The shear capacity of the column should be greater than the elastic shear load (CHBDC 2010), which determines the required tie spacing. The limit state varies with tie spacing for the same ductility demand, however, in FBD, there is no iteration for adjusting the response modification factor with tie spacing to attain the required limit state. The longitudinal reinforcement in column C1 is 92-35 mm bars with 12 mm tie bar at 55 mm c/c. Minimum 1% reinforcement governs in columns C2 and C3 for design of longitudinal reinforcement. Maximum 150 mm spacing has been provided for 10 mm tie bar, used for the longitudinal bars of 25 mm, at each alternate longitudinal bar. According to CHBDC 2010 the maximum tie spacing is the smallest of six times the longitudinal bar diameter or 0.25 times the minimum component dimension or 150 mm and tie should cover every longitudinal bar.

In displacement-based approach, a target displacement and effective stiffness of the bridge have to be determined to calculate the base shear demand. The procedure described in Priestley et al. (2007) has been adopted in this study. In DDBD a number of design solutions are possible by choosing column section size and level of ductility. Therefore, the designer has options in choosing column size and tie spacing. In this study 3 m × 1.5 m section is chosen, which has been designed for damage control limit state. Since column C1 is the shortest column, the yield displacement and target displacement of the bridge are governed by this column. From section size, the yield displacement has been found to be 22.9 mm. Confinement reinforcement in C1 is provided as 12 mm tie @ 135 mm c/c, the target displacement for damage control limit state is found to be 111.8 mm. Therefore, the corresponding ductility for C1, C2, and C3 are 4.89, 0.54, and 1.22, respectively. Hence, the equivalent viscous damping ratio (ξ_eq) of an individual column is computed as (Priestley et al. 2007)

Here, the first part of the equation is for 5% material damping of concrete and the second part is for hysteresis damping associated with ductility (μ). The equivalent viscous damping ratios for C1, C2, and C3 have been found to be 0.162, 0.05, and 0.076, respectively. The equivalent viscous damping ratio for the whole system ξ_sys is derived according to eq. (2) (Priestley et al. 2007).

where V_i is the distributed base shear in each column and ξ_i is its corresponding equivalent viscous damping ratio, where m is the number of columns. Base shear distribution in the column is proportional to the inverse of the column height in DDBD. The equivalent viscous damping of the bridge has been found to be 11.83%. The 5% damped displacement spectrum has been derived from SeismoSignal (2010) at 5% damped acceleration spectrum for Vancouver. The displacement spectra R_ξ for Vancouver at 11.83% damping has been derived by applying the spectral reduction factor (eq. (3)) to the 5% damped displacement spectrum, as shown in Fig. 4 (Priestley et al. 2007).

The effective time period (T_e) of the system for the target displacement of 111.83 mm is 1.747 s, which is determined from the displacement response spectrum of Vancouver. The effective weight (W_e) of the bridge is 101 043 kN. The effective stiffness K_e of the structure has been determined according to eq. (4).

Total base shear demand has been calculated as 14 912 kN by multiplying the effective stiffness and the target displacement. This base shear is distributed to each column in inverse proportion to the height of the column. Therefore, columns will be subjected to equal bending moments, which leads to equal design longitudinal reinforcement. The design base shear and bending moment in each column is given in Table 2. The design reinforcement is provided in Table 3. Design longitudinal reinforcement for each column is 92-35 mm bar. The shear capacity of the column should be greater than 1.6 times the base shear corresponding to the design moment (Wang et al. 2008). Tie bar for C1 is 12 mm at 135 mm c/c as mentioned previously. The required lateral reinforcement in C2 and C3 are 12 mm at 600 mm and 400 mm, respectively. Due to lower ductility demand and lower base shear demand, the required lateral reinforcements in C2 and C3 are low. However, tie spacing more than 300 mm is not very common. The maximum tie spacing allowed in CHBDC 2010 is 150 mm whereas there is no strict guideline for DDBD. Therefore, the comparison between DDBD and FDB are as follows:

For bridges with irregular column heights, the distribution of total base shear demands for flexure and shear design of columns are different in DDBD and FBD. In DDBD the total base shear in an individual column is inversely proportional to the column height, resulting equal bending moment demand in each column, which can be expressed as

where V_i is the base shear of an individual column, V is the total base shear, and H_i is the height of the column. Therefore, the longitudinal reinforcement for each column is the same. However, in FBD, the base shear distribution is inversely proportional to the cube of the column heights (eq. (6)); resulting design moments in the columns are inversely proportional to the square of the column heights

As per FBD, the required amount of longitudinal reinforcement is significantly less in longer columns than shorter columns compared to that of DDBD. In the case of this bridge, the amount of longitudinal reinforcement in the longer columns is governed by the minimum steel–concrete ratio of 1%. Therefore, longitudinal reinforcement in the longer columns is higher than required. The main differences in the two design methods include the following:

To evaluate the dynamic performances of the three DDBD bridges (defined in Section 2.1) and one FBD bridge finite element models have been generated in SeismoStruct (2010), which is based on the fibre modeling approach. This software has been validated under reversed cyclic loading of RC elements (Alam et al. 2008), and shake table tests of RC columns (Alam et al. 2008) and RC frames (Alam et al. 2009). The cross section of a member is divided into a number of fibres and the uniaxial response of the individual fibre is obtained from the nonlinear stress–strain behavior of the material. These responses of the fibres are integrated to get the sectional response of the member. However, this model does not take shear deformation into account. Concrete has been modeled using the nonlinear constant confinement concrete model, which was initiated by Madas (1993) following the constitutive relationship proposed by Mander et al. (1988) and the cyclic rules proposed by Martinez-Rueda and Elnashai (1997). Confinement factor has been determined according to Park et al. (1982). Steel has been modeled using the model of Monti and Nuti (1992) with an additional memory rule (Fragiadakis and Papadrakakis 2008) for higher numerical stability under transient seismic loading. Here, the deck column joints are considered to be rotationally fixed. Abutments are free to move in the longitudinal direction and fixed in the transverse direction. Nonlinear time history analyses have been conducted with these bridge models to compare the performance of the bridges designed in displacement-based and force-based approach. Uniformly distributed mass of 11.1 t/m and 20.2 t/m have been assigned to the column and deck element, respectively. Modal analysis has been performed in SeismoStruct and the fundamental periods for the bridges designed in DDBD and FBD have been found to be 0.692 and 0.696 s, respectively, which are very close. The scale factors have been determined at the period of 0.69 s.

The structural response depends on the ground motion properties of the earthquake, which can vary widely in terms of predominant period, peak ground acceleration (PGA), peak ground velocity (PGV), and duration. Since, the upcoming earthquake characteristics in any area is truly unpredictable, a set of earthquake ground motions is generally selected containing different ground motion characteristics to predict the worst structural response. In this study, an ensemble of 17 ground motion records has been selected. The properties of these ground motions are provided in Table 4 and the acceleration spectra are plotted in Fig. 5. The predominant periods of the ground motions vary from 0.09 to 4.55 s, whereas, the PGA varies from 0.22g to 0.73g. Since, the bridges have been designed for Vancouver involving firm soil with 2% probability of exceedance in 50 years, the original earthquake ground motions need to be scaled to fit the 5% damping Vancouver’s design spectra (Fig. 3). Geometric mean scaling, spectrum-matching scaling, and fundamental period scaling are conventional scaling techniques. Since the risk computation of the bridge will be in longitudinal direction, this can be considered as a single degree of freedom system. Therefore, spectrum-matching system is not suitable for scaling the ground motions (Huang et al. 2011). Fundamental period scaling gives better results than geometric mean scaling. The frequency content of the ground motions remains the same in the fundamental period scaling method. In this study, the geometric dimensions and the initial fundamental period of the bridge designed in two methods match, however the reinforcement in the columns differs. Since the bridge designed in the two methods has the same initial fundamental period and the system is SDOF, performance computation with a set of ground motions scaled at fundamental period will give reasonable results. Therefore, the earthquake ground motions have been scaled at the fundamental period of the structure (Shome et al. 1998) (Fig. 6).

Longitudinal responses of the bridge have been simulated by NTHA using SeismoStruct (2010) for 17 scaled ground motions. Collapse of bridge in time history analysis is defined for instability or exceeding the 5% drift (Dutta and Mander 1999) of the column. The FBD bridge collapsed for the scaled ground motion of eq. (2). The DDBD bridge collapsed in eqs. (2) and (9). The DDBD bridge with limited tie spacing and equal tie spacing of 144 mm survived all 17 earthquakes. Maximum displacement demand, residual displacement, and energy dissipation have been set as the parameters to compare the seismic performance of the bridges. To compare the four cases, mean (μ) and standard deviation (σ) have been determined for these parameters excluding eqs. (2) and (9), since collapse occurred in some of these cases in these two earthquake ground motions. Mean plus standard deviation and mean plus twice standard deviation represents 68% and 95% data are within the limit.

Figure 7 shows the displacement demand comparison between the four cases and the maximum, μ, μ + σ, and μ + 2σ values for the ground motion excitations. Among the three DDBD bridges, the bridge with limited tie spacing has the lowest displacement demand in terms of μ, μ + σ, and μ + 2σ. The displacement demand of the FBD bridge is lower than that of the DDBD bridges, although, it is very close to the original DDBD bridge and the DDBD bridge with limited tie spacing. The DDBD bridge with equal tie spacing is the worst among the four bridges.

Figure 8 shows the residual displacement of bridges under ground motion excitations and the maximum, μ, μ + σ, and μ + 2σ values for the ground motion excitations. The DDBD bridge with limited tie spacing performs the best with respect to residual displacement among the DDBD bridges in terms of μ, μ + σ, and μ + 2σ. The residual displacement of the DDBD bridge is less than half of that of the DDBD bridge with limited tie spacing, which indicates that, the performance of DDBD bridge can significantly be improved by imposing the maximum allowable tie spacing rule as per Canadian code. Residual displacement is the lowest in the case of FBD bridge, which implies the better capability of the FBD bridge to restore its original position.

The energy dissipation under seismic loading is maximum in the case of FBD bridge and the maximum, μ, μ + σ, and μ + 2σ values for the ground motion excitations. The minimum energy dissipation in the case of DDBD bridge with limited tie spacing in terms of μ, μ + σ, and μ + 2σ, which is shown in Fig. 9. The FBD bridge shows the highest energy dissipation capacity. However, variation in energy dissipation among the four cases is very low (less than 6%) for μ, μ + σ, and μ + 2σ.

Figures 10 to 12 show the base shear demand in columns C1, C2, and C3, respectively, for the ground motion excitations. Base shear demands in each column have been found lower than the shear capacity, therefore, no shear failure has been observed. Since FBD takes the elastic base shear for shear design, the shear capacity of 7 m column in the FBD bridge is more than twice that of the base shear demand; however, the design moment is 23% lower than the demand. Both of the design base shears for shear design and base shear corresponding to design moment for 14 m and 21 m column are lower than those of the base shear demands from time history analyses. Since the provided longitudinal and transverse reinforcement in 14 m and 21 m columns are governed by the code specified minimum amount, the actual capacities of these columns are higher than the demand. Therefore, these columns did not fail in the time history analyses. The FBD is not accurate in determining the design load in shorter and longer columns. It was observed that the shear capacity of 7 m column in DDBD bridge is 21% higher than the base shear demand. The design moment of 7 m column in DDBD is almost equal to that of FBD; however, FBD experienced higher demand by 20% compared to that of DDBD. In longer columns, unlike FBD, the design moment as per DDBD is equal to that of shorter column, whereas it is smaller by 24% and 19% in the case of 14 m and 21 m FBD columns compared to that of 7 m FBD columns, respectively. The shear capacities of 14 m and 21 m columns in DDBD bridges are 21% and 29% higher than those of the demand base shear, respectively. Hence, the shear capacity for all short and long columns in DDBD is above the demand, and its moment capacity is slightly lower than the demand, however, this method ensures an even distribution of base shear to the columns of different heights.