Numerical investigation of seismic rehabilitation of mass concrete piers of a railway bridge using post-tensioning technique

Abstract

Lightly reinforced mass concrete piers are seismically vulnerable due to their excessive weight and inadequate reinforcement. The present paper aims to outline an innovative method for seismic rehabilitation of such piers using post-tensioning technique. In the numerical case study carried out in the paper, it is shown that a prescribed compressive stress about five percent of the pier axial strength can promote the lateral load-carrying capacity by 60 percent. Moreover, the piers with originally brittle behavior show proper ductility applying adequate prescribed compression. The capability of the rehabilitated piers to reveal stable non-linearity is resulted in a minimum seismic response modification factor of 3.0, approximately. This value is comparable with unity considered for the existing piers. The nonlinear analysis conducted under cyclic loading confirms that the efficiency of this method does not considerably degrade during the loading excursions. Finally, a simplified method is proposed to estimate the amount of compression needed to achieve required strength and ductility.

Keywords

Seismic rehabilitation mass concrete pier post-tensioning technique static nonlinear analysis cyclic behavior

1 Introduction

Bridge mass concrete piers with no or improper reinforcement are highly susceptible to brittle failure during a major excitation. Such piers can undergo the earthquake actions while the overturning moment of the lateral forces does not dominate resisting moment due to gravitational loads. The equilibrium condition is often not met owing to the excessive weight of the piers, and its subsequent earthquake forces. Accordingly, the rehabilitation of this sort of pier is mandatory, especially in the high risk seismic zones.

To improve the seismic behavior of mass concrete piers, two strategies may be traced. First, strengthening of the pier to reach more lateral load-carrying capacity; and second, increasing the piers capability to dissipate a part of input energy induced by earthquake. These strategies may be resulted through traditional methods such as steel and concrete jacketing [14]. However, since the dimensions of mass concrete piers are commonly large, the jacketing methods do not work efficiently unless massive details are employed. Such details are often expensive and difficult to use.

In the first method, the pier should be so strong to tolerate any probable severe excitations. The second method is aligned to the current design codes in which the piers are allowed to experience inelastic deformation during severe earthquakes. There is a major drawback with this philosophy. The drawback is unavoidable damage of piers after earthquake, leading to high expenditure needed to bring the bridge into the service. This has encouraged researchers to find more sophisticated methods.

One recent improvement in this regard is the Damage Avoidance Design (DAD) philosophy. This concept was first introduced as Rocking Systems [1 , 13]. In the original usage of this system in bridges, the piers acted as a rigid block, freely rotating about their base toe. Mander and Cheng [9] showed that the performance of this system can be improved using post-tensioned anchorage connecting the pier to its foundation. This method has been improved using supplemental energy dissipation devices [7 , 16].

The DAD concept may well be adapted to precast piers in which a separation interface exists between the pier and its foundation. For the case studied in this paper there is no possibility to provide a separation surface. However, the main idea may still be employed. The main idea is to apply a prescribed compressive stress on the piers to prevent the concrete from unstable cracks. In this condition the pier demonstrates more ductility. Moreover, the method leads the pier performance from a flexure-dominated field to a compression-dominated one, resulting in more moment resistance. In essence, the pier attains more strength and ductility simultaneously. Wilson and Panian [19] and Walsh [18] reported use of this method for rehabilitation of two different bridges. However, no computational and technical details were provided in their publications.

Furthermore, a simplified procedure is developed to predict nonlinear behavior of mass concrete piers rehabilitated using post-tensioning technique. The performance of the method is confirmed through the static and cyclic nonlinear analyses. The numerical results imply that the ultimate strength and ductility of the pier are promoted considerably.

2 Research program

A numerical investigation was carried out on the seismic performance of a series of typical railway bridges. The general specifications of the bridges are shown in Fig. 1. The piers are of mass concrete sections reinforced with six I-section vertical elements as well as horizontal L-section ties distributed along the height. The piers have the gravel facet which was, in essence, used as a framework for concreting. The piers high 10 to 15 meters from the ground surface. The above-ground portion of the piers weighs 3150 kN to 5250 kN, approximately.

Fig.1

General specifications of the bridges.

The superstructure of the bridges includes two steel plate-girders as shown in Fig. 1. The timber cross-ties are connected to the girders directly without using slab and ballast. Therefore, the superstructure possesses notably light weight. The superstructure weighs about 17 kN per unit length. Therefore, the concentrated dead load transmitted from the superstructure to the pier is 306 kN ( $= 17_{/ m}^{kN} x 18^{m}$ ). This load is significantly less than the pier weight.

The laminated elastomeric bearing type EF175-18 (FIP products) has been provided between the superstructure and the pier. This type of bearing has a shear stiffness of 4000 kN/m, approximately. This value is negligible as compared with the pier lateral stiffness. In next sections, it is shown that the elastic lateral stiffness of the pier is 220000 kN/m, approximately.

The pier massive weight and the type of bearings denote that the dynamic response of the bridge system is mostly dominated by the piers behavior. Consequently, the rehabilitation of the piers to reach more resistance and ductility will guarantee the seismic response of the whole bridge system.

3 Soil-pier interaction

The piers were constructed on bedrock and embedded within a thick layer of dense soil. Since a considerable part of the piers are in contact with the soil, a separate analysis was performed to account the soil-pier interaction. The modeling and analysis were performed using LUSAS finite element software [8]. The modeling of the pier and soil is done in the state of plane strain using element QPN8 in LUSAS software. This element is an 8-node quadrilateral element with two translational degrees-of-freedom in each node.

In this phase, a linear material behavior assigned to the concrete pier, while a nonlinear one to the soil. The concrete modulus of elasticity was considered 21 GPa. The soil mechanical properties were determined based on the individual geotechnical studies employed in situ. According to the tests, the soil friction angle was 35 degrees and the corresponding cohesion 200 kPa. In the analysis, the soil properties were addressed employing the Mohr-Coulomb failure surface.

A push-over analysis was done on a typical model. To this end, the pier was incrementally loaded on its top. Figure 2 illustrates the plastic strain contour occurred in the soil. As observed, the soil superficially excesses the elastic limit. On the top right of the figure, the lateral deformation along the pier height is shown at three different times of loading. This figure proves that the soil effectively restrains the pier from lateral deformation; and in consequence the pier deformation is assumed to be negligible beneath a depth of 3-m from the ground surface.

Fig.2

Plastic strain within the soil and pier deformation profile.

For next analyses, the effect of the near-surface soil on the behavior of the pier is taken into account using a series of nonlinear spring elements. According to the soil-pier interaction analysis, the force-displacement relation for the compression springs is drawn as Fig. 3. The figure was depicted based on the nodal force and displacement obtained for the nodes in the pier-soil interface. Furthermore, for simplicity in next numerical analyses, a bilinear curve was best fitted to the obtained force-displacement relation.

Fig.3

Force-displacement relation of spring.

In case of cyclic loading, the soil springs were used in both sides of the pier (see Fig. 4). However, a compression-only action was assigned to the soil springs in order the springs not to be activated in tension. The compression-only spring had a negligible stiffness in tension and bilinear behavior in compression. According to Figs. (3) and (4), the elastic compression stiffness, K_ec, and the plastic one, K_pc, are 1300 and 270 kN/mm, respectively.

Fig.4

Soil spring modelling for cyclic loading.

4 Nonlinear behavior of existing piers

The seismic vulnerability of the bridge as a whole was first evaluated using the linear equivalent earthquake analysis based on FHWA [6]. According to the geotechnical case study, the bridges lie in the seismic design category D (SDC D) [3]. The evaluation showed that the piers were vulnerable about their minor axis due to the bridge longitudinal excitation while they had adequate strength about their major axis. Some important results obtained from this analysis are given in Table 1. The C/D ratios (Capacity/Demand ratio) for the bending about the minor and major axes are 0.63 and 2.46, approximately.

Table 1
Results from linear equivalent earthquake analysis

Direction of excitation Axial Force (kN) Moment (kN.m) T

P _c P _D C/D M _c M _D C/D (sec)

Longitudinal 396000 3150 125 17000 26780 0.63 0.17

Transverse 66000 26780 2.46 0.05

Direction of excitation	Axial Force (kN)	Moment (kN.m)	T
Longitudinal	396000	3150	125	17000	26780	0.63	0.17
Transverse				66000	26780	2.46	0.05

In addition that the piers are weak about their minor axis, they have not been reinforced appropriately. The minimum ratio of longitudinal reinforcement for wall-type piers (in SDC D) is 0.005 according to AASHTO [3] while this ratio is 0.002 in the existing piers.

To achieve more accurate assessment with respect to the seismic behavior of piers, a push-over analysis was performed on three piers with various heights ranging from 10 to 15 meters. The push-over analysis was based on the pier first mode of vibration. For this purpose, the top of piers was incrementally pushed to reach their ultimate ductility. It should be noted that the effect of gravitational loads, P-Δ effect, was involved during the push-over analysis.

The numerical models designate Model-1, Model-2 and Model-3 with the height of 10, 12 and 15 meters, respectively (Fig. 1). The modeling is performed in the state of plane stress using element DPM4M in LUSAS software. This element is a 4-node quadrilateral element with two translational degrees-of-freedom in each node.

A nonlinear material based on the multi-crack concrete model assigned to the pier concrete. The specifications required for nonlinear analysis are including: the minimum specified compressive strength taken as 21.2 MPa obtained through testing on concrete cores extracted from the piers; the tensile strength of concrete assumed to be 2.1 MPa; and the strain at the end of softening curve considered to be 0.003. For linear analysis, the concrete modulus of elasticity and the poison’s ratio are taken as 21 GPa and 0.2, respectively. An elastic-perfect plastic model was assumed for the steel material used for I-sections inside the piers. The steel yield stress, modulus of elasticity and the poison’s ratio are considered to be 240 MPa, 210 GPa and 0.3, respectively.

The applied loads consist of the pier self-weight of the pier (automatically applied), gravitational loads transmitted from the superstructure, and an incremental lateral load applied on the top of the pier.

The nonlinear performance of existing piers can obviously be inferred from the force-displacement of the pier top shown in Fig. 5. The model-1 revealed an elastic behavior up to a 1500-kN tip force. Then, the geometric cracks gradually formed, leading to a slight reduction in the stiffness. This stage was so limited, followed by the ultimate strength of the pier at about 1800-kN lateral force. Hereafter, the resisting forces could not limit the crack propagation, resulting in deterioration of the load-carrying capacity of the pier. The models Model-2 and Model-3 represented a more brittle behavior. No significant inelastic range is observed in the latter models.

Fig.5

Force-displacement relation of the piers in the existing situation.

Based on FEMA 356 [5] if the maximum to elastic displacement ratio is less than two, the member behavior shall be classified as brittle. As shown in Fig. 5, this ratio is about 1.4 (= 10/7.4) for Model-1. For the other models this ratio is relatively smaller. Therefore, the existing piers shall be treated as brittle members.

5 Rehabilitation technique and its effects

The present study showed that the existing piers suffer from two major problems; inadequate strength and brittle behavior. Among many methods to overcome these problems, the external post-tensioning technique is highlighted because it can solve these problems instantaneously. Although the post-tensioning is a conventional technique in the design of bridge girders, it is not widely used as a mean for seismic rehabilitation of the piers. In fact, this method leads the pier performance from a flexure-dominated field to a compression-dominated one, leading to more moment resistance. Moreover, the prescribed compressive stress prevents the pier from the unstable growth of the cracks. In this condition the pier will demonstrate more ductility.

The prescribed compression shall be evaluated so that the following inequality is satisfied:

$\begin{matrix} M_{D} \leq M_{C} = \\ φ_{s} A_{s} F_{y} (d - \frac{1}{2} \frac{φ_{s} A_{s} F_{y} + P_{G} + P_{PT}}{b_{w} (0.85 φ_{c} f_{c})}) \\ + \frac{1}{2} F_{w} (t_{w} - \frac{φ_{s} A_{s} F_{y} + P_{G} + P_{PT}}{b_{w} (0.85 φ_{c} f_{c})}) \end{matrix}$ (1) where M_D= moment demand; M_C= moment capacity of the pier; φ_s and φ_c=strength reduction factors for steel and concrete, respectively; F_y and F_c= steel yield stress and concrete ultimate strength, respectively; A_s= cross-sectional area of tension steel; P_G= super-structure gravitational loads; P_PT= total compression introduced by the tendons; F_w= pier axial force including the compression introduced by the tendons. The geometric properties d, b_w and t_w are shown in Fig. 6. It should be noted that in this paper, the strength reduction factors, φ_s and φ_c, are taken as 1.0 as prescribed by FEMA 356 [5] for the rehabilitation of existing structures.

Fig.6

Geometric properties of the pier section.

The minimum required compression introduced by the tendons may be evaluated using trial and error method on Equation 1. According to this equation the minimum compression needed for the rehabilitation of the pier, is 14 MN, approximately. More computational details are given in Table 2 for the shortest pier whose moment capacity is the least.

Table 2

Moment demand and moment capacity at two levels

Level	d (m)	t_w (m)	b_w (m)	A_s (m²)	F_y (MPa)	f_c (MPa)	Axial Force (kN)		M_c (kN.m)	M_D (kN.m)	C/D
							Pier weight	Tendon force
+46.3	1.95	2.2	8.5	0.0207	240	21.2	3150	14000	26960	26780	1.01
+50.3	1.58	1.88	7.9				1722	5600	14200	12760	1.11

In the design process, such a force is applied using ten tendons of type 7/0.6^′^′, each of them pre-tensioned by 1.4 MN. In the numerical models, a linear behavior was assigned to the tendons. The tendons are assumed to be of high-strength steel with the modulus of elasticity of 210 GPa and the ultimate strength of 1800 MPa.

To prevent stress concentration, the tendons were cut at two levels depending on the moment demand along the piers height. A typical arrangement of the tendons is illustrated in Fig. 6. The C/D ratios at these levels are noted in Table 2. As given in the table, the number of tendons is chosen so that the C/D ratio is slightly greater than 1.0.

The bottom end of the tendons is fixed to the own pier where the stress level is significantly decreased due to the soil passive stiffness and the increase of pier cross-sectional area. In order the tendons not to punch the concrete support, a corbel-like element was provided above the stepped section as shown in Fig. 7.

Fig.7

Arrangement of tendons in model-1.

An appropriate steel bearing supports the top end of the tendons. Since the concrete material of the piers was highly prone to be locally destroyed by the bearings, a proper confinement was provided using transverse bars connecting two opposite bearings. The transverse bars were pre-stressed to prevent the concrete from the localized crushing. The pre-stressed transverse bars provide friction performance for the bearing system. In this condition, the post-tensioning forces are more proper transmitted to the concrete.

The rehabilitated piers were numerically modeled and investigated in the same manner used for the existing piers. In this state, the models showed more resistance and ductility, as expected. The force-displacement behavior of the rehabilitated piers are shown in Fig. 8 and compared with those before rehabilitation. The models reached, approximately, a maximum lateral force of 2800 kN, and exceeded a lateral displacement of 30 mm without degradation in the load-carrying capacity. It is noteworthy that the analyses were stopped eventually due to the numerical convergence difficulties, and not necessarily for the models failure. Furthermore, Table 3 compares the concrete compressive stress before and after rehabilitation with the compressive strength limits of ACI 318-05 [2] using φ_c=1 per FEMA 365 [5]. Even if the strength limits assuming φ_c=1 is not satisfied, the authors believe that this rehabilitation is satisfactory in terms of engineering judgement. Because, the rehabilitation design is conceptually based on the structural performance in which the ductility plays a key role.

Fig.8

Force-displacement relation of the piers before and after rehabilitation.

Table 3

Concrete compressive stress

Model	S₁ (MPa)	S₂ (MPa)	S₃ (MPa)	φ_cf_c (MPa)
Model-1	6.6	6.7	16.3	21.2
Model-2	5.0	6.5	15.6	21.2
Model-3	6.3	6.9	17.0	21.2

φ_c taken as 1.0 for the existing piers. S₁ = concrete stress before rehabilitation at a 10-mm lateral displacement. S₂ = concrete stress after rehabilitation at a 10-mm lateral displacement. S₃ = maximum concrete stress after rehabilitation. f_c = concrete minimum specified compressive strength.

It should be noted that the prescribed compressive stress is less than five percent of the piers axial strength. This designates that a compressive stress about five percent of the pier axial strength promoted the lateral load-carrying capacity by 60 percent. The comparison between the performance of the pier before and after rehabilitation will be useful in another point of view. The cracks grew rapidly through the section of the existing piers. Whereas, after rehabilitation, the compression stress does not permit the cracks to be opened easily, resulting in reduction of the rate of crack growth. Accordingly, under a same lateral displacement, the cracks are much more superficial in the rehabilitated piers. These can be observed in Fig. 9. The figure demonstrates the crack pattern at a same lateral displacement (10 mm) for the piers before and after rehabilitation. For more displacement, the existing pier lost its load-carrying capacity while the rehabilitated piers exceeded a lateral displacement of 30 mm.

Fig.9

Crack pattern before and after strengthening at a 10-mm lateral displacement.

6 Evaluation of nonlinear behavior of rehabilitated piers

In the following a simplified method is presented to predict the key issues influencing the nonlinear behavior of rehabilitated piers:

Step 1. Calculation of the prescribed compression introduced by tendons using Equation 1.

Step 2. Evaluation of the displacement at the stage of general yielding, Δ_y, the plastic displacement, Δ_p, and the maximum displacement, Δ_max. The following equations may be employed:

$\begin{matrix} Δ_{y} = \frac{M_{C}}{h} \int_{0}^{h} \frac{x^{2}}{E_{c} I_{e} (x)} dx; \\ Δ_{p} = M_{C} (h - h_{p}) \int_{0}^{h_{p}} \frac{dx}{E_{c} I_{u} (x)} \end{matrix}$ (2) $Δ_{max} = Δ_{y} + Δ_{p}$ (3) where h= pier height; h_p= height of plastic hinge which is taken as half of the pier thickness [5]; E_c= concrete modulus of elasticity. I_e and I_u= moment of inertia about minor axis in service and ultimate states, respectively.

Step 3. Calculation of the ductility ratio (μ = Δ_max/Δ_y).

Step 4. Evaluation of the seismic response modification factor, R: In the elastic design of structures, the capability of the structure in energy dissipation may be taken into account using the seismic response modification factor, R. As previously noted, the existing piers are not able to reveal ductility, and as a consequence, the modification factor should be considered unity. In contrast, the rehabilitated piers attained energy dissipation capability. Accordingly, the actual seismic forces may be considered less than elastic forces. The seismic response modification factor is defined as below [17]: $R = R_{μ} . \frac{S_{u}}{S_{s}}$ (4) where R_μ= ductility modification factor; S_ps and S_u= initial yield resistance and ultimate strength, respectively. The ductility modification factor is estimated as follows for the structures constructed on bedrock [10]:

$\begin{matrix} R_{μ} = \\ \frac{μ - 1}{1 + 1 / (10 T - μ T) - 1 / (2 T) exp (- 1.5 {(ln T - 3 / 5)}^{2})} \\ + 1 \geq 1 \end{matrix}$ (5) in which T= first natural period of the structure. It should be noted that the first natural period T used in this equation is nearly the same as that of pier before rehabilitation. Because, as previously mentioned, the post-tensioning force is negligible as compared with the pier axial strength.

The following relationships are proposed for estimation of S_s and S_u:

$S_{u} = M_{c} / H; S_{s} = \frac{2 I_{e}}{{Ht}_{w}} (\frac{P_{G} + P_{PT}}{b_{w} t_{w}} + f_{r})$ (6) in which the concrete modulus of rupture, f_r, is assumed to be 0.63√f_c according to ACI 318-05 [2].

In Table 4 the seismic response modification factor of the models are calculated. This factor is more than 3.0 for the models.

Table 4

Seismic response modification factor of rehabilitated pier

Model	S_s (kN)	S_u (kN)	Δ_max (mm)	Δ_y (mm)	μ	T (sec)	R _μ	R
Model-1	1340	2800	30	12	2.5	0.17	1.8	3.8
Model-2	1350	2650	30	10	2.5	0.17	2.1	4.1
Model-3	1275	2430	37	12	2.5	0.17	1.8	3.5

7 Efficiency of post-tensioning technique in cyclic loading

Capability of the tendons to keep their effectiveness under the cyclic loading was another concern which must be studied. To this purpose, the rehabilitated model was analyzed applying a cyclic lateral displacement on the top of the model Model-1.

To determine the efficiency of the tendons during the cyclic loading, the forces acting on the tendons installed at both sides of the pier are evaluated. The results are illustrated in Fig. 10. It should be noted that each tendon has initially been tensioned up to 1400 kN. Figure 10 implies that during the loading excursions, the maximum increase of the tendon force is about 140 kN, and maximum degradation is 90 kN, approximately. These values denote ten percent increase and six percent decrease of the tendon initial force during the cyclic loading. However, as shown in the figure, the average tension in the tendons has been relatively constant.

Fig.10

Variation of the tendons force.

The above slight changes in the tendons force during the cyclic loading as well as the long-term effects such as the tendons relaxation and the concrete creep should be considered in determination of the required pre-stressing force.

It should be declared that the effect vertical excitation was not addressed in this analysis. However, it is not supposed that the tendons effect is considerably influenced by vertical seismic excitation. Because, the seismic load is a percent of the pier weight which is itself about 20 percent of the whole compression applied by the tendons. In case of lower prescribed compression, the adverse effect of vertical vibration should be taken into account. For this purpose, the minimum required compression introduced by the tendons may be evaluated using Equation 1 providing that the vertical seismic load is subtracted from F_w.

8 Conclusions

This paper dealt with the nonlinear behavior of bridge mass concrete piers and rehabilitation of which using external post-tensioning technique. Although the post-tensioning is a conventional technique in the design of bridge girders, it is not famous as a mean for seismic rehabilitation of the piers. The notable results are pointed out as follows:

The typical mass concrete piers studied in this paper were not able to meet sufficient strength. They, indeed, show a brittle behavior with respect to lateral loading.

The post-tensioning technique can efficiently increase the pier strength. For the models studied in the present paper, a prescribed compressive stress about five percent of the piers axial strength promoted the lateral load-carrying capacity by 60 percent.

The rehabilitated piers show more ductility and stable nonlinear behavior.

The capability of the rehabilitated piers to show stable non-linearity was resulted in a seismic response modification factor of 3.0, approximately. This value is comparable with unity considered for the existing piers.

The nonlinear behavior of the rehabilitated piers under cyclic loading confirmed that the efficiency of the post-tensioning technique does not considerably degrade during the loading excursions.

A simplified method was proposed to help designers to well select the prescribed compression needed to achieve required strength and ductility.

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Numerical investigation of seismic rehabilitation of mass concrete piers of a railway bridge using post-tensioning technique

Abstract

Keywords

1 Introduction

2 Research program

Table 1 Results from linear equivalent earthquake analysis Direction of excitation Axial Force (kN) Moment (kN.m) T P c P D C/D M c M D C/D (sec) Longitudinal 396000 3150 125 17000 26780 0.63 0.17 Transverse 66000 26780 2.46 0.05

References

Table 1
Results from linear equivalent earthquake analysis

Direction of excitation Axial Force (kN) Moment (kN.m) T

P _c P _D C/D M _c M _D C/D (sec)

Longitudinal 396000 3150 125 17000 26780 0.63 0.17

Transverse 66000 26780 2.46 0.05