A parametric study on the axial behaviour of elastomeric isolators in multi-span bridges subjected to horizontal seismic excitations
- 1.6k Downloads
- 5 Citations
Abstract
This paper investigates the potential tensile loads and buckling effects on rubber-steel laminated bearings on bridges. These isolation bearings are typically used to support the deck on the piers and the abutments and reduce the effects of seismic loads and thermal effects on bridges. When positive means of fixing of the bearings to the deck and substructures are provided using bolts, the isolators are exposed to the possibility of tensile loads that may not meet the code limits. The uplift potential is increased when the bearings are placed eccentrically with respect to the pier axis such as in multi-span simply supported bridge decks. This particular isolator configuration may also result in excessive compressive loads, leading to bearing buckling or in the attainment of other unfavourable limit states for the bearings. In this paper, an extended computer-aided study is conducted on typical isolated bridge systems with multi-span simply-supported deck spans, showing that elastomeric bearings might undergo tensile stresses or exhibit buckling effects under certain design situations. It is shown that these unfavourable conditions can be avoided with the rational design of the bearing properties and in particular of the shape factor, which is the geometrical parameter controlling the axial bearing stiffness and capacity for a given shear stiffness. Alternatively, the unfavourable conditions could be reduced by reducing the flexural stiffness of the continuity slab.
Keywords
Bridges Seismic isolation Steel-laminated elastomeric bearings Tensile stress Buckling Shape factor1 Introduction
Seismic isolation with rubber-steel laminated bearings is used extensively in contemporary bridge engineering as a means of mitigating the effect of earthquake loads by reducing the forces transmitted to the substructures (Lee et al. 2001; Chen and Duan 2003; Tubaldi and Dall’Asta 2011). Bridge isolation bearings are designed mainly to sustain the compressive loads, transmitted from the deck, while accommodating horizontal and rotational deflections. However, under some design situations they may be exposed to the possibility of uplift and to tensile loads during an earthquake. These loadings may be induced by the seismic motion of the deck along the longitudinal (Mitoulis 2014) and/or the transverse direction of the bridge (Katsaras et al. 2009), when the bearings are fixed to the superstructure and the substructures through bolted connections, as often suggested by seismic codes (EN1337-3 2005; EN15129 2009). The occurrence of tensile forces in isolation bearings has been reported in many reconnaissance surveys such as those following the 2011 Tohoku earthquake (Buckle et al. 2012) and also in experimental shaking-table tests on isolated structures, as discussed in Yang et al. (2010).
Rubber-steel laminated bearings are vulnerable to tensile loads. In fact, it has long been known that hydrostatic tensile stress causes internal local ruptures in the rubber known as cavitation (Gent and Lindley 1959; Gent 1990; Pond 1995; Dorfmann and Burtscher 2000). Void nucleation and the growth of microcavities in rubber is a complex process that involves breakage of bonds in the polymer network, fracture of filler clusters and detachments of rubber chains from reinforcing particles. While in compression the rubber can easily withstand high pressures without exhibiting damage (Gent 1990), the tensile stresses, at which cavitation initiates, are very low. According to Gent and Lindley (1959), cavitation occurs at a negative pressure of about 2.5 G, where G is the shear modulus of the bearing, obtained experimentally from testing at moderate shear strain (between 100 and 200 %) under nominal axial loads (Kumar et al. 2014). The load deformation of the isolators under pure tensile loads has been described by Constantinou et al. (2007) and Warn et al. (2007). Yang et al. (2010) tested bearings that had a shear modulus of 0.55 MPa under pure tensile loads and identified that the bearings exhibit cavitation when the tensile strains exceeded 1.0 % (corresponding to tensile stress between 1 and 2 MPa). The potential of bearing uplift, i.e. tension, is also evident throughout most bridge design codes (AASHTO 2012; CalTrans 1999; EN1998-2 2005; JRA 2002), providing rules to limit the likelihood of occurrence.
In order to adequately describe the behaviour of laminated rubber-steel isolation bearings in bridges, accurate models are required, which are capable of simulating the response under a combined state of stress following the imposition of shear and vertical loadings. Kumar et al. (2014) recently developed models for describing the combined axial, rotational, and shear response of different types of laminated steel-rubber bearings, including low damping rubber bearings, high-damping rubber bearings, and lead rubber bearings. The proposed models have been implemented in OpenSees (McKenna et al. 2006) and can simulate the coupling of vertical and horizontal deflection, cavitation and post-cavitation behaviour in tension and strength degradation in cyclic tensile loading due to cavitation.
In this paper, the case of high-damping natural rubber (HDNR) bearings is considered and a bridge model representative of typical bridge design practice is analysed first to shed light on the coupled horizontal-vertical bearing behaviour. For this purpose, the model developed for Kumar et al. (2014) is employed to accurately simulate the HDNR bearing response and the results of tests carried out at TARRC on HDNR double-shear specimens are used to calibrate the bearing model parameters.
An extensive parametric study is then carried out to identify under which design situations the uplift effect is critical, i.e. for which properties of the superstructure, the substructures, the bearings, and the pier-to-deck connection (i.e. the eccentricity of the bearings with regards the axis of the pier), the uplift effect is more likely to occur. It is noteworthy though that the self-weight and the additional variable loads of the deck induce compression on the bearings, which protect them against cavitation. However, the same loads increase the vulnerability of the isolators to buckling and to other limit states as described by EN15129 (2009), EC8-2 (2005), and of EN1337-3 (2005). For this reason, the bearings are checked against all these limit state conditions, thus ensuring their reliable behaviour under both the seismic and non-seismic loading conditions. Furthermore, emphasis is placed on the effect of the shape factor, which controls the vertical behaviour and stiffness of the bearing for a given horizontal stiffness. The results of the parametric study provide useful information on the sensitivity of the uplift mechanism to the properties of the bridge and the bearing and sheds light on the most critical limit states of eccentrically-placed bearings.
2 Bridge modelling and seismic input
2.1 Description of the reference bridge
The deck comprises three simply-supported spans of length L_{sp} = 30 m, each consisting of five precast I-beams and by a cast-in situ slab, whose width is 13.45 m, whereas the carriageway width is considered to be 11.5 m. The deck spans are connected by cast-in situ continuity slabs reinforced with ordinary reinforcement. This connection of the adjacent spans provides a continuous deck surface, thus avoiding the use of expansion joints at the piers. Despite the connection of the adjacent segments, the bridge behaves, under the vertical loads, as if it consisted of a series of simply-supported beams, provided that the slab is sufficiently flexible to accommodate the rotations. The class of the concrete employed for the deck is C30/40 whereas the class of steel is S355 for the ordinary reinforcement and S1570/1770 for the prestressed.
The deck I-beams sit on the piers through two lines of five steel-laminated isolation bearings, as shown in Fig. 2b and on one line of five bearings on the abutments, i.e. a total of ten bearings are used on the piers and five bearings on each abutment. The eccentricity of the axis of the isolators with respect to the axis of the pier is e_{b} = 0.8 m. The deck has a depth of 2.02 m and comprises the precast beams of depth equal to 1.75 m and a slab 0.27 m thick. The eccentricity of the neutral axis of the deck with respect to the bottom fibre of the deck is H_{dg} = 1.47 m. The slab is continuous along the total length of the bridge, whilst two expansion joints are used at the abutments. The eccentricity of the continuity slab with respect to the deck neutral axis is e_{cs} = 0.4 m.
The piers are circular solid sections with diameter D_{p} = 2.5 m. The height of the pier H_{p} is equal to 10 m. The cap beam of the pier has a depth of 1.5 m. The longitudinal dimension of the cap beam is equal to the diameter of the pier plus 0.5 m, the transverse dimension is 5 m. The longitudinal pier reinforcement consists of 93 rebars of diameter 26 mm, whereas the transverse reinforcement consists of rebars of 14 mm with a spacing of 100 mm. The class of the concrete is C30/40 whereas the class of steel is S355.
The foundations of the bridge piers consist of 3 × 3 piles and the soil profile consists of a deformable soil layer overlying a very dense sand deposit. The soil type is classified as C according to EC8-1 (2005), corresponding to a soil factor S = 1.15 and the peak ground acceleration (PGA) expected at the site is PGA = 0.4 Sg, where g denotes the gravity constant.
2.2 Finite element model of the benchmark bridge
The superstructure and the piers are modelled by a spine of linear beam-column elements with lumped nodal masses, spanning between successive nodes along the elements length. The spine represents the geometrical centre of the modelled structural components. The prestressed deck is modelled by linear elastic frame elements with uncracked stiffness properties, whereas the piers are modelled by linear elastic frame elements with cracked effective stiffness properties evaluated based on the moment–curvature analysis of the base section under the axial force induced by the permanent loads. The effective cracked stiffness (secant to the yielding point) is 50 % of the gross stiffness and it is described in the model by reducing the second moment of area of the transverse pier section. The value of the bending moment demand is monitored during the analysis to check whether the yield strength of the piers is exceeded or not. A set of lumped parameter models (LPMs), as shown in Fig. 3b, is used to describe the soil-structure inertial interaction between the soil and the foundations (Makris et al. 1994; Dezi et al. 2013; Lesgidis et al. 2015), as discussed more in detail in the next section. The Young’s modulus of the reinforced concrete piers and of the deck is assumed equal to 30GPa. The viscous damping of the system, which represents the energy dissipation sources other than that of the isolators, is taken into account through the “region command”, i.e. by assigning a Rayleigh damping to the piers nodes. The parameters of the Rayleigh damping model are evaluated by considering a damping ratio of 5 % in correspondence of the vibration frequencies of the two higher modes which involve the flexure of the piers in the longitudinal direction. In the reference bridge configuration and for the fixed base condition, these circular vibration frequencies are 50.3 rad/s and 241.0 rad/s.
2.3 High damping rubber isolators model description
The isolators are described by employing the high damping rubber (HDRB) model developed by Kumar et al. (2014). This model consists of a two-node, twelve degrees-of-freedom discrete element, where the two nodes are connected by six springs representing the mechanical behaviour along the six main degrees of freedom of the bearing. The HDRB element permits to accurately describe both the nonlinear amplitude-dependent behaviour in shear of the isolator and the vertical (tensile or compressive) behaviour, as well as the coupling between the horizontal and vertical responses.
The axial (i.e., vertical) stiffness K_{v} of the HDRB model is obtained from the two-spring model of Koh and Kelly (1987), which has been validated experimentally by Warn et al. (2007). The coupling of horizontal and vertical behaviour is due to: (1) the variation of the shear stiffness with axial load and (2) the dependence of axial stiffness on the magnitude of the lateral displacement (Kumar et al. 2014). The shear stiffness reduces progressively for compressive axial loads approaching the critical buckling load. The value of the latter is considered to decrease as the bearing effective area, i.e., the overlapping area between the top and bottom internal plates, reduces, but, in first approximation, a residual buckling resistance of the bearing at zero overlap area is considered (Kumar et al. 2014). Also, the vertical stiffness of the isolator reduces with increase in the horizontal displacement. It is worth pointing out that the large lateral displacements experienced by the isolators, corresponding to shear strains of the order of 100–200 %, can potentially lead to substantial reductions in the vertical load-carrying capacity and the vertical stiffness of the HDRB bearings (Cardone and Perrone 2012), and the employed bearing model is capable of reproducing this behaviour.
Parameters of the HDRB model in OpenSees (McKenna et al. 2006) for the reference bridge case
G_{eff}(kN/m^{2}) | K_{bulk}(kN/m^{2}) | a_{1}(kN/m) | a_{2}(kN/m^{3}) | a_{3}(kN/m^{5}) | b_{1}(kN/m) | b_{2}(1/m^{2}) | b_{3}(1/m) |
---|---|---|---|---|---|---|---|
700 | 2,000,000 | 646.485 | −3586.12 | 24015.72 | 30.69 | 402.96 | 32.65 |
c_{1}(1 m^{−3}) | c_{2}(1 m^{−3}) | c_{3}(−) | c_{4}(1 m^{−3}) | K_{c}(−) | f_{M}(−) | a_{c}(−) | |
---|---|---|---|---|---|---|---|
0 | 0 | 1 | 0 | 0.02 | 0.5 | 1 |
Period, modal participation mass factor (MPMF), and shape of most relevant vibration modes
No# | Period (s) | Direction | MPMF (%) | Mode shape |
---|---|---|---|---|
1 | 2.02 | Longitudinal deck displacement | 78.6 | |
2 | 0.28 | Vertical deck displacement | 17.7 | |
4 | 0.25 | Vertical deck displacement | 47.8 | |
5 | 0.13 | Longitudinal pier displacement | 10.4 |
2.4 Seismic input description and modelling of the soil-structure interaction effects
The seismic input is described by the EC8-1 (2005) soil type C spectrum for a PGA of 0.4 g and soil type C. The seismic assessment of the bridge is performed by carrying out non-linear time history analyses of the structure under a set of seven natural records describing the record-to-record variability effects. This set of ground motion records is compatible with the response spectrum considered for the design and has been selected by using the software Rexel v3.5 (Iervolino et al. 2010). The vertical component of the earthquake, albeit important for the safety evaluation of the isolators, was not studied thoroughly in this paper, as emphasis was placed on the evaluation of the effects of the coupled horizontal-vertical behaviour on the axial force demand in the bearings as observed due to horizontal excitation only.
With regards to the soil-structure interaction (SSI) effects, LPMs placed at the base of the piers describe the inertial SSI effects (Fig. 3b). These LPMs consist of a set of translational and rotational springs, dampers and masses that permit simulation of the frequency-dependent compliance of the soil-foundation system for analyses in the time domain. It is noteworthy that this study does not account for the kinematic effects on the foundation input motion and thus the free-field motion can be directly used as the input motion. Since isolated bridges are expected to be excited by relatively low frequencies, the error resulting from neglecting the kinematic SSI effects is expected to be negligible (Dezi et al. 2013; Olmos and Roesset 2012; Ucak and Tsopelas 2008).
The properties of the LPMs are derived by employing the approach outlined by Dezi et al. (2013), based on simplified formulae calibrated from results of extensive non-dimensional parametric analyses considering head-bearing pile groups. The proposed approach allows accurately simulation of the compliance of pile foundations and important features of the soil-foundation system behaviour, such as the coupled rotational-translational response. The properties of the LPMs are consistent with the considered soil type and the geometrical and mechanical properties of the foundation. The piles are fully embedded in the soil, socketed into the sand deposit and connected at the heads by a cap (Fig. 2b). The concrete piles have a Young’s modulus of 30 GPa and a density of 2.5 ton/m^{3}. They have a length of 18 m, with circular cross sections of 0.8 m diameter and a spacing of three diameters (center to center). The deformable deposit has a depth of 15 m, a shear wave velocity V_{s1} = 200 m/s and a density ρ_{s1} = 1.7 ton/m^{3}. The dense sand deposit has shear wave velocity V_{s2} = 800 m/s and density ρ_{s2} = 2.5 ton/m^{3}. Poisson’s ratio is considered to be v_{s} = 0.4 and material hysteretic damping ξ_{s} = 10 %, which is compatible with the design level of strain in the soil.
3 Parametric study
This section investigates the likelihood of the occurrence of bearing uplift, buckling, or of other relevant limit states, as prescribed in the Appendix of this paper, for the bridge configuration under consideration. First, the reference bridge model is analysed in detail to show some important features of the bridge response. In particular, the following response parameters are monitored, since they provide information useful to assess the performance of the bridge components: (a) the isolator translational and rotational deflections and forces along the horizontal and vertical directions; (b) the internal actions on the piers; (c) the displacements and the rotations of the pier cap with respect to the ground; (d) the horizontal and vertical displacements and rotations of the deck with respect to the ground; (e) the displacements and rotation of the continuity slab with respect to the ground. Particular emphasis is placed on the response of the isolators, which mainly depends on the displacements and rotations of the deck and the pier cap.
Successively, an extensive parametric study is carried out to evaluate the performance of a set of realistic bridge models obtained by varying critical design parameters of the reference model, which are related to the properties of the superstructure, the substructures and the isolators. The aforementioned critical design parameters were defined based on a preliminary sensitivity analysis, which has identified which design choices influence significantly the bearing vertical response.
Then, analyses are carried out for two different load combinations, i.e. the ultimate limit state (ULS) combination for non-seismic actions and the seismic design combination corresponding to the earthquake input described previously. The adequacy of the isolation system to sustain the design loads is assessed on the basis of the checks provided in EN1337-3 (2005) for the ULS design combination of actions, EN15129 (2009) and EC8-2 (2005) for the seismic load combinations.
The code prescriptions that need to be satisfied by the bearings are given in the Appendix of this paper. These prescriptions are expressed in the form of inequalities based upon demand-to-capacity ratios (D/C), where the demand is the value of the response parameter of interest for the limit state being monitored, evaluated by structural analysis, whilst the capacity is the maximum allowable value for the relevant parameter, as prescribed by the codes. It is noteworthy that in calculating the D/C ratios, the mean value of the peak response parameters obtained for the seven natural seismic motions considered are used. A value of the ratio higher than one, i.e. D/C > 1, implies that the limit state is not satisfied, whereas a value less than 1 implies that that the design satisfies the relevant code requirement.
In addition to the assessment of the bearing performance, the performance of the piers is monitored to make sure that they do not yield under the combination of the axial loads and bending moments and that the shear demand does not exceed their shear capacity.
It is noteworthy that for all bridge models investigated, the isolation bearing system is designed through the procedure outlined in the following section to achieve a target period of 2.0 s. This value is significantly higher than the value of 0.46 s corresponding to the fixed-base configuration.
3.1 Procedure for the design of the isolation system
In the second step, the design displacement d_{Ed} is calculated based on the damped mean record spectrum as given in Fig. 5a for the records considered in this study. In the third step, for a fixed value of the shear strain γ_{Ed} the total thickness of the elastomer T_{r} is calculated. The bearing area A_{r} and diameter D_{r} can also be computed based on the knowledge of the rubber shear effective modulus \(G_{eff}\) at the design shear strain γ_{Ed}. The last step of the procedure involves fixing the value of the shape factor of the isolator S_{r}, denoting the ratio between the loaded area and the force-free area for a circular bearing of diameter D_{r}. This provides the thickness t_{r} and the number n_{r} of the rubber layers in each bearing.
The parameters T_{is}, T_{r}, and S_{r}, the mechanical (effective) properties of the bearings, G_{eff} and ξ_{is}, and the seismic input spectral ordinate S_{d}(T_{is}) define unequivocally the geometry of the isolation bearings with the exception of some parameters such as the thickness of the steel plates and of the side cover layer of rubber for which additional design rules are given in the codes. In particular, values of the thickness of the steel plates significantly higher than the minimum value required to avoid steel yielding are chosen, since very flexible plates have been found to affect significantly the stability of the bearings (Muhr 2006, 2007).
The design procedure employed in this study is not intended to cover all the aspects related to the bearing design and may lead to bearing properties not consistent with those available in manufacturer catalogues. Although alternative design procedures and criteria could have been employed for the design (e.g. Cardone et al. 2009, 2010), the proposed one was chosen for its simplicity, as it requires no iterations and also allows to obtain and control directly all the properties required for the calibration of the HDNR bearing model.
3.2 Seismic response of the reference bridge with emphasis on the response of the bearings
The geometry of the reference bridge considered for the in-depth analysis of the seismic response is described by the following parameters: span length L_{sp} = 30 m, pier height H_{p} = 10 m, cap beam height H_{cb} = 1.35 m, bearing eccentricity e_{b} = 0.8 m, continuity slab length L_{cs} = 0.5 m. The design of the bearing is carried out by following the procedure outlined above for a target vibration period of T_{is} = 2.0 s, corresponding to a displacement demand in the fundamental mode of vibration of 0.264 m and a damping ratio ξ_{is} = 10.8 %. The initial value of the design shear deformation under the seismic input is γ_{Ed} = 1.5. For the assumed effective shear modulus G_{eff} = 700 kPa, this corresponds to bearings with a total rubber height T_{r} = 176 mm and a rubber diameter D_{r} = 490 mm. The value of the bearing shape factor is S_{r} = 15, leading to a thickness of single rubber layer of t_{r} = 8 mm, and number of rubber layers n_{r} = 22. The assumed value of the shim plate thickness is t_{s} = 5 mm.
Bearing material, geometric, and mechanical properties for reference bridge
Diameter D [m] | 0.4908 | Moment of inertia of bearing I [m^{4}] | 0.0030 |
Bearing cover t_{c} [m] | 0.0050 | Adjusted moment of inertia I_{s} [m^{4}] | 0.0047 |
Rubber thickenss t_{r} [mm] | 0.0082 | Adjusted shear area A_{s} [m^{2}] | 0.3057 |
Steel thickenss t_{s} [mm] | 0.0050 | Compressive modulus E_{c} [kN/m^{2}] | 579,754.6 |
Number of rubber layers [−] | 22 | Rotational modulus E_{r} [kN/m^{2}] | 193,251.5 |
Effective shear modulus G_{eff} [kN/m^{2}] | 700 | Initial vertical stiffness K_{v0} [kN/m] | 621,960.7 |
Bulk modulus of rubber K [kN/m^{2}] | 2,000,000 | Cavitation force F_{c} [kN] | 405.4 |
External diameter D_{e} [m] | 0.4958 | Euler buckling load P_{e} [kN] | 110,316.2 |
Total rubber height T_{r} [m] | 0.1800 | P_{s} = G_{eff}A_{s} [kN] | 213.991 |
Total bearing height h [m] | 0.2850 | Critical buckling load P_{cr} [kN] | 4858.67 |
Shear area A_{s} [m^{2}] | 0.1931 | Rotational stiffness K_{r} [kNm] | 5043.5 |
Shape factor S_{r} [−] | 15.000 | Initial horizontal stiffness K_{h0} [kN/m] | 751.0 |
Figure 7 also reports the time histories of the deck and the pier cap displacements obtained by neglecting SSI effects, i.e., by considering a fixed base condition. SSI effects influence significantly only the pier displacement demand. In fact, the maximum pier top displacement obtained when accounting for SSI effect is 0.032 m and thus it is significantly higher than the 0.020 m value obtained for the fixed-based condition.
Influence of SSI effects on deck and pier and response
Record | SSI | Fixed base | ||||
---|---|---|---|---|---|---|
d_{hd,max}(m) | d_{hp,max}(m) | r_{p,max}(rad) | d_{hd,max}(m) | d_{hp,max}(m) | r_{p,max}(rad) | |
#1 | 0.309 | 0.032 | 0.003 | 0.303 | 0.020 | 0.003 |
#2 | 0.211 | 0.025 | 0.003 | 0.202 | 0.020 | 0.003 |
#3 | 0.148 | 0.017 | 0.002 | 0.148 | 0.009 | 0.001 |
#4 | 0.335 | 0.031 | 0.004 | 0.329 | 0.024 | 0.003 |
#5 | 0.173 | 0.016 | 0.002 | 0.171 | 0.013 | 0.002 |
#6 | 0.317 | 0.036 | 0.004 | 0.312 | 0.021 | 0.003 |
#7 | 0.463 | 0.064 | 0.008 | 0.460 | 0.041 | 0.006 |
Average | 0.280 | 0.031 | 0.004 | 0.275 | 0.021 | 0.003 |
Figure 10b compares the time histories of the vertical displacements of the top and bottom nodes connected by the HDNR bearing element at the first support line, respectively denoted to as d_{vd,1} and d_{vp,1} and representing the vertical motion of the deck and the pier cap. The motion of the bottom node can also be obtained as the pier cap rotation times the bearing eccentricity, i.e., d_{vp,1=}r_{p}·e_{b}.
It can be seen in Fig. 10b that although the axial bearing deflection is mainly controlled by the pier rotation, it is also influenced by the deck motion, since d_{vb,1} = d_{vd,1}−d_{vp,1}. It is also noteworthy that both these vertical displacements are developed due to the horizontal seismic actions only, as no vertical component of the seismic action is considered in this study.
Influence of SSI effects on bearing axial displacements
Record | SSI | Fixed-base | ||||||
---|---|---|---|---|---|---|---|---|
d_{vb1,max}(mm) | d_{vb1,min}(mm) | d_{vb2,max}(mm) | d_{vb2,min}(mm) | d_{vb1,max}(mm) | d_{vb1,min}(mm) | d_{vb2,max}(mm) | d_{vb2,min}(mm) | |
#1 | −0.50 | −5.50 | −0.13 | −3.60 | −0.51 | −5.10 | −0.20 | −3.37 |
#2 | −0.18 | −4.20 | 0.14 | −2.60 | −0.47 | −4.20 | −0.06 | −2.54 |
#3 | −0.96 | −2.70 | −0.85 | −2.80 | −1.00 | −2.40 | −0.91 | −2.42 |
#4 | −0.17 | −7.10 | −0.29 | −5.10 | −0.36 | −6.80 | −0.54 | −4.89 |
#5 | −0.47 | −3.30 | −0.40 | −2.70 | −0.48 | −2.90 | −0.53 | −2.74 |
#6 | 0.43 | −6.30 | 0.27 | −3.90 | 0.24 | −5.40 | −0.03 | −3.55 |
#7 | −0.48 | −17.20 | 0.18 | −7.20 | −0.66 | −14.10 | −0.42 | −5.37 |
Average | −0.31 | −6.77 | −0.16 | −4.04 | −0.46 | −5.95 | −0.41 | −3.58 |
On average, the SSI effects are found to induce an increase of the absolute values of the bearing axial deflection demand. This highest relative increase is about 40 % for the maximum deflection, and of 24 % for the minimum deflection. Similar observation can be drawn for the bearing axial forces, not reported due to space constraint.
3.3 Influence of shape factor on pier response and bearing capacity
Based on the results of the analysis of the reference bridge model, it is evident that there is a strong coupling between the horizontal and the vertical response of the isolators, as a result of the eccentricity of the bearings and of their axial stiffness, which results in a force couple, i.e. a bending moment forming at the pier top. Since for a given target design period and, thus, for a given bearing translational stiffness the parameter that governs the vertical behaviour and capacity is the shape factor S_{r} of the isolator, the benchmark bridge is re-analysed for values of the shape other than the value 15 employed for the reference case. The rest of the bridge properties and bearing design parameters are kept constant and equal to the reference values. Smaller shape factors correspond to axially and rotationally flexible bearings, which are also more prone to buckling under compression, whilst the opposite is valid for larger shape factors.
The safety margin with respect to the different limit states related to the bearing performance is given by the demand-to-capacity ratios (D/C) defined also in Appendix. In summary, \(\left( {D/C} \right)_{{\gamma_{tot} }}\) is related to the check that the maximum local shear strain in the isolator is smaller than 7, \(\left( {D/C} \right)_{Pcr}\) refers to the check of the stability under seismic actions, \(\left( {D/C} \right)_{Pcav}\) refers to the tensile stress of the bearing that should be kept under 2 G, \(\left( {D/C} \right)_{{\gamma_{ULS} }}\) refers to the design check with respect to the shear induced by the ULS load combination (Manos et al. 2012), and \(\left( {D/C} \right)_{\alpha }\) is related to the check on the rotation limit for the bearing under the ULS load combination.
Figure 12b shows the same results obtained by considering the vertical component of the seismic excitation in addition to the horizontal longitudinal component. As expected, considering this effect results in increased values of the D/C ratios corresponding to the buckling and cavitation condition.
4 Parametric study results
This section describes the parametric study carried out to evaluate the sensitivity of the bearing response and capacity with respect to the bridge model parameters.
In particular, the influence of the parameters describing the deck span length, the eccentricity of the isolators, the stiffness of the continuity slab, the stiffness of the deck, and the pier flexibility is investigated by observing how the D/C ratios change by changing these parameters one at a time for different values of the shape factor.
4.1 Influence of the deck-span
In the first case analysed, the span length is assumed equal to L_{sp} = 20 m instead of 30 m, whereas the values of all the other parameters are kept unchanged with respect to the reference bridge. By this way, both the total isolated mass and the bearing vertical force due to vertical loads reduce. The bearing design is carried out by assuming T_{is} = 2 s, ξ_{is} = 10.8 %, G = 700 kN/m^{2} and γ_{Ed} = 1.5 as for the reference bridge. For S_{r} = 15, this yields a total rubber height T_{r} = 0.18 m, a rubber diameter D_{r} = 0.40 m, a single rubber layer thickness t_{r} = 6.7 mm, and a number of layers n_{r} = 27. The shim plate thickness is again assumed as t_{s} = 5 mm. The mean values of the pier top and deck displacement obtained through the analyses are 0.026 and 0.278 m respectively. The vertical pressure due to the deck weight is similar to that of the reference bridge model.
For a given S_{r} value, the values of the moment ratio (denoted by “L_{sp} = 20 m” in Fig. 11) are slightly lower than the corresponding values observed in the reference case, because of the lower bearing diameter, which results in a reduced vertical stiffness. For low S_{r} values, the behaviour is very close to that of a cantilever since both the horizontal and vertical stiffness of the bearings reduce significantly during the seismic action due to nonlinear geometric effects.
4.2 Influence of the eccentricity of the isolators
In this case, the bearing eccentricity is assumed as e_{b} = 1.0 m instead of e_{b} = 0.8 m, and also the continuity slab is assumed to have a length L_{cs} = 0.9 m instead of L_{cs} = 0.5 m. The bearing design yields the same bearing properties as in the reference bridge configuration, since the deck mass is unchanged.
For a given value of the shape factor, the values assumed by the moment ratio (denoted by “e_{b} = 1 m” in Fig. 11) are higher than the corresponding values observed in the reference case, because of the stiffer rotational constraint provided by the eccentric bearings.
4.3 Influence of the stiffness of the continuity slab
In order to investigate the influence of the continuity slab, the analyses are carried out again by assuming a zero stiffness value for the element representing it. This is equivalent to assuming that there are expansion joints in place of the continuity slabs between the adjacent deck spans. For this particular case the potential pounding interaction between the adjacent spans is not taken into account, as it was considered that adequate expansion joints prevent the interaction between the spans. The bearing design yields the same bearing properties as in the reference bridge.
Figure 15b shows the D/C ratios obtained by considering also the vertical component of the seismic excitation. Although this component results in increased values of the axial loads on the bearings, it is still possible to find S_{r} values in the range between 17 and 20 for which all the code safety checks are satisfied.
4.4 Influence of the deck stiffness
In order to investigate the influence of the deck flexural stiffness on the bridge response, the analyses were carried out again by assuming a very high stiffness value for the elements representing the deck and the continuity slab. This is only a hypothetical case and may be thought of as representing a continuous multi-span stiff box girder deck supported on the piers through two lines of bearings. The bearing design yields the same bearing properties as in the reference bridge configuration.
4.5 Influence of the pier height
The height of the piers is assumed to be equal to H_{p} = 20 m, instead of 10 m. The bearing design yields the same bearing properties as in the reference bridge configuration since the pier flexibility is not taken into account during the design process.
By looking at the results plotted in Fig. 11 (denoted by “H_{p} = 20 m” in Fig. 11) it can be observed that for a given S_{r} value the moment ratio is higher in the case of more flexible piers because of the higher degree of rotational restraint.
5 Conclusions
This paper has analysed the potential of occurrence of different limit states in high damping natural rubber (HDNR) bearings employed for isolating multi-span simply-supported isolated bridges. These bridge have the isolators placed eccentrically with respect to the pier axis and this induces significant axial force variation that may lead to the unsatisfactory conditions of cavitation or buckling of the bearings under certain design situations.
In order to shed light on these potential mechanisms and their dependence on the properties of the bearings and of the bridge, a set of bridges representative of design practice has been considered. Finite element models of the bridges have been developed in OpenSees and the behaviour of the isolators has been described through an advanced model, which allows the accurate description of the horizontal and vertical responses as well as their interaction. After investigating in detail the most important characteristics of the seismic response of the reference bridge model, an extensive parametric study has been carried out to identify under which design situations the uplift effect is critical, i.e. for which properties of the superstructure, the substructures, bearings, and pier-to-deck connection (i.e. the eccentricity of the bearings with regards the axis of the pier), the uplift effect is more likely to occur. The performance of the bridge models has been checked against a set of limit states related to the bridge performance and consistent with current codes for earthquake resistance.
- 1.
The vertical response of the bearings is influenced by many factors including the rotation of the pier top and the vertical motion of the deck, which can be significant even if the vertical motion of the seismic input is not considered. Significant coupling is also observed between the horizontal and vertical motion of the bearing, which is accurately described by the bearing model utilised in this paper. SSI effects are also been found to be important for both the horizontal and the vertical response of the isolators, since they influence the deformations of the piers and the bearings.
- 2.
Higher modes to the piers are excited and contribute to the pier top rotation and, thus, influence the vertical deformations of the bearings.
- 3.
The occurrence of different limit states related to the bearing performance is strongly affected by the bearing design, and in particular by the value assumed for the shape factor. In most of the cases buckling of bearings is found to be more critical than bearing cavitation, except for the case of piers of great heights and high values of the bearing shape factor.
- 4.
Cavitation can be avoided in most cases by a proper choice of the value of the shape factor during the design stage. The most effective design solution against bearing cavitation and buckling may be the use of a very flexible continuity slab or to avoid it altogether by using expansion joints between adjacent spans. This design solution appears also effective when considering the effects of the vertical component of the seismic action in the analysis, which leads to increased values of the axial force demand in the bearings.
While the authors do not expect the obtained results to be overturned for other types of laminated bearing (such as lead-plug bearings) or bearing models (such as those considering an elasto-plastic response in shear uncoupled from the vertical response), such matters are being addressed in future studies, which will need also to address more deeply the effect of the vertical component of the seismic excitation and the optimal design of the geometrical and mechanical bearing properties.
Notes
Acknowledgments
Funding for the project Uplift of Elastomeric Bearings in Isolated Bridges, provided by Innovate-UK, is gratefully acknowledged.
References
- American Association of State Highway and Transportation Officials (AASHTO) (2014) Guide specifications for seismic isolation design, 4th edn. AASHTO, WashingtonGoogle Scholar
- Buckle I, Yen W-H, Marsh L, Monzon E (2012) Implications of bridge performance during Great East Japan Earthquake for US seismic design practice. In Proceedings of the international symposium on engineering lessons learned from the 2011 Great East Japan Earthquake, Tokyo, 1–4 Mar 2012, pp 1363–1374Google Scholar
- California Department of Transportation (CalTrans) (1999) Bridge memo to designers (20-1)—seismic design methodology. CalTrans, SacramentoGoogle Scholar
- Cardone D, Perrone G (2012) Critical load of slender elastomeric seismic isolators: an experimental perspective. Eng Struct 40:198–204CrossRefGoogle Scholar
- Cardone D, Dolce M, Palermo G (2009) Direct displacement-based design of seismically isolated bridges. Bull Earthq Eng 7(2):391–410CrossRefGoogle Scholar
- Cardone D, Palermo G, Dolce M (2010) Direct displacement-based design of buildings with different seismic isolation systems. J Earthq Eng 14(2):163–191CrossRefGoogle Scholar
- Chen WF, Duan L (eds) (2003) Bridge engineering: seismic design (principles and applications in engineering). CRC Press, Boca RatonGoogle Scholar
- Constantinou MC, Whittaker AS, Kalpakidis Y, Fenz DM, Warn GP (2007) Performance of seismic isolation hardware under service and seismic loading. MCEER-07-0012, Multidisciplinary Center for Earthquake Engineering Research, University at Buffalo, New YorkGoogle Scholar
- Dezi F, Carbonari S, Leoni G (2013) Lumped parameter model for the time-domain soil-structure interaction analysis of structures on pile foundations. ANIDIS, PadovaGoogle Scholar
- Dorfmann A, Burtscher SL (2000) Aspects of cavitation damage in seismic bearings. J Struct Eng 126:573–579CrossRefGoogle Scholar
- European Committee for Standardization (ECS) (2005a) Structural bearings—Part 3: elastomeric bearings (EN 1337-3). ECS, BrusselsGoogle Scholar
- European Committee for Standardization (ECS) (2005b) Eurocode 8: design of structures for earthquake resistance, Part 1: general rules, seismic actions and rules for buildings (EN 1998-1). ECS, BrusselsGoogle Scholar
- European Committee for Standardization (ECS) (2005c) Eurocode 8: design of structures for earthquake resistance, Part 2: bridges (EN 1998-2). ECS, BrusselsGoogle Scholar
- European Committee for Standardization (ECS) (2009) Anti-seismic devices (EN 15129. ECS, BrusselsGoogle Scholar
- Gent AN (1990) Cavitation in rubber: a cautionary tale. Rubber Chem Technol 63:49–53CrossRefGoogle Scholar
- Gent AN, Lindley PB (1959) Internal rupture of bonded rubber cylinders in tension. Proc R Soc Lond A Math Phys Eng Sci 249(1257):195–205CrossRefGoogle Scholar
- Grant DN, Fenves GL, Whittaker AS (2004) Bidirectional modelling of high-damping rubber bearings. J Earthq Eng 8:161–185Google Scholar
- Iervolino I, Galasso C, Cosenza E (2010) REXEL: computer aided record selection for code-based seismic structural analysis. Bull Earthquake Eng 8(2):339–362CrossRefGoogle Scholar
- Japan Road Association (JRA) (2002) Chapter 1: seismic design specifications for highway bridges. International Institute of Seismology and Earthquake Engineering, Japan Road Association, TokyoGoogle Scholar
- Kappos AJ, Saiidi MS, Aydinoglu MN, Isakovic T (2012) Seismic design and assessment of bridges, inelastic methods of analysis and case studies. In: Geotechnical, geological and earthquake engineering. SpringerGoogle Scholar
- Katsaras CP, Panagiotakos TB, Kolias B (2009) Effect of torsional stiffness of prestressed concrete box girders and uplift of abutment bearings on seismic performance of bridges. Bull Earthq Eng 7(2):363–375CrossRefGoogle Scholar
- Kelly JM (1997) Earthquake-resistant design with rubber. Springer, LondonCrossRefGoogle Scholar
- Koh CG, Kelly JM (1987) Effects of axial load on elastomeric isolation bearings. EERC/UBC 86/12, Earthquake Engineering Research Center, University of California, Berkeley, p 108Google Scholar
- Kumar M, Whittaker AS, Constantinou MC (2014) An advanced numerical model of elastomeric seismic isolation bearings. Earthq Eng Struct Dyn 43:1955–1974CrossRefGoogle Scholar
- Lee GC, Kitane Y, Buckle IG (2001) Literature review of the observed performance of seismically isolated bridges. In: Research progress and accomplishments: multidisciplinary center for earthquake engineering research, pp 51–62Google Scholar
- Lesgidis N, Kwon OS, Sextos A (2015) A time-domain seismic SSI analysis method for inelastic bridge structures through the use of a frequency-dependent lumped parameter model. Earthq Eng Struct Dyn 44(13):2137–2156CrossRefGoogle Scholar
- Makris N, Badoni D, Delis E, Gazetas G (1994) Prediction of observed bridge response with soil–pile-structure interaction. J Struct Eng (ASCE) 120(10):2992–3011CrossRefGoogle Scholar
- Manos GC, Mitoulis SA, Sextos A (2012) A knowledge-based software for the design of the seismic isolation system of bridges. Bull Earthq Eng 10(3):1029–1047CrossRefGoogle Scholar
- McKenna F, Fenves GL, Scott MH (2006) OpenSees: open system for earthquake engineering simulation. Pacific Earthquake Engineering Center, University of California, BerkeleyGoogle Scholar
- Mitoulis SA (2014) Uplift of bearings in isolated bridges subjected to longitudinal seismic actions. Struct Infrastr Eng. doi:10.1080/15732479.2014.983527 Google Scholar
- Muhr AH (2006) Effect of thickness of reinforcing plates on rubber-steel laminated bearings. In: 6th World conference on joints and bearings and seismic systems for concrete structures, Halifax, 17–21 Sept 2006Google Scholar
- Muhr AH (2007) Mechanics of Elastomer-Shim laminates. CMC 5(1):11–30Google Scholar
- Olmos B, Roesset J (2012) Inertial interaction effects on deck isolated bridges. Bull Earthq Eng 10(3):1009–1028CrossRefGoogle Scholar
- Pond TJ (1995) Cavitation in bonded natural rubber cylinders repeatedly loaded in tension. J. Nat Rubber Res 10(1):14–25Google Scholar
- Tubaldi E, Dall’Asta A (2011) A design method for seismically isolated bridges with abutment restraint. Eng Struct 33(3):786–795CrossRefGoogle Scholar
- Ucak A, Tsopelas P (2008) Effect of soil-structure interaction on seismic isolated bridges. J Struct Eng 134(7):1154–1164CrossRefGoogle Scholar
- Warn GP, Whittaker AS, Constantinou MC (2007) Vertical stiffness of elastomeric and lead-rubber seismic isolation bearings. J Struct Eng 133(9):1227–1236CrossRefGoogle Scholar
- Yang QR, Liu WG, He WF, Feng DM (2010) Tensile stiffness and deformation model of rubber isolators in tension and tension-shear states. J Eng Mech 136:429–437CrossRefGoogle Scholar
Copyright information
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.