Revised: 13 September 2022
Accepted: 19 September 2022
DOI: 10.1002/eqe.3751
RESEARCH ARTICLE
Optimization of the seismic response of bridges using
variable-width joints
Ioannis G. Mikes1,2
Andreas J. Kappos1
1 Department
of Civil Infrastructure and
Environmental Engineering, Khalifa
University, Abu Dhabi, UAE
Abstract
An aspect of seismic design of bridges that has hardly received proper attention
2 Department of Civil Engineering,
University of London, London, UK
so far is the appropriate selection of joint gaps. End gaps define the boundary conditions of the bridge and affect its dynamic response; their proper design can lead
Correspondence
Ioannis G. Mikes, Department of Civil
Engineering, City, University of London,
London, UK.
Email: ioannis.mikes@city.ac.uk
to an improved structural performance under dynamic actions. The idea of the
‘Dynamic Intelligent Bridge’ is explored here, wherein current bridge joints that
Funding information
Khalifa University, Grant/Award Number:
FSU-2020-15
have a fixed width are substituted by variable-width joints and, under seismic
loading, the joint gap is optimised either with a one-off adjustment, or continuously (in real time) through semi-active control. In all cases, a novel device is
used that permits this improved behaviour of the joints, the moveable shear key
(MSK), a device for blocking the movement of the bridge deck, which has the
possibility to slide, hence varying the size of the existing joint gap. In this context, the effect of gap size on the seismic response of bridges is assessed herein
and a methodology is put forward for optimising this size, using a number of
criteria such as maintaining the functionality of the bridge for moderate earthquakes, and ensuring the safety of the bridge and its users under earthquakes
stronger than that used for design.
KEYWORDS
bridges, dynamic intelligent structures, performance-based optimum design, seismic joints
1
INTRODUCTION
Seismic design of a bridge entails ensuring adequate performance of all its critical components, namely (i) the deck that
has to remain elastic during the design seismic action; (ii) the piers, which in most cases are the main source of seismic
energy dissipation (ductile pier design); (iii) the bearings that should not exceed the code-prescribed deformations; (iv)
the isolators (special bearings and supplemental energy dissipation devices), which are the primary energy dissipation
component when a passive system (seismic isolation) is used; (v) the shear keys, which are typically designed with a view
to protecting the foundation elements; (vi) the foundations that, as a rule, are capacity protected, through the design of
the shear keys and/or the backwall of seat-type abutments; (vii) the abutments, which are generally designed for elastic
response. Code provisions (e.g.1,2 ) adequately address the design of all these components, although the resulting design
is not an optimal one, particularly in the case of a passive system (see i.a.3,4 ). On the contrary, the issue of bridge joints,
accounting for seismic action effects, is not covered in the same depth, as further discussed in the following.
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the
original work is properly cited.
© 2022 The Authors. Earthquake Engineering & Structural Dynamics published by John Wiley & Sons Ltd.
Earthquake Engng Struct Dyn. 2023;52:111–127.
wileyonlinelibrary.com/journal/eqe
111
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Received: 20 May 2022
MIKES and KAPPOS
American seismic design practice as represented by Caltrans2 , requires that ‘seismic joints shall accommodate the
required horizontal movements and rotations while maintaining their full functionality with little or no damage under
the Functional Evaluation Earthquake (FEE)’; however, no specific provision on the joint size is included.
Eurocode 8 Part 2 (EC8-2)1 includes a specific provision for the size (dEd ) of ‘expansion’ joints between the deck and the
abutments, which is a function of the seismic displacement (dE ), the displacement from permanent and quasi-permanent
actions (dG ), including prestressing after losses, shrinkage and creep in concrete bridges, and the displacement due to
thermal movements (dT )
𝑑𝐸𝑑 = 𝛼𝑑𝐸 + 𝑑𝐺 + 0.5𝑑𝑇
(1)
wherein the fraction (α) of dE is chosen ‘based on a judgement of the cost-effectiveness of the measures taken to prevent
damage’. When the abutment backwalls are designed as ‘sacrificial’ elements α = 0.4, hence closing of the joint is indeed
envisaged under the design earthquake. If the designer decides that the abutments should be treated as ‘critical’ elements
and their backwalls are no more deemed as ‘sacrificial’ α = 1.0 is recommended. The same value (1.0) is also taken to
design overlap length between the deck and the abutment seat (as, obviously, unseating is a ‘critical’ failure mode).
Despite being more detailed than those of other codes, the EC8-2 provisions do not provide an end joint gap that leads to
an optimum performance of the bridge. Indeed, in practice, European designers usually opt for gaps substantially larger
than the code-prescribed ones, apparently to avoid the difficulties of accounting for gap closure in design. When elastic
analysis methods are used and gaps are expected to close, Caltrans2 recommends ‘dual’ analysis, considering the scenarios
of free movement and full restraint at one end of the deck.
The recent years having witnessed a broad endorsement of performance-based design (PBD), the idea of optimising
this design has emerged as attractive, and indeed, some recent studies in the USA have put forward the concept of
performance-based optimum seismic design (PBOSD);5 is a notable contribution in this direction adopting a probabilistic
PBOSD approach and applying it to the design of a seismically isolated bridge.
The idea of the ‘Dynamically intelligent bridge’ (DIB) was recently coined in an (essentially conceptual) paper by the
senior author6 . The key concept was that current bridge joints that have a fixed width are substituted by variable-width
joints, with a view to optimising the response of the bridge under different ‘loading scenarios’, using a novel device that
permits varying the joint gap, the moveable shear key (MSK). When performance under dynamic loading is the main
concern, the joint gaps in the DIB are optimised either with a one-off adjustment (passive system), or continuously (in
real time) through semi-active control.
The basic aspects of the MSK are presented in Section 2 of this paper. The effect of joint gap size on the seismic performance of a class of bridges, which is related to the feasibility of the DIB, is studied in Section 3, while Section 4 puts
forward a methodology for optimising the gap size using selected performance criteria (limit states), as a function of the
level of seismic action to which the bridge is subjected.
2
2.1
THE MOVABLE SHEAR KEY
Goals and envisaged configurations
The MSK6 is a stopper, arranged either externally (Figure 1A), or internally (Figure 1B) to the deck, which (unlike the
usual shear keys) is not permanently fixed to the seat of the abutment but can slide, opening a previously closed gap or
closing an existing gap between the deck and the substructure (abutment). Vertical support to the deck during the lateral
displacement is provided either by a system of (the commonly used in modern bridges) elastomeric bearings or, preferably,
by friction pendulum bearings that have the advantage that they restore the bridge to its initial position when the dynamic
action causing the horizontal displacement ceases.
The performance sought by varying the joint gap size depends on the main objectives of the bridge design, i.e.:
1. Whenever low-probability, high-amplitude, dynamic loads (e.g. strong earthquakes or hurricanes) are a key consideration, an open gap in the transverse direction of the bridge (d1t in Figure 1A, B) reduces the stiffness and may result
in a more favourable behaviour under, e.g. medium-intensity earthquakes, whereas a closed gap may be preferable for
strong earthquakes under which control of displacements and prevention of unseating are the primary performance
requirements. The longitudinal gap (d1l ) is also variable, as shown in Figure 1D, E). Since the idea here is to control the
seismic performance through varying a bridge property (end gap) in real time, the concept may be termed the ‘Dynamic
Intelligent Bridge’ (DIB), and is further explored in the remainder of the paper.
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112
113
F I G U R E 1 Possible configurations of bridges with adaptive boundary conditions: MSKs in the transverse direction (top); in the
longitudinal direction (bottom). Details of MSKs given in Figure 2.
2. When the durability of the bridge and the cost of maintenance are the key considerations (e.g. in low seismicity zones),
MSKs are provided in the longitudinal direction of the bridge, at one abutment only (Figure 1C), while the other one is
monolithically connected to the deck (fully integral). The gap (d0 , typically equal to d1l ) remains closed under normal
temperature variations and contractions caused by shrinkage, creep, and prestress in concrete decks; opening of the
gap takes place only if an ‘extreme’ temperature scenario occurs. The key goal here is to further extend the range of
spans permitted in integral bridges by allowing a gap to form at the end of the bridge where the MSKs are located when
environmental actions exceed an appropriately optimized threshold. As the bridge is typically behaving as integral but
can be transformed to a jointed one under the ‘extreme’ scenario, this new type of bridge may be termed the ‘Hybrid
Bridge’. Exploring this concept is beyond the scope of this paper.
2.2
Approaches for the design of bridges with MSKs
Two different solutions for bridges with MSKs are envisaged6 :
(i) An adaptive passive system, wherein the optimum gap size is applied to the bridge by displacing the shear keys, as
soon as an early warning system and/or a measure of ground acceleration in the area of the bridge is transmitted to
the control system of the MSKs. This is the solution that is explored in the remainder of this paper.
(ii) A semi-active system that entails closed-loop feedback control wherein the position of the shear key can be adjusted
in (almost) real-time to obtain the optimum dynamic response of the bridge. This option is primarily meant for major
bridges, due to the higher costs associated with installing and maintaining the control devices; in this case, movement
of the MSK can only be achieved by a piston or equivalent mechanism, which is feasible but clearly increases the cost
of the device. This challenging solution will be explored in future research endeavours.
3
3.1
EFFECT OF JOINT GAP SIZE
Selection and modelling of bridges
The effect of gap size on seismic performance is, of course, different in each bridge. In general, the effect of the abutmentbackfill system is more significant in shorter bridges7 and hence the effect of end joint gap is expected to be more significant
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MIKES and KAPPOS
MIKES and KAPPOS
FIGURE 2
Views of alternative shear key configurations.
in these bridges. In view of this, a parametric study of the effect of gap size was carried out for typical overpass bridges
with a length of around 100 m, as described in the following.
The reference bridge is an actual three-span overpass of Egnatia Motorway (N. Greece), shown in Figure 3, consisting of
a 45 m central span and two 27 m outer spans. The deck consists of a 10 m wide prestressed concrete box girder section. The
two piers of the bridge have a circular cross section, with a diameter of 2 m and are monolithically connected to the deck;
their clear heights are 5.9 and 7.9 m, their longitudinal reinforcement consists of two layers of 48⌀25 each, separated by
a space of 0.014 m, and their transverse reinforcement of ⌀16/75 circular ties and ⌀14/75 spirals (Figure 3). The deck ends
are supported on seat-type abutments (Figure 3D), which include backwall, wingwalls, and external shear keys, through
two elastomeric bearings with dimensions (mm) equal to 350 × 450 × 136. The total heights of the abutments are equal
to 5.6 and 5.7 m, while the height of both backwalls is 2.4 m. The deck is separated from the seat-type abutments with
joints of 100 mm in the longitudinal direction and 150 mm in the transverse direction; it is noted that providing gaps in
the transverse direction is a rather common practice in modern bridges in S. Europe. Surface footings are used (due to the
relatively firm soil), 9.0 m × 8.0 m × 2.0 m for the piers, and 12.0 m × 4.5 m × 1.5 m for the abutments. The mean yield and
ultimate strengths of the reinforcement steel are fy = 550 MPa and 742.5 MPa, respectively, and its ultimate strain is εsu
= 0.095. The concrete used for the construction of the piers and the deck roughly corresponds to C30/37 of Eurocode 28 ,
while the concrete of the footings and the abutments to C25/30. The soil in the bridge area mainly consists of moderately
stiff clay formations, corresponding to soil class C according to Eurocode 81 .
In addition to the basic configuration, two more bridges were considered with equal pier heights, one with two ‘short’
piers (5.9 m) and one with two ‘long’ piers (7.9 m), with a view to identifying any significant effects of the irregularity in
pier height.
3.1.1
Modelling of the bridges
The finite element model of the reference bridge set up in OpenSees9 is shown in Figure 4; it is a spine model, consisting
of 3D beam-column elements and a number of nonlinear spring systems that account for material nonlinearities and
the presence of joint gaps. In the bridge itself inelastic response is accounted for at the pier ends (taking into account
confinement effects), while the deck is expected to remain essentially in the elastic range, as also confirmed in a previous
study10 ; a gap element is separating the top of the backwall (see Figure 3D) from the deck. Pier foundation compliance
is accounted for using the standard frequency-independent springs reported i.a. in11 . The elastomeric bearings at the end
of the deck were modelled as bilinear springs under horizontal shear and as elastic springs under flexure and axial load,
following the relationships given in12 .
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114
FIGURE 4
(A) Model of the overpass bridge; (B) modelling of the abutment area.
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F I G U R E 3 (A) Overpass bridge taken as basic configuration; (B) pier cross section; (C) bridge section at pier; (D) abutment section
(measured dimensions in m).
115
MIKES and KAPPOS
MIKES and KAPPOS
F I G U R E 5 Fitting of the resistance curve of the
studied abutment-backfill system to the constitutive
model of HyperbolicGap material.
The abutment-backfill system, which is an important part of the model when the effect of gap closure is to be studied,
was modelled using two approaches:
A (relatively) complex one shown in Figure 4B, wherein the entire abutment is modelled using beam-column elements
(with the possibility of plastic hinge formation at the base of the backwall), the backfill is modelled using a number of
springs and dashpots along both the stem wall and the backwall, and the shear keys are modelled through two nonlinear
springs in the transverse direction, separated from the deck through gap elements; in OpenSees, the two are merged into
one nonlinear element with flat initial part until gap closure, an option that prevents numerical instabilities caused by the
use of separate gap elements13 ). A more involved model including an extra spring (in series with that of the shear keys)
to model the transverse stiffness of the embankment was also explored13 , but the effect on the bridge response was small
and it was dropped in the parametric analyses. The constitutive law for the nonlinear springs representing the backfill
behaviour (Figure 4B) included the envelope proposed in14 , and the compression-only hysteresis loops of the backfill were
based on the unloading/reloading stiffness values suggested in15 for various types of backfill soil. The effect of radiation
damping was also considered in the longitudinal direction with the use of a dashpot at each end of the deck. The dashpot
coefficient was calculated in line with11 , which, however, refers to surface footings rather than abutments, therefore an
adjustment to take into account the fact that there is no soil above the backwall was made in the same way as in16 , where
comparisons with other approaches were also made.
A ‘simple’ model wherein the abutment-backfill system is modelled as a nonlinear spring in the longitudinal direction,
whose backbone curve is derived from a pushover analysis of the abutment-backfill system. To use the HyperbolicGap
‘material’ of OpenSees, fitting of this backbone to a hyperbolic curve is required. Specifically, the resistance curve derived
from the pushover analysis is fitted to the hyperbolic force-deflection relationship of the HyperbolicGap material for initial
gap size dgap = 0 (Equation 2)17 :
𝐹 (𝑥) =
𝑥
1
𝐾𝑚𝑎𝑥
+ 𝑅𝑓
𝑥
(2)
𝐹𝑢𝑙𝑡
where F(x) is the resistance of the abutment-backfill system as a function its displacement x, Kmax is the initial stiffness
of the system, Fult is the ultimate resistance of the system and Rf is the failure ratio, which is the ratio of Fult over the
asymptote of the hyperbolic curve. The calibration is achieved with an algorithm that implements curve fitting via optimization of the parameters Kmax and Rf of equation 2 in MATLAB18 . The sum of squared errors (SSE) was used as the
objective function for the optimisation; i.e., the backbone curve used for the ‘simple’ model is the curve for which the SSE
is minimised. The backbone curve of the ‘simple’ model of the studied abutment-backfill system, derived following the
described methodology, is shown in Figure 5 along with the resistance curve of the system.
Modelling of the transverse direction of the ‘simple’ model is the same as that of the complex model.
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116
FIGURE 6
Force-displacement curve of the shear key according to19 and trilinear approximation.
FIGURE 7
Spectral matching of the suite of the artificial ground motions to elastic design spectrum of EC8.
117
For the nonlinear springs representing the shear keys (Figure 4B), the shear force versus displacement relationship
based on the test results reported in19 was simplified as shown in Figure 6.
3.1.2
Seismic action
A set of seven spectrum-compatible artificial accelerograms were used (Figure 7), scaled to various amplitudes of the
design spectrum. The lowest level of action was selected at PGA = 0.08 g (50% of the design earthquake Ed) and the
highest at PGA = 0.96 g (six times the design earthquake Ed). The Eurocode 81 spectrum for Soil class C was considered,
with corner period TD (the period value delineating the beginning of the constant displacement response range of the
spectrum) equal to 4s in lieu of TD = 2s, as a more appropriate value for high seismicity areas (see e.g.20,21 ), especially
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MIKES and KAPPOS
MIKES and KAPPOS
when displacements are a key output parameter. It is clarified that the PGA values shown in the legend of Figure 7 and
in all presented results of the analyses (Section 3.2) are the reference PGAs at bedrock. In Eurocode 8 (and several other
codes) these are multiplied by the ‘soil factor’ (S), which in our study was 1.15, the S factor for Ground type C. To be noted
that although the highest seismic actions associated with performance objectives in PBD typically have a probability of
exceedance of 2% in 50 years, which roughly corresponds to 2Ed for high seismicity areas, scaling to values corresponding
to more than 2Ed is needed for research purposes (particularly for fragility analysis). In the cases studied, the results of
the nonlinear response history analysis (NRHA) showed that pier yielding occurred for input motions scaled to around
2Ed , clearly indicating that the piers were overdesigned (i.e. presented significant overstrength). As a result, verification
of limit states involving heavy damage (highly nonlinear response of the bridge) had to be made for input motions with
substantially higher amplitudes than the design earthquake.
3.1.3
Gap sizes
The selected sets of joint gap sizes were 0, 25, 50, 75, 100, 125 and 150 mm in the longitudinal and 0, 50, 100, 150, 200 and
250 mm in the transverse direction, i.e. values higher as well as lower than those used in the actual bridge (i.e. 100 mm
in the longitudinal and 150 mm in the transverse direction) were considered. The joint gap sizes calculated according to
EC8-2 treating the elements of the abutment either as ‘sacrificial’ or as ‘critical’ (see Section 1) also fall within the range
of the selected sets of joint gap sizes in both directions. Using equation 1 with α = 0.4 (abutments designed as ‘sacrificial’)
results in dEd = 55 mm in the longitudinal direction (out of which 45 mm are due to ‘non-seismic’ actions) and dEd = 35
mm in the transverse direction. Taking α = 1 (i.e. avoiding gap closure under the design seismic action) leads to dEd = 70
mm in the longitudinal and dEd = 85 mm in the transverse direction. The zero-gap case in the longitudinal direction was
analysed mainly for completeness, as there is typically a non-zero longitudinal gap to account for ‘non-seismic’ actions
in seat-type abutments. Note that there is no need to consider gaps larger than the maximum values considered herein,
as the expected displacement of the bridge under the highest seismic action used in the analysis will just close the gaps.
The previous remarks offer a useful guide for the selection of gaps, i.e. the latter may vary from a minimum needed for
‘non-seismic actions’ to a maximum corresponding to the displacement resulting for twice the design seismic action.
In designing the longitudinal joint gap size, a reasonable ‘scenario’ of simultaneous actions has to be adopted. Referring
to concrete bridges, it is appropriate to assume that long-term contractions due to shrinkage and creep (and, wherever
applicable, prestressing) have developed, while temperature (which varies during the year) may be either ignored or
accounted for as in equation (1). Under this scenario, the initial opening of the longitudinal gaps will be equal to dG
in Equation (1), i.e. 36 mm in the studied bridges, when dT is ignored.
3.2
Results of analysis
Figure 8 shows the maximum displacements at the ends of the deck of the reference bridge, calculated for various levels
of seismic action (from Ed to 3Ed ) and different gap sizes; in addition to the OpenSees results, the values calculated using
a similar model set up on SAP 200022 are also reported. It is seen that the results are quite similar in both cases, the largest
discrepancies being found for higher levels of earthquake, and for the zero longitudinal gap case; they are attributed
primarily to minor differences in the nonlinear laws used in each program. This is one form of ‘cross-verification’ of
the models set up; it should be reminded here that individual aspects of the models and constitutive laws used were
verified against test results in previous studies; for instance, the nonlinear constitutive law used for the backfill system
(one of the more uncertain components to be modelled) was that proposed in14 , based on findings from large-scale tests
of backwall-backfill systems.
A number of modal analyses were also conducted, not only to cross-verify the models that were set up in OpenSees and
SAP 2000 but also to obtain a first estimation of how much the boundary conditions affect the response of the bridge
in either horizontal direction subsequent to joint gap closure. The first two models that were set up included deck ends
restrained in the vertical direction only (roller supports) (Model 1) and also deck ends supported by the elastomeric bearings of the studied bridge (Model 2). To assess the effect of gap closure in the longitudinal direction, a variation of Model
2 was set up (Model 3) that accounted for the contribution of the abutment-backfill system considering its secant stiffness
at δ50 (displacement when 50% of the ultimate resistance of the abutment-backfill system is activated). For the studied
abutment-backfill system δ50 ≈ 16 mm, whereas the peak strength is reached at about 120 mm; therefore, the considered
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118
119
F I G U R E 8 Maximum displacement at the top of the end bearings in the longitudinal direction for three levels of ground motion and
various gap sizes, calculated with SAP 2000 and OpenSees.
F I G U R E 9 Modal shapes (bird-eye view), natural periods, and mass participation factors of the predominant mode in the transverse
direction for Models 1, 2, 4 and 5.
TA B L E 1
Effect of gap size on maximum pier drift – Longitudinal excitation
0.08 g
0.16 g
0.32 g
0.48 g
0.64 g
0.80 g
0.96 g
0
0.10%
0.22%
0.52%
0.94%
1.42%
1.91%
2.36%
PGA:
dgap (mm)
25
0.16%
0.31%
0.61%
1.00%
1.46%
1.92%
2.36%
50
0.16%
0.31%
0.68%
1.07%
1.47%
1.91%
2.38%
75
0.16%
0.31%
0.68%
1.14%
1.52%
1.95%
2.41%
100
0.16%
0.31%
0.68%
1.15%
1.60%
2.00%
2.48%
125
0.16%
0.31%
0.68%
1.15%
1.62%
2.10%
2.54%
150
0.16%
0.31%
0.68%
1.15%
1.62%
2.15%
2.63%
secant stiffness is relatively high and hence tends to overestimate the effect of the backfill system. To assess the effect of
gap closure in the transverse direction, two variations of Model 2 were studied. Namely, Model 4 included a full restraint
in the transverse direction and Model 5 accounted for the contribution of the shear keys considering their secant stiffness
at δ = 250 mm. This very large δ roughly corresponds to the transverse displacement of the deck for input motion scaled
to 0.96 g (6Ed ) as derived from the nonlinear response history analysis; thus, the considered secant stiffness is the lowest
possible.
Modal analysis of the aforementioned models revealed that gap closure significantly affects the response of the studied
bridge in the transverse direction, even for Model 5 that underestimates the effect of gap closure. This effect is clearly
seen in Figure 9, which illustrates the predominant modes in the transverse direction for Models 1, 2, 4 and 5 with their
respective natural periods and modal mass participation factors. On the contrary, the effect is minor in the longitudinal
direction since the predominant modes in the longitudinal direction of Models 1, 2 and 3 had similar shapes and modal
mass participation factors, while the respective natural periods were very similar (0.41–0.43 s).
Some key results of the nonlinear response history analysis are presented in Tables 1–4. Table 1 summarises the maximum pier drifts (average of 7 motions) calculated for all the longitudinal gap sizes considered, as a function of the level
of seismic action, while Table 2 reports the same results for the transverse direction. Top drift is an important indicator
of the performance of the bridge piers and the results of the NRHAs clearly indicate that in both directions small gap
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MIKES and KAPPOS
TA B L E 2
MIKES and KAPPOS
Effect of gap size on maximum pier drift – Transverse excitation
0.08 g
0.16 g
0.32 g
0.48 g
0.64 g
0.80 g
0.96 g
0
0.20%
0.39%
0.87%
1.29%
1.64%
2.05%
2.65%
PGA:
dgap (mm)
TA B L E 3
50
0.23%
0.47%
0.96%
1.32%
1.77%
2.18%
2.57%
100
0.23%
0.45%
0.99%
1.38%
1.75%
2.15%
2.60%
150
0.23%
0.45%
1.00%
1.47%
1.83%
2.25%
2.76%
200
0.23%
0.45%
1.00%
1.45%
1.86%
2.35%
2.83%
250
0.23%
0.45%
1.00%
1.45%
1.87%
2.41%
2.89%
Effect of gap size on maximum displacement at the top of the backwall (in m)
0.08 g
0.16 g
0.32 g
0.48 g
0.64 g
0.80 g
0.96 g
0
0.007
0.016
0.040
0.067
0.096
0.124
0.151
25
N/C1
N/C
0.020
0.045
0.073
0.099
0.125
50
N/C
N/C
N/C
0.024
0.048
0.074
0.101
PGA:
dgap (mm)
1
75
N/C
N/C
N/C
N/C
0.025
0.050
0.078
100
N/C
N/C
N/C
N/C
0.002
0.026
0.056
125
N/C
N/C
N/C
N/C
N/C
N/C
0.035
150
N/C
N/C
N/C
N/C
N/C
N/C
0.003
N/C: no gap closure occurred in the analysis.
TA B L E 4
Effect of gap size on maximum displacement of the shear key (in m)
0.08 g
0.16 g
0.32 g
0.48 g
0.64 g
0.80 g
0.96 g
0
0.004
0.007
0.014
0.036
0.110
0.172
0.245
50
N/C1
0.009
0.025
0.055
0.093
0.134
0.172
PGA:
dgap (mm)
1
100
N/C
N/C
0.012
0.025
0.044
0.076
0.102
150
N/C
N/C
N/C
0.011
0.024
0.041
0.063
200
N/C
N/C
N/C
N/C
N/C
0.022
0.041
250
N/C
N/C
N/C
N/C
N/C
N/C
0.013
N/C: no gap closure occurred in the analysis.
sizes may alter the response of the bridge significantly, mainly in the range of intensities between 0.16 g and 0.64 g. The
resulting pier drifts had fairly close values for all the used input motions since the coefficient of variation (COV) of the
drift varied from 4% (for PGA = 0.80 g and dgap = 50 mm, transverse excitation) to 15% (for PGA = 0.96 g and dgap =
25 mm, longitudinal excitation). As expected, the abutment-backfill system is activated more when the gap size is small,
and this emerges as beneficial for the piers (following the gap closure, a substantial part of the seismic force is resisted
by the abutment-backfill system). It is also beneficial for the bearings whose performance is critical with regard to the
functionality of the bridge and which sustain less damage when gaps are small. Bearings are affected more than piers,
particularly in the transverse direction, where for PGA = 0.32 g, the displacement at their top decreases by 90% in the case
of closed gap compared to the case of the gap size that was originally used in the case study bridge (150 mm).
Table 3 summarises the maximum displacement at the top of the backwall (average of seven motions) in the longitudinal
direction, which is an important indicator of the performance of the abutment-backfill system; whenever no gap closure
occurs, there is no backwall deformation. In the same manner as the pier drifts, the maximum displacements at the top
of the backwall did not show significant variation with the input ground motion; the COV reached up to 16% in cases
with dgap = 0. It is seen that in the range of very high intensities, small gaps may be detrimental since the capacity of the
abutment-backfill system is almost reached and hence, it does not further contribute to the response of the bridge, while
it is also heavily damaged.
Table 4 reports the maximum displacement of the shear keys (in the transverse direction); when no gap closure occurs,
there is no shear key deformation. As expected, the most critical gaps for the shear keys are the small ones; however, even
for very high levels of earthquake, no shear key failure is expected. In the case of initially closed gaps (dgap = 0), the results
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F I G U R E 1 0 Longitudinal direction, ground motion scaled to PGA = 0.96 g: Nonlinear dynamic response at the bottom of the pier for
dgap = 125 mm (A) and dgap = 0 (B), and force-deflection diagram of the abutment-backfill system for dgap = 0 (C).
F I G U R E 1 1 Transverse direction, ground motion scaled to PGA = 0.96 g: Nonlinear dynamic response at the bottom of the pier for
dgap = 250 mm (A) and dgap = 0 (B), and force-deflection diagram of the shear key for dgap = 0 (C).
varied significantly depending on the input ground motion in the range of PGAs from 0.48 g to 0.80 g, where the COV of
the resulting displacements reached 40%. For the other ranges of the examined PGAs, it was less than 20%. From Figure 6
that shows the nonlinear constitutive law used for the shear key (red line), it is seen that in the worst case of zero gap and
PGA of 0.96 g the shear key is still in the second branch of its force-displacement relationship. Overall, both the shear keys
and the bridge in general were clearly overdesigned, as will also be seen in the next section.
The moment-curvature diagrams of the plastic hinge at the bottom of the shorter pier (for ground motion ‘1’) (Figure 7)
scaled to 0.96 g in the longitudinal direction are depicted in Figure 10A and 10B for the cases of dgap = 125 mm and dgap
= 0, respectively. In Figure 10C, the force-deflection diagram of the ‘simple’ model of the abutment-backfill system in the
same direction for dgap = 0 is shown. It is clear that although the reduction of the pier curvature in the case of closed
gap was noticeable, 34% less than in the case of dgap = 125 mm, it occurred at the cost of reaching the point of maximum
resistance of the abutment-backfill system (i.e. more damage in this component).
The moment-curvature diagrams for the longer pier for the cases of dgap = 250 mm and dgap = 0 and the force-deflection
diagram of the shear key for ground motion ‘1’ (Figure 7) scaled to 0.96 g in the transverse direction are illustrated in
Figure 11A–11C, respectively. The reduction of the curvature at the bottom of the pier reached 24%, while the shear key
displacement was well below the point of its peak strength. Given that even complete damage of the shear keys is not
deemed critical for the collapse of the entire bridge, which is the only reasonable requirement for such strong seismic
motions, having an initially closed transverse gap proved to be a beneficial factor for the performance of the bridge.
It is noted that for both Figures 10 and 11, the depicted results were derived from the ground motion that led to the
highest values of pier curvatures and abutment displacements for the largest gaps considered in the longitudinal and the
transverse direction, respectively. However, the results were similar for all the seven artificial ground motions.
The parametric study also included different configurations of the reference bridge (equal pier heights)23 . The effect
of pier configuration may be inferred from Figure 12, which shows the maximum longitudinal shear force in the most
critical abutment (average of 7 motions) for the three pier configurations and various gap sizes. It is seen that although the
values of the shear force are different in each bridge (the lowest values are for the bridge with two equal ‘short’ piers), the
general effect of the gap size is the same in all bridges, i.e. the maximum force always develops for zero gap (deck fixed to
the abutment seat) and reduces in a similar way as the joint gap increases. Overall, the tendencies observed in the equal
pier configurations (with respect to the gap size) were similar for most of the critical parameters; hence, it appears that
they are representative of a broader class of (relatively) short bridges/overpasses.
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MIKES and KAPPOS
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F I G U R E 1 2 Horizontal force in the abutment (kN), for various bridge configurations and gap sizes; ground motion scaled to EC8
spectrum for 0.32 g.
4
OPTIMIZATION OF GAP SIZE BASED ON PERFORMANCE CRITERIA
The parametric studies presented in Section 3 pave the way for a proper selection of the gap size with a view to optimising the seismic performance of a bridge. This can be done solely on the basis of performance criteria, or adding also cost
considerations. It is important to clarify the ‘scenario’ envisaged in each case, as different situations may be encountered
in practice. In this section, the basic scenario is the one described in the Introduction, i.e. in an existing bridge the optimum gap size is selected (and implemented in the bridge joints through the MSKs described in Section 2) whenever the
intensity of an incoming earthquake is known (e.g. through an early warning system) and its frequency content is similar
to that of the design spectrum; clearly, other spectral shapes may also be addressed (also in a probabilistic way), but the
procedure becomes significantly more involved. It is noted that other gap selection scenarios are also worth exploring;
i.e. a single optimum gap size may be selected at the design stage, by designing the bridge for a number of reasonable
gap sizes (see Section 3.1.3); the quasi-optimum gap may be selected as the one that leads to the minimum reinforcement
(while satisfying all code requirements) or, more comprehensively (but also more time-consumingly) when it minimises
the life-cycle cost of the bridge.
4.1
Steps of the proposed methodology
For the case that the joint gaps are adjusted (through the MSKs) in ‘quasi-real time’ to optimise the performance of the
bridge during the incoming earthquake the following steps can be applied.
Step 1: NRHA for each level of seismic action and joint gap size in either direction
A number of end gap sizes are selected in each direction of the bridge, e.g. starting from code recommendations like
those of Eurocode 8-2 (Equation 1 with α = 1 or α = 0.4) and including higher and lower values (see Section 3.1.3). While
for the longitudinal direction ‘non-seismic’ requirements (temperature, and shrinkage, creep and prestress in concrete
bridges) will result in a minimum gap size, in the transverse direction the case of zero gap (shear keys blocking any transverse displacement) should also be included. The bridge is modelled for NRHA, including an appropriate representation
of the joint between the deck and the abutment and accounting for the contribution of the abutment-backfill system in
either horizontal direction, to properly capture the effect of joint gap closure. A full NRHA is in order in major/important
bridges, while simpler models accounting only for gap closure may be envisaged in other cases. Analyses are conducted
in the longitudinal and the transverse direction separately for appropriately selected sets of accelerograms, scaled to the
pertinent design spectrum and for various amplitudes (both lower and higher than that of the ‘design earthquake’), so
that several limit states can be checked; analyses are repeated for each selected joint gap size. Recall that in a ‘standard’
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122
TA B L E 5
123
Limit state definitions for the critical components of a bridge
Component
Limit
state
Threshold value
Description
Piers
LS2
Drift: 1%
Concrete cracking, spalling; seismically designed pier
LS4
Drift: 5%
Pier collapse; seismically designed pier
Bearings (and unseating)
LS2
Shear deformation: 100%
Initiation of slipping; visible damage; yield of steel shims
LS4
Displacement at top equals
abutment seat width
Deck unseating (longitudinal direction only)
Backwall-backfill (Longitudinal
direction)
LS2
Displacement: δ(Fmax /2)
Stiffness reduction of the abutment-backfill system
LS4
Displacement: 3×δ(Fmax )
Ultimate deformation of abutment-backfill system
Shear keys (Transverse direction)
LS2
Displacement: 4×δy,SK
Significant damage of the shear key
LS4
-
LS4 not controlled by shear key
PBD procedure up to four levels of seismic action are considered. So long as the procedure is automatised, more levels can
be considered in the analysis.
Step 2: Derivation of component-based safety factor matrices
A database is compiled with the max response quantities (‘engineering demand parameters’ EDPi) from each of the
analyses carried out in Step 1, to be subsequently used for gap optimization. An appropriate number of limit states are
defined for each selected bridge component I in either horizontal direction; the suggested limit state thresholds (LSTi) for
two LS that are deemed appropriate for gap selection are provided in Table 5; LS2 is the functionality LS (no closure of the
bridge) and LS4 is associated with Life Safety. A component-based safety factor (CSFi) is calculated for each state, level of
seismic action, and joint gap size by dividing each LST by the respective maximum response (EDPi,max), retrieved from
the database; hence, the CSF of a bridge component i is CSFi = LSTi/EDPi,max (obviously the lower CSFi will be used
if a component includes multiple elements, e.g. piers or bearings). Using these safety factors, the so-called CSF matrices
are set up. Such a matrix is produced for each limit state, EDPi and bridge direction considered; hence, the number of
CSF matrices is 2 × m × n, where m is the number of the considered limit states and n is the number of the EDPs used
for checking each LS. The size of a CSF matrix is j × k, where j is the number of the considered gap sizes in the studied
direction and k is the number of the considered levels of seismic action.
Step 3: Global safety factor matrices
The lowest safety factor among the considered bridge components for a certain direction, limit state, level of seismic
action, and joint gap size is defined as the global safety factor (GSF) of the bridge. The GSF values are selected in this
manner, based on the assumption that the critical bridge components form a series system for the evaluation of the performance of the bridge, i.e. it is assumed that exceedance of a limit state in at least one component implies exceedance of
the corresponding global limit state for the bridge (cf.3, 24 ). In this step, a single j × k matrix, denoted as GSFLSi matrix,
is set up for each considered limit state LSi, consisting of the GSFs for each of the j gap sizes and the k levels of seismic
action.
Step 4: Selection of quasi-optimum gap sizes
The quasi-optimum gap size in either direction of the bridge and for each level of seismic action is defined as the joint
gap size that maximises the GSF of the bridge for a certain limit state. As multiple LS are considered, the final optimum
gap size for a certain level of seismic action is the one that results in the maximum GSF for the limit state associated with
the highest performance (i.e. here with LS2), provided that this GSF is larger than 1, i.e. that the bridge response does not
exceed the adopted limit state threshold. Note that these values are generally different for different seismic intensities.
In the practical implementation of the ‘passive system’ the aforementioned optimum gap sizes (as functions of the reference PGA) are stored in an online system and, as soon as an early warning system and/or a measure of ground acceleration
recorded by a sensor in the area provide the PGA of the incoming ground motion, the end gaps of the bridge are adjusted to
the ‘optimum’ size through movement of the MSKs (Figure 1). This, of course, requires a battery-operated device (such as
a small piston) for displacing the MSKs. Clearly, simplifications can be made, particularly for low importance bridges; i.e.,
if it is not deemed to be cost-effective to adjust in real time the gap size, the optimum gaps for the design earthquake level
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MIKES and KAPPOS
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TA B L E 6
CSFi and GSFLSi (shown in bold) for ground motions scaled to PGA = 0.24 g and PGA = 0.96 g – Longitudinal direction
Component
dgap (mm)
0
Pier
LS4 / 0.96 g
0.42
2.12
2.05
10.28
0.42
2.12
1.87
9.35
0.40
2.02
0
1.49
20.26
0.29
3.92
25
1.11
15.18
0.29
3.92
100
1.03
14.04
0.27
3.74
0
0.57
13.20
0.11
2.45
25
1.30
30.15
0.13
2.95
100
N/C1
N/C
0.28
6.56
N/C: no gap closure occurred in the analysis.
TA B L E 7
Component
Pier
Bearing
Shear key
2
LS2 / 0.96 g
14.06
25
Backwall-backfill
1
LS4 / 0.24 g
100
Bearing
1
LS2 / 0.24 g
2.81
CSFi and GSFLSi (shown in bold) for ground motions scaled to PGA = 0.32 g and PGA = 0.96 g – Transverse direction
dgap (mm)
LS2 / 0.32g
LS4 / 0.32g
LS2 / 0.96 g
LS4 / 0.96 g
0
1.15
5.73
0.38
1.89
50
1.04
5.20
0.39
1.94
250
1.00
5.01
0.35
1.73
0
3.19
N/A2
0.18
N/A
N/A
50
0.60
N/A
0.20
250
0.33
N/A
0.16
N/A
0
4.15
N/A
0.25
N/A
50
2.36
N/A
0.35
N/A
250
N/C1
N/A
4.47
N/A
N/C: no gap closure occurred in the analysis.
N/A: no corresponding limit state thresholds.
(e.g. Tr = 475 years) may be selected. It should also be noted that the outlined method is based on the (typically adopted)
assumption that the incoming earthquake is well represented by the design spectrum (code-prescribed or site specific). In
principle, the method can be extended to account for other types of ground motion (that have lower probability to occur
in the area of the bridge), but it is doubtful that such an extra effort is worth in a practical context.
4.2
Pilot study of optimum gap size
A pilot application of the afore-described methodology was carried out for the 99 m overpass shown in Figure 3, which
has joint gaps in both the longitudinal and transverse directions and was designed for the code spectrum for PGA = 0.16
g. For the selection of a quasi-optimal gap size for each joint of the bridge, the two limit states defined in Table 5 were
utilised (LS2 and LS4), as these were deemed to be the most pertinent for PBD.
After carrying out the NRHA for the selected seismic actions and joint gap sizes presented in Section 3.1 (Step 1),
the CSF matrices were produced (Step 2). Indicatively, the CSFi for each considered critical component (pier, bearing,
backwall-backfill system and shear keys) for the selected PGAs, gap sizes, and limit states are shown in Table 6 (longitudinal direction) and Table 7 (transverse direction). In Table 7, the LS4-related CSF (CSFLS4 ) for the shear keys and the
bearings were not calculated, since LS4 was deemed not to be controlled by these components in the transverse direction, as shown in Table 5 (no unseating was predicted for any case). The global safety matrices (GSF) for each considered
direction, limit state, seismic action level and gap size were then derived (Step 3). The GSF derived from the CSF that are
shown in Tables 6 and 7 are marked in bold in these tables (note that for each PGA three GSF are reported, one for each
of the three gap sizes shown in the tables).
The results of Tables 6 and 7 show that for the studied bridge, seismic action levels of PGA = 0.24 g and PGA = 0.32
g governed the optimization with respect to LS2 in the longitudinal and the transverse direction, respectively, since the
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125
F I G U R E 1 3 Quasi-optimum joint gap sizes for each considered level of seismic action and limit state (a) longitudinal direction, (b)
transverse direction.
CSFLS2 of at least one EDP2 took values smaller and larger than 1, depending on the selected joint gap size. On the other
hand, no CSFLS4 lower than 1 was found, even for seismic motion scaled to 0.96 g (6Ed ), a clear indication of the overdesign of the bridge; notwithstanding this, the values of CSFLS4 for different gap sizes varied significantly. Another note
on the results in Tables 6 and 7 related to the lowest considered seismic actions is that while in the transverse direction
an initially closed gap relieved the piers and the bearings at the expense of relatively small damage in the shear keys, in
the longitudinal direction this was achieved only at the expense of significant damage in the backwall-backfill system.
Moreover, it is seen in Table 7 that having an initially open gap in the transverse direction did not necessarily lead to less
damage in the shear keys than having an initially closed gap, as it would perhaps be assumed intuitively, meaning that the
response of the shear keys should be estimated in all cases. This can be attributed to the complex behaviour of the response
of the studied bridge in the transverse direction and, as shown in the results of the various modal analyses illustrated in
Figure 9, to its dependence on the state of the transverse gaps, which constantly changes during a seismic excitation.
Figure 13 shows the quasi-optimum gap sizes (i.e. the best among those selected) for the two limit states (LS2 and LS4)
as a function of the level of seismic action (PGA) for the longitudinal and transverse joint gaps (Step 4); the lowest PGA
considered was 0.08 g, i.e. one half the design seismic action (0.5Ed ). The solid part of the lines denoting the optimum
gap for each limit state are those for which this LS governs the selection of optimum gap; the remainder of the lines are
shown as dashed and are included for completeness only; i.e. referring to Figure 13A, for 0.24 g (1.5Ed ), the bridge will
remain functional (a clear indication of how overdesigned it is) and the optimum gap is 25 mm (blue solid line for LS2);
for 0.32 g (2Ed ) LS4 (Life safety) governs, and the optimum gap is 0 (solid red line on the x-axis). It is worth noting that,
for the specific bridge configuration, the optimum gaps are the smaller ones, i.e. 0 or 25 mm. This can be interpreted as
a confirmation of the important role of the abutment-backfill system in carrying part of the seismic action effects in the
bridge. It should also be reminded herein that in the longitudinal direction of this jointed bridge an initial open gap of
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36 mm is expected due to ‘non-seismic’ actions and, in a practical design context, this should be taken as the minimum
gap size to be considered.
5
CONCLUSION AND RECOMMENDATIONS FOR FUTURE RESEARCH
A methodology was put forward for defining ‘quasi-optimum’ joint gaps in bridges equipped with novel devices (movable
shear keys) that can adjust the gap size. The methodology is based on performance criteria (exceedance of selected limit
states) and was applied to an existing, seismically designed bridge. The performance of the bridge with quasi-optimum
joint gaps was compared to that of the actual bridge. Some interesting conclusions were drawn from this study:
∙ The size of the longitudinal and the transverse end joint gaps can generally affect the dynamic response of the bridge.
In the case study presented herein, this effect was more prominent in the transverse direction.
∙ In the longitudinal direction, the effect of the joint gap size was negligible in the cases where the ‘Operationality’ limit
state (LS2) was the prevailing performance criterion. However, when the ‘Life safety’ limit state (LS4) prevailed, the
effect of gap size was important. For ground motions scaled to the code spectra for 0.24 g – 0.40 g even the mode of
failure changed due to the joint gap size; namely, in the case of the actual bridge, the piers governed the dynamic
response of the bridge, while an initially closed gap made the abutment-backfill system the critical bridge component,
eventually leading to a much higher GSF.
∙ In the transverse direction, the joint gap size significantly affected the performance of the bridge against the ‘Operationality’ limit state. The bearings, which were the weakest components in terms of ‘operationality’ in the actual bridge,
were completely relieved when the transverse gap was initially closed, and the GSF of the bridge increased to the point
that LS2 was not exceeded even for input motion scaled to twice the design earthquake intensity.
∙ An initially closed gap was the preferred option in most cases (also in the longitudinal direction); this is indeed a viable
option in fairly short bridges. However, in the case that the ‘Operationality’ limit state was used as the performance
criterion, having an initially closed gap in the longitudinal direction was unfavourable for the response of the bridge
since it caused damage to the abutment-backfill system without relieving the other bridge components.
It appears that there is still substantial room for further efforts towards integrating the effect of joint gaps in the design
of bridges. Both the option of selecting a practically optimum set of gaps (longitudinal and transverse) at the design stage
and the (more complex) option of varying the gap sizes during an earthquake with a view to optimising response, are
worth further exploration. The pilot case study presented herein indicates that such options may result in a substantial
improvement of the performance of the bridge, especially in the transverse direction, at least for bridges similar to the
studied overpasses. Clearly, more bridge configurations have to be considered, in particular more flexible than those of
the overpasses addressed herein.
Another issue that has to be addressed is the uncertainty in the limit state definition of the abutment-backfill system,
which, as is evident from the results of the presented case study, is critical for the assessment of the bridge response in
cases of small gaps. There has been little research focus over the past years on the appropriate definition of limit state
thresholds for the components of the abutment-backfill system, particularly regarding the transverse direction. This issue
is further discussed in25 , where the same bridge was used as case study and different definitions of the abutment-based
limit state thresholds resulted in substantially different assessments of the performance of the bridge for various gap sizes.
The design and fabrication of the MSK is also a challenge in itself. The forces to which it is subjected are substantial
and several options have to be explored for the connection of the concrete block to the steel rail needed for sliding and
the steel base (fixed to the abutment seat) on which the rail is supported (see Figure 2) and which should limit the sliding
displacement to avoid falling off the seat.
AC K N OW L E D G E M E N T
The authors gratefully acknowledge the financial support by Khalifa University, Abu Dhabi [research project reference
number: FSU-2020-15, Principal Investigator: A. Kappos].
D A T A AVA I L A B I L I T Y S T A T E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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127
ORCID
Ioannis G. Mikes https://orcid.org/0000-0002-9210-6093
Andreas J. Kappos https://orcid.org/0000-0002-5566-5021
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How to cite this article: Mikes IG, Kappos AJ. Optimization of the seismic response of bridges using
variable-width joints. Earthquake Engng Struct Dyn. 2023;52:111–127. https://doi.org/10.1002/eqe.3751
10969845, 2023, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/eqe.3751 by CochraneUnitedArabEmirates, Wiley Online Library on [12/12/2022]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
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