Chapter 3 Modelling
3.2 Modelling
3.2.3 Modelling of spring-dashpot system
For the spring-dashpot model, the soil domain below the foundation is replaced by suitable springs for modelling the behaviour of the foundation soil. The force- displacement curves for the nonlinear springs in the near-field zone are assigned to represent lateral load bearing capacity, vertical skin friction and pile tip axial load carrying capacity (API rp2a, 2000). . To incorporate soil-pile interaction, spring-dashpots have been modelled to represent the near-field and the far field effects of the soil domain.
In the near field soil, nonlinear damping has been taken care by hysteresis damping. The springs in the lateral and the vertical directions are modelled in parallel configuration to represent lateral load bearing capacity and skin friction of pile surface respectively.
Although soil mass has not been considered in the spring-dashpot model, far field soil stiffness and radiation damping have been modelled from the coefficients provided in Gazetas and Dobry (1984). Far field spring-dashpots have been modelled in series with near field nonlinear spring-dashpots (Fig. 3.3(a)). Far field spring and dashpots are modelled in parallel configuration. At the end of far field spring-dashpots, free field motions have been applied at the corresponding depth of piles. Abutment backfill interaction (ABI) is modelled by assigning nonlinear springs (Fig. 3.3(b)) behind the abutment wall. The abutment has been considered as ‘frame type’ and the corresponding
Soil layer
Elastic properties Nonlinear properties Maximum
shear modulus (MPa)
Poisson’s ratio
Total unit weight
(t/m3)
Undrained strength,
(MPa)
Shear modulus ratio (Figs. 3.3(b)
and (c))
Plasticity Index
OL/SM 76 0.45 1.9 0.030 Cyan Line 10
SP/SM 171 0.45 1.9 0.012 Yellow Line 0
CL 288 0.45 1.8 0.100 Orange Line 30
SP 525 0.45 2.1 0.053 Brown Line 0
springs are modelled at a spacing of 1m along the height of the abutment. The nonlinear force-deformation curves of the springs are obtained from BA 42/96 (2003). In the analysis, Rayleigh damping is considered as 5% at the frequencies of 0.5Hz and 5.0Hz (Fig. 3.3(c)). From past studies (Granas, 2016; Brødbæk et al., 2009; Monkul, 2008), it has been proven that API force-displacement curves overestimate the soil stiffness. So, implementing API curves in bridge with spring-dashpot model tends to enhance the imposed forces on the piles and other bridge members, leading to a conservative estimate of the response. Thus, to reduce this error, 10% Rayleigh damping has been used in spring-dashpot models. In Fig. 3.4, at different depth of piles and abutment backwall, force-displacement curves are plotted to implement SPI and ABI in the spring-dashpot models built in OpenSees. The outer pile and the middle pile from the intermediate pile group are denoted as Pile_1 and Pile_2, respectively.
Under a suitable set of input ground motions from past earthquake records, the impact is studied on the overall structural response of the following modelling assumptions: (i) free field input at the base of fixed base bridge (only B_FF); (ii) SSI input at the base of fixed base bridge (only B_SSI); (iii) full scale SSI model which includes soil domain, foundation and bridge with and without backfill soil (full SSI with BA model and full SSI no BA model, respectively) and (iv) bridge model with foundation and spring–dashpots elements to represent soil with and without backfill (FB_SD with BA model and FB_SD no BA model, respectively). In Table 3.5, all the modelling assumptions with proper specifications are given for different types of models considered for this study.
(a)
(b)
1.5 3.0 4.5 6.0
0.0 2.5 5.0
Fundamental frequency full SSI no BA B_FF
FB_SD no BA
Rayleigh Damping (%)
frequency, Hz (c) Fig. 3.3 Schematic diagram of (a) soil-pile interaction , (b) abutment-backfill interaction
and (c) Rayleigh damping considered for analysis (For the notations refer to Table 5.1).
-0.10 -0.05 0.00 0.05 0.10 -800
-400 0 400 800
skin friction, kN
(c) (d)
(b) depth
1m 2m 3m 4m 5m
lateral force, kN
(a)
0.00 0.01 0.02 0.03 0.04
0 50 100 150
Pile_2 Pile_1 Abutment pile
Pile_2 Pile_1 Abutment pile
0.00 0.05 0.10 0.15 0.20
0 200 400 600 800
end bearing capacity, kN
vertical displacement, m lateral displacement, m
pile tip displacement, m 0.0 0.1 0.2 0.3 0.4 0.5
0 20000 40000 60000 80000
lateral load, kN
lateral displacement, m depth
1m 5m 12m
Fig. 3.4 Variation of (a) lateral force with lateral displacement for Pile_2, (b) skin friction with vertical displacement at the depth of 5.2 m, (c) pile tip end bearing load
with vertical displacement at the depth of 5.2 m and (d) the lateral force with lateral displacemnt at abutment-backfill interface for different depths.
Table 3.5 Comparison between different modelling approaches
Model Boundary conditions (BC) of vertical members
Soil Abutment
backfill
Input Motion only B_FF Base of the bridge’s vertical
members are fixed
No soil Not present Free field soil surface motion at
the base of piers only B_SSI Base of the bridge’s vertical
members are fixed in translation and rotation
No soil Not present Input motion at base of piers from
the base of piers of SSI model full SSI no
BA
Base of the bridge’s vertical members have same BC in
horizontal and vertical direction to pile cap
connecting node
Linear or nonlinear continuum soil
domain
Not present Rock outcrop motion at the base
of soil domain
full SSI with BA
Base of the bridge’s vertical members have same BC in
horizontal and vertical direction to pile cap
connecting node
Linear or nonlinear continuum soil
domain
Continuum backfill soil
Rock outcrop motion at the base
of soil domain
lin str+nl soil no BA model
Same as full SSI no BA model (piers are linear)
nonlinear continuum soil domain
Not present Rock outcrop motion at the base
of soil domain nl str+lin
soil no BA model
Same as full SSI no BA model (piers are nonlinear)
Linear continuum soil domain
Not present Rock outcrop motion at the base
of soil domain FB_SD no
BA
Base of vertical members have no restraints (piers are
nonlinear)
Soil stiffness and damping represented
by equivalent nonlinear/linear springs and dashpots
Not present Free field input motion at different
depth of piles
FB_SD with BA
Base of vertical members have no restraints (piers are
nonlinear)
Soil stiffness and damping represented
by equivalent nonlinear/linear springs and dashpots
Equivalent springs represent backfill soil
Free field input motion at different
depth of piles and abutment backwall