6.4 Soil-Structure Interaction under Seismic Loading
6.4.3 Application to Liquefaction Modelling: Structure Founded on Stratified Soilon Stratified Soil
In the following, a two-dimensional computational model of a soil-structure system is anal-ysed using the FE package PANDAS. The seismic excitation appears in form of vertically incident shear waves, which is applied at the bottom of the profile, i. e. along the soil-bedrock interface boundary. The geometry and the boundary conditions are illustrated in Figure 6.39, where the abbreviations in Table 6.1 are still valid for this example.
0000000000000000000000000000 0000000000000000000000000000 1111111111111111111111111111 1111111111111111111111111111 000000 111111
00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000
11111111 11111111 11111111 11111111 11111111 11111111 11111111 11111111
x1
x2
30 m 25 m
20m
10m10m
p= 0
uS1= 0 vS1= 0 vF1= 0 uS2= 0 vS2= 0 vF2= 0 clayey silt
1 loose sand
2 moderate-dense sand
seismic excitation stiff clay
bedrock
single-mass structure structural load
top layer weight A2(2, -4)
A1(-2, -4)
B(2, 2)
Figure 6.39: Geometry and boundary conditions of the soil-structure interaction problem under seismic loading
In analogy to the liquefaction-prone Wildlife Refuge site strata in Figure 5.4, the chosen
soil layers of the considered problem provide an appropriate environment to liquefaction events. Here, the present layers are illustrated in Figure 6.39 and can briefly be demon-strated as follows: (1) A clayey silt surface layer, which is replaced in the numerical model by a uniformly distributed load (50.0 kN/m2). This layer provides permeable up-per boundaries during the consolidation and the excess pore-pressure dissipation phases.
(2) A liquefiable sand layer (10 m thick), on which the structure is installed. The ex-pected behaviour of this layer is nonlinear elastoplastic. Therefore, it is modelled using the elasto-viscoplastic constitutive model with isotropic hardening as introduced in Sec-tion 3.2. Here, in order to capture the two significant liquefacSec-tion events of flow liquefac-tion and cyclic mobility, the parameters in Table 5.1 for very loose and moderate-dense sands are exploited. (3) A stiff clay layer (10 m thick) under the sand layer, which is characterised by a low permeability parameter and expected to behave linear elastic. (4) A bedrock that marks the bottom boundary of the considered IBVP, at which the seismic load is applied.
In the current treatment, the modelling of soil is confined to the case of a saturated, materially incompressible solid-fluid aggregate. Moreover, the domain of the boundary-value problem is chosen sufficiently wide in order to avoid the influence of the lateral boundaries on the response of the region of interest under the structure. Herein, the applied loads during the whole numerical treatment are illustrated in Figure 6.40, left.
00000000000000000000 00000000000000000000 11111111111111111111 11111111111111111111
0 1000 5016 5500
50 150
3000 5000
time [s]
load[kN/m2 ]
top layer weight structure weight
seismic load post-liquefaction consolidation
x1
x2
rigid structure
soil
F=600 kN A2
A1
Figure 6.40: Applied loads to the soil-structure interaction problem (left) and illustration of the rigid structure at the top of the soil domain (right)
The structure is assumed to be made of a very stiff material, e. g. concrete with material parameters given in Table 6.9, which can be approximated by a single-mass oscillator with very high stiffness, cf. Figure 6.40, right. Here, depending on the nature of the foundation soil, the structure might undergo settlement, uplifting, or overturning as will be discussed throughout the example. The parameters of the elasto-viscoplastic sand layer are given in Tables B.1 and B.2 with kF = 10−5 m/s, whereas the parameters of the elastic, stiff clay layer are presented in Table 6.2 with kF = 10−8m/s.
In the following investigation, the aim is to reveal the flow liquefaction and the cyclic mobility in saturated soils under extreme dynamic loadings and not to model a particular seismic event. Therefore, the given earthquake excitation is multiplied by amplification factors depending on the initial density of the soil. In this connection, two cases of the ini-tial density and amplification factors are considered for the sand layer under the structure:
Firstly, the case of flow-liquefaction-prone very loose sand, where the seismic excitation
6.4 Soil-Structure Interaction under Seismic Loading 141
is multiplied by a factor of 15. Secondly, the case of moderate-dense sand, where the earthquake data is magnified by a factor of 20 to manifest the cyclic mobility behaviour.
In the IBVP, the earthquake velocity (Figure 6.37, right) is used as an input data, which is compatible with the structure of the governing balance relations. In this connection, the treatment proceeds from the set of equations uvp (2) with the primary variables uS, vS, vF, andp, cf. DAE (4.42). Moreover, an implicit monolithic time-stepping algorithm using the Backward Euler (BE) scheme is exploited to solve the problem, cf. Box (4.50) . The choice of the implicit BE is mainly due to the fact that the BE is easy to implement and needs less calculation time in comparison with the TR or the TR-BDF2 schemes.
However, as has been shown in Section 6.2, the BE method suffers from an artificial nu-merical damping, which can be reduced by choosing smaller time steps. This damping should be taken into account if one compares the numerical solution with reference or benchmark solutions, which is not the case in this treatment.
Case (1): Very Loose Sand Layer −→ Flow Liquefaction
Starting with the case of a very loose sand layer according to the definitions and pa-rameters in Section 5.2, Figure 6.41 shows exemplary contour plots of the solid plastic volumetric strain evolution εVSp.
sand sand
clay clay
t= 5006.70 s t= 5006.84 s
sand sand
clay clay
t= 5006.97 s t= 5007.08 s
0.14
0.07
0.02
|εVSp|
Figure 6.41: Time sequence of solid plastic volumetric strain contour plots for the case of very loose saturated sand layer under the structure (scale factor 10)
This type of soil collapse is known as seismic-induced flow liquefaction, which leads to a punching shear failure in the loose foundation soil12.
12Bearing capacity failure happens when the shear stresses in the soil exceed its shear strength. Herein, depending on the foundation soil properties, three modes of bearing capacity failure can be recognised, cf. Day [41]: (1) Punching shear failurewhich occurs for loose foundation soils. In this case, no general shear surface is generated, and the main deformations happen in the soil directly below the structure’s
Unlike the perfectly undrained CU triaxial tests, the behaviour in the considered IBVP is partially undrained with possible excess pore-pressure dissipation during and after the dynamic loading. Herein, the excess pore pressure firstly accumulates in certain sand zones with high plastic volumetric strain and then migrates due to the pore-pressure gradient into neighbouring zones of lower accumulated pore pressure. In this regard, Figure 6.42 shows exemplary time sequence contour plots of the excess pore-water pressure p with deformed mesh (scale factor 10) of the considered 2-d problem.
sand sand
clay clay
t= 5006.70 s t= 5007.08 s
0.08
0.01
−0.04 p [ MN/m2]
Figure 6.42: Time sequence of pore pressure contour plots for the case of very loose soil layer under the foundation (scale ×10)
Following this, one distinguishes between the oscillatory pore pressure that appears in the elastic clay layer and the accumulative pore pressure in the sand layer under the foundation. In the latter, it is clear that the development of the plastic volumetric strain in certain zones coincides with the pore-pressure build-up. Figure 6.43, left, depicts the interplay between the mean effective stress and the pore pressure (cf. (5.5)) at pointB(2, 2) in the soil domain.
-0.04 0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 0.32 0.36
5000 5002 5004 5006 5007
timet[s ]
p,trT
S E
[MN/m2 ]
3p
−trTS
E
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2
5000 5002 5004 5006 5007
timet[s]
εV Sp[10−3 –]
εVSp
Figure 6.43: Pore pressure and mean effective stress time history atB(2, 2)(left), and solid plastic volumetric strain time history at point B (right) during the seismic load, case (1) footing. (2) Local shear failure which is normally seen in soils of medium dense nature. This type of failure is an intermediate state between punching and general shear collapse, where a partial shear surface can be distinguished immediately below the footing. (3)General shear failureusually happens in soils of dense or hard state and involves total rupture of the soil with a continuous and distinct shear surface.
6.4 Soil-Structure Interaction under Seismic Loading 143
Figure 6.43, left, shows that the flow liquefaction takes place due to the reduction of the mean effective stress and the build-up of the pore pressure until most of the applied stress is carried by the pore fluid (trTSE tends to become zero). Moreover, the fast increase of the pore pressure at a certain stage of the loading is accompanied by a drastic increase of the plastic volumetric stain. In this case, the continuous increase of the pore-water pressure under deviatoric stress conditions is associated with a contraction tendency of the loose sand layer (εVSp<0), cf. Figure 6.43, right.
During the seismic excitation, Figure 6.44 illustrates how the structure undergoes vertical as well as horizontal deformations. Moreover, shortly before the collapse, a rapid increase of the horizontal and vertical deformations as well as a small inclination of the structure is demonstrated.
-0.24 -0.2 -0.16 -0.12 -0.08 -0.04
5000 5002 5004 5006 5007
-0.24 -0.12
5006.9 5007 5007.1
timet[s ] displ.uS2[m]
pointA1
pointA2
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
5000 5002 5004 5006 5007
timet[s ]
pointA1
displ.uS1[10−2 m]
Figure 6.44: Vertical displacement time history at pointsA1 (-2, -4)andA2 (2, -4) (left), and horizontal displacement time history at point A1 (right) during the seismic load, case (1)
It is worth mentioning that in the case of a very loose foundation sand layer with mag-nified seismic excitation by a factor of 15, the collapse takes place during the earthquake excitation, and thus, the calculation terminates before the post-liquefaction (dissipation) phase, cf. Figure 6.40, left.
Case (2): Moderate-Dense Sand −→ Cyclic Mobility
For a moderate-dense sand layer underlying the structure (see the classifications in Sec-tion 5.2 and the parameters in Table 5.1), Figure 6.45 shows exemplary contour plots of the solid plastic volumetric strain evolution εVSp at different times during the earthquake loading.
The behaviour in Figure 6.45 represents the seismic-induced cyclic mobility, where a limited accumulation of the pore pressure takes place and the effective stress can never reach a zero value. In this connection, Figure 6.46, left, depicts the interplay between the effective stress and the pore pressure at pointB(2, 2)during the application of the seismic excitation. Here, a slight build-up in the excess pore pressure can be seen till t≈5004 s, which is followed by a decrease in the pore pressure and an increase in the mean effective
sand sand
clay clay
t= 5007.52 s t= 5008.48 s
sand sand
clay clay
t= 5009.63 s t= 5011.78 s
0.009
0.006
0.003
|εVSp|
Figure 6.45: Time sequence of solid plastic volumetric strain εVSp contour plots for the case of moderate-dense saturated sand layer under the structure (scale×30)
stress. Figure 6.46, right, shows that an immense increase in the solid plastic volumetric strain (εVSp >0 → dilative) occurs when the seismic excitation reaches its peak value at t ≈ 5006.6 s leading to clear features of the plastic shear band under the structure (cf.
Figure 6.45).
-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25
5000 5004 5008 5012 5016 5020
timet[s ]
p,trT
S E
[MN/m2 ]
3p
−trTS
E
post-liquefaction phase
0.05 0.1 0.15 0.2 0.25 0.3 0.35
5000 5004 5008 5012 5016 5020
timet[s]
εV Sp[10−2 –]
εVSp
post-liquefaction phase
Figure 6.46: Pore-pressure and mean effective stress time history at point B(2, 2) (left), and solid plastic volumetric strain time history at B (right) during the seismic excitation
and dissipation for case (2) of moderate-dense sand
Figure 6.47, left, depicts the vertical displacement time history at points A1(-2, -4) and A2(2, -4) at the top of the structure. It is clear that the vertical settlement in the case of moderate-dense sand is less than that in the very loose sand layer case. Moreover, at the end of the earthquake loading, a small inclination of the structure can be observed.
Figure 6.47, right, shows the horizontal motion of the structure during the seismic excita-tion. Here, a residual horizontal displacement can be detected at the end of the dynamic loading.
6.4 Soil-Structure Interaction under Seismic Loading 145
-6 -5.5 -5 -4.5 -4 -3.5 -3 -2.5
5000 5004 5008 5012 5016 5020
-6 -5.6
5014 5016
timet[s ] displ.uS2[10−2 m]
pointA1
pointA2
post-liquefaction phase
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
5000 5004 5008 5012 5016 5020
timet[s ]
pointA1
displ.uS1[10−3 m] post-liquefaction phase
Figure 6.47: Vertical displacement time history at pointsA1 (-2, -4)andA2 (2, -4) (left), and horizontal displacement time history at point A1 (right) during the seismic excitation and
dissipation for case (2) of moderate-dense sand
Remark: In general, the flexibility of the foundation in the SSI system reduces the peak deformations induced by the ground motion. However, under certain circumstances, the natural frequency of the SSI system and that of the excitation are found in a state that leads to a large response (i. e. resonance phenomenon). In such cases, the inertial force due to structural oscillations may lead to large inelastic deformations in both the structure and the foundation soil.