Examples • Example Applications
Wharf Subjected to Earthquake Loading (FLAC3D)
Problem Statement
Note
The project file for this example is available to be viewed/run in FLAC3D. The project’s main data files are shown at the end of this example.
This is an illustrative example of a wharf subjected to earthquake loading. The soil profile with three layers (4.5, 6.5 and 11.0 m) is shown in Figure 1. The water table is 2 m below the slope ground. The wharf and its piles are shown in Figure 2 in threedimension. The slope contains sand that may spread laterally during an earthquake. This lateral spreading could push the piles and potentially compromise the safety of the wharf. However, the piles also have a spinning effect that can reduce the lateral spreading of the slope. This is an example of soilstructure interaction.
The liquefiable part of the middle layer is modeled using the P2PSand model (colored by green in Figure 2 and Figure 3) and other soils are modeled using the MohrCoulomb model plus hysteretic damping.
Several analysis stages are required:
setup initial stress and porepressure;
install structures;
assign P2PSand model; and
dynamic analysis.
Setup initial stress and porepressure
This model requires configures of fluid and dynamic (model configure
fluid dynamic
. At this stage, both fluid and dynamic are off. This model follows the standard boundary conditions: fixed degrees of freedom at the bottom and roller boundaries on the side surfaces. Take advantage of the command zone face skin
, the boundary conditions can be
zone face apply velocityx 0 range union group 'west' group 'east'
zone face apply velocityy 0 range union group 'north' group 'south'
zone face apply velocity (0,0,0) range group 'bottom'
At this stage, all soils are assumed MohrCoulomb. The soil matrix material properties are list in Table 1.
Soil 
Density 
Bulk 
Shear 
Friction 
Cohesion 
Mg/m^{3} 
kPa 
kPa 
deg. 
kPa 

#1 
1.7 
5.10e5 
2.35e5 
40 
4 
#2 
1.5 
1.36e5 
6.30e4 
35 
2 
#3 
1.4 
1.36e5 
6.30e4 
30 
2 
The fluid related material properties are water density \(\rho_f\) = 1e3 \(kg/m^3\), porosity = 0.3. Gridpoint above the water table is set to zero saturation while others are 1 by default. Note that the input density should be dry density since the fluid is configured.
The pore pressures are initialized statically using a gradient. The static water pressure should be applied to the top surfaces below the water table. These can be achieved with two simple commands:
zone gridpoint initialize porepressure 200 ...
gradient 0 0 10 range positionz 0 20
zone face apply stressnormal 200 gradient 0 0 10 ...
range group 'top' positionz 10.99 20
The model is cycled to equilibrium by the command:
model solve elastic convergence 1
Install Structures
This stage sets up the wharf structures.
Before installation of structures, the model should be reinitialized to zero displacements and velocities and be cleared of all yield states because any previously calculated displacement/velocity and yield states are solely provide an initial field of stress and porepressure for subsequent stages. These operations are performed with the commands:
zone gridpoint initialize velocity (0,0,0)
zone gridpoint initialize displacement (0,0,0)
zone initialize state 0
The platform is modeled by shell elements and the foundations are modeled by pile elements. The thickness of the shell is 0.3 m. The diameter of the piles is 0.6 meter. The structure geometry and properties are summarized in Table 2. The friction between the pile and the soil is assumed to be 30 degrees.
Structure 
Density 
Young 
Poisson 
Thickness 
Diameter 
Mg/m^{3} 
kPa 
m 
m 

Shell 
2 
2e8 
0.2 
0.3 

Pile 
2 
2e7 
0.2 
0.6 
To account for the endbearing effect of the piles, an additional set of links are created at the pile ends. This is necessary because the default pile links are frictional. If a node requires multiple links, the links can be created on a different ‘side’. In this model, the ‘side=2’ designation is used for endbearing links (grouped under the name ‘End’) between the piles and the soil. The endbearing links are assumed to be of the normyield type.
Note that the link displacement is defined as the target displacement minus the node displacement. The (downward) driving is regarded as the ‘tension’ of the link, while the (upward) pulling is considered the ‘compression’ of the link. Therefore, the yieldcompression of the link is set to zero, while the yieldtension is assigned a large value to ensure consistency with the effect of endbearing. The commands associated with the endbearing links are:
structure link create side=2 target zone group 'End' range positionz 5.9 6.1
structure link attach x=normalyield y=free z=free range group 'End'
structure link attach rotationx=free rotationy=free rotationz=free range group 'End'
structure link property x area=[math.pi*d^4/64] stiffness=1e6 ...
yieldcompression=0 yieldtension=1e20 range group 'End'
To make sure the pile heads are rigidly linked to the shell, the following command is used
structure node join range group 'wharf'
Again, the model is cycled to equilibrium for the installation of the structures.
Assign P2PSand model
The P2PSand model is applied to certain zones within the middle soil layer (colored by green in Figure 2 and Figure 3). To simplify the analysis and demonstration, the internally calibrated material parameters for the P2PSand model are utilized. The primary parameter, relativedensityinitial, is set at 0.50 and 0.75 to examine the impact of sand densities on the dynamic behavior of the slope and wharf. The reference pressure is assigned as pressurereference = 100, as the model’s unit for stress/pressure is in kPa.
Because the P2PSand model is a stressdependent model, the initial stress will be utilized to determine the initial moduli. A small FISH function is adopted here:
fish operator ini_stress(z, modelname)
if zone.model(z) == modelname then
local pp = zone.pp(z)
zone.prop(z,'stressxxinitial') = zone.stress.xx(z) + pp
zone.prop(z,'stressyyinitial') = zone.stress.yy(z) + pp
zone.prop(z,'stresszzinitial') = zone.stress.zz(z) + pp
zone.prop(z,'stressxyinitial') = zone.stress.xy(z)
zone.prop(z,'stressyzinitial') = zone.stress.yz(z)
zone.prop(z,'stressxzinitial') = zone.stress.xz(z)
endif
end
[ini_stress(::zone.list, 'p2psand')]
Note that here the keyword operator
is used in place of the usual define
, declaring it a function to be used with multithreading. The argument ‘z’ is a zone pointer and ‘modelname’ is a constitutive model name. The command [ini_stress(::zone.list, ‘p2psand’)] runs the FISH function “ini_stress” in a multithreaded mode.
After assigning the P2PSand model, the model requires another reinitialization of displacement/velocity and yield state to zero. This time, structure nodes require similar reinitialization as well:
structure node initialize displacement (0,0,0)
structure node initialize displacementrotational (0,0,0)
To this point, all necessary constitutive models and structures with allowable stress, porepressure, and structural forces forces are set up. The model is ready for dynamic analysis.
Dynamic Analysis
The dynamic configure needs to be turned on for dynamic analysis. For quicker calculation, the ‘multistep’ is turned on as well. Because the earthquake can be considered a quick loading, the response of the soil is approximately considered undrained process. The fluid modulus is assigned a nonzero value 2e5 Pa so that the deformation from the soil matrix can cause the change of porepressure. Note that a smaller water bulk modulus 2e5 Pa rather than the real water bulk modulus 2e6 Pa is assigned to compromise the effect the free surface of water table (water is not completely confined). The commands are:
model dynamic active on
zone dynamic multistep on
zone gridpoint initialize fluidmodulus 2.0e5
The base of the model is considered compliant, so the input motion at the base should be the upward motion, or half of the outcrop motion. The input outcrop motion (labeled Nahanni) before scaling is shown in Figure 4:
In practice, the adopted input motion is usually be scaled to abide by some design codes or guidelines. In this example, a scaling factor of 5 is used so that the maximum acceleration is around 0.75 g.
At the base, quiet boundaries are used to accommodate the compliant base. The input acceleration motion is integrated into a velocity motion and then into a shear stress history. Shear stress can be calculated from velocity by:
where \(\rho\) is density at the base, \(G\) is the shear modulus at the base, and \(v_s\) is the shear wave velocity, and \(C_s = \sqrt{G / \rho}\) is the speed of swave propagation at the base.
The commands are
zone gridpoint free velocity
table 'acc' import "Nahanni.acc"
[table.as.list('vel') = table.integrate('acc')]
; since g=10, scaling factor=5
[global mf = 1.8*math.sqrt(2.35e5/1.8)*10*5.0]
zone face apply stressdip [mf] table 'vel' range group 'bottom'
zone face apply stressstrike 0.0 range group 'bottom'
zone face apply stressnormal 0.0 range group 'bottom'
zone face apply quiet range group 'bottom'
For zones that use the MohrCoulomb model, hysteretic damping is incorporated to account for the \(G/G_{max}\) effect. In contrast, zones that use the P2PSand model have an intrinsic energy dissipation in their constitutive relation, so additional energy dissipation is not strictly necessary. However, a small amount of Rayleigh damping is added to zones in order to reduce highfrequency noise. Since the Rayleigh damping is very small, the dynamic timestep is rarely affected.
For structures, the situation is different as the inclusion of Rayleigh damping can significantly reduce the timestep. Therefore, this model uses Maxwell damping with a target damping ratio of 5% for structures (see Maxwell Damping):
zone dynamic damping hysteretic hardin 0.2 range cmodel 'mohrcoulomb'
zone dynamic damping rayleigh 0.0015 1.0
structure dynamic damping maxwell 0.0385 0.5 0.0335 3.5 0.052 25.0
Freefield boundary is then applied following the basic the guideline for dynamic analysis:
zone dynamic freefield on
The results for sand \(D_r\) = 0.50 (50%) are plotted in Figure 5 to Figure 7. Figure 5 plots porepressure histories at selected points (35,25,13), (46,25,14) and (58,25,15). Apparent excess porepressure buildsup are observed. Figure Figure #wharfssdr50
plots the contour of the maximum shear strain. Figure 7 plots the Xdisplacement histories of platform, slope top and the base. Interestingly, the lateral displacement of the platform is less than that of the slope, which is a common effect of soilstructure interaction. Specifically, the lateral spreading of the soil pushes the pile foundation, while the pile foundation reinforces the stability of the slope to some extent.
Similar results for sand \(D_r\) = 0.75 (75%) are plotted in Figure 8 to Figure 10. Excess porepressure buildsup are observed in Figure 8. Lateral displacements of platform and slope shown in Figure 10 are less than those in Figure 7 which is apparently due to the sand in Figure 10 being denser.
break
Data Files
wharfini.dat
model new
model configure fluid dynamic
model largestrain off
model fluid active off
model dynamic active off
model gravity 0 0 10
;
zone import 'grid.flac3d'
zone face skin
;
zone face apply velocityx 0 range union group 'west' group 'east'
zone face apply velocityy 0 range union group 'north' group 'south'
zone face apply velocity (0,0,0) range group 'bottom'
;
zone cmodel assign mohrcoulomb
zone property density=1.8 bulk=5.10e5 shear=2.35e5 ...
cohesion=4 friction=40.0 range group 'wharf:soil 1'
zone property density=1.6 bulk=1.36e5 shear=6.30e4 ...
cohesion=2 friction=35.0 range group 'wharf:soil 2'
zone property density=1.5 bulk=1.36e5 shear=6.30e4 ...
cohesion=2 friction=30.0 range group 'wharf:soil 3'
;
zone fluid cmodel assign isotropic
zone initialize fluiddensity 1.0
zone fluid property porosity 0.3
zone gridpoint initialize saturation 0 range positionz 20 100
zone gridpoint initialize porepressure 200 ...
gradient 0 0 10 range positionz 0 20
zone face apply stressnormal 200 gradient 0 0 10 ...
range group 'top' positionz 10.99 20
;
model solve elastic convergence 1
;
zone gridpoint initialize velocity (0,0,0)
zone gridpoint initialize displacement (0,0,0)
zone initialize state 0
;
structure pile create byline 42 20.5 6 42 20.5 22.1 segments 8 id 1
structure pile create byline 45 20.5 6 45 20.5 22.1 segments 8 id 2
structure pile create byline 42 23.5 6 42 23.5 22.1 segments 8 id 3
structure pile create byline 45 23.5 6 45 23.5 22.1 segments 8 id 4
structure pile create byline 42 26.5 6 42 26.5 22.1 segments 8 id 5
structure pile create byline 45 26.5 6 45 26.5 22.1 segments 8 id 6
structure pile create byline 42 29.5 6 42 29.5 22.1 segments 8 id 7
structure pile create byline 45 29.5 6 45 29.5 22.1 segments 8 id 8
;
[global d = 0.6]
structure pile property density 2 young 2e7 poisson 0.2 ...
crosssectionalarea [math.pi*d^2/4] ...
torsionconstant [math.pi*d^4/32] ...
moiy [math.pi*d^4/64] moiz [math.pi*d^4/64] ...
couplingstiffnessshear 1e5 couplingcohesionshear 0 ...
couplingfrictionshear 30 couplingstiffnessnormal 1e5 ...
couplingcohesionnormal 0 couplingfrictionnormal 30 ...
perimeter [math.pi*d] couplinggapnormal off
; Include endbearing effect
structure link create side=2 target zone group 'End' range positionz 5.9 6.1
structure link attach x=normalyield y=free z=free range group 'End'
structure link attach rotationx=free rotationy=free rotationz=free range group 'End'
structure link property x area=[math.pi*d^4/64] stiffness=1e6 ...
yieldcompression=0 yieldtension=1e20 range group 'End'
;
zone create brick size 3 5 1 point 0 39 17.5 21.1 point 1 48 17.5 21.1 ...
point 2 39 32.5 21.1 point 3 39 17.5 22.1 group 'wharfdeck'
structure shell create byzoneface id 9 ...
range positionx 39 48 positiony 17.5 32.5 positionz 22.0 22.2
structure shell group 'wharf'
zone delete range group 'wharfdeck'
;
structure shell cmodel assign elastic
structure shell property density 2 young 2e8 poisson 0.2 ...
thickness 0.3 range id 9
structure node join range group 'wharf'
;
model solve convergence 1
model save 'static'
wharfp2pdr75.dat
model restore 'static'
;
zone cmodel assign p2psand range group 'wharf:soil 2' ...
positiony 5 45 positionx 0 70
zone property relativedensityinitial 0.75 pressurereference 100 ...
frictioncritical 35.0 range cmodel 'p2psand'
;
fish operator ini_stress(z, modelname)
if zone.model(z) == modelname then
local pp = zone.pp(z)
zone.prop(z,'stressxxinitial') = zone.stress.xx(z) + pp
zone.prop(z,'stressyyinitial') = zone.stress.yy(z) + pp
zone.prop(z,'stresszzinitial') = zone.stress.zz(z) + pp
zone.prop(z,'stressxyinitial') = zone.stress.xy(z)
zone.prop(z,'stressyzinitial') = zone.stress.yz(z)
zone.prop(z,'stressxzinitial') = zone.stress.xz(z)
endif
end
[ini_stress(::zone.list, 'p2psand')]
;
model solve convergence 1
;
zone gridpoint initialize displacement (0,0,0)
zone gridpoint initialize velocity (0,0,0)
zone initialize state 0
structure node initialize displacement (0,0,0)
structure node initialize displacementrotational (0,0,0)
;
model dynamic active on
zone dynamic multistep on
zone gridpoint initialize fluidmodulus 2.0e5
;
zone gridpoint free velocity
table 'acc' import "Nahanni.acc"
[table.as.list('vel') = table.integrate('acc')]
; since g=10, scaling factor=5
[global mf = 1.8*math.sqrt(2.35e5/1.8)*10*5.0]
zone face apply stressdip [mf] table 'vel' range group 'bottom'
zone face apply stressstrike 0.0 range group 'bottom'
zone face apply stressnormal 0.0 range group 'bottom'
zone face apply quiet range group 'bottom'
;
zone dynamic damping hysteretic hardin 0.2 range cmodel 'mohrcoulomb'
zone dynamic damping rayleigh 0.0015 1.0
; target 5% for structure elements (& transferred to deformabale links)
structure dynamic damping maxwell 0.0385 0.5 0.0335 3.5 0.052 25.0
;
zone dynamic freefield on
;
history delete
history interval 10
model history name 'time' dynamic timetotal
program call 'excpphist.f3fis'
zone history name 'pp1' porepressure source gridpoint position 35 25 13
zone history name 'pp2' porepressure source gridpoint position 46 25 14
zone history name 'pp3' porepressure source gridpoint position 58 25 15
structure node history name 'dispx1' displacementx componentid 86
zone history name 'dispx2' displacementx position 50 24 0
zone history name 'dispx3' displacementx position 48 24 22
;
model dynamic timestep maximum 1.00e4
model solve timetotal 21
;
history export 'dispx1' vs 'time' table 'disp1'
table 'disp1' export 'disp1dr75' truncate
;
model save 'wharf_dr75'
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