Examples • Example Applications
Wharf Subjected to Earthquake Load
Problem Statement
Note
View this project in FLAC2D:
Pick “ExampleApplications”
Pick “Wharf”
Select “wharf.prj” to load
The project’s data files is shown at the end of this example.
This is an illustrative example of a wharf subjected to earthquake load. This model can be run in demonstration mode without a license. 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 that slope ground. There is sand in the slope which may have later spreading during an earthquake. The slope spreading will push the piles and may affect the safety of the wharf. On the contrary, the piles have effect of spinning which will reduce the slope lateral spreading. This is an example of soilstructure interaction.
The liquefiable part of the middle layer will be modeled by the P2PSand model and other soils will be modeled by the MohrCoulomb model plus hysteretic damping.
Several analysis stages are required:
Build model
Setup initial stress and porepressure;
Install structures;
Assign P2PSand model;
Dynamic analysis.
Build model
Sketch is used to build the model geometry as described in the following steps:
Create a new Sketch Set
Open the Slope Wizard and enter the parameters as shown in Figure 2
Draw a horizontal line from the toe of the slope to the right side. Draw another horizontal line at y=17.5.
To ensure a structured mesh, draw a vertical line from the toe of the slope to the bottom. The slope should appear as in Figure 3.
Note that when drawing the vertical line, it will attempt to snap to the grid and will create a slightly nonvertical line. This is OK, but if you want the line to be perfectly vertical, turn off grid snapping for this operation, or manually change the coordinates after drawing the line.
Set the zone length to 1.25 and mesh all blocks.
Select the block representing the top layer and assign it the group name “soil 3”. Set the middle layer to “soil 2” and the two blocks comprising the bottom layer to “soil 1”.
Generate the zones. The resulting model in the model pane will appear as in Figure 4.
At this stage, all soils are assumed MohrCoulomb. The soil matrix material properties are list in Table 1. Select the entire model and set the constitutive model to MohrCoulomb. Then select each layer and assign the properties as shown in the table. It may help to change the slot of the groups being shown to “Construction”.
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 
Finally, click the icon to “Assign group names to faces automatically”. Check the box to “Ignore existing group names”.
Save the project and the model state. It is also good practice to click on the State Record and save the commands as a data file.
Setup initial stress and porepressure
From this point forward, data files are used to perform the analysis. See data files for details.
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.
The boundary conditions can be assigned using the face group names created from the Model Pane.
zone face apply velocityx 0 range union group 'West' group 'East'
zone face apply velocity (0,0) range group 'Bottom'
The fluid related material properties are water density \(\rho_f\) = 1e3 \(kg/m^3\), porosity = 0.3. Gridpoints above the water table are set to zero saturation while others are 1 by default. Note that the input soil density should be dry density since the fluid is configured.
The pore pressures are initialized by specifying a water table. The static water pressure should be applied to the soil surfaces below the water table to account for the weight of the water. These can be realized by two simple commands:
zone water plane origin 0,20 normal 0,1
zone face apply stressnormal 200 gradient 0 10 ...
range group 'top' positiony 10.99 20
The model is cycled to equilibrium by the command:
model solve elastic convergence 1
Install Structures
This stage is to set up the wharf structures.
Before installation of structures, the model should be reinitialized to zero displacements and velocities and be cleared all yield states because any previously calculated displacement/velocity and yield states are just to obtain an initial field of stress and porepressure for subsequent stages. These can be realized by three simple commands:
zone gridpoint initialize velocity (0,0)
zone gridpoint initialize displacement (0,0)
zone initialize state 0
The platform is modeled by beam elements and the foundations are modeled by pile elements. The thickness of the beam is 0.3 m. The diameter of the piles is 0.6 m. The structure geometry and properties are summarized in Table 2. The friction between the piles and the soil is assumed to of 30 degrees.
Structure 
Density 
Young 
Poisson 
Thickness 
Diameter 
\(Mg/m^3\) 
kPa 
m 
m 

Beam 
2 
2e8 
0.2 
0.3 

Pile 
2 
2e7 
0.2 
0.6 
To include the endbearing effect of the piles, anther set of links are created at the pile ends because by default the pile links are frictional. If a node must have multiple links, the links can be created by a different ‘side’. Here ‘side’ is a generalized concept similar to ‘slot’. In this model ‘side=2’ is used for endbearing links (with a group name ‘End’) between the piles and soil. The endbearing links are assumed normyield type. Note that the link displacement is defined as target displacement minus node displacement, the (downward) driving is considered ‘tension’ of link and the (upward) pulling is considered ‘compression’ of link so the link’s yieldcompression is set to zero and yieldtension is set to a large value to be consistent to the effect of endbearing. The commands on endbearing links are:
struct link create side=2 target zone group 'End' range positiony 5.9 6.1
struct link attach x=normalyield y=free range group 'End'
struct link attach rotx=free roty=free range group 'End'
struct 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 beam, the following command is used
structure node join range group 'wharf'
Note that there must be coincident nodes to be able to join them.
Again, the model is cycled to equilibrium for the installation of the structures.
Assign P2PSand model
P2PSand model is reassigned to some zones in the middle soil layer (colored by green in Figure 6). In this analysis for simplicity and demonstration, we just adopt the internally calibrated material parameters for the P2PSand model. The primary parameter relativedensityinitial is input as 0.50 and 0.75 to analyze the effect of sand densities on the dynamic performance of the slope and the wharf. The reference pressure is assigned as pressurereference = 100 since the model’s unit on stress/pressure is kPa.
Since the P2PSand model is a stressdependent model, the initial stress will be used 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)
endif
end
[ini_stress(::zone.list, 'p2psand')]
Note that herein a keyword ‘operator’ (instead of ‘fish’) is used for multithreading. The argument ‘z’ is a zone pointer and ‘modelname’ is a constitutive model name. The command [ini_stress(::zone.list, ‘p2psand’)] is to run the FISH function “ini_stress” in a multithreaded mode
After assigning the P2PSand model, the model needs another reinitialization of displacement/velocity and yield state to zero. This time, structure nodes need the similar reinitialization as well:
structure node initialize displacement (0,0)
structure node initialize displacementrotational 0
So far, we have set up all necessary constitutive models and structures with allowable stress, porepressure and structure forces. It is ready for dynamic analysis.
Dynamic Analysis
The dynamic configure needs to be turned on for dynamic analysis. Because the earthquake can be considered a quick loading, the response of the soil is approximately considered an undrained process. To achieve this, fluid calculations are left off (fluid does not flow), but a nonzero fluid bulk modulus is assigned so that changes in pore volume will cause pore pressure changes. The fluid modulus is assigned a nonzero value 2e5 kPa so that the deformation from the soil matrix can cause the change of porepressure. Note that a smaller water bulk modulus 2e5 kPa rather than the real water bulk modulus 2e6 kPa is assigned to compromise the effect the free surface of water table (water is not completely confined). There commands are:
model dynamic active 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 (1/2) of the outcrop motion. The input outcrop motion (Nahanni) before scaling is shown in Figure 7
Note that the input acceleration in g, needs to be multiplied by 10 to convert to \(m^3/s\). Also, in practice, the adopted input motion is usually 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 0.75g.
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, \(G\) is the shear modulus, and \(v_s\) is the shear wave velocity.
The commands are
zone gridpoint free velocity
[table.as.list('vel') = table.integrate('acc')]
[global mf = 1.8*math.sqrt(2.35e5/1.8)*10*5.0]
zone face apply stressxy [mf] table 'vel' range group 'bottom'
zone face apply quiet range group 'bottom'
For zones with the MohrCoulomb model, the hysteretic damping is used to include the \(G/G_{max}\) effect (Hysteretic Damping). For zones with the P2PSand model, the constitutive relation has intrinsic energy dissipation so that theoretically extra energy dissipation is unnecessary. However, in order to reduce the high frequency noises, a small amount of Rayleigh damping is added to zones (0.15%). The dynamic time step is hardly decreased because the Rayleigh damping is very small. For structures, things are different because Rayleigh damping will significantly reduce the time step. Instead, Maxwell damping with a target 5% damping ratio is used for structures in this model (see Maxwell Damping):
zone dynamic damp hysteretic hardin 0.2 range cmodel 'mohrcoulomb'
zone dynamic damp rayleigh 0.0015 1.0
structure dynamic damp maxwell 0.0385 0.5 0.0335 3.5 0.052 25.0 ;target 5%
Freefield boundary is then applied following the basic guidelines for dynamic analysis (FreeField Boundaries).
zone dynamic freefield on
Results
The results for sand \(D_r\) = 0.50 (50%) are plotted in Figure 8 to Figure 10. Figure 8 plots porepressure histories at selected points (35,13), (46,14) and (58,15). Apparent excess porepressure buildsup are observed. Figure 9 plots the contour of the maximum shear strain. Figure 10 plots the Xdisplacement histories of platform, slope top and the base. It is interesting to observe that the lateral displacement of the platform is less than the lateral displacement of the slope, which is generally the effect of soilstructure interaction: lateral spreading of the soil pushes the pile foundation and the pile foundation somewhat reinforces the stability of the slope.
Similar results for sand \(D_r\) = 0.75 (75%) are plotted in Figure 11 to Figure 13. Excess porepressure buildsup are observed in Figure 11. Lateral displacements of platform and slope shown in Figure 13 are less than those in Figure 10 which is apparently due to the sand in Figure 13 being denser.
break
Data Files
wharfini.dat
model new
; Commands from Sketch and Model Pane
program call 'geometry.dat'
model configure fluid dynamic
model largestrain off
model fluid active off
model dynamic active off
model gravity 0 10
;
zone face apply velocityx 0 range union group 'West' group 'East'
zone face apply velocity (0,0) range group 'Bottom'
;
; fluid
zone fluid cmodel assign isotropic
zone water density 1.0
zone fluid property porosity 0.3
zone gridpoint initialize saturation 0 range positiony 20 100
; assume water table is 2 m below the surface (y=20)
zone water plane origin 0,20 normal 0,1
;zone gridpoint initialize porepressure 200 ...
; gradient 0 10 range positiony 0 20
; stress applied to the slope to account for the weight of water
zone face apply stressnormal 200 gradient 0 10 ...
range group 'top' positiony 10.99 20
;
model solve elastic convergence 1
model save 'wharfini'
wharfstruct.dat
model restore 'wharfini'
zone gridpoint initialize velocity (0,0)
zone gridpoint initialize displacement (0,0)
zone initialize state 0
;
struct pile create byline 42,6 42,22.1 segments 8 id 1
struct pile create byline 45,6 45,22.1 segments 8 id 2
;
[global d = 0.6]
structure pile property density 2 young 2e7 poisson 0.2 ...
crosssectionalarea [math.pi*d^2/4] ...
moi [math.pi*d^4/64] ...
couplingstiffshear 1e5 couplingfrictionshear 30 ...
couplingstiffnormal 1e5 couplingfrictionnormal 30 ...
perimeter [math.pi*d] couplinggapnormal off
; Include endbearing effect
struct link create side=2 target zone group 'End' range positiony 5.9 6.1
struct link attach x=normalyield y=free range group 'End'
struct link attach rot=free range group 'End'
struct link property x area=[math.pi*d^4/64] stiffness=1e6 ...
yieldcompression=0 yieldtension=1e20 range group 'End'
;
; create deck
structure beam create byline 39,22.1 48,22.1 segments 3 id 3 group 'wharf'
;
; thickness is 0.3 m
structure beam cmodel assign elastic
structure beam property crosssectionalarea 0.3 ...
density 2 young 2e8 poisson 0.2 ...
moi [0.3^4/12.0] range id 3
structure node join range group 'wharf'
;
model solve convergence 1
model save 'static'
wharfp2pdr75.dat
model restore 'static'
;
zone cmodel assign p2psand range group 'soil 2' ...
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)
endif
end
[ini_stress(::zone.list, 'p2psand')]
;
model solve convergence 1
==========================================================
; Dynamic analysis
;
zone gridpoint initialize displacement (0,0)
zone gridpoint initialize velocity (0,0)
zone initialize state 0
structure node initialize displacement (0,0)
structure node initialize displacementrotational 0
model dynamic active on
zone gridpoint initialize fluidmodulus 2.0e5
;
; free boundary
zone gridpoint free velocity
; get earthquake history as acceleration
table 'acc' import "Nahanni.acc"
; integrate to get velocity
[table.as.list('vel') = table.integrate('acc')]
; shear stress = 2*sqrt(G/density)*density
; input motion is half of the outcrop motion
; multiplied by 10 to convert from g to m/s2
; multiplied by a scaling factor of 5
[global mf = 1.8*math.sqrt(2.35e5/1.8)*10*5.0]
zone face apply stressxy [mf] table 'vel' range group 'bottom'
zone face apply quiet range group 'bottom'
;
zone dynamic damp hysteretic hardin 0.2 range cmodel 'mohrcoulomb'
; 0.15% damping. Center frequency 1 Hz.
zone dynamic damp rayleigh 0.0015 1.0
; target 5% for structure elements (& transferred to deformable links)
structure dynamic damp maxwell 0.0385 0.5 0.0335 3.5 0.052 25.0
;
zone dynamic freefield on
;
model history name 'time' dynamic timetotal
program call 'excpphist.fis'
zone history name 'At (35,13)' porepressure source gridpoint position 35 13
zone history name 'At (46,14)' porepressure source gridpoint position 46 14
zone history name 'At (58,15)' porepressure source gridpoint position 58 15
struct node history name 'Platform' displacementx position 39,22.1
zone history name 'Base' displacementx position 50 0
zone history name 'Slope Top' displacementx position 48 22
;
model dynamic timestep maximum 1.00e4
model solve timetotal 21
;
model save 'wharf_dr75'
Was this helpful? ...  Itasca Software © 2022, Itasca  Updated: Mar 09, 2023 