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Coupled Consolidation of a Spherical Ball
1
Introduction
This example is about the analysis of a saturated spherical ball. The phenomenon in which the porepressure in a particulate medium can become bigger than the applied surface pressure has been known for
a long time. It was first described by Mendel in 1953 for a triaxial soil sample, and later by Cryer in 1963
for a spherical soil sample. It is known as the Mendel-Cryer effect..
In the 1990’s GEO-SLOPE developed a coupled consolidation formulation, which was incorporated into
SIGMA/W. The highlights of the formulation and related SIGMA/W results were published in the
Canadian Geotechnical Journal by T. Wong, D.G. Fredlund and J. Krahn (A Numerical Study of Coupled
Consolidation in Unsaturated Soils; CGJ, 1998, Vol. 35, No. 6, pp. 926-937).
Numerical tests were performed to demonstrate that the formulation and implementation exhibited the
Mendel-Cryer effect, and to check that the results compared with earlier closed-form analytical solutions
provided by Cryer.
This example repeats the analysis published by Wong, Fredlund and Krahn, and shows that the
SIGMA/W coupled-consolidation formulation gives results that match analytical and earlier published
solutions.
2
Feature highlights
GeoStudio feature highlights include:
3

Using the axisymmetric option to simulate the consolidation of a sphere

Using one analysis to induce the excess pore-pressure due to the surface loading and a second
analysis to dissipate the excess pore-pressures

Performing a coupled analysis

Applying both displacement and hydraulic boundary conditions
Geometry and boundary conditions
Figure 1 shows the problem configuration. In this 2D view it is a semi-circle. It consists of two Regions
where the outer curved surface has a large number of Points numerically specified so that they fall on the
perimeter of a circle.
The specified normal pressure boundary is shown in Figure 1. The hydraulic boundary condition is
specified as a Pressure Head with an action of zero.
The excess pressure is induced using an analysis with one time step with a very short duration of 0.01 sec.
This analysis becomes the “Parent”, or initial conditions analysis, for the excess pore-pressure dissipation
phase.
All the other example details, like material properties, can be examined and viewed in the related
GeoStudio file.
SIGMA/W Example File: Cryers Ball.docx (pdf) (gsz)
Page 1 of 5
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File Name: Cryers Ball.gsz
Coupled axisymmetric
2.0
Applied pressure 100 kPa
1.9
Two stages
- loading stage
- dissipation stage
1.8
1.7
1.6
1.5
1.4
1.3
Elevation - m
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Figure 1 The Cryer Ball problem setup
4
Results
Figure 2 shows the pore-pressure distribution at the center of the sphere. As expected, the pore-pressure
rises immediately after the load is applied, reaches a peak and then begins to dissipate.
The peak computed value is 118.789 kPa, or 1.188 times the applied pressure. Wong et al. indicate that
the analytical hand-calculated peak value is 1.192.
SIGMA/W Example File: Cryers Ball.docx (pdf) (gsz)
Page 2 of 5
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PWP versus time
120
Pore-Water Pressure (kPa)
100
80
60
40
20
0
0
2
4
6
8
10
12
14
Time (sec)
Figure 2 Pore-pressure distribution at center of ball
Wong et al. normalized the pore-pressure relative to the initial applied pressure and computed a Square
Root of Time factor, and presented the plot in Figure 3. This view makes it better to look at the early rise
and fall in pore-pressure.
Figure 3 Time variation of non-dimensionalized pore-press (after Wong et al.)
The data from the graph in Figure 2 was cut and pasted into EXCEL, modified and re-plotted as in Figure
4. Visual inspection and comparison of Figure 3 and Figure 4 indicate that the current version of
SIGMA/W gives the same results as was published earlier.
SIGMA/W Example File: Cryers Ball.docx (pdf) (gsz)
Page 3 of 5
GEO-SLOPE International Ltd, Calgary, Alberta, Canada
www.geo-slope.com
1.200
1.000
U / Uo
0.800
0.600
0.400
0.200
0.000
0.00
0.20
0.40
0.60
0.80
1.00
Square root of time factor, sqrt (T)
Figure 4 Pore-pressure ratio versus the T time factor
Figure 5 shows the pore-pressure distribution along a horizontal line at the center of the ball. It is another
way to view the initial rise in pore-pressure and then the subsequent dissipation.
PWP distribution
120
Pore-Water Pressure (kPa)
100
80
60
40
20
0
0.0
0.2
0.4
0.6
0.8
1.0
X (m)
Figure 5 Pore-pressure distribution with time along a horizontal line at the center
Figure 6 shows a contour plot of the pore-pressure distribution within the ball at 4.84 seconds after
application of the applied load. The contours are nice and smooth except for the 2 kPa contour; it has
some squiggles at the top and bottom of the ball. This is a mesh discretization issue. Refining the mesh in
this area will eliminate the squiggles.
SIGMA/W Example File: Cryers Ball.docx (pdf) (gsz)
Page 4 of 5
GEO-SLOPE International Ltd, Calgary, Alberta, Canada
www.geo-slope.com
File Name: Cryers Ball.gsz
Coupled axisymmetric
2.0
Applied pressure 100 kPa
1.9
1.8
1.7
Two stages
- loading stage
- dissipation stage
1.6
1.5
1.4
1.3
Elevation - m
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
Figure 6 pore-pressure contour at t = 4.84 seconds
5
Conclusion
A SIGMA/W coupled consolidation analysis of a spherical ball exhibits all the correct behavior, and the
results are essentially the same as obtained from analytical solutions. All this confirms that the SIGMA/W
formulation and implementation is correct.
SIGMA/W Example File: Cryers Ball.docx (pdf) (gsz)
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