1. No category

13 Compression of a Poroelastic Sample (Mandel’s Problem)
13.1 Problem Description
This verification considers an infinitely long rectangular block with two rigid frictionless plates
at the top and bottom. A distributed load of 10 kN/m2 is applied on the model. The geometry of
the model is shown in Figure 13-1.
2F = 10 kN/m2
z
2b = 10 m
x
2a = 20
m
Figure 13-1: Geometry of the model
13.2 Model Properties
A liner is used at the face under normal load to resemble rigid plates. The initial state for the
stresses and pore water pressure are all zero. The mesh and boundary conditions are illustrated in
Figure 13-2. Input parameters are listed in Table 13-1.
Zero Pore Water Pressure
(Drainage from the Sides)
Zero Pore Water Pressure
(Drainage from the Sides)
Z Restraint
(Front and Back)
YZ Restraint
(Bottom)
Figure 13-2 : Mesh and boundary conditions
Table 13-1 : Table of input parameters
Parameter
Young's Modulus (E)
Poisson's Ratio (v)
Permeability (k)
Fluid Bulk Modulus
Value
20000 kPa
0.2
0.001 m/s
2200000 kPa
13.3 Results
Figure 13-3 shows the relation between normalized pressure that is evaluated as
versus
the normalized horizontal distance evaluated as
where a is half the width of the model, p
is the total pore water pressure, F is the normal distributed load divided by two and x is the
horizontal distance from the origin at the center of the model.
Figure 13-4 shows the relation between normalized effective vertical stress evaluated as
versus normalized horizontal distance where
is the effective vertical stress.
Figure 13-5 shows the relation between normalized horizontal displacement evaluated as
versus normalized horizontal distance.
Figure 13-6 shows the relation between normalized vertical displacement evaluated as
versus normalized time evaluated as
. d is the generalized consolidation coefficient
defined in Equation 1.
B=1, vu = 0.5
k is the coefficient of permeability, G is the shear modulus and v is the Poisson’s ratio.
(1)
0.6
t = 0.00
Normalized Pressure
0.5
0.4
t = 0.01
t = 0.10
t = 0.50
0.3
Ref. [1]
RS3
t = 1.00
0.2
0.1
t = 2.00
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Horizontal Distance
Figure 13-3 : Normalized pressure versus normalized horizontal distance
1.1
t = 2.00
Normalized Effective Vertical Stress
1
t = 0.10
0.9
t = 0.50
0.8
Ref. [1]
0.7
RS3
t = 1.00
0.6
t = 0.01
0.5
t = 0.00
0.4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Normalized Horizontal Distance
Figure 13-4 : Normalized effective vertical stress versus normalized horizontal distance
1
0.00035
t = 0.00
t = 0.01
0.00025
t = 0.10
0.0002
t = 0.50
Ref. [1]
0.00015
RS3
t = 1.00
0.0001
t = 2.00
0.00005
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Normalized Horizontal Distance
Figure 13-5 : Normalized horizontal displacement versus normalized horizontal distance
-0.0005
Normalized Vertical Displacement
Normalized Horizontal Displacement
0.0003
-0.0006
-0.0007
Ref. [1]
RS3
-0.0008
-0.0009
-0.001
0
0.4
0.8
1.2
1.6
Normalized Time
Figure 13-6 : Normalized vertical displacement versus normalized time
2
13.4 References
1. Abousleiman, Y., Cheng, A. H.D, Cui, L., Detournay, E., & Roegiers, J.C. (1996). Mandel's
problem revisited. Geotechnique, 46(2), 187-195.
13.5 Files
The input data file Verification 013 (Mandel's Problem).rs3 can be found in the RS3
installation folder.