PART 1: (12 Marks in total)
1) Consider the sandy soil region shown in Figure 1. The water head on the right side is
0.0 m and that on the left side is a constant 10.0 m. The lower and upper edges are
impermeable. The coefficients of permeability of the soil are kxx = kyy = 25×10-5
m/s. The region has been discretised by four two-dimensional linear triangular
elements of equal size with unit thickness. Determine (a) the water head distribution,
(b) the velocity of the water flow and (c) the volumetric flow rate in the elements
using Finite Element Method. (3 Marks)
Figure 1
2) Consider the following second-order differential equation in the domain 0 1 x .
Derive the approximate solution using the methods of moments, Galerkin,
School of Civil and Environmental
Engineering
Semester 2, 2017
2
collocation, sub-domains and least squares. Use the approximate function
2 3
01 2 3 u a ax ax ax . (4 Marks)
2
2
2 0, ( 0) 0, ( 1) 1 d u du
u x ux x
dx dx
3) Derive the weak form of the following governing equation for the seepage flow,
2 0 v
w
k
p
where p is the excess pore water pressure generated in the soil due to loading, w is the
unit weight of water, k is the coefficient of permeability of the soil, v is the volumetric
strain rate, and
222
2
222
ppp
x y z
. The boundary conditions are expressed as
on :Undrained boundary
0 on :Drained boundary
u
w
d
k p S
p S
v
where p is the gradient of the excess pore water pressure and v is the known velocity
of the pore water through the boundary surface u S . Use the weight function w p to
construct the weak form of the seepage flow equation. (5 Marks)
3
PART 2: (14 Marks)
The computer program “FEM2D_STRIP_FOOTING” has been developed in MATLAB
for the plane strain analysis of a flexible strip footing, with a breadth of B, resting on a
uniform soil (see Figure 2). The soil has a Young’s modulus of E, Poisson’s ration of ,
and subjected to a vertical load of F. The program uses 3-node linear triangular elements
for the discretisation of the domain. As the displacement boundary conditions, the side
boundaries are supported horizontally, while the bottom surface is constrained in both x
and y directions. Due to the symmetricity of the problem, only half of the domain is
modeled in horizontal direction. The input data for the simulations can be determined
based on your zID as
B = Sum of all digits of zID/10 (m)
E = The last 4 numbers of zID (kPa)
= 0.3
F = The last 2 numbers of zID + 30 (KN)
h = 5 (m).
The horizontal boundary should be selected at least 3B away from the middle of the
footing. You are required to consider an appropriate finite element mesh for the analysis.
1) Investigate the convergence speed of the discretisation method. (4 Marks)
Note: This is related to how small the elements need to be in order to ensure that the
numerical results are not affected by changing the size of the mesh. You can plot the
vertical displacement of the node at x=0 and y=h versus the number of the elements
considered in the analysis.
2) Extend the program for the analysis of the strip footing resting on a multi-layered
soil. Conduct the simulations for the following two cases. Plot the vertical nodal
displacements versus depth at x=0, i.e. the settlement along the centre line. Plot the
distributions of the vertical stress and strain in the medium. (5 Marks)
Case 1: The strip footing resting on a two-layered soil (top layer: h1=2.5m and E1=
E, bottom layer: h2=2.5m and E2= 2E)
Case 2: The strip footing resting on a three-layered soil (top layer: h1=2m and E1=
E, mid layer: h2=1m and E2= 1.5E, bottom layer: h3=2m and E3= 2.5E)
3) Extend the program for the analysis of a rigid footing subjected to vertical loading.
Plot the vertical nodal displacements versus depth at x=0, i.e. the settlement along the
centre line x=0. Compare the results with the ones of the flexible footing under the
same load. (5 Marks)
4
4) Extend the program for the nonlinear analysis using
0.5
0
p Ep E
p
where ( )/3 xyz p is the mean stress, ( , , ) i i xyz are the normal
stresses, and 0 p is the initial value of p . Consider 0 ( ) d p h y where
3 d 16KN/m is the unit weight of the soil. Plot the vertical nodal displacements
versus depth at x=0, i.e. the settlement along the centre line. Plot the distributions of
the mean stress and the volumetric strain v xyz in the medium. (5 Bonus
Marks)