DEFLECTION OF THE REINFORCED ZONE OF MSE WALLS Dr. Huabei Liu, Soufiane Nezili

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DEFLECTION OF THE REINFORCED ZONE OF MSE WALLS

Dr.
Huabei

Liu,
Soufiane

Nezili

Abstract

This paper examines the deflection of the reinforced soil zone under the lateral earth pressure
employing the deep beam theory. In this research, deep beam theory will be used to derive the
analyzing method for the deflection of the reinforced soil zone under the lateral earth pressure
behind the reinforced soil zone. A computer code will be developed based on the theory and
method used to analyze the deflection of real structures, the results of which will be compared to
the measured ones to validate the methodology
.

Introduction

Mechanically Stabilized Earth (MSE) walls are used extensively for various applications. This type of
earth structure is relatively new but it is aesthetic in appearance, low
-
cost in construction and
exhibits good performance under static and dynamic loadings. MSEW are considerably cost
-
effective
compared to the conventional reinforced concrete or gravity type walls that have traditionally been
used to retain soil. These include bridge abutments and wing walls as well as areas where the right
-
of
-
way is restricted. Additional uses of MSEW include dams and seawalls, retaining walls, marine
walls, etc. MSE walls offer significant technical and cost advantages over reinforced concrete
retaining structures at sites with poor foundation conditions. In such cases, the elimination of costs
for foundation improvements such as piles and pile caps, that may be required for support of
conventional structures, have resulted in cost savings of greater than 50 percent on completed
projects. This research is a continuation of a former study which investigated the lateral deformation
of the reinforced soil zone.



Methods

To determine the lateral deformation of MSEW, the analysis was based on the shear beam theory.
The MSEW under consideration is assumed to be a 2D deep beam of width B and Length H, under a
varying static load. A Cartesian coordinates system (
x, y
) is used as shown in Figure 1
.




Figure 1 Structural Model of the
MSEW









For MSEW that consist of cohesive material, it is acceptable to assume a constant value for the shear
modulus (
Haroun
). The forces acting on the element shown in Figure 2 include the load F, and the
shear force
Qy
. Equilibrium of the forces acting on this element on the x
-
direction yields the
following equation.







Figure 2 Details of Element from
MSWE













Eq.1


Analysis

The reinforced soil zone was divided into finite elements where the deflection was assumed linear
and the method of finite
-
difference could be used. Figure 3 shows the details of the analysis. In each
layer, the deflection is calculated by solving the second order differential equation, Eq.1, using the
finite difference method
.



Figure 3 Analysis Model







The
backfill was modeled using a nonlinear elastic soil model with a variable Young’s
modulus and a constant Poisson’s ratio (Ling), the initial Young’s modulus, the initial shear
modulus, and the shear failure are expressed as follows
:





Eq.2

𝐸
𝑖
=

initial Young’s modulus



𝐾
=

modulus number



𝑛
=

modulus exponent


Eq.3
𝑅
𝑓
=

failure ratio




𝑝
𝑎
=

atmospheric pressure


Eq.4
𝜎
=

principal stress



The
developed code was generated using
MatLab

to run a loop that would do the following.

First the principal stresses are computed and
stored,
then the initial shear modulus and the
shear failure
𝜏
𝑓
are computed using Eq.2 and the following equation:




After storing these two values, Eq.4 is manipulated to determine the value of strain at half
strength. The calculations below demonstrate this procedure.









Since
both stress and strain are determined, the value of the Young’s modulus E can be
computed as shown in Figure 4 using the following relation:










Figure 4 Shear Stress
-
Strain Relationship


Once
E is determined, G can be computed using Eq.3, and Eq.1 can be solved numerically
using finite
-
difference method as described below
.








To verify the results of this research, a thesis prepared by Barry Rodney Christopher, Ph.D.
from the University of Perdue was used to compare the findings. The thesis discussed
deformation response and wall stiffness in relation to reinforced soil design.

This paper only focuses on two types of soils namely gravel and silt
.


Results


Gravel

To be conservative, the lateral pressure was computed using the following equation


𝑃
=
𝐾
0
𝛾ℎ

Eq.5

Facing displacement for this type of soil was computed for comparison. The code
would

store the initial deflection of each layer as initial (j) in the code, and then calculates
the deflection of every layer due to the placement of the last layer. Then, the facing
displacement will be the difference between these two values for every layer. Figure 5
shows the details of facing displacement.




Figure
5 Details of Facing Displacement.









Figure
6 shows the facing displacement of the reinforced soil zone obtained from the code using the soil
data given in the thesis and compared to the results from the two theories. Wall 1 & 3 in the figure
represents the results obtained from the case study led Barry Rodney Christopher, Ph.D. the only
difference between wall 1 & 3 is the type of reinforcement used, where wall 1 consists of ribbed metal
strips (8 layers @ 14ft) and wall 3 had a bar mat (8 layers @ 14ft) as reinforcement.

Using:

𝐾
=
460



=
40°


𝛾
=
130

𝑙
/
𝑓𝑡
3


=
0


𝑅
𝑓
=
0
.
7

𝑝
=
0
.
5




Figure
6 Facing Displacement of Gravel Soil








Silt

Similar to Gravel, Facing displacement for Silt was computed and compared to the result from the case
study as shown in Figure 7.

Where

𝐾
=
200

𝛾
=
130

𝑙
/
𝑓𝑡
3



=
35°


=
50


𝑅
𝑓
=
0
.
7

𝑝
=
0
.
6

Wall 5 in the figure represents the results obtained from the case study led Barry Rodney Christopher,
Ph.D.



Figure 7 Facing Displacement of Silt Soil








Conclusion

A study on the deflection of the reinforced soil zone was performed based on deep beam theory using a
program code. The shear modulus was varied at each layer to accommodate the effect of changing
stresses due to depth. The results shown are conservative and close to those obtained from the case
study.


References

Christopher, Barry Rodney.
Deformation Response and Wall Stiffness in Relation to Reinforced Soil Wall
Design
. PhD Thesis. Ann Arbor: University Microfilms International A Bell & Howell Information Company,
1993.

Haroun
,
Medhat

A. "Seismic Response Analysis of Earth Dams Under Differential Ground Motion."
Bulletin
of the Seismological Society of America

October 1987: 1514
-
1529.

Ling, H.I.,
Cardany
, C.P., Sun, L
-
X. and
Hashimito
, H., 2000,. "Finite Element Study of a
Geosynthetic
-
Reinforced Soil Retaining Wall with Concrete Block Facing ."
Geosynthetic

International
2 April 2000: 137
-
162.



GB
P
y
u



2
2
n
a
a
i
p
Kp
E
)
(


)
1
(
2



E
G
0
1
G
R
f
f
f











sin
1
sin
2
cos
2



c
f
)
5
.
0
1
(
5
.
0
5
.
0
)
5
.
0
1
(
1
5
.
0
5
.
0
1
5
.
0
0
@
0
0
0
f
f
th
halfstreng
f
f
f
f
f
f
f
R
G
G
R
G
R
G
R






















th
halfstreng
f
E
@
5
.
0



GB
P
h
y
u
h
y
u
h
h
y
u
h
GB
P
h
y
u
h
y
u
h
y
u
GB
P
y
u















)
(
1
)
(
2
)
(
1
)
(
2
)
(
)
(
2
2
2
2
2
2
B
Q
Given
dy
y
Q
F
dy
y
Q
Q
Q
F
Fx
y
y
y
y
y
y
*
:
)
(
0
)
(
0















B
G
Q
y


dy
y
u
GB
F
2
2



0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0
5
10
15
20
25
Facing Displacement "in"
# of Layers


Calculated Values
Wall 1 (Ribbed Metal Strips)
Wall 3 (Bar Mat)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
5
10
15
20
25
Facing Displacement "in"
# of Layers


Calculated Values
Wall 5 (Bar Mat)