COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Plane Strain Finite Element Analysis of a Piled Bridge Abutment on
Soft Ground
H. T. Wang1*, Z. P. Chen2, L. J. Xiao2,
1
2
Department of Civil Engineering, South China University of Technology, Guangzhou City, 510640 China
Guangdong Guansheng Civil Engineering & New Tech Corporation, Guangzhou City, 510000 China
Email: [email protected]
Abstract In order to predict the behaviour of a piled bridge abutment constructed on soft ground, a plane strain finite
element model is developed by using commercial finite element code ABAQUS. With the help of the finite element
model, the loading and displacement of the structure and embankment at any time after the construction are readily
obtained. In the present model, the consolidation of soil is realized by employing ‘couple pore fluid diffusion and stress
procedure’ in ABAQUS which models soil as saturated porous media and uses pore pressure element at the same time.
Critical state plasticity model provided by ABAQUS, which is an extension of the modified Cam-clay model, is used to
describe the inelastic behavior of clay. Vertical, horizontal sand-drain and major construction process are all taken into
account. For better conforming the intrinsic 3D nature of the structure (especially piles) and soil-structure interaction,
some measures are taken when constructing the plane strain model.
Key words: finite element, soil-structure interaction, consolidation, piled bridge abutment, soft clay
INTRODUCTION
The construction of a piled bridge abutment on soft ground is a long term process. A general construction procedure is
summarized as followed. The first step is the construction of sand-drain and sand-mat. Sand-drain and sand-mat make
up a vertical and horizontal drainage system, and this system can accelerate the consolidation process of soft clay. Then
the embankment will be filled. In order not to cause the instability of the underlaid soft clay, the filling process should
be scheduled according to the strength of the soft clay and be monitored to predict the possible soft clay instabilities.
When the embankment is considered to reach the designed height stably, a part of embankment will be removed, and
then the piled bridge abutment will be constructed. At last, the road and bridge will be constructed.
During the service lifetime, piled bridge abutments are subject to lateral soil-structure interaction resulting from the
time dependent movement of backfills behind abutments and of the underlaid soft clay. In order to get an economical
and reliable design, a thorough understanding of lateral deformations of abutments as well as stresses in piles is
needed. Furthermore, because the behavior of soils is highly dependent on their stressed history, every step in the entire
construction process should be taken into account.
In present paper, by using commercial finite element code ABAQUS, a plane strain finite element model is developed
to analyze a piled bridge abutment which is constructed on soft ground in Zhujiang delta. With such model, the loading
and displacement of the structure and embankment at any steps of construction and during the whole service lifetime
are readily obtained. In the present model, the consolidation of soil is realized by employing ‘couple pore fluid
diffusion and stress procedure’ in ABAQUS which models soil as saturated porous media and uses pore pressure
element at the same time. Critical state plasticity model provided by ABAQUS, which is an extension of the modified
Cam-clay model, is used to describe the inelastic behavior of clay. Vertical, horizontal sand-drain and major
construction process are all taken into account. For better conforming the intrinsic 3D nature of the structure
(especially piles) and soil-structure interaction, some measures are taken when constructing the plane strain model.
OUTLINE OF THE PILED BRIDGE ABUTMENT
The piled bridge abutment belongs to Zhongjiang expressway and is located in Zhujiang delta with network
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of waterways. The representative sequence of subsoil at the site where the piled bridge abutment is constructed is:
embankment with a thickness of 1.5m, silt with a thickness of 13.5m, mild clay with a thickness of 21m, gravel with a
thickness of 13m, intense weathering sandrock with a thickness of 1m and moderate weathering sandrock with a
thickness of 5-8m on weak weathering sandrock.
The construction process of the piled bridge abutment can be outlined in two stages—the preload of the soft ground
and the construction of the piled bridge abutment.
The Preload of the soft ground First of all, sand-drain with a depth of 21m and interval of 1.2m are constructed. Then
a sand-mat layer with a thickness of 0.6m is followed. Sand-drain and sand-mat make up a vertical and horizontal
drainage system, and this system can accelerate the consolidation process of soft clay. The last procedure in this stage
is the filling of the embankment to the height of 6m.
The construction of the piled bridge abutment When the observed settlement of the top of the embankment is less than
3cm per month, the embankment is considered to reach the designed height stably. Then a part of embankment will be
removed, and the piled bridge abutment will be constructed thereafter. At last, the road and bridge will be constructed.
The dimension of the abutment is 8.5m in width, 0.7m in thickness and 6m in height; the dimension of the bearing
platform is 1.5m in height, 5.7m in length. The whole abutment is supported by four reinforced concrete piles with
1.2m in diameter. And all piles are driven to moderate weathering sandrock layer.
FINITE ELEMENT MODELLING
Generally speaking, in finite element analysis, a piled bridge abutment can be simplified as a longitudinal plane strain
model. In such a model, the effect of the side slope of an embankment can not be taken into account directly;
furthermore, the intrinsic 3D nature of soil-pile interaction has to be ignored. Such drawbacks can be avoided if a full
3D model is employed, but much more complexities arise in a 3D model, therefore the computational expenses rocket
up. In order to balance the drawbacks of traditional plane strain models and full 3D models, a plane strain model
proposed by Ellis and Springman [1], in which 3D effects associated with soil-pile interactions are incorporated, is
employed in present paper. With this model, a fully coupled consolidation analysis can be accomplished in a personal
computer. The method which introduces 3D effects associated with soil-pile interaction into a plane strain model will
be stated below.
In this section, the details of modeling of materials, meshes and boundary conditions in the finite element model are
described. No in situ experiment was done to measure the material properties for this piled bridge abutment. And the
material properties are directly obtained or computed from the database of the entire project.
1. The modeling of materials
(1) Embankment: All embankments are regarded to be composed of the same material.
Constitutive model: Drucker-Prager model with non-associated flow rule;
Unit weight: 18.8KN/m3;
Parameter of strength: By using Eq. (1) [3], cohesion parameter d and angle of internal friction β for Drucker-Prager
model with non-associated flow rule are computed from cohesion parameter c and angle of internal friction φ for
Mohr-Coulomb model;
tan β = 3 sin φ, d / c = 3 cos φ
(1)
Elastic modulus: the elastic modulus was assumed to be E50, the secant modulus at the stress of 50% of the strength of
the material. The strength of the material is obtained from consolidated drained triaxial tests.
(2) Sand-mat: Constitutive model: Drucker-Prager model with non-associated flow rule;
Unit weight: 20KN/m3;
Parameter of strength: please refer to item (1);
Elastic modulus: please refer to item (1).
(3) Abutment: Constitutive model: linear elastic model;
Material properties: elastic modulus, unit weight and Poisson’s ratio are set to be the design values of reinforced
concrete in Chinese codes for reinforced concrete bridge design.
(4) Piles: Equivalent sheet pile wall: As shown in Fig.1, in a plane strain analysis, the structural behavior of a pile row
can be replaced by an ‘equivalent sheet pile wall’ according to Ellis and Springman [1]. The wall has the same flexural
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Figure 1: Pile row and equivalent sheet pile wall
stiffness per unit width as the piles and soil it replaces. When we compute the total flexural stiffness of a pile row, the
effect of soil is usually excluded, because the soil contributes little to the total flexural stiffness. The equivalent sheet
pile wall can be simulated by plane strain elements or beam elements in finite element analysis. In present paper, beam
elements are adopted. In order to simulate soil-pile interaction, sheet pile wall are treated ‘floating’ on the soil and link
elements are used to incorporate soil-pile interaction (Fig. 2). The mechanical properties of link element will be
determined through another finite element computation specified later.
Figure 2: Sheet pile wall ‘floating’ on the soil
Constitutive model: the equivalent sheet pile wall is treated as linear elastic wall;
Material properties of piles: elastic modulus, unit weight and Poisson’s ratio of piles are set to be the design values of
reinforced concrete in Chinese codes for reinforced concrete bridge design.
(5) Soft clay (including silt and mild clay): Constitutive model: Critical state plasticity model with non-associated flow
rule;
Unit weight: please refer to Table 1;
Parameters to calibrate critical state plasticity model: CC (compression index), CS (swelling index) and pc
(pre-consolidation pressure) obtained from oedometer tests are available in the database of the project. Before their
uses in ABAQUS finite element models, these parameters have to be transformed according to [3]. Since the pore
water pressure was not measured in the consolidated-undrained triaxial shear tests, the effective angle of internal
friction φ′ is not available in the database of the project. We take the value of φ′ according to [2] as 30 degree, therefore
the critical state factor M can be calculated by using:
M = 6sin φ′ /(3 − sin φ′)
(2)
The effective Poisson’s ratio is assumed to be 0.2 for all clay soils.
(6) Gravel and intense weathering sandrock: Constitutive model: Drucker-Prager model with non-associated flow rule;
Unit weight: please refer to Table 1;
Parameter of strength: please refer to item (1);
Elastic modulus: please refer to item (1).
The material properties for finite element analysis are summarized in Table 1 to Table 3.
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Table 1 Material properties for critical state plasticity models
Subsoil layers
γ (kN/m3)
CC
CS
e0
pc(kPa)
k (m/s)
silt
16.1
0.2114
0.0173
1.557
58.0
1.0E-9
Mild clay
16.6
0.2353
0.0149
1.281
126.2
1.0E-9
Where γ is the unit weight; CC, CS and pc are respectively compression indices, swelling indices and pre-consolidation
pressure obtained from oedometer tests; e0 is void ratio and k is the order of magnitude of coefficient of permeability.
Table 2 Material properties for Drucker-Prager elasto-plasticity models
Subsoil layers
E (kN/m2)
ν
φ (degree)
C (kN/m2)
γ (kN/m3)
k (m/s)
embankment
5000
0.30
30
10
18.8
1.0E-9
Sand-mat
5000
0.30
30
0
20
1.0E-4
gravel
intense
weathering
sandrock
60,000
0.30
30
0
18
1.0E-5
60,000
0.30
30
0
18
1.0E-5
Where E is the elastic modulus; ν is Poisson’s ratio; φ is the angle of internal friction and C is cohesion parameter.
Table 3 Material properties for linear elasticity models
Reinforced concrete
γ (kN/m3)
E (kN/m2)
ν
k (m/s)
24.5
2.5E+7
0.17
1.0E-20
Figure 3: Fundamentals for soil-pile interaction
2. The modeling of sand-drain In order to accelerate the process of consolidation, sand-drain is constructed. In the
present finite element model, three equivalent plane strain sand-drain are adopted following the method proposed by
Ellis and Springman [1].
3. The modeling of soil-pile interaction In the present analysis, soil-pile interaction is simulated by using link
elements (refer to item (4) in section 1). In order to reflect the intrinsic 3D nature of soil-pile interaction, an ad hoc
finite element analysis is preformed to compute the mechanical properties of link elements.
The fundamentals for computing the mechanical properties of link elements are shown in Fig. 3, and this case is used
for analyzing an isolated pile. Plane strain section in the figure represents a layer of subsoil. When the pile is subjected
to a specified displacement, reaction force are generated. The mechanical properties of link element can be extracted
from the relationship between the displacement and the reaction force. For an isolated pile, the dimensions of the plane
strain section in Fig. 3 are at least 50d×50d, where d is the diameter of the pile and the pile is placed at the centroid of
the section.
In the present analysis, the abutment is supported by four piles and the relationship between the displacement and the
reaction force of piles is computed by finite element method. Two of the piles are considered in the model due to the
symmetry of the problem. To incorporate pile-soil-pile interaction in the model, the finite element mesh shown in
Fig.4. is employed.
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Figure 4: The finite element mesh used for the analysis of pile-soil-pile interaction
4. The modeling of boundary conditions In the present analysis, subsoil layers above moderate weathering sandrock
layer and a 100m long longitudinal segment (50m segment before the abutment and 50m segment behind the abutment)
are modeled in the finite element mesh. The boundary conditions for the bottom of the model are set as all vertical
displacement constrained and undrained. This is a result of neglecting the vertical deformation and permeabilities of
underlaid sandrock layer. Horizontal displacements are fixed at both sides of the model. Although several different
subsoil layers are considered in the present model, displacements and pore water pressure are assumed continuous
across the interface of two different layers. Soil-abutment interactions are also simulated by using link elements, all
link elements are set to have very large stiffness to resist pressure and have zero stiffness to resist tension. At last, all
soil-structure interfaces are assumed can not transfer shear stress.
NUMERICAL RESULTS
The settlement deformation of the soft clay is a complex and long term process. It involves the interaction among the
solid phase, liquid phase and gas phase of soils. After the soft clay is subject to extra loads, the settlement deformation
of soft clay usually will last years, tens of years or even over a hundred years. Generally speaking, the designed service
lifetime for a bridge is seventy years or one hundred years in China. So the entire settlement process will have effects
on the usage of the bridge and the loading and the deformation history during this period of time are of designers’
concerns. Therefore consolidation analysis is preformed; the time span is from the construction of sand-drain to the
time when the settlement becomes stable.
In this section, selected numerical results are listed. The converged finite element mesh for the present consolidation
analysis is shown in Fig. 5. There are 9372 plane strain elements, beam elements and link elements and 9423 nodes in
the mesh. And fifty years after being constructed, the computed settlement of the embankment becomes stable.
Figure 5: Converged finite element mesh for consolidation analysis
1. Settlement of the embankment 50 years after being constructed Fig. 6 shows the settlement of the
embankment 50 years after being constructed. The black part in Fig.6 represents embankment, and the white
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part represents abutment. It is shown clearly in the figure that after fifty years of settlement, there exist a step
between the abutment and the embankment. The computed height of the step is 14cm in the present analysis.
This step will cause the road unevenness at the connection of abutment and embankment. Further measures
should be taken to solve this problem.
Figure 6: Settlement of Embankment 50 years after being constructed
2. Settlement of the original ground surface 50 years after being constructed Fig. 7 shows the settlement of the
original ground surface 50 years after being constructed. The position 50m on horizontal axis represents the location of
the centre line of the abutment, 0m—50m on the horizontal axis represents the segment where the bridge locates, and
50m—100m on the horizontal axis represents the embankment segment.
Position (meter)
25
50
0
Settlement (meter)
0
75
100
-0.5
-1
-1.5
Figure 7: Settlement of the original ground surface 50 years after being constructed
3. The history of settlements of two points on original ground surface Fig. 8 shows the history of settlements of
two points on original ground surface. The distance between point A and the central line of abutment is 5.35m. Point B
is a little farther than point A to the abutment and the distance is 10.35m. From the figure, the removal of part of the
embankment and the construction of piled bridge abutment have little influence on the settlement of these points.
4. Moment in piles 50 years after being constructed Fig. 9 shows the moment in piles 50 years after being
constructed. The bridge abutment is supported by four piles. ‘Bridge side’ in the caption of the picture represents one
of the piles on the bridge side and ‘embankment side’ represents one of the piles on the embankment side. Only part of
the computed results is shown in the picture and two moment peak can be found in all piles. Furthermore, the values
and the positions of maximum moments are almost identical in all of them.
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Time (log(day))
100
0
101
102
103
104
Point A
Point B
Settlement (meter)
-0.4
-0.8
-1.2
Figure 8: History of settlements of two points on the original ground surface
bridge side
embankment side
-1.5
-1
Position of the cross section (m)
50
-0.5
40
30
20
0
10
0.5
Moment in piles (MN*m)
Figure 9: Computed moment in piles 50 years after being constructed
CLOSURE
In order to get a service lifetime prediction of the behaviour of a piled bridge abutment constructed on soft ground in
Zhongjiang expressway, a plane strain finite element model is developed by using commercial finite element code
ABAQUS. In the present model, consolidation analysis is employed to reflect the interaction among the solid phase,
liquid phase and gas phase of soils; critical state plasticity model provided by ABAQUS, which is an extension of the
modified Cam-clay model, is used to describe the inelastic behavior of clay. Vertical, horizontal sand-drain and major
construction process are all taken into account. For better conforming the intrinsic 3D nature of the structure
(especially piles) and soil-structure interaction, some measures are taken when constructing the plane strain model.
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REFERENCES
1. Ellis EA, Springman SM. Modeling of soil-structure interaction for a piled bridge abutment in plane strain FEM
analyses. Computers and Geotechnics, 2001; 28: 79-98.
2. Hara T, Yu Y, Ugai K. Behaviour of piled bridge abutment on soft ground: a design method proposal based on
2D elasto-plastic-consolidation coupled FEM. Computers and Geotechnics, 2004; 31: 339-355.
3. ABAQUS Theory Manual. Version 6.3. Hibbitt, Karlsson & Sorensen, Inc.
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