Three-Dimensional Finite Element Modeling and Validation of Combined Pile-Raft Foundation under Static Loading
Authors: Tushar K. Mandal*, Girish R. Patil, Apurba Mondal, Indrajit Ray, Roghupati Roy Nuclear Power Corporation of India Limited (NPCIL), Mumbai-400094 * Corresponding author:
[email protected] Abstract: A methodology for analyzing pile-soil interaction under static load has been discussed. Three dimensional mathematical modeling of pile-soil system has been carried out in finite element analysis package Abaqus considering surface to surface interactions at the interface. Soil and pile both have been modeled using 8-noded brick elements (C3D8R). Soil strata have been idealized with elastic-plastic Mohr-Coulomb model whereas piles have been idealized to behave linearly elastic. An analytical basis of calculating elastic slip has been suggested. Some important modeling techniques and analysis sequences have been suggested for better numerical convergence for this nonlinear solid to solid contact problem. Vertical loadsettlement behaviour of single pile is obtained from analysis and compared with published field test result, showing a reasonable agreement. Load displacement behaviour of another field test comprising of single pile and pile-group subjected to lateral loads is modeled and compared with test results. Keywords: Soil-pile interaction, elastic slip, numerical convergence, nonlinear finite element analysis
1.
Introduction
Pile foundation is one of the most popular forms of deep foundation. Combined pile-raft foundation has become an engineering solution for soil sites for heavy construction. Modeling and analysis of combined pile-raft foundation is one of the richest fields of engineering research which involve pile to soil, pile to pile, raft to soil, pile to raft interactions. In general we can say it is basically a solid to solid interaction problem, if water isn’t modeled explicitly. This involves solid to solid contact problem. Behaviour of two interacting solid surfaces can be transformed into behaviour of surface into two mutually perpendicular planes: normal and tangential. Normal behaviour of the surface depends on overburden pressure (p), overclosure (h)whereas the tangential behaviour depends on shear strength of the interface (τ) and critical elastic slip (γcrit) which is the relative displacement at the interface at which the tangential shear capacity get completely mobilized. To define to the normal behaviour for soil-pile interaction problem is quite effortless since very few parameters are required to specify. Tangential behaviour of the interface is based on the concept of Coulomb friction stick-slip model (Simulia, 2014).Shear strength of the interface usually estimated from the shear strength of the soil. But there are issues with estimating the elastic slip, γcrit which depends on type of soil (Anne Lemnitzer, 2008). As per our study there is no theoretical approach available for estimating the elastic slip, experimental data ( (Palmer A.C., 2005), (Khare M.G.)) is only option. Elastic slip affects the overall response of pile-soil system much or less depending on type of soil. Whatever, it may be very important parameter in response of the entire structure since it may change the behaviour of the foundation system. There are some parameters for modeling which may affect numerical convergence of analysis of this nonlinear contact problem. In this study, an attempt has made towards the theoretical estimation of elastic slip for pilesoil interface from the basic soil properties: cohesion (C), angle of internal friction (ϕ), Young modulus (E), Poisson’s ratio (ν) etc. The parameters responsible for numerical nonconvergence issues have been identified and suitable values for convergence have been suggested. Analysis has been carried out using commercial FEA package ABAQUS. Pile-soil interface has been modeled as master-slave surface (pile as master and soil as slave). Published experimental results have been compared with results obtained through analysis in ABAQUS. Pile-raft is a reinforced concrete structure. Modeling reinforcement in reinforced concrete structure is a tedious work and its analysis also requires a lot of computational effort. In a
problem like soil-structure interaction involving nonlinear contact, modeling reinforcement makes the problem computationally more expensive. In this study it is shown where incorporating reinforcement in FE modeling is needed. 2.
Theoretical estimation of critical elastic slip
As discussed in previous section, theoretical estimation of critical elastic slip is very much necessary, but no such approach available still now. Principle of continuum mechanics has been used to reach at the relationship between skin friction (τ) and elastic slip (γ). Shear stresses are assumed to decay radially (Mosher R L, 2000) in the soil according to some polynomial, 𝑹𝑹𝒑𝒑 𝒏𝒏
𝝉𝝉 = 𝒇𝒇 � 𝒓𝒓 � ………………………………(1)
Where,
τ=shear stress in soil 𝑓𝑓 = skin friction at pile soil interface
r= radial distance from the pile centerline G=the soil shear modulus of elasticity. Rp= radius of pile n= order of decay function If radial deformations of the soil are ignored, the shear strain at any point in the soil may be expressed as (using Eqn.1) 𝝐𝝐 =
𝜹𝜹𝜹𝜹
𝒇𝒇 𝑹𝑹 𝒏𝒏
𝝉𝝉
= 𝑮𝑮 = 𝑮𝑮 � 𝒓𝒓 � ……………………………………(2) 𝜹𝜹𝜹𝜹
Hence, shear deformation of soil compared to pile at the interface can be calculated as: 𝒇𝒇
𝒓𝒓
𝑹𝑹 𝒏𝒏
𝜸𝜸 = 𝑮𝑮 ∫𝑹𝑹 𝒎𝒎 � 𝒓𝒓 � 𝒅𝒅𝒅𝒅 ……………………………(3)
Where, rm=limiting radial distance beyond which deformations of soil mass are negligible and it can be approximated as 2Lρ(1-ν) (Mosher R L, 2000)where ρ is a factor to incorporate vertical non-homogeneity of soil layers.
We are interested in calculate the shear deformation at the interface at which the interface reaches its capacity i.e. shear strength of the interface. Hence, γ = γcrit when 𝑓𝑓 = τmax . Shear stress in soil may vary linearly or nonlinearly, hence numbers of trials have been carried out and corresponding critical elastic slip has been formulated as follows:
(4)
⎧ ⎪ ⎪
γcrit =
γcrit =
𝜏𝜏 𝑚𝑚𝑚𝑚𝑚𝑚 𝐺𝐺
𝜏𝜏 𝑚𝑚𝑚𝑚𝑚𝑚 𝐺𝐺
𝑟𝑟
𝑅𝑅𝑝𝑝 𝑙𝑙𝑙𝑙 �𝑅𝑅𝑚𝑚 � … . . . . . . . . . . . . for linear [n = 1] (4a) 2
1
𝑝𝑝
1
𝑅𝑅𝑝𝑝 �𝑅𝑅 − 𝑟𝑟 � … . . . . . for quadratic [n = 2] (4b)
𝜏𝜏 𝑚𝑚𝑚𝑚𝑚𝑚
𝑝𝑝
𝑚𝑚
1 1 𝑅𝑅𝑝𝑝 3 �𝑅𝑅 2 2 𝑝𝑝
1 ⎨ γcrit = 𝐺𝐺 − 𝑟𝑟 2 � … for cubic [n = 3] (4c) 𝑚𝑚 ⎪ ⎪ 𝜏𝜏 𝑚𝑚𝑚𝑚𝑚𝑚 1 1 1 = 𝐺𝐺 𝑅𝑅𝑝𝑝 4 �𝑅𝑅 3 − 𝑟𝑟 3 � … for 4th order variation[n = 4] (4d) γ 3 ⎩ crit 𝑝𝑝 𝑚𝑚
Now, shear strength of the interface (𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚 )can be estimated from basic soil parameters as (M,
1971):
(𝜙𝜙))…………………..…………….…(5) 𝜏𝜏𝑚𝑚𝑚𝑚𝑚𝑚 = 𝛼𝛼(𝐶𝐶 + 𝜎𝜎tan
Where, α is called as adhesion coefficient (~0.8) for soil-concrete interface C=cohesion of soil ϕ = angle of internal friction σ =pressure onto pile-soil interface=K0*vertical pressure at that depth of soil K0=Coefficient of earth pressure at rest (usually expressed as a function of Poisson’s ratio of soil) Thus, critical elastic slip (γcrit ) can be estimated from basic soil properties shear modulus, cohesion, angle of internal friction, Poisson’s ratio and diameter of pile, depth of soil etc(using equation 4 and 5). Broms, B.et. al (B, 1979)has carried out the field test to calculate the pile displacement required to mobilize the skin friction & reported a range of value for elastic slip i.e. 1-8 mm. With the same data, the pile displacement is calculated using above equations and the values obtained are given in Table 1.It is found out that the pile displacement corresponding to cubicshear stress distribution is in line with the results obtained from the field tests conducted by Brom B. et. al. Hence, equation 3c may give a good estimate for critical elastic slip.
Table 1 Pile elastic displacement for different shear stress variation (B, 1979)
Lp
20
3.
E
C 2
(t/m )
2
(t/m )
500
0.3
μ
0.3
φ
20
Densit 3
y (t/m )
1.8
Coeff.
Max. elastic relative displacement of
of EP,
pile and soil (mm)
Ko
Linear
0.65
68.7
Quadratic Cubic 11.4
5.7
4th order 3.8
Developing analysis steps and setting modeling parameters
Main problem with modeling soil medium is state of stress due to self-weight need to be modeled without allowing any settlement in it. This phenomenon is known as geostatic condition and it is modeled in ABAQUS using Geostatic analysis. This is an iterative nonlinear analysis and hence having non-convergence issues depending on type of problem and how it is modeled mathematically. Since, pile-soil interaction problem involves modeling nonlinear interaction between soil-pile, initiation of contact is usually accompanied by nonconvergence issues. Hence, these two phenomena should not be modeled simultaneously (to avoid the non-convergence problems); instead it should be carried out one by one: Geostatic analysis is carried out without soil-pile interaction and then interactions are initiated at the interface. A boundary condition constraining the soil mass not to fall into the openings created for pile in soil medium need to be present during Geostatic analysis and to be replaced by the interactions just after this step. Contact initiation itself involves huge non-convergence issues if a force-based analysis is carried out. So a displacement based analysis is proposed for a very negligible displacement onto pile and then regular force-based analysis can be carried out without big trouble. The modeling steps with suitable boundary condition and interactions are shown in Table 2. Table 2 Analysis steps for soil-pile interaction problem Analysis step
Boundary condition
Interactions
Initial
All physical boundary condition including boundary No interactions condition at the interface
Geostatic
All physical boundary condition including boundary No interactions condition at the interface
Contact-
All physical boundary condition only. Boundary Contact initiated
initiation
condition at the interface is removed
Pile Load
All physical boundary condition only
Contact propagated
There are some more parameters which are found extremely important in resolving the nonconvergence issues in soil-pile interaction problem. These parameters come in picture during defining a new interaction and plays significant role in contact iterations. These are listed below: Sliding formulation: Since pile-soil interaction problem involves very small amount of displacement compared to the size of structure, small sliding should be used to avoid more complex formulation for general case of finite sliding. Slave adjustment: Adjusting the slave surface to remove the overclosure is found to satisfactory. Surface smoothing: Uneven surface artificially created due to discretization should be removed automatically when applicable. 4.
Parametric study of contact parameters for single pile under vertical loading
Behaviour of pile predicted from model incorporating slip at soil-pile interface is quite different as compared to that from continuum model (Lee C. J., 2002). This prediction significantly depends on contact parameters used for modeling the problem mathematically. Among different parameters required to define soil-pile interface property, coefficient of friction and maximum shear strength can be calculated directly from the basic soil properties directly and is fixed for a type of soil. In this study, critical elastic slip has been estimated using variation of shear stress distribution in the soil. So it depends on what is the assumed distribution of stress in soil. A vertical pile field load test (Naveen B P, 2011) has been modeled with pile and soil details as given in Table 3 and Table 4. Interaction properties for the soil-pile interface have been calculated using the method discussed in section- 2 assuming different variation of shear stress in soil. Calculated critical elastic slip changes along the depth of pile and there is a sudden change where soil layer changes (Table 5). The resulted load settlement behaviour of the pile is quite different for different assumption on shear stress distribution (Figure 1). Estimated initial stiffness of pile-soil system changes quite significantly with the assumed distribution of shear stress in soil.
20
21
Mohr-
6-20
2
soft weathered rock
1e+06
0.33
50
25
22
Coulomb
(ν)
model
30
(kN/m2) Material
0.3
friction (ϕ) Density
Possons ratio
4e+04
internal
E (kN/m2)
clay
Angle of
Soil type
1
(kN/m2)
Layer
0-6
Cohesion
Depth (m)
Table 3 Soil properties for vertical pile test
Table 4 Details of pile Length, m
Diameter, m
E (kN/m2)
Possons ratio (ν)
Material model
15
1.2
30e+06
0.2
Linear Elastic
Coefficient of friction μ
Maximum shear strength (τmax), kN/m2
0-2
1
0.6
27
2-4
1
0.6
32
4-6
1
0.6
39
6-9
2
0.6
70
9-12
2
0.6
81
Critical elastic slip (γcrit), mm
Critical elastic slip, mm 0.00
3.00
6.00
0.00 3.00 Depth, m
Depth (m)
Layer
Table 5 Interaction properties of soil-pile interface
6.00 9.00
Linear Quadratic
12.00
12-15
2
0.6
93 15.00
Cubic
9.00
Load, kN 0
1000
2000
3000
4000
5000
0
Settlement, mm
0.5 1 1.5 2 2.5
Linear, K=1.84e+06 kN/m Quadratic, K=2.33e+06 kN/m
3 Cubic, K=2.43e+06 kN/m 3.5 Figure 1: Load settlement of pile tip assuming different variation of shear stress in soil from pile-soil interface 5.
Validation of single pile behaviour under vertical loading
A single pile with soil, pile and interface related information as given in Table 3,Table 4 and Table 5 has been modeled using the methodology discussed in Section-3. A single pile (which is modeled as modeled as concrete only) fully embedded in soil has been loaded vertically using standard vertical loading platform (Figure 2). Behaviour of the pile estimated from finite element analysis is matching significantly with field test result (Figure 3).
0
1000
2000
Load, kN 3000 4000
5000
6000
0 Settlement, mm
0.5 1 1.5 2 2.5 3 Figure 2 Vertical pile load test set-up (Naveen B P, 2011)
Field test, K=2.5e+06 kN/m FE Analysis, K=2.43e+06 kN/m
3.5 Figure 3 Load displacement behaviour of single pile from field test and finite element analysis
6.
Validation of single pile and pile-group-raft behaviour under lateral loading
Piles have been tested under lateral load at structural and geotechnical laboratory (UCLA) for a highway project (Stewart J.P., 2007). Three scaled specimen were tested: (a) A flagpole pile (b) A fixed-head single pile (c) A 3x3 pile group with raft. These three tests have been modeled in Abaqus with soil, pile and interface properties as given in Table 6, Table 7 and Table 8. Each specimen has been modeled with/without: (1) reinforcement (2) incorporating material nonlinearity. Since flagpole specimen is subjected to high bending moment, pile becoming nonlinear at ground level, hence modeling reinforcement and incorporating material nonlinearity in concrete and reinforcement becomes necessary(Figure 4(a)). In this study reinforcements are modeled as embedded layers in concrete, without allowing relative displacement at the reinforcement-concrete interface (no bond slip). On the other hand, single fixed-head pile and pile group are fully embedded in soil and hence subjected to lesser displacement and moments; hence modeling material nonlinearity in soil (only) is sufficient to represent the behaviour of foundation under lateral load. It is also found that modeling reinforcement also isn’t necessary for these two cases (Figure 4(b),(c)). These observations may alter for very soft soil site; modeling reinforcements along with material nonlinearity may be required for all three cases.
Possons ratio (ν)
Cohesion (kN/m2)
Angle of internal friction (ϕ)
Density (kN/m2)
1
Rubble and fill
4.75e+04
0.3
0
45
19
1.5-6.4
2
Silty Clay
1.14e+05
0.3
80
1
20
6.4-7.3
3
3.95e+04
0.3
0
38
20
7.3-14.6
4
Silty Clay
1.38e+05
0.3
170
1
20
14.6-20.0
5
Medium sand
3.15e+04
0.3
0
38
20
Soil type Medium to fine grained silty sand
Material model
Layer
0-1.5
E (kN/m2)
Depth (m)
Table 6 Soil properties for lateral pile test (Stewart J.P., 2007), (Anne Lemnitzer, 2008)
MohrCoulomb
Table 7 Details of single pile and pile-raft for lateral test (Stewart J.P., 2007), (Anne Lemnitzer, 2008) Pile
Diameter(m)
Length (m)
Concrete
0.6
7.6 (Added 4.0m above
Unit Weight
Young Modulus
(kN/m3)
(kN/m2)
25
30×106
ground for flagpole specimen) Pile-raft
No of pile
Raft thickness (m)
Concrete
3x3
1.83
Raft width
Raft length (m)
(m) 4.88
5.48
Table 8 Interaction properties of soil-pile interface Depth (m)
Layer
Coefficient of friction μ
Maximum shear
Critical
strength (τmax),
elastic slip
kN/m
(γcrit), mm
2
0-1.5
1
0.73
27
0.040
1.5-6.4
2
0.50
32
0.282
6.4-7.3
3
0.59
39
0.154
7.3-14.6
4
0.50
70
0.418
14.6-20.0
5
0.58
81
0.282
140
1400
120
1200
100
1000
9000 8000
Load, kN
7000 6000
80
800
5000
60
600
4000
I 40 II 20
I II 200 III
50 100 150 200 Displacement, mm
(a) Single flagpole pile
2000
II
1000
III
0
0 0
3000
400
III 0
I
0
15 30 45 Displacement, mm
(b) Single fixedhead pile
0
10
20
30
Displacement, mm (c) 3x3 Pile-group
I: Field test (Stewart J.P.,
I: Field test (Stewart J.P.,
I: Field test (Stewart J.P., 2007),
2007), K=1030 kN/m
2007), K=5.5e+04 kN/m
K=4.9e+05 kN/m
II: Elastic pile (concrete and
II: Elastic pile (excluding
II: Elastic pile (excluding
reinforcement both modelled),
reinforcement), K=5.3e+04kN/m
reinforcement), K=5.9e+05 kN/m
K=1600 kN/m
III: Nonlinear pile (concrete and III: Nonlinear pile (concrete and
III: Nonlinear pile (concrete and
reinforcement both modelled) reinforcement both modelled) pile,
reinforcement both modelled)
pile, K=5.7e+04 kN/m
K=5.9e+05 kN/m
• Modelling reinforcement is
• Modelling reinforcement is NOT
pile, K=1350 kN/m • Modelling reinforcement is necessary. • Incorporating material nonlinerity is necessary.
NOT necessary. • Incorporating material
necessary. • Incorporating material
nonlinerity is NOT necessary.
nonlinerity is NOT necessary.
Figure 4 Load displacement behaviour of pile under lateral load in different configurations from field test and finite element analysis
7.
Observations and conclusions
A methodology for soil-pile interaction problem for static load has been compiled and validated for pile foundation under vertical and lateral load individually. An analytical basis for calculating elastic slip at the pile-soil interface has been introduced. Some of the analysis steps have been suggested for better convergence for this nonlinear solid to solid contact problem. Followings are the some of the conclusive observations: •
Cubic variation of shear stress distribution in soil found to be good representation for estimating elastic slip at soil-pile interface.
•
Boundary condition need to be defined in soil at the soil-pile interface for Geostatic analysis to avoid numerical non-convergence issues.
•
Contact should be initiated in a displacement controlled analysis step for better convergence.
•
Modeling of reinforcements is required for flagpole type piles. For other piles and pile-raft reinforcement modeling is not required for assessing general load displacement behaviour under static loading for general soil sites.
8.
Acknowledgement
We like to thank NPCIL (Nuclear Power Corporation of India Ltd., India) for giving the opportunity to involve in such an interesting developmental work.
Bibliography Anne Lemnitzer, E. R.-T. (2008). Experimental Testing of a Full-Scale Pile Group Under Lateral Loading. The 14th World Conference on Earthquake Engineering. Beijing: WCEE. B, B. (1979). Negative skin friction. Proc. 6th Asian Regional Conf. Soil Mech. Found. Engng. Asian regional conference, (pp. 41-75). Siingapore. Khare M.G., G. S. (n.d.). Behaviour of coated piles under dragload. IIT Madras . Lee C. J., B. M. (2002). Numerical modelling of group effects on the distribution of dragloads in pile foundation. Geotechnique , 325-335. M, T. (1971). Some effects of pile driving on skin friction. conference proc on behaviour of piles (pp. 107-114). London: ICE. Mosher R L, D. W. (2000). Theoretical Manual for Pile Foundations. USA: US Army Corps of Engineers. Naveen B P, S. T. (2011). Numerical simulation of vertically loaded pile. Proceedings of Indian Geotechnical Conference. Kochi: Indian Geotechnical Society. Palmer A.C., W. D. (2005). Uplift resistance of buried submarine pipelines: comparison between centrifuge modelling and full-scale tests. Geotechnique , 338-340. Simulia. (2014). Abaqus users' manual 6.14. Dessults System. Stewart J.P., T. E. (2007). Full Scale Cyclic Large Deflection Testing of Foundation Support Systems for Highway Bridges.Part I: Drilled Shaft Foundations. Los Angeles: Structural and geotechnical engineering Lab., UCLA.
