The 12th International Conference of International Association for Computer Methods and Advances in Geomechanics (IACMAG) 1-6 October, 2008 Goa, India

Consolidation Analysis Using Finite Element Method

Krishnamoorthy Dept. of Civil Engineering, Manipal Institute of Technology, Manipal, Karnataka, India

Keywords: Consolidation, finite element method, non linear, strip footing

ABSTRACT: Consolidation analysis using nonlinear finite element method is performed to study the behaviour of a footing resting on soil mass. Four noded isoparametric plane strain element with two translational degrees of freedom is used to model the soil deformation. Pore pressure is also modeled using four noded isoparametric element. Behaviour of soil is considered as nonlinear and is modeled using the hyperbolic relationship proposed by Duncan and Chang. The displacement of footing and pore pressure in soil are coupled and the resulting equations are solved to obtain the displacement of soil and footing and pore pressure in soil at various time interval. The analysis is used to model the laboratory consolidation test with double drainage. The displacement obtained from the analysis are compared with the displacement obtained from the laboratory consolidation test. The applicability of the analysis is also demonstrated to study the behaviour of a strip footing resting on soil mass.

1 Introduction Consolidation plays an important role in soil engineering problems such as analysis of footings, pile foundations, embankments etc. It has received greater attention after Terzaghi published his consolidation theory and principle of effective stress. Consolidation settlement were obtained in most cases using Terzaghi’s one dimensional consolidation theory. Biot (1941) developed a more general theory for three dimensional consolidation coupling the soil deformation and the pore pressure. Conventional theories for consolidation analysis often neglected the nonlinearity of soil. However, the behavior of soil is nonlinear from the beginning, it depends on stress path and dilatancy and the analysis considering the behaviour of soil as nonlinear is essential to model the consolidation behaviour of soil. Equations of nonlinear consolidation of soil are complex and in order to solve problems of any complexity it is necessray to resort to a numerical approach. Due to the availability of high speed large storage computers, a numerical method known as finite element method became popular due to its versatility and is widely used for solution of engineering problems. Conventional finite element method has proved to be an extremely powerful analytical tool for the solution of many engineering problems. Sandhu and Wilson (1969) formulated analysis for coupled consolidation problem using finite element method. Formulation of the nonlinear consolidation problem was first proposed by Lewis et al. (1976). Hyperbolic stress-strain relationship proposed by Duncan and Chang (1970) was used to model the behaviour of soil. Manoharan and Dasgupta (1995) studied the consolidation behaviour of strip footing modeling the behaviour of soil as elastic – perfectly plastic satisfying the Mohr-Coulumb yield cretirion. In the present work the consolidation behaviour of a strip footing resting on soil mass is analysed using nonlinear finite element method. The behaviour of soil is modeled using a hyperbolic stress-strain relationship proposed by Duncan and Chang (1970). The settlement at centre of the footing and pore pressure with in the soil mass at various time interval are obtained. The analysis is also used to model the laboratory consolidation test. The settement obtained from the analysis are compared with the settlemet obtained from the laboratory consolidation test.

2 Analysis The displacement u and excess pore pressure p within the finite element can be related to nodal displacement vector {un} and the nodal pore pressure vector {pn} as u=[Ns]{un} (1) and p= [Nf]{pn} (2) Ns is the shape functions defining the displacement of the soil element while Nf is the shape functions defining the pore pressure distribution. The elemental equation of consolidation proposed by Zienkiewicz (1977) can be

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expressed in matrix form as

Ks L un 0 0 n.u f

0 H pn + LT 0 .np = 0 (3)

Ks and H are the stiffness and fluid conductivity matrices respectively and L is the coupling matrix which is formed from the equation

L = dsNN fs

Ts

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

∂∂∂∂

∫y

x (4)

2.1 Nonlinear Analysis If Δf is the change in load between successive times, the incremental form of equation 3 can be written as

Ks L Δun 0 0 n.uΔ Δf

(5)

0 H Δpn +

LT 0 n

.pΔ

= 0

Δun and Δpn are the resulting changes in displacement and excess pore pressure respectively. The displacement ui and excess pore pressure pi at the end of ith time step is

ui = ui-1 + Δun

pi = pi-1 + Δpn (6)

Load is applied in increments at each time interval and the corresponding displacement, excess pore pressure and stresses in soil are obtained for each time step. To model the nonlinear behaviour of soil a stress-strain relationship proposed by Duncan and Chang (1970) is used. A tangent modulus of elasticity Et is obtained for each time step using the equation

2

Et = E sinφ2σ2ccosφ

)σφ)(σsin (1R1

3

31f

+−−

− (7)

c and φ are the effective cohesion and angle of internal friction of soil. σ 1 and σ 3 are the effective major and minor principal stresses respectively. The value of Et obtained above for each time step is used to modify the stiffness matrix Ks of soil and is used for next time step.

2.2 Solution of the equation Several procedures were adopted for solving the finite element time dependent problem. Lewis et al. (1976) solved the equation using finite difference approximation while Manoharan and Dasgupta (1993) solved the equation in iterative form using finite difference approximation. In the proposed analysis, the equation 5 in incremental form is solved using Newmarks method. Owing to its unconditional stability, the constant average acceleration scheme (with β = 1/4 and γ = 1/2) is adopted.

3 Example problems The analysis explained above is used to model the laboratory consolidation test with double drainage. The displacement obtained from the analysis at various time interval is compared with the displacement obtained from the laboratory consolidation test. The analysis is also used to study the behaviour of a strip footing resting on soil mass.

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3.1 Modeling of laboratory consolidation test Figure 1 shows the finite element model of laboratory consolidation test. The diameter and height of the soil sample is 38 mm. Drainage is allowed at top surface AC and bottom surface BD of the soil sample and the horizontal displacements are restrained along the sides AB and CD. Vertical displacement is restrained along the bottom surface BD. Material properties considered for the anaysis are modulus of Elasticity E=2.0 kN/m2 and poissons ratio μ = 0. Coefficient of permeability of soil K = 3.9x10-10 m/sec. Time is expressed in terms of a dimensionless time factor T

2d

tvcT =

in which Cv is the coefficient of consolidation and t is the actual time. d is the maximum length of the drainage path. Uniformly distributed load of intensity 50 kN/m2 is applied at top porous stone and is held constant there after. Displacement obtained at centre of the top porous stone and the pore pressure at centre of the soil sample at various time factor T are shown in figure 2.The displacement obtained from the laboratory consolidation test is also shown in the same figure. It can be observed from the figure that the displacement obtained from the analysis agree well with the displacement obtained from laboratory consolidation test. Pore pressure at the beginning of the test is nearly equal to 50 kN/m2 and becomes almost zero at time factor equal to 10

3.2 Consolidation analysis of a strip footing resting on soil Figure 3 shows the finite element descretization of a strip footing resting on soil mass. Since the problem is symmetry, only one half of the soil and footing are considered for the analysis. Footing is considered as smooth and impervious. Half width of the footing considered for the analysis is 3.0 m. Bottom of the model is considered as rough and both vertical and horizontal movements are restrained along the base BD. Horizontal displacements are restrained along the sides AB and CD. Seepage is permitted only along the load free surface and the sides and bottom surface are considered as impervious. Soil is assumed as homogeneous and weightless. The various material properties considered for the analyses are:

Modulus of elasticity of soil = 2000 kN/m2 Poissons ratio of soil = 0.3 Effective cohesion = 10 kN/m2 Effective angle of internal friction = °20 Coefficient of permeability of soil = 0.00001 m/day

O O O O O O

B D 38 mm

O O O O O O O

A C

38

mm

0

0.2

0.4

0.6

0.8

1

0.001 0.01 0.1 1 10 100

Time Factor (T)

Dis

plac

emen

t (m

m) Analysis

Experiment

0

10

20

30

40

50

0.001 0.01 0.1 1 10

Time Factor (T)

Pore

Pre

ssur

e (k

N/m2 )

Figure 2 Variation of pore pressure and dispalcement with time

Figure 1 Finite element model of consolidation test

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Modulus of elasticity of footing = 20000 kN/m2

Poissons ratio of footing = 0.20 Uniformly distributed load of intensity 2.0 kN/m2 is applied in increments at time interval Δt = 2 days until it reaches the maximum of 76 kN/m2 at time t = 38 days ( T= 0.0076) and is held constant there after. Settlement at the centre of the footing and pore pressure in soil at point E and point F are obtained at various time intervals for elastic analysis and nonlinear analysis. Settlement at centre of the footing obtained from elastic and nonlinear analysis is shown in figure 4. Maximum settlement obtained from non linear analysis is considerably more than the maximum settlement obtained from elastic analysis. However, the rate of settlement of the footing after the application of load (T>0.0076) is almost same for both non linear and elastic analysis. Pore pressure in soil at point E and at point F for various time factor T is shown in figure 5. It can be observed from the figure that the pore pressure obtained from elastic analysis increases with increase in time during the application of load, reaches a maximum value at time equal to 38 days (T=0.0076) and then starts decreasing. Pore pressure obtained from non linear analysis is almost same as the pore pressure obtained from elastic analysis at point E where as there is a considerable difference in pore pressure obtained from elastic and nonlinear analysis at point F. Pore pressure at F obtained from elastic analysis is positive at all times wheras the pore pressure at F obtained from non linear analysis increases with increase in load upto the time factor is less than 0.005 and decreases suddenly with increase in load and becomes negative at T=0.0076. It again starts increasing there after and reaches a maximum value at time facor T=0.1 and then starts decreasing with time. It can also be observed from the figure that the maximum pore pressure obtained from nonlinear analysis is more than the maximum pore pressure obtained from elastic analysis. However, the time taken to dissipate the excess pore pressure is almost same for both elastic and nonlinear analysis at point E and F.

30 m

O O O O O O O B

O O O O O O O

D

Figure 3. Finite element idealization of strip footing resting on soil

o

A C

3.0 m

15

m

footing soil

.F

Figure 4. Variation of displacement with time at centre of strip footing

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0001 0.001 0.01 0.1 1 10 100

Time Factor (T)

Settl

emen

t (m

)

NonlinearElastic

E.

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4 Conclusions Nonlinear consolidation analysis using finite element method is developed. Consolidation behaviour of strip footing resting on soil is studied using the above analysis. The analysis is also used to model the laboratory consolidation test. Based on the analysis it is concluded that the settlement of the footing obtained from nonlinear analysis is considerably more than the settlement obtained from elastic analysis. Pore pressure in soil becomes negative at some points in the soil mass and the maximum pore pressure obtained from nonlinear analysis may be more than the maximum pore pressure obtained from elastic analysis. However, the time taken to disscipate the excess pore pressure is almost same for both elastic and nonlinear analysis. Finite element method using appropriate constitutive relation ship to model the behaviour of soil may be used to obtain realistic consolidation behaviour of footing resting on soil.

5 References Biot, M.A. 1941. General theory of three dimensional consolidation, Journal of Applied Physics, 12, 155-164 Duncan J.M., Chang C.Y. 1970. Nonlinear analysis of stress and strain in soils, Journal Soil Mechanics Division, ASCE, 96,

1629 – 1653 Lewis R.W., Roberts G.K., Roberts., Zienkiewicz O.C. 1976. Nonlinear flow and deformation analysis of consolidated problems.

Proc. 2nd Int. Conf. on Numerical Methods in Geomechanics, 1106-1118. Manoharan N., Dasgupta S.P. 1995. Consolidation analysis of elasto-plastic soil, Computers and Structures, 54(6). 1005-1021. Sandhu R.S., Wilson E.L. 1969. Finite element analysis of seepage in elastic media, Journal of Engineering Mechanics, ASCE,

95(3), 641-652. Zienkiewicz, O.C. 1977. The Finite element method, McGraw - Hill Book Company, UK.

Figure 5. Variation of pore pressure with time at point E and point F

-20-15-10-505

1015202530

0.0001 0.001 0.01 0.1 1 10 100

Time Factor (T)

Pore

Pre

ssur

e (k

N/m2 )

NonlinearElastic

05

10152025

303540

4550

0.0001 0.001 0.01 0.1 1 10 100

Time Factor (T)

Pore

Pre

ssur

e (k

N/m2 )

NonlinearElastic

At point E At point F

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