Pile Soil Interaction—Moment Area Method

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PILE SOIL INTERACTION—MOMENT AREA METHOD

D.M. Dewaikar Professor, Department of Civil Engineering, IIT Bombay, Mumbai–400 076, India. E-mail: dmde@civil.iitb.ac.in P.A. Patil Former Postgraduate Student, Department of Civil Engineering, IIT Bombay, Mumbai–400 076, India. E-mail: m3poonam@civil.iitb.ac.in J. Dixit Research Scholar, Department of Civil Engineering, IIT Bombay, Mumbai–400 076, India. E-mail: p9jagabandhu@civil.iitb.ac.in

ABSTRACT: Pile foundations of fixed offshore structures are subjected to high magnitudes of lateral loads and moments. The analysis of laterally loaded piles is a complex soil-structure interaction problem. This paper presents pile-soil interaction for laterally loaded pile through the hyperbolic p–y curves developed for soft clay coupled with moment area method. The proposed p–y model is validated by analyzing a long, flexible, unrestrained vertical pile in soft clay subjected to static lateral loads. In the moment area method, the pile is discretized into small segments of equal length. A node is considered at the center of each segment and the soil reaction is assumed to be uniformly distributed over the segment. This method is used to estimate the pile displacements and moments for different applied horizontal loads and moments without using any finite difference approach. The results obtained using the proposed p–y curves are comparable to earlier analytical and field test results. 1. INTRODUCTION

Pile foundations of fixed offshore structures are often subjected to a variety of lateral loads and moments of considerable magnitudes due to environmental loading such as wind loads and wave loads on the supported structure in addition to the axial loads. Analysis of pile foundation under the action of vertical loads is performed as per the conventional methods. However, analysis of laterally loaded piles in order to design for overall stability and safety of structure is a complex soil-structure interaction problem due to inherent nonlinear behaviour of soil. All external forces and moments applied to the pile-soil system are introduced at the top of the pile at one point. The reactions that are generated in the soil are taken consistent with the pile deflections and must satisfy the static equilibrium condition. The response of a pile to lateral load depends on stiffness of pile, its length and size, and stiffness of soil along the depth. Flexural stresses, axial stresses and bending moments in the pile must be reliably predicted to determine required pile sizes. Pile-soil interaction is considered here through moment area method. This method for the analysis of laterally loaded pile is coupled with hyperbolic p–y curves for soft clay.

2. MOMENT AREA METHOD

Sawant & Dewaikar (1994) introduced moment area method for the analysis of pile-soil interaction. This method is combined with sub-grade reaction approach and equations are generated to obtain soil pressures, pile displacements and

moments. Its main advantage is that it does not require the use of finite difference method. Sub-grade reaction approach treats the laterally loaded pile as a beam interacting with surrounding soil. The soil behaviour is modeled on the basis of Winkler’s hypothesis, in which the pressure p and deflection y at a point are assumed to be related through the modulus of sub-grade reaction Es. Thus,

.sp E y= (1)

The moment area method is a matrix approach, in which the pile is discretized into small segments of equal length and a node is considered at the center of each segment as shown in Figure 1.

Fig. 1: Discretization of a Pile

IGC 2009, Guntur, INDIA

Pile Soil Interaction—Moment Area Method

915

The soil reactive pressure is assumed to be uniformly distributed over each segment (Fig. 2). The p–y curve for lateral soil reaction, p per unit length along the pile as a function of corresponding horizontal deflection, y is developed for each node (Fig. 1). The p–y curves give reasonable estimates of the deflection and bending moments when used for modelling the lateral pile behavior. Equations are formulated using conventional bending theory of beam for the evaluation of nodal displacements including the displacement and rotation at pile head. Secant modulus approach (Fig. 3) is used for analysis in which the applied predetermined load is analyzed repeatedly. First, the analysis is carried out with an assumed value of soil modulus; usually the initial tangent modulus values (Fig. 3). In the next analysis, the assumed value of soil modulus Es is revised to become consistent with the evaluated deflection. The procedure is repeated until the calculated deflections between two successive analyses vary within a permissible limit. The reactions on a pile and the corresponding displacements are shown in Figure 2. In Figure 2, each element is acted upon by a uniform horizontal pressure, p assumed constant. The jth element is assumed to be acted upon by a uniform horizontal stress pj across the pile width.

Fig. 2: Pile-Soil Interaction

The pile is considered as a single free headed floating pile in the form of a thin rectangular strip of width (diameter) d, length L and flexural rigidity EI.

Soil reaction matrix [p] is related to the nodal displacement matrix [yp] for the estimation of soil displacement as,

[p] = [Es]×[yp] (2)

Palmer & Thompson (1948) method is widely used for the estimation of Es, which is expressed as,

m

s sLx

E EL

= × (3)

Where, EsL is the value of Es at pile tip (at x = L) and m is an empirical index dependent on soil type, which is taken as 1.0 in the present analysis.

Hence, the diagonal matrix for variation of modulus with depth for soft clay, [Es], can be obtained in a iterative manner (Fig. 3) defined as,

0.5( , )

m

s sLi

E i i En

− = × (4)

Fig. 3: Iterative Analysis Using a p–y Curve

The tangential deviation (ytAi) of node point i with respect to tangent at top point A is given by the moment about point i of moment area diagram between A and node point i (Fig. 4). The bending moment diagrams due to applied load H, moment M and soil pressures are shown in Figure 5.

Fig. 4: M/EI Diagram of a Pile Segment

The tangential deviations (ytA) of nodal points with respect to tangent at top point A are given by,

[ ][ ] [ ]4 3

tAd H

y X p BEI EIδ δ

= + (5)

Where, d = L/n and [ytA] represents tangential deviation matrix. [X] and [B] are non-dimensional matrices for pile

(a) Soil reactions on pile (b) Pile displacements

Pile Soil Interaction—Moment Area Method

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deflection due to soil pressures (internal loads) and applied external loads respectively. The first and second terms in Eqn. 5 are due to soil pressures and external loads respectively. Figure 5 shows the bending moment diagram by parts comprising moments due to soil pressures and external loads. The diagonal elements X (i, i) of pile deflection due to soil pressures are given as,

X (i, i) = –1/384, i = 1, n (6)

Fig. 5: Bending Moment Diagram by Parts

The upper triangular elements of this matrix are zero and lower triangular elements are given as,

2 31( , ) 2 4(2 2 1) 3(2 2 1) (2 2 1)

48X i j i j i j i j

− = + − − + − − + − − (7) Where, j = 1, n–1 and i= (j+1), n

The matrix B (i) due to applied loads is given as,

3 21( ) (2 1) (2 1)

48 8e

B i i i= − + −δ

(8)

Where, e = M/H (eccentricity of lateral load) M = applied moment at top and H = applied horizontal load at top

If ? and ? are displacement and rotation of pile head, the displacement matrix, yp at any point i (Fig. 6) can be written as,

[ ] [ ] [ ]1p tAy y U L UL = + ∆ − θ (9)

Where, [U1] represents a column matrix with all elements equal to unity and [UL] represents a column matrix, with elements,

0.5( )

iUL i

n−

= (10)

After calculating the tangential deviation of pile at point i (located at a depth z) with respect to top point A, the lateral displacement yp of the pile at point i (Fig. 6) is calculated. Substituting the matrix [ytA] and simplifying the equations, following matrix equation is obtained.

[ ][ ] [ ] [ ] [ ]4 3

1s pd H

I X E y U L UL BEI EI

δ δ − − ∆ − θ = (11)

[I] represents a unit matrix. The above equation provides n equations for n + 2 unknowns, i.e. y1 to yn, ? and ?.

Fig. 6: Lateral Pile Displacements

In addition to above n equations, other two equations are required to be generated to obtain the solution for n+2 unknowns using equilibrium conditions, namely horizontal equilibrium and moment equilibrium.

Horizontal equilibrium equation can be expressed as,

[ ] [ ]1 2 3 ... n s pH

p p p p E y Bd

+ + + + = = δ (12)

In the above equation, [Es] is a row matrix for modulus of sub-grade reaction as given by Es (j) = Es (j, j) and it can also be expressed as,

[ ]3

1s pH

E y aEIδ × = (13)

Where, a1 is expressed as,

1 4EI

ad

=δ

(14)

The moment equilibrium equations can be expressed as,

1 2( 0.5) ( 1.5) ... 0.5 nH e

n p n p p nd

− + − + + = + δ δ (15)

[ ] 1M pH e

K y a nd

= + δ δ (16)

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Where, [KM] is a row matrix expressed as,

( ) ( 0.5) ( , )M sK j n j E j j= − + (17)

Hence, for the computation of y1 to yn, ? and ?, following matrix relation can be written.

[] [ ][ ] [ ] [ ][ ]

[ ]

[ ]4

1

1

1

0 00 0

( )

sn

s

M

dI X E U L ULy BEI

E aK e

a n

δ − − ∆ = θ + δ

(18) The moments can be subsequently evaluated after the computation of soil pressures, pile top displacement, ? and rotation, ?. Moment at any node point can be computed by following expression,

[ ] [ ][ ] [ ][ ]2MZ d CM p H CP M= δ + δ (19)

Where, [MZ] = moment at a node point. [CM] is the moment coefficient matrix (lower triangular) for soil pressure loading as given below,

[ ]

0.1251.0 0.1252.0 1.0 0.125

0.125( 1) ( 2) ( 3) 1.0 0.125

CM

n n n

− − − = − − − − − − − − − − − −

(20) [CP] is the column matrix for node point i. The elements of [CP] are given as,

() 0.5CP i i= − (21)

Poulos (1971) analyzed a floating pile in a soil with linearly increasing soil modulus subjected to a lateral load and a moment and presented a detailed parametric study to bring out major factors such as the effects of length to diameter ratio (L/d) and flexibility factor KR that influences local yielding for relatively flexible piles. The load displacement relation was found to be influenced by the distribution of yield resistance of soil along the pile. The pile flexibility factor is given as,

5p p

Rh

E IK

L=

η (22)

Where, EPIP is the stiffness of the pile and ?h represents the coefficient of soil modulus variation with depth.

The displacement of pile (Fig. 6) at the ground surface is expressed in terms of dimensionless influence factors. The lateral displacement (y) and rotation (?) for the free headed pile are given in terms of influence factors as,

2 3tA yH yMh h

H My I I

L L= +

η η (23)

3 4tA H Mh h

H MI I

L Lθ θθ = +

η η (24)

IyH, IyM = elastic influence factors for displacement caused by applied horizontal load and moment respectively. I?H and I?M are the respective influence factors for rotation.

3. HYPERBOLIC P–Y CURVES FOR SOFT CLAY

According to Matlock (1970), a p–y relationship is influenced by variation of soil properties with depth, the general form of the pile deflection, load history, state of stress and the corresponding strain. The analysis of a complex pile-soil interaction is reduced at each depth to a simple p–y curve. A hyperbolic model is proposed here for development of p–y curves for soft clay. The p–y curves relate unit soil resistance to pile deflection. The slope of p–y curve at any deflection represents tangent soil stiffness at that deflection. However, the ratio p/y at any deflection represents secant soil stiffness at that deflection. The unit soil resistance results from the mobilization of strength of soil surrounding the pile.

Offshore piles are generally large and driven into soils that exhibit highly non-linear stress strain behaviour, even at low levels of applied loads. Use of nonlinear p–y curves to represent static soil resistance is the common approach for analyzing the response of laterally loaded piles. A simple and improved method for the construction of hyperbolic p–y curves for soft clay is proposed here for static loading conditions. The initial stiffness of p–y curve is taken to be increasing with depth which depends entirely on the soil properties and ultimate lateral resistance of soil according to the expression suggested by Poulos & Davis (1980).

s hE z= η (25)

Where, z represents depth below ground surface and ?h is coefficient of soil modulus variation with depth.

A set of p–y curves for soft soil are generated for static loading conditions using following three methods.

3.1 Method A—Matlock (1970) p–y Curves

For soft clay, Matlock’s method (1970) for construction of p–y curves for a chosen depth is expressed as,

13

500.5

u

p yp y

= ×

(26)

Where, p = soil pressure (kN/m) at a displacement, y. pu = ultimate soil pressure (kN/m), dependent upon

chosen depth, pile diameter and un-drained cohesion of clay.

y50 = deflection at one half the ultimate resistance corresponding to 50% of deviator failure stress in an un-drained tri-axial test.

Matlock’s p–y curves are used for analysis of pile loads as shown in Figure 7.

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Fig. 7: p–y Curve for Soft Clay under Static Load

(Matlock 1970)

3.2 Method B—Georgiadis et al. (1992) p–y Curves

The p–y curve as proposed by Georgiadis et al. (1992) is shown in Figure 8. The method for construction of p–y curves by this approach is expressed as,

1

u

yp

yk p

=+

(27)

Where, parameter pu is ultimate soil pressure, dependent upon chosen depth, pile diameter and un-drained cohesion of clay. k is initial stiffness of the p–y curve expressed as (Vesic 1961),

412

20.65

1s sE d E

kEI

= ×− µ

(28)

Where, Es and µ are the Young’s modulus and Poisson’s ratio of soil, d is pile diameter and EI is flexural rigidity of pile section.

Fig. 8: p–y Curve for Soft Clay under Static Load

(Georgiadis et al. 1992)

3.3 Method C—Proposed Hyperbolic p–y Curves

Another method for construction of p–y curves for a chosen depth as proposed by Dewaikar & Patil (2001) is expressed as,

1

s u

yp

yE p

=+

(29)

pu = ultimate lateral soil resistance (Matlock 1970) Es = initial stiffness of p–y curves increasing with depth (Eq. 25)

The coefficient of soil modulus variation with depth, ?h is generally taken in the range 160–3450 kN/m3 (Reese and Matlock 1956).

The hyperbolic p–y curves proposed here for soft clay are shown in Figure 9 and these curves are coupled with moment area method for the analysis of laterally loaded pile.

Fig. 9: Proposed p–y Curve for Soft Clay

Under Static Load

4. RESULTS AND DISCUSSIONS

The results obtained using the proposed hyperbolic p–y curves coupled with moment area method are compared with field test (Matlock 1970) and analytical results (Matlock 1970) as shown in Figures 10 to 12. The results computed using methods A and C are in close agreement with Matlock’s both

Fig. 10: Comparison of Moments at various Depths of

Embedment (P = 35.6 kN)

Fig. 11: Comparison of Moments at various Depths of

Embedment (P = 80.07 kN)

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Fig. 12: Comparison of Load vs. Maximum Moment

field and analytical results. The values computed using method B show a divergence at higher embedment depths from the field test results. At the point of maximum moment, the results obtained using methods A, B and C show a good agreement with field test results with a difference within 0.5%, 3% and 1% respectively.

The moments developed along the pile length for applied horizontal loads and moments for a pile in a soil with linearly increasing soil modulus are computed using moment area method (Figs. 13–14). These results are compared with those obtained using Poulos’s (1971) solution as shown in Figures 13–14. A close agreement is seen between the two, which establishes the validity of the proposed method.

5. CONCLUSIONS

This paper presents an iterative scheme in which, hyperbolic p–y curves for soft clay are coupled with moment area method for the analysis of a laterally loaded pile. The distinguishing feature of this method is that it requires no finite difference equations. The proposed p–y model is validated by analyzing a long, flexible, unrestrained vertical pile in soft clay subjected to static lateral loads. The predicted results are found to be in close agreement with field test results and analytical results reported by Matlock (1970) and Poulos (1971).

Fig. 13: Comparison of Moments Along the Pile Length for

Applied Horizontal Load

Fig. 14: Comparison of Moments Along the Pile Length for

Applied Moment

REFERENCES

Dewaikar, D.M. and Patil, D.S. (2001). “Behavior of Laterally Loaded Piles in Cohesionless Soil under One-Way Cyclic Loading”, The New Millennium Conference, 1, 97–100.

Georgiadis, M., Anagnostopoulos, C. and Saflekou, S. (1992). “Cyclic Lateral Loading of Piles in Soft Clay”, Journal of Geotechnical Engineering, ASCE, 23 (GT1), 47–59.

Matlock, H. (1970). “Correlation for Design of Laterally Loaded Piles in Soft Clay”, 2nd Annual Offshore Technology Conference, Houston, Texas, USA, OTC 1204(1), 577–595.

Palmer and Thomson (1948). “The Earth Pressure and Deflection Along the Embedded Lengths of Piles Subjected to Lateral Thrust”, Proceedings 2nd Int. Conference on Soil Mechanics and Foundation Engineering, Rotterdam, 5, 156–161.

Poulos, H.G. (1971). “The Behavior of Laterally Loaded Piles I: Single Piles”, Journal of Soil Mechanics and Foundation Engineering Division, ASCE, 97(SM5), 711–731.

Poulos, H.G. and Davis, E.H. (1980). Pile Foundation Analysis and Design, John Wiley and Sons, New York.

Reese, L.C. and Matlock, H. (1956). “Non-Dimensional Solutions for Laterally Loaded Piles with Soil Modulus Assumed Proportional to Depth”, Proceedings 8th Texas Conference on Soil Mechanics and Foundation Engineering, Special Publication No. 29, Bureau of Engineering Research, University of Texas, Austin.

Sawant, V.A. and Dewaikar, D.M. (1994). “Analysis of Laterally Loaded Pile by Moment Area Method”, National Seminar on Numerical and Analytical Methods in Geotechnical Engineering: Developments, Applications and Future Trends, IIT Delhi, II, 3.1–3.9.

Vesic, A. (1961). “Bending of Beams Resting on Isotropic Elastic Solid”, Journal of Engineering Mechanics Division, ASCE, 87 (EM2), 35–53.