A Simplified Approach for Piles Due to Soil Movement

A Simplified Approach for Piles Due to Soil Movement Metodología Simplificada para Análisis de Pilotes Sometidos a Movimientos del Suelo Wei Dong Guo Griffith University, Gold Coast 9726, Queensland, Australia

Abstract Predicting pile-soil response due to soil movement is generally based on numerical approaches. Although relevant closed form solutions are available, they are not related with quantity of soil movement. In this paper, a correlation between an ‘equivalent load’ and values of soil movement is proposed, which allows the response of the piles due to soil movement to be analyzed readily using the solutions developed for laterally loaded piles. A case study has been provided herein to show the validity and conditions for the current method, where a good comparison for the pile response with the numerical predictions and field measured data is noted.

Resumen La predicción de la respuesta del sistema pilote-suelo basada en el movimiento del suelo se funda generalmente en análisis numéricos. Existen soluciones cerradas relevantes, sin embargo, estas no se relacionan con la cantidad de movimiento del suelo. En este artículo se propone una correlación entre una ‘carga equivalente’ y valores del movimiento del suelo. Esto permite que la respuesta de los pilotes debido al movimiento del suelo pueda ser analizado utilizando las soluciones desarrolladas para pilotes cargados lateralmente. Un caso histórico es presentado en donde se demuestra la validez y las condiciones del método actual. En este caso se logra una buena comparación entre la respuesta del pilote estimada con las predicciones numéricas y los datos medidos en campo.

1 INTRODUCTION The available solutions for predicting pile-soil response due to soil movement are generally based on numerical approaches (Byrne, et. al, 1984; Poulos, et. al, 1995; Chen and Poulos, 1997). These kinds of solutions are, however, normally limited to specialists. In many cases, during preliminary design, an expedite analysis often requires a simple and yet sufficiently accurate approach, e.g. via closed form solutions (Stewart, et. al., 1994; Rajani, et. al., 1995). The existing simple solutions, however, cannot generally account for the quantity of soil movement or predict the detail profiles of a pile response, and are valid only for soil flow around a pile (Fig. 1), e.g. those by Ito and Matsui (1975); De Beer and Carpentier (1977); or for a rigid pilesoil interaction, e.g. by Viggiani (1981). Therefore, a new model seems necessary. In this paper, a concept of equivalent load is proposed to analyse the response of the piles due to soil movement, using the elastic-plastic

solutions developed for piles due to lateral loading (Guo, 2003). Relationship between ‘equivalent load’ and the corresponding soil movement is provided. Case study shows that the current simple solutions offer good predictions when they are compared with either numerical predictions, or the field data. The approach may be modified to predict pile response in other forms of failure modes. 2 PILE RESPONSE DUE TO LOAD 2.1 Coupled Load Transfer Approach Applying variational approach to the coupled pile-soil system (Fig. 1(a)), the following conclusions achieved previously (Guo and Lee 2001) will be directly adopted herein: (1) A pile is defined as ‘infinite long’, should the pile-soil relative stiffness, (Ep/G*) be less than a critical ratio: (Ep/G*)c, where (Ep/G*)c = 0.052(L/ro)4/(1+ 0.75νs), G* = (1+3νs/4)Gs, Gs = soil shear modulus, νs= Poisson’s ratio of the soil; L and ro = pile length and radius, respectively. Ep = Young’s

modulus of an equivalent solid pile. Lateral piles generally behave as if infinite long except for very short ones. (2) Each Winkler spring has a subgrade modulus, k given by 2   3πGs  K1 (γ ) 2   K1 (γ )    2γ −γ  − 1  k=     2 K o (γ ) K (γ )   o  

(1)

where Ki(γ) = modified Bessel function of second kind of i-th order (i = 0, 1). (3) The fictitious tension, Np of the strectched membrane linking the springs is given by

  K (γ )  2   − 1 N p = πr G s   1    K o (γ )    2 o

(2)

(4) The factor γ for a free-head, infinite long pile is given by

γ = k1 (E p G * )

−0.25

(3)

where k1 = 1.0 for a lateral load applied at ground level, and 2.0 for a pure moment loading. For a lateral load applied above ground level, k1 is about 1.0 ~ 2.0. And finally (5) Eqs. (1) and (2) may be readily estimated using a spreadsheet operating in EXCELTM (Guo and Lee 2001) P Plastic zone, x p

Normalised force, p/pu

Plastic zone 1

Transition zone

0.5

Elastic zone

Spring, k

Membrane, Np

Elastic zone

0

0 2 Normalised deflection, w/d

2.2 Limiting Force Profiles The net limiting force per unit length, pu around a laterally loaded pile may be expressed as pu = AL ( x + α o )

2.3 Response Under Lateral Load When in elastic state, the soil is treated as homogenous medium; while the pile-soil relative slip develops, the force on the pile will be estimated by Eq. (4). The governing equations for the pile may thus be written as

E p I p wAIV = − AL ( x + α o ) q

(0≤ x ≤ xp)

E p I p wBIV − N p wB'' + kwB = 0 (0≤ z ≤ L-xp)

(4)

where AL = gradient of the limiting force profile [FL-1-q]; x = depth below ground level [L]; αo = a constant to include the force at x = 0; pu = the limiting force per unit length [FL-1]; q = power of

(5) (6)

where wA = displacement of the pile within the upper zone. Ep, Ip = Young’s modulus, and moment of inertia of an equivalent solid pile. z = x - xp; xp = a transition depth from plastic to elastic state of the pile-soil interaction; wB = pile displacement within the elastic state. Under a lateral load P, exerted at the pile-head level, it has − Q(0) = E p I p w 'A'' (0) = P − M (0) = E p I p w 'A' (0) = 0

Figure 1 Coupled load transfer analysis

q

the sum of the depth x and the constant αo. The depth at which the interface force between the pile and soil just touches the pu, is the critical depth, xp at which the pu is generally higher than that induced above the depth, and it may be taken as the maximum value. Eq. (4) is valid to the critical depth, below which the soil actually behaves elastically. Values of ‘AL’, ‘αo’ and ‘q’ may be directly estimated by matching Eq. (4) with available profiles, or indirectly back-estimated through matching theoretically predicted pile response (e.g. displacement, and maximum bending moment under a given load) with measured one. The back-estimation is actually adopted in the later case studies.

(7)

•

Plastic zone Pile response may be expressed as shown below, such as the force Q(x), the moment M(x), the displacement wA and the rotation ωA induced in the pile at depth x:

− Q( x) = AL (− G (1, x) + G (1,0) + P AL ) −

M ( x)  − G (2, x) + G (2,0) +   =  AL  (G (1,0) + P AL )x 

ωA =

 AL  − G (3, x) + G (2,0) x   + C3 2  E p I p  + 0.5(G (1,0) + P AL )x 

(8) (9)

(10)

wA =

AL EpI p

 − G (4, x) + 0.5G (2,0) x 2     + (G (1,0) + P A ) x 3 6  + C 3 x + C 4   L

(11) where the constants C3, and C4 are determined using the boundary conditions at the depth of transition from elastic to plastic state. C3 =

 4 AL  G (3, x) − G (2,0) x −  kλq −1  0.5(G (1,0) + Pλq +1 AL )x 2 

(12)

− αC 5 + βC 6   x  G (4, x) − G (2,0) −  4A 2 −C x +C C 4 = qL  3 5 3 λ kλ  x  q +1 AL  G (1,0) + Pλ  6   2

(

)

( (

)

'' ' ''  e −αz  wP + 2αwP cos βz +  2  2λ  αwP''' + α 2 − β 2 wP'' sin βz β 

(21)

 wP'' cos βz +   − M B ( z ) = E p I p e −αz  '''  w + αw '' sin βz β  P  P 

(22)

−ωB =

(

(

C6 =

EpI p kβ

((α

2

)

(

Relevant parameters have been shown previously. The variable herein is z rather than x. 3 RESPONSE DUE TO SOIL MOVEMENT Remove this part L1

(14) (15)

where when m ≥ 1, ( x + α o λ ) q+m G (m, x) = (q + m)(q + m − 1)...(q + 1)

(

 − G (2, x) + G (2,0) wP'' kλq = 4λ2   + G (1,0) + Pλq +1 A AL L 

(

  x 

)

Np

β = λ2 −

4E p I p

Np 4E p I p

)

wB =

(

(

) )

3.1 Expressions for Critical Pile Response Lateral load is correlated with the normalised slip depth x , by

(18)

(

)

)

Pλq +1 AL = −G (1,0) +  G (2, x) − G (2,0)    x + α λ  + G (1,0) α λ + 0.5( x + α o λ ) q  1

(24)

3.2 Profiles for Pile Response

(19)

 2αw P''' + 3α 2 − β 2 w P'' cos βz   EpI p    α 2 − β 2 w P'''  sin βz  (20)  k  +    + α α 2 − 3 β 2 w ''  β  P    

(

Stable soil

(17)

Elastic Zone Within elastic state, the pile response may be given by : e

L2

(16)

•

−αz

Equivalent P

(a) The problem (b) The imaginary pile Figure 2 Simplified Analysis of a Pile due to Soil Movement

where x = λxP with λ = 4 k 4 E p I p . And

α = λ2 +

Sliding soil L L2

) )

− β 2 wP''' + α α 2 − 3β 2 wP''

wP'''kλq = 4λ3 − G (1, x) + G (1,0) + Pλq +1 AL AL

)

 wP''' cos βz −    −αz − QB ( z ) = E p I p e  sin z β  (23) ''' 2 ''  (αwP + 2λ wP ) β   

(13) with the constant C5 and C6 being given by EpI p C5 = (2αwP''' + (3α 2 − β 2 )wP'' ) k

) )

To predict the pile response shown in Fig. 2(a), due to soil movement, the pile-soil system may be replaced by an ‘imaginary’ free-head pile embedded in the lower stable layer (Fig. 2(b)). The upper part of the pile shown in Fig. 2(a) in the moving soil is removed and its effect is compensated by an equivalent load, P, thus it is shorter than the real pile by a length of the thickness of the upper layer. Therefore analysis becomes a problem of estimating the load P using a known soil movement profile, which may be

undertaken using the following iterative procedure. With known soil parameters, the pile load exerted at the level of sliding interface may be calculated using the following equation, dAL P= (Ls + α o )q +1 − (xs + α o )q +1 (25) 1+ q

(

)

where Ls = thickness of the moving soil layer; xs = thickness of the soil layer within which the soil displacement is less than the pile displacement, which is dependent on the load P. Thus an iterative procedure is required. Initially, the thickness may be taken as that of the upper moving layer. (1) The estimated load in Eq. (25) is applied at the head of an imaginary free-head pile embedded in the lower stable layer. (2) The displacement wt and rotation θt at the head level (or sliding interface) of the imaginary pile may be calculated using Eqs. (11) and (10), or Eqs. (20), (21). Therefore, the soil movement, ws may be assessed as ws = wt + θ t (Ls − xs ) (26) This is because that the upper part (above slide interface) of the pile in the moving soil is assumed to move rigidly around the head of the ‘imaginary pile’. With the estimated head displacement and -rotation of the imaginary pile, the pile displacement profile in the upper moving soil may be readily estimated; (3) With the estimated pile profile from step (2) and a given soil movement profile, a new thickness of xs may be estimated. If this new depth agrees with the assumed depth of xs within a desired accuracy, the xs is accepted as the real value; otherwise, a new depth, xs may be assumed, the step (1)- (3) is repeated. Normally, only a few times of iterations are needed to obtain the results. The above procedure works well for predicting all the pile response, except for the moment induced in the pile. Since due to the dragging of the upper moving layer, the moment in the pile at the interface of sliding may be not zero, as noted in field tests (e.g. Esu and D’Elia, 1974). This means that the assumption adopted in Fig. 2(b) may be slightly different from a real case. A possible solution for this is to set the imaginary pile-head at a depth of zero moment rather than at the sliding surface. The depth of the zero moment, xo may be readily estimated, for instance, using the following equation derived from a rigid pile at q = 0. Because the moment is

generally independent of the pile rigidity, as long as the plastic response dominates. xo =

 P Ls 1 − AL1 dLs 2 

2 −1

  

(27)

3.3 Pile Failure Modes The mobilised force on a pile depends on the slip depth thus the pile rigidity. However, modes of pile failure are normally controlled by the limiting bending moment (e.g. Viggiani, 1981), which is independent of pile stiffness. Therefore, relevant failure modes may be determined using the results for rigid piles, as long as slip is fully developed along the pile-soil interface. The following failure modes may be observed similar to those for a rigid pile (Viggiani, 1981). (a) Mode A, the pile-soil interaction attains the yield value only below the slip surface; the whole pile translates together with the sliding soil, ripping the underlying stable soil. (b) Mode B, soil failure occurs both above and below the slip surface; the whole pile undergoes a rigid rotation. (c) Mode C, the pile fixed in the stable soil and the sliding soil flows around it. The current solution is proposed for Modes A and B, although it may be modified for Mode C. In the case of constant limiting force (q = 0), it was shown that (Viggiani, 1981), Mode B occurs, if

λF

2 + 2λ F − 1 1 + 2λ F

≤

Ls ≤ λ F + 2λ F + 2λ2F L2

(28)

where λF = AL1/AL2, AL1, AL2 = the limiting force (for the case of q = 0) on the pile for the upper moving and the lower stable layer, respectively. Ls, L2 = the length of the pile embedded in the upper moving soil and the lower stable layer, respectively. Expression for q > 0 are much more complicated, thus they are not pursued herein. 3.4 Determination of Parameters The expressions for the following critical state may be obtained from Eq. (4): The depth for zero moment, xo; the depth, xmax at which the maximum moment occurs, and the maximum P bending moment, M max as well. If a measured moment profile is available, the three known parameters allow the parameters (αo, q, and P) to be estimated as well.

4 CASE STUDY

Pile deflection (mm) - 10

10

30

50

Depth (m)

0 5

(a)

Measured Poulos Winkler model Current prediction

10 15

Shear force (kN ) -250

0

250

500

Depth (m)

0 5 10

(b)

15

Pile rotation (rad) -0.015 -0.01 -0.005 0

0.005

Depth (m)

0 5 (c)

10

Esu and D’Elia (1974) described a field test where a reinforced concrete pile was installed in a sliding slope consisting mainly of clay. Lateral movement of the slope took place in the upper 7.5 m layer. The pile was instrumented to determine the shear forces, bending moments, deflections and rotations. The test pile was 0.79 m in diameter with a length of 30 m. The bending stiffness of the pile (EpIp) was 360 MNm2. Poulos (1994) analyzed this problem using boundary element approach, and the following assumptions: (1) The soil Young’s modulus was assumed to increases from zero at the surface to 16 MPa at the pile toe; and (2) The soil movement was assumed to be 110 mm, and uniformly distributed from the ground surface to the slide surface (i.e. 7.5 m below the ground surface). The undrained soil strength cu is about 40 kPa, and the values of the lateral limit pressure was assumed as 120 kPa and 320 kPa for the upper moving soil layer and the lower stable layer. The results of the analysis were plotted together with the measured data, as shown in Fig. 3. In current analysis, with the above-mentioned data and the parameters given in Table 1, analysis was performed for the pile in soil of three different values of q. For instance, in the Case I, the equivalent load exerted on the pile at the sliding surface was calculated to be 456.0 kN. Table 1 Input and Output for the case study

15

Depth (m)

Bending moment, kN-m -1000 - 750 -500 -250 0 -1 4 (d)

9 14

Figure 3 Comparison among various predictions and the measured data

Case q AL(kPa/mq) αo (m) Gs (MPa) xs (m) xp (m) P (kN) wt (mm) θt (×10-3)

I 0 120 0 22.4 4.81 6.01 456.0 49.5 -12.64

II 0.5 51 0 28.6 3.25 6.34 324.6 63.0 -14.0

III 1.0 25 0 14. 2.325 6.09 291.01 74.1 -15.5

The measured data (Fig. 3) demonstrates that the upper part of the imaginary pile moves rigidly, thus, the part of the pile in the moving soil should rotate rigidly against the head of the imaginary pile as stipulated previously. In view of this finding, the displacement profile for the part in moving soil is readily calculated. Comparing this pile displacement profile with the input soil movement profile of 110 mm uniformly within the upper moving soil, only at the part where the soil

displacement is less than the soil movement offers resistance to the sliding. Thus, the length of the active part, xs was estimated to be 4.81m, with the equivalent load of 456.0 kN for q = 0. This length, and the equivalent load satisfy Eq. (25), thus the calculated results are final. The xo is evaluated approximately to be 1.10 m using Eq. (27). Thus, the depth of zero moment may be taken as 1.1 m above the sliding surface, which matches the measured data very well. With the above estimated equivalent load, the response of the ‘imaginary’ pile in the lower stable layer can be calculated readily using the current solutions. The calculated deflection, shear force, rotation and the bending moment profiles for the imaginary pile are illustrated in Fig. 3(a – d), together with the measured data. It shows that the current prediction compares well with the measured data. In the current analysis, taking ‘Np = 0’ in Eq. (19) for calculating the factors, α and β, the analysis will be reduced to an analysis based on ‘Winkler model’. This analysis has also been shown in the Fig. 3. As shown previously, for a lateral pile in a homogenous soil, the limiting force is not constant with depth. The effect of the distribution of the limiting force on the prediction is explored, using the data given in Table 1. Some results are provided in the table as well. It demonstrates (not shown herein) that using a value of q = 1, the shear force is better predicted comparing with the measured, but the maximum bending moment is underestimated slightly. The upper part of the pile in the moving soil may be treated as a rigid pile, for which relevant prediction has been elaborated previously by Viggiani (1981), thus it is not repeated herein. 5 CONCLUSIONS

In this paper, elastic-plastic solutions are developed, based on a theoretical load transfer approach proposed recently by the author. The solutions can be reduced to the available solutions for some simple cases. With the solutions, a new approach is proposed for analysing slope induced pile response. The key for the approach is the pile failure mode. For the generalised failure mode, an example study shows that the current simple solutions offer good predictions when compares with either numerical predictions or the field data. ACKNOWLEDGMENTS

The work reported here forms part of the research into displacement-based beam analysis due to soil movement undertaken in Griffith

University currently sponsored by Australian Research Council through the Discovery Project (DP0209027). This financial assistance is gratefully acknowledged. REFERENCES De Beer, E. and Carpentier, R. (1977). "Methods to estimate lateral force acting on stabilising piles.” By Ito, T., and Matsui, T. (1975). Soil and Foundations 16 (1), 68-82. Byrne, P. M., Anderson, D. L. and Janzen, W. (1984). "Response of piles and casings to horizontal free-field soil displacements." Can. Geotech. J., 21(?), 720-725. Chen, L. T., and Poulos, H. G. (1997). "Piles subjected to lateral soil movements." J. of Geotech. & Geoenviron. Engrg. Div., ASCE, 123(9), 802-811. Esu, F., and D’Elia, B. (1974). "Interazione terreno-struttura in un palo sollecitato dauna frana tipo colata." Rivista Italiana di Geotechica., 111, 27-38. Guo, W. D. (2003), ‘On critical depth and lateral pile response’ J. of Geotech. And Geoenviron. Engrg. Div., ASCE, (tentatively accepted, June 11, 2002). Guo, W. D., and Lee, F. H. (2001). "Theoretical load transfer approach for laterally loaded piles. " Int. J. Num. & Analy. methods in Geomechanics, 25(11): 1101-1129. Ito, T., and Matsui, T. (1975). "Methods to estimate lateral force acting on stabilising piles.” Soil and Foundations 15 (4), 43-59. Poulos, H. G., Chen, L. T., and Hull, T. S. (1995). "Model tests on single piles subjected to lateral soil movement." Soil and Foundations 35 (4), 85-92. Rajani, B. B., P. K. Robertson, and N. R. Morgenstern (1995). "Simplified design methods for pipelines subject to transverse and longitudinal soil movements." Can. Geotech. J., 32(?), 309-323. Scott, R. F. (1981). Foundation analysis. Prentice Hall, Englewood Cliffs, N. J. Stewart, D. P., Jewell, R. J., and Randolph, M. F. (1994). "Design of piled bridge abutments on soft clay for loading from lateral soil movements." Geotechnique, 44 (2), 277-296. Viggiani, C. (1981). "Ultimate lateral load on piles used to stabilise landslide." Proc. 10th Int. Conf. Soil Mechanics and Foundation Engrg., 3, Stockholm, Sweden, 555-560.