Soil Interaction Of Laterally Loaded Free Head Long Pile Embedded In Layered Sand Medium

Soil Interaction Of Laterally Loaded Free Head Long Pile Embedded In Layered Sand Medium

Sumit Kumar Biswas 1, Prof. S. P. Mukherjee 2, Prof. S. Chakrabarti 3, Moyukh De 4

1. Assistant Professor of Civil Engineering, Jadavpur University, Kolkata-32, India

2. Professor of Civil Engineering, Jadavpur University, Kolkata-32, India

3. Professor of Civil Engineering, Jadavpur University, Kolkata-32, India

4. Assistant Engineer, PHED, Government of West Bengal, India

Abstract

In this paper, some aspects of behavior of laterally loaded free head single pile in cohesionless soil (with both homogeneous and layered sand beds) have been studied based on experiments and numerical methods. The experimental results are supplemented by adopting numerical techniques. The analysis has been done by PLAXIS-3D Foundation version 2.1 software for single pile with variation of slenderness ratio and embedment ratio of pile (ratio of embedded length of pile in a particular soil layer to the pile length) and also stiffness of soil layers. The results are found to agree very well with the experimental results. Further a numerical study has been carried out by finite difference approach to evaluate the coefficient of modulus of horizontal sub-grade reaction (h). Its variation with other soil-pile parameters, such as slenderness ratio and embedment ratio of pile and relative density of sand (used as foundation medium) were also studied.

Key words: Laterally-loaded pile, layered soil- medium, finite difference, finite element, modulus of sub-grade reaction

1. Introduction

   In civil engineering practice piles are subjected to a variety of loading conditions due to earthquake, wind, sea wave and the like. Thus it is understood that response of piles under lateral loading has a great importance in analysis and design of piles. Analysis of lateral resistance of vertical piles with non dimensional relative stiffness factors has been done to predict the behavior of piles (Reese and Matlock 1956). Elastic analysis has been extended to piles (Davisson and Gill 1963) and subsequently design charts for prediction of lateral response of piles using theoretical and experimental studies were created (Broms 1965). Biswas et al (2012, 2013) carried out an experimental and numerical analysis using cylindrical hollow model

   cast iron piles of slenderness ratio of 15, 20 and 25 respectively, where the experimental model test results were found to be in close proximity (within 10%) with the 3D finite element models which were created in PLAXIS 3D FOUNDATION version 2.1 software. The PLAXIS 3D results pertaining to piles of higher slenderness ratios (30, 42, 50 and 55) were subsequently used to find the modulus of horizontal sub-grade reaction (h) with relevant finite difference formulations.

   With this in view, a numerical study has been carried out in which the value of h and its normalized form that is the flexibility ratio (Barber 1953) is found out for laterally loaded piles in homogenous and layered cohesionless soil medium and their variation with other soil-pile parameters, such as relative density, slenderness ratio of pile for homogenous soil and ratio of top layer thickness to embedded pile length for Layered Soil. The soil intrinsic parameters and material properties of cast iron (material used in hollow cylindrical model pile) were taken accordingly to those considered by Biswas et al (2012, 2013).

2. Numerical Study

   The numerical study was carried out by PLAXIS 3D finite element software. Variation of deflection and moments of pile at different depths were obtained for different 3D finite element models. Further finite difference formulations were used to evaluate the co- efficient of modulus of sub grade reaction (h) from the above results. (Biswas et al 2013)

   Using PLAXIS-3D same results would have been obtained indirectly by evaluating the stress at the Gaussian integration points or stress points as done earlier by (Kim and Jeong 2011). However, this would have taken a larger computational time than the method adopted for the present study. Thus the numerical part of the study comprises of two analyses:

   1. Finite Element Analysis by PLAXIS-3D Foundation version 2.1 software.

   2. Application of Finite Difference Method on data obtained from PLAXIS Analysis.

1. Soil modelling by PLAXIS 3D

   The whole soil continuum has been divided into a number of 15-noded wedge elements to form the 3D finite element mesh. The 15-noded elements were constructed with 8-noded quadrilaterals in vertical direction and 6-noded triangles in horizontal direction. Consequently interface elements were modeled as 16- node (consisting of eight pairs of nodes) interface elements. The depth of soil continuum was taken as 1.7 times the length of the pile (Kim and Jeong 2011) for numerical modelling (Fig1). These dimensions were considered adequate to eliminate the boundary effects on the performance of pile.

   Fig1 Schematic diagram of the model

   To model the behaviour of soil surrounding the pile, Mohr-Coulomb failure criteria have been used. This elasto-plastic model depends on the soil parameters, friction angle ´ and cohesion intercept c´. Young's modulus E´, change in elastic modulus with depth Eincrement and Poissons ratio ´. These parameters helped to obtain stress-strain behaviour of the soil and they were taken according to Biswas et al. (2012). The pile was modeled with linear elastic elements following Hooke's law. The linear elastic model included two elastic stiffness parameters viz. Young's modulus E and Poisson's ratio which were also considered from Biswas et al. (2012). A total number of 54 cases of 3D Finite Element models were considered for analysis with variation in slenderness ratio and relative density for sand and ratio of top layer thickness to embedded pile length for Layered Soil exist.

2. Finite Difference Method

From the analysis of PLAXIS 3D models the variation of moments with respect to the depth of pile and the displacement profile of the pile for gradual increment of horizontal load are obtained in each case of analysis. On these results the finite difference approach is further infused to obtain horizontal modulus of sub-grade reaction (h). The process has been illustrated as follows:

The pile was divided into number of nodes (n) having the node to node length of h units; and the boundary conditions used in this study [Poulos and Davis (1980)] are briefly outlined below.

1. For long floating free head pile the value of end moments that is M1 (moment at extreme top of the pile) and Mn+1 (moment at extreme bottom end of the pile) will be equal to zero.

2. It is known that the soil induced stress in cohesionless soil medium varies with the depth as the node 1 is situated at the ground level so it can be assumed that the soil reaction per unit length of the pile at node 1 will be negligible, so w1=0 (as no overburden is present).

   Another boundary condition incorporated in the present analysis is furnished below:

3. For pile under consideration at the pile head shear force at the node 1 will be equal to the horizontal load acting on it.

The coefficient of modulus of subgrade reaction kh having units of force/lengtp at a node n can be obtained from the expression given below.

(1)

The coefficient modulus of sub-grade reaction kh was obtained from Equation (1) through computer program developed for the purpose which was coded in C++ programming language. The flowchart provided by Biswas et al. (2012) was followed during the writing of computer codes.

The value of kh was calculated for each incremental load up to the assumed fixity point (Chin et al. 2010). The fixity point was calculated for each incremental load corresponding to a depth where horizontal deflection was negligible.

It was observed that the values of kh obtained from finite difference calculations were found in close proximity (at a variation of 10%) with those obtained indirectly from PLAXIS 3D. The proximity was further increased by increasing the number of nodes in which the pile was divided.

2.4 Failure Load Criteria

Both for experimental and PLAXIS models a common failure criterion was considered which is explained as follows.

The value of kh corresponding to each depth down to the fixity point for each incremental load was calculated to the failure load. (Reese and Matlock, 1960) proposed an empirical equation to calculate horizontal pile head deflection corresponding the ultimate soil resistance of the pile. The equation is given as:-

(2)

Where, b = Width of the pile (in inch) and Yu

= Ultimate deflection (in inch).

As the above deflection corresponds to ultimate soil resistance the load corresponding to it has been taken as the ultimate failure load for the present study.

3( 24 )

Y 25.4

In all the cases of homogeneous and layered soil systems it is observed that kh increases with depth and the variation is linear. The curves appear to be much dispersed from the average straight line at larger depth. From the average straight line the coefficient of modulus of horizontal sub-grade reaction for sand h is calculated in accordance with the method suggested by Poulos and Davis (1980):

(3)

Where, h is coefficient of modulus of subgrade reaction in case of sand (Force/lengtp), z is depth of the pile (length) and B is breadth of the pile or

the diameter of the pile (length), which in this case is the pile diameter.

5.2 Variation of Coefficient of Horizontal Modulus of Sub-grade Reaction (h) With Soil-Pile Parameters

The variation of coefficient of horizontal modulus of

So,

u 80

= 0.0354 inch = 0.9 mm

sub-grade reaction (h) is studied with respect to the following parameters both for homogeneous and layered soil:-

(This ultimate deflection accounts about 3.75% of the pile diameter)

1. Results and Discussion

   5.1 Soil interaction parameters: kh and h

   From the results of PLAXIS 3D model analysis of all the 54 cases an attempt has been made to study the variation of the soil interaction parameter for the cohesionless soil medium with different soil-pile parameters.

   Main soil interaction parameter is the horizontal sub-grade reaction, K (force/lengtp), however the ratio of the former with the diameter or width of pile is known as coefficient of horizontal sub-grade reaction, kh (force/lengtp). Further the product of kh with the ratio of diameter or width to the depth is known as coefficient of modulus of horizontal sub-grade reaction,

   h (force/lengtp). The modulus of horizontal sub- grade reaction, h is used as an integral soil interaction parameter in case of cohesionless soil medium as it has a constant value for a particular cohesionless soil medium.

   Using the data obtained from the PLAXIS 3D models and subsequently with the finite difference approach kh vs. depth graphs were obtained. After obtaining the graph an average line of variation for values of kh with respect to depth is drawn as was previously done by Biswas et al. (2013).

   1. Relative density (33%, 44%, 62%, 67%, 74%

   and 90%)

   1. Slenderness ratio (30, 42, 50 and 55)

   2. Ratio of top layer thickness to embedded pile length (10%, 20%, 33%, 50%, 67% and 100%).

      The figures between the parentheses indicate the values of the parameters considered in this study.

      1. Homogenous Soil

         1. Coefficient of horizontal modulus of subgrade reaction (h) vs. Relative Density From the numerical results coefficient of horizontal modulus of sub-grade reaction (h) vs. relative density graphs have been plotted as shown in Fig 2 in case of homogeneous soil continuum for different values of slenderness ratio.

            From the graph it is noted that in case of loose to medium compactness of the sand medium the values of

            h do not vary much as the relative density decreases. The variation is not prominent for change in slenderness ratio within the range of study. This is due to the fact that as less relative density decreases the stiffness of the soil pile system. With lesser stiffness of the system the effect of slenderness ratio is not much pronounced whereas for denser soil the stiffness of the system is more and the effect of slenderness ratio appears to be more pronounced.

            Fig 2 Represents a typical graph showing variation of Relative density with h for homogeneous soil medium.

         2. Coefficient of horizontal modulus of sub-grade reaction (h) vs. Slenderness ratio

            A combined graph showing the change of coefficient of horizontal modulus of sub-grade reaction (h) vs. slenderness Ratio (L/D) for different compactness of sand is presented in Fig 3.

            Fig 3 Typical graph showing Slenderness ratio vs.

            h for homogeneous soil medium

            The figure shows that with the increase in slenderness ratio, decrease in the value of h takes place but the rate of decrease depends on the compactness of sand. Sand with dense to very dense compactness shows an appreciable rate of decrease whereas sand having medium or low compaction shows very little rate of decrease. From the figure it is clear that as the slenderness ratio increases the h decreases. With decrease of the stiffness of the soil the values of

            h become lower since the stiffness of the system reduces.

      2. Layered Soil

      In this study two types of sand layer have been used,

      i.e. Sand A and Sand B.

      Sand A: Relative density for top layer is 44 and for bottom layer is 74%.

      Sand B: Relative density for top layer is 44 and for bottom layer is 90%.

      5.2.2.1 Coefficient of horizontal modulus of subgrade reaction (h) vs. Ratio of top layer thickness to embedded pile length for Layered Soil

      1. For Sand A and Sand B when slenderness ratio is fixed:

         Variation for coefficient of horizontal modulus of subgrade reaction (h) with Ratio of top layer thickness to embedded pile length for two combinations of density variations that is the medium to dense and medium to very dense compactness is represented respectively in Fig 4.

         Fig 4 Typical graph showing Percentage for thickness of top layer for two different layer cases vs. h

         ZONE1 and 2 marked in figure represents the difference in the nature of slopes. It is observed from Fig4 that as the percent top layer thickness with respect to total embedded length of pile increases h decreases. This is due to the fact that the weaker top layer thickness is increasing which decrease the system stiffness represented by h. Finally the two curves (for Sand A and Sand B) approach each other when the value of percent top layer thickness is increasing and the soil is tending to be homogeneous.

      2. For different values of slenderness ratio with fixed relative densities of upper layer and lower layers.

      Using the numerical results another graph (Fig

      5) is plotted which shows a comparison of variation for coefficient of horizontal modulus of sub-grade reaction (h) with the change in thickness of the top layer for varied slenderness ratio for Sand A. In this case also the variation is linear and separate zones 1 and 2 have

      identical. It indicates that the decrease in slenderness ratio of pile decreases the system stiffness in such away that even the effect of increase of sand density is becoming insignificant.

      1. Normalized Form of h

         The normalized form of h proposed by Barber 1953 has been considered in this study is given by:

         been marked in the figure indicating the variation of

         slope of the lines. It is noted from the Fig 5 that the magnitude of h decreases as the upper layer of weaker

         Flexibility Ratio (FR) = h L

         5

         5

         Ep I p

         (4)

         soil thickness increases. However this rate of increase

         is much predominant in case of ZONE1.

         Fig 5 Typical graph showing Percentage for thickness of top layer for varied L/D vs. h

         It can also be observed that as the compactness of the bottom layer increases from dense to very dense (Fig 4) the rate of increase in coefficient of horizontal modulus of subgrade reaction (h) exhibits a higher rate of increase in ZONE 1. It reveals that if the top layer thickness is such that the fixity point enters well inside the bottom layer then there will be decrease in overall head deflection since the pile undergoes bending like a cantilever which is considered to be fixed at the fixity point. The head deflection of pile depends on the position of the fixity point. If it is within the strong stratum rate of increase of h with decrease of ratio of top layer thickness to pile length will be more. Otherwise if the fixity point lies inside the weaker stratum the rate will not appreciably increase. In Fig 5 it can be observed that as the slenderness ratio of pile decreases, due to the decrease in pile stiffness, the slope of the graph between ZONE 1 and 2 tends to become

         Ep and Ip are the elastic modulus and the moment of

         inertia of model pile respectively. An attempt has been made to examine the variation of flexibility ratio with different parameters like relative density, slenderness ratio, and ratio of top layer thickness to embedded length.

         1. Homogeneous Soil

            1. Flexibility Ratio vs Relative Density

               From the numerical results described earlier flexibility ratio vs. relative density with varied slenderness ratio have been plotted in Fig 6. It appears from the figure that for each slenderness ratio and for both ranges of sand densities, flexibility ratio increases with slenderness ratio. This is quite justified since the factor by which h has been multiplied represents the flexibility of pile. Increase of slenderness ratio results in increase in flexibility, hence variation is as observed in the figure.

               Fig 6 Variation of h with increase in relative density

            2. Flexibility Ratio vs Slenderness Ratio

      The graph shown in Fig 7 represents a the variation of flexibility ratio vs. slenderness ratio for different sand densities which are indicative of its

      degree of compaction, e.g. medium, dense, very dense respectively of the sand medium considered.

      Fig 7 Variation of Flexibility Ratio with increase in Slenderness Ratio

      It is observed from the graph that for each compactness (i.e. density) flexibility ratio increases with increase of slenderness ratio. This is attributed to increase in flexibility of pile with increase of slenderness ratio. Further it is observed that for a fixed slenderness ratio flexibility ratio increases with sand density. This is due to the fact that h increases the flexibility ratio when pile parameters remain the same as is seen from equation (4).

      5.3.2 Layered Soil

      Two types of sand layer (Sand A and Sand B) have been used in this study.

      Sand A: Relative density for top layer is 44 and for bottom layer is 74%.

      Sand B: Relative density for top layer is 44 and for bottom layer is 90%.

      5.3.2.1 Flexibility Ratio vs. Ratio of top layer thickness to embedded pile length

      The variation of flexibility ratio vs. the ratio of top layer thickness to embedded pile length has been shown in Fig 8. It is observed from Fig 8 that there is a marked difference between the curves for Sand A and Sand B in Zone 1 (stiffer slope), but in case of Zone 2 (flatter slope) for Sand A and Sand B the curves are becoming closer to each other and tends to meet at a point. This indicates that the effect of stronger sand of bottom layer is becoming less and less as percent of pile length in layer 1 increases with respect to embedded pile length. However the differences in the slope between the zones are not as sharp as was

      observed in Fig 5 since flexibility ratio is non dimensional form of h involving other parameters.

      Fig 8 Flexibility ratio vs. the variation of thickness of top layer graphs have been plotted for a fixed slenderness ratio

      Fig 9 Comparison of variation for Flexibility ratio due to the change in thickness of the top layer for varied Slenderness Ratio

      Flexibility ratio vs. percent of top layer thickness to embedded pile length has been plotted in Fig 9. It can be inferred from the figure that for a constant slenderness ratio the value of flexibility ratio decreases with increase in thickness of top layer. But this rate of decrease of the flexibility ratio increases with the increase in slenderness ratio. If the curves representing different values of slenderness ratio as shown in Fig 9 is extrapolated beyond 100% (as shown

      by the dotted lines), then it can be observed that at 110% the value of flexibility ratio shown by them corresponds to the value of h which is in fact the value corresponding to a homogeneous layer of medium degree of compactness. If the dotted line as shown in the figure is followed corresponding to the line showing the variation for slenderness ratio 42, the value of h obtained at 110% is 18195, using Equation 1, the h is 12434.6 KN/m3 which according to Fig 6 represents the value corresponding to the Relative Density of 44% (medium degree of compaction), which appears to be reasonable. This indicates that with a virtual thickness of top layer equal to 110% of pile embedded length, the effect is same as that of a homogeneous soil layer with relative density of 44%. Thus the bottom layer looses to produce any effect in soil pile stiffness when the top layer thickness is quite large.

2. Conclusions

The following conclusions may be drawn from the present study:

1. The value of coefficient of horizontal modulus of sub-grade reaction (h) increases with the increase in the compactness of the sand.

2. With the increase in slenderness ratio a decrease in the value of coefficient of horizontal modulus of sub-grade reaction (h) takes place. However the rate of decrease depends on the compactness of sand.

3. Coefficient of horizontal modulus of sub- grade reaction (h) increases as the upper layer thickness of weaker sand decreases in case of a two layered soil system. However, the rate of increase in the magnitude of coefficient of horizontal modulus of sub-grade reaction (h) depends on the slenderness ratio.

4. The Flexibility Ratio (normalized form of h) increases with the increase in the compactness of the sand.

5. With the increase in slenderness ratio there is a decrease in Flexibility Ratio. However the rate of decrease depends on the compactness of sand.

6. The Flexibility Ratio increases as the upper layer thickness of weaker sand decreases in case of a two layered system. However, the rate of increase in the magnitude of coefficient of horizontal modulus of sub-grade reaction (h) depends on the slenderness ratio of pile.

7. References

1. Biswas, S.K., Mukherjee, S., and De, M., (2012): A Study on Response of Laterally Loaded Piles Embedded in Layered Cohesionless Soil. Proceedings of IGC-2012 at Delhi held on December 13-15, 2012, Delhi, India.

2. Biswas, S.K., Mukherjee, S., and De, M., (2013): Study of Soil Stiffness Parameter in Case of Laterally Loaded Free Head Long Pile. Indian Science Cruiser, Vol.27, No.1. January 2013), pp.10-19.

3. Broms, B. B., (1965): Design of Laterally Loaded Piles, Journal of the Soil Mechanics and Foundations Division, Vol.

   -91, pp 79-99.

4. Chin Tan Yean, Sew Gue See and Chung Fong Chew (2010): Interpretation of Subgrade Reaction from Lateral Load Tests on Spun Piles in Soft Ground, G & P Geotechnics Sdn Bhd, Kuala lumpur, Malaysia

5. Davisson, M.T. and Gill, H.L., (1963): Laterally loaded piles in layered soil system, Journal of Soil Mechanics Foundation Division. ASCE, 89(3): pp.63-94.

6. Kim Byung-Tak and Kim Young-Su, (1999): Back Analysis for Prediction and Behavior of Laterally Loaded Single Piles in Sand, KSCE journal of Civil Engg.Vol.3, No.3.Sept (1999), pp.273-288.

7. Kim Byung-Tak and Kim Young-Su, (2001): Prediction of Lateral Behavior of Single and Group Piles using Artificial Neural Networks, KSCE journal of Civil Engg., Vol.5, No.2.June(2001), pp.185-198.

8. Kim Youngho and Jeong Sangseom, (2011): Analysis of soil resistance on laterally loaded piles based on 3D soilpile interaction, Computers and Geotechnics 38 (2011), pp. 248 257.

9. Nihar Ranjan Patra and Prabhakar Jagannath Pise, June, 2001. ©ASCE, Ultimate lateral resistance of pile groups in sand is part of the Journal of Geotechnical and Geo environmental Engineering, Vol. 127,No. 6.

10. Poulos H.G and Davis E.H, (1980). "Pile Foundation Analysis and Design", Publication John Wiley & Sons, The University of Sydney

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

12. Reese, L. C., and Matlock, H., (1960): Generalised solutions for laterally-loaded piles, Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, vol. 86, no. SM3, 1960, pp 63-91.