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**1 Soil Response Modelled by p-y Curves
**

In order to properly analyze a laterally loaded pile foundation in soil/rock, a nonlinear

relationship needs to be applied that provides soil resistance as a function of pile deflection. The

drawing in Figure 1-1a shows a cylindrical pile under lateral loading. Unloaded, there is a

uniform distribution of unit stresses normal to the wall of the pile as shown in Figure 1-1b. When

the pile deflects a distance of y1 at a depth of z1, the distribution of stresses looks similar to

Figure 1-1c with a resisting force of p1: the stresses will have decreased on the backside of the

pile and increased on the front, where some unit stresses contain both normal and shearing

components as the displaced soil tries to move around the pile.

Figure 1-1: Unit stress distribution in a laterally loaded pile

**When it comes to this type of analysis, the main parameter to take from the soil is a reaction
**

modulus. It is defined as the resistance from the soil at a point along the depth of the pile divided

by the horizontal deflection of the pile at that point. RSPile defines this reaction modulus (Epy)

using the secant of the p-y curve, as shown in Figure 1-2. p-y curves are developed at specific

depths, indicating the soil reaction modulus is both a function of pile deflection (y) and the depth

below the ground surface (z). More information will be given on the p-y curves used in a later

section.

Figure 1-2: Generic p-y curve defining soil reaction modulus

This requires a change to the third term of Equation 1 given by Equation 5: 𝐸𝑝𝑦 (𝑦 − 𝑦𝑠𝑜𝑖𝑙 ) Equation 5 The modified form of the differential equation now becomes 𝑑4𝑦 𝑑2 𝑦 Equation 6 𝐸𝑝 𝐼𝑝 + 𝑃𝑥 + 𝐸𝑝𝑦 (𝑦 − 𝑦𝑠𝑜𝑖𝑙 ) − 𝑊 = 0 𝑑𝑥 4 𝑑𝑥 2 where soil reaction modulus (𝐸𝑝𝑦 ) is found from the p-y curve using the relative pile soil movement (𝑦 − 𝑦𝑠𝑜𝑖𝑙 ) instead of only the pile deflection (𝑦). must be solved for implementation of the p-y method.2 Governing Differential Equation The differential equation for a beam-column. as derived by Hetenyi (1946). The conventional form of the differential equation is given by Equation 1: 𝑑4𝑦 𝑑2𝑦 Equation 1 𝐸𝑝 𝐼𝑝 + 𝑃𝑥 + 𝐸𝑝𝑦 𝑦 − 𝑊 = 0 𝑑𝑥 4 𝑑𝑥 2 Where 𝑦 = Lateral deflection of the pile 𝐸𝑝 𝐼𝑝 = Bending stiffness of pile 𝑃𝑥 = Axial load on pile head 𝐸𝑝𝑦 = Soil reaction modulus based on p-y curves 𝑊 = Distributed load down some length of the pile Further formulas needed are given by Equations 2 – 4: 𝑑3𝑦 𝑑𝑦 Equation 2 𝐸𝑝 𝐼𝑝 3 + 𝑃𝑥 = 𝑉 𝑑𝑥 𝑑𝑥 𝑑2𝑦 𝐸𝑝 𝐼𝑝 = 𝑀 Equation 3 𝑑𝑥 2 𝑑𝑦 =𝑆 𝑑𝑥 Equation 4 Where V = Shear in the pile M = Bending moment of the pile S = Slope of the curve defined by the axis of the pile In the case where the pile is loaded by laterally moving soil. . the soil reaction is determined by the relative soil and pile movement.

A moment. which start from 0 at the pile head to n at the pile toe with two imaginary nodes above and below the pile head and toe.Using a spring-mass model in which springs represent material stiffness. especially for relatively small values of 𝑃𝑥 . required for the p-y method. and the soil reaction (𝐸𝑝𝑦 ) varied with pile deflection and depth down the pile. The assumption made that the axial load (𝑃𝑥 ) is constant with depth is not usually true. This provides the benefit of having the bending stiffness (𝐸𝑝 𝐼𝑝 ) varied down the length of the pile. p y-ysoil Soil Lateral Resistance (p) p Lateral component of moving soil y-ysoil p y-ysoil Pile Bending Stiffness (EI) p y-ysoil Sliding Surface p y p y Figure 2-1: Spring mass model used to compute lateral response of loaded piles 3 Finite Difference Method The finite difference form of the differential equation formulates it in numerical terms and allows a solution to be achieved by iteration. axial. as shown in Figure 3-1. These imaginary nodes are only used to obtain solutions. . and soil movement load are also shown. Nodes along the pile are separated by these segments. numerical techniques can be employed to conduct the load-deflection analysis (Figure 2-1). With the method used. shear. respectively. still holds true. 𝑃𝑥 . However. The value of 𝑃𝑥 also has little effect on the deflection and bending moment (aside from cases of buckling) and therefore it is concluded that this assumption is generally valid. in most cases the maximum bending moment occurs at a relatively short distance below the ground surface at a point where the constant value. the pile is discretized into n segments of length h.

and pipe piles.e. the yield stress of the steel is required from the user. This error however. Since only two equations can be defined at each end of the pile. Eccentric loads are not considered. is thought to be small. Deflections due to shearing stresses are small. Currently. Error is involved in using this method when there is a change in bending stiffness down the length of the pile (i. The analysis is done by performing a balance of forces (tension and compression) in n slices of the pile cross section parallel to the bending axis. the stiffness of each pile node (𝐸𝑝 𝐼𝑃 ) is calculated by multiplying the elastic modulus by the moment of inertia of the pile. tapered or plastic piles): The value of 𝐸𝑝 𝐼𝑃 is made to correspond with the central term for 𝑦 (𝑦𝑚 ) in Figure 3-1. The pile is geometrically straight. slope 𝑀 (𝑆). the shear can be defined based on this user defined function. moment (𝑀). This is . 2. The two boundary conditions that are employed at the toe of the pile are based on moment and shear. rectangular. rotational stiffness ( 𝑆 ). 3. y ym-2 h ym-1 h ym h ym+1 h ym+2 x Figure 3-1: Pile segment discretization into pile elements and soil elements The imaginary nodes above and below the pile head are used to define boundary conditions. 4 Pile Bending Stiffness For elastic piles. Assuming information can be developed that will allow the user to define toe shear stress (𝑉) as a function of pile toe deflection (𝑦). and deflection (𝑌). The assumptions made for lateral loading analysis by solving the differential equation using finite difference method are as follows: 1. Therefore moment is set to zero at the toe. the engineer has the ability to define the two that best fit the problem. plastic analyses can be performed on uniform cylindrical. 4. Transverse deflections of the pile are small. Five different boundary equations have been derived for the pile head: shear (𝑉). In analyzing a plastic pile. The case where there is a moment at the pile toe is uncommon and not currently treated by this procedure.

2nd Edition. the stiffness remains in the elastic range until yielding occurs. Reese and W. As shown. usually at a fairly high moment value. Some guidance and specific suggestions are presented in the text. This book also provided the tables presented in the different types of soil that follow.5 𝑐𝑢 = undrained shear strength at depth z (kPa) . Selection of soil models (p-y curves) to be used for a particular analysis is the most important problem to be solved by the engineer. sand. Van Impe.done at many different values of bending curvature (𝜙). The relation between moment and stiffness will look like Figure 4-1. Equation 7 and 8 are used in order to find the bending stiffness based on the moment and curvature. When the forces in the slices balance around the neutral axis. M Figure 4-1: Pile segment discretization into pile elements and soil elements 5 Soil Models Recommendations are presented for obtaining p-y curves for clay. the moment can be computed. by L. A list of the variables used in the equations that follow can be found below: 𝑏 = diameter of the pile (m) 𝑧 = depth below ground surface (m) 𝛾′ = effective unit weight (kN/m3) 𝐽 = factor determined experimentally by Matlock equal to 0. Single Piles and Pile Groups Under Lateral Loading. Pile Stiffness. All are based on the analysis of the results of full scale experiments with instrumented piles. EI Moment. ℇ = 𝜙𝜂 Equation 7 𝑀 𝐸𝐼 = 𝜙 Equation 8 Where ℇ = Bending Strain 𝜙 = Bending curvature 𝜂 = Distance from the neutral axis M = Bending moment of the pile The stiffness of the pile is then checked against the moment value at the node for each iteration. and weak rock.

010 Stiff 96-192 0. 20. the user will need the strain corresponding to one-half the maximum principal stress difference ℇ50.005 *Peck et al. The development of the p-y curve for soft clay is presented in Figure 5-1. Figure 5-1: p-y curve for Soft Clay 𝑝𝑢𝑙𝑡 is calculated using the smaller of the values given by the equations below. 𝛾′ 𝐽 𝑝𝑢𝑙𝑡 = [3 + 𝑧 + 𝑧] 𝑐𝑢 𝑏 Equation 9 𝑐𝑢 𝑏 𝑝𝑢𝑙𝑡 = 9𝑐𝑢 𝑏 Equation 10 . Additionally.020 Medium 48-96 0. the user must obtain the best estimate of the undrained shear strength and the submerged unit weight. pg.1974. Table 5-1: Representative values of ℇ50 for normally consolidated clays Consistency of Clay Average undrained shear strength (kPa)* ℇ50 Soft <48 0. 1970) To complete the analysis for soft clay.𝑐𝑎 = average undrained shear strength over the depth z (kPa) 𝜑 = friction angle of sand 5.1 p-y curves for soft clay with free water (Matlock. Some typical values of ℇ50 are given in Table 5-1 according to undrained shear strength.

Table 5-2: Representative values of ℇ50 for overconsolidated clays Average undrained shear strength 50-100 100-200 200-400 (kPa)* ℇ50 0. As is a coefficient based on the depth to diameter ratio according to Figure 5-3.2 p-y curves for stiff clay with free water (Reese. Figure 5-2: p-y curve for Stiff Clay with water .005 0.5.007 0. It should be defined as half the total maximum principal stress difference in an unconsolidated undrained triaxial test. et al. and the variable ks mentioned above is used to define the initial straight line portion of the p-y curve. The development of the p-y curve for submerged stiff clay is presented in Figure 5-2.004 Table 5-3: Representative values of kpy for overconsolidated clays Average undrained shear strength 50-100 100-200 200-400 (kPa)* kpy (static) MN/m3 135 270 540 kpy (cyclic) MN/m3 55 110 540 *The average shear strength should be computed from the shear strength of the soil to a depth of 5 pile diameters.. Typical values of ℇ50 according to undrained shear strength can be found in Table 5-2. 1975) The analysis of stiff clay with free water requires the same inputs as soft clay as well as the value ks with some representative values presented in Table 5-3.

should you not have an available stress- strain curve.83𝑐𝑎 𝑧 Equation 11 𝑝𝑐 = 11𝑐𝑢 𝑏 Equation 12 Figure 5-3: Values of constants As and Ac (Reese & Van Impe. The larger value is more conservative. The development of the p-y curve for dry stiff clay is presented in Figure 5-4.005 as given in Table 5-2. 1972) The input parameters for stiff clay without free water are the same as for soft clay. Figure 5-4: p-y curve for Stiff Clay without water .3 p-y curves for stiff clay without free water (Welch & Reese. 𝑝𝑐 = 2𝑐𝑎 𝑏 + 𝛾 ′ 𝑏𝑧 + 2. but the soil unit weight will not be submerged and the value for ℇ50. 2011) 5.𝑝𝑐 is calculated using the smaller of the values given by the equations below. should be 0.01 or 0.

. ym. If yk is greater than yu then the p-y curve is linear from the origin to yu. multiplied by coefficients 𝐴 and 𝐵 from Figure 5-6. and pile deflection.3 34 Table 5-5: Representative values of kpy for sand above the water table (Static and Cyclic) Relative Density Loose Medium Dense Recommended kpy (MN/m3) 6. Figure 5-5: p-y curve for Sand 𝑝𝑢 and 𝑝𝑚 are calculated using the smaller of the values given by 𝑝𝑠 in the equations below. The value kpy is also required and some values are given in Table 5-4 and Table 5-5. 𝛾′ 𝐽 𝑝𝑢𝑙𝑡 = [3 + 𝑧 + 𝑧] 𝑐𝑎 𝑏 Equation 13 𝑐𝑎 𝑏 𝑝𝑢𝑙𝑡 = 9𝑐𝑢 𝑏 Equation 14 5. pk. . 1974) To achieve a p-y curve for sand. pu. et al. soil unit weight (buoyant unit weight for sand below.4 p-y curves for sand above and below water (Reese. and total unit weight for sand above the water table). The variable kpy mentioned above is used to define the initial straight line portion of the p-y curve. the user must obtain values for the friction angle.4 61 The development of the p-y curve for sand is presented in Figure 5-5. pu.4 16.𝑝𝑢𝑙𝑡 is calculated using the smaller of the values given by the equations below. Table 5-4: Representative values of kpy for submerged sand Relative Density Loose Medium Dense Recommended kpy (MN/m3) 5. are calculated from pm. yk.8 24. The soil resistance. and yu.

When it comes to intermediate materials (rock and strong soil). reaction modulus of rock. 1978) For the design of piles under lateral loading in rock. and a strain factor krm ranging from 0. . 2011) 5. special emphasis in necessary in the coring of the rock. designers may wish to compare analysis performed by stiff clay and this method. 𝛽 = 45 + 2 . It is noted that bending stiffness of the pile must reflect non-linear behavior in order to predict loadings at failure. The input parameters required for this method are. 𝑝𝑚 = 𝐵𝑠 𝑝𝑠 Figure 5-6: Coefficients for soil resistance versus depth (Reese & Van Impe. 𝜑 𝜑 𝜑 𝛼 = 2 . Designers must address the potential weakness of the rock in a case by case manner.00005. therefore reliance on the method presented is limited. the rock quality designation (percent of recovery). uniaxial compressive strength. 𝐾0 = 0. krm can be taken as the compression strain at fifty percent of the uniaxial compressive strength.0005 to 0.5 p-y curves for weak rock (Reese & Nyman.4. 𝐾𝑎 = tan2 (45 − 2 ) 𝐾0 𝑧 tan 𝜑 sin 𝛽 tan 𝛽 𝑝𝑠 = 𝛾𝑧 [ + (𝑏 + 𝑧 tan 𝛽 tan 𝛼) Equation 15 tan(𝛽 − 𝜑) cos 𝛼 tan(𝛽 − 𝜑) + 𝐾0 𝑧 tan 𝛽 (tan 𝜑 sin 𝛽 − tan 𝛼) − 𝐾𝑎 𝑏] Equation 16 𝑝𝑠 = 𝐾𝑎 𝑏𝛾𝑧(tan8 𝛽 − 1) + 𝐾0 𝑏𝛾𝑧 tan 𝜑 tan4 𝛽 𝑝𝑢 = 𝐴𝑠 𝑝𝑠 .

2 𝑅𝑄𝐷% 𝛼𝑟 = 1 − ( ) 3 100% 𝑧𝑟 Equation 17 𝑝𝑢𝑟 = 𝛼𝑟 𝑞𝑢𝑟 𝑏 (1 + 1. the integrals of the ultimate soil resistance over depth are equated for the two layers with z1 as the depth of the top layer. The values of pult are computed from the soil properties as noted above. Figure 5-7: p-y curve for Weak Rock 𝑝𝑢𝑟 is calculated using the smaller of the values given by the equations below. is calculated from the input parameters. pur.The development of the p-y curve for weak rock is presented in Figure 5-7. To find the equivalent depth (z2) of the layer existing below the top. The ultimate resistance for rock. Since p-y curves are developed based on the depth into the soil. The linear portion of the curve with slope Kir defines the curve until intersection with the curved portion defined in the figure. z1) moving down as you would until another layer is hit in terms of actual depth. this is very important. This concept can be carried down for the rest of the existing layers the pile is in .2𝛼𝑟 𝑞𝑢𝑟 𝑏 Equation 18 6 Layered Soil Profile: Method of Georgiadis The method of Georgiadis is based on the determination of the “equivalent depth” of every soil layer existing below the top layer. 𝑧1 𝐹1 = ∫ 𝑝𝑢𝑙𝑡1 𝑑𝑧 0 𝑧2 Equation 19 𝐹1 = ∫ 𝑝𝑢𝑙𝑡2 𝑑𝑧 0 The p-y curves of the second layer are computed starting at z2 (actual depth.4 ) 𝑏 𝑝𝑢𝑟 = 5.

The depth independent equations for ultimate resistance (Equations 10. 12. 𝛾′ 𝐽 cos 𝜃 𝑝𝑢𝑙𝑡 = [3 + 𝑧 + 𝑧] 𝑐𝑢 𝑏 𝑐𝑢 𝑏 √2 cos(45 + 𝜃) Notice the equations are the exact same as Equation 9 for soft clay with additional terms on the end. Sign convention is clockwise positive for both values as shown in the “Sign Convention” document in the Help menu. 𝛾′ 𝐽 1 𝑝𝑢𝑙𝑡 = [3 + 𝑧 + 𝑧] 𝑐𝑢 𝑏 𝑐𝑢 𝑏 1 + tan 𝜃 The ultimate soil resistance at the back of the pile is. The same end terms can be applied to Equation 11 for submerged stiff clay and Equation 13 for dry stiff clay.7 Ground Slope and Pile Batter RSPile allows the input of a ground slope and a pile batter angle. . 𝐾0 𝐻 tan 𝜑 sin 𝛽 tan 𝛽 𝑝𝑠𝑎 = 𝛾𝐻 [ (4𝐷13 − 3𝐷12 + 1) + (𝑏𝐷2 + 𝐻 tan 𝛽 tan 𝛼 𝐷22 ) tan(𝛽 − 𝜑) cos 𝛼 tan(𝛽 − 𝜑) + 𝐾0 𝐻 tan 𝛽 (tan 𝜑 sin 𝛽 − tan 𝛼)(4𝐷13 − 3𝐷12 + 1) − 𝐾𝑎 𝑏] The ultimate soil resistance if the pile is deflecting up the slope is. If there is an effective slope in sand. these equations go back to their original form. 𝐻 = depth below ground surface (m) cos 𝜃−√cos2 𝜃−cos2 𝜑 𝐾𝑎 = cos 𝜃 . 𝐷2 = 1 − 𝐷1 . and 14) remain the same and the smaller of the values is still used. Equation 16 remains the same and the smaller of the two values is still used. 𝐾0 𝐻 tan 𝜑 sin 𝛽 tan 𝛽 𝑝𝑠𝑎 = 𝛾𝐻 [ (4𝐷33 − 3𝐷32 + 1) + (𝑏𝐷4 + 𝐻 tan 𝛽 tan 𝛼 𝐷42 ) tan(𝛽 − 𝜑) cos 𝛼 tan(𝛽 − 𝜑) + 𝐾0 𝐻 tan 𝛽 (tan 𝜑 sin 𝛽 − tan 𝛼)(4𝐷33 − 3𝐷32 + 1) − 𝐾𝑎 𝑏] Where. This effective slope alters the calculation of the soil resistance for both clay and sand. the effective slope (𝜃) is calculated from the difference between the pile batter angle and the ground slope. Note: if the effective slope is equal to 0. In order to incorporate this into the analysis. 𝐷4 = 1 + 𝐷3 These equations for ultimate soil resistance in sand replace Equation 15 when there is an effective slope. cos 𝜃+√cos2 𝜃−cos2 𝜑 tan 𝛽 tan 𝜃 𝐷1 = tan 𝛽 tan 𝜃+1 . the ultimate soil resistance if the pile is deflecting down the slope is. the ultimate soil resistance in front of the pile is. tan 𝛽 tan 𝜃 𝐷3 = 1−tan 𝛽 tan 𝜃 . If there is an effective slope in clay.

Foundation engineering. Houston. 2nd Edition. Reese.. London: Taylor & Francis Group. H. Single Piles and Pile Groups Under Lateral Loading. Houston.C. Figure 7-1: Proposed factor for modifying p-y curves for battered piles (Reese & Van Impe.C. W. Reese. Houston. Suggested values are provided in Figure 7-1. they can choose appropriate p-multipliers to alter the p-y curves. Proceedings of the II Annual Offshore Technology Conference. Thorburn 1974. Texas. the user either has the option to use the method above by entering ground slope and batter angle.E. 3. 5. & W. L. Field testing and analysis of laterally loaded piles in stiff clay. 4.C.B.When modelling piles that are in a slope and/or battered.C. Austin. & L. . Proceedings of the VII Annual Offshore Technology Conference. Field testing and analysis of laterally loaded piles in sand.C.D. 1970. Hanson & T. R. University of Texas. L. Koop 1975. Matlock.R. 2.. Peck. Cox & F. R. L. Cox & F.F. or more commonly. W. Proceedings of the VI Annual Offshore Technology Conference. Reese. W.R. (OTC 1204): 577-594.. Research Report 3-5-65-89. Texas. 2(OTC 2312): 672-690. Van Impe 2011. Texas. Center for Highway Research. 2nd edn. 2011) 8 References 1. Welch. 6. Laterally loaded behavior of drilled shafts. Koop 1974. Reese 1972.D. a responsible engineer may wish to request full scale testing. New York: Wiley. Correlations for design of laterally loaded piles in soft clay. 2(OTC 2080): 473-485.H. however on important projects.

L. Georgiadis. ASCE: 536-545. Construction. 1983. Clearwater. Proceedings of the Geotechnical Practice in Offshore Engineering. Nyman 1978.7. Isenhower. . and Testing of Deep Foundations. (OTRC): 150-164. Florida. Reese.C. Austin.M. Analysis. W. 1999. 8. Florida. Field load test of instrumented drilled shafts at Islamorada. Texas. & K. 9. Development of p-y curves for layered soils. M. Analysis of pile groups subjected to deep-seated soil displacements.J. Design. A report to Girdler Foundation and Exploration Corporation (unpublished).

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