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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

. the soil reaction is determined by the relative soil and pile movement. 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. must be solved for implementation of the p-y method. as derived by Hetenyi (1946). 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).2 Governing Differential Equation The differential equation for a beam-column.

and the soil reaction (𝐸𝑝𝑦 ) varied with pile deflection and depth down the pile. 𝑃𝑥 . especially for relatively small values of 𝑃𝑥 . Nodes along the pile are separated by these segments. required for the p-y method. This provides the benefit of having the bending stiffness (𝐸𝑝 𝐼𝑝 ) varied down the length of the pile. shear.Using a spring-mass model in which springs represent material stiffness. 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. still holds true. However. numerical techniques can be employed to conduct the load-deflection analysis (Figure 2-1). With the method used. A moment. the pile is discretized into n segments of length h. respectively. These imaginary nodes are only used to obtain solutions. The assumption made that the axial load (𝑃𝑥 ) is constant with depth is not usually true. 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. and soil movement load are also shown. axial. . as shown in Figure 3-1. 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. 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 shear can be defined based on this user defined function. and deflection (𝑌). Assuming information can be developed that will allow the user to define toe shear stress (𝑉) as a function of pile toe deflection (𝑦). the engineer has the ability to define the two that best fit the problem. moment (𝑀). 2. The pile is geometrically straight. the stiffness of each pile node (𝐸𝑝 𝐼𝑃 ) is calculated by multiplying the elastic modulus by the moment of inertia of the pile. and pipe piles. The two boundary conditions that are employed at the toe of the pile are based on moment and shear. 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. Error is involved in using this method when there is a change in bending stiffness down the length of the pile (i. Therefore moment is set to zero at the toe. 4 Pile Bending Stiffness For elastic piles. Eccentric loads are not considered.e. rotational stiffness ( 𝑆 ). 4. Since only two equations can be defined at each end of the pile. 3. 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 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. Five different boundary equations have been derived for the pile head: shear (𝑉). In analyzing a plastic pile. Deflections due to shearing stresses are small. rectangular. the yield stress of the steel is required from the user. Transverse deflections of the pile are small. The case where there is a moment at the pile toe is uncommon and not currently treated by this procedure. plastic analyses can be performed on uniform cylindrical. slope 𝑀 (𝑆). This error however. This is . The assumptions made for lateral loading analysis by solving the differential equation using finite difference method are as follows: 1. is thought to be small. Currently.

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. Some guidance and specific suggestions are presented in the text. 2nd Edition. usually at a fairly high moment value. EI Moment. When the forces in the slices balance around the neutral axis. This book also provided the tables presented in the different types of soil that follow. sand. the moment can be computed. All are based on the analysis of the results of full scale experiments with instrumented piles.done at many different values of bending curvature (𝜙). and weak rock. 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. by L. Pile Stiffness. Equation 7 and 8 are used in order to find the bending stiffness based on the moment and curvature. Van Impe. As shown. 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. 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 stiffness remains in the elastic range until yielding occurs. The relation between moment and stiffness will look like Figure 4-1.5 𝑐𝑢 = undrained shear strength at depth z (kPa) . Single Piles and Pile Groups Under Lateral Loading. Reese and W.

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

et al. Figure 5-2: p-y curve for Stiff Clay with water .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.005 0. 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. Typical values of ℇ50 according to undrained shear strength can be found in Table 5-2. 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. and the variable ks mentioned above is used to define the initial straight line portion of the p-y curve.5. It should be defined as half the total maximum principal stress difference in an unconsolidated undrained triaxial test. Table 5-2: Representative values of ℇ50 for overconsolidated clays Average undrained shear strength 50-100 100-200 200-400 (kPa)* ℇ50 0. The development of the p-y curve for submerged stiff clay is presented in Figure 5-2..007 0.

𝑝𝑐 is calculated using the smaller of the values given by the equations below. 𝑝𝑐 = 2𝑐𝑎 𝑏 + 𝛾 ′ 𝑏𝑧 + 2. should you not have an available stress- strain curve.005 as given in Table 5-2. should be 0.83𝑐𝑎 𝑧 Equation 11 𝑝𝑐 = 11𝑐𝑢 𝑏 Equation 12 Figure 5-3: Values of constants As and Ac (Reese & Van Impe. The development of the p-y curve for dry stiff clay is presented in Figure 5-4. 2011) 5. The larger value is more conservative. but the soil unit weight will not be submerged and the value for ℇ50.01 or 0. Figure 5-4: p-y curve for Stiff Clay without water .3 p-y curves for stiff clay without free water (Welch & Reese. 1972) The input parameters for stiff clay without free water are the same as for soft clay.

. are calculated from pm. et al. pk.𝑝𝑢𝑙𝑡 is calculated using the smaller of the values given by the equations below.4 16. The soil resistance. yk.4 p-y curves for sand above and below water (Reese. soil unit weight (buoyant unit weight for sand below. pu. If yk is greater than yu then the p-y curve is linear from the origin to yu. The variable kpy mentioned above is used to define the initial straight line portion of the p-y curve. 1974) To achieve a p-y curve for sand. Table 5-4: Representative values of kpy for submerged sand Relative Density Loose Medium Dense Recommended kpy (MN/m3) 5. 𝛾′ 𝐽 𝑝𝑢𝑙𝑡 = [3 + 𝑧 + 𝑧] 𝑐𝑎 𝑏 Equation 13 𝑐𝑎 𝑏 𝑝𝑢𝑙𝑡 = 9𝑐𝑢 𝑏 Equation 14 5.8 24. The value kpy is also required and some values are given in Table 5-4 and Table 5-5. multiplied by coefficients 𝐴 and 𝐵 from Figure 5-6. . the user must obtain values for the friction angle.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. and total unit weight for sand above the water table). and yu. and pile deflection. ym. 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.4 61 The development of the p-y curve for sand is presented in Figure 5-5. pu.

2011) 5. 𝐾0 = 0. reaction modulus of rock. and a strain factor krm ranging from 0. Designers must address the potential weakness of the rock in a case by case manner. 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 . 1978) For the design of piles under lateral loading in rock. When it comes to intermediate materials (rock and strong soil). 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. uniaxial compressive strength. krm can be taken as the compression strain at fifty percent of the uniaxial compressive strength. the rock quality designation (percent of recovery). designers may wish to compare analysis performed by stiff clay and this method. 𝛽 = 45 + 2 . .00005. 𝑝𝑚 = 𝐵𝑠 𝑝𝑠 Figure 5-6: Coefficients for soil resistance versus depth (Reese & Van Impe.4. therefore reliance on the method presented is limited. 𝜑 𝜑 𝜑 𝛼 = 2 . special emphasis in necessary in the coring of the rock.0005 to 0.5 p-y curves for weak rock (Reese & Nyman.

z1) moving down as you would until another layer is hit in terms of actual depth. 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.4 ) 𝑏 𝑝𝑢𝑟 = 5. pur. The values of pult are computed from the soil properties as noted above.The development of the p-y curve for weak rock is presented in Figure 5-7. The ultimate resistance for rock. This concept can be carried down for the rest of the existing layers the pile is in . Figure 5-7: p-y curve for Weak Rock 𝑝𝑢𝑟 is calculated using the smaller of the values given by the equations below. the integrals of the ultimate soil resistance over depth are equated for the two layers with z1 as the depth of the top layer. 2 𝑅𝑄𝐷% 𝛼𝑟 = 1 − ( ) 3 100% 𝑧𝑟 Equation 17 𝑝𝑢𝑟 = 𝛼𝑟 𝑞𝑢𝑟 𝑏 (1 + 1. To find the equivalent depth (z2) of the layer existing below the top.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. is calculated from the input parameters. 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. this is very important.

the ultimate soil resistance in front of the pile is. . these equations go back to their original form. Note: if the effective slope is equal to 0. If there is an effective slope in clay. 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. In order to incorporate this into the analysis. tan 𝛽 tan 𝜃 𝐷3 = 1−tan 𝛽 tan 𝜃 . The depth independent equations for ultimate resistance (Equations 10. Equation 16 remains the same and the smaller of the two values is still used. 𝐻 = depth below ground surface (m) cos 𝜃−√cos2 𝜃−cos2 𝜑 𝐾𝑎 = cos 𝜃 . cos 𝜃+√cos2 𝜃−cos2 𝜑 tan 𝛽 tan 𝜃 𝐷1 = tan 𝛽 tan 𝜃+1 . 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. This effective slope alters the calculation of the soil resistance for both clay and sand. The same end terms can be applied to Equation 11 for submerged stiff clay and Equation 13 for dry stiff clay. Sign convention is clockwise positive for both values as shown in the “Sign Convention” document in the Help menu. 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. If there is an effective slope in sand. the effective slope (𝜃) is calculated from the difference between the pile batter angle and the ground slope. and 14) remain the same and the smaller of the values is still used. the ultimate soil resistance if the pile is deflecting down the slope is. 12. 𝐷2 = 1 − 𝐷1 . 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. 𝐷4 = 1 + 𝐷3 These equations for ultimate soil resistance in sand replace Equation 15 when there is an effective slope.7 Ground Slope and Pile Batter RSPile allows the input of a ground slope and a pile batter angle.

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

C. Design.7. (OTRC): 150-164. A report to Girdler Foundation and Exploration Corporation (unpublished). Florida. Georgiadis. Reese.M. Austin. Field load test of instrumented drilled shafts at Islamorada. Proceedings of the Geotechnical Practice in Offshore Engineering. Development of p-y curves for layered soils. Construction. Analysis. Florida. Nyman 1978. . Analysis of pile groups subjected to deep-seated soil displacements. 1983. and Testing of Deep Foundations.J. Isenhower. Clearwater. L. & K. W. 8. ASCE: 536-545. Texas. 9. M. 1999.

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