Technical Documentation for the Pile Action Module.

New features in this update:

Elastic-plastic pile sections with user-defined moment capacity (Section 6)
Translational and rotational pile head spring with user-defined stiffness (Section 7)

Technical Documentation for the Pile Action Module (version 0.0.15, updated on 26/08/2024)

1. Introduction

This is the technical document for the Pile Action module.

The Pile Action module is a Design Module in the Apeiron Design Studio for lateral pile analysis using the py method, which uses non-linear py curves to represent soil behaviour. The calculation is powered by OpenSees and is done on the cloud.

2. Sign conventions

Fig 2.1 Sign convention in the Pile Action module

A 3D model is built in OpenSees, with 3 degrees of freedom (x, y and z).

z increases in magnitude with increasing depth.

z = 0.0 m at the ground surface, i.e. the reference level is always the ground surface.

The analysis is done in the xz plane only, therefore any displacement in the y direction and rotation about the x (out of xz plane rotation) and z direction (torsion) is not allowed in the model.

Shear force is positive if it’s caused by a force applied from left to right on the xz plane.

The bending moment is negative if it’s caused by a force applied from left to right on the xz plane. Note that this means the bending moment is plotted on the tension side, as per example output Fig 2.2 when a 100 kN top lateral load is applied at the top of the pile.

Positive applied moments rotate the pile in the antilock-wise direction, which bends the pile to the left-hand side. This is because a positive applied moment causes an increase in the bending moment. When the bending moment is positive, it's plotted on the tension side of the pile.

Fig 2.2 Example output for sign convention

3. Assumptions

3.1 General assumptions

The following assumptions are made:

The pile is initially straight.
The axial load caused by the pile's weight is neglected.
The pile has a longitudinal plane of symmetry, and all loads and reactions are within this plane (xz plane).
The pile's section materials are homogeneous.
The pile material's structural capacity is not exceeded.
The behaviour of an elastic pile section remains constant in tension and compression throughout the analysis.
The pile is not subjected to dynamic loading. When the cyclic loading mode is enabled, cyclic soil springs are used when available to capture the softening effect of cycling loading. Cyclic springs may have different shapes compared to the static springs.
Z-direction translation is fixed at the pile's bottom, but rotation is still permitted. The only vertical support comes from the pile's bottom.
No plastic behaviour of soil springs or damping is considered. All py curves are elastic, meaning the loading and unloading cycles follow the exact same path.
Grouping effects for closely spaced piles are captured by user-specified p-multipliers and y-multipliers. The p and y multipliers are multiplied by the soil spring resistance p and pile soil relative displacement y before the soil springs are connected to the pile. For example, a p-multiplier of 0.8 across the entire pile length means the resistance of all soil springs is reduced by 20% for the entire pile length.
The pile is divided into 201 nodes connected by 200 elements in the analysis. If the pile is 20 m long, each element is 0.1 m.
The equilibrium tolerance used in the analysis is 1e-4 m, i.e. a displacement difference less than 0.1 mm is considered to be in equilibrium.

3.2 Loading assumptions

Overlapping loads are not allowed. Only one of each type of load is applied per loading scenario.
Multiple point loads can be specified along the pile. However, if the point load applied is between two nodes, the point load is applied to the nearest point along the pile.

e.g. if the pile extends between 0 - 40 m.b.g.l., consisting of 200 nodes, each element is 0.2 m long. The depth of the top two pile nodes are 0.0m and 0.2m respectively. If a point load is applied at 0.15 m, it will be applied to the node at 0.2 m.

For loads applied exactly in the middle between two points, it will be applied to the upper node at a shallower depth. e.g. if a point load is applied at 0.1 m between two nodes at 0.0 m and 0.2 m depth respectively, the load is applied to the node at 0.0 m.

Applied pile displacement follows the same principles.
When a distributed load is applied over a range, it is split into multiple point loads to be applied to each node within range based on the tributary length of each node. When the specified range does not coincide perfectly with the node locations, the extra length in the specified loading range is added to the tributary length of the edge nodes, such that the total amount of load applied is always exactly the same as the load applied, regardless of the specified range. Distributed soil displacement follows the same principles.

Example of how distributed loads are applied:

Consider a case where the pile nodes are at 0.5 m increments between 0 - 3 m.b.g.l., and the specified distributed load is 30 kN/m from 0.5 m to 2.0 m. The tributary length of the nodes and the distributed load applied per node are summarised in the table below:

Node depth (m.b.g.l.)	Tributary length (m)	Applied load from distributed load (kN)
0.5	0.5	5
1.0	1.0	10
1.5	1.0	10
2.0	0.5	5

Pile nodes and applied distributed load coinciding perfectly

Now if the distributed load is shifted up by 0.1 m:

Untitled 1.png

Pile nodes and applied distributed load not coinciding perfectly

The pile node at 2.0 m.b.g.l. is no longer within range and therefore is not loaded. However the tributary area of the nodes at 0.5 m.b.g.l. and 1.5 m.b.g.l. both increased. The distributed load applied per node based on the new tributary lengths are summarised in the table below:

Based on this method of distributing loads, the total applied load on the pile nodes is always equal to the load intensity multiplied by the distance (and for linearly varying distributed loads, it’s always the area under the free body diagram).

Node depth (m.b.g.l.)	Tributary length (m)	Applied load from distributed load (kN)
0.5	0.6	6
1.0	1.0	10
1.5	1.4	14
2.0	0	0

3.3 Soil spring specific assumptions

Soft Clay (Matlock et al., 1970)

When using the default initial secant modulus ks or kc correlation, the average undrained shear strength at a point is equal to the input undrained shear strength at that point.

Stiff Clay without Free Water (Welch and Reese, 1972)

When using the default initial secant modulus ks or kc correlation, the average undrained shear strength at a point is equal to the input undrained shear strength at that point.

Stiff Clay with Free Water (Welch and Reese, 1972)

When using the default initial secant modulus ks or kc correlation, the average undrained shear strength at a point is equal to the input undrained shear strength at that point.
Based on Fig. 15 in Reese et al. (1975), we’ve used the following formula to calculate As:

4. Calculation procedure

Elastic pile sections are represented as displacement-based beam-column elements in OpenSees. As OpenSees operates on a finite element basis, all loadings must be applied incrementally over small steps to analyse the final pile response. Equilibrium must be achieved before progressing to the subsequent step.

In the Pile Action module, loads are applied in two stages. However, loading sequences are not currently available. There are two types of loading: force-based and displacement-based. In the current version, force-based loadings are always applied first. If there is any displacement loading, it is applied after the force-based loading is complete.

4.1 Force-based loading

Applied point loads, distributed loads, and soil displacements are categorised as force-based loading. These loads are incrementally applied in 200 steps, with each step constituting 0.5% of the total load. Each step involves a maximum of 50 iterations. If equilibrium is not achieved within these iterations, an error is returned.

Please note that in the current release, loading stages are not available. Therefore, all force-based loads are applied simultaneously. For instance, if you have a point load, a distributed load, and a soil displacement load, each step of the analysis applies 0.5% of each of these three loads to the model. By the end of 200 steps, all loads reach 100% loading, completing the force-based analysis.

The Pile Action module uses the norm of the displacement increment to determine if equilibrium is achieved. The equilibrium tolerance used in each step of the analysis is 1e-4 m.

4.2 Displacement-based loading

Applied pile displacements fall under the category of displacement-based loadings. This feature enables you to push any point of the pile in a specified direction by a defined displacement. At present, only one displacement-based loading can be applied per model.

When applying a displacement-based loading, the displacement is gradually implemented in 0.5 mm increments until the desired displacement is achieved. Consequently, larger specified displacements will require more steps to complete the analysis. Any load applied during the force-based loading stage remains constant throughout the displacement-based loading process.

4.3 Analysis tips

Just like any other finite element calculations, the Pile Action module may run into convergence errors during the calculations. Convergence error means a calculation fails to reach equilibrium within a certain number of steps.

Running into convergence error does not necessarily mean your analysis is incorrect, it may well be that the soil py curves simply cannot provide sufficient lateral capacity.

There are many reasons why a calculation runs into convergence error. If you run into convergence errors, we recommend the following:

Try to increase the strength of the py curves defined, or select a py curve model with stiffer springs and higher resistance
Try to reduce the applied load on the pile
Try to reduce the number of loads applied to the pile i.e. only specify one load at a time to see if a particular load is pushing the analysis to failure
Try to use a higher elastic modulus and second moment of inertia for your pile.

5. Benchmarking results

5.1 Structural response

In this section, the results from the Pile Action module is compared to those by hand calculations using traditional beam deflection formulae.

Note that all results from Pile Action were read off from the results diagram. More accurate results are available in the downloaded Excel sheets.

Note that for the structural response cases, the pile is assumed to have a fixed base, and this is achieved by specifying very stiff soil springs for the portion of the pile below the ground surface. As a result, the structural actions immediately below the ground surface may spike to very large values due to the stiff springs.

For all structural response cases tested, we have used a 10 m long pile with a fixed base, using 100 nodes. Each segment is about 0.1 m long. The difference between Pile Actions and traditional analytical solutions is less than 2% under these settings. In practical use cases where the free length of the piles is less, the error is expected to be less than 2%.

Untitled 2.png

Case 1: Cantilever Beam with a point load at the free end

This is equivalent to having a fixed-based pile with a lateral load applied at the pile top.

Example input:

P = 10.0 kN

Pile length above ground surface = 10.0 m

Top axial load = 0.0 kN

Elastic modulus = 35,000,000 kPa

Poisson’s ratio = 0.25

Second moment of inertia = 32206233600 mm^4

Analytical solution:

Pile Action results:

Untitled 3.png

Max bending moment: 100 kNm

Maximum shear force: 10 kN

Maximum deflection: 3.0 mm

Case 2: Cantilever Beam with a point load at any point

This is equivalent to having a fixed-based pile with a lateral load applied at any point along the pile above the ground surface.

Example input:

P = 100.0 kN

Pile length above ground surface = 10.0 m

Point load applied at 5 m below pile top

Top axial load = 0.0 kN

Elastic modulus = 35,000,000 kPa

Poisson’s ratio = 0.25

Second moment of inertia = 32206233600 mm^4

Pile Action results:

Untitled 4.png

Maximum shear force: 100 kN

Maximum bending moment: 500 kNm

Maximum deflection: 9.4 mm

Case 3: Cantilever Beam with uniformly distributed load

This is equivalent to having a fixed-based pile with a lateral uniformly distributed load.

Example input:

w = 100 kN/m

Pile length above ground surface = 10.0 m

Top axial load = 0.0 kN

Elastic modulus = 35,000,000 kPa

Poisson’s ratio = 0.25

Second moment of inertia = 32206233600 mm^4

Pile Action results:

Untitled 5.png

Maximum shear force: 1000 kN

Maximum bending moment: 5020 kNm

Maximum deflection: 112.0 mm

Case 4: Cantilever Beam with uniformly varying load

This is equivalent to having a fixed-based pile with a lateral uniformly varying distributed load.

Example input:

w = 0 kN/m at top of pile, 100 kN/m at bottom of pile, uniformly varying

Pile length above ground surface = 10.0 m

Top axial load = 0.0 kN

Elastic modulus = 35,000,000 kPa

Poisson’s ratio = 0.25

Second moment of inertia = 32206233600 mm^4

Pile Action results:

Untitled 6.png

Maximum shear force: 498kN

Maximum bending moment: 1666 kNm

Maximum deflection: 29.9 mm

Case 5: Super-imposing cases 2 and 4

If we applied the loads from case 2 and case 4 together, the bending moment, shear force and deflection profiles can be calculated by super-imposing the results from case 2 and 4.

Example input:

Applied point load:

P = 100.0 kN

Applied distributed load:

w = 0 kN/m at top of pile, 100 kN/m at bottom of pile, uniformly varying

Pile length above ground surface = 10.0 m

Point load applied at 5 m below pile top

Top axial load = 0.0 kN

Elastic modulus = 35,000,000 kPa

Poisson’s ratio = 0.25

Second moment of inertia = 32206233600 mm^4

Expected results from superposition:

Pile Action results:

Untitled 7.png

Maximum shear force: 600kN

Maximum bending moment: 2180 kNm

Maximum deflection: 39.6 mm

5.2 FWHA report cases

Record Number 19: Matlock Soft Clay

This case comes from the paper “Synthesis of Load-Deflection Characteristics of Laterally Loaded Large Diameter Drilled Shafts: Technical Report”, a technical report published by the Texas A&M Transportation Institute in collaboration with FHWA and Texas Department of Transportation.

This report provides both the original experimental results that were used to develop the Matlock soft clay model, as well as the results generated by LPile.

The results from Pile Action are matching the LPile results provided in the report. However, the report did not mention the distance between the pile head and the ground surface. Fig 2 in the original paper by Matlock (1970) suggests that the load is applied 1 ft above the ground surface, however the text in the paper suggests that the lateral load is applied at the mud line (in the “Research Program” Section). Other sources also mentioned that the lateral load was applied at 0.0635 m (2.5 inches) above the ground surface. Therefore, we’ve assumed the lateral load is applied at the ground surface, which gave us the best agreement with the FHWA report. If the load was to be applied at 0.0635 m above the ground surface, the difference is still within 5%.

Soil properties and pile properties adopted in record case 19

The input used in Pile Action:

Pile Section:

Pile diameter	0.324 m
Elastic modulus	199948040 kPa
Poissons ratio	0.25
Second moment of inertia	174400967.5 mm^4
Pile top depth	0 m.b.g.l.
Pile bottom depth	12.2 m.b.g.l.

Soil Model:

Soil model	Matlock Soft Clay
Unit weight	20 kN/m^3
Undrained shear strength	26 kPa
Strain factor	Default (taken as 0.02)

Loading Option:

Applied load type	Applied pile top displacement
Analysis setting	Static
Applied displacement	52 mm
Applied displacement depth	0 m

Untitled 9.png

Top lateral shear force vs. top lateral deflection

Record Number 13: Sand (Reese)

This report provides both the original experimental results that were used to develop the Matlock soft clay model, as well as the results generated by LPile.

The results from Pile Action are within 15% of the LPile results provided in the report. However, the distance between the pile head and the ground surface was not mentioned in the report or the original paper. We’ve assumed the pile is at the ground surface. The analysis results from Pile Action is slightly closer to the experimental results compared to the LPile results provided in the report, but we are unsure what assumptions were made in the LPile model based on the information provided in the report.

Soil properties and pile properties adopted in record case 13

The input used in Pile Action:

Pile Section:

Pile diameter	0.356
Elastic modulus	15857948 kPa
Poissons ratio	0.25
Second moment of inertia	786677395.1 mm^4
Pile top depth	0 m.b.g.l.
Pile bottom depth	21.0 m.b.g.l.

Soil Model:

Soil model	Reese (Sand)
Unit weight	20 kN/m^3
Friction angle	28°
Initial modulus, ks or kc	Default

Loading Option:

Applied load type	Applied pile top displacement
Analysis setting	Static
Applied displacement	85 mm
Applied displacement depth	0 m

Untitled 11.png

Top lateral shear force vs. top lateral deflection

5.3 LPile py curve benchmarking cases

Py curves generated using the LPile software were compared to those generated by Pile Actions.

Soft Clay (Matlock 1970)

The Matlock Soft Clay model proposed by Matlock (1970) has been verified against the results produced by the LPile software.

Note on the applicability of this py curve model:

The Matlock soft clay model is only based on tests conducted in soils in soft clay near Lake Austin, Texas and Sabine Pass, Texas.
The range of undrained shear strength in these tests is between 14.4 kPa to 38 kPa. Caution should be exercised when using an undrained shear strength outside the test ranges.
The strain factor, e_50, is the strain of the soil at 50% of the maximum stress on a laboratory stress-strain curve. It can be determined by dividing the shear strength, c, by an estimated secant modulus of elasticity, E_c. Matlock (1970) quotes Skempton’s approach, the ratio of E / c generally falls between 50 - 200 for most clays, therefore a value of 0.005 to 0.02 is recommended in Matlock (1970).

Matlock Soft Clay - Static springs

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 35 GPa, dia = 0.75 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Undrained shear strength (kPa)	Strain factor, e50	Static or cyclic
0 - 20	Matlock soft clay	18 top and bottom	30 top and bottom	default	Static

If we were to overlay the graphs, the py curves generated by Pile Action at the same y locations are 100% matching those generated by LPile.

Note that in actual calculation, the values of soil/pile relative displacement y may vary from those generated by LPile. For this benchmarking exercise, the input y values were set to be the same for comparison purposes.

Matlock Soft Clay - Cyclic springs

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 45 GPa, dia = 0.9 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Undrained shear strength (kPa)	Strain factor, e50	Static or cyclic
0 - 20	Matlock soft clay	16 - 18, linearly interpolated	10 - 35, linearly interpolated	default	Cyclic (5 cycles)

If we were to overlay the graphs, the py curves generated by Pile Action at the same y locations are 100% matching those generated by LPile.

Stiff Clay without free water (Welch and Reese 1972)

The Stiff Clay without free water model proposed by Welch and Reese (1972) has been verified against the results produced by the LPile software.

Stiff Clay without free water - Static springs

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 35 GPa, dia = 0.75 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Undrained shear strength (kPa)	Strain factor, e50	Static or cyclic
0 - 20	Stiff Clay without free water	18 top and bottom	50 top and bottom	default	Static

If we were to overlay the graphs, the py curves generated by Pile Action at the same y locations are 100% matching those generated by LPile.

Stiff Clay without free water - Cyclic springs

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 45 GPa, dia = 0.9 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Undrained shear strength (kPa)	Strain factor, e50	Static or cyclic
0 - 20	Stiff Clay without free water	16 - 18, linearly interpolated	50- 100, linearly interpolated	default	Cyclic

If we were to overlay the graphs, the py curves generated by Pile Action at the same y locations are 100% matching those generated by LPile.

Note that in actual calculation, the values of soil / pile relative displacement y may vary from those generated by LPile. For this bench marking excercise, the input y values were set to be the same from comparison purposes.

Stiff Clay with free water (Reese et al. 1975)

The Stiff Clay with free water model proposed by Reese et al. (1975) has been verified agasint the results produced by the LPile software.

Stiff Clay without free water - Static springs

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 35 GPa, dia = 0.75 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Undrained shear strength (kPa)	Strain factor, e50	Static or cyclic
0 - 20	Stiff Clay with free water	18 top and bottom	50 top and bottom	default	Static

There are three main differences between Pile Action and LPile:

The difference in ultimate soil resistance and displacement required to reach the final straight line segment was caused by the difference in As values calculated.
The bottom curve at depth = 0.0 m.b.g.l has a different peak resistance. The py curve computed in LPile was set to have a resistance of 0 kN/m for all displacements, but Pile action still calculated some resistance for the spring at the ground surface. The difference caused by this is negligible due to the small tributary area of an individual spring in the calculations.
The initial stiffness is also slightly different. Pile Action generates the initial linear section, the parabolic sections, the third linear section then the final straight line section separately before joining them together at intersecting points, ensuring the initial stiffness from the point of origin matches that calculated using the user input initial modulus value k. The difference is also considered negligible.

Stiff Clay without free water - Cyclic springs

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 45 GPa, dia = 0.9 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Undrained shear strength (kPa)	Strain factor, e50	Static or cyclic
0 - 20	Stiff Clay without free water	16 - 18, linearly interpolated	50- 100, linearly interpolated	default	Cyclic

If we were to overlay the graphs, the py curves generated by Pile Action at the same y locations are within 95% compared to those generated by LPile.

The minor differences between Pile Action and LPile are due to the following reasons:

The empirical adjustment factor Ac is rounded to 3 dp in LPile but is not rounded in Pile Action. This creates a minor difference in the calculation of yp, hence the small difference in the residual strength of the cyclic springs.
The initial modulus kc from correlation are the same (i.e. the correlation relationship between input undrained shear strength and the initial modulus kc is the same), but does not appear to be always enforced in LPile. The user defined or correlated kc values from Pile Action are the exact same as those generated by LPile, but is always enforced so that the initial stiffness of the py curves match 100% of what the user entered or what the correlation gives. This created a minor difference in initial stiffness of the py curves.
Pile Action uses the cumulated average undrained shear strength for soil springs within 5 times the diameter of the pile at that depth for e50 and kc correlations. If the current depth is greater than 5 times the diamter of the pile, the undrained shear strength at that depth is used. These assumptions appear to be consistent with the original paper (Reese 1975) and LPile.
The bottom curve at depth = 0.0 m.b.g.l has a difference peak resistance. The py curve computed in LPile was set to have a resistance of 0 kN/m for all displacements, but Pile action still calculated some resistance for the spring at ground surface. The difference caused by this is negligible due to the small tributary area of an individual spring in the calculations.

Sand (Reese et al. 1974)

Sand (Reese et al. 1974) - Static springs

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 35 GPa, dia = 0.75 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Effective friction angle, phi	Initial modulus, ks	Static or cyclic
0 - 20	Sand (Reese et al. 1974)	18 top and bottom	34 top and bottom	default	Static

The difference between Pile Action and LPile is minimum and can be considered the same from a practical perspective.

However there’s a < 5% difference in the benchmarking case, which comes from how the factor As is functionalised in Pile Action and LPile. This is considered practically negligible for analysis.

Sand (Reese et al. 1974) - Cyclic springs

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 45 GPa, dia = 0.9 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Effective friction angle, phi	Initial modulus, ks	Static or cyclic
0 - 20	Sand (Reese et al. 1974)	16 - 18, linearly interpolated	28 - 42, linearly interpolated	default	Cyclic

If we were to overlay the graphs, the py curves generated by Pile Action at the same y locations are 100% matching those generated by LPile.

The difference between Pile Action and LPile is minimal and can be considered the same from a practical perspective.

However there’s a < 5% difference in the benchmarking case, which comes from how the factor As is functionalised in Pile Action and LPile. This is

Weak rock (Reese and Nyman 1978)

Weak Rock case 1

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 35 GPa, dia = 0.75 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Uniaxial Compressive Strength, UCS (kPa)	Etrain factor, e_ir	E_rm, rock mass modulus (kPa)	RQD (%)	Static or cyclic
0 - 20	Weak Rock	18	1500	0.005	200000	40%	Static

If we were to overlay the graphs, the py curves generated by Pile Action at the same y locations are 100% matching those generated by LPile.

The difference between Pile Action and LPile is minimal and can be considered the same from a practical perspective.

However there’s a < 5% difference in the benchmarking case, which comes from how the factor As is functionalised in Pile Action and LPile.

Weak Rock case 2

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 45 GPa, dia = 0.9 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Uniaxial Compressive Strength, UCS (kPa)	Etrain factor, e_ir	E_rm, rock mass modulus (kPa)	RQD (%)	Static or cyclic
0 - 20	Weak Rock	16 - 18, linearly interpolated	1500- 5000, linearly interpolated	0.005 - 0.0005, linearly interpolated	200000 - 400000, linearly interpolated	40% - 80%, linearly interpolated	Cyclic

If we were to overlay the graphs, the py curves generated by Pile Action at the same y locations are 100% matching those generated by LPile.

The difference between Pile Action and LPile is minimal and can be considered the same from a practical perspective.

However there’s a < 5% difference in the benchmarking case, which comes from how the factor As is functionalised in Pile Action and LPile.

Massive rock (Liang et al. 2009)

Massive rock case 1

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 35 GPa, dia = 0.75 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Uniaxial Compressive Strength, UCS (kPa)	Hoek Brown material index, m_i	Geological Strength Index, GSI	Poisson’s ratio of rock mass, v	Static or cyclic
0 - 20	Massive rock (correlated Em)	18	5000	17	50	0.3	Static

If we were to overlay the graphs, the py curves generated by Pile Action at the same y locations are 100% matching those generated by LPile.

The difference between Pile Action and LPile is minimal and can be considered the same from a practical perspective.

However there’s a < 5% difference in the benchmarking case, which comes from how the factor As is functionalised in Pile Action and LPile.

Massive rock case 2

Pile input:

Load case	Pile properties	Pile Head Constraints
Single pile, 20 m embedment, 2 m above ground	Elastic non-yielding section, E = 45 GPa, dia = 0.9 m	Free pile head

Soil model input:

Depth	py curve model	Unit weight (kN/m^3)	Uniaxial Compressive Strength, UCS (kPa)	Hoek Brown material index, m_i	Geological Strength Index, GSI	Poisson’s ratio of rock mass, v	Static or cyclic
0 - 20	Weak Rock	16 - 18, linearly interpolated	1500- 5000, linearly interpolated	17	30 - 50, linearly interpolated	0.3	Cyclic

If we were to overlay the graphs, the py curves generated by Pile Action at the same y locations are 100% matching those generated by LPile.

The difference between Pile Action and LPile is minimal and can be considered the same from a practical perspective.

However there’s a < 5% difference in the benchmarking case, which comes from how the factor As is functionalised in Pile Action and LPile.

6. Structural Sections

Pile sections can be defined in the 'Pile' tab. Each pile can consist of multiple pile sections of different properties and types.

6.1 Elastic Sections

Elastic pile sections are fully elastic in all loading directions, with infinite moment capacity. The moment in the section is proportional to the curvature in the pile element. The stress/strain relationship and the moment/curvature relationship for an elastic section is shown below:

6.2 Elastic Perfectly Plastic Sections

Elastic Perfectly Plastic sections behave elastically until the user-defined moment capacity is reached. After reaching the moment capacity, the sections are perfectly plastic, i.e. no strain hardening occurs after reaching the moment capacity, and the moment capacity remains constant.

Since no strain hardening occurs, if a particular load is applied to the pile resulting in an Elastic Perfectly Plastic section reaching its moment capacity, further rotation in that section does not generate any additional resistance against the load. Therefore, when Elastic Perfectly Plastic Sections are used, we recommend applying a pile displacement rather than loads to increase the likelihood of reaching convergence.

7. Pile Head Restraints

There are currently three types of pile head translational restraints and three types of pile head rotational restraints.

7.1 Pile Head Translation

Pile Head Fixity	Description
Free	Pile head displacement in the x direction is allowed with no restriction.
Fixed	Pile head displacement in the x direction is not allowed; the fixed pile head will provide a force to restrict any pile head movement in the x direction.
Restrained	Pile head displacement in the x direction is restricted by a translational spring with a user-defined translational stiffness in units of kN/m. The resistance of the spring depends on the displacement of the pile head from its original neutral x axis position. i.e. if we have a translational spring of stiffness 10,000 kN/m, and we applied a pile head displacement of 100 mm (0.1 m), the resistance applied to the pile head due to the translational spring is 0.1 * 10,000 = 1000 kN, in a direction opposite to the applied pile head displacement.

7.2 Pile Head Rotation

Pile Head Fixity	Description
Free	Pile head rotation about the y axis (into the page) is allowed with no restriction.
Fixed	Pile head rotation about the y axis is not allowed; the fixed pile head will provide a moment to restrict any pile head rotation about the y axis.
Restrained	Pile head rotation about the y axis is restricted by a rotational spring with a user-defined rotational stiffness in units of kNm/rad. The resistance of the spring depends on the rotation of the pile head from its original neutral rotation about the y axis. i.e. if we have a rotational spring of stiffness 10,000 kNm/rad, and we applied a pile head displacement that causes the pile head to rotate by 0.1 radian, the rotational resistance applied to the pile head due to the rotational spring is 0.1 * 10,000 = 1000 kNm, in a direction that acts against the resultant rotation.

7.3 Summary of Pile Head Fixity

Based on the pile head fixity options, there are 9 possible combinations of pile head fixities:

Case	Translation	Rotation	Description
1	Free	Free	Both pile head translation and rotation are allowed.
2	Free	Fixed	Pile head translation is allowed, but pile head rotation is fixed to be vertical.
3	Fixed	Free	Pile head rotation is allowed, but pile head translation is fixed to be 0 mm.
4	Fixed	Fixed	Both pile head translation and rotation are not allowed. Pile head translation is fixed at 0 mm, and pile head rotation is fixed to be vertical.
5	Restrained	Free	Pile head rotation is allowed, but pile head translation is retrained by a translational spring of user-defined stiffness.
6	Restrained	Fixed	Pile head rotation is fixed to be vertical, but pile head translation is retrained by a translational spring of user-defined stiffness.
7	Free	Restrained	Pile head translation is allowed, but pile head rotation is retrained by a rotational spring of user-defined stiffness.
8	Fixed	Restrained	Pile head translation is fixed to be 0 mm, but pile head rotation is retrained by a rotational spring of user-defined stiffness.
9	Restrained	Restrained	Pile head translation is restrained by a translational spring, and pile head rotation is retrained by a rotational spring. Both springs have user-defined stiffness values.

References

Structural analytical solution formula:
https://home.engineering.iastate.edu/~shermanp/STAT447/STAT Articles/Beam_Deflection_Formulae.pdf

Convergence criteria:
https://opensees.berkeley.edu/wiki/index.php/Norm_Displacement_Increment_Test

1. Introduction

2. Sign conventions

3. Assumptions

3.1 General assumptions

3.2 Loading assumptions

3.3 Soil spring specific assumptions

Soft Clay (Matlock et al., 1970)

Stiff Clay without Free Water (Welch and Reese, 1972)

Stiff Clay with Free Water (Welch and Reese, 1972)

4. Calculation procedure

4.1 Force-based loading

4.2 Displacement-based loading

4.3 Analysis tips

5. Benchmarking results

5.1 Structural response

Case 1: Cantilever Beam with a point load at the free end

Case 2: Cantilever Beam with a point load at any point

Case 3: Cantilever Beam with uniformly distributed load

Case 4: Cantilever Beam with uniformly varying load

Case 5: Super-imposing cases 2 and 4

5.2 FWHA report cases

Record Number 19: Matlock Soft Clay

Record Number 13: Sand (Reese)

5.3 LPile py curve benchmarking cases

Soft Clay (Matlock 1970)

Matlock Soft Clay - Static springs

Matlock Soft Clay - Cyclic springs

Stiff Clay without free water (Welch and Reese 1972)

Stiff Clay without free water - Static springs

Stiff Clay without free water - Cyclic springs

Stiff Clay with free water (Reese et al. 1975)

Sand (Reese et al. 1974)

Weak rock (Reese and Nyman 1978)

Weak Rock case 1

Weak Rock case 2

Massive rock (Liang et al. 2009)

Massive rock case 1

Massive rock case 2

6. Structural Sections

6.1 Elastic Sections

6.2 Elastic Perfectly Plastic Sections

7. Pile Head Restraints

7.1 Pile Head Translation

7.2 Pile Head Rotation

7.3 Summary of Pile Head Fixity

References

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