21 - Pile Head Stiffness Matrix and Spring Constants

1.0 Introduction

This tutorial covers how to obtain the Pile Head Stiffness Matrix and Simplified Spring Constants of a reinforced concrete pile under various loading conditions in RSPile.

Topics Covered:

• Pile Head Stiffness Matrix
• Simplified Spring Constants

Finished Product

The finished product of this tutorial can be downloaded here.

1.1 Problem description

Say we have a reinforced concrete rectangular pile with:

• B = 0.8 m
• W = 1.2 m

The section is reinforced with 32 mm bars. There are 7 peripherally distributed bars on each width side (x’) and another 3 intermediate 32mm bars at each of the breath sides, giving a total of 20 bars.

• f’c=50MPa
• Es=200000MPa
• Fy=420MPa

The pile will be 20m long in a competent stiff clay, with an undrained shear strength Su=250kPa, and the adhesion factor for skin friction = 0.5.

Required:

We will use RSPile to obtain the following:

1. The pile head stiffness matrix for the following loading conditions
   
   Fx = 1000 kN
   My = 750 kN.m
   Fy = 500 kN
   Mx = -750 kN.m
   Fz = -10 000 kN
   using 10 equal steps of load levels.
2. The direct spring constants using the simplified spring constants option.

2.0 Model Pile Head Stiffness Matrix

2.1 Project Settings

Begin by defining your Project Settings:

1. Select Home > Analysis > Project Settings
2. Go to the Pile Analysis Type tab.
3. Select Individual Pile Analysis = Axial/Lateral Loading
4. Check the box for Include P-Delta Effect
   
5. Click OK to close the dialog.

2.2 Soils

1. Select Soils > Soil > Define Soil Properties
2. For Soil Property 1, select the Lateral tab.
3. Enter:
   1. Soil Type = Dry Stiff Clay
   2. Undrained shear strength = 250kPa
      
4. Select the Axial tab.
5. Enter:
   1. Soil Type = Drilled Clay
   2. Ultimate Shear Resistance = 125 kPa (assuming adhesion=0.5*cohesion)
   3. Enter Ultimate End Bearing Resistance = 2250 kPa (maximum end bearing is 9*250 = 2250kPa)
      
6. Click OK to close the dialog.

2.3 Piles

2.3.1 Define Pile Section Properties

1. Select Piles > Pile Properties > Pile Sections
2. For Pile Section 1, select Section Type = Reinforced concrete.
3. Enter Compressive Strength f’c = 40MPa
4. For the Cross Section enter:
   1. Cross Section = Rectangular
   2. Section Width = 1.2 m
   3. Section Depth = 0.8 m
5. Open the Concrete Designer by clicking Design…
6. In the Reinforcement tab, click Add to add a Reinforcement pattern.
7. Enter:
   1. Rebar Size = Europe > CEB 32 mm
   2. Pattern Type = Rectangle
   3. Check the Peripheral bars box for peripheral distribution.
   4. Number of Bars in X’ = 7
   5. Number of Bars Y’ = 5
   6. Min Cover Depth = 87 mm
8. Click OK to close the Concrete Designer. Your section is ready.
9. Click OK again to close the Pile Section dialog.

2.3.2 Pile Types

1. Select Piles > Pile Properties > Pile Types
2. For Pile Type 1:
   1. Section Property = Pile Section 1
   2. Length = 20m
3. Click OK to close the dialog.

2.3.3 Define Loading Property

1. Select Piles > Add Piles > Single
2. In the Loading tab, click Add to add a Loading Property.
3. Change the Loading Direction to General.
4. Click the Add button at the bottom of the dialog to add the following loads:
   1. Shear X (kN) = 500
   2. Moment Y (kNm) = 375
   3. Shear Y (kN) = 250
   4. Moment X (kNm) = -375
   5. Force Z (kN) = - 5000

The loads are as shown above (as required to be half of the maximums given in the problem data).

2.3.4 Stiffness Matrix Calculator

1. In Advanced tab, check the box for Stiffness Matrix Calculator.
2. A dialog will open. Enter Number of Steps = 10 and Load Division = Evenly Spaced. Fill in the loads as below:
   1. X Direction
      1. Shear X (kN) = 1000
      2. Moment Y (kNm) = 750
   2. Y Direction
      1. Shear Y (kN) = 500
      2. Moment X (kNm) = -750
   3. Z Direction
      1. Force Z (kN) = -10 000

Here you filled the maximum loads given in the problem data.

3. Click OK to close the dialog and return to the Advanced tab.

2.3.5 Simplified Spring Constants

1. In the Advanced tab, check the box for Simplified Spring Constants. A dialog will open asking for the loads and the steps.
2. Enter Number of Steps = 10 and then enter the loads as shown below:
   1. X-Direction
      1. Shear X (kN) = 1000
      2. Moment XZ (kNm) = 750
   2. Y-Direction
      1. Shear Y (kN) = 500
      2. Moment YZ (kNm) = -750
   3. Z-Direction
      1. Axial Force (kN) = -10 000
         
3. Click OK to close Simplified Spring Constants dialog.
4. Click OK to close the Add Pile dialog and add the pile.

3.0 Save and Compute

3.1 Number format

1. Before computing, select View > Settings > Contour
2. Click Number Format…
3. Select Decimal and enter Decimal places = 3.
4. Click Defaults… and select Make current settings the default. Click OK to save the new default.
5. Click OK and then Done to close the Contour options dialog.

3.2 Save the model

1. Select File > Save and save the model in the desired name and folder.

The program is ready to compute now.

3.3 Compute

1. Select Results > Results > Compute
   

4.0 Results

After the run is finished, you can view the results for pile head stiffness matrix and your spring constants by selecting Results > Graphs > Other.

4.1 Pile Head Stiffness Matrix

1. Select Results > Graphs > Other
2. Choose Pile head stiffness matrix from the drop down.
3. A small square cursor will appear. Put the cursor on the pile, left click to choose the pile, and press Enter.

The results will be shown as 9 graphs as follows:

The data can then be exported to an excel sheet by selecting:

1. Chart > File > Export to Excel
2. In the drop down, select Export Data to Excel.

The results for the nine graphs will appear in 9 sheets similar to the following:

The above is for K11. There are 8 more sheets for K22, K23, K32, K33, K44, K45, K54, and K55.

4.2 Spring Constants

If simple spring constants are required, the results may be obtained the same above.

1. Return to the modeller view.
2. Select Results > Graphs > Other
3. Choose Pile Spring Constants from the drop down.
4. A small square cursor will appear. Left click to choose the pile, and press Enter.
   

Using the same approach, we shall get 5 graphs this time:

In this method the stiffness matrix is made of diagonals only. Again, you may export the data into an Excel spread sheet by selecting:

1. Chart > File > Export to Excel
2. In the drop down, select Export Data to Excel.

Below are the spring constants for the fifth loading step: