FINITE DIFFERENCE MODELLING OF LATERALLY LOADED PILES GEEN501 – Advanced Geotechnics Coursework (Group 13) Abstract This report aims to provide a design analysis for the client for a concrete pile that is subject to an active load of 45kN and a passive load generated by an embankment. In order to carry out the design analysis, a finite difference method was used in conjunction with a number of logical assumptions to generate a matrix and from a series of iterations, the deflection of the pile under passive and active loading was obtained. The lateral deflection of the concrete pile was found to be 0.0258m which generated a bending moment of 182.3426kNm. The deflection was deemed to be too great for the assumed usage of the pile and so suitable remedies to this problem were suggested. A suitable pile design was determined from this moment to be 0.6m, reducing the lateral deflection by 0.016m. Daniel Wilkinson – Student ID: 10342869 Lawrence Taylor – Student ID: 10395608
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Abstract This report aims to provide a design analysis for the client for a concrete pile that is subject to an active load of 45kN and a passive load generated by an embankment. In order to carry out the design analysis, a finite difference method was used in conjunction with a number of logical assumptions to generate a matrix and from a series of iterations, the deflection of the pile under passive and active loading was obtained. The lateral deflection of the concrete pile was found to be 0.0258m which generated a bending moment of 182.3426kNm. The deflection was deemed to be too great for the assumed usage of the pile and so suitable remedies to this problem were suggested. A suitable pile design was determined from this moment to be 0.6m, reducing the lateral deflection by 0.016m.
Daniel Wilkinson – Student ID: 10342869 Lawrence Taylor – Student ID: 10395608
1.1 Statement of the Problem A pile is required near an embankment to support a lateral load. The head of the pile is to
be fixed so as not to provide any rotation to the attached structure. Either the embankment
will not be built until the pile has been embedded or the ground conditions relating to the
passive displacement of the earth has not been collected. Therefore, the conditions of the
embankment must be estimated using graphs to produce the lateral displacement of the
soil. It is unknown what the pile will be supporting but possible structures include a bridge
abutment, supporting an inclined column or supporting a retaining structure etc. (The
Institution of Structural Engineers, 2013, p. 142).
A report must be compiled which not
only analyses the conditions that the
pile will be subjected to using Finite
Difference Modelling, but also
produce a critical analysis of the
results and produce a conclusion to
determine whether the pile solution is
appropriate.
The report is structured as follows:
Section 2 outlines the method
employed to model the pile and the boundary conditions used. Section 3 provides the initial
results of the pile solution. Section 4 critically analyses and discusses the results providing
an alternative solution to the problem & Section 5 presents the conclusions of the report. All
calculations relating to the report can be found in Appendices A & B.
1.2 Soil and Loading Conditions Loading conditions:
The head of the pile is to be subjected to an active (Design) load of 45kN at the head of the
pile. The pile will also be subjected to passive lateral loading via an adjacent embankment of
which the loading will only affect the portion of the pile embedded in the top soil. The
maximum passive soil displacement due to the embankment is assumed to be
approximately 20mm.
Pile attributes:
The diameter of the pile is 0.52m. The head of the pile is to be fixed (to allow for horizontal
translation but no rotation) and the tip of the pile is to be modelled as rigidly embedded, as
it is embedded in a hard stratum by 1.5m. The elastic modulus of the concrete is, E =
2.2x107kN/m².
Soil conditions:
Depth of top soil is 6.7m. The Kh value of the top soil is to 5700yz0.72kN/m³ and the Kh value
of the hard stratum is to 3300yzkN/m³ (subgrade reaction moduli).
Figure 1: Diagram showing type of problem to be
solved.
3
2.0 Method
2.1 Determination of Boundary Conditions The boundary conditions of the numerical model must conform to those of the pile being
modelled. As the pile has a fixed head, it can be assumed that there is no rotation at node 1,
so that the pile will only deflect through translation. This means the first fixed head
boundary condition is (Azizi, 2013, p. 290):
𝑦−1 = 𝑦2
As the pile head is fixed, a moment will be generated at the head of the pile when it is
subjected to a lateral force. The equation for the free head pile is modified to include a
moment and through rearrangement it can be seen that the second boundary condition for
the fixed head pile is as follows (Azizi, 2013, p. 290):
𝑦−2 = 𝑦3 −2𝐻𝑜(∆𝑧)³
𝐸𝑝𝐼𝑝
For this problem, the pile is to be embedded into a hard stratum. For this reason, it was
assumed that the pile would be rigidly embedded. Although the depth of embedment is not
a design parameter, it is assumed that for the use of a numerical model that the pile is
flexible (Azizi, 2013, p. 281).
In order to implement a computational molecule in correspondence with these boundary
conditions, it is necessary to have two imaginary nodes above the fixed head pile. These
imaginary nodes form part of the boundary conditions by using the assumptions that the
pile head is fixed and the pile tip is rigidly embedded in the hard stratum.
As the pile is rigidly embedded, it can be assumed that there is neither deflection nor
rotation at the tip and according to Azizi (2013, 2013, p. 290) this creates only one imaginary
node at the base of the pile. These conditions conform to the boundary conditions below:
𝑦𝑛 = 0
𝑦𝑛+1 = 𝑦𝑛−1
2.2 Determination of Pile Deflection To allow for a full analysis of the pile, the deflection must be determined by using the
second derivative of the beam equation. This involves the use of a computational molecule
in conjunction with a suitably sized finite difference mesh. A mesh with a value of ∆𝑧 = 0.05
is used as too finer a mesh would introduce truncation errors (Azizi, 2013, p. 300). In total,
this gives 165 nodes which is deemed an appropriate number of nodes for the solution.
As the problem involves two layers of soil, there is a need for two separate molecules, one
for each soil layer. In addition to this, a third computational molecule is used for node 𝑦165
to incorporate the additional imaginary node placed at the tip of the pile. In total three
computational molecules are used in the realisation of the problem based on the finite
difference formulas (Azizi, 2013, p. 285):
(1)
(2)
(4)
(3)
4
𝑦𝑖−2 − 4𝑦𝑖−1 + (6 +𝐾ℎ𝑑
𝐸𝑝𝐼𝑝(∆𝑧)4) 𝑦𝑖 − 4𝑦𝑖+1 + 𝑦𝑖+2 =
𝐾ℎ𝑑
𝐸𝑝𝐼𝑝(∆𝑧)4𝑔(𝑧)
𝑦𝑖−2 − 4𝑦𝑖−1 + 6𝑦𝑖+1 (𝐾ℎ𝑑
𝐸𝑝𝐼𝑝
(∆𝑧)4 − 4) 𝑦𝑖 + 𝑦𝑖+2 =𝐾ℎ𝑑
𝐸𝑝𝐼𝑝
(∆𝑧)4𝑔(𝑧)
Computational molecules are then devised for each soil layer and one for the rigidly
embedded boundary condition at the pile tip.
These computational nodes are placed systematically on each node along the pile shaft and
a separate nodal equation is used for each of the 165 nodes. The first molecule is used to a
depth of 6.7m of top soil and the second is used in the hard stratum for the remaining 1.5m.
The third modified molecule is used to enforce the boundary conditions at the rigidly
embedded tip of the pile.
The first iteration uses an assumed value for y, the deflection. A value of 𝑦 = 0.05𝑚 is used
for an initial deflection assumption. Subsequent interactions use the deflections from the
previous matrix iteration. Iterations are then performed until the difference between each
iteration was less than or equal to 1 × 10−4𝑚.
2.3 Determining of Slope, Bending Moment, Shear Force and Soil Reaction From the deflection values determined from matrix iterations, the slope, bending moment,
shear force and soil reaction can then be determined (equations 7, 8, 9 & 10).
𝑆𝑙𝑜𝑝𝑒 =𝑦𝑖+1−𝑦𝑖−1
2(∆𝑧)
𝐵𝑒𝑛𝑑𝑖𝑛𝑔 𝑀𝑜𝑚𝑒𝑛𝑡 =𝐸𝑝𝐼𝑝
(∆𝑧)2(𝑦𝑖−1 − 2𝑦𝑖 + 𝑦𝑖+1)
1
1
-4
-4
6 + 0.000000234613yz0.72
1
1
-4
-4
6 + 0.000000135829yz
1
1
-4
0.000000135829yz - 4
6
(5)
(6)
(7)
(8)
Figure 2: Computational molecules.
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𝑆ℎ𝑒𝑎𝑟 𝐹𝑜𝑟𝑐𝑒 =𝑀𝑖+1−𝑀𝑖−1
2(∆𝑧)
𝑆𝑜𝑖𝑙 𝑅𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = −𝐾ℎ𝑑. 𝑦
These values are plotted to obtain the graphs shown in the results section below.
03.0 Results
Figure 3: Graph showing lateral deflection of pile with and
without passive loading.
Figure 4: Graph showing bending moment of pile with and without passive loading.
(9)
(10)
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Figure 5: Graph showing shear force of pile with and without passive loading.
Figure 6: Graph showing Soil Reaction of pile with and
without passive loading.
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Table 1: Table showing maximum values for 0.52m diameter pile actively and passively loaded and
only actively loaded.
4.0 Calculations (Design of Concrete Pile) The pile has been modelled as a beam. To convert the rectangular beam to a circular
concrete pile the Charles Whitney method (McCormac and Brown, 2015) has been used