To derive the interaction equation, first consider a single pile as illustrated in the figure below.
Assume that the deflected shape of the pile is very nearly in a plane containing the axis of the pile. This assumption is valid if:
1. The pilehead torque does not influence the lateral deflection.
2. The resultant pilehead bending moment is about an axis perpendicular to the direction of the resultant pilehead lateral force.
Note: The reasons for this assumption will be addressed later in the discussion.
The first of these conditions may be accepted based on the usual small displacement restriction of structural analysis.
The usual conditions under which offshore structures (and indeed most other structures) operate produce resultant pilehead bending moments and lateral forces that nearly satisfy condition 2. Note that it is not assumed that all of the piles deform in the same plane, but only that each pile deforms in a plane. That plane, however, may be different from pile to pile.
Plots can be developed relating any pilehead force (or moment) component to any pilehead displacement (or rotation) component for fixed values of axial load and the other displacement or rotation components. A typical plot may have the general appearance of Figure 3. The slope of the curve at a point such as “A”, is defined as the stiffness coefficient relating the force or moment to the displacement or rotation at that point “A”. It is a function of displacement, rotation, or axial load.
31 The equation of the F vs. δ curve may be written in the form:
(1)
where K and FO are functions of δ, θ, and P.
These considerations are generalized to 6 pilehead degrees of freedom and the results written in matrix form:
(2)
where {F}, {δ}, and {FO} are 6 × 1 matrices (column vectors) and [K] is a 6 × 6 matrix. In addition, [K] and {FO} are functions of δ, θ, and P.
32 Figure 4 is a schematic sketch of a jacket supported by piles. The nonlinear piles are symbolically represented by the spring-like elements at the pilehead joints. External forces are applied over the jacket including, perhaps, at the pilehead joints. The jacket consists of the pile interface degrees of freedom (designated by subscript I) and the “free”
degrees of freedom (designated by the subscript F). The Force-Displacement relationship for the jacket-pile combination can be written in partitioned matrix notation as:
(3)
In equation 3, the terms FF and FI are the external force vectors applied to the structure at the “free” and interface degrees of freedom respectively and DF and DI are the corresponding displacement vectors. KP is the assembled nonlinear stiffness matrix of the piles at the interface degrees of freedom, and FO is the column vector of the pile
“intercept” forces. As discussed previously, both KP and FO depend on the interface displacement vector DI. All other stiffness coefficients are independent of the displacements and can be evaluated once at the start of the problem.
Figure 5 (above) shows the free bodies of the jacket and piles. The forces acting in these bodies include the equal and opposite interface force vector, FI. The force-displacement relationships for the piles and jacket respectively are:
(4)
(5)
33 Equations 4 and 5 are simply a breakdown of equation 3 into the contribution from the nonlinear pile and linear structure respectively. Combining these two equations yields equation 3.
Equation 5 can be expanded, resulting in:
(6)
(7)
Equation 6 is solved for DF and the result is substituted into equation 7, which is then rearranged to give:
(8)
Equation 8 is a matrix equation whose order is equal to the number of interface degrees of freedom of equation 4.
Adding these two equations eliminates the internal interface vector FI.
(9)
The terms in this equation can be grouped into those that depend on DI and those that do not. The like terms are collected and the equation are rearranged resulting in:
(10) where:
Equations 4 and 10 are the basis for the iterative solution. One can do an analysis of each pile using the current pilehead displacement vector as its boundary condition. The pilehead force and moment are calculated, then a second pile analysis is done with an increment added to the displacements, resulting in new forces and moments. The stiffness coefficients then are the ratios of each of the pilehead force (or moment) increments to each of the displacement (or rotation) increments. The pilehead intercept force (or moment) components are then calculated using equation 4.
This process can be repeated for each iteration at each pilehead and for each load case. This approach, although theoretically sound, can require a large number of pile analyses.
The PSI program uses a more efficient approach. Instead of doing pile analyses at each pile for each iteration of each load case, a number of pile analyses are done at the outset to produce a set of pilehead force vs. displacement curves similar to Figure 3. Values for pilehead axial load (or deflection), lateral deflection, and rotation that span the range of values expected in the final solution are used. The program performs a pile analysis for each combination of these loads and rotations and stores the results. For each iteration, the pilehead displacements are used to determine the resulting pilehead stiffness coefficient and intercept forces from the curves. This procedure is continued until a preliminary convergence is met. Upon converging, PSI continues iterating but now performs a complete pile stiffness analysis for each iteration. This fine tuning procedure continues until the force tolerance or maximum number of iterations is met.
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