Introduction
The objective of this project was to develop a method where sophisticated soil-structure interaction (SSI) effects such as soil layering, frequency dependency and embedment can be accurately represented in a general purpose structural analysis program and also in the popular earthquake engineering programs DRAIN2DX and DRAIN3DX.
It was thought that the best way to do this would be to develop a similar program to the soil structure interaction program CLASSI, but one which could also accurately model an embedded foundation, and then convert the calculated impedance functions directly into lumped parameter systems in the time domain. These lumped parameter systems would be more sophisticated than what is normally understood as 'lumped parameter SSI' and the method is called the 'advanced lumped parameter method'.
Generation of Impedance Functions
Surface Foundations
Firstly a program which can calculate the frequency-dependent dynamic stiffness of a single rigid arbitary-shaped rigid surface foundation on a horizontally layered site called BASEMAT was developed. The foundation area is represented using equal square subregions. The Greens function calculation method used in the program is based on a more up-to-date method developed by Hisada which is supposedly more accurate than the Aspel & Luco method in CLASSI for situations where source and receiver locations are at identical or very similar depths (which is the case for foundations). However no particularly significant difference in graphical comparisons of impedance functions between the program and published results for CLASSI have been noticed.
Embedded Foundations
An extended version of the surface foundation program which could model a single rigid embedded foundation, BASEMAT_EMBED, was developed next. The embedded foundation is represented using a stack of parallel sub-meshes. The excavated soil volume is handled by subtracting the dynamic stiffness of the soil mass. Full welded contact between the foundation and surrounding soil is assumed. For some reason a working embedded foundation version of CLASSI was never publicly released, but it is possible to compare the results of this program with results in the literature given by Aspel.
View BASEMAT_EMBED verification material
Multiple Foundations
This program, called BASEMAT_MULT, is an extension of the original surface foundation version of the program to model two separate rigid surface foundations. The soil-structure-soil interaction (SSSI) effect predicted by the program may be exaggerated due to the assumption of linear behaviour of the soil medium. The ASCE-4 Standard allows SSSI effects to be ignored, arguing that these are a secondary effect which can be covered by considering soil property variations for a single foundation SSI analysis. Results of the program were compared (for the smallest foundation separation case available in the literature) with published results by Wong & Luco.
Conversion of impedance functions into lumped parameter system
The impedance functions can be converted into a lumped parameter system by calculating pole and residue parameters which give a least square fit to the function, and then use formulae to convert the pole and residue parameters into a network of soil springs and dampers.
The conversion is carried out using a suite of programs called LUMP_PARAM_SSI. In earlier development work the curvefitting was carried out using the Rational Fraction Polynomial method used in experimental modal analysis, but it was found that it could be difficult to retain a good curvefit after stabilising or discarding the unstable poles generated with this method. Unstable poles cannot be carried over into the time domain and the issue seems to become more problematic when higher levels of soil material material damping are present in the impedance functions. It was decided to switch to a more up-to-date curvefitting method used in electrical engineering applications called 'vector fitting'. This vector fitting method is more well-suited to producing curvefits where unstable poles have to be avoided (the method is claimed to eliminate unstable poles but this has not been observed when applied to SSI impedance functions).
The networks of soil springs and dampers are very likely to include some negative valued springs and dampers, which may be rejected by some structural analysis programs. Use of soil masses to represent frequency dependency is deliberately avoided as this is not compatible with the way that structural analysis programs tend to handle base excitation problems.
The conversion of impedance functions to a lumped parameter system has only been tested on single foundations, and the method may not be workable for multiple foundation systems.
SSI version of PCSAP4
An SSI version of PCSAP4, called PCSAP4SSI, was developed to provide a test platform for the method. The normal version of SAP4 only allows a single pair of Rayleigh damping parameters, alpha and beta, to be used for a direct time history analysis of the whole model, and the damping matrix is not stored. In the modified version of the program, multiple values of alpha and beta can be entered and a damping matrix is stored. The beta damping capability is available for three element types: truss elements, beam elements and plate/shell elements. SAP4 does not formally include spring elements but a truss element can be used to represent a translational spring and the torsional degree of freedom of a beam element can be used to repesent a rotational spring.
The Wilson-theta integrator was replaced by a Newmark-beta integrator partly as a precaution that an unconditionally stable time history integrator might be needed for negative valued springs and dampers.
Demonstration of advanced lumped parameter method using Lotung SSI experiment
There is not much experimental data regarding soil-structure interaction available in the public domain to demonstrate the method. One exception is that the 1986 Lotung Soil-Structure Interaction experiment is used as the basis for an example problem in the SASSI2000 User Manual.
This example problem was run using BASEMAT_EMBED and LUMP_PARAM_SSI to generate the network of soil springs and dampers. DRAIN2DX was chosen as the 'off the shelf' structural analysis program to generate structural response data that can be compared with the experimental results.
View Lotung test simulation material
Input files for the DRAIN2DX analyses are also provided.
Drain2dx input file for 1% soil material damping
Drain2dx input file for 17% soil material damping
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