Site Response Analysis of a Layered Soil Column (Total Stress Analysis)
Example posted by: Christopher McGann, University of Washington
This article describes the OpenSees implementation of a site response analysis for a layered soil profile using total stress analysis. A single soil column is modeled in two-dimensions and is subject to an earthquake ground motion in a manner which accounts for the finite rigidity of the underlying medium.
Provided with this article is the main input file needed to execute this analysis in OpenSees, freeFieldDamp.tcl, along with several additional necessary and/or helpful files:
- ELCENTRO.EQ, the acceleration time history for the considered ground motion
- forceHistory.out, the path timeSeries information needed to run the analysis
- getForceHistory.m, a Matlab script which computes the timeSeries information from the acceleration time history
- getAccel.m, accelPlots.m, and depthPlots.m, Matlab scripts which produce the plots included in this article from the recorded results
Download them all in a compressed file: siteResponseSand.zip
To run this example, the user must download the files freeFieldDamp.tcl and forceHistory.out and place them in a single directory. Once this has been done, the user can then type "source freeFieldDamp.tcl" into the interpreter of the OpenSees.exe application to run the analysis. The Matlab scripts and the acceleration time history files are not essential to the analysis, however, they are provided to demonstrate how an alternative acceleration time history can be converted into the analysis and how certain plots can be obtained from the recorded output.
Model Description
The site response analysis is performed for a soil profile with three layers of cohesionless soil. It is assumed that there is no groundwater, therefore, total stress analysis is used in this example. The soil is modeled in two-dimensions with two degrees-of-freedom using the plane strain formulation of the quad element. The nDMaterial model, PressureDependMultiYield, is used as a constitutive model for the cohesionless soil, and each layer is assigned separate material properties. A schematic of the model is shown in Fig. 1. In this example, the horizontal direction is the first degree-of-freedom and the vertical is the second. The soil node, element, and layer numbering schemes all begin at the bottom.
To account for the finite rigidity of the underlying medium (assumed to be bedrock in this example), a Lysmer-Kuhlemeyer (1969) dashpot is incorporated at the base of the soil column using a zeroLength element and the Viscous uniaxial material. The Lysmer-Kuhlemeyer (1969) dashpot is assigned a dashpot coefficient equal to the product of the mass density and shear wave velocity of the underlying bedrock layer. The soil column is excited at the base by a horizontal force time history which is proportional to the known velocity time history of the ground motion. Further information on this modeling approach can be found in Joyner and Chen (1975) and Lysmer (1978) among others.
The horizontal force time history is applied as a Path timeSeries object using the file, forceHistory.out. This force time history has been precomputed from the acceleration time history detailed in the file ELCENTRO.EQ. The provided Matlab script, getForceHistory.m, shows how this computation was accomplished, and can be modified to produce the required force time history for any given acceleration time history.
Soil Profile Geometry
There are three layers of cohesionless soil in this example. The user can specify the thickness of each layer. The default layer thickness values are:
- uppermost layer, 2 m thick
- middle layer, 8 m thick
- lower layer, 40 m thick
Mesh Geometry
In this example, each of the soil elements is a one meter square. This is controlled by defining the element size in the x-direction and the number of elements in each layer. These are the only required inputs in this section. The number of nodes and the total number of elements are computed automatically.
Soil Nodes
The soil nodes are created automatically from the input geometry and meshing information. As shown in Fig. 1, the node numbering scheme is left-to-right, top-to-bottom. Nodes with even numbers fall on the y-axis, and the odd-numbered nodes are spaced horizontally by the input horizontal element size (1 m in this example).
Dashpot Nodes
A single zeroLength element is used to define the Lysmer-Kuhlemeyer (1969) dashpot, therefore, only two nodes are required. These nodes are arbitrarily assigned numbers 2000 and 2001. If the user has modified the meshing information such that there are more than 2000 nodes in the soil column, the dashpot node numbers will need to be changed.
Boundary Condition and Equal Degrees-of-Freedom
The nodes at the base of the column are fixed against displacements in the y-direction in accordance with the assumption that the soil layers are underlain by bedrock. The remaining soil nodes are then tied together using the equalDOF command in order to achieve a simple shear deformation pattern. This is done by declaring equalDOF for every pair of nodes which share the same y-coordinate.
One of the dashpot nodes is fully fixed (node 2000), while the other is fixed only against displacements in the y-direction (node 2001). To incorporate the dashpot element into the total model, equalDOF is again used, this time linking the horizontal degrees-of-freedom of the partially fixed dashpot node and one of the nodes at the base of the soil column.