Dynamic Analysis

 

 



Introduction

The dynamic response analysis of soils has generated much active research in geotechnical design in last few decades. Some causes of dynamic loading are outlined as follows:

The pioneering researcher of dynamic response analysis of soils was Dr. Bolton Seed7,8,9 and his colleagues at the Earthquake Engineering Research Centre at the University of California at Berkeley. Dr. Liam Finn2,3 from the University of British Columbia, Vancouver, Canada has also lead the way in the field of dynamic response analysis.

A fundamental assumption made in the majority of the earthquake response analysis techniques is that soil deformations are a result of vertically propagating shear wavelets. Dr. Seed and his colleagues had implemented the wave equation with the assumption of harmonic vertically propagating shear waves. Dr. Seed and his colleagues also assumed a linear viscoelastic system when numerically modeling soil response characteristics to dynamic loading10. The Seed et al. Technique is often referred to as the equivalent linear method. The equivalent linear method is capable of giving an accurate solution only when the shear strain's generated are less than 1 percent6.

To obtain deformations at larger shear stains, researchers2,3,8,11, such as Dr. Finn, treat the soil profile as a spring-mass-dashpot (SMD) system and implement step-by-step integrations procedures in deriving soil response characteristics.



Single Degree of Freedom System

The term Single Degree of Freedom (SDF) system refers to the dynamic analysis of a homogeneous and heterogeneous soil profile (i.e. half space). Figure 1 schematically illustrates the SDF system and its equivalent SMD representation.

















The generally idea shown in figure 1 is that an earthquake acceleration, ÿB, at the base of a homogeneous soil profile of Mass M can be represented as a second order mass-spring-system with elastic spring constant k and viscous damping c. The SMD system is a linear second order system, with a characteristic natural frequency and damping factor. Equation 1 outlines the continuous mathematical model of a second order system with acceleration as an input.



(1)



In equation (1) k/m = wn2 and c/m = 2 * damping factor * wn. These second order parameters are related to the soil profile characteristics as follows:

k = GA/H, ~ ~ ~ ~ ~ ~(2)

where G is the shear modulus, and A and H are the area and height of the soil profile, respectively.

The damping factor of the soil profile is represented as a percentage (e.g. for critical damping = 100% or 1). There are many ways to derive this parameter (e.g., spectral ratio and corner frequencies of displacement spectra from full seismic waveforms). Another standard technique in deriving soil damping is from Ln decrement curves. Figure 2 illustrates a soil's seismic response to excitation (e.g. resonant column test). The calculation of from the Ln decrement curve is outlined as follows:

Damping Factor = Ln (x1/x2) / (2 pi) ~ ~ ~ ~ ~ ~(3)





As stated previously, the standard technique in solving equation (1) is by step-by-step numerical integration. In the numerical integration technique it is assumed that the input acceleration, ÿB, is constant over a sample or intergation interval. In addition, the Talyor series for the soil displacement, x, is taken out to the second term (i.e. acceleration). In the numerical integration, the strain dependent values of the soil properties, k and c, are updated at each new increment5,8. Ishihara8 carried an extensive investigations into soil properties at differing strain levels. He found that the use of the hyperbolic model combined with the Masing rule can only be used for medium levels of shear strains of up to 1 percent. For larger strain levels, the implementation of the Ramberg-Osgood model had provided satisfactory dynamic analysis results.

Baziw Consulting Engineers has been having success in implementing a state-space formulation of equation (1) so that it fits into a Kalman Filter formulation.



Kalman Filter Formulation

The Kalman, KF, is a method for estimating a state vector x from measured data z. The state vector may be corrupted by a noise vector w and the measurement vector is corrupted by a noise vector v. The KF is applicable for systems that can be described by a first order differential in x and a linear (matrix) equation in z1. The state and measurement equations are assumed to be in the following canonical form:

(4)
In equation (4), F(t) is defined as the space matrix (i.e. state-space) and H(t) is the measurement matrix. G(t) defines how the state vector (i.e. x ) is effected by the input. If we compare the definition defined by equation (4) with dynamics described by equation (1), we see that the SMD system is a perfect candidate to fit into the KF formulation.

The KF is desirable numerical analysis tool because ittakes into account the statistics of the measurement and state errors, and the apriori model information provides for optimal use of any number, combination, and sequence of external measurements. The KF can be applied to problems that can be described or approximated by linear time-varying equations, with non-stationary system and measurement input noise statistics. Physical problems with nonlinearities can often be adequately handled by linearizing the system and measurement equations. Furthermore, the KF formulation can then be readily designed for estimation, smoothing, and prediction of the physical parameter of interest.

Another advantage of a Kalman Filter is that it allows one to use "state noise" to compensate for errors in the mathematical model. The use of state noise causes the filter to apply less weight to measurements made in the distant past, and to apply more weight to state vector estimates based on more recent measurement data. Table 1 outlines the governing equations for a Discrete Kalman Filter as follows4 :



Table 1. Summary of Discrete Kalman Filter Equations



























The discrete model for the continuous second order system outlined in equation (1) is outlined as follows:

(5)
The vector x k+1 in Eq. 5 is defined as

(6)
The computational sequence for the Discrete KF is outlined as follows:

 

Download Single Degree of Freedom Demo



Multi Degree of Freedom System



The term Multi Degree of Freedom (MDF) system refers to the dynamic analysis of a multi-layered soil half space. Figure 3 schematically illustrates the MDF system and its equivalent SMD representation.



















The equivalent continuous state-space representation of the MDF system is outlined as follows:



(7)



In equation (7), states x1(t) and x3(t) represent displacements xa and xb outlined in figure 3, respectively.



References

1. Baziw, E.J.: Applications of Digital Filtering Techniques for Reducing and Analysing In-Situ Seismic Time Series, MASc Thesis, Dept. Of Civil Engineering, University of British Columbia, Vancouver, BC August 1988.

2. Finn, W.D. Liam (1984). Dynamic Response Analysis of Soils in Engineering Practice, Mechanics of EnGineering Materials. John Wiley & Sons Ltd. Chapter 13.

3. Finn, W.D. Liam, Byrne, Peter M., and Martin, R. Martin, "Seismic Response and Liquefaction of Sands", Journal of the Geotechnical Engineering Division, ASCE, Vol. 102, No. GT8, August, 1976.

4. Gelb, A., APPLIED OPTIMAL ESTIMATION, MIT Press, Cambridge, Mass., 1974.

5. Hardin, Bobby O., and Drnevich, Vincent P., "Shear Modulus and Damping in Soils: Design Equations and Curves", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 98, No. SM7, July, 1972.

6. ISHIHARA,K., "Evaluation of soil properties for use in earthquake response analysis", Internatrional Symposium on Numerical Models in Geomechanics", Zurich, 13-17 September 1982, pp. 237-259.

7. Seed, H. B., Idriss, I.M., and Keifer, F.W., "Characteristics of Rock Motions During Earthquakes", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 95, No. 5, pp. 1199-1218, 1969.

8. Seed, H. B., and Idriss, I. M., "Soil Moduli and Damping Factors for Dynamic Response Analyses", Earthquake Engineering Research Center, College of Engineering, University of California - Berkeley California, Report No. EERC 70-10, December 1970.

9. Seed, H. B., and Botton., "Evaluation of Soil Liquefaction Effects on Level Ground During Earthquakes", State-of-the-Art Paper, ASCE, National Convention, Philadelphia, 1976.

10. Seed, H. B., et al., "SHAKE - A Computer Program for Earthquake Response Analysis of Horizontally Layered Sites", Earthquake Engineering Research Center, College of Engineering, University of California - Berkeley California, Report No. EERC 72-12, December 1972.

11. Wilson, Edward L., and Clough, Ray W., "Dynamic Response by Step-by-Step Matrix Analysis", Symposium on the Use of Computers in Civil Engineering, Laboratorie Nacional de Engenharia Civil, Lisbon - Portugal, 1-5 October, 1962, pp. 45.1 - 45.14.