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# THEORETICAL AND EXPERIMENTAL INVESTIGATION OF EFFECTIVE DENSITY AND PORE FLUID INDUCED DAMPING IN SATURATED GRANULAR MATERIALS

#### THEORETICAL AND EXPERIMENTAL INVESTIGATION OF EFFECTIVE **DENSITY AND PORE FLUID INDUCED DAMPING IN SATURATED ****GRANULAR MATERIALS**

# ABSTRACT

In current geotechnical engineering research and practice, two assumptions are generally made regarding the dynamics of saturated soil. The first is that pore fluid induced damping during shear wave excitations is negligible. The second is that saturated density can be used to calculate shear modulus based on measured shear wave velocity. The validity of these assumptions depends on the magnitude of fluid motion relative to solids during shear wave excitations. For soils with low permeability (e.g., silts and clays) and under low-frequency excitations (e.g., seismic waves), these assumptions are generally valid. However, relative fluid motion may be important for soils with high permeability (e.g., sands and gravels) and under high-frequency excitations, rendering the above mentioned assumptions questionable. This study presents an experimental investigation of the concept of effective density for propagation of small strain shear waves through saturated granular materials. Bender element tests and resonant column tests were conducted on various granular materials in dry and saturated conditions. Values of small-strain shear modulus measured for the dry condition are compared to corresponding values measured for the saturated condition using saturated density and effective density. Analysis of test results indicates that effective density instead of saturated density should be used to calculate small-strain shear modulus. For bender element tests, the use of saturated density produced errors in shear modulus as high as 28%; whereas the use of effective density resulted in errors generally less than 5%. For resonant column tests, errors in shear modulus obtained using saturated density were smaller than those for bender element tests due to the lower range of excitation frequency. This study presents two analytical solutions for Biot flow induced damping in saturated soil specimens in resonant column tests based on the half-power bandwidth and free vibration decay methods. These solutions are compared with a closed-form analytical solution readily available in literature. The solutions indicate that Biot flow induced damping may provide an important contribution to total soil damping in coarse sand and gravel, but can be practically neglected for less permeable soils (e.g., fine sand, silt, and clay). The solutions also indicate that Biot flow induced damping increases as porosity increases and decreases considerably as the ratio of the mass polar moment of inertia of the loading system to the specimen increases. It is concluded that Biot flow induced damping is suppressed by the boundary condition of typical resonant column apparatuses and is hence difficult to be measured. The solution from the free vibration decay method is compared to RC test results of various granular materials at dry and saturated conditions. The comparison suggests that the validity of this analytical solution is inconclusive, which is largely due to the very small magnitude of Biot flow induced damping in RC tests. In addition, a theoretical investigation of energy dissipation in a nearly saturated poroviscoelastic soil column under quasi-static compressional excitations, which is applicable to slow phenomena (e.g., consolidation), is also presented in this study. Different components of the energy dissipation are evaluated and compared. This investigation indicates that the magnitude of pore fluid induced energy dissipation is primarily a function of a normalized excitation frequency . For small values of , a drained soil column is fully relaxed and behaves essentially as a dry column with negligible pore pressure. In this case, fluid induced energy dissipation is negligible and the total damping ratio of the column is essentially the same as that of the solid skeleton. For very high values of , a drained soil column is fully loaded and the excitationgenerated pore pressure decreases as the fluid becomes more compressible. In this case, the fluid pressure gradient only exists in a thin boundary layer near the drainage boundary, where drainage occurs and fluid induces energy dissipation; whereas the rest of the column is essentially undrained. Significant fluid induced energy dissipation occurs for moderate values of due to a combination of moderate fluid pressure, pressure gradient and fluid relative motion throughout the soil column. The effects of boundary drainage condition, saturation, porosity, and skeleton damping ratio on fluid induced energy dissipation are discussed. ** ****TABLE OF CONTENTS ** List of Figures ………………………………………………………………………………………………. viii List of Tables ………………………………………………………………………………………………… xi Acknowledgements ………………………………………………………………………………………… xii Chapter 1 Introduction ……………………………………………………………………………………. 1 1.1 Background ……………………………………………………………………………………….. 1 1.2 Objectives of Research ……………………………………………………………………….. 4 1.3 Organization of Dissertation ………………………………………………………………… 5 References ………………………………………………………………………………………………. 6 Chapter 2 Effective Soil Density for Small Strain Shear Waves in Saturated Granular Materials …………………………………………………………………………………… 7 2.1 Introduction ……………………………………………………………………………………….. 7 2.2 Experimental Program and Results ………………………………………………………. 10 2.2.1 Granular Materials …………………………………………………………………….. 10 2.2.2 Bender Element Tests ………………………………………………………………… 11 2.2.3 Resonant Column Tests ……………………………………………………………… 25 2.2.4 Quick Chart ……………………………………………………………………………… 32 2.3 Discussion …………………………………………………………………………………………. 35 2.4 Conclusions ……………………………………………………………………………………….. 39 Notations ………………………………………………………………………………………………… 41 References ………………………………………………………………………………………………. 42 Chapter 3 Biot Flow Induced Damping in Saturated Poroviscoelastic Soil Specimens in Resonant Column Test …………………………………………………………. 46 3.1 Introduction ……………………………………………………………………………………….. 46 3.2 Governing Equations ………………………………………………………………………….. 49 3.3 Analytical Solutions for Biot Flow Induced Damping …………………………….. 52 3.3.1 Spectral Response ……………………………………………………………………… 53 3.3.2 Damping from Half-Power Bandwidth Method …………………………….. 57 3.3.3 Damping from Free Vibration Decay Method ………………………………. 59 3.4 Resonant Column Test ………………………………………………………………………… 63 3.5 Results and Discussion ……………………………………………………………………….. 64 3.6 Conclusions ……………………………………………………………………………………….. 70 Notations ………………………………………………………………………………………………… 71 References ………………………………………………………………………………………………. 75 Chapter 4 Energy Dissipation in Nearly Saturated Poroviscoelastic Soil Column during Quasi-Static Compressional Excitations …………………………………………… 80 4.1 Introduction ……………………………………………………………………………………….. 80 4.2 Governing Equations ………………………………………………………………………….. 83 4.3 Analytical Solutions of Steady-State Motion …………………………………………. 86 4.3.1 General Solution ……………………………………………………………………….. 86 4.3.2 No-Drainage (ND) Case …………………………………………………………….. 89 4.3.3 Top-Drained (TD) Case ……………………………………………………………… 90 4.3.4 Double-Drained (DD) Case ………………………………………………………… 91 4.3.5 Normalization …………………………………………………………………………… 93 4.4 Energy Dissipation and Damping …………………………………………………………. 94 4.5 Results and Discussions ………………………………………………………………………. 100 4.6 Illustrative Example ……………………………………………………………………………. 114 4.7 Conclusions ……………………………………………………………………………………….. 116 Notations ………………………………………………………………………………………………… 117 References ………………………………………………………………………………………………. 120 Chapter 5 Conclusions and Recommendations …………………………………………………… 124 5.1 Conclusions ……………………………………………………………………………………….. 124 5.2 Recommendations for Future Work ……………………………………………………… 127 Appendix Data from Resonant Column Tests …………………………………………………… 129 Group 1 ………………………………………………………………………………………………….. 129 Group 2 ………………………………………………………………………………………………….. 140 Group 3 ………………………………………………………………………………………………….. 161 **Chapter 1 **** **** **** **

# Introduction

## 1.1 Background

Shear modulus and damping are two important soil dynamic properties. These two parameters play crucial roles in ground motion analyses in geotechnical earthquake engineering. For example, Fig. 1-1 shows a deposit of homogeneous soil layer on top of bedrock. The fundamental frequency, *f _{o }*, of the soil layer can be estimated as

*V*

*(1-1)*

^{s}*f*

* 4*

_{o }*H*where

*H*is the soil layer thickness and

*V*is shear wave velocity which can be calculated as

_{s}*G*

*V*

* (1-2) where is the soil density and*

_{s }*G*is shear modulus. Eq. (1-1) indicates that

*f*depends on the shear wave velocity and layer thickness. For a typical earthquake ground motion, the dominant frequency,

_{o}*f*

*, is generally in the range of 1 – 5 Hz (Kramer 1996). If*

_{g }*f*

*is close to*

_{g}*f*, dynamic amplification will occur and large ground motions (e.g., acceleration, velocity, and displacement) will be expected. On the other hand, if

_{o }*f*

*is significantly different than*

_{g}*f*and other modes of natural frequencies, deamplification may occur and the resulted ground motions will be small. As the seismic wave travels between the bedrock and ground surface as shown in Fig. 1-1, wave energy is dissipated through soil damping. Higher soil damping will result in smaller ground motions as more energy is dissipated during the wave propagation. Therefore, shear modulus and damping are considered as input parameters for ground motion analyses.

_{o}**Fig. 1-1.**Seismic wave propagation in homogeneous soil layer In the current geotechnical engineering research and practice, shear modulus is generally calculated based on shear wave velocities measured from various field and laboratory tests involving shear waves, such as the seismic cone penetration tests (SCPT), bender element (BE) tests, and resonant column (RC) tests, using the following equation

*G*

*V*

_{s}^{2}(1-3) This equation is based on the theory of elasticity in a continuum, where is the density of the continuum (single-phase) under all conditions. Soil is a multi-phase system, consisting of a solid phase, liquid phase (e.g., water), and gas phase (e.g., air). For dry soil, dry density

*is used in Equation (1-3); for saturated soil, saturated density *

_{d}*is generally used. However, the use of *

_{sat}*for saturated soil assumes no relative motion between pore fluid and solid skeleton. The validity of this assumption depends on the magnitude of fluid motion relative to solids during shear wave excitations. For soils with low permeability (e.g., silts and clays) and under low-frequency excitations (e.g., seismic waves), this assumption is generally valid. However, relative motion may be important for soils with high permeability (e.g., sands and gravels) and under high-frequency excitations based on Biot theory (Biot 1956), rendering this assumption invalid. Qiu and Fox (2008) proposed the concept of “effective soil density”, *

_{sat}*, that is related to the fraction of pore fluid that moves with solid skeleton during shear wave propagation. This effective density is always between *

_{eff }*and *

_{d}*and is the theoretically correct value to use in Equation (1-3) to calculate shear modulus based on measured shear wave velocity. Qiu and Fox (2008) provided analytical solutions of *

_{sat}*based on Biot theory (Biot 1956). However, this analytical solution has not been rigorously validated against laboratory test data for different soils. Damping is a consequence of energy dissipation due to sliding and rolling at particle contacts, and the loss and creation of particle contacts when there is particle rearrangement. This form of energy dissipation is generally considered as “skeleton damping” (Ellis et al. 2000) and is the only source of material damping in dry soil. For saturated soil, in addition to skeleton damping, energy is also dissipated due to the relative motion and viscous drag between pore fluid and solid skeleton (i.e., viscous coupling). Therefore, saturated soils exhibit higher damping than the same soils in their dry condition. This has been experimentally observed by various researchers (e.g., Hall and Richart 1963; Bolton and Wilson 1990; Ellis*

_{eff}*et al*. 1998 and 2000). In geotechnical engineering research and practice, however, pore fluid induced damping is generally neglected due to the lack of quantitative assessment of its values in various soils. Qiu and Fox (2006) and Qiu (2010) provided analytical solutions of pore fluid induced damping in saturated soils during shear wave excitation. These studies suggest that pore fluid induced damping depends on soil types and may have significant contribution to the total damping for coarse sands and gravels, in particular at small strain levels. However, these findings have not been validated by any experimental test data.

## 1.2 Objectives of Research

The objectives of this research are to quantify effective density and pore fluid induced damping in granular materials for small strain shear waves using BE and RC tests, and to conduct additional analytical study on pore fluid induced damping in saturated soils under quasi-static compressional excitations. This study can potentially improve the accuracy of how small strain shear modulus and damping are evaluated, especially in highly permeable granular materials (e.g., coarse sands and gravels) under highfrequency excitations (e.g., BE tests), which may improve the accuracy of current ground motion analyses in geotechnical earthquake engineering. The findings of this study will be of significant value to geotechnical earthquake engineering and soil dynamics. Ultimately, the benefit will be the reduction of losses to society as a result of earthquakes.

## 1.3 Organization of Dissertation

Chapter 2 presents effective density for small strain shear waves in saturated granular materials. It presents an experimental investigation consisting of RC and BE tests on various granular materials in dry and saturated conditions for the concept of effective density. This chapter is based on a manuscript submitted to the *Journal of Geotechnical **and Geoenvironmental Engineering, *ASCE. Chapter 3 presents analytical solutions and their comparison with RC test results for pore fluid induced damping in saturated granular materials. It presents two analytical solutions and compares them with a closed-form analytical solution readily available for RC test in literature. Furthermore, solution of pore fluid induced damping based on the free vibration decay method is compared with RC test results of various granular materials in dry and saturated conditions. This chapter is based on a manuscript submitted to the *Soil Dynamics and Earthquake Engineering. * Chapter 4 presents energy dissipation in nearly saturated soil columns during quasistatic compressional excitations, which is of particular relevance to slow phenomena (e.g., consolidation). Different components of energy dissipation in a saturated soil column are derived and compared. This chapter is based on a paper published in the *Journal of Engineering Mechanics*, ASCE. Chapter 5 draws final conclusions of this study and presents suggestions for future work.

## References

Biot, M.A. (1956). “Theory of Propagation of Elastic Waves in a Fluid-Saturated Porous Solid. I. Low-Frequency Range. ІІ. Higher Frequency Range.” *J. Acoust. Soc. Am.*, 28(2), 168-191. Bolton, M. D., and Wilson, J. N. (1990). “Soil stiffness and damping.” *Structural dynamics*, W. B. Kratzig, D. E. Beskos, and I. G. Vardoulakis, eds., Balkema, Rotterdam, The Netherlands, 209–216. Ellis, E. A., Soga, K., Bransby, M. F., and Sato, M. (1998). “Effect of pore fluid viscosity on the cyclic behavior of sands.” *Proc., Centrifuge 98*, T. Kimura, O. Kusakabe, and J. Takemura, eds., Balkema, Rotterdam, The Netherlands, 217–222.. Ellis,E.A., Soga, K., Bransby,M.F. and Sato, M. (2000). Resonant Column Testing of Sands with Different Viscosity Pore Fluids, *J. Geotech. Geoenviron. Eng.*, 126(1), 10-17. Hall, J.R. and F.E. Richart (1963). “Dissipation of Elastic Wave Energy in Granular Soils.” *J. Soil Mech. and Found. Div.*, 89(6), 27-56. Kramer, S.L. (1996). *Geotechnical Earthquake Engineering*, Prentice Hall, Upper Saddle River, NJ. Qiu, T. and Fox, P.J. (2006). “Hydraulic damping of saturated poroelastic soils during steady-state vibration.” *J. Eng. Mech.*, 132(8), 859-870. Qiu, T. and Fox, P.J. (2008). “Effective Soil Density for Propagation of Small Strain Shear Waves in Saturated Soil.” *J. Geotech. Geoenviron. Eng.*, 134(12), 1815-1819. Qiu, T. (2010). “Analytical Solution for Biot Flow-Induced Damping in Saturated Soils during Shear Wave Excitations.” *J. Geotech. Geoenviron. Eng.*, 136(11), 1501-1508.