Analysis of Soil-structure Interaction Effects on Tall Buildings

1. Introduction

Earthquake causes ruptures of fault and leads to collapse or large deformation of structure and this causes human deaths, the deformation or damage of structure is caused by the lateral inertial forces induced by dynamics vibration and soil deformation. The seismic excitation of the structure depends on the earthquake source, travel path effects, local site effects and soil-structure Interaction effects. The first 3 factors only consider a free-field ground motion, whereas SSI consider how structure response to the free-field ground motion.

The effects of soil-structure interaction is important for structure founded on soft deposit subjected to earthquake, it is believed that strong ground motion was caused by soft deposit from the recordings obtained during the famous 1985 Mexico and 1989 Lima Prieta earthquakes. Structure on a flexible foundation responds differently compared to structure on a rigid base when subjected to a dynamic load since the response not only depends on the dynamic modes of the structure but also the interaction between structure and soil.

1.1.             Aims and Objectives

The aim of this paper is to investigate the importance of SSI as well as examine the range of applicability of SSI method.

Objectives:

• Understanding of Soil-Structure Interaction, the role of SSI on seismic response on structure
• Different approaches of soil structure analysis model
• Examine various earthquake of different frequency content
• Assess suitability of analytical relations comparing with finite element

2. Literature Review

The first part of this literature review will look into the effects of SSI and its role in the seismic response of structure. The second part will look into different methods of analysing SSI and the limitation of each method.

2.1.             Soil-Structure Interaction Overview

2.1.1.         What is SSI

When rupture of fault occurs, seismic waves propagate through soil causing vibration on soil and structure. This leads to displacement to both structure and soil. The ground motion of a free field is found to be different compared to the ground motion occurring at the base of the structure. The mutual effects between structure, foundation and soil is called soil-structure interaction. The difference between 2 motions is caused by the 2 phenomena of SSI:

I)                    
Kinematic Interaction, since the foundation is stiffer then the soil, there is a resistance of matching the distortion of free-field motion induced by the seismic waves. By assuming a massless structure, kinematic interaction effect are described by a frequency dependent transfer function defined as the ratio of the foundation motion to the free field ground motion. (Torabi & Rayhani 2014)

II)                   Inertial Interaction, mass of structure transmits inertial force induced by kinematic interaction to the foundation, base shear and moment effects cause soil to further deform and changes the modal frequency and damping factors of the structure. Frequency dependent foundation impedance functions is used to describe the flexibility of the foundation and damping associated with the interaction. (Torabi & Rayhani 2014)

The characteristics of SSI depends on several factors, e.g. the intensity of seismic waves, soil stratigraphy, stiffness and damping of soil layers, geometry and rigidly of the foundation, embedment depth of the structure and inertia characteristics of the superstructure etc. (Encyclopaedia of earthquake engineering)

2.1.2.         Effects of SSI

Structure founded on deformable soil has different vibrational characteristics compared with a corresponding rigidly supported structure. Firstly, the fundamental period of a flexible base structure is longer than a fixed-based structure. Secondly, in a flexible base structure, since part of the vibration energy from the structure is dissipated into the soil through wave radiation and hysteretic soil damping, the effective damping ratio is higher than a fixed-base structure as such effects do not occur in a rigidly supported structure. Figure 3 shows two single degree of freedom models for fixed-base and flexible-base structure, where the flexible foundation medium is represented by a translational and rotational springs.

The role of SSI in seismic response of structures were re-explored by Mylonakis and Gazetas. SSI is conventionally considered as a beneficial effects on seismic response of structure. Most of the design codes treat it as a conservative simplification that allow designers to reduce base shear of buildings. An idealised smooth design spectra with a constant acceleration up to a certain period and decreases monotonically with period is used in most of the design codes. Since SSI leads to longer fundamental period and effective damping ratio of a structure, the design spectra suggests a reduction in seismic response. The reduction is shown in Figure 4 where period of fixed-base T is increased to $\tilde{T}$

, therefore, SSI effects is recognised as beneficial in seismic provisions. (Mylonakis and Gazetas 2000)

Effects of SSI is not always beneficial, soft deposit can elongate the period of seismic waves, and the longer fundamental period of structure due to SSI can lead to resonance with long period ground motion which means an increased seismic response of structure. The results of resonance of soft deposit can be seen in Figure 5 comparing with the design spectra from NEHRP-97 seismic code. (Mylonakis and Gazetas 2000)

The 1985 Mexican earthquake is famous for the unexpected heavy damage.  The Mexico City basin used to be a lake which was buried artificially causing a soft deposit of soil in the city. Since the ground in some location of Mexico City has a long fundamental period, the earthquake motion was amplified. Figure 5 shows a resonance at long period ground motion.                                          

2.2.             Methods of analysis of SSI

Two general approaches are commonly used to analyse SSI, direct method and sub-structure method. In direct method, the entire soil and structure system is treated together and analysed in a single step, it can be modelled in two or three dimension using finite element and finite differences method. In substructure method, system is divided into different components, and the response of each component is calculated separately.

2.2.1.         Finite Element Analysis

Finite element can be used to model both direct and sub-structure approaches. In direct approaches, the entire soil-foundation-structure system is modelled. Free-field input motion has to be specified at the base of the model and the system response can be computed. Appropriate boundary conditions and equation solution algorithms are required to perform a nonlinear and dynamic analysis.

2.2.2.         Simple analytical models – spring-mass-dashpot oscillator

In sub-structure approach, the 2 primary causes of soil structure interaction, kinematic and inertial interaction is used, the calculation can be performed in 3 steps, since the method rely on superposition, this can only be applied on linear behaviour.

The 3 steps is shown in Figure 7 (Kramer and Stewart, 2004):

1. Determine the foundation input motion (FIM) which is the seismic motion occurring at an assumed rigid massless foundation. FIM depends on the stiffness and geometry of the soil and foundation. FIM represents the kinematic interaction effects.
2. Determine the foundation dynamic impedance, the impedance function describes the stiffness and damping of the foundation-soil system.
3. 
   
   Determine the seismic response of structure and foundation by analysing a structure supported by springs and dashpot with stiffness obtained from impedance function and subjected to the foundation input motion calculated in 1^st step.

2.2.3.         Foundation Impedances Function

Figure 8 shows a superstructure on a flexible foundation modelled as a single-degree of freedom system with a height h, mass m, stiffness k and damping c. ${\bar{k}}_h$

and ${\bar{k}}_r$

represents a frequency dependent and complex-valued translational and rotational spring for each mode of vibration. $c_h$

and $c_r$

represents pair of frequency-dependend dashpots modelling energy dissipation in the soil due to wave radiation and hysteretic damping. This system can be view as a single story structure or the first mode of vibration of a multi-story structure.

2.2.4.         Static Stiffness

Hsieh and Lysmer discover that the dynamic behaviour of a vertically loaded massive foundation can be represented by a mass-spring-dashpot oscillator shown in Figure 8.

Lysmer suggested that frequency-independent coefficients can be used to approximate the response in low and medium frequency:

$K_v=\frac{4\mathit{GR}}{1–v}; {\mathit{ C}}_v=\frac{3.4R^2}{1–v}\sqrt{G_p}$

Which, $K_v$

= spring constant; $C_v$

= dashpot constant; $R$

= radius of circular loading area;

$G$

= shear modulus

Richart and Whitman later extended that all modes of vibration can be studied by selecting proper frequency-independent parameters. The vertical and torsional oscillation of the system can be described by:

$m\ddot{x}+C\dot{x}+\mathit{Kx}=P\left(t\right)$

Where $m$

, $C$

, $K$

is the mass, effective damping and effective stiffness of the system, $\ddot{x},\dot{x},x$

is the acceleration, velocity and displacement of the mass

Table 1 shows the frequency-independent coefficients of circular foundations on elastic halfspace presented by Richart, Woods and Hall and analytical solutions for circular, strip and rectangular foundations are presented by Luco and Gazetas.

2.2.5.         Dynamic Stiffness

Springs and dashpots for each degree of freedom can be condensed to complex-valued impedances which the real part of the impedance function represents a linear spring and the imaginary part represents a dashpot for wave radiation. Impedance can be written in:

$K_a\left(\omega \right)=K_{a1}\left(\omega \right)+i{K}_{a2}\left(\omega \right)$

Which $\omega$

is the angular frequency, real and imaginary components are both functions of $\omega$

. Dynamic impedance of a single degree of freedom oscillator can be expressed as a product of static stiffness and $k+i{a}_{0}c$

which is the dynamic characteristic of the system

$K=K(k+i{a}_{0}c)$

Which $a_0$

is dimensionless frequency factor, $a_0=\frac{\mathit{\omega B}}{V_s}$

, $B$

is the critical foundation dimension, $V_s\mathit{ is\; the\; soil\; shear\; wave\; velocity }$

; $c$

is the variation of damping coefficients with frequency,  $c=c_s\frac{V_s}{B}$

2.3.             References

• Gazetas, G. (1991). “Foundation Vibrations.” Foundation Engineering Handbook, Ed. H. Fang, Kluwer Academic Publishers, Norwell, Massachusetts.
• Gazetas, G. (1983). Analysis of machine foundation vibrations: State of the art. International Journal of Soil Dynamics and Earthquake Engineering, 2(1), pp.2-42
• Dobry, R. (2014). Simplified methods in Soil Dynamics. Soil Dynamics and Earthquake Engineering, 61-62, pp.246-268.
• Kramer, S. L. (1996). “Geotechnical Earthquake Engineering.” Prentice Hall, Upper Saddle River, New Jersey.
• Mylonakis, G., and Gazetas, G. (2000). “Seismic Soil-Structure Interaction: Beneficial or Detrimental?.” Journal of Earthquake Engineering, 4(3), 277-301.
• Mylonakis, G., Nikolaou, S. and Gazetas, G. (2006). Footings under seismic loading: Analysis and design issues with emphasis on bridge foundations. Soil Dynamics and Earthquake Engineering, 26(9), pp.824-853.
• Mekki, M., Elachachi, S., Breysse, D., Nedjar, D. and Zoutat, M. (2014). Soil-structure interaction effects on RC structures within a performance-based earthquake engineering framework. European Journal of Environmental and Civil Engineering, 18(8), pp.945-962.
• National Institute of Standards and Technology (NIST). (2012). “Soil-Structure Interaction for Building Structures.” NIST GCR 12-917-21, Gaithersburg, Maryland.
• Stewart, J. P., Fenves, G. L., and Seed, R. B. (1999a). “Seismic Soil-Structure Interaction in Buildings. I: Analytical Aspects.” Journal of Geotechnical and Geoenvironmental Engineering, 125, GEOBASE, 26-37.
• Torabi, H. and Rayhani, M. (2014). Three dimensional Finite Element modeling of seismic soil–structure interaction in soft soil. Computers and Geotechnics, 60, pp.9-19
• Towhata, I. (2011). Geotechnical earthquake engineering. Berlin: Springer, pp.252-254.