1. Dynamic Soil-Structure Interaction
Compliance and Impedance Functions

2. Soil-Structure Interaction (SSI): The Problem
In structures founded on non-rigid formations, the soil flexibility influences structural response and vice versa. This phenomenon is called SSI.
SSI is an important factor in many applications such as machine foundation vibrations and seismically loaded structures.
SSI is one of several factors influencing response to seismic loads that include source effect, travel path effect and site-response effect.

3. Soil-Structure Interaction (SSI): The Problem
The problem is tackled by modelling the system in two approaches:
Entire Structure-soil system as a unit (full FE modelling necessary)
Substructure approach (FE-Analytical: hybrid)
The substructure approach has multiple advantages (see figure):
Easier to understand the mechanics
Less computational effort
Enables to make parametric studies
Encourages professional interaction between geotechnical and structural engineers

4. Soil-Structure Interaction (SSI): The Problem
SSI problems involve the interaction of three interlinked systems:
The structure
The foundation
The geological medium
SSI comprises two physical phenomena:
Kinematic interaction: The mere presence of the stiff foundation element, regardless of its mass, causes foundation motion to deviate from the free- field motion
Inertial interaction: Inertia of the structure during vibration causes base shear and moment, which in turn cause additional soil deformation with respect to the free-field motion

5. Soil-Structure Interaction (SSI): The Problem
Kinematic Interaction:
Can be handled through the use of a complex-valued transfer function that relates the free-field and the foundation motions
Inertial Interaction:
can be treated with the use of complex-valued compliance or impedance functions that relate the resultant base forces and moments to the respective displacements and rotations
Compliance function: Ratio of the foundation displacement/rotation to the respective resultant base force/moment
Impedance function: Ratio of the resultant base force/moment to the respective foundation displacement/rotation
Impedance functions are complex valued and thus directly give complex-valued foundation stiffness that can be split into real-valued spring and dashpot coefficients. These are introduced at the structure base in order to analyse the structure.

6. Soil-Structure Interaction (SSI): The Problem
Ideal Case: perfectly rigid geological medium supporting the foundation. In this case,
No relative motion occurs; the impedance function has an infinite real part (infinite stiffness) and zero imaginary part (no damping);
The transfer function has only a real part of unit value implying that the free-field motion is the same as the foundation motion.
It thus implies that the structure can be modelled as a fixed-base system subjected to the free-field motion (see figure)
All other cases: SSI effect exists which may be accounted for by introducing mechanical elements to the structure subjected to modified ground motions as shown.

7. SSI: Compliance functions-Historical Development
The basis for inertial SSI is the relationship between foundation displacements/rotations and foundation forces/moments
Lamb, in 1904, became pioneer in this aspect of foundation dynamics by studying the response of an elastic half space subjected to a vertical (and a horizontal) vibrating point loads – a problem generally considered as the dynamic Boussinesq’s problem.

8. SSI: Compliance functions-Historical Development
Lamb has actually studied the problems of both the concentrated and the line load acting on the surface and inside the body of the elastic half-space (Richart et al, 1970)
This challenging mathematical problem at the time involves solving Navier’s dynamic equilibrium equations for the known boundary conditions with due account for the elastic material law (Hooke’s Law).

9. SSI: Compliance functions
Reissner, in 1936, analysed the response of the elastic half space subjected to a harmonic dynamic vertical load acting on a flexible circular area (massless flexible plate) at the surface causing a uniform stress of
The solution was obtained by integrating Lamb’s solution for a vertical point load.

10. SSI: Compliance functions
Reissner obtained the vertical displacement of the centre of the flexible circle as
Where 𝑓 1 + 𝑖𝑓 2 is a frequency-dependent complex-valued function known as compliance function, which is the ratio of the displacement function to the force function
𝑄 𝑡 = 𝑄 0 𝑒 𝑖𝜔𝑡 : is the resultant dynamic force
The amplitude of the displacement follows easily as the modulus of the complex displacement function after Euler’s expansion of the exponential function as

11. SSI: Compliance functions
The real and imaginary parts of Reissner’s compliance function as functions of the dimensionless frequency parameter and Poisson’s ratio, adjusted later by Bycroft (1956) for a rigid foundation, are as shown
At zero-frequency (i.e. static loading), f2 is zero and f1 assumes the value
With this inserted in the displacement amplitude, one obtains the static response (and thereby the static stiffness):

12. SSI: Compliance functions
Reissner also solved the ideally flexible foundation problem by considering its weight, W, or mass, m. He obtained for the displacement amplitude of the centre
Where
Thisequation simplifies to the previous one when m=0; ρ is the density of the half-space material.
Reissner’s theory forms the basis for all subsequent analytical studies, and his work is regarded as the classic paper in the field (Richart et al, 1970).

13. SSI: Compliance functions
Quinlan (1953) and Sung (1953) considered the problem of a dynamic load that causes, in addition to a uniform pressure, non-uniformly distributed contact pressure, which is pertinent to foundations with some degree of rigidity.
The uniform and parabolic contact stresses resulted in a displacement increasing from edge to centre, whereas the rigid-base distribution gave uniform displacement.
Sung (1953) found results of similar form for the centre displacement as Reissner’s problem with differences reflected only on the variation of the compliance function with frequency.

14. SSI: Compliance functions
In all cases, the plots of the normalized displacement amplitude, w0, of the mass-bound foundation against the frequency ratio, a0, can be easily prepared for different values of the mass ratio, b.
Such plots for a constant amplitude force are as shown here (left)
This may be compared with the plots of the displacement magnification factor of a lumped-parameter SDF oscillator as shown (right) under similar loading.

15. SSI: Compliance functions
These two sets of plots have a striking similarity in the way the vibration amplitudes vary.
It is evident that the mass ratio in the block foundation is related to the damping of the SDF model, in which decreasing values of the mass ratio, b, implies increasing damping.
Looking at the expression for b (𝑏= 𝑚 𝜌 𝑅 3 ) this indicates that for a given mass of the foundation-system, enlarging the contact area increases damping. This is attributed to the generation of more waves over the increased area that carry away energy.

16. SSI: Impedance functions – Hseih’s Analog
Hseih (1962), after observing the similarity of the works described above, sought a SDF mechanical analog to Reissner’s problem of massless foundation (flexible circular area)
The equation of motion of the SDF for m=0 is
The harmonic response is easily obtained as
This ratio, which is the inverse of the compliance function defined above, is also a complex-valued function and is called impedance function

17. SSI: Impedance functions – Hseih’s Analog
By rearranging Reissner’s solution, one can re-write it as
By equating the two expressions, he obtained
Hence, the soil-foundation system can indeed be represented by a SDF model with real- valued, but frequency dependent, spring and dashpot coefficients, 𝑘 𝑧 and 𝑐 𝑧 .
The impedance function represents the force- displacement ratio in contrast to the compliance function, which represents the displacement-force ratio: one is the inverse of the other.
Since no material damping was considered in Reissner’s study, the damping 𝑐 𝑧 can only be a result of geometric/radiation damping alone that evolved from the theoretical analysis.

18. SSI: Impedance functions – Hseih’s Analog
The above finding that the vibration of the half space can be modelled by a SDF oscillator has a significant implication in the vibration analysis of real foundations with mass.
A rigid footing (e.g. rigid block machine foundation) of mass m subjected to a vertical dynamic force Q(t) can be represented by the simple mechanical model shown and its equation given by
The solution can be obtained in the usual manner once the spring and damping coefficients are known as presented above.

19. SSI: Impedance functions – Lysmer’s Analog
Lysmer (1965), redefined the compliance functions as
Then he observed that the plots of the real and imaginary parts of the modified function collapse into almost a single curve in each case as shown in the figure (being nearly independent of Poisson’s ratio)
He also introduced a modified dimensionless mass ratio given by

20. SSI: Impedance functions – Lysmer’s Analog
This led him to proposing frequency-independent spring (static) and dashpot coefficients given by
Alternatively, using the definition of the critical damping coefficient, the damping ratio can be determined as
With these parameters, the SDF vibration, thus that of the foundation, can be completely evaluated
Note that Lysmer’s dynamic spring coefficient is identical to the static spring coefficient found earlier.

21. SSI: Impedance functions – Horizontal Vibration
Arnold, Bycroft and Warburton (1955) followed a similar approach and provided solutions for the impedance functions and Hall (1967) developed an analog with frequency-independent coefficients for horizontal vibration given by

22. SSI: Impedance functions – Rocking Vibration
Arnold, Bycroft and Warburton (1955) also provided solutions for the impedance functions for rocking vibration and developed an analog with frequency-independent coefficients given by
𝐼 𝜃 is the mass moment of inertia with respect to the y-axis through O given by

23. SSI: Impedance functions – Torsional Vibration
Reissner (1937) solved also the torsional problem, and later Reissner and Sagoli (1944) proposed frequency-independent parameters
𝐼 𝛼 is the mass moment of inertia with respect to the vertical axis of rotation.
In all the above models, the spring coefficients correspond to the static case (zero frequency).

24. SSI: Impedance functions
A lot of results has already been obtained published in different outlets.
Some efforts have been made to compile them
The main outcomes have made their way to SSI provisions of standard codes (International Building Code (IBC), American Society of Civil Engineers (ASCE), etc)
The above discussion shows numerous factors influence the impedance functions:
Elastic parameters of the soil
Foundation size
Frequency
Mode of vibration
Further factors include
Foundation shape
Embedment depth
Foundation type (footing, raft, pile)
Soil stratification
For this reason, the determination of impedance functions for different combinations of these factors has been the subject of decades of research until recently

25. SSI: Impedance functions -Generalization
In the most general case, a rigid foundation unit on the surface of an elastic half space has six degrees of freedom.
The force-displacement relationship of the foundation can be expressed in a unified manner in matrix form as
Where
{P} is the vector of foundation (base) forces and moments;
{u} is the vector of foundation displacements and rotations;
𝑘 is the complex impedance matrix.
The six-by-six impedance matrix is dominated by diagonal terms with only two off-diagonal terms coupling the rocking and horizontal motions
The off-diagonal terms are, however, neglected in practice as they are comparatively small for a small foundation thickness, which is normally assumed in the derivation of the impedance functions.

26. SSI: Impedance functions
In seismic analyses of structures the impedance functions for the horizontal direction considered and the associated rocking around the perpendicular axis are the most important ones. Neglecting the small coupling term, the corresponding force-displacement relationship may be expressed as
As shown earlier using a SDF model, the dynamic complex stiffness for any mode of vibration can be expressed as
Where j stands for either the translational or rotation degree of freedom
On the other hand, the theoretical complex- valued impedance functions are available for various combinations of the different factors listed above.
The most commonly employed are those provided by Veletsos and Wei (1971) and Veletsos and Verbic (1973) for rigid circular foundations over a wider frequency range than the earlier versions.
The recent results are commonly available in form of

27. SSI: Impedance functions
The static stiffnesses are summarized below for the various modes of vibration as obtained from elastic theory:
The coefficients αi and βi are coefficients that represent the frequency dependence of the impedance functions. They are provided graphically in the next slide for a representative Poisson’s ratio of 0.4
For large frequencies ( 𝑎 0 >2), the dynamic multipliers may be considered constant, whereas for smaller frequencies, ( 𝑎 0 <2), the variation could generally be significant, especially of the damping
By equating the impedance function of the model oscillator and the foundation, one obtains the real-valued foundation stiffness and damping coefficients as
Where the Ki is the static stiffness for the ith mode of vibration

29. SSI: Impedance functions
For a rectangular foundation of B by L the equivalent radius may be used that is obtained by equating pertinent areas and moment of inertias:
Translation:
Rocking (around the long side)
Torsion:
The equivalent radius approach gives very good results for L/B up to 2 (Richart et al, 1970).
This is an alternative to the direct use of impedance functions available for rectangular foundations due to the simplicity of those presented above for circular foundations

30. SSI: Impedance functions
For foundations with shapes other than circular and rectangular, use of the impedance functions available for rectangular foundations is possible using the dimensions of the enclosing rectangle as shown.
The equivalent-rectangle approach gives good results for L/B up to 4, where L and B are the length and width of the rectangle enclosing the arbitrary-shaped foundation (Gazetas, 1983, 1991).

31. SSI: Impedance functions – non-uniform soil profile
The case of a rigid circular foundation on the surface of the elastic half space discussed thus far is considered as the basic case.
The solutions to this basic case have been recently validated using experimental measurements (Stewart et al 1998).
Important other factors that influence the values of the impedance functions are considered next
The first one is non-uniform soil profile
Two cases may be identified:
Gradual increase in stiffness with depth
Soft layer overlying a much stiffer one
Case 1: Gradual increase in stiffness
This case was studied by Roesset (1980) and validation work conducted by Stewart et al (1998).
The conclusion is that properties at a depth of 0.5R give half space impedances that are representative of the variable profile.

32. SSI: Impedance functions – non-uniform soil profiles
Case 2: Soft layer over much stiffer half space
Increased static stiffness and modified dynamic multipliers are proposed
The static stiffnesses are proposed by Kausel (1974)
Where
𝐾 𝑢 and 𝐾 𝜃 are the static stiffnesses for the uniform half space and 𝑑 𝑠 is the thickness of the softer, upper finite layer
The dynamic multipliers of the half space solution may be used unmodified in most cases with the following changes on damping:
Half space damping may be used for frequencies exceeding the soil layer fundamental frequency; a transition to zero radiation damping may be made with decreasing frequencies from this threshold.

33. SSI: Impedance functions – foundation embedment
To account for embedment, Elsabee and Morray (1977) suggested the following static stiffnesses for a circular foundation embedded a depth e in a layer of thickness ds.
The geometric quantities are as shown in the figure

34. SSI: Impedance functions – on foundation shape
Gazetas and Dobry (1981) confirmed the conventional practice that equivalent circular foundation equivalence may be used for any arbitrary shape, provided that the aspect ratio, L/B, of the enclosing rectangle is less than 4:1.
However, they pointed out a notable difference in the dashpot coefficients for the rocking mode
Their results presented in the plots (next slide) show that the equivalent disc approach can seriously undermine the damping of slender foundations for rocking.
The reason for this is that with increasing L/B, during rocking, the opposite edges behave like independent sources of waves with reduced destructive interference so that increasingly more energy is dissipated.
This tendency is particularly true for small frequencies where the difference between the conventional approach and the rigorous theoretical results can reach up to 100%
Thus, it is suggested to determine the damping for rocking by multiplying the value for the equivalent circle by the ratio of 𝐶 𝑟 𝐿 𝐵 / 𝐶 𝑟 𝐿 𝐵 =1 read from the plots corresponding to the fundamental frequency of the structure

35. SSI: Impedance functions – on foundation shape

36. Thank You