STIFFNESS AND DAMPING PROPERTIES OF EMBEDDED MACHINE FOUNDATIONS

In this study, a dynamic analysis of machine foundations under vertical excitations is carried out. The effect of embedment and foundation geometry has been taken into account. The stiffness and damping of soil are considered as frequency dependents. A computer program ( CPESP ) in FORTRAN POWER STATION has been coded to evaluate the stiffness and damping coefficients depending on excitation frequency and embedment depth. Results have shown that increasing the embedment depth leads to increasing the resonant frequency and decreasing the amplitude of vibration.


INTRODUCTION
Most of the solution methods treat the machine foundation as a block resting on the surface of an elastic soil.The real footings are usually embedded and this considerably affects the dynamic response of footing, Barken.D.D (1962).The rigorous analytical solution of embedded footings has many mathematical difficulties.The most promising way of studying this problem is the finite element analysis as had been used by many researchers such as Lysmer.J(1979) and by kaldjian .M.J (1969) for static analysis.Nevertheless, there is a need for alternative approximate solutions that would be able to predict the motion and to evaluate the stiffness and damping characteristics of embedded footings.

EQUATION OF MOTION
By applying de Alembert's principle, the equation of motion can be written as; Fig. ( 1) (1) Where:-= m Total mass.
Elastic force.For harmonic loading with an excitation frequency ofω , the steady state solution can be assumed as: Dividing both sides of the equation by ) exp( t iω and separating real and imaginary parts, the amplitude of motion z A will be: Multiplying the numerator and denominator of eq.(3) by , the amplitude can be written as:- A into eq.( 2) the steady state solution becomes:- The real part of the amplitude of vibration is:- where:-Eq.( 5) gives the dynamic response of the foundation in vertical vibration and for an exciting force of constant amplitude o P .The natural frequency of the undamped free vibration is:- In this study a rigid foundation will be studied which is located at depth D below the ground surface.This foundation is subjected to a steady-state vibration by a harmonic vertical force, , having an amplitude of o P and a circular frequencyω , and acting through the centeroid of the base.This dynamic force is resisted by normal soil stresses against the base and by shear stresses along the vertical foundation sides.The rotational oscillations that may occur due to the lack of complete symmetry in the soil reactions at the base and especially at the foundation sides are ignored in this study.The steady-state response of the foundation is thus described by the vertical dynamic settlement  (11)   The corrected dynamic stiffness, ) (β K and the dynamic damping coefficient, are given by:- Where: β = frequency independent damping ratio.For most soils β ranges typically from 0.02 to 0.05, Richart.F.E. (1970).
Both the effective dynamic stiffness and the radiation damping coefficient of the soil-foundation system are functions of the frequencyω .It is convenient to express dy sur K ) as a product of the static stiffness, sur K of the system times a dynamic stiffness coefficient STATIC STIFFNESS OF SURFACE FOUNDATIONS For a surface foundation of an arbitrary shape, the vertical static stiffness sur K , is given by Dominguez,J(1978): Where:-L =Semi-length of a rectangle circumscribed to base surface.
z S =Vertical static stiffness parameter.
For non-rectangular base, sur K may be obtained as follows, Prakash,S(1988 The equivalent circle approximation predicts z S as follows (10) :- The equivalent circle approximation gives good results for L/B 2 to 3 as calculated by Dobry and Gazetas (1986).The factors that modify the foundation stiffness are the "trench" and "sidewall contact" effects, that tend to increase the stiffness of the embedded foundation.These two effects are to be explained with the aid of Fig. (3).

Trench Effect
Even in perfectly homogenous soil a rigid footing will settle less if it is placed at the bottom of an open trench.The normal and shear stresses resulting from the overlying soil restricts the vertical movement and thus reducing the settlement of the foundation base by increasing its vertical stiffness.
The trench effect suggested by Gazetas and Dobry (1986) is:- Where:tre K is the vertical static stiffness of an embedded foundation mat with no sidewall contact.

Sidewall Effect
Part of the applied load is transmitted to the ground through shear stresses along the vertical sides of the footing when the sides are in contact with the surrounding soil.As a result, the overall stiffness of an embedded foundation emb K is larger than tre K stiffness corresponding to a foundation with the same depth of embedment but without side effect , Ricardo.D (1985).τ Experimental studies, such as those of Lysmer.j (1969), offer valuable guidance in this direction.Combining eqs.13 and 14 lead to: Based on test results the following empirical equations had been derived:- ) also increases.This trend is more pronounced for the case of a square foundation (L/B=1).The foundation static stiffness ( emb K ) for a full embedment case is:- These equations were obtained by Gazetas and Dobry (1986), Where:-) (ω K :is a dimensionless frequency dependent factor given in Table (1).Hence the dynamic stiffness of an embedded foundation can be written as:- Where: - as given in eq. ( 18).
The factor e F of eq. ( 19) is the effective embedment factor.

DAMPING COEFFICIENT
The coefficient of damping is a measure of vibration energy transmitted into the soil and carried away by spreading waves.These waves are generated at every point on the soil-foundation interface so that in general Where:- : Coefficient of dynamic damping as given in Table (2).The static embedment stiffness will be:emb K =8.268*10 6 kN/m From Table (1) and using eq.(18a) then the effective embedment factor is equal to 0.713.(3) shows the final results for applications (1) and (2) .The Lysmers analog velocity using eq.( 20) is:- From Table (2) and using ( o a =1.5) then the dynamic damping coefficient is :- The dynamic damping of soil using eq.( 21) is:-C = 0.726*10 6 kN.m -1 .sec The corrected dynamic stiffness and damping using eq.( 7  A convergence in results is obvious when the depth ratio will be about 0.50.This means that the reduction in dynamic displacement will be less pronounced when the depth ratio is to be increased higher than 0.50.

CONCUSIONS
The effect of embedment upon vertical forced vibration of a rigid footing was investigated theoretically.
The conclusions can be summarized as follows: 1-The use of equivalent circle approach to estimate the dynamic stiffness and damping factors can cause errors as the aspect ratio of the foundation (L/B )and the soil Poisson's ratio (υ ) being increased.The error will generally be increased at higher frequencies.2-Embedment of foundations has a significant effect on the dynamic response.It causes an increase in the dynamic stiffness and damping coefficients and leads to increase the resonant frequency and to decrease the dynamic response of foundation.A convergence in results is obvious when the depth ratio will be about 0.50.This means that the reduction in dynamic displacement will be less pronounced when the depth ratio is to be increased higher than 0.50.3-The dynamic displacement in the vertical direction is smaller for the case of square foundations as compared to those of rectangular foundations for the same weight and contact soil pressure.
The results indicate a reduction in the dynamic displacement in a range of (15% -17% )as compared to those of the rectangular foundation.

Fig
Fig. (3) effects of embedment on vertical static stiffness of foundation (a) settlement due to surface foundation (b) trench effect (c) combined trench and sidewall effects.

A
=Base area of foundation.s A =Sides area of foundation.Fig (4) shows that as (D/B) increases the ratio of ( sur tre K K

Fig.( 5 )
Fig.(5) shows that as (D/B) increases the ratio ( Fig. (6) Shows the variation of this factor with the normalized frequency parameter( o a ).The relationships have been obtained in the present study by coding the above equations through a short computer program.
area of contact.The contact surface for a vertically oscillating embedded foundation consists of a horizontal base and vertical sides.The base transmits to the underlying ground compression-extension waves in propagation velocity close to the Lymers (1969) analogy .
Lysmer's analog" velocity   On the other hand the sides transmit mainly shear waves through the surrounding soil.The two types of waves generated at the base and at the sides of an embedded foundation are independent.Summing up the respective radiated energies.

Table ( 3
) dynamic stiffness of embedded foundation using the present study and approximate methods.

Table ( 3
) compares the results of the present study and the equivalent circle approximation and the maximum discrepancy is about 3%.
The obtained coefficients in this study Table(1 and 2) are used to study the dynamic response of machine foundation under vertical dynamic load by using SAP 2000.The analysis parameters are:-

Table ( 4
) results obtained from the analysis of SAP2000(application 3)The same foundation has been analyzed for different embedment ratios (D/B) and the results for the displacement-time output are shown in Fig.(8).It is evident that when the depth ratio increases the vertical displacement decreases.