Dynamic response of machine foundations considering soil damping and embedment Indrajit Chowdhury, Petrofac Int.Ltd. ,Sharjah UAE, Email :
[email protected] ABSTRACT: Foundations subjected to dynamic loading are usually designed as per IS code, ignoring damping and embedment effect of soil. This generally makes the foundation more expensive and difficult to design, especially in brown field project when foundation frequency is in the proximity of operating frequency of the machine, while there is no space available to modify the foundation footprint. Present paper proposes a method based on which a number of such deficiencies as cited above can be circumvented.
INTRODUCTION Technology adapted for design of machine foundations under harmonic load as per IS-2974 Part IV (1979), is as proposed by Barkan (1962). This has been in practice for last 40 years or more though far more advanced and realistic soil models are available in advanced countries as well as industry (ACI 351.3R-04). Some major limitations that can be attributed to Barkan’s method are as follows: •
•
• •
•
Barkan’s model does not take damping into consideration.It has been observed from field instrumentation data that damping plays a significant role in overall response of foundation, especially when operating frequency of the machine is low. It does not take into account embedment effect of surrounding soil which could play a significant role on the magnitude of soil stiffness and damping. It does not take into cognisance virtual mass of soil which vibrates in same phase with machine and the foundation. Barkan suggested spring value (usually the coefficient of uniform elastic compression) of the soil to be obtained from dynamic plate load test (carried out with a plate of size 300mmx300 mm). This may only give correct values for a shallow depth below ground surface and may not be valid for layered soil, especially when contact area of the foundation is large for big machines. It ignores transient part of the excitation. This can become critical for high speed machine, especially for pipe flanges that are connected rigidly to the equipment and can undergo fatigue failure due to transient shocks.
Present paper proposes a technique by which a number of shortcomings as cited above can be overcome based on other type of soil models that are also in vogue in the industry. The technique is also computationally efficient. Barkan’s method for dynamic analysis As a prelude to the proposed technique, Barkan or IS-2974 method is briefly explained as hereafter. In this method Barkan assumed the block foundation, shown in Fig. 1 as a rigid lumped mass (i.e. he assumed the concrete block to have infinite stiffness in comparison to the soil and neglected any internal deformation of the concrete block itself) having three degrees of freedom as shown below. Pz sinωmt
M0sinωmt H Zc
m &x&
P0sinωmt m &z& φ x0
H
Fig-1. Barkan’s Model for vertical and coupled motion The soil medium is idealised as linear springs which he defined in terms of soil parameter c z , cτ & cφ which are otherwise known as coefficient of elastic uniform compression, coefficient of elastic uniform shear, coefficient of elastic non-uniform compression respectively. In the vertical direction the spring constant is considered as
k z = c z .A f
(1)
Where, k z = equivalent spring in vertical direction; c z = coefficient of elastic uniform compression, and A f = plan area of foundation. Natural frequency of the foundation in vertical direction is given by k (2) ωz = z m amplitude of vertical vibration is given by ( P / k ) sin ω t δz = z z 2 m (3) 1− r where, r= ω m ω z ; ωm = operating frequency of the machine. For coupled horizontal and rocking mode, when a foundation has horizontal force along its minor axis the foundation undergoes sliding and rocking simultaneously. When the foundation starts vibrating, resistance is mobilised in the soil in terms of forces H and MR. The resistive force may thus be expressed as
∫
H = Cφ ldA
∫
Where, l = distance between rotation axis and the element of area dA; φ = angular rotation of the machine foundation; IA= second moment of area of the foundation contact surface with respect to the axis passing through centroid of the area and perpendicular to the plane of vibration. For laterally applied force Pxsinωmt horizontal resistive force, H can be expressed as H= cτ A f x0 = cτ A f ( x − Z cφ )
machine-foundation block about minor axis of rotation including the machine installed over it. From eqns. (7) and (8), we see that they contain both x and φ , so a coupled sliding and rocking motion will develop along this direction. Using the above equations and considering free vibration, Barkan developed following equations for calculation of the frequencies. 4
ω −
J 0 (ωφ 2 + ω x 2 )ω 2
+
J xφ
ωφ 2 ω x 2 J 0
Here J 0 = J xφ + mZ c 2 ; and ω x 2 =
J xφ
ωφ 2 =
(7)
where m is mass of the machine foundation and machine . Similarly for moment equation about minor axis of the foundation we have
=0
(9)
cφ I A − WZ c J0
cτ A f
. m Based on above, two principal frequencies for coupled vibration is given by
J = 0 2J xφ
⎡ωφ 2 + ωx 2 ± ⎢ ⎢ ⎢ ω 2 +ω 2 x φ ⎢ ⎣
(
)
2
−
4J xφ ωφ 2 ωx 2 J0
⎤ ⎥ ⎥ (10) ⎥ ⎥ ⎦
Considering forced vibration, the amplitudes Ax , Aφ may be expressed as Ax =
(cφI A −WZc + cτ Af Zc2 − Jxφωm2 )P0 ± cτ Af ZcM0
Aφ =
mJxφ(ω12 − ωm2 )(ω22 − ωm2 )
c τ A f Z c P0 ± (c τ A f − mωm 2 )M 0 mJ xφ(ω12 − ωm 2 )(ω2 2 − ωm 2 )
sinωmt
sin ωm t
(11)
(6)
where, Af= area of base contact; Zc, x, xo etc are shown in Fig.1. Now applying D’Alembert’s equation for dynamic equilibrium, we have m&x& + H = P0 sin ω m t or m&x& + cτ A f ( x − Z cφ ) = P0 sin ω mt
where J xφ = mass moment of inertia of the
ω1,2 2
(5)
(8)
= M 0 sin ωmt
(4)
and resistive moment is expressed M R = Cφ l 2φdA = Cφ I Aφ
J xφ φ&& − cτ A f Z c x + φ (cφ I A − WZ c + cτ A f Z c 2 )
For torsional mode, again the foundation considered is a lumped mass having single degree of freedom when frequency and amplitude are given by Kψ T sin ωm t / kψ ωψ = and ψ = (12) Iψ 1− r2
(
)
where, Kψ = cψ Iψ ; r = ω m / ωn . Method as elaborated above is recommended by IS 2974 “Code of practice for design and construction of machine foundation” and still remains the most popular method for vibration analysis of block foundations in Indian industry.
PROPOSED METHOD CONSIDERING SOIL DAMPING AND EMBEDMENT It is apparent from above that IS code does not take into cognizance damping as well as embedment effect of soil. For vertical direction the equation considering soil damping becomes that of a lumped mass having single degree of freedom when m&z& + C z z& + K z z = P0 sin ωmt (13) The natural frequency remains same as equation (2) the damped amplitude of vibration can be expressed as (Chowdhury and Dasgupta 2008) as
z = e − Dωnt (C1 cos ω D t + C 2 sin ω D t ) +
(
P0 sin ω m t
(14)
)
K z 1− r + (2 Dr )2 C1 and C2 are integration constants that need to be derived from appropriate boundary conditions. First part of the expression represents transient response and the second part depicts steady state response. 2 2
2
Here, ω D = ω n 1 − D , here ωD= Damped natural frequency of the system and D= Damping ratio expressed as C z / 2 mK z Considering the two boundary conditions as 1) at t=0; z=0 and 2) at t=0; dz/dt=0 we finally have z=
− P0 sin ωm t
(
Kz 1− r2
)
2
+ (2Dr)2
⎡ r sin ω t.e −Dωnt D ⎢1 − ⎢ sin ω t. 1 − D 2 m ⎣
⎤ ⎥ (15) ⎥ ⎦
In equation (15) P0sinωmt is to be considered negative as it is assumed to be acting in negative z direction In the above equation values of Kz and Cz are vertical spring stiffness and damping of soil based on Lysmer and Richart’s model (1966) and as furnished in Table-A1, A2 etc. in the appendix. The embedment effect can be considered by multiplying the stiffness and damping of soil by factors as furnished in Table A3 & A4 respectively. The values are as per Whitman (1972). For analysis of coupled horizontal and rocking motion let us write Barkan’s expressions as K x = cτ A f and K xφ = cφ I A
(16)
Then, equations of equilibrium are defined as
m&x& + cτ A f ( x − Z cφ ) = P0 sin ω mt , and J xφ φ&& − cτ A f Z c x + φ (cφ I A − WZ c + cτ A f Z c 2 ) = M 0 sin ωmt
(17)
Substituting values of K x and Kφ , we have m&x& + K x ( x − Z cφ ) = P0 Sinω m t and J xφ φ&& − K x Z c x + φ ( Kφ − WZ c + K x Z c 2
(18) = M 0 sin ωmt Equation (18) in matrix form can be represented as − Kx Zc ⎡m 0 ⎤⎧&x&⎫ ⎡ Kx ⎤⎧x⎫ ⎢ 0 J ⎥⎨&&⎬ + ⎢− K Z K + K Z 2 −WZ ⎥⎨ ⎬ φ φ xφ ⎦⎩ ⎭ ⎣ x c x c c ⎦⎩φ⎭ ⎣ (19) ⎧ P0 ⎫ = ⎨ ⎬sinωmt ⎩M0 ⎭ Since above equation is based on D’Alembert’s equation, the equations are said to be statically coupled, when stiffness and damping matrix have the same form (Meirovitch 1967). Thus, the damped equation of motion in coupled rocking and sliding mode becomes −C x Z c ⎤ ⎧ x& ⎫ ⎥⎨ ⎬ Cφx + C x Z c 2 − WZ c ⎦⎥ ⎩φ&⎭
⎡ m 0 ⎤ ⎧ &x&⎫ ⎡ C x ⎥ ⎨ &&⎬ + ⎢ ⎢ ⎣⎢ 0 J xφ ⎦⎥ ⎩φ ⎭ ⎣⎢− C x Z c ⎡ Kx +⎢ ⎢⎣ − K x Z c
− K x Zc Kφx + K x Z c
2
⎤ ⎧ x ⎫ ⎧ P0 ⎫ ⎥ ⎨ ⎬ = ⎨ ⎬ sin ωmt − WZ c ⎥⎦ ⎩φ ⎭ ⎩M 0 ⎭
(20) Above equations constitute the complete equation of motion for coupled sliding and rocking mode considering the damping effect of soil in generalized form, and having soil stiffness as Kx and Kφ. Actually for all practical calculations for finding out the dynamic response of foundation, the term –WZc may be neglected, for it has been observed that unless the foundation is very massive and deep, the term WZc has no significant effect on overall response of the system. Based on above argument equation (20) reduces to − C x Z c ⎤ ⎧ x& ⎫ ⎡m 0 ⎤ ⎧ &x&⎫ ⎡ C x ⎢ 0 J ⎥ ⎨ &&⎬ + ⎢− C Z C + C Z 2 ⎥ ⎨ &⎬ φx xφ ⎦ ⎩φ ⎭ ⎣ x c x c ⎦ ⎩φ ⎭ ⎣ − K x Z c ⎤ ⎧ x ⎫ ⎧ P0 ⎫ ⎡ Kx +⎢ ⎬ sin ωmt 2 ⎥⎨ ⎬ = ⎨ ⎣− K x Z c Kφx + K x Z c ⎦ ⎩φ ⎭ ⎩M 0 ⎭
(21)
Equation (21) above looks elegant, but has a serious catch in it, for this damping matrix of soil is not proportional to either the mass or stiffness of the soil. Moreover as they are coupled by the term Zc , as such do not de-couple on orthogonal transformation. This forms a major headache to a
designer as he is not in a position to guess the soil damping ratio at the outset or resort to a modal analysis. The most appropriate technique in such case is then to resort to time history analysis (like say Wilson-θ method (Bathe-1996)) for a correct answer. However, many engineers find time history too intensive in terms of calculation, and prefer to use modal response technique as a tool for analysis of the same. Of course, easiest way out is to neglect soil damping and argue that the design is conservative (but conveniently overlooking the fact that it becomes more expensive as heavier mass is to be used to restrict the amplitude)! But this need not be done, for it is possible to by pass this problem of orthogonal de-coupling, even when damping matrix is non-proportional which though not exact would give still give a designer a reasonable value to estimate a more realistic amplitude of vibration (it is surely a better value than no damping considered) and is considered hereafter. Approximate analysis to de-couple equations with non-proportional damping Based on equation (21) the natural frequencies are obtained from eigen value analysis when undamped equation becomes − K x Zc ⎡ K x − mλ ⎤ ⎢−K Z ⎥ = 0 (22) 2 K K Z J λ + − x c φ x c φx ⎦ ⎣ Solving the above equations we find out eigen values vis-à-vis natural frequencies of the foundation system. Let the eigen values be λ1 and λ2 respectively. Let corresponding normalized eigen vectors be
< φxx φxφ >
T
and
< φ φx
φ φφ > T
respectively, when the complete eigen vector ⎡φ xx φφx ⎤ matrix is expressed as, ⎢ ⎥ ⎣φ xφ φφφ ⎦ Since the eigen vectors are known separately for each mode we find out the damping ratio separately for each mode as follows. As a first step we perform the operation {φ}T [C]{φ } for each mode. For the first mode, we have ⎡ Cx ⎢− C x Z c ⎣
< φ xx φ xφ > ⎢
− CxZc ⎤ ⎥ Cφx + C x Z c 2 ⎥ ⎦
⎧φ xx ⎫ ⎨ ⎬ ⎩φ xφ ⎭
This gives C xφ xx − C x Z cφ xφ ⎧⎪ ⎫⎪ < φ xx φ xφ > ⎨ ⎬ 2 ⎪⎩− C x Z cφ xx + (Cφx + C x Z c )φ xφ ⎪⎭
=
C xφ xx 2 − 2C x Z cφ xφ φ xx + (Cφx + C x Z c 2 )φ xφ 2
(23) It should be realised that the above is a unique value and we also know that the operation {φ}T [C]{φ} breaks up the equation to form 2 Diωi where i is the degrees of freedom of the system. Now considering, 2 Diωi = C xφ xx 2 − 2C x Z cφ xφφ xx + (Cφx + C x Z c 2 )φ xφ 2 ,
For first mode D1 =
Cxφxx2 − 2Cx Zcφxφφxx + (Cφx + Cx Zc 2 )φxφ 2 2ω1
(24)
where D1 = damping ratio for first mode and; ω1 = first natural frequency of the foundation. Similarly, for second mode proceeding in same manner it can be proved that D2 =
Cφxφφx 2 − 2Cx Zcφφxφφφ + (Cφx + Cx Zc 2 )φφφ2
(25) 2ω2 Once the damping ratios are identified we assume, [C] = α [M ] + β [K ] and performing the operation {φ}T [C]{φ} = α {φ }T [M ]{φ } + β {φ}T [K ]{φ} (26)
We have, for two degrees of freedom 2D1ω1 = α + βω12 and 2D2ω2 = α + βω2 2 (27) Thus, we have two equations with two unknowns, α and β, and solving the above two equations we get values of α and β, which may be expressed as. ⎧ 1 ⎫ D2ω2 ⎤ ⎬+ Ω ⎥⎦ ⎩ Ω⎭ ⎣ 2(D1ω1 − D 2 ω 2 ) β= Ω ⎡
α = 2⎢ D1ω1 ⎨1 −
(28) (29)
(30) Here Ω = ω1 2 − ω 2 2 Once these values are known one can obtain an equivalent proportional soil damping from the
[]
ˆ = α[M ] + β [K ] which is now quite operation C suitable for modal response technique (Chowdhury et al 2002). The corrected modal damping matrix can now be expressed as
[Cˆ ] = α ⎡⎢m0
Torsional mode In this mode the block foundation is again considered as a lumped mass having single degree of freedom, natural frequency and the torsional rotation, ψ is given by
− K x Zc ⎤ 0 ⎤ ⎡ Kx + β⎢ ⎥ 2 ⎥ (31) J xφ ⎦ ⎣− K x Zc Kφx + K x .Zc ⎦
⎣ Equation (21) in matrix form considering ˆ as expressed in corrected modal damping C
[]
ωψ =
Kψ
[M ]{X&& }+ [Cˆ ]{X& }+ [K ]{X } = {Px }sin ω m t
[ ]
[ ]
[ϕ ]T
[ϕ ]T [M][ϕ ]{ξ&&}+ [ϕ ]T [Cˆ ][ϕ ]{ξ&}+ [ϕ ]T [K][ϕ ]{ξ } = [ϕ ]T {Px }sin ω m t
we
(33)
[]
ˆ = α[M ] + β [K ] , i.e. a In equation (33) as C converted equivalent Rayleigh type damping, it decouples into two equations of the form
[ ] + [ω ]ξ
ξ&&1 + [2 D1ω1 ]ξ&1 + ω1 2 ξ1 = p x sin ω m t ξ&&2 + [2 D2ω 2 ]ξ&2
2
(34)
= m x sin ω m t (35) Solution to equation (34) and (35) are expressed as ⎡ −D1ω1t ⎤ − px sinωmt ⎢1− r1 sinω1Dt.e ⎥ (36) ξ1 = ⎢ ⎥ 2 2 λ1 1− r12 + (2D1r1 )2 ⎣ sinωmt. 1− D1 ⎦
(
2
2
)
And
ξ2 =
(
− mx sinωmt
)
λ2 1− r22 + (2D2r2 )2 2
⎡ −D2ω2t ⎤ ⎢1− r2 sinω2Dt.e ⎥ (37) ⎢ sinω t. 1− D 2 ⎥ m 2 ⎦ ⎣
The coupled motion in the global co-ordinate is then expressed as
[X ] = [ϕ ][ξ ] Where [X ] = {x
(38)
φ }T
and
(1 − r ) + (2Dψ r ) 2 2
2
⎡ r sin ω t.e − Dψ ω n t ⎤ Dψ ⎢1 − ⎥ ⎢ 2 ⎥ − sin ω t . 1 D m ψ ⎥ ⎢⎣ ⎦
(40)
(32)
Let λ1 and λ2 be the eigen values and ϕ 2 X 2 be the normalized eigenvectors. Such that {X } = ϕ {ξ } where {ξ } = Displacement vectors in the decoupled coordinate Multiplying equation (32) by the term have
Iψ
− T sin ωmt
ψ =
equation (31) can now be expressed as
Kψ
(39)
where Kψ = 16Grψ 3 / 3 , Dψ is the damping ratio in the torsion mode and r is the ratio between the natural frequency of the foundation in torsion mode and the operating frequency of the machine. RESULTS AND DISCUSSIONS To determine how the procedure works a real life gas turbine foundation having following data is analyzed as a benchmark problem.
Geometric data • Length of foundation =16.1 m • Width of foundation = 6.77 m • Depth of foundation = 3.6 m • Depth of embedment =3.0 m Soil Data • Bearing capacity of soil = 200 kN/m2 • Shear wave velocity of soil = 125 m/sec • Poisson’s Ratio =0.25 • Density of soil =20 kN/m3 Machine Data • Center line height of shaft over machine foundation = 2.0m • Operating frequency = 2250 rpm • Allowable amplitude at top of foundation =0.2mm • Static weight of machine = 4760 kN The natural frequencies based on various methods are as presented hereafter
Analysis ωv ωx ωφ based on (rpm) (rpm) (rpm) Barkan 2779 1738 3858 Lysmer & 2378 2301 5004 Richart Wolf 3945 2091 3800 Lysmer 2636 2034 4001 &Richart(with embedment) Maximum Amplitude of vibration based on various methods are as presented hereafter X(mm)
φ(rad)
0.00285 0.00156*
0.0083 0.0055
0.00143* 0.00103*
0.0045 0.0032
0.00001 0.000006 0.000004 0.000002
Barkan Rot Wolf Rotation 0.71
0.66
0.62
0.57
0.52
0.48
0.43
0.38
0.33
0.29
0.24
0.19
0.1
0.14
-0.000002
0
0 0.05
Rocking amplitude(rad)
0.000008
Lysmer Rot.
-0.000004 -0.000006 -0.000008 -0.00001 Time steps
Fig-4 Time history plot of rotational amplitude Comparison of amplitude based on time history versus corrected damping 0.000015 0.00001
not steady state vibration
amplitude, which is much less
Amplitude
Analysis Zv(mm) Based on Barkan 0.0015 Lysmer 0.0010 &Richart Wolf 0.0010 Lysmer 0.00094 (emebeded) * This is transient peak and
Comparison of rocking amplitude
0.000005 0 -0.000005
1 23 45 67 89 111 133 155 177 199 221 243 265
Displacement with non proportional damping Displacement with corrected proportional damping
-0.00001 -0.000015 Time steps
Fig-5 Comparison of amplitude based on non proportional damping and corrected modal
Amplitude of vertical vibration 0.015
Lysmer spring with damping Barkan undamped
5.8
5.4
4.9
4.5
4.0
3.6
3.1
2.7
2.2
1.8
1.3
0.9
0
0 0.4
Disp lacemen t(mm)
0.01 0.005
-0.005 -0.01 -0.015 Time steps(secs)
Fig-2 Vibration Amplitude in vertical direction Comparison of translational amplitude
0.00003 0.00002 Barkan Disp.
0.00001
Wolf Disp.
0 -0.00001
0 0 .0 4 0 .0 8 0 .1 2 0 .1 6 0 .2 0 .2 4 0 .2 8 0 .3 2 0 .3 6 0 .4 0 .4 4 0 .4 8 0 .5 2 0 .5 6 0 .6 0 .6 4 0 .6 8 0 .7 2
D is p a lc e m e n t (m m )
0.00004
Lysmer Disp.
-0.00002 -0.00003 Time steps (secs)
Fig-3.Time amplitude
history
plot
of
translational
Figure 2 to 4 clearly shows that Barkan’s theory overestimates the amplitude and does not take into cognizance the transient peak which the foundation experiences during starting and stopping a machine. This can be critical for high frequency machines when it passes through the natural frequency zone of the foundation. Lysmer and Wolf’s model considering soil damping gives comparable results. While embedment effect further enhances the frequencies and reduce the amplitude. Adapting Barkan’s theory for design of foundation in brown field plants can put a designer in significant difficulty especially when the foundation is in resonant zone while there is no space available to modify the dimension of the foundation footprint. In such cases if we consider the soil damping, we can very well let the foundation be within the resonance zone so long as we can prove that it does not harm the functional behaviour of the machine as amplitude and velocity of foundation is within acceptable limit- this is of great technical as well as commercial advantage. Figure-5 shows that modal analysis based on corrected damping matrix and time history analysis, considering non-proportional damping are in excellent agreement. Though at the stage
when foundation reaches the steady state in the long run, it shows a phase difference however as far a magnitude is concerned there is hardly any difference between the two analysis. This gives significant computational advantage as one can circumvent the expensive numerical analysis that can be cost wise justified only for very big and expensive machines only. CONCLUSION A comprehensive mathematical model is proposed herein for dynamic analysis of block foundation which is mathematically more realistic and takes into consideration soil damping and embedment effect which the IS code ignores presently. REFERENCE
1. 2. 3. 4.
5.
ACI 351.3R-04 “Foundations for Dynamic Equipment” Report of ACI committee 351. Bathe K.J. (1996), Finite element procedures in engineering; Prentice Hall, New Delhi, India. Barkan D.D. (1962), Dynamics of Bases and Foundations; Mçgrawhill Publications NY USA. Chowdhury I, Ghosh B & Dasgupta, S.P. (2002), “Analysis of Hammer Foundations considering soil damping “ Advances in Civil Engineering ACE 2002, Indian Institute of Technology Kharagpur India Vol-II pp-1019-1028. Chowdhury, I. & Dasgupta, S.P. (2008), Dynamics of structure and foundation- a
unified approach, Volume I & II, Taylor and Francis, Leiden Holland. 6. IS-2974 Part IV (1979) – Code of practice for design and construction of Machine foundations Bureau of Indian Standard New Delhi, India. 7. Lysmer J and Richart F.E. (1966) “Dynamic response of footings subjected to vertical Loading” J. of Soil Mechanics and Foundation Div. ASCE Vol. 92 # SM1 pp 65-91 8. Meirovitch L (1967), “Analytical Methods in Vibration analysis” Macmillan Company UK. 9. Whitman, R.V. (1972), “Analysis of soil structure interaction – a state of the art review” Soil Publication # 300 MIT USA. 10. Wolf John P.(1988) “Dynamic soil structure interaction in time domain” Prentice Hall Engelwood Cliff New Jersey USA
APPENDIX Table-A1 Values of Soil Springs as per Lysmer &Richart (1966)Model
Sl No 1)
Direction Vertical
2)
Horizontal
3)
Rocking
3.1)
Rocking
4)
Twisting
Spring value 4Grz Kz = (1 − υ)
Equivalent radius
32(1 − υ)Grx (7 − 8υ)
Kx =
K φx = K φy =
8Grφx 3
3(1 − υ ) 8Grφy 3
3(1 − υ )
16Grψ 3
Kψ =
rz =
LB π
rx =
LB π
rφx = 4
LB3 3π
rφy = 4
L3B 3π
rψ = 4
3
Remarks This is in vertical Z direction This induce sliding in horizontal x or y direction This produces rocking about Y axis This produces rocking about X axes This produces twisting about vertical Z axis
L3B + BL3 6π
Table-A2 Values of Soil Damping as per Lysmer & Richart’s(1966) Model.
Sl No
Direction
1)
Vertical
2)
Horizontal
Mass ratio( B) Bz =
0.25m(1 − υ)g
Bx =
ρ s rz
3
(7 − 8υ)mg 32(1 − υ )ρ s rx 3
Damping Ratio Damping Value 0.425 , ζz = Bz
and
This is damping value is in vertical Z direction.
C z = 2ζ z K z m ζx =
0.288 Bx
This damping value is in lateral X or Y direction
,
C x = 2ζ x K x m
3)
Rocking
Bφx =
0.375(1 − υ)J φx g ρ s r φx
5
ζ φx =
0.15
(1 + Bφx )
Bφx
,
C φx = 2ζ φx K φx J φx
Rocking
Bφy =
0.375(1 − υ)J φy g ρ s rφy
5
ζ φy =
0.15
(1 + Bφy )
Bφy
C φy = 2ζ φy K φy J φy
4)
Twisting
Bψ =
J ψg ρ s rψ
5
ζψ =
Remarks
0.5 , 1 + 2B ψ
C ψ = 2ζ ψ K ψ J ψ
,
This damping value is for rocking about Y direction This damping value is for rocking about Y axes This damping value is valid for twisting about vertical Z axis.
Table-A3. Embedment Coefficients for Spring Constants as per Whitman (1972).
Sl No 1)
Direction Vertical
2)
Horizontal
3)
Rocking
Coefficient
Equivalent Radius
h η z = 1 + 0.6(1 − υ) rz
rz =
LB π
η x = 1 + 0.55(2 − υ)
h rx
rx =
LB π
ηφx = 1 + 1.2(1 − υ)
h rφx
rφx = 4
LB3 3π
h rφy
rφy = 4
L3B 3π
⎛ h + 0.2(2 − υ)⎜ ⎜ rφx ⎝
3.1)
4)
Rocking
Twisting
⎞ ⎟ ⎟ ⎠
This induce sliding in horizontal x or y direction This produces rocking about Y axis
3
ηφy = 1 + 1.2(1 − υ) ⎛ h ⎞ ⎟ + 0.2(2 − υ)⎜ ⎜ rφy ⎟ ⎠ ⎝ None available
Remarks This is in vertical Z direction
This produces rocking about X axes
3
T
Table-A4. Embedment Coefficients for Soil damping ratio Whitman (1972)
Sl No 1)
Direction Vertical
Coefficient αz =
2)
Horizontal
1 + 1.9(2 − υ)
Rocking α φx =
Rocking 1 + 0.7(1 − υ) α φy =
4)
Twisting
LB π
h rx
rx =
LB π
ηx
1 + 0.7(1 − υ)
3.1)
rz =
ηz
αz =
3)
Equivalent Radius
h 1 + 1.9(1 − υ) rz
None available
3
⎛ h + 0.6(2 − υ)⎜ ⎜ rφx rφx ⎝ ηφx
⎞ ⎟ ⎟ ⎠
⎛ h + 0.6(2 − υ)⎜ ⎜ rφy rφy ⎝ ηφy
3
h
h
⎞ ⎟ ⎟ ⎠
rφx = 4
LB3 3π
rφy = 4
L3B 3π
Remarks η z is value as obtained as coefficient for soil spring constant
η x is value as obtained as coefficient for soil spring constant
ηφx is value as obtained as coefficient for soil spring constant
ηφy is value as obtained as coefficient for soil spring constant
Table- A5. Soil spring constants as per Wolf (1988).
Mode
Spring Stiffness
γ0
µ0
Vertical
4Gr0 1− υ 8Gr0 1− υ
0.58
0.095
0.85
0.27
Horizontal Rocking
Torsion
8Grθ 3 3(1 − υ)
1+
16Grψ 3
0.3 3(1 − υ )m 8rθ 5ρ
0.433 2m 1+ 5 rψ ρ
3
0.24
⎛ m ⎞ ⎜ ⎟ ⎜ r 5ρ ⎟ ψ ⎝ ⎠
0.045
2
In which, C =
r ⎛ r ⎞ kγ 0 and m = ⎜ ⎟ kµ0 Vs ⎝ Vs ⎠
where r= equivalent radius and shall be r0 , rθ , rψ as the case may be; G= Dynamic shear modulus of the
soil; ρ = mass density of the soil; vs= shear wave velocity of the soil; m= mass of the soil participating in the vibration with the machine and the block foundation, and C= damping of the soil.
