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Dynamic-table_eqip Foundation Design
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Following tables and graphs are copied from different text books available in the market. Before using these tables and graphs please check with text books available with you / library. Civildesignhelp.info is not responsible for any error in foundation analysis and design.
LIST OF SYMBOLS
a Maximum cantilever length (inches); measured from the face of block to edge of foundation
d Maximum deflection (inches), for estimating the fundamental natural frequency of beam or column e Rotor eccentricity for centrifugal machines f1 First coupled natural frequency
f2 Second coupled natural frequency
f0 Machine operating speed (cps or rpm)
fd Damped resonant frequency (cps or rpm)
fn Undamped natural frequency (cps or rpm)
fc Concrete compressive strength
fy Yield strength of reinforcement
fnx Uncoupled natural frequency in translational x direction
fny Uncoupled natural frequency in translational y direction
fnz Uncoupled natural frequency in translational z directions
fn Uncoupled natural frequency , rocking about X fn Uncoupled natural frequency , rocking about Y
fn Uncoupled natural frequency , rocking about z, Torsional fx,1
Parameters for calculating single pile stiffnesses in horizontal (X),
fz,1 Parameters for calculating single pile stiffnesses in vertical (Z),
f,1 Parameters for calculating single pile stiffnesses in rocking () modes fx,2
Parameters for calculating single pile damping coefficients in horizontal (X),
fz,2 Parameters for calculating single pile damping coefficients in vertical (Z)
f,2 Parameters for calculating single pile damping coefficients in rocking () modes, g Acceleration of gravity (32.2 ft/sec2 or 386.4 in/sec2)
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h Foundation embedment depth, the distance between the CG of a machine foundation system and the foundation base. hi Thickness of ith soil layer
l Embedded length of a pile
m Total mass of a machine foundation system (machine+foundation) me Rotor mass for centrifugal machines
r Frequency ratio, defined as f0/fd (or f0/fn)
r0 Equivalent radius of foundation; equivalent radius of concrete
pile, pA t Thickness of mat foundation or pile cap A0 displacement amplitude
Ai Stress influence area (spreading below the foundation at a ratio of 2 vertical to 1 horizontal) measured at the centroid of ith soil layer
Ap Cross sectional area of concrete pile
B Foundation width, parallel to the y-axis Bx
Mass ratios for calculating the soil geometrical damping ratios in horizontal X directions
By Mass ratios for calculating the soil geometrical damping ratios in horizontal Y directions
Bz Mass ratios for calculating the soil geometrical damping ratios in vertical Z directions
BBMass ratios for calculating the soil geometrical damping ratios in rocking about X () directions
BBMass ratios for calculating the soil geometrical damping ratios in rocking about Y () directions
BBMass ratios for calculating the soil geometrical damping ratios in torsional about Z () directions
Cx Damping coefficients for a single pile in horizontal (X) direction
Cz Damping coefficients for a single pile in vertical (Z) direction
C Damping coefficients for a single pile in rocking about X () directions
D System damping or equivalent modal damping ratio for a coupled vibrational motion Dx Soil or soil-pile damping ratios in horizontal X directions
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Dy Soil or soil-pile damping ratios in horizontal Y directions
Dz Soil or soil-pile damping ratios in vertical (Z) directions
D Soil or soil-pile damping ratios in rocking about X() directions D
Soil or soil-pile damping ratios in rocking about Y ()directions
DSoil or soil-pile damping ratios in torsional about Z () directions
Ep Young's modulus of concrete pile
F0 Machine unbalanced force for centrifugal machines
Fn Unbalanced force at natural frequency of the foundation system
Fx Unbalanced force in horizontal x direction
Fy Unbalanced force in horizontal y direction
Fz Unbalanced force in vertical z direction
G, Gs Dynamic soil shear modulus
Gavg Weighted average soil shear modulus
Io Mass moment of inertia at the C.G. of a machine foundation
Ip Area moment of inertia of single pile
IMass moments of inertia at the foundation base in rocking about X () direction
IMass moments of inertia at the foundation base in rocking about Y () direction
IMass moments of inertia at the foundation base in torsional about Z () direction
Ks Soil dynamic subgrade reaction (lbs/in3)
Kx Soil spring constants or single pile spring constants in translational X direction.
Ky Soil spring constants or single pile spring constants in translational Y direction.
Kz Soil spring constants or single pile spring constants in translational Z direction.
K Soil spring constants or single pile spring constants in rotational modes K Soil spring constants or single pile spring constants in rotational modes
K Soil spring constants or single pile spring constants in rotational modes
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(Ki)g Pile spring constants for a pile group
L foundation length, parallel to the X-axis . M Unbalanced moments about X (), M Unbalanced moments about Y (),
M Unbalanced moments about Z (), N Number of soil layers; number of piles Ni Dynamic amplification factors at ith vibrational motion
V0 Maximum velocity amplitude
Vp Compression wave velocity of pile, g/E pp Vs Shear wave velocity of soil s sG / W Total weight of a machine foundation system (m x g) We Rotor weight
X Cartesian coordinates in horizontal X (parallel to L) Y Cartesian coordinates in horizontal Y (parallel to B ) Z Cartesian coordinates in vertical Z x Coefficients for soil damping calculation due to embedment effect in horizontal (X), z Coefficients for soil damping calculation due to embedment effect in vertical (Z) Coefficients for soil damping calculation due to embedment effect in rocking about X () mode Coefficients for soil damping calculation due to embedment effect in rocking about Y () mode 0I
I or 0I
I
x Geometrical coefficients for soil spring calculation for rectangular foundation in translational X direction. y Geometrical coefficients for soil spring calculation for rectangular foundation in translational Y direction. z Geometrical coefficients for soil spring calculation for rectangular foundation in translational Z direction. Geometrical coefficients for soil spring calculation for rectangular foundation in rotational about X () modes. Geometrical coefficients for soil spring calculation for rectangular foundation in rotational about Y () modes. Correction factor for soil damping calculation in rocking mode
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s Weight density of soil (Ds x g) p Weight density of concrete pile s Mass density of soil Poisson's ratio of soil Machine operating speed (rad/sec) x Coefficients for soil spring calculation due to embedment effect in horizontal X direction. y Coefficients for soil spring calculation due to embedment effect in horizontal Y direction. z Coefficients for soil spring calculation due to embedment effect in vertical Z direction. Coefficients for soil spring calculation due to embedment effect rocking about X () Coefficients for soil spring calculation due to embedment effect rocking about Y () rotation about vertical Z-axis (torsion) rotation about horizontal X-axis (rocking) rotation about horizontal Y-axis (rocking)
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Table 1
GENERAL MACHINERY - VIBRATION - SEVERITY DATA
Horizontal Peak Velocity
(in./sec)
Machine Operation
0.630 Very rough
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Table 2 ROTOR ECCENTRICITIES
Machine Operating Speed f (RPM) Eccentricity
e (MILS)
Centrifugal Compressors
and Turbines
f 3000 1.8 _ 107
2f
(or Fo = 0.51 We)
Centrifugal Compressors
and Turbines
f > 3000
12000f
Gear Units f 8000 0.8 Gear Units f > 8000 0.5
Motors f 1500 1.5 Motors 1500 < f < 3000 1
Motors f 3000 0.5
Fo : Machine unbalanced force We : Rotor weight
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Table 3
SOIL SPRING CONSTANTS FOR RECTANGULAR FOUNDATION
Mode of Vibration Soil Spring Constants
Horizontal (X) xxx BLGK ) + 2(1 =
Horizontal (Y) y y yK G BL = 2(1 + )
Vertical (Z) z z zK
G BL = (1 - )
Rocking (about X)
KG
B L = (1 - ) 2
Rocking (about Y)
KG BL = (1 - )
2
Torsional (about Z)
KG r = 16 3
o3
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Table 4
SOIL SPRING COEFFICIENTS FOR SOIL SPRING CALCULATION INCLUDING EMBEDMENT EFFECT
Mode of
Vibration Equivalent Radius (ro)
Soil Spring Coefficient
Horizontal
(X & Y) BL
Vertical (Z)
BL
zo
= 1 + 0.6(1 - ) hr
Rocking (about X)
3
43
B L
= 1+1.2(1- ) hr + 0.2(2 - )
hro
3
o
Rocking (about Y)
43
3BL
rh)-0.2(2 +
rh)-1.2(1+1 =
o
3
o
Torsional (about Z)
422
6) + (
LBBL
N/A
* h = embedment depth
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Table 5
GEOMETRICAL DAMPING RATIOS FOR RECTANGULAR FOUNDATION (INCLUDING THE EMBEDMENT EFFECT)
Mode of Vibration
Mass Ratio Damping Ratio (D)
Vertical
(Z) z
sB
mr
= (1 - )4
o3
z
zzD
B = 0.425
Horizontal (X or Y)
xs
Bmr
= (7 - 8 )32(1 - )
o3
x
xxD
B = 0.288
Rocking (about X)
BIrs
= 3(1 - )8
o5
D B B = 0.15
(1 + )
Rocking (about Y)
BIrs
= 3(1 - )8
o5
D B B = 0.15
(1 + )
Torsional (about Z)
BIrs
= o5
D
B = 0.5
1 + 2
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Table 6
DAMPING COEFFICIENTS FOR SOIL DAMPING CALCULATION INCLUDING EMBEDMENT EFFECT
Mode of Vibration Damping Coefficient
Vertical (Z) z
z
hr
= 1 1 + 1.9(1 - )
o
Horizontal (X or Y) x
x
hr
= 1 1 + 1.9(2 - )
o
Rocking (about X)
=
1 1 + 0.7(1 - ) + 0.6(2 - ) o
3
o
hr
hr
Rocking (about Y)
) - 0.6(2 + ) - 0.7(1 + 1 1 = o
3
o rh
rh
Torsional (about Z)
N/A
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Table 7
CORRECTION FACTOR FOR SOIL DAMPING CALCULATION IN ROCKING MODE
5 1.079
3 1.110
2 1.143
1 1.219
0.8 1.251
0.5 1.378
0.2 1.600
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Table 8
PILE STIFFNESSES FOR A SINGLE PILE
Mode of Vibration Pile Stiffness
Horizontal (X or Y)
x y p p xK K E I f = = /,1 o3r
Vertical (Z)
z p p zK E A f = /,1 or
Rocking (about X or Y)
K K E I fp p = = /,1 or
Torsional (about Z)
N/A
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Table 9
DAMPING COEFFICIENTS FOR A SINGLE PILE
Mode of Vibration Damping Coefficient
Vertical (Z) z p p zC E A f V = ,2 s
Horizontal (X or Y) x p p xC E I f r = (,2 o
2
Rocking (about X or Y) C E I f Vp p s = ,2
Torsional (about Z)
N/A
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Table 10
PILE STIFFNESS AND DAMPING PARAMETERS fx,1 , fx,2 , f,1 AND f,2
s/ p
s
p
VV
Stiffness Parameters
fx,1 f,1
Damping Parameters
fx,2 f,2
0.4
0.7 (Concrete)
0.01 0.0036 0.202 0.0084 0.139
0.02 0.0100 0.285 0.0238 0.200
0.03 0.0185 0.349 0.0438 0.243
0.04 0.0284 0.403 0.0674 0.281
0.05 0.0397 0.450 0.0942 0.314
0.25 0.7 (Concrete) 0.01 0.0032 0.195 0.0076 0.135
0.02 0.0090 0.275 0.0215 0.192
0.03 0.0166 0.337 0.0395 0.235
0.04 0.0256 0.389 0.0608 0.272
0.05 0.0358 0.435 0.0850 0.304
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Table 11
PILE STIFFNESSES FOR A PILE GROUP
Mode of Vibration Pile Group Stiffness
Vertical (Z)
( ) = =1
z gi
n
izK K
Horizontal (X or Y) ( ) = ( ) =
=1x g y g
i
n
ixK K K
Rocking (about X)
( ) = ( + )=1
i2
K K K ygi
n
i iz
Rocking (about Y)
( ) = ( + )=1
i2 K K K xg
i
n
i iz
Torsional (about Z)
( ) = ( + )=1
i2
i2
K K x ygi
n
ix
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Table 12
DAMPING RATIOS FOR A PILE GROUP
Mode of Vibration Damping Ratios
Vertical (Z) z z
z g
D CK m
= 2 ( )
Horizontal (X or Y) x x
x g
D CK m
= 2 ( )
Rocking (about X) [ ]
D
C C y
K I
z
g
= +
2 ( ) i2
Rocking (about Y) [ ]
D
C C xK I
z
g
= + 2 ( )
i2
Torsional (about Z)
D C x
yK I
x
g
= ( + )2 ( )
i2
i2
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Table 13
CALCULATION OF UNCOUPLED NATURAL FREQUENCIES
Mode of Vibration Uncoupled Frequency (CPS)
Vertical (Z)
fnz = 1
2Km
z
Horizontal (X) fnx =
12
Km
x
Horizontal (Y)
fny = 1
2Km
y
Rocking (about X)
fn = 1
2
KI
Rocking (about Y)
fn = 1
2
KI
Torsional (about Z)
fn = 1
2
KI
* Use pile group stiffness (K)g for pile-supported foundation
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Table 14
CALCULATION OF DISPLACEMENT AMPLITUDES BASED ON UNCOUPLED MODES OF VIBRATION
Mode of Vibration Displacement Amplitudes
Vertical (Z)
z z zz
= d N FK
Horizontal (X)
x x xx
= d N FK
Horizontal (Y)
y yy
y = d N
FK
Rocking (about X)
= NMK
Rocking (about Y)
= NMK
Torsional (about Z)
= N MK
Dynamic amplification factors: Nx, Ny, Nz, N, N, N, are to be calculated as follows:
( )r D2 + )r-(11 = N
ii222
i
i , where, ri = 0ni
ff and Di = Damping ratio at ith
mode.
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Table 15
CALCULATION OF TOTAL DISPLACEMENT AMPLITUDE BASED ON UNCOUPLED MODES OF VIBRATION
Direction Displacement Amplitude
Vertical (Z)
Z = d + ( Y) + ( X)Z Horizontal
(X) X = d + ( Y) + ( Z)X
Horizontal (Y)
Y = d + ( X) + ( Z)Y
* The total displacement Ao is then obtained as follows:
o2 2 2 = + + A x y z
* x, y and z are the distances from the center of foundation base to the point where the amplitude is determined.
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Figure 1: Criteria for Vibrations of Rotating Machinery
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Figure 2 - General Limits of Displacement Amplitude for a Particular Frequency of Vibration
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Figure 3 - Geometrical Coefficients for Rectangular Foundation
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Figure 4: Vertical Pile Stiffness and Damping
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Figure 5: Vertical Pile Stiffness and Damping Parameters for Friction Piles
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COUPLED HORIZONTAL AND ROCKING MOTIONS
Couple mode frequencies, Sliding along X and rocking about Y
( )12 nx2 n2 2nx2 n2 nx2 n2f = 12 f + f + f + f - 4 f f
( )22 nx2 n2 2nx2 n2 nx2 n2f = 12 f + f - f + f - 4 f f
Where, fnx =
12
Km
x
; uncoupled sliding frequency along X
fn = 1
2 K
I
; uncoupled rocking frequency about Y
Kx: horizontal soil/pile stiffness K: rocking soil/pile stiffness
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m: total mass of a machine foundation system I: Io + mh2, the mass moment of inertia at the foundation base Io: the mass moment of inertia at the C.G. of a machine
foundation system h: the distance between combined C.G. and the base of a
machine foundation
= 0II
Note that the coupling effect becomes insignificant and can be neglected when
the two coupled frequencies are determined to be away from the machine operating speed by at least 40%.
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