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1. The University of Western Ontario
DEPARTMENT OF CIVIL AND
ENVIRONMENTAL ENGINEERING
ENVIRONMENTAL ENGINEERING
Instructor
Instructor
M. Hesham El Naggar, Ph.D., P. Eng
M. Hesham El Naggar, Ph.D., P. Eng., MASCE, FEIC
., MASCE, FEIC
es a agga , , g
es a agga , , g , SC , C
, SC , C
Professor and Research Director
Professor and Research Director
Geotechnical Research Centre
Geotechnical Research Centre
1

2. Chapter 4
RESPONSE OF FOUNDATIONS TO HARMONIC EXCITATION

3. RESPONSE OF RIGID FOUNDATIONS IN ONE DEGREE OF FREEDOM
The governing equation of the motion is obtained by considering the equilibrium of the
excitation force and the system reaction forces including elastic, damping and inertia
forces, i.e.:
(4.1)
t
i
Pe
kv
v
c
v
m 


 


Because the excitation force is complex, the response is also complex. The imaginary
part of the solution will be labeled by i and can be deleted if only the real part of the
excitation force is of interest. The particular integral that gives the steady‐state
solution is:
solution is:
v(t) = vceit (4.2)
where vc is a complex amplitude, vc = v1+iv2. Substitution of Eq. 4.5 in Eq. 4.4
yields
vc(‐2m + ic + k) = P 4.3)
3

4. From this, the complex amplitude of the response is
1
where
(4.4)
 P
i
c
i
m
k
vc 






 2
1
1 (4.5)
 
c
i
m
k
i






 2
1
is known as the admittance (or transfer function) of the system.
The reciprocal of (i) defines the impedance of the system.
(4.6)
 
c
i
m
k
i






 2
1
4

5. The particular solution for the governing equation is
(4 .7)
 
)
(
2
2
2
2
)
( 





 t
i
t
i
i
e
k
P
Pe
r
e
t
v
If the real vibration amplitude is denoted by
  2
2
2
2

 
 c
m
k
r
P
(4.8a)
the real response described by the real part of Eq 4 7 is
  2
2
2
2
c
m
k
P
v

 


the real response, described by the real part of Eq. 4.7, is
v(t) = v cos (t+) (4.8b)
5

6. th l lit d f th b itt
the real amplitude of the response can be rewritten as:
(4.9)



st
v
D
k
P
v 








2
2
2
2
4
1
1
in which vst = P/k = the static displacement and  = dynamic amplification (or
magnification) factor


D







 2
0
2
2
0
4
1
magnification) factor,
(4.10)
2
2
2
2
2
2
4
1
1



D









This factor is equal to the ratio of the amplitude of the response to the static
displacement
2
0
2
0
4
1


D







displacement.
6

7. The phase shift 
The phase shift
(4.11)
2
0
1
2
arctan















D
The particular solution describing the steady‐state response to the real excitation
force given by Eq. 4.1 is
0




force given by Eq. 4.1 is
v(t) = v cos (t + ) (4.12)
The complete solution, including the transient part is
( ) ( ) t i ( ’  ) (4 13)
v(t) = v cos (t + ) + v0 e‐t sin (’
0t + 0) (4.13)
in which v0 and 0 are integration constants depending on initial conditions.
7

8. Dimensionless Response to Harmonic Loads
(dynamic amplification factor and phase shift)
Fig. 1a) constant force excitation
8

9. Fig.1 b) quadratic excitation
9

10. Example for Machine Foundation Response Analysis
Fig. 2 Example Used with Piles and Without Piles in the analysis (1ft = 0.3048m)
10

11. Fig. 3 Vertical Response of (A) Pile Foundation, (B) Embedded Pile Foundation, (C)
Shallow Foundation, and (D) Embedded Shallow Foundation
11

12. Fig. 4 Vertical Response of a (1) Embedded Pile Foundation with Parabolic Soil
Profile, and (2) Embedded Pile Foundation with Homogeneous Soil Profile and (3)
Shallow Foundation Without Embedment
12

13. Fig 5 Torsional Response of (1) Pile Foundation and (2) Shallow Foundation
Fig. 5 Torsional Response of (1) Pile Foundation and (2) Shallow Foundation
13

14. COUPLED RESPONSE OF RIGID FOUNDATIONS IN 2 DOF
Fig. 6 Rigid foundations’ motion in space is described by three translations and three
rotations, i.e., they have six degrees of freedom (generally coupled).
14

15. Examples of Coupled Response
Fig. 7 Horizontal Component of Coupled Footing Response to Horizontal Load. ( (Bx
/ R3 5 81 B I / R5 3 46 ( ) d l l i )
= m /pR3
x = 5.81, B = I / pR5
 = 3.46; (+) = modal analysis)
15

16. Fig. 8 Rocking Component of Coupled Footing Response to Horizontal Load.
16

17. Fig. 9 Horizontal and Rocking Components of Coupled Response of Embedded
Foundation for Different Shear Wave Velocities of Soil (e /R = 0.4, granular soil)
17

18. Fig. 10 Effect of Soil Stiffness on Horizontal Component of Coupled
Response of Pile Foundation
18

19. DESIGN PROCEDURE
It is a trial and error procedure which proceeds as follows:
It is a trial‐and‐error procedure, which proceeds as follows:
1) Estimate the dynamic loads.
2) Establish the soil profile and determine the soil properties
2) Establish the soil profile and determine the soil properties
required for the analysis (Shear modulus, mass density,
Poisson’s ratio and material damping ratio).
3) Select the type and trial dimensions of the foundation and
with clients input, establish the performance criteria.
4) Compute the dynamic response of the trial foundation (step
3) supported by the given soil profile (step 2) due to the
estimated load (step 1) and compare the response with the
estimated load (step 1) and compare the response with the
performance criteria. If the response is not satisfactory,
modify the dimensions of the foundation (step 3), repeat
the analysis until satisfactory design is achieved.
19

20. DESIGN INFORMATION
To carry out the design of a foundation system to support a
vibration producing equipment, certain loading and site
parameters must be known or evaluated The information
parameters must be known or evaluated. The information
required for the design can be generally categorized into three
main groups:
1 machine properties
1. machine properties,
2. soil and foundation parameters, and
3. environmental requirements.
20

21. Machine Properties
The machine properties required include:
The machine properties required include:
1) Outline drawing of machine assembly
2) Weight of machine and its rotor components (or head for
hammers)
3) L i f f i b h i ll d
3) Location of center of gravity both vertically and
horizontally
4) Speed ranges of machine and components or frequency
4) Speed ranges of machine and components or frequency
of unbalanced primary and secondary forces
5) Magnitude and direction of unbalanced forces, vertically
5) Magnitude and direction of unbalanced forces, vertically
and horizontally, and their points of application
6) Limits (tolerance) of deflection (total or differential) &
vibration amplitudes to satisfy the machine functions.
21

22. Arrangement of Equipment and Masses
The total mass of motor (23,360 Kg) and the compressor (frame, crankshafts,
dampeners and other components, 300,00 Kg) have been provided by the client.
Fig. 11 Elevation and Side View of equipment layout
22

23. Fig 12 Equipment Layout Plan
Fig. 12 Equipment Layout Plan
23

24. Dynamic Unbalanced Loads
Dynamic unbalanced loads are provided by the client and are applied as harmonic
loads. The response at the bearing points and some other check points are obtained
and reported.
Different components of dynamic unbalanced forces are shown below. These
components are Driving Torque (MT), Horizontal Force (FH), Vertical force (FV), Vertical
components are Driving Torque (MT), Horizontal Force (FH), Vertical force (FV), Vertical
Couple (MV), Horizontal Couple (MH) and Motor eccentric forces (ECCx and ECCz). The
amplitudes of these components are summarized in Table 1.
24
Fig. 13 Compressor and Motor loads

25. Compressor Unbalanced (Dynamic) Loads
Compressor Speed 360 rpm
1st Order loads act at 360 rpm
1 Order loads act at 360 rpm
2nd Order loads act at 720 rpm
Table 1 Booster Compressor Unbalanced Loads
25

26. Motor (Centrifugal) Loads
Table 2 motor properties
The ECCx and ECCz are motor centrifugal forces that are generated due to the
eccentricity of its rotor mass (the two components are 90 out of phase)
eccentricity of its rotor mass (the two components are 90 out of phase).
The rotor eccentricity is assumed to be according to balance quality Q = 6.3 mm/s
for the primary frequency (360 rpm or 6 Hz). The mass of the rotor is 13,199 kg,
for the primary frequency (360 rpm or 6 Hz). The mass of the rotor is 13,199 kg,
and the surface factor, Sf = 2, thus:
ECCx or ECCz =P = me Q ω Sf
= 13199 x 6.3x10‐3 (2π x 6) x 2 = 6269 N
26

27. Soil and Foundation Parameters
Knowledge of the soil formation (soil profile) and its properties
is required for the dynamic analysis. The information is to be
b d f f ld b ( d ) d l b
obtained from field borings (or soundings) and laboratory
tests. The following parameters are required for the dynamic
analysis:
y
1) shear modulus of soil, G, at several levels of strain.
1) shear modulus of soil, G, at several levels of strain.
2) material damping ratio, D, at several levels of strain.
3) Poisson’s ratio, .
4) d it f il d it
4) density of soil, , or mass density, .
27

28. Fig. 14 Site General Layout and Borehole Locations
28

29. Summary of Subsurface Conditions
Depth (m) Description of PEBH-11*
0.0 – 4.6 Dense to very dense poorly graded sand with silt
4.6-6.0 Very dense poorly graded sand with silt
6.0-16.5 Very dense clayey sand
16.5-25.0 Very dense silty sand with clay and occasional gravel
Depth (m) Description of PEBH-12*
0.0 – 2.4 Medium dense silty sand
2.4 – 4.6 Dense poorly graded sand with silt
4.6 – 11.0 Very dense poorly graded sand
y p y g
11.0 – 21.3 Very dense clayey sand
21.3 – 25.0 Hard sandy fat clay
29

30. Table 3 Seismic Cross‐Hole Testing PEBH11
Depth Vp Vs (vp/vs)2 µ γt G E
D (m)
tcomp/x
(m/s)
tshear/x
(m/s) (kN/m³) (kN/m²) (kN/m²)
0 75 500 00 230 77 4 69 0 36 18 9 77E+04 2 67E+05
0.75 500.00 230.77 4.69 0.36 18 9.77E+04 2.67E+05
1.50 800.00 352.94 5.14 0.38 18 2.29E+05 6.30E+05
2.25 400.00 176.47 5.14 0.38 18 5.71E+04 1.58E+05
3.00 410.96 187.50 4.80 0.37 19 6.81E+04 1.86E+05
3.75 400.00 200.00 4.00 0.33 20 8.15E+04 2.17E+05
4.50 428.57 200.00 4.59 0.36 20 8.15E+04 2.22E+05
5.25 600.00 250.00 5.76 0.39 20 1.27E+05 3.55E+05
6.00 750.00 326.09 5.29 0.38 20 2.17E+05 6.00E+05
6.75 500.00 230.77 4.69 0.36 20 1.09E+05 2.96E+05
7.50 1000.00 428.57 5.44 0.39 20 3.74E+05 1.04E+06
8.25 750.00 375.00 4.00 0.33 20 2.87E+05 7.65E+05
9.00 750.00 333.33 5.06 0.38 21 2.38E+05 6.55E+05
9.75 789.47 333.33 5.61 0.39 21 2.38E+05 6.62E+05
10 0
10.50 750.00 333.33 5.06 0.38 21 2.38E+05 6.55E+05
11.20 937.50 416.67 5.06 0.38 21 3.72E+05 1.02E+06
12.00 909.09 447.76 4.12 0.34 21 4.29E+05 1.15E+06
12.70 937.50 480.00 3.81 0.32 21 4.93E+05 1.30E+06
13.50 857.14 375.00 5.22 0.38 21 3.01E+05 8.32E+05
14.20 857.14 428.57 4.00 0.33 21 3.93E+05 1.05E+06
15 00 882 35 428 57 4 24 0 35 21 3 93E+05 1 06E+06
15.00 882.35 428.57 4.24 0.35 21 3.93E+05 1.06E+06
15.70 1000.00 468.75 4.55 0.36 21 4.70E+05 1.28E+06
16.50 1000.00 517.24 3.74 0.32 21 5.73E+05 1.51E+06
17.20 1034.40 517.24 4.00 0.33 21 5.73E+05 1.53E+06
18.00 1111.10 555.56 4.00 0.33 21 6.61E+05 1.76E+06
18.70 1071.40 500.00 4.59 0.36 21 5.35E+05 1.46E+06
19.50 967.74 476.19 4.13 0.34 21 4.85E+05 1.30E+06
20.20 1111.10 521.74 4.54 0.36 21 5.83E+05 1.58E+06
21.00 937.50 428.57 4.79 0.37 21 3.93E+05 1.08E+06
21.70 857.14 422.54 4.12 0.34 21 3.82E+05 1.02E+06
22.50 937.50 491.80 3.63 0.31 21 5.18E+05 1.36E+06
23.20 1000.00 461.54 4.69 0.36 21 4.56E+05 1.24E+06
24.00 1071.40 566.04 3.58 0.31 21 6.86E+05 1.79E+06
24 70 1363 60 697 67 3 82 0 32 21 1 04E+06 2 76E+06
24.70 1363.60 697.67 3.82 0.32 21 1.04E+06 2.76E+06
25.50 1000.00 491.80 4.13 0.34 21 5.18E+05 1.39E+06
30

31. Depth Vp Vs (vp/vs)² µ γt G E
D (m)
tcomp/x
(m/s)
tshear/x
(m/s) (kN/m³) (kN/m²) (kN/m²)
0 75 375 00 176 47 4 52 0 36 18 5 71E+04 1 55E+05
Table 4 Seismic Cross‐Hole Testing PEBH12
0.75 375.00 176.47 4.52 0.36 18 5.71E+04 1.55E+05
1.50 434.78 171.43 6.43 0.41 18 5.39E+04 1.52E+05
2.25 500.00 230.77 4.69 0.36 18 9.77E+04 2.67E+05
3.00 413.79 181.82 5.18 0.38 19 6.40E+04 1.77E+05
3.75 500.00 240.00 4.34 0.35 20 1.17E+05 3.17E+05
4.50 600.00 200.00 9.00 0.44 20 8.15E+04 2.34E+05
5.25 750.00 250.00 9.00 0.44 20 1.27E+05 3.66E+05
6.00 576.92 260.87 4.89 0.37 20 1.39E+05 3.81E+05
6.75 937.50 240.00 15.20 0.46 20 1.17E+05 3.44E+05
7.50 857.14 375.00 5.22 0.38 20 2.87E+05 7.92E+05
8.25 625.00 230.77 7.34 0.42 20 1.09E+05 3.09E+05
9.00 600.00 272.73 4.84 0.37 21 1.59E+05 4.36E+05
9.75 600.00 250.00 5.76 0.39 21 1.34E+05 3.73E+05
10.50 1000.00 500.00 4.00 0.33 21 5.35E+05 1.43E+06
11.25 1000.00 428.57 5.44 0.39 21 3.93E+05 1.09E+06
12.00 1071.43 447.76 5.73 0.39 21 4.29E+05 1.20E+06
12.75 600.00 272.73 4.84 0.37 21 1.59E+05 4.36E+05
13.50 833.33 357.14 5.44 0.39 21 2.73E+05 7.58E+05
14.25 750.00 352.94 4.52 0.36 21 2.67E+05 7.24E+05
15 00 882 35 428 57 4 24 0 35 21 3 93E 05 1 06E 06
15.00 882.35 428.57 4.24 0.35 21 3.93E+05 1.06E+06
15.75 909.09 234.38 15.04 0.46 21 1.18E+05 3.44E+05
16.50 1000.00 428.57 5.44 0.39 21 3.93E+05 1.09E+06
17.25 1111.11 480.00 5.36 0.39 21 4.93E+05 1.37E+06
18.00 1071.43 555.56 3.72 0.32 21 6.61E+05 1.74E+06
18.75 1034.48 500.00 4.28 0.35 21 5.35E+05 1.44E+06
19 50 967 74 476 19 4 13 0 34 21 4 85E+05 1 30E+06
19.50 967.74 476.19 4.13 0.34 21 4.85E+05 1.30E+06
20.25 1111.11 280.37 15.71 0.47 21 1.68E+05 4.93E+05
21.00 1200.00 500.00 5.76 0.39 21 5.35E+05 1.49E+06
21.75 1034.48 422.54 5.99 0.40 21 3.82E+05 1.07E+06
22.50 1000.00 491.80 4.13 0.34 21 5.18E+05 1.39E+06
23.25 1000.00 461.54 4.69 0.36 21 4.56E+05 1.24E+06
24 00 1111 11 566 04 3 85 0 32 21 6 86E+05 1 82E+06
24.00 1111.11 566.04 3.85 0.32 21 6.86E+05 1.82E+06
24.75 750.00 333.33 5.06 0.38 21 2.38E+05 6.55E+05
25.00 1071.43 491.80 4.75 0.37 21 5.18E+05 1.42E+06
31

32. Table 5 Dynamic Soil Properties Adopted in Modelling
Depth LB AVE UB unit wt Poisson Damping
(m) Vs (m/s) Vs (m/s) Vs (m/s) N/m3
(m) Vs (m/s) Vs (m/s) Vs (m/s) N/m
0.75 170 192 230 18000 0.36 0.02
1.5 180 224 360 18000 0.38 0.02
2.25 170 196 230 18000 0.38 0.02
3 180 194 220 19000 0.37 0.02
3.75 190 281 370 20000 0.33 0.02
4.5 170 196 220 20000 0.36 0.02
5.25 220 274 350 20000 0.39 0.02
6 230 318 360 20000 0.38 0.02
6.75 210 326 440 20000 0.36 0.02
7.5 380 440 510 20000 0.39 0.02
8.25 220 392 490 20000 0.33 0.02
9 240 430 600 21000 0.38 0.02
9 230 06 20 21000 0 39 0 02
9.75 230 406 520 21000 0.39 0.02
10.5 340 428 500 21000 0.38 0.02
11.25 410 437 470 21000 0.38 0.02
12 440 454 480 21000 0.34 0.02
12.75 270 410 500 21000 0.32 0.02
13.5 280 350 380 21000 0.38 0.02
14 25 360 439 500 21000 0 33 0 02
14.25 360 439 500 21000 0.33 0.02
15 300 382 430 21000 0.35 0.02
15.75 210 428 540 21000 0.36 0.02
16.5 415 495 550 21000 0.32 0.02
17.25 440 520 600 21000 0.33 0.02
18 510 538 560 21000 0.33 0.02
18 75 460 482 530 21000 0 36 0 02
18.75 460 482 530 21000 0.36 0.02
19.5 450 482 500 21000 0.34 0.02
20.25 240 464 600 21000 0.36 0.02
21 350 447 500 21000 0.37 0.02
21.75 420 470 520 21000 0.34 0.02
22.5 450 462 470 21000 0.31 0.02
23.25 430 486 570 21000 0.36 0.02
32
24 470 505 550 21000 0.31 0.02
24.75 300 482 550 21000 0.32 0.02
25.5 400 474 500 21000 0.34 0.02

33. Foundation Requirements
1) i i d h f f d i
1) minimum depth of foundation.
2) base dimensions for the machine and other components
attached to it
attached to it.
3) type of foundation system to be used (recommended by
the geotechnical consultant).
the geotechnical consultant).
4) configuration and layout of the foundation (width, length
and depth). For piled foundations, the number of piles,
p ) p p
pile geometry (diameter or width and cross‐sectional
area), pile length and spacing between piles are required
on top of the configuration of the foundation block
on top of the configuration of the foundation block.
5) material properties of the foundation (unit weight of the
concrete or steel, the Poisson’s ratio and elastic modulus).
concrete or steel, the Poisson s ratio and elastic modulus).
33

34. General Foundation Concept
p
• The foundation‐pile system advocated is a pile cap 21.00 x
p y p p
21.00 x 2.0m deep.
• The cap is supported by 36 (6x6 arrangement) piles 0.8m
di t 20 l
diameter 20m long.
2 m
2H:1V
2H:1V
Fill
Fill
Pile cap 21.0 m
2 m
2 m
34
Fig. 15 Backfill configuration

35. General Foundation Layout
35
Fig. 16 Isometric view of compressor pedestal

36. Fig 17 Foundation Layout Plan
Fig. 17 Foundation Layout Plan
36

37. Fig. 18 Critical Points for Response
Fig. 18 Critical Points for Response
37

38. h hi d ib i h l h
Environmental Requirements
The machinery produces vibrations that may travel to the
neighboring vicinity. If vibration amplitudes are significant,
measures are taken to minimize the environmental impact of
p
the machine (especially, for shock producing equipment).
In some situations, the machine is installed in the vicinity of
, y
vibration sources (quarry blasting, vehicular traffic) or in a
seismic active area. In this case, the information requested
include the character of vibration and attenuation at the
include the character of vibration and attenuation at the
installation site.
The effects of seismic forces are addressed considering
The effects of seismic forces are addressed, considering
ground response and SSI analyses.
38

39. Vibration Criteria
Fig. 19 Vibration criteria: British Standard CP 2012‐1:1974: Code of Practice for
Foundations for Machinery
1 d i f i i hi
39
Part 1: Foundations for reciprocating machines

40. Vibration Criteria
Table 6 Vibration limit criteria
Vibration limit criteria
Booster/Primary
Compressor
Note
Machinery, cylinder gas connections
(compressor as a rigid body) m pk 50. 10^-6
(compressor as a rigid body) m pk
Foundation, m pk (from CP2012 since not
vendor-specified), 1st order 75. 10^-6
Location furthest
from CoG
Equivalent Foundation Velocity, mm/s rms 1.75
40
Running speed (1st order), Hz 6.0

41. Fig. 20 General Configuration of Foundation from DYNA6 3DF Module
41

42. Fig. 21 Plan View from DYNA6 3DF Module
42

43. Fig. 22 Elevation View from DYNA6 3DF Module
43

44. Fig. 23 Side View from DYNA6 3DF Module
Fig. 23 Side View from DYNA6 3DF Module
44

45. Fi 24 Pil L f P i C f DYNA6
Fig. 24 Piles Layout for Primary Compressor from DYNA6
45

46. Modeling Pile Stiffnesses
The primary and the secondary forces of the compressor act at frequencies of 360 and
720 rpm. Thus, two sets of stiffness and damping are calculated using DYNA6, one for
each loading frequency. These values are assigned as equivalent LINK stiffness and
damping at the pile heads.
damping at the pile heads.
46

47. Table 7 Stiffness and Damping Constants for Hyper Compressor Foundation
Pile Group Stiffness LB Vs
Freq Kx Cx Ky Cy Kz Cz
RPM N/m N/m/s N/m N/m/s N/m N/m/s
50 3 90E+09 1 47E+08 3 90E+09 1 47E+08 1 53E+10 5 28E+08
50 3.90E+09 1.47E+08 3.90E+09 1.47E+08 1.53E+10 5.28E+08
360 4.00E+09 1.49E+08 4.00E+09 1.49E+08 1.44E+10 6.99E+08
720 6.22E+09 1.67E+08 6.22E+09 1.67E+08 6.77E+10 5.97E+08
Pile Group Stiffness BE Vs
Freq Kx Cx Ky Cy Kz Cz
RPM N/m N/m/s N/m N/m/s N/m N/m/s
50 4 97E+09 1 55E+08 4 97E+09 1 55E+08 1 85E+10 5 08E+08
50 4.97E+09 1.55E+08 4.97E+09 1.55E+08 1.85E+10 5.08E+08
360 5.27E+09 1.51E+08 5.27E+09 1.51E+08 1.62E+10 5.87E+08
720 6.85E+09 1.73E+08 6.85E+09 1.73E+08 4.93E+10 7.41E+08
Pile Group Stiffness UB Vs
Freq Kx Cx Ky Cy Kz Cz
RPM N/m N/m/s N/m N/m/s N/m N/m/s
50 6 40E+09 1 65E+08 6 40E+09 1 65E+08 2 21E+10 5 00E+08
50 6.40E+09 1.65E+08 6.40E+09 1.65E+08 2.21E+10 5.00E+08
360 6.61E+09 1.54E+08 6.61E+09 1.54E+08 1.97E+10 5.26E+08
720 6.87E+09 1.76E+08 6.87E+09 1.76E+08 3.02E+10 7.14E+08
Pile Group Rotational Stiffness LB Vs
Freq Krx Crx Kry Cry Krz Crz
RPM N/m N/m/s N/m N/m/s N/m N/m/s
50 1 64E+12 4 54E+09 1 65E+12 4 57E+09 8 16E+11 4 88E+09
50 1.64E+12 4.54E+09 1.65E+12 4.57E+09 8.16E+11 4.88E+09
360 1.27E+12 2.03E+10 1.27E+12 2.04E+10 6.37E+11 1.10E+10
720 2.56E+12 2.50E+10 2.56E+12 2.50E+10 6.90E+11 1.35E+10
Pile Group Rotational Stiffness BE Vs
Freq Krx Crx Kry Cry Krz Crz
RPM N/m N/m/s N/m N/m/s N/m N/m/s
50 2 01E 12 4 31E 09 2 01E 12 4 35E 09 9 74E 11 5 61E 09
50 2.01E+12 4.31E+09 2.01E+12 4.35E+09 9.74E+11 5.61E+09
360 1.57E+12 1.64E+10 1.57E+12 1.64E+10 7.95E+11 1.08E+10
720 2.00E+12 2.50E+10 2.01E+12 2.50E+10 8.31E+11 1.36E+10
Pile Group Rotational Stiffness UB Vs
Freq Krx Crx Kry Cry Krz Crz
RPM N/m N/m/s N/m N/m/s N/m N/m/s
50 6 67E 10 1 55E 08 6 67E 10 1 56E 08 3 56E 10 1 51E 08
47
50 6.67E+10 1.55E+08 6.67E+10 1.56E+08 3.56E+10 1.51E+08
360 5.39E+10 3.89E+08 5.39E+10 3.92E+08 2.94E+10 2.76E+08
720 5.28E+10 6.14E+08 5.28E+10 6.14E+08 2.72E+10 3.78E+08

48. Natural Frequencies Obtained from DYNA6
DYNA6 was used to calculate the foundation response over a frequency range of
50 rpm to 750 rpm (0.8Hz to 12.5Hz). The natural frequency was identified as the
location of peak on the plot of response at C.G. with frequency. The observed
natural frequencies for different vibration modes are presented Below. The results
h h h l f i i h h i l di i f h P i
show that the natural frequencies in the horizontal direction for the Primary
compressor foundation are close to the operating speed. However, the calculated
responses were found to be satisfactory due to the existence of sufficient damping.
Natural Horizontal Horizontal Vertical Z Rocking Rocking Y Torsion
Table 8 Natural frequencies obtained from DYNA6 analyses
Freq X Y X
Soil
Condition
Hz Hz Hz Hz Hz Hz
Condition
LB 350 350 590 650 650 870
BE 400 400 620 700 700 980
48
UB 430 430 650 740 740 1060

49. Table 9 Coordinates of Critical Points for Primary Compressor
Joint Area GlobalX GlobalY GlobalZ
Text Text m m m
Text Text m m m
1822
Cap Floor
‐2.25 18.75 0
3940 18.75 ‐2.25 0
4001 18.75 18.75 0
4001 18.75 18.75 0
14951 ‐2.25 ‐2.25 0
14793 5.565 5.395 5.1475
14823 7.565 5.395 5.1475
Comp
Cylinders
Pads
14823 7.565 5.395 5.1475
14853 9.65 5.395 5.1475
14877 6.035 11.105 5.1475
14907 8.035 11.105 5.1475
14937 10.1 11.105 5.1475
14489 Comp
Floor
5.25 6.5 5.3425
14495 5.25 10 5.3425
14577
Motor
Floor
11.15 5.85 5.3425
14587 11.15 10.65 5.3425
14605 14.605 5.85 5.3425
49
14612 14.605 10.65 5.3425

50. Response Results from DYNA6
The calculated total responses for the Primary compressor and Hyper
The calculated total responses for the Primary compressor and Hyper
foundations from the DYNA6 analyses are obtained by directly adding (taking
no account of phase) the responses to the first and second order load
components. The response was calculated at critical points representing
p p p p g
support points for the compressor, the motor and the corners of the pile cap.
In DYNA6 analyses, however, the response was calculated at only a selected no.
of these points. However, the response was calculated at all noted critical
points using the FE analysis. The maximum vibration amplitude for the Primary
compressor was calculated at the top of the compressor pedestal (Joint 14489
bl ) h h l h h f d b
in Table 9) as 48 microns, which is less than the specified vibration criterion.
50

51. SAP2000 Analyses
The foundation is modeled in SAP2000 using SOLID elements for the pile cap and
the tables. Piles are modeled using zero‐length Linear Link element attached to
the corresponding joints at the bottom of the pile cap LINEAR LINK has 3 stiffness
the corresponding joints at the bottom of the pile cap. LINEAR LINK has 3 stiffness
(Kx, Ky, Kz) and 3 damping (Cx, Cy, Cz) parameters.
These parameters are determined utilizing the software DYNA5 The LINK local
These parameters are determined utilizing the software DYNA5. The LINK local
coordinate system is related to the GLOBAL coordinate system as shown below:
R t ti Sh ft
3
1 Rotating Shaft
Y
Z
2
LOCAL GLOBAL
Fig 25 Global and local coordinate systems
51
Fig. 25 Global and local coordinate systems

52. Dynamic Unbalanced Loads
Dynamic unbalanced loads are provided by the client and are applied as harmonic
loads. The response at the bearing points and some other check points are obtained
and reported.
Different components of dynamic unbalanced forces are shown in Figure 6. These
components are Driving Torque (MT), Horizontal Force (FH), Vertical force (FV), Vertical
Couple (MV), Horizontal Couple (MH) and Motor eccentric forces (ECCx and ECCz). The
p ( ), p ( ) ( )
amplitudes of these components are summarized in Table 8.7.
52
Fig. 26 Load directions for compressor and motor

53. SAP2000 MODEL
X
Y Z
53
Fig. 27 Finite element model

54. Table 10 Unbalanced Harmonic Force Amplitudes
Load
Components
First Order
f1=360 rpm
Second Order
f2=720 rpm
MT (KN-m) 1217.547*Cos(ω1t) 0.00
FH (KN) 119 08*C ( t+120 9°) 18 997*C ( t 36 1°)
FH (KN) 119.08*Cos(ω1t+120.9°) 18.997*Cos(ω2t-36.1°)
FV (KN) 0.038*Cos(ω1t -115.4°) 0.00
MV (KN-m) 138.326*Cos(ω1t -30.0°) 0.00
MH (KN-m) 635.744*Cos(ω1t -87.6°) 49.238*Cos(ω2t-54.5°)
ECCx (KN) 6.269*Sin(ω1t) 0.00
ECCz (KN) 6.269*Cos(ω1t) 0.00
( ) ( 1 )
54

55. Other Loading Conditions
St ti L d
Static Loads
Static loads include the weight of concrete, the weight of major parts of the
Primary compressor and the motor.
The short circuit load is applied as vertical forces at the Motor bearings in Z
direction where +ve Z is downward. Two cases were considered:
Case 1: loads of ‐184kN, 362kN, 16kN, and 104kN are applied at pads A, B, C and
, , , pp p , ,
D as marked on the vendor’s drawing denoted “Preliminary Compressor Motor
Loads”.
Case 2: Loads of 362kN, ‐184kN, 16kN, and 104kN are applied at pads A, B, C and
D as marked on the vendor’s drawing denoted “Preliminary Compressor Motor
Loads”.
S i i L d
Seismic Loads
The seismic load is considred in the analysis using the design spectra specified by
the client. The seismic load is applied in the horizontal plane‐ both X and Y axes.
55

56. Load Cases and Load Combinations
The following load cases are used in the SAP2000 model:
• DEAD (static) = Includes weights of all of elements and concentrated masses
• 360 rpm (dynamic) = The unbalanced dynamic loads from the vibration of
the machines at the frequency of 6Hz (360 rpm).
• 720 rpm (dynamic) = The unbalanced dynamic loads from the vibration of the
machines at the frequency of 12Hz (720 rpm).
• SHORT CIRCUIT (static) = Includes short circuit couple.
E h k X Y (d i ) S i i l d i X Y di i
• Earthquake‐X or Y (dynamic) = Seismic loads in X or Y direction.
56

57. Modeling Piles
Piles are modeled using zero‐length Linear Link elements attached to the
corresponding joints at the bottom of the pile cap. LINEAR LINK has 3 stiffness (Kx,
Ky, Kz) and 3 damping (Cx, Cy, Cz) parameters. These parameters are determined
utilizing the software DYNA6.
If dynamic forces of the compressor act at 2 different frequencies, two sets of
stiffness and damping are calculated using DYNA6, one for each loading frequency
These values are assigned as equivalent LINK stiffness and damping at the pile
These values are assigned as equivalent LINK stiffness and damping at the pile
heads.
57

58. SAP2000 Analysis Results – Natural Frequencies
The 3D model has more than 30000 modes.
However only the first 6 modes have important contributions to the
However, only the first 6 modes have important contributions to the
response and are considered in the analysis. The modal information for the
model is given here including: modal participation factors, period and
frequency. The lowest two natural frequencies are 5.7 Hz (this value is very
frequency. The lowest two natural frequencies are 5.7 Hz (this value is very
close to the values evaluated from the DYNA6 analyses), which is close to the
primary loading frequency of 6 Hz. However, the calculated responses were
found to be satisfactory due to the existence of sufficient damping. In
addition, a damping safety factor was considered in the analysis, ensuring
the damping is not overestimated.
58

59. Table 11 SAP2000 Analysis Results – Natural Frequencies
LB Period Freq UX UY UZ RX RY RZ
Mode No Sec Hz Unitless Unitless Unitless Unitless Unitless Unitless
1 0.176134 5.68 0.0000 1.0000 0.0000 0.0020 0.0000 0.3800
2 0.175973 5.68 1.0000 0.0000 0.0000 0.0000 0.0022 0.3400
3 0.103007 9.71 0.0000 0.0004 0.0000 0.0002 0.0000 0.2700
4 0.094334 10.60 0.0000 0.0000 1.0000 0.6800 0.7100 0.0000
5 0.059096 16.92 0.0026 0.0000 0.0009 0.0014 0.2800 0.0007
6 0.057999 17.24 0.0000 0.0026 0.0000 0.3100 0.0034 0.0014
BE Period Freq UX UY UZ RX RY RZ
Mode No Sec Hz Unitless Unitless Unitless Unitless Unitless Unitless
1 0.1540 6.49 0.0000 1.0000 0.0000 0.0024 0.0000 0.3900
2 0.1538 6.50 1.0000 0.0000 0.0000 0.0000 0.0025 0.3400
3 0.0925 10.82 0.0000 0.0004 0.0000 0.0002 0.0000 0.2700
4 0.0893 11.20 0.0000 0.0000 1.0000 0.6800 0.7100 0.0000
5 0 0540 18 52 0 0033 0 0000 0 0007 0 0017 0 2800 0 0009
5 0.0540 18.52 0.0033 0.0000 0.0007 0.0017 0.2800 0.0009
6 0.0530 18.87 0.0000 0.0033 0.0000 0.3000 0.0037 0.0018
UB Period Freq UX UY UZ RX RY RZ
Mode No Sec Hz Unitless Unitless Unitless Unitless Unitless Unitless
1 0 1380 7 25 0 0001 1 0000 0 0000 0 0026 0 0000 0 3900
1 0.1380 7.25 0.0001 1.0000 0.0000 0.0026 0.0000 0.3900
2 0.1378 7.26 1.0000 0.0001 0.0000 0.0000 0.0028 0.3400
3 0.0817 12.25 0.0000 0.0000 1.0000 0.6800 0.7100 0.0000
4 0.0804 12.44 0.0000 0.0004 0.0000 0.0001 0.0000 0.2700
5 0.0495 20.19 0.0038 0.0001 0.0008 0.0018 0.2700 0.0010
59
6 0.0487 20.55 0.0001 0.0038 0.0000 0.3000 0.0039 0.0022

60. Joint Responses to Machinery Vibrations
SAP2000 Analysis Results – Dynamic Response
The response of some selected control points are given here. These responses include
the displacement at some control points under the machine vibrations. Control points
include the top of the tabletop at locations of support pads for motor and
compressor and at the C G of the Motor and Compressor The results show the
compressor and at the C.G of the Motor and Compressor. The results show the
maximum amplitude in each direction. They do not necessarily occur at the same
time.
The results show that the maximum vibration amplitude is 49 μm in the horizontal
direction. The allowable vibration amplitude is 50 μm. This represents satisfactory
performance.
60

61. Table 12 SAP2000 Analysis Results – Vibration Response
Response to 1st Order
Loads
Response to 2nd Order
Loads
Total Response
Loads Loads
Joint U1 U2 U3 U1 U2 U3 U1 U2 U3
No μm μm μm μm μm μm μm μm μm
1822 0.50 32.0 8.0 0.05 1.5 0.7 0.55 33.5 8.7
3940 0.50 33.0 8.1 0.07 1.5 0.7 0.57 34.5 8.8
4001 0.49 33.0 7.8 0.08 1.5 0.7 0.57 34.5 8.5
14951 0.40 32.0 8.0 0.06 1.5 0.7 0.46 33.5 8.7
14793 1 45 46 0 12 0 0 21 1 9 1 2 1 66 47 9 13 2
14793 1.45 46.0 12.0 0.21 1.9 1.2 1.66 47.9 13.2
14823 1.30 45.0 11.0 0.18 1.9 1.1 1.48 46.9 12.1
14853 1.55 45.0 11.0 0.22 1.9 1.1 1.77 46.9 12.1
14877 0.94 46.0 12.0 0.15 1.9 1.2 1.09 47.9 13.2
0 9 6 0 0 0 5 9 09 9 3
14907 0.95 45.0 11.0 0.15 1.9 1.1 1.1 46.9 12.1
14937 1.19 45.0 11.0 0.19 1.9 1.0 1.38 46.9 12
14489 0.98 47.0 6.6 0.13 2.0 0.7 1.11 49 7.3
14495 0.56 47.0 7.0 0.09 2.0 0.7 0.65 49 7.7
14577 1.60 45.0 7.2 0.22 1.9 0.6 1.82 46.9 7.8
14587 1.15 45.0 6.8 0.19 1.9 0.6 1.34 46.9 7.4
14605 1 64 44 0 7 1 0 22 1 9 0 6 1 86 45 9 7 7
61
14605 1.64 44.0 7.1 0.22 1.9 0.6 1.86 45.9 7.7
14612 1.19 44.0 6.4 0.19 1.9 0.6 1.38 45.9 7

62. DESIGN CRITERIA
h b i l d d i h d i i
Factors that may be included in the design requirements.
1) Static requirements for bearing capacity and settlement.
2) Dynamic behaviour
) y
• limiting vibration amplitude
• limiting velocity
• limiting acceleration
• maximum dynamic magnification factor
• maximum transmissibility factor
3) Possible modes of vibration vertical; horizontal; torsional;
• maximum transmissibility factor
• resonance conditions
3) Possible modes of vibration vertical; horizontal; torsional;
rocking; pitching and possibility of coupled modes.
4) Possible fatigue failures in the machine, in the structure,
or in connections.
62

63. 5) Environmental considerations
• physical and physiological effects on people
physical and physiological effects on people
• effects on nearby sensitive equipment
• possible resonance of structural components
id ti f f d ti i l ti
• consideration of foundation isolation
6) Economy
• initial cost
• initial cost
• maintenance costs
• down time costs
• replacement costs
63

64. DESIGN CHECKLIST FOR MACHINE FOUNDATIONDS
After the response of the proposed foundation is predicted from the dynamic
analysis, it is checked against certain design requirements including:
1 the usual check of bearing capacity and settlement and structural strength of
1. the usual check of bearing capacity and settlement, and structural strength of
the foundation under static loads.
2. the maximum bearing pressure (static + dynamic) should be less than 75% of
th ll bl f th il F il d f d ti th i l d f
the allowable pressure of the soil. For piled foundations, the maximum load for
any pile (static + dynamic) should be less than 75% of the design capacity of the
pile.
3. comparison to tolerance for dynamic behaviour which includes a)
maximum vibration amplitudes; b) maximum velocity ( x displacement
amplitude) and acceleration (2 x displacement amplitude) ; c) maximum
magnification factor, should be less than 1.5 at resonance; d) possible
resonance conditions, the operating frequency of the machine should not
be within  20% of the resonance frequency (damped or undamped).
64

65. 4. Consideration of possible fatigue failure in the machine components
and connections.
5. Consideration of environmental requirements and physiological effects
on workers.
Sometimes, a factor of safety (FS) could be used to account for the
y ( )
relative importance of the machine to overall plant operation. The
predicted amplitude is multiplied by a service factor (safety factor) FS
(1.5‐2) to obtain an effective vibration amplitude.
65

66. Table 13 Summary of Maximum Pile Loads for Different Load Types
Pile Load Case P V2 V3 T M2 M3
No N N N N‐m N‐m N‐m
1 DEAD ‐770939 0 0 0 ‐1625 98679
1 Siesmic (X) 9807 46639 6 67 14 64845
1 Siesmic (Y) 9797 656 45989 7207 64542 24
1 BaseLL ‐95901 0 0 0 492 ‐12455
1 MotorRatedDL ‐5626 0 0 0 2755 26515
1 MotorShortCircuit (+MX) ‐1986 0 0 0 ‐17488 24024
1 MotorShortCircuit (‐MX) ‐7296 0 0 0 17488 24023
66

67. SAP2000 Results ‐ Section Forces
(‐2.25,‐2.25,0.0)
Positive Signs
Convention and
Resultants
Location
Fig. 28 Example for section cut (Cap‐D) for calculation of bending moment and
67
shear forces for the Primary Compressor foundation due to different loading
conditions

68. Table 14 Section forces for the Primary compressor at section Cap‐ D
SectionCut OutputCase FX FY FZ MX MY MZ
SectionCut OutputCase FX FY FZ MX MY MZ
Location Loading Condition N N N N‐m N‐m N‐m
CAP ‐ D Left
DEAD ‐498624 32986 2793272 ‐28339 1246017 ‐4402
CAP ‐ D Left
Siesmic (+ve X) 146484 1323 103165 23150 141380 97783
CAP ‐ D Left
Siesmic (+ve Y) 879 159539 494 246542 4900 228637
CAP ‐ D Left
BaseLL ‐111476 ‐643 246714 13911 168445 ‐3473
CAP ‐ D Left
MotorRatedDL ‐5207 ‐777 5270 4113 ‐8544 3603
CAP D Left
CAP ‐ D Left
MotorShortCircuit +MX ‐2287 18920 7130 ‐69532 ‐8408 ‐17184
CAP ‐ D Left
MotorShortCircuit ‐MX ‐8010 ‐15459 2264 58719 ‐7261 18605
C i h
DEAD 347045 ‐83247 2948598 1510550 ‐1097434 ‐518059
CAP ‐ D Right
DEAD 347045 83247 2948598 1510550 1097434 518059
CAP ‐ D Right
Siesmic (+ve X) 88259 4581 68598 8593 145241 125036
CAP ‐ D Right
Siesmic (+ve Y) 1151 92298 8767 302984 8401 307016
B LL 108304 1495 255988 161717 165906 68560
CAP ‐ D Right
BaseLL 108304 1495 255988 161717 ‐165906 ‐68560
CAP ‐ D Right
MotorRatedDL 20317 ‐1628 96710 66426 ‐21364 ‐14895
CAP ‐ D Right
MotorShortCircuit +MX 15674 ‐25377 80460 ‐220086 ‐15827 7961
CAP ‐ D Right
MotorShortCircuit ‐MX 22082 16516 92530 270183 ‐22183 ‐29536
68

69. The following figures and tables may be used to check the compliance of the
vibration amplitudes with different design requirements.
1. Figure 4.33 shows dynamic response limits in terms of limiting “single
amplitude” vibration at any frequency. The figure has 5 zones of sensitivity
shown by persons (standing and subjected to vertical vibration).
2. Figure 4.34 may be used to establish permissible horizontal vibration
amplitudes for rotating machinery.
3 Figure 4 35 shows the vibration standards for high speed machines
3. Figure 4.35 shows the vibration standards for high‐speed machines.
4. Figure 4.36 shows the vibration limits for foundations supporting
turbomachinery.
5. Table 4.3 gives suggested limits of peak velocities for various categories of
operation.
69

70. + Steady state vibrations
 Steady state vibrations
 D bl i
 Due to blasting
 Sh d d li t li it
 Shaded line represents limits
for safe operation of machines
and foundations (not for
ti f t ti )
satisfactory operation).
 Dotted lines are limits
associated with blasting. Do not
apply to steady state vibration.
Figure 29 General limits of vertical vibration amplitudes
70

71. E Dangerous, shut it down immediately
D Failure is near, Correct very quickly.
C Faulty, correct quickly.
B Minor faults.
A No faults, typical of new equipment
Figure 30 Vibration performance of rotating machines
A No faults, typical of new equipment
Figure 30 Vibration performance of rotating machines
71

72. Figure 31 Vibration standards of high‐speed machines
Figure 31 Vibration standards of high speed machines
72

73. Figure 32 Turbomachinery bearing vibration limits
73

74. Table 15 General machinery‐vibration‐severity data
Horizontal Peak Velocity
(in/sec)
Machine Operation
0 005 E l h
< 0.005 Extremely smooth
0.005-0.010 Very smooth
0.010-0.020 Smooth
0.020-0.040 Very good
0.04-0.080 Good
0.080-0.160 Fair
0.160-0.315 Slightly rough
0.315-0.630 Rough
>0.630 Very rough
74