1. MathCAD - Embedded Footing - Combined (JCB-edited).xmcd Page 1 of 10
Motion of an Embedded Footing
Fz
Fy
2B
D
Compressor Properties and Characteristics
Wcomp 100000 lb g 32.17
ft
sec
2

mcomp
Wcomp
g

Radius of compressor
(assumed to be
cylindrical in shape)
rcomp 8 ft
f
450
60
1
sec
 f 7.5s
1

Ω 2 π f Ω 47.124 s
1

Fz 4000 lb Fy 3000 lb
Foundation Geometry and Properties
B 5 ft L 10 ft
L
B
2 D 2.5 ft
D
B
0.5
Assume partial contact of sidewall: d 2.0 ft
γconcrete 150
lb
ft
3

Wfound 4 B L D γconcrete mfound
Wfound
g
 m mcomp mfound
Soil Properties and Characteristics
Vs 600
ft
sec
 ν 0.48 γsoil 110
lb
ft
3
 G
γsoil
g
Vs
2

Determine parameters needed to calculate the foundation stiffness
Ab 4 B L Area of the base in contact with the soil
χ
Ab
4 L
2

 Shape factor
VLa
3.4
π 1 ν( )
Vs Lysmer's analog velocity
Dr. Glenn Rix Web Site

2. MathCAD - Embedded Footing - Combined (JCB-edited).xmcd Page 2 of 10
Area moment of inertia of the
foundation-soil contact area about
the x axis
Ibx
2 L( ) 2 B( )
3

12

Area moment of inertia of the
foundation-soil contact area about
the y axis
Iby
2 B( ) 2 L( )
3

12

Aw d 4 B 4 L( ) Area of the sidewall contact
Aws 2 d 2 B Area of sidewall in shear for horizontal motion
Awce 2 d 2 L Area of sidewall in compression for horizontal motion
a0
Ω B
Vs
 a0 0.393 Dimensionless frequency
Height of centroid of
foundation-compressor
system above the
ground surface
zc
mfound
D
2
 mcomp D rcomp 
mfound mcomp
 zc 78.429 in
Mass moment of inertia of
the foundation-compressor
system about the x axis
Iox
1
12
mfound 2 B( )
2
D
2
  mfound zc
D
2






2

1
2
mcomp rcomp
2
 mcomp D rcomp zc 2


Resolve the horizontal force into an equivalent horizontal force acting at the centroid of the
foundation-compressor system
Mx Fy D rcomp zc  Mx 1.427 10
5
 in lb
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3. MathCAD - Embedded Footing - Combined (JCB-edited).xmcd Page 3 of 10
Stiffness and damping for vertical motion
Stiffness
Kzsur
2 G L
1 ν






0.73 1.54 χ
0.75
  Kzsur 6.493 10
6
 in
1
lb
Kzemb Kzsur 1
1
21
D
B
 1 1.3 χ( )





 1 0.2
Aw
Ab






2
3











 Kzemb 7.708 10
6
 in
1
lb
Kzemb
Kzsur
1.187
The dynamic stiffness and dashpot coefficients are frequency and foundation geometry (L/B)
dependent. Thus, they must be selected for each frequency and foundation geometry of interest.
kzsur 0.95 from Table 15.1, Gazetas (1991)
kzemb kzsur 1 0.09
D
B






3
4
 a0
2









 kzemb 0.942
kztre kzsur 1 0.09
D
B






3
4
 a0
2









 kztre 0.958
Interpolate to obtain the dynamic stiffness coefficient for partial embedment
kzpar kztre
d
D
kzemb kztre  kzpar 0.945
Damping
cz 1.0 from Table 15.1, Gazetas (1991)
Czsur
γsoil
g
VLa Ab cz Czsur 7.117 10
4
 in
1
s lb
Czemb Czsur
γsoil
g
Vs Aw Czemb 9.168 10
4
 in
1
s lb
Czemb
Czsur
1.288
κz Kzemb kzpar i Ω Czemb
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4. MathCAD - Embedded Footing - Combined (JCB-edited).xmcd Page 4 of 10
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5. MathCAD - Embedded Footing - Combined (JCB-edited).xmcd Page 5 of 10
Stiffness and damping for horizontal motion
Stiffness
Kysur
2 G L
2 ν






2.0 2.50 χ
0.85
  Kysur 4.572 10
6
 in
1
lb
Kyemb Kysur 1 0.15
D
B







 1 0.52
d
B
Aw
L
2







0.4









 Kyemb 7.017 10
6
 in
1
lb
Kyemb
Kysur
1.535
The dynamic stiffness and dashpot coefficients are frequency and foundation geometry (L/B)
dependent. Thus, they must be selected for each frequency and foundation geometry of interest.
kyemb 1.0 1.0 from Table 15.2, Gazetas (1991)
Damping
cysur 1.0 from Table 15.1, Gazetas (1991)
Cysur
γsoil
g
Vs Ab cysur Cysur 3.419 10
4
 in
1
s lb
Cyemb Cysur
γsoil
g
Vs Aws
γsoil
g
Vs Awce Cyemb 5.471 10
4
 in
1
s lb
Cyemb
Cysur
1.6
κy Kyemb kyemb i Ω Cyemb
Dr. Glenn Rix Web Site

6. MathCAD - Embedded Footing - Combined (JCB-edited).xmcd Page 6 of 10
Stiffness and damping for rocking
Stiffness
Krxsur
G
1 ν
Ibx
0.75

L
B






0.25
 2.4 0.5
B
L






 Krxsur 2.335 10
10
 in lb
Krxemb Krxsur 1 1.26
d
B
 1
d
B
d
D






0.2

B
L














 Krxemb 3.86 10
10
 in lb
Krxemb
Krxsur
1.653
The dynamic stiffness and dashpot coefficients are frequency and foundation geometry (L/B)
dependent. Thus, they must be selected for each frequency and foundation geometry of interest.
krxsur 1 0.20 a0
krxemb krxsur
Damping
crxsur 0.15 c1 0.25 0.65 a0
d
D






a0
2


D
B






1
4


Crxsur
γsoil
g
VLa Ibx crxsur
Crxemb
4
3
γsoil
g
 VLa B
3
 L crxsur
4
3
γsoil
g
 VLa d
3
 L c1
4
3
γsoil
g
 Vs B d B
2
d
2
  c1 4
γsoil
g
 Vs B
2
 d L c1
Crxemb 6.137 10
7
 in s lb Note: this is for a rectangular footing only
Crxemb
Crxsur
4.791
κrx Krxemb krxemb i Ω Crxemb
Dr. Glenn Rix Web Site

7. MathCAD - Embedded Footing - Combined (JCB-edited).xmcd Page 7 of 10
Coupled stiffness and damping between rocking and horizontal motion
Stiffness
Kyrxemb
1
3
d Kyemb Kyrxemb 5.613 10
7
 lb
kyrxemb 1.0
Damping
Cyrxemb
1
3
d Cyemb Cyrxemb 4.377 10
5
 s lb
κyrx Kyrxemb kyrxemb i Ω Cyrxemb
Dr. Glenn Rix Web Site

8. MathCAD - Embedded Footing - Combined (JCB-edited).xmcd Page 8 of 10
Stiffness and damping for torsion
Stiffness
Ktsur G Ibx Iby 0.75
 4 11 1
B
L






10







 Ktsur 5.167 10
10
 in lb
Ktemb Ktsur 1 1.4 1
B
L







d
B






0.9








Ktemb 9.924 10
10
 in lb
Ktemb
Ktsur
1.921
ktsur 1 0.14 a0
ktemb ktsur
Damping
ctsur 0.25 c2
d
D






0.5 a0
2
a0
2
0.5
L
B






1.5


Ctsur
γsoil
g
Vs Ibx Iby  ctsur
Ctemb
4
3
γsoil
g
 Vs B L B
2
L
2
  ctsur
4
3
γsoil
g
 VLa d L
3
B
3
  c2 4
γsoil
g
 Vs d B L B L( ) c2
Ctemb 2.083 10
8
 in s lb Note: this is for a rectangular footing only
Ctemb
Ctsur
4.061
κt Ktemb ktemb i Ω Ctemb
Dr. Glenn Rix Web Site

9. MathCAD - Embedded Footing - Combined (JCB-edited).xmcd Page 9 of 10
Form the complex stiffness matrix
Κ
κz Ω
2
m
0
0
0
κy Ω
2
m
κyrx κy zc
B
0
κyrx κy zc
B
κrx Ω
2
Iox 2 κyrx zc κy zc
2

B
2



















Form the force-moment vector Note that some of the terms in the stiffness
matrix and force-moment vector are divided
by a "characteristic length" to make the matrix
and vector dimensionally correct.
F
Fz
Fy
Mx
B













Solve the system of equations
u Κ
1
F
u
4.323 10
4
 2.974i 10
4

1.713 10
3
 4.649i 10
4

9.726 10
4
 1.603i 10
4













in
The magnitude and phase of the individual motions are:
uz u
0
 uz 5.248 10
4
 in
Vertical motion
ϕz arg u
0  ϕz 0.603
δy u
1
 δy 1.775 10
3
 in
Horizontal motion
ϕy arg u
1  ϕy 0.265
Dr. Glenn Rix Web Site

10. MathCAD - Embedded Footing - Combined (JCB-edited).xmcd Page 10 of 10
θrx
u
2
B
 θrx 1.643 10
5

Rocking motion
ϕrx arg
u
2
B






 ϕrx 0.163
The natural frequency and fraction of critical damping for each motion are:
fz
1
2 π
Kzemb kzemb
m
 fz 20.144 s
1

Vertical motion
βz
Czemb
2 Kzemb kzemb m
 βz 0.799
fy
1
2 π
Kyemb kyemb
m
 fy 19.801 s
1

Horizontal motion
βy
Cyemb
2 Kyemb kyemb m
 βy 0.485
frx
1
2 π
Krxemb krxemb
Iox
 frx 17.909 s
1

Rocking motion
βrx
Crxemb
2 Krxemb krxemb Iox
 βrx 0.097
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