CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
DYNAMICALLY LOADED MACHINE FOUNDATION
THEORY, ANALYSIS, AND DESIGN
A graduate project submitted in partial satisfaction
of the requirements for the degree of Master of Science
in
Engineering
by
Henry S. Wu
May 1987
Q
The Graduate Project of Henry S. Wu is approved:
Dr. Willis Downing, Jr.
D~James Roberts
'
Dr. Edward Dombourian, Chair
California State University, Northridge
JI
•
Acknowledgement
The
author
appreciation
to
Dombourian for
would
the
like
committee
his helpful
to
express
his
sincere
chairman,
Dr.
Edward
guidance and
recommendations.
Appreciation is
also extended to Dr. James Roberts and Dr.
Willis Downing,
Jr. for their helpful participation in the
advisory committee.
][
TABLE OF CONTENTS
Page
III
ACKNOWLEDGEMENT
LIST OF TABLES
-------- ------ ------
v
LIST OF SYMBOLS
VII
ABSTRACT
XII
CHAPTER
1. INTRODUCTION
-
2. DESIGN CONDITIONS AND MODELING TECHNIQUES
-
-
1
- 5
3. THEORY OF ELASTIC HALF-SPACE
- - - 15
4. FOUNDATION DESIGN FOR VIBRATING MACHINES
- - - 34
5. DETERMINATION OF APPROPRIATE SOIL PROPERTIES
FOR DESIGN
- - -
- - - - - -
- - - - 59
6. DESIGN EXAMPLES FOR MACHINE FOUNDATION
- - - - 71
APPENDIX A. DYNAMIC SOIL PROPERTIES
REFERENCE
- - - - - - - - - 115
- - - - - - - - - - - - - - - - - - - - - - 124
LIST Of TABLES
Table
4-7-1
Page
Summary of derived expressions for a singledegree-of-freedom system ------------------------ 44
4-7-2
General machinery-vibration-severity data ------- 46
4-7-3
Equivalent damping ratio for rigid circular
and rectangular footing ------------------------- 48
4-7-4
Typical values of internal damping in soils ----- 49
4-7-5
Equivalent spring constant for rigid
circular and rectangular footing ---------------- 49
5-2-1
Recommend design values for subgrade
coefficient k~
5-3-1
--------------------------------- 61
Some typical values of internal damping
in soils ---------------------------------------- 64
5-3-2
Spring constants for rigid circular footing
resting on elastic half-space ------------------- 66
5-3-3
Spring constants for rigid rectangular
footing resting on elastic half-space ----------- 66
6-2-1
Mass moments of inertia of common geometric
shapes
------------------------------------------81
6-2-2
Embedment coefficients for spring constants ----- 91
6-2-3
Effect of depth of embedment on damping ratio --- 92
6-2-4
Values of nt for various of ~~ ------------------ 92
LIST OF SYMBOLS
A
=
Displacement amplitude
A1 ,A 2 = Constant of integration
a. = With of section i, ft.
1
B
=
Length of rectangular foundation block, ft.
Bz,Bx,B~,B8
= Mass (or inertia) ratio; vertical,
horizontal, rocking and torsional
vibration mode.
b.1 = Depth of section i, ft.
c 1 ,c 2 =
Constants of integration
c = Damping coefficient
c c = Critical damping coefficient
cz,cx,c~,c 6 ,c;
= Damping coefficients in the vertical,
horizontal, rocking, twisting, and
pitching modes
=
D = Damping ratio
Dz,Dx,D~,De
cfcc
= Damping ratio, vertical, horizontal,
rocking and torsional modes
Di
=
Internal damping ratio
E = Tension-Compression modulus of elasticity
e
=
Eccentricity of unbalanced mass to axis of rotation at
operating speed, in.
e = Eccentricity of the machine's unbalanced mass, in.
F = Exciting force
F
= Amplitude of exciting force, lbs.
0
F ,F
X
=
f
Z
= Maximum horizontal, vertical force, lbs.
Operating speed of the machine, rpm
fc
=
Critical speed of machine, rpm
fn
=
Nature frequency, rpm
fm
=
Resonance frequency for constant force-amplitude
excitation, rpm
fmr = Resonant frequency for rotating mass type excitation,
rpm
fmx'fm~'fm;
= Resonant frequency in the horizontal, rocking
and pitching direction, rpm
fnx'fnz'fn~'fn'
=
Natural frequency in horizontal,
vertical,
rocking and pitching
direction, rpm
G
=
Shear modulus, psi
g = Acceleration of gravity, ft/secZ
H = Dynamic horizontal force, lbs
h = Effective foundation embedment depth, ft.
I~,r 9 ,r9
= Mass moment of inertia in the rocking, twisting
and pitching, lbs.
i
= Segments
k
= Spring
k.l
=
se~ft
(1, 2, 3, - - -)
constant, kips/ft.
Distance from center of mass to base of footing for
segment i, ft.
kz,kx,k~,ke,kp
= Equivalent spring constants: vertical,
horizontal, rocking, torsional and
pitching modes.
= Width
L
M
r
=
at base of machine foundations block, ft.
Magnification factor
M = Dynamic Magnification factor
2
m = Mass, lbs.sec/ft
me = Unbalanced mass
mi
n
= Mass of segment i
= Number of segments
=
n~,llf
Rocking and pitching mass ratio factors for
geometric damping
P 0 = Force transmitted through spring mounts
=
PH,PV
Force transmitted in the horizontal and vertical
direction, lbs.
PT~'PT¢
Pbdyn
=
=
Transmitted rocking, pitching moment, ft.lbs.
Bearing pressure due to transmitted dynamic force,
psf
Rhx'Rhy= Horizontal distance from center to edge of footing
in x and y direction, ft.
R
v
= Vertical distance from base to center of rotor axis (or
to horizontal machine load in the case of reciprocating
machine), ft.
r = Ratio of operating frequency to nature frequency, f/fn
=
r
0
Sall
T
0
=
Equivalent radius for rectangular footing, ft.
=
Unbalanced torque, ft-lbs.
T~,Tp
TR
=
Allowable soil pressure, ksf
=
Unbalanced rocking and pitching moment, ft-lbs.
Transmissibility factor
TR(P)
=
Transmissibility factor for primary operating
frequency
TR(S)
=
Transmissibility factor for secondary machine
frequency
t
=
Time, sec
W = Total weight of machine plus foundation, lbs.
WB,WC,WR,WT,WF
= Weight
of base plate, compressor , rotor,
turbine and foundation, lbs.
WM =Total machine weight= Wc+WT+w6 , lbs.
X = Displacement response in the horizontal direction, in.
Xt = Total displacement response in the horizontal
direction, in.
x
=
Displacement, in.
xa,xb = Complementary and particular solution, in.
x
=Velocity, in./sec
x =Acceleration,
in./sec2
Y = Displacement amplitude,. in.
z = Displacement response in the vertical z direction, in.
zt = Total displacement response in the vertical z
direction, in.
~z'~,J~,~ = Damping ratio embedment factor, vertical,
horizontal rocking and pitching modes.
~
=
Phase angle, rad
~z'~'~~~~¢
=
Spring coefficients; vertical, horizontal,
rocking and pitching modes.
X
iJz'1lx'7Jrp'7J1 =
Spring constant embedment factors; vertical,
horizontal, rocking and pitching modes.
V = Poisson's ratio of soil
C = Damp1ng
'
'
ra t 10
"'9
o
=
f
2
=Mass density of soil=-%- lbs.sec jft4
-~C
GU = Frequency of excitation force, rad/sec
UJn
=
Natural circular frequency, radjsec
UJd
=
=
Damped natural circular frequency, rad/sec
UJf
Forced circular frequency, rad/sec
:xr
ABSTRACT
VIBRATION ANALYSIS AND DESIGN OF
DYNAMICALLY LOADED MACHINE FOUNDATION
BY
HENRY S. WU
MASTER OF SCIENCE IN ENGINEERING
The design
loads
is
of machine foundations supporting dynamic
rather
complicated
interaction of
structural and
well as
expertise of
machine
some
foundations
is
because
it
involves
the
geotechnical engineering as
machine design.
generally
The
determined
form of
by
the
information provided by the geotechnical consultant and the
machine manufacturer.
The design engineer therefore has to
utilize
of
the
theory
vibration
analysis,
modeling
techniques and soil dynamics to meet the design criteria.
This paper investigates the response of foundations to
the vibrations resulting from two types of machines used in
industry-Reciprocating and
introductory chapter
Rotary machines. It includes an
which explains
the coverage
of
the
contents. Chapter
Two describes
dynamically loaded
systems while
analyze the
response of
of
loadings.
dynamic
information necessary
procedures.
examples
Finally,
of
Appendix A
is added
list of
techniques
Chapters Three
for
and Four
foundations under different kinds
The
for
Fifth
design
Chapter
foundations
properties. A
modeling
chapter
and
Six
supporting
for the
considers
the
lists
the
design
contains
two
actual
dynamic
explanation of
machines.
dynamic soil
symbols is included at the front of
this paper to provide a convenient reference.
CHAPTER 1
INTRODUCTION
The
purpose
of
this
vibrations developed
project
by two
machineries (Reciprocating
design of
addition to
the size
of the
suitable to
and Rotary
the static
loads instead
criteria. However,
the
of operating
machines)
for
the
to resist these dynamic
is
foundation for
of strictly
investigate
loads. In some cases, if
machine involved
design the
to
different kinds
appropriate foundations
loads in
is
small,
it
may
be
equivalent
static
vibration
design
applying the
once the design engineer has recognized
the need for a vibration analysis, it is necessary that the
designer possess a clear understanding of the followings:
1. Design Conditions
In
order to
design the
structure to
support dynamic
machines, a
number of information are required to be known
even before
preliminary sizing
completed.
geometrical
supported
These
design
constraints
but
also
of the
information
of
the
include
1
structure
include
actual
detail
can
not
machine
knowledge
be
only
to
of
be
the
2
structural
supports
and
environmental
requirements.
The
first half
of the Chapter Two has a thorough explanation of
these design conditions and requirements.
2. Modeling Techniques
The
second half
of the
Chapter Two
described
a
few
practical considerations which are commonly used in modeling
of structure
An
system consists
equivalent
supported by
of springs and lumped masses.
mathematical
model
of
block-type foundation
the investigation
horizontal and
of three
vibrating
is also
dynamic mode
rocking modes.
machine
developed with
shapes- vertical,
The results
will be used in
Chapter Six for solving actual design problems.
3. Theory of Elastic Half-Space
The
most widely accepted model to be used in predicting
the response
of the
"elastic half-space
circular footing
halfspace (the
is
model". This
rests
upon
and
can
be
theory
the
soil) extending
homogeneous
properties
soil to imposed dynamic loading is the
surface
by
of
an
that
a
elastic
to an infinite depth, which
isotropic
defined
presumes
and
two
whose
stress-strain
elastic
constraints,
and Poisson's
ratio ( V). After
usually shear
modulus (G)
accepting the
assumptions involved in considering a footing
resting on the elastic half-space, it is possible to develop
mathematical solutions
thus supported.
Several
for the
dynamic response of footing
solutions
which
demonstrate
the
,,
'
3
importance
of
the
exciting forces
basis for
geometrical
are presented
the design
variables
in Chapter
procedures which
and
types
Three to
will be
of
form a
treated in
Chapter Four.
4. Design Procedures
In the design of machine foundations or other dynamically
loaded foundations, successive correction are used to arrive
at the
final physical
are assumed
and then
conditions are
primarily to
to
analyzed to
determine if
the design
satisfied. If they are not, then some of the
physical parameters
The design
system. A set of physical parameters
are varied and the process is repeated.
procedures
discussed
the dynamic
steady-state
in
Chapter
Four
relate
response of foundations subjected
vibrations
or
transient
loadings.
The
emphasis of that chapter lies in the procedures for carrying
out the
dynamic analyses
after the
design conditions have
been established.
5. Soil Properties
The
proper
characteristics is
evaluation
very
of
important
soil
for
properties
the
design
and
of
a
foundation. These soil properties will be different for each
construction site.
The dynamic
response of
a
given
soil
depends upon the loading conditions and strain distributions
developed in
the soil mass. The primary emphasis in Chapter
4
Five is
directed toward
the response of soils to vibratory
stresses.
6. Design Examples
The
analytical procedures
behavior of
a foundation
for establishing the dynamic
relate the
applied forces,
soil
properties, foundation weights and geometry to the response.
By successive
corrections of
analytical procedures
dynamic response
design limits.
the
provide a
of the
Two actual
design
method
foundation which
parameters,
for
developing
the
a
falls within the
design problems are developed in
Chapter Six by an easy-to-follow step-by-step routine.
CHAPTER
2
DESIGN CONDITIONS AND MODELING TECHNIQUES
2.1 General
When a dynamically loaded structure is to be designed,
it
requires
parameters
that
be
structure can
certain
known
site
before
be made.
conditions
preliminary
and
sizing
loading
of
the
Generally, we could classify these
design conditions and requirements into three categories as
follows:
1. Machine properties and requirements
2. Soil parameters
3. Environmental requirements
2.2 Machine Properties and Requirements
Three groups of machines usually can be classified to
cause dynamic loads on a structure:
1. Reciprocating machines
2. Centrifugal machines
3. Impact causing machines
5
6
The machine's geometrical and performance factors are
always needed
for structure
design. We
can obtain
these
information from machine manufacturers, sales catalogues or
engineering handbooks.
Following are
the required machine
properties and parameters:
a. Outline Drawing of Machine Assembly:
The size
dimensions for
of the
foundation depends
the machine
minimum dimensions
base.
and locates
The
on the required
outline
specifies
areas needed to be cleared
for machine attachments.
b. The Foundations of Machine
It includes information on the critical nature and over
all purpose
of the
machine, plus
the importance
of
the
machine is to the overall operation.
c. The Weight of the Machine and Its Rotor Components:
The weights
machines
of rotors
determine
the
and the speeds in centrifugal
magnitude
of
possible
machine
unbalanced forces.
d. The Coordinates of the Center of Gravity of the Machine:
This information
calculated
or
assumed
is often
whenever
provided. It
it
is
not
can also be
available.
Generally, the machine is set on the foundation in a way to
avoid eccentricities
between the
results of all the loads
and the support center of resistance.
7
e. The Speed Range of Machine and Components or Frequencies
of Unbalanced Primary and Secondary Forces:
These are
required in
dynamic analysis
in order to
check for possible resonance. The operating frequencies are
of primary
interest to
available, a
the designee.
temporary resonance
If high
caused
by
damping is
start-up
or
shout-down of the machine may be tolerated.
f. The Magnitude and the Direction of Unbalanced Forces
both Vertically and Horizontally and Their Points of
Application:
Even through
some
centrifugal machines
years of
the use
will develop.
manufacturers
are perfectly
due to
that
their
balanced, after
a few
normal wear, some eccentricities
For reciprocating
large magnitude
claim
of unbalanced
machines,
forces are
the
generally
provided by the
machine manufacturer.
g. The Limits Imposed on the Foundation with Respect to
Differential Deflection between points on the Plane Area
of Foundation:
There are
other components
cause of
set to avoid possible damage to piping and
that connect
machines with
to the
very rigid
machine. Under
attached
piping,
the
the
0.0001 in. differential deflection limit usually is set for
high pressure (50,000 psi.) piping.
8
h. Foundation Requirements:
Foundation requirements
refer to
minimum
depth
of
foundation, as indicated by the following factors:
1. expansive soil
2. frost action
3. fluctuating water table
4. piping clearance
5. pavement elevation
The geotechnical
the design
weathered
process
soil
is
foundation. Also,
soil may
consultant usually is needed during
for
suggestions.
not
recommended
the bearing
The
for
top
layer
supporting
strength required
for
of
the
the
dictate placing the bottom of the foundation at a
deeper level for structure.
2.3 Soil Parameters
The following
for static
soil parameters are generally required
and dynamic
analysis
of
soils
under
loaded
structure:
a. Density of soil
r
and Poisson Is Ratio
v:
These two parameters are usually reported by the
geotechnical consultant.
9
b. Shear modulus of soil G, at several levels of strain (or
magnitudes of bearing pressures):
This parameter is a controlling factor in the
calculation of the half-space spring constants, so the
most reliable value is to be expected. The answer has to
be the result of field test. As was discussed in the
previous chapter, the soil strain level has an important
effect on the value of shear modulus.
c. Coefficient of subgrade reaction of soil, if the above
parameters are not accurately known:
This coefficient may be used instead of shear
modulus G or used as a check on the order of
magnitude of soil moduli as calculated using the
half-space theory.
d. The foundation depth and the bearing pressure at which
the above parameters are applicable:
The bearing pressure and foundation depth should be
designed corresponding to specific values we used for
the above parameters. Since dynamic loads often produce
low strain levels and the shear moldulus in so sensitive
to strain levels.
e. Other information required for the static design of the
footing:
p
•
10
These information include relative economy,
settlement, bearing _capacity of the soil, vibration
isolation and the level of underground water table.
2.4 Environmental Conditions
The design
engineer must
consider
if
the
machine
installation is in the vicinity of vibration sources. If it
is necessary,
asked to
a vibration measurement consultant should be
help determine the character of the vibration and
the attenuation
of the
amplitude of
the vibration at the
installation site.
2.5 Modeling Technique for Dynamic Systems
An idealized
springs and
the actual
the
structure does.
is very
equivalence
element, and
structural system consists of
lumped masses which perform in the same way as
system parameters
of
model of
of
The proper
selection
of
the
important for the determination
the
idealized
spring,
damping
lumped mass in the model which will result in
equivalent displacements
prototype structure.
at same points of interest in the
11
A few practical considerations which are commonly used
in modeling are:
1. The Lumping of Mass:
The location
of the
equivalent lumped
mass in a model
must be at
(a) a point where dynamic force or load is acting
(b) a point where vibration response is desired
(c) a point where maximum static deflection will occur
(d) the intersection of a beam and a column
(e) the
nodal point of finite elements in a continuous
system
(f) the
center of gravity of all masses, when a single
degree of freedom system is employed
2. Elastic·Spring Constant
This represents
and the
a relationship between the applied load
displacement of
spring constant
the mass.
usually is
structural stiffness
of
the
The
derived by
elastic
value
of
the
determining the
medium
existing
between oscillating masses or between a mass and another
infinitely stiff support.
3. Damping Ratio
Generally, the
viscous type
structural system.
of damping
The following
needed to be considered:
is used in the
assumptions are
also
12
(a) the
internal damping
concrete and
1.0 to
ratio is
5.0 % for
2.0 to
7.0 percent for steel structures
and is normally neglected.
(b) one
of the
soils is
two kinds
the internal
magnitude in
of damping associated with
damping. Its value is small in
all modes
of oscillation
except in the
rocking mode. Geometric damping is the other one which
has quite
system
an effect
and
usually
on the
is
dynamic response
included
in
the
of the
model
representation.
4. Forcing Function
This is
often treated
force applied
as
an
equivalent
concentrated
at point where masses are lumped. Torques
are applied at mass point either in concentrated form or
are converted into an equivalent force couple.
2.6 Mathematical Model of Vibrating Machine Supported by
Block-type Foundation
13
From the
models developed
by Arya,
Pincus,
and
O'Neill
(REf. 9)
there three types of dynamic mode shape should be
considered for this kind of
Vibrating Machine
Supported by
Block-Type Foundation
The vertical
of
the
Vertical Mode
mode is generally acts on its own. The mass m
machine
concentrated on
and
the
foundation
the vertical
is
assumed
to
be
axis. Spring constant of the
soil k 2 , damping coefficient in the soil c 2 , inertia of
mass m1 and the forcing function F 1 of the machine should
have their line of action coincide with the vertical axis.
Equation of motion:
mz
+ c~z + kzz
= Fz(t)
Horizontal Mode
14
An approximation
the horizontal
has to
mode. The
be made for the representation of
masses do
not lie
on the
same
horizontal axis nor the action of the forces coincide. This
mode is
normally coupled
with the rocking mode because of
these reasons.
Equation of motion can be represented by:
my+ cyy + kyy = Fy(t)
Rocking Mode
The rocking
mode is
dynamic behavior
solution is
of
hard to
a better
the
representation of
structure.
obtained because
But
the
the true
analytical
of the coupling
of
the rocking and horizontal motions. Especially when we have
a machine
that is located high above the foundation level,
this coupling effect should be carefully investigated.
The equation of motion can be shown to be:
CHAPTER
3
THEORY OF ELASTIC HALF-SPACE
3.1 General
In
foundations,
subjected
has
will
achieve
dynamic
a
successful
response
of
design
such
of
foundations
to steady-state vibrations or transient loadings
of foundations
be presented
the
supported by
an
elastic
medium
and discussed, then it will be followed
introduction
laboratory
to
the
to
to be carefully analyzed. In this chapter, the dynamic
behavior
by
order
of
appropriate
theory
and
both
and in the field test which have been developed
provide an
estimate
of
the
dynamic
response
of
a
particular foundation.
3.2 Vertical Oscillation of Footing Resting on the Surface
of an Elastic Half-Space
In
theory
footing
1936, E.
Reissner (Ref.
for evaluating
as it
24) tried
to develop a
the dynamic response of a vibrating
was influenced
iS
by properties
of the soil.
~
16
The
soil
was
mass
homogeneous,
represented
by
a
semi-infinite
isotropic, elastic body (elastic half-space).
Three parameters
1. the poisson's ratio
2. the shear modulus G
3. the mass density P/g
were
He
used to
used an
describe the properties of the elastic body.
oscillating mass
vertical
pressure which
circular
area of
space
used
was
evenly
the vibrating
the mathematical
produced
on the
radius ro
to represent
which
a
distributed
surface of
periodic
over
the half-
footing. Sung (Ref. 25)
treatment to represent the vertical
displacement by the equation:
P 0 exp(iCIJt)
-------------G r
- - - -(3-2-1)
0
in which
P0
=
amplitude of the total force applied to the circular
contact area,
W
= frequency of force application,
G
= shear modulus of the half-space,
r 0 = radius of the circular contact area,
fl'f = Reissner's "displacement functions", positive in
2
downward direction.
a 0 = wr 0
where
ao
=
J-;-
=
a
~~Q-
- - - - - -
s
dimensionless frequency
(3-2-2)
.
17
vs
= velocity of propagation of the shear wave in
elastic body
A
=
JG/P
second dimensionless term called the
"mass ratio" b was
also established by Reissner as
m
b
= ----3pro
m = total
- - - - - -(3-2-3)
mass of the vibrating footing and the
exciting mechanism which rests on the surface
of the elastic half-space. {see Fig. 3-2-1)
Rotating Mass
Oscillator _ __..,
Total
Weight= W
Moss= m= Y:L
g
Elastic Body
G, 11,p
+z
Fig. 3-2-1, Rotating mass oscillator with circular
footing resting on semi-infinite elastic body.
(Ref. 5)
The
amplitude of
oscillator motion
can then be expressed
by:
A
Oo
z = -----
- - - - -{3-2-4)
18
and
tan
- - - - - - - -(3-2-5)
where
~=phase angle between the external force, Q-Q 0exp(iwt),
and the displacement
z0 .
Input power required is as follow
Oo 2
PR
- - - -(3-2-6)
= -~~2JG~~~--
where
Q = external force amplitude, which can be constant or a
0
function of the frequency of excitation expressed
as
- - - - - - - - - - - - - - - - - - -(3-2-7)
for
a rotating-mass-type
acting
exciter
with
a
total
mass
m
at a radius designed as the eccentricity. The total
eccentric mass on Fig. 3-2-1 is equal to 2m.
Lyster
and Richart
(Ref. 32)
also introduced
a modified
dimensionless mass ratio
1-11
m
- - -(3-2-8)
4
19
3
- - Half-space Theory
<tO0
Nl
.... ...
(v=
0-
sl
- - - - Simplified analog
C>l
'¢=:;
A
2
:::'!:
Q =Ooeiwt
,,L1~ J ,,
G,p,v ~
.:
0
u0
LL.
c
.2
0
.!.:!
·c:0'
0
:::'!:
0
0.5
1.5
Dimensionless Frequency, 0 0
Fig. 3-2-2, Response of rigid circular footing to
vertical force developed by constant force
excitation. (Ref. 32)
Fig 3-2-2 shows the influence of a change in
poisson's
also
for
ratio on
found that
the response
of the
footing. It
was
a lumped-system analogue can be developed
vertical vibration
of a
rigid circular
footing with
the motion equation:
mz•• +
3.4
2
r 0 JGP
-----( 1- V)
if
0 < ao
3
k
z =
4Gr 0
-----1- )I
1- Jl
z =
Q
- - - - -
-(3-2-9)
< 1.0, then
Jp G
____
3_. 4_r
__o__
(1-V)
4Gr
0
z + ------
- - - - - - - - - - - - - - - -(3-2-10)
- - - - - - - - - - - -
Now the critical damping is
~(3-2-11)
20
c =
and
=
2Jkm
J-i~~Q.- -------------(
2
3-2-12)
0.425
c
D =
= -------
K
cc
0.85
or
D
1
= -----
footing.
- - - - - - - - - - - - - - -(3-2-14)
,/1- JJ ,[b
The
- - - - - - - - - - - - -(3-2-13)
above analysis
If the
of
dimension
was considered
footing
x
a
b,
for
is
rectangular
the
equivalent
a
in
circular
plan,
radius
with
may
be
determined from the relationship:
In
detailed
the next
section (3.3)
there
will
be
a
more
discussion concerning the vertical oscillation of
a rigid rectangular footing.
3.3 Vertical Oscillation of Rigid Rectangular Footing
The
circular
oscillating
the
suggested
the
the
loads on
by
loads
will be
Elorduy,
rectangular base
on
3.2 as
solutions
based
for
on
a
vertical
a rectangular zone of the surface of
elaborated here.
Nieto,
solutions for
surface of
in section
Analytical
footing
elastic half-space
published
rigid
case discussed
the
and
vertical
Szekely
A paper
(Ref.
oscillation
28)
of
a
with a length 2c and a width 2d on
the elastic half-space. If we assumed that
a
square
footing
which
were
uniformly
21
distributed
the
should produce
pressure distribution
displacement
was
uniform
displacements.
required to
proportional
to
keep that
the
Also,
uniform
frequency
of
oscillation.
The
solid line
in Fig. 3-3-1 shows the distribution
functions
for
of pressure and the displacement functions f 1 ,f 2
a square and rectangular loading area on an elastic
half-space for which the Poisson's Ratio was 1/4.
22
0.20.-------------------------.
0.16----
~=1
----~
....... ~f,
....................
0.12
~
...............
~-
~
'
...0
1
~:;...-"
.: 0.08
::.><
.........
----- ,
(a)
~
.........._
- .........
......_........._
~
............................
..............
0.04
Elorduy et al.- Rigid Rect.} '--.
- - - - Bycroft- Rigid Circular
a-=
- - Sung- Rigid Circular
-t
0
0.5
1.5
LO
00
=
wd
v,
0.24.------------------------.
0.20
~
S
.
-
(b)
. -1-
'.
Reclongle
..k.
d
=2
1.0
0
L5
Fig. 3-3-1, Displacement function for vertical
vibration of rigid rectangular and rigid
circular footing. (Ref. 29)
Also
developed
seen on Fig. 3-3-1 are the corresponding curves
by Sung
(Ref. 27)
and Bycroft
(Ref. 29) after
23
the
radius
circular,
these
a
was
adjusted
square, and
curves are
circular
rectangular
rigid
to
yield
the
same
rectangular footings.
area
for
Because
all
so close to each other, the solution for
base
of
footings is
the
same
area
representing
a ratio of cfd of up
adopted for
to 2.0.
3.4 Torsional Oscillation of Circular Footings on the
Elastic Half-Space:
In
1944, Reissner
analytical
circular
solutions
and Sagoci
for
footing resting
(Ref.
torsional
on the
30)
presented
oscillation
surface of
the
of
elastic
half-space as follows:
a. The tangential shearing stress
z
developed by the
applied torque Te is
3
'lze
=
~r
-4rr- -~~1];~~:;~-
for 0 < r < ro - -
(3-4-1)
b. The static spring constant under static condition is
derived from the relation between applied torque
and resulting rotation
16
3
G r 03
.
-
- -
a
- - - - - - -(3-4-2)
c. The mass ratio for torsional oscillation is
- - - - - - - - - - -(3-4-3) in which
24
I
= the mass moment of inertia of the footing about
the footing about the axis of rotation
d. Dynamic magnification factor for constant torque
excitation is
Mem
= -~@~e
-- - - - - - - - - - -(3-4-4)
s
e. The exciting torque for the excitation by a rotatingmass system is
Te = meexw
2
-
-
(3-4-5)
in which x= the horizontal-moment arm of the
unbalanced weights from the center
of rotation.
f. The peak amplitude of motion in this excitation is
Aem =
me ex .....
______
Ie
Merm - - - - - - - -
(3-4-6)
Values of Mem' Merm' and a m are given in Fig. 3-4-1
8
as function of Be.
25
T= T0
e;...'
~
II
...
ID
~
4
2
2.0
0
-•l"'o
-(b)
~
•
tl
Mem or Merm
Fig. 3-4-1, Torsional oscillation of rigid circular
footing on elastic half-space (a) Mass ratio vs.
dimensionless frequency at resonance. (b) Mass ratio
vs. magnification factor at resonance.
If the footing is rectangular (ax b), thy equivalent
radius r 0 may be determined by
26
- - - - - - (3-4-7)
3.5 Rocking Oscillation of Footing Resting on the
Elastic Half-Space
For the rocking problems studied by Hall (Ref. 31), it
was
required to evaluate the damping and
spring constants
in the following equation of motion:
Itt'~+
ctpcf
+ kcptp
=
Ttpexp(iwt)
- - - - - - - -
(3-5-1)
in which
Itp = the mass moment of inertia of the footing about the
center of rotation,
ro
=
radius of cylindrical footing,
h
=
height of cylindrical footing.
For a uniformly distributed mass
2
= -------- (----- + ----)
g
4
3
~ro
Itp
2
hr
ro
2
h
The spring constant K
KljJS =
so
-=~'l's
is
3 ( 1-V)
but
lfJs
=
- - - - - -(3-5-2)
-------8
Ttp
----3Gr 0
8Gr 0 3
-------3 ( 1-1.1)
where the damping constant is expressed by
- - (3-5-3)
27
o.aor 0 4,JGP
------------( 1-V) ( 1 +Bc.p)
- - - - - - - - - - - -(3-5-4)
(a)
Constant Force Excitation
6
3(1-v) T +
A.;m = - 3 Mojtm
8
Gr0
Rotating Moss Excitation
(b)
Fig. 3-5-1, Rocking of rigid circular footing on
elastic half-space. (a) Mass ratio vs. dimensionless
frequency at resonance. (b) Mass ratio vs.
magnification factor at resonance. (ref. 5)
28
Where from the damping term through Eq. 3-5-4 and Fig.
3-5-1
(b), the
maximum amplitude of rocking motion or the
maximum dynamic magnification factor could be obtained.
This result can be checked by through the following
equations:
a. Critical damping for the mass-spring-dashpot system
c,
=
c
2JK lfJ rev
- - - - - - - --(3-5-5)
b. Damping ratio
0.15
c
D
'P
--~--
=
c
- - -
=
(3-5-6)
tpc
c. The maximum magnification factor for rocking in small
damping is
M
4'm
~
__I__
Hall
frequency
space
be
- - - - - - - - ( 3-5-7)
2DljJ
also found
to agree
theory an
added to
that in order to force the resonant
with the value obtained from the half-
additional mass
the real
value for
moment of inertia has to
the rocking
footing. It
follows that
- - - - - - - where the value of
TJtJ~
(3-5-8)
is as below:
~f---~------:------~------~-----~:~----~:~----~:~-1.110 1.143 1.219 1.251 1.378 1.600
~~ 1.079
If the footing is not circular, then
r 0
~ ~-
- - - - - - - - - - -(3-5-9)
29
3.6 Sliding Vibrations According to the Elastic Half-Space
Theory
(Ref. 31) considered the mass for the analog to
Hall
be
equal to the mass resting on the half-space to describe
the
slide motion
expression
of a
for the
rigid circular disk. The resulting
equivalent spring and damping are then
as follows:
32(1-11)
- - - - - - -(3-6-1)
7-8&1
and
18.4(1-V)
- -(3-6-2)
Cx = ----------7-8 v
The corresponding mass ratio (B
and the dimensionless
frequency factor are defined as follows.
m
7-8V
B
X
- - - - -(3-6-3)
= --------32(1-V)
and
ao = wro Jfi
- -(3-6-4)
Now the critical damping is given by
Cc
=
2~
, .......
Cc =
f.
- - - - - - - - - - - - -(3-6-5)
m32(1-11)
----------Gr 0
(7-8V)
c
=
so
t.
or
- - - - - - - -(3-6-6)
0.2875
=
- - -(3-6-7)
30
Fig. 3-6-1 shows the response curves for the
horizontal translation of the footing under a pure sliding
force.
- - Exact Solution
- - - Analog Solution
3
2'
~
..:
0
u
&
c
.f'
2
0I.)
;
·c:
C7'
0
~
1.5
0
ao
Fig. 3-6-1, Magnification factor vs. dimensionless
frequency relations for pure sliding of rigid circular
footing on elastic half-space. (Ref. 31)
If the footing is not circular, the equivalent radius
r
0
can be determined by
31
- - - - - - - - - -(3-6-8)
3.7 Coupled Rocking and Sliding of the Rigid circular
Footing on the Elastic Half-Space
Coupled motion of rocking and sliding most frequently
happens
rigid
in the
machine foundation.
circular footing
elastic half-space.
which rests
Fig.
3-7-1
shows
a
on the surface of the
32
T+
+'if
~··
+P
X
-rR+
(b)
to)
I
I
~
-,
I
I
I
I
I
I
I
~0
=
+
I
--1 I..- X
Ii
Q
(c)
xb= x 9 - h0 1jr
(d)
pl
=-c
X
dxb
-dl -k Xxb
dljr
R+ =- c+ dt- k+ljr
Fig. 3-7-1, Notation for
of vibration. (Ref. 5)
rocking and sliding mode
The translation of the base of the footing is
- - - - -(3-7-1)
The horizontal force on the base of the footing is
-(3-7-2)
The resistance of the half-space to rocking of the footing
•
R~p=- C'Plp- Ktplp
- - - - - - - - - - - - -(3-7-3)
33
The
equation of
motion for
horizontal translation of the
center of gravity of the footing is
••
•
mxg = Px = -cxxb - kxxb
- - - - - - - - - -(3-7-4)
Substituting Eq. 3-7-1 and rearranging terms, Eq. 3-7-4
becomes
mxg + cxxg + kxxg
hocxti'- hokx'P
=
0
- - -(3-7-5)
The equation of motion for rotation about the e.G. is
••
- -(3-7-6)
IgtfJ = T<p + Rtp- hOPx
in which
Ig
=
the moment of inertia of the footing about the
e.G.
Substituting R , Px and xb, transforms Eq. 3-7-6 to
••
2
•
2
•
Igf.JJ+ (etp + h 0 Ci t.p+ (ktp+h 0 kx)'P- h 0 cxxg- hokxxg
= T rp
- - - - - - - - - - - - - - - ( 3-7-7)
Substitution of the three equations
xg
=
=
Ttp
into
-(3-7-8)
Ax 1 Sinwt + Ax 2 eoswt
AX1 Sinwt + AX2 eoswt
unknowns.
--
-
--
-
-
-
- - - -
= Ttpo Sinwt
Eq. 3-7-5
-
-
-(3-7-9)
(3-7-10)
and 3-7-7, leads to four equations in four
The damping
frequency-dependent
and
the
spring
therefore,
'
values
of
are
the
coefficients
must be
It
concluded that coupling takes place only when
could be
the
above
calculated for
the
coefficients
vertical coordinate
of the
any given frequency.
e.G. of
the footing lies
the line of action of the horizontal force Px of the
half-space.
If h 0 in Fig. 3-7-1 is zero, there will be no
coupling presented.
CHAPTER
4
FOUNDATION DESIGN FOR VIBRATING MACHINE
4.1 General
In its complete sense the word foundation also includes
the
soil
underneath.
A
satisfactory
foundation
must
therefore meet the three requirements below:
a. It must be placed at an adequate depth to prevent frost
damage, undermining by scour, or damage from future
construction nearby.
b. It must be safe against breaking into the ground.
c. It must not settle enough to disfigure or damage the
structure.
For machine foundation, in addition to static loads due
to the
weight of
acting on
such
general, a
foundations
and the
are
foundation weighs
machine. Also
parts of
the machine
a dynamic
a machine
static load.
dynamic
load associated
loads
nature.
as much
with the
In
as a
moving
small as compared to its
foundation,
34
in
several times
is generally
In machine
foundation,
a
dynamic
load
is
35
applied repetitively
its magnitude
over a
is small,
very long
period of time but
and it is necessary that the soil
behavior be elastic, or else deformation will increase with
each cycle
of loading
unacceptable. The
amplitude of
operating frequency
determined in
to
until the
is the
designing a
determining
the
soil becomes practically
motion of a machine at its
most important parameter to be
machine foundation, in addition
natural
frequency
of
a
machine
foundation soil system.
4.2 Degrees of Freedom of A Block Foundation
Under the action of unbalanced forces, the rigid block
undergoes displacements and oscillation as follows (Fig. 42-1):
a. Translation along
z
axis
b. Translation along X axis
c. Translation along y axis
d. Rotation about
z
axis
e. Rotation about X axis
f. Rotation about y axis
Hence, the rigid block has six degrees of freedom and thus
six natural frequencies.
36
Vertical
Fig. 4-2-1. Degrees of freedom of a block
foundation. (Ref. 22)
4.3 Vertical Vibrations of A Block
From Fig. 4-3-1 and Newton's Second Law, we get
..
mz + kz•
=
- - -
Po sinwt
- - - -
-
- - - - -
-
-(4-3-1)
m
=
the mass of the machine and the foundation,
k
=
C m is the equivalent spring constant of the soil in
vertical direction for a base area A of the foundation,
Cu = coefficient of elastic uniform compression.
~ n = J-f.u!L
m
The amplitude of motion X is given by
Xz =
Po sinwft
-----~-----~--
m(wn
- - - - - - - - -
- Wf )
Maximum amplitude of motion X is given by
~
- -(4-3-2)
37
X
z =
Po
- - - - - - - - - - - -(4-3-3)
-----~-----~--
m(Wn
-Wf )
tP
0
t
sinwt
P0 sin wt
m
D,
(a)
Area A
HXN\,
I
m
I
(b)
m
k,
(c)
(d)
Fig. 4-3-1, Block foundations under vertical
vibrations(a) Block resting at depth D . (b) Block
resting on surface of ground. (c) Soil replaced by
equivalent spring K . (d) Equivalent spring-mass
system for analysis.
4.4 Sliding Vibrations of A Block
In practice, rocking and sliding occur simultaneously.
But if the vibrations in the rocking can be neglected, then
only horizontal
displacement of
the foundation would take
place under an exciting force P 0 sinwft on the block of area
A.
~
.
38
.. Po S;o Ulft.,..
Fig. 4-4-1, Block foundation under sliding
vibration.
The corresponding equation of motion is this
~:: f-~: P:s}:~!~- -
- -(4-4-1)
-(4-4-2)
with maximum
Px
X
=------~-----~--
m(u> n
-w f
- - - - - - - -(4-4-3)
)
4.5 Rocking Vibration of A Block
From Fig. 4-5-1, the moment due to the soil reduction
can be obtained as
= -JAdRdAl = -clrf - - - - - - - - - - -(4-5-1)
dR = soil reaction acting over small area dA,
M
1
= distance of dA from center of rotation,
= angular displacement of block,
where
@ •
39
=
I
moment of inertia of contact area about an axis
passing
through the centroid of the base contact area.
dR=
assume
Mw
My
- - - - - - - - - -(4-5-2)
=
••
Mmo<f
M0 sinWft
- - - - -(4-5-3a)
= 'l.M
- - - - - - -(4-5-3b)
where
=
Mmo
mass moment of inertia of the machine and foundation
about the axis of rotation,
..
1
= angular acceleration of the block,
W
= weight of the block foundation,
L
=
distance from contact area to the
e.G.
therefore
M0 sin ft - C0 Io + WLo
= Mmof
••
- - - - - -(4-5-4)
Mmo1 +1(C1I - WL)
Wn~~
xp
- - - - - - - - - - - - -(4-5-5)
Mo
-------2-----2--
=
Mmo<Wn
or
- - - - - - - - - -(4-5-6)
- Wf )
Because C'I is many times WL, hence Eq. 4-5-5 can be
rewritten
J-~~~-~-
Wn =
f nT
m = --'-21T
where
K~
-(4-5-7a)
-~--~Mmo
=
CPI
and I
=
-~~12
-(4-5-7b)
so
1
211
- - - - - - - - -(4-5-8)
12
40
The amplitude of the vertical motion of the edge of the
footing is
A
=
a
2
X
A<f'
- - - - -(4-5-9)
Fig. 4-5-1, Block foundation under rocking vibration.
4.6 Trial Sizing of A Block Foundation
The following guidelines for initial trial sizing have
been found to result in acceptable configurations:
41
1). The bottom of the block foundation should be above the
water table when possible. It should not be resting on
previously backfilled soil nor on a specially sensitive
(to vibration) soil.
2). The following items are recommended by Arya, O'Neill,
and Pincus (Ref. 9) which can be applied to block-type
foundations resting on soils:
a. A rigid block type foundation resting on soil should
have a mass of two to three times the mass of the
supported machine for centrifugal machines. However,
when the machine is reciprocating, the mass of the
foundation should be three to five times the mass of
the machine.
b. The top of the block is usually kept 1 ft above the
finished floor or pavement elevation to prevent damage
from surface water runoff.
c. The vertical thickness of the block should not be less
than 2 ft, or as dictated by the length of anchor
bolts used. The vertical thickness may also be
governed by the other dimensions of the block in order
that the foundation be considered rigid. The thickness
is seldom less than one fifth the least dimension or
one tenth the largest dimension.
42
d. The foundation should be wide to increase damping in
the rocking mode. The width should be at least 1 to
1.5 times the vertical distance from the base to the
machine centerline.
e. Once the thickness and width have been selected, the
length is determined according to (a) above, provided
that sufficient plan area is available to support the
machine plus 1-ft clearance from the edge of the
machine base to the edge of the block for maintenance
purposes.
f. The length and width of the foundation are adjusted so
that the center of gravity of the machine plus
equipment coincides with the center of gravity of the
foundation. The combined center of gravity should
coincide with the center of resistance of the soil.
g. For large reciprocating machines, it may be desirable
to increase the embedded depth in soil such that 50 to
80% of the depth is soil embedded. This will increase
the lateral restraint and the damping ratios for all
modes of vibration.
43
h. Should the dynamic analysis predict resonance with the
acting frequency, the mass of the foundation is
increased or decreased, so that, generally, the
modified structure is overtuned or undertuned for
reciprocating and centrifugal machines, respectively.
4.7 Design Procedure
All the
machinery must
foundations designed
to
support
vibrating
be considered under both static and dynamic
loading condition.
The static loading includes:
a. Static bearing capacity: proportional to the footing
area for 50% of the allowable soil pressure.
b. Static settlement: it must be uniform, the
e.G.
of
footing and machine loads should be within 5% of each
linear dimension.
The dynamic loading design conditions include:
1. The calculation of vibration amplitude at operational
frequency:
The maximum single amplitude of motion of the foundation
system as calculated from Table 4-7-1 should lie in zone
A or B of Fig. 4-7-1 for a given acting frequency where
unbalanced forces are caused by machines operating at
44
different frequencies, the total displacement amplitudes
to be compared at the lower acting frequency, are taken
as the sum of all displacement amplitude.
Table 4-7-1, Summary of Derived Expressions for a
Single-Degree-of-Freedom System. (Ref. 9)
Expression
Constant Force Excitation
F 0 Constant
Rotatinl1, Mass-type Excitation,
Fo = m;e w 2
1
Magnification factor
M = --::;:;:::::;;:::::;::;:::;
.J(l-r 2 ) 2
(2 Dr)'
+
Amplitude at frequency f
Y
=
i\,[ (Fo/k)
= -:;:;==;;:;;;:=;:::;;:;;:::;:
.J(l-r2)2
.Jl-2D2
(m;e/m)
(Fo/k)
Ymu
=----
Ymal:
2D.Jl- D 2
Transmissibility factor
+ (2Dr)•
Y = M, (m;e/m)
/.
fmr=---
Resonant frequency
Amplitude at resonant frequency f,
:Mr
+ (2 Dr) 2
-::;::=:::::;~:::;::::;:=;;
.J(l-r 2 )2 + ('2 Dr) 2
.J 1
Tr =
where r = w/w.
"'• (Undamped natural circular frequency) = .J(k/m)
D (Damping ratio) = C/C.
C. (Critical Damping)= '2.J km
T, = Force transmitted/ Fo
f; = Force transmitted/m; ew 2 •
=---2D.J~
+
_
.Jl
(2Dr) 2
T, = --::;:;:::::;:;::::::::;::;::;
.;(1-r•)•
(2Dr)2
r2
+
Q •
45
\
\~
~
E
\\..
\"9,
\
~
\-;,
\
1000
10,000
Frequency, cpm
EXPLANATION
OF
E
DANGEROUS.
D
FAILURE
TO
C
IS
AVOID
FAULTY.
CASES
SHUT
MINOR
A
NO
NOW
NEAR. CORRECT
TO
WITHIN
AVOID
TWO
DANGER.
DAYS
BREAKDOWN.
CORRECT
MAINTENANCE
8
IT DOWN
[0 DAYS
TO
SAVE
DOLLARS .
FAULTS.
FAULTS •
WITHIN
CORRECTION
TYPICAL
NEW
WASTES
DOLLARS.
EQUIPMENT.
Fig. 4-7-1, Vibration performance of rotating machines.
(Ref.
9)
46
2. Velocity:
Velocity of machine is equal to
2~(cps)
x (Displacement
Amplitude as calculated in the condition above). This
velocity should be compared to the limit values
indicated by Table 4-7-2 and Fig. 4-7-1 at least for the
case of "good" conditions. The resultant velocity where
two machines operate at different frequencies is
calculated by the RMS (root mean square) method.
where
V =resultant velocity, in/sec.,
wl,w2 = operating frequencies for machines 1
&
2
respectively, rad/sec.
A1 ,A 2 =vibration displacement, in., for machine 1 & 2
respectively.
Table 4-7-2, General Machinery-Vibration-Severity
Data. (Ref. 9)
Horizontal Peak Velocity
(in./sec.)
<0.005
0.005-0.010
O.OHHJ.020
0.02D-D.040
0.04D-D.080
O.OSD-D.160
0.16D-D.315
0.315-0.630
>0.630
:Machine Operation
Extremely smooth
Very smooth
Smooth
Very good
Good
Fair
Slightly rough
Rough
Very rough
47
3. Acceleration:
The acceleration
of the
displacement amplitude
is only
necessary if
satisfied. The
machine is
equal to
4Uf 2
x
as determined before. This check
two previous
conditions are
acceleration should
not
fall in zone "A" or
"B" of Fig. 4-7-1.
4. Magnification Factors:
M, represents
mass at
the ratio of the dynamic amplitude of the
a given
frequency to the deflection that would
be obtained
if the
static load.
For a
given excitation
which has
a constant
amplitude F 0
frequency
, the expression is
M
A
tV n
=
A
-----Fojk
=
dynamic force
were amplified
as a
force F = F 0 sinwt
and
acts
with
a
1
-----------~-~-------------~~
~[1-(W/Wn) ]
where
+ [2D(W/Wn)]
= dynamic amplitude,
= natural frequency,
D = damping ratio (combined total of Table 4-7-3 and
4-7-4.
k
= spring constant (Table 4-7-5)
Note: f/fn
For a
= / n
dynamic system
in which
the excitation force is
created by a rotating member with unbalanced masses, the
amplitude of
the force
and is given by:
F
. ..~
= mee 2 Slnw~
is a
function of the frequency
48
In this
expression, meew 2 is
the unbalanced
of this
the centrifugal force of
mass me' and mee~sin t is the component
force in
the
magnification factor
direction
for this
of
the
motion.
type of excitation force
is given by:
where
JUe
=
jU =
free amplitude,
me/m
= ratio a unbalanced mass to total mass,
e = eccentricity of unbalanced mass to the axis of
rotation,
M = dynamic magnification factor for the constant
-force-amplitude case.
Table 4-7-3, Equivalent Damping Ratio for Rigid
Circular and Rectangular Footings. (Ref. 5)
~lode of
Yibr:uion
~lass
(or Inertia) Ratio
II"
B,=--
Dampin!). Ratio D
(1-v)
\·ertic:d
1ro3
4
Horizont31
7-8v
IV
B,=--32{1-v) yro 3
0.425
D~
=-a,
-iF..
0.288
D~
=-a,
../B.
3(1-v) I~
Rocking
B.;=--8
pro 5
Torsional
Bo=-
0.50
Do=--
pro5
1+2Bo
Io
The
49
Table 4-7-4, Typical Values of Internal Damping in
Soils. (Ref. 5)
Equivalent
D
Type Soil
Dry sand and gravel
Dry and saturated
sand
Dry sand
Dry and saturated
sands and gravels
Cay
Silty sand
Dry sand
0.03-{1.07
0.01-0.03
0.03
0.0~.06
0.02-0.05
0.03-0.10
0.01~.03
Table 4-7-5, Equivalent Spring Constant for Rigid
Circular and Rectangular Footings. (Ref. 39)
Mode of
Vibration
Circular
Rectan~tular
Footin~t
G
4Gro
Vertical
k.
Footing
= --'1·
k. = -{3. ,; BL11.
1-v
1-~
32(1-v)Gro
Horizontal
k, =
7-8v
k. = 2(l+v)G{3, vBL11.
'I•
G
8Gr 0 3
ky=--1].
3(1-v)
Rocking
k.:.
1-·
16 Gro 3
Torsional
= -{3yBL2'1~
No solution available;
ke=-3
5. Resonance Frequency:
Resonance frequency
fm is
used when a forcing function
of a sinusoidal nature is acting on a damping system. It
is
defined
as
the
amplitude occurs,
frequency of
amplitude,
at
which
the
maximum
and is less than the undamped natural
the
and
frequency
foundation
greater
for
the
for
constant-force-
rotating
mass
type
50
excitation. The
forces and
expression for
the relative
fm for
these types
of
value of dynamic magnification
are as follows:
Dynamic Force
Resonance Frequency Dynamic Magnification
Factor
fm
= fnJ1 - 2D 2
fmr= fn/J1 - 2D
The acting
least
a
frequencies of
difference
of
the machine
+ 20%
should have
with
the
at
resonance
frequencies, that is f < o.8fm or, f > 1.2fm.
6. Transmissibility Factor, TR:
Machines are
often mounted
transmission of
transmitted to
forces to
on springs
the
the foundation
to minimize the
foundation.
The
force
is the sum of the spring
forces and the damping force. The ratio of the amplitude
of the
force transmitted to that of the impressed force
is called
the transmissibility
factor and is given for
sinusoidal loads as follows:
Force Type
Transmissibility Factor
TR
=
TR
j
=
1 + (2Dr) 2
---------~-~--------,-----
~1 - r )
+ (2Dr)
51
in which, P0 is the force transmitted through the
springs and r =W/Wn· These value of transmissibility to
be obtained should be less than 3%.
7. Resonance of Individual Structure Components (SuperStructure without the Footing):
The
resonance
frequency can
condition
be avoided
with
the
lowest
by maintaining
natural
the frequency
ratio either less than 0.5 or greater than 1.5.
All modes
of vibrations
resonance with
the
should be checked for possible
operating
frequency
and
critical
speeds of the machine.
(a). Vertical Oscillation: This mode is possible if the
force acts in this direction.
(b). Horizontal Oscillation: This mode is possible if
the force acts in this direction.
(c). Rocking Oscillation: This mode is possible when the
point of application of horizontal force is above
mass center of foundation.
52
t
F( I)
X
• !-1 (t)
(A) Vertical Excitation
(B) Horizontal Translation
----.--,:fj?_j;J
-~> Rockl~~
(D) Torsional Excitation
Excitation
TRANSLATION
--..........
-~L
I
_
r
I
. i__fl
,
--) I.,. __
_, ___
'I
,....
ROCKING
'
~
~~-{--
'- .:...1'rr-..J~
.I
(E) Coupled Horizontal Translation & Rocking
Oscillation
Fig .. 4-7-2, Modes of oscillation. CRe-r:. 9)
i
53
(d). Torsional Oscillation: This mode is possible when
the horizontal forces form a couple in the
horizontal plane.
(e). Coupled Mode: The horizontal translation and
rockingoscillation are usually coupled. But if
It nx 2 + f n~2 / f nx f n~ ~~ 2/3 f, then the coupling
effect may beignored.
N
The dynamic design of the machine foundation must also take
into account any possible fatigue failure of the machine
components, the connections and the supporting structure.
1. Machine Components:
Limits stated
be
in Fig.
followed.
delicate, then
with
an
added
In
4-7-3 and for Table 4-7-2 are to
case
machine
components
are
very
the machine should be mounted on springs
inertia
block
which
can
limit
the
vibration amplitude at the foundation base to much lower
values.
54
<t
1
0.1
z
<t
-
0.01
w
0
:::>
....
0.005
::::i
ll.
:IE
<t
0.002
!Zw
0.001
w
u
<t
0.0005
:IE
..J
a.
Ill
i5
0. 0 0 01 ....___,____,__,_..J....I...w.J.L.>....-.1..-~.L..J..~.....
100 200
500 1000 2000 5000 10,000
FREQUENCY
I
+
FROM
REIHER
o
FROM
RAUSH (1943}-(STEADY STATE
/::,. FROM
AND
CRANDELL
MEISTER
(1949}
CPM
( 1931}- (STEADY
(DUE TO
STATE
VIBRATIONS}
VIBRATIONS}
BLASTING}
Fig. 4-7-3, General limits of vibration amplitude
for a particular frequency. (Ref. 5)
2. Connections:
These should
machine
be given
component
the same
coitions
stresses using the AISC Code.
considerations as
above
and
checked
the
for
55
3. Supporting Structure:
For steel
structures, the
connections connection above
should be observed.
For concrete
place and
footing, if
reversal
of
stresses
takes
the amplitude is very high such that the peak
stress reversal is over 50% of the allowable stress, the
main and
the shear
reinforcement (if
any)
should
be
designed for the stress reversal condition.
There are
environmental demands
structure and
these may
placed
on
the
include physiological
over-all
effects on
persons. Possible undesirable effects on adjacent sensitive
equipment, damage
to the
individual structural
structure or
members with
resonance
the machine
of
the
frequency
should be checked.
a. Physiological Effects on Persons:
If the machine is inside a building 1 the procedure given
in the
transmissibility factor conditions above and the
limits indicated in Fig. 4-7-2 through Fig. 4-7-6 should
be adopted
. The
supporting
structure
amplitude of
concept of
is
vibration in
physical isolation of the
another
any
alternative.
direction
should
The
fall
below the zone "troublesome to persons" for the specific
acting frequency as determined from Fig. 4-7-3.
56
b. Psychological Effects on persons:
The above
be used when the machine is
procedure should
located close
to
people
not
connected
with
machine
operations, an acoustical barrier may be necessary.
c. Resonance of Structure Components (Superstructure above
the Footing):
Resonance could
be
avoided
with
the
lowest
natural
frequency by keeping the ratio of operating frequency to
natural frequency less than 0.5 or greater than 1.5.
d. Damage to Structure:
The limits indicated in Fig. 4-7-3 and 4-7-4 should be
used to avoid structural damage.
e. Sensitive Equipment Nearby:
The support system should be physically isolated from
the sensitive equipment.
57
Frequency, cps
Fig. 4-7-4, Response spectra for allowable vibration
at facility. (Ref. 9)
Fig. 4-7-5, Vibration standards of high-speed machines.
(Ref. 9)
58
0
0
0
..
0 s~~~~w~~~~_.~~~~s~o~._,~~~~~~~
CPS
Fig. 4-7-6, Turbomachinery bearing vibration limits.
(Ref.
5)
CHAPTER 5
DETERMINATION OF APPROPRIATE
SOIL PROPERTIES FOR DESIGN
5.1 Basic Considerations
In general
the amplitude
of vibration of the machine
foundation endangers the adjoining structures. Amplitudes of
vibration of
the foundation may be large enough to make the
foundation loose
its stability
nonuniform settlement
due to the possibility of a
which may prohibit the normal work of
the machine and thus create a dangerous vibration of machine
connections.
5.2 Dynamic Subgrade Reaction
Elastic Subgrade
the deflection
replacing the
that
produce
Reaction: This
of a
is a method for estimating
loaded structure
resting on
soils by
soil with a set of independent elastic spring
an
equivalent
reactive
force
displacement developed as shown in Fig. 5-2-1.
59
to
the
60
0
Fig. 5-2-1, Springs replacing soil support
to provide "dynamic subgrade reaction".
The theory gives useful results only for the undamped
natural frequency of vibration. The coefficient of subgrade
reaction (lb
=
K
I
ft) is
-(5-2-1)
K'A
in which
K'= pressure per unit displacement, lb
2
A = foundation contact area. ft
I
3
ft .
In 1955, Terzaghi (Ref. 33) suggested a procedure to be
used when both the model footing and the prototype footing
produce equivalent stresses in similar soils.
For cohesive soils:
K'
For cohesiveless soils:
z = K' zl ---K'
z=
-(5-2-2)
2d+l 2
K'z1(--4ct--) - - - -(5-3-3)
in which
2d
K'
=
width of a beam, or least dimension of foundation base,
z =coefficient of vertical subgrade reaction for base of
least dimension of 2d (lb
I
ft ),
61
K'
z1
=
coefficient of vertical subgrade reaction for base of
J
least dimension of 1 f t (lb
Barken (Ref.
various of
6) developed
ft ).
the spring
constants
for
modes of vibration of rigid foundations based on
the concept of elastic subgrade reaction.
For vertical motion
Kz
For horizontal motion
K
For rocking motion
K(/J
For torsional motion
Ke
X
=
=
K' z A
K'
X
A
= K''I'A
= K' eA
----
- -
-(5-2-4)
in which
I' =the second moment of contact area about a horizontal
axis normal to the plane of rocking passing through
the centroid,
I" == the second moment of contact area about a vertical
axis passing through the centroid.
The coefficients of subgrade reaction are as follows:
K'x ~ 0.5 K'z
K'o/ ~ 2.0 K'z
K'e ;:::::.1.5 K'z
1
- - - - - - - - - - - ~-- -(5-2-5)
The value of K' z is shown in Table 5-2-1 for different kinds
of soils:
62
Table 5-2-1, Recommended Design Value for
Subgrade Coefficient K . (Ref. 6)
Soil Group
Allowable Static
Bearing Stress
(ton/ft•)
Coefficient k;
(ton/ft")
1.5
95
1.5-3.5
95-155
Weak soils (clay and silty clays with sand,
in a plastic state; clayey and silty sands)
· Soils of medium strength (clays and silty
clays with sand, close to the plastic
limit; sand)
Strong soils (clay and silty clays with
sand, of hard consistency; gravels and
gravelly sands, loess and loessial soils)
Rocks
155-310
310
3.5-5
5
5.3 Lumped-Parameter Vibrating Systems
In this system, the foundation is represented by a mass,
spring, and dashpot. These three parameters are then used to
represent the motion of rigid foundations.
Fig. 5-3-1 illustrates typical equivalent lumped systems
for
foundations
torsional exciting
chapters, the
motions with
to
forces. As
vertical and
a single
excitation produces
and sliding.
subjected
vertical,
was
mentioned
torsional
degree
of
horizontal
in
previous
excitations
freedom
but
and
produce
horizontal
a coupled motion involving both rocking
63
EQUIVALENT SYSTEM
Rigid Block having
Equivalent Moss
ACTUAL FOUNDATION
+
Vertical
fExcitotion
~
Equivalent
Damping
~Equivalent
Spring
"
.o
Rigid Block having Equivalent
Mass and Moment of Inertia
about Horizonal Axis
Equivalent
Equivalent
Horizontal Spring ..---'----,Horizontal Damping
-
Horizontal
Excitation
..;=1:;\:?:~~y;-r~J~~
~---,.-,--,-+-H
Equivalent
Rototionar Spring
Torsional
E>d~~~g~
Equivalent
Rotational Damping
Rigid Block having Equivalent
Moment of Inertia about
Vertical Axis
Fig. 5-3-1, Typical equivalent lumped systems.
The equation of motion for a single-degree-of-freedom with
lumped parameters is
mz
+
cz
+ kz
=
Q(t)
in which
m
= equivalent mass,
c
= effective damping constant,
k
= effective spring constant,
• ••
z,z,z
= displacement, velocity, and acceleration of the
mass in the direction of the chosen coordinate,
Q(t) = time dependent exciting force or the forcing
function.
64
a. The lumped mass is chosen as the mass of the foundation
and supported machinery.
b. There are two types of damping in the real system
1). Use the loss of energy through propagation of
elastic waves away from the immediate vicinity
of the
footing.
2). Use the internal energy losses within the soil due
to hysteric and viscous effect.
The
lumped
damping
foundation-soil system
damping described
type of
soil to
from Table
for
any
particular
will include both the effects of the
above. If we take value from a particular
represent a typical internal-damping ratio
5-3-1, then
the geometrical
parameter
damping
this value
from
Fig.
should be compared with
5-3-2
to
obtain
contributions of both.
Table 5-3-1, Some Typical Values of Internal Damping
in soils. (Ref. 5)
Type Soil
Dry sand and gravel
Dry and saturated
sand
Dry sand'
Dry and saturated
sands and gravels
Oay
Silty sand
Dry sand
Equivalent
D
O.Q3-0.Q7
0.01-0.03
0.03 .
0.05-0.06
0.02-0.05
0.03-0.10
0.01-0.03
the
65
1.0.------.-----,.------.----~.------.----~
0
Fig. 5-3-2, Equivalent damping ratio for oscillation
of rigid circular footing on the elastic half-space.
(Ref. 5)
c. The spring constant is a critical parameter in the
dynamic analysis for lumped-parameter system. It can be
obtained through the theory of elasticity for circular
and rectangular footing resting on the surface of the
elastic half-space. Table 5-3-2 and 5-3-3 (Ref. 5) show
the spring constants for circular and rectangular
footing respectively.
66
Table 5-3-2, Spring Constants for Rigid Circular
Footing on Elastic Half-Space. (Ref. 5)
'
Motion
Vertical
Horizontal
Rocking
Torsion
(Note: G = l(I
Spring Constant
4Gr.
k.=-1- v
32(1- v)Gr.
k.,=
7- 8v
8Gr3
k
'I'
3(1 - v)
ka
¥Gr!
=--·=
~ v))
Table 5-3-3, Spring Constants for Rigid Rectangular
Footing Resting on Elastic Half-Space (Ref. 5)
Motion
Spring Constant
G
..,;= --P.
4cd
1- v
Vertical
k.
Horizontal
= 4(1 + v)Gp., V cd
G
k = --P Sed•
op
1- v op
Rocking
k.,
The value for ~z' ~x and ~~are given in Fig. 5-3-4.
67
3
-
1.5
p; .
I
l--2d--J
"''i
.
co.
...
0
.....
0
01
.!:0
Ill·-"'
·;;; u
T
1.0
u
N
c:r&
~·
j_
co."
co.
die
Fig. 5-3-3, Coefficient
footings. (Ref. 39)
p ,f3 ,
x
z
and~(/' for rectangular
The effect of spring constants is increased because the
soil
resistance
increase with
vertical spring
to
the
depth. Fig.
motion
of
the
5-3-4 shows
constants for
foundation
the change
in
will
the
circular footings along with
the increasing of the depth of embedment.
68
Depth to Radius Ratio, H/r 0
Fig. 5-3-4, Effect of depth of embedment on the
spring constant for vertically loaded circular
footing. (Ref. 40)
5.4 Methods for Decreasing Vibrations of Existing
Foundations
a. Counterbalancing of existing loads imposed by machines:
Two methods have been utilized before, they are as
follows:
1. Counterbalancing completely a component in the
direction perpendicular to piston motion and partly a
component in the direction of piston motion.
,, .
69
2. The dimensions of counterweights and their distances
from the axis of rotation may be selected to
counterbalance completely the first harmonic of the
component exciting forces in the direction of the
piston motion. In this case, the component in the
perpendicular direction will increase. This method
usually is applied in a horizontal reciprocating
machine because it is accompanied by rocking and
sliding vibrations.
b. Chemical stabilization of soils:
Chemical or cement stabilization of the soil under the
foundation may be used to decrease the vibration of
structure. This is because it will result in an increase
in the rigidity of the base and cause an increase in the
natural frequencies of the foundation. Consequently, the
difference between the frequency of natural vibrations
and the operating frequency of the machine becomes large
enough to avoid resonance.
c. Structural measure:
This method is achieved by increasing the foundation
contact area as well as by increasing the rigidity of
its base. Sometimes we can increase the foundation mass
without inducing changes in the frequency of foundation
vibrations which will result a decrease in th amplitudes
70
of vertical vibrations. This method may be very costly
as
it entails a temporary discontinuity of operations.
CHAPTER
6
DESIGN EXAMPLES FOR MACHINE FOUNDATION
Foundation design
for two
types of
machineries
Reciprocating,(b) Rotary
machines
are
chapter. These
use
theory
examples
the
presented
and
in
(a)
this
information
developed in previous chapters. The selected foundations are
typical and
commonly used
in many
industrial plants.
The
examples include a series of steps so that a thorough design
is accomplished without missing any check.
6.1 Foundation Design for Reciprocating Compressors
Reciprocating compressors
heavy machines.
Even
at
as shown
low
operating
in Fig. 6-1-1 are
frequencies
they
generate high vibrating forces. The operating frequencies of
these
machines
are
generally
frequencies of
the
foundation
modes. Because
the
magnitude
close
in
of
resonance condition
become a
should be
the favorable
taken of
the
the
various
vibration
natural
vibrating
amplitude
at
critical criterion. Advantage
effects of
geometrical damping during oscillation.
71
to
internal and
72
IJ'.l£A OER CYLI ~ OER
INSPECTION PLATE
Fw\DATION
OIL GAUGE
Fig. 6-1-1, Cutaway view of a typical reciprocating
air compressor.
Foundation blocks for reciprocating compressors have in
the
past
been
using
the
concept
of
dynamic
subgrade
reaction.
Concept of Dynamic Subgrade Reaction:
Theory:
The loaded
on a
structure resting on the soil is assumed to rest
set of
independent
elastic
springs.
These
springs
provide an equivalent soil reactive force in response to the
displacement of the foundation.
Disadvantages:
a. This theory disregards damping effects on the response of
the vibrating
system, and
provide useful
results on
resonate frequencies.
as a
consequence, it
the amplitude
does
not
of motion at near
73
b. The
evaluation of
soil requires
the dynamic
subgrade reaction of the
testing of a modal footing to obtain reliable
results. This is generally expensive and time consuming.
Elastic Half-Space Theory:
Theory:
In this
theory, the
footing is
assumed
to
rest
on
the
surface of the elastic half-space and have geometrical areas
of contact such as a circle or a rectangular.
Advantages:
a.
This
theory
permits
"geometrical damping"
the
dissipation
and yields
of
energy
calculations of
by
a finite
amplitude of vibration at the "resonance frequency".
b. The
method is an analytical procedure for evaluating the
spring and
damping constants
to be
used
in
the
lumped-
parameter, mass-spring-dashpot vibrating systems.
Definition of Reciprocating Machines:
Machinery
such
as
piston-type
compressor,
pumps
internal-combustion engines produce reciprocating forces.
and
74
Design Criteria:
a. Vibration modes:
A rigid
block foundation supporting a vibrating machine can
exhibit as
many as
six modes of vibration as shown in Fig.
6-1-2.
_, .·"
' 'C"£'':
~
lh
<(
M> [>WOSTONG OR
~
A:ONG)
Fig. 6-1-2, Six vibration modes of a block-type foundation
(Translation
modes:
vertical,
longitudinal,
lateral.
Rotational modes: twisting, rocking, pitching).
The figure
shows three
lateral 3)
longitudinal
twisting (yawing)
vertical
and
independent.
translatory modes:
and
2) rocking
twisting
three
3)
rotational
pitching
vibration
1) vertical
modes
modes:
(rolling).
are
2)
1)
The
usually
Q
'
Q '
75
b. Failure criteria: As we mentioned before, the foundation
block is investigated for:
1). Static and dynamic pressure intensity on the soil.
2). Static and dynamic settlement of the footing.
3). Vibration amplitude of displacement, velocity and
acceleration for independent and coupled modes at the
foundation level.
4). Resonance conditions and the resulting magnification
of the static deflections and static reactions.
5). Effect of vibration on machine components connecting
piping and anchorage system.
c. Environmental Conditions:
An isolation system should be included in the vicinity of
the foundation to prevent transmission of vibrations to
nearby systems.
Design Procedure:
Formulation of the mathematical model:
Design of a block type foundation subjected to dynamic
loads is initiated by developing a mathematical model of
the real life foundation.
76
6.2 Example: Design of Foundation Block for a Reciprocating
Compressor.
A. Machine Parameters:
The following information is provided by the machine
specification
(XLE 2MC from Ingersoll-Rand).
Compressor weight:
27,700 lbs
Motor weight:
17,000 lbs
Gas coolers and others:
9,500 lbs
Total machine weight:
54,200 lbs
Compressor speed:
Primary:
595 RPM
secondary:
1,200 RPM
Primary:
1,900 lbs
FzSecondary:
800 lbs
Horizontal force:
Fx
Primary:
940 lbs
Rocking:
T~
Primary: 12,109 lb-ft
Pitching:
T~
Primary: 35,300 lb-ft
Vertical force:
Fz
T secondary: 13,450 lb-ft
1
Motor speed:
595 RPM
77
B. Soil Parameters:
Soil is Medium Dense Sand
Soil density:
= 120 pcf
Shear modulus: G
=
Poisson's ratio:
V=
15,000 psi
0.4
Soil internal damping ratio
D~;
= 0.05
Static allowable bearing capacity Sa = 3.5 ksf
Permanent Settlement of
soil
= o.JS Jn at .3.5 KSf
C. Selection of a Foundation Configuration:
1). The bottom of the block foundation should be above
the water table whenever possible.
2). The top of the block foundation is usually kept 1 f t
above the finished floor or pavement elevation tO
prevent damage from surface water runoff.
3). At least 80% of the footing thickness should be in
the soil to restrain the translation movement of the
footing.
4). Footing trial outline ( see Fig. 6-2-1)
78
•
,----~---'-·:-t'\1
~·
·---~----~
---·-.- - +
-------
I
- Ln II
~
- - - - - - - - - AXIS -OF ROCKING
-- -- OSCILLION
~ID
I
-('\1
---~--'-'=~--r-:.:..p;.,.._..::L-.....lo.L-
---~~
,0
II
~-J-----.,,L--__........,___.....__
=. . .•. .•.
:~;1St PllCHING
m2
z
=12.11B ·
(l 2 =28' ABOUT ~
b 2::: s.o'
- - ----- OSCILLATION, l.f'
K2
=2.5'
a.z·=t6'ABOUT
Fig.
6-2-1
I .
79
5). The with should be at least 1 to 1.5 times the
vertical distance from the base to the machine center
to increase damping in the rocking mode.
=
So selected thickness
5 ft
width = 16 ft
6). Weight of footing WF
footing weight
=
=
3
(28 x 16 x 5) 150 lb/ft
336,000 lb
336,000
----------------- = --------- =
machine weight
54,200
6.2 > 5.0
Total static load (W) = Machine wt + Footing wt
= 54,200 + 336,000
= 390,200 lbs
7). Actual soil pressure
w
390,000
= ----------- =
(28) (16)
A
871 psf < (0.5 Sa
=
1750 psf)
The size of footing is good.
D. Dynamic Analysis:
Force generated
result in
pitching
by the
(1) vertical
oscillation
generated by
footing in
the motor
the (a)
reciprocating compressor will
(2) lateral
of
the
(3)
footing,
only will
tend
rocking
which
to
the
and
(4)
forces
oscillate
the
vertical (b) longitudinal (c) torsional
and (d) pitching modes. However, the magnitude of the forces
generated by the motor are small and the resulting vibration
80
response is
negligible. Therefore,
a dynamic
analysis
is
only performed for the compressor forces in this example.
1). Mass and mass moment of inertia:
a). Vertical excitation Mz = Ji..= 390,200 I 32.2 = 12,118
9
2
lb sec 1 ft.
b). Horizontal excitation Mx
=
390,200
I
2
1
lb sec
c). Rocking excitation
m
m;
I
12
=~[---
I~=
!~(machine)
32.2
=
12,118
ft
+!~(footing)
( a 2 • + b2 .I) + m2 i k2i 1
I
336,000
Ilf'(footing) =
32.2
2
2
28 + s
5 2
(----------) + ( ---)
12
2
= 768,696 lbs.sec~ft
54,200
(5 + 2.5) 2 =94,682 lbs.set-ft
32.2
I~=
!~(footing)
+
!~(machine)
2
= 863,378 lbs.sec-ft
,,
81
I
2
SLENDER ROO
Iy= Iz =12 mt
THIN RECTANGULAR
PLATE
I y-j2m
- I 02
2
I--..Lmb
z 12
RECTANGULAR PRISM
THIN DISK
-Iz-.!.4 mr 2
I y-
Ix=-kma 2
CIRCULAR
CYLINDER
Iy=Iz=~m(3o 2+~)
l;t=·Iz+ m~/4
CIRCULAR CONE
SPHERE
Table 6-2-1, Mass Moment of Inertia of
Common Geometric Shapes. (Ref. 9)
'
82
d). Pitching excitation:
I
I
I
2
2
336,000
16 + s
5 2
(footing) - --------- [(---------) + (---) ]
12
2
32.2
=309,565 lbs.sec 2 -ft
54,200
2
2
2
(machine) =--------(5 + 2.5 ) = 52,601 lbs.sec -ft
32.2
= 309,565 + 52,601 = 362,166 lbs.sec 2 -ft.
2. Spring Constant:
a). Equivalent radius (rectangular footing), r :
=
Considering embedment height h
Table 6-2-2).
J-~J-
=
i). r
L
=
ii). r
J-~~~~-
= 11.94 ft
=A-~~-=
=
~~-
13.89 ft
= 10.50 ft
where L = 16, B = 28.
b). Embedment factor for spring constant
Effect height h
i) .
where
= 11.94 ft
28, B = 16.
iii). r
iv). r
=
3'-0' ', (Refer to
= 4.5
'Y) z = 1 + 0 . 6 ( 1
.
- 1.5 = 3 (From Table 6-2-2)
-}I)
fa
= 1 + 0 . 6 (1 -
o. 4)--a
__
ll.gq.
= 1. 09
ii).
1J X=
1 + 0.55 (2
-V)
_fl~
1 + 0.55 (2- 0.4)11.9~
3
=
=1.22
iii).
11'~''.n
·I r
= 1
+ 1.2
c1
= 1 + 1. 2 <1
:3
<-r:3~e§- >
-Y> ..h.+
t"
0
o.2 (2
-r3-:389-
o. 4 >
2
-v> c-2->
•o
+ o. 2 <2 - o. 4 >
83
=
1.156 + 0.015
= 1.171
3
= 1 + 1.2 (1
iv).
0.4)-,3~89 + 0.2 (2 -
0.4)
( ___3_E"_)
=
10.;;,
1. 232
c). Spring constant coefficient(~):
From Table 4-7-5, we get
i). L/B
{3z
= 28/16 = 1.75
ii). L/B = 1.75
---------
iii). L/B = 1.75
----
iv). L/B = 16/28 = 0.57
-----
=
2.15
~X= 0.95
f3cp =
C3cp=
0.58
0.45
d). Equivalent spring constant (Rectangular footing):
From Table 4-7-5, we can get
i). kz
= -~~v- ~z ,[BLT]z
15000
..-~---= (144) - : ~ - (2.15) J16 X 28 X (1.09)
1 04
= 178.57 X 106 1bjft
ii). k
=
2 (1+ )I) G ~,J3L1Jx
= 2 (1 + 0.4) (144)
X
=
X
16
1.22
=148.37
iii) . k
(15,000) (0.95) J28
X
6
10 lb/ft
-~~ii- ~'f!Bl~tp
= -l~,gQ o __ (144) (0.58) (16)
' o. 4 ro
= 3.07 x 10
lbs-ft/rad
(28)2
(1.171)
84
= -~~D- ~~ BL'7Jcp
iv). k
2
= -\~~~ij~ (144) (0.45) (28) (16 ) (1.232)
= 1. 43 x 1010 lbs-ft/rad
3. Damping Ratio:
a). Embedment factor:
From Table 6-2-3, we can find
1+1.9(1-0.4)3/11.94
1+1. 9 ( 1-ll)h/r
i) .
J. z = ------l11~-----0
=
1.09
= 1. 232
ii).
r/..'j..
1+1.9(2-V)h/r
= ---------------0 =
1=1.9(2-0.4)3/11.94
A17x
1.22
1. 60
1+0.7(1-V)(h/r )+0.6(2-V)(h/r (
iii) .
0
0
cAlf= ---------------------------------J 'I cp
3
1+0.7(1-0.4)(3/13.89)+0.6(2-0.4)(3/13.89)
= ------------------------------------------
)1. 71
= 1. 016
1+0.7(1-0.4)(3/10.5)+0.6(2-0.4)(3/1o.5r
iv).
c<p = ------------------- --Ji. 232
---~------~---~---
= 1.029
b) . Mass ratio:
Refer to Table 4-7-3
i). Bz
1-0.4
390,200
(1-JI)
w
= -------(------) = ---4---<-12a~11~943->
4
rr;
=o.287
85
7-8
v
--------32 ( 1-V)
ii) • Bx =
7-8(0.4)
(W/'Kr; ) =
---------32 ( 1- V)
390,200
-12o(11~94r
= 0.378
iii). Btp
5
3(1-11)
= ---------(I'f/Pr
8
0
863,378
3(0.6)
=
)
---------------------------8
(120/32.2)(13.89)
= 0.101
3(0.6)
iv). Be=
-------8
362,166
-----------------(120/32.2)(10.5)
= 0.171
c). Effective damping coefficient:
From Table 6-2-4, we got
E.p=
iii).
~'P= 1.6
because
iv).
~cp=
because ~= 0.171
1.6
0.101
d). Geometrical damping ratio:
According to Table 4-7-3
i). Dzg = _g.:g.2..s_ 'z = _D.dJ.g.5_(1.232) = o.977
!l'!n D\.
,/ Q.ZB1
2
ii). o
= -2~ ~8-~x = -D·g~~--(1.60) = 0.749
X9
o'/1 =
iii> ·
iC5X
-rT=n3~1i)7s,f=ry~-
= 0.321
iv).
0
tl 0. 318
=
-cr+co:-,-o,ft,:r~JiOj~l>ci.b1-
4 = -rT+coJ1T)o~2v~7o~11>cl:6)--
=
0 225
'
e). Internal damping:
From soil data, we have
i). Dz.j
= 0.05
ii).
Dx; =
o.o5
86
iii). D . = 0.05
iv).
o/1
= 0.05
octr
f) . Total damping:
Total damping is equal to (d) + (e)
i) . oz = Ozg + ozi = 0.977 + 0.005 = 1. 027
=
ii). ox= oxg+ ox;
iii).
0~
iv).
.:1X )
0~
D~5
=
O~g
=
+
+
D~;
O~J
0.749 + o.o5
= 0.799
= 0.321 + 0.05 = 0.371
= 0.225 + 0.05 = 0.275
•
a). Natural frequency (rpm):
i). fnz =
ii). fnx
f
=
-t~-j~
-~Q-J-118~~J-~1Q~
= 1159.2 rpm
2.1T
I Z,l I 8
= -i~-J-L4~z7t~~~- = 1056.6 rpm
_6Q_J-~~rrJ..X.lO.!.C:_
= 1800 69 rpm
2TT
863,318
.
0
iv). f II'}= -~fr ~-11if,;ioJ -- = 1897.65 rpm
iii)
•
fn"' =
'1"
b). Resonance frequency (rpm):
From Table 4-7-1,
(rotating mass type excitation
2
F 0 =m;ew).
i).
f
= ___fo•-- =~
Jl-202
r
m
= ,J !~g2
=~
=~ fm~l- 202 =
1
frnt-
iii) . f
... 11-.202- 1159.2
m,~
= fnz
202 > 1, resonance is not possible.
=~
ii) .
f
1156. 6
= fnx
20 2 > 1, resonance is not possible.
,
mT11
iv). fm~
r
= ___fa!£'
=
11-2.02.
___ f1l.!fi__
Ar- :z. oa.
-
== --l~_Q.QL6.9____ ~ frnm= 2115. 2
Jf-2(0•37/)2
== __ J..a.s.'l.s.§_S____
J 1- z co. z 1s )2.
T
~ ~~ = 2059.8
r
87
c). Vibrating force (Maximum absolute amplitude);
From machine parameters:
o,
i). Fz(P) = 1900 lbs. @Wt =
=
=
Fz(P)
Fz(P)
0
@ Wt
BOO lbs.
@Wt
=
FX(P)
= (2n+l)1T/2
=
0, nff/2
@Wt = ( 2n+l rrr/2
= 940 lbs.
ii). Fx(P)
2nn
@evt = n 11'-
0
=
iii). Primary @ wt
T~p(P)
(2n+1)11'/2
F~(P)x
+
=
12,109
=
0
= 940
X
B.17
----------~-------------19,7BB.8 lbs.-ft
Secondary
@
wt = ( 2n+1 )'lt/2
Fz(s)z
iv).
9
=
BOO X 2.75
=
2200 lbs.-ft
T~(P)
= 35,300 lbs.-ft
@wt
=
0, 2nrr
T¢(S)
=
@Wt
=
0, ntr/2
13,450 lbs.-ft
d) • Magnification factor:
= -~ = -f~- =
i). rz(P)
and Dz
Mz(P)
= 1.207
0.51
then
= ____ , ___ t ______ _
,J(l-r
--
2
)
2
+ ( 2Dr )2
J------------------------Jkl-0.512)2+(2xl.027x0.51)
= 0.7B @ 595 rpm
= -~~- = _fe_ = --1~~-~- =
and Dz
=
Wn
1.027
fn
I 159.Z
the
1.035
88
= ______________ ! _______________ _
M (S)
J(1-1 . 0 35z )
2
= 0.47
+ ( 2 X 1 . 0 2 7X 1 · 0 3 5
r
@ 1200 rpm
Mz milX is not possible
ii). rX(P)
= --~~-1056.6 = 0.563 and Dv
,. = 0.799
MX(P) =
--------------1---------------J
22
2
~(1-0.563
) +(2x0.799x0.563)
= 0.885 @ 595 rpm
Mx max is not possible
iii). rlf'(P) = -l~~-~~1f = 0.33 and D~ = 0.371 then
Mtp(P) =
_______________ t ____________ _
I
22
2
~(1-0.33 ) +(2x0.33x0.371)
= 1.082 at 595 rpm
= -~~~1t~91 = 0.666 and D~ = 0.371 then
= ________________ ! _____________ _
J
2 2
2
1-0.666 ) + ( 2x0. 666x0. 371 )
= 1.344
iv) .
@ 1200 rpm
= -~~~~6-5 = 0.314 and D~ = 0.275
= ________________ ! _____________ _
)< 1-0.314 f +(2x0. 314x0. 275 )
2
= 1.09
2
@ 595 rpm
= -~~~~65- = 0.632 and Df = 0.275
= _______________ 1______________ _
J< 1-0.6322 )2 +( 2x0. 632x0. 275 )2
= 1.44
@ 1200 rpm
p '
89
e). Displacement response:
From Table 4-7-1, we found Y = M(F /k) and
0
~= Mf/' ( Tcp/k!p)
0.78x1900+0.47x800
-------------------178. 57x10
6
-6
= 10.405 X 10 ft
-3 in
= 0.125 x 10
= -----------=
6
148.37x10
5.607 X 10- ft =0.067 X 10 in
1.082x19788.8+1.344x2200
iii). ~= ---------3~o7~1o'o
_______ _ =
1.09x35300+1.44x13450
iv).t=
3
6
0.885x940
ii). X
----------------------1.43x1o10
-6
0.794 x 10 rad
= 4.405 x 10- 6 rad
f). Total displacement response:
= z + fiJ * Rhx + p* Rhy
-3
-6
~
=0.125x10 +0.794x10 x(14x12)+4.405x10x(8x12)
=
0 .125x10
=0.68x10
ii).
Xt =X
-3
3
+0 .133x10-.3 +0. 422x10-
in
+~Rv
=0.067x10
=
3
-3
-6
+0.794x10
-3
0.138x10
in
Yt =lfRv
-6
= 4.405x10
=
0.364x10-
3
x(7.5x12)
in
x(7.5x12)
90
g). Transmissibility factor:
From Table 4-7-1, we found Tt
i).
Dz = 1.027,
rz(P) = 0.51
2
Jl + ( 2 ( 1. 0 2 7 ) ( 0 . 51 U
Tr(P)
= -J<i=o~512)2~(2;1~o27;o~51)i- =
rz(S) = 1.305,
ii).
Tt(S) = 1.104
ox= o.799
rx(P) = o.563 ===>
Tr(S)
iii). Dtp
r
41
T~(P)
= 1.19
= 0
= 0.371
===>
Tr(P) = 1.114
= 0.666 ===>
Tt(S) = 1.499
(P) = 0.33
r'f(S)
iv).
1.13
D<f = 0.275
rcf>(P) == 0.314 ===> 1.106
rf(S) = 0.632 =====> 1.525
h). Transmitted force:
i) . Pv
= l: T t-Fzo = 1.13x1900+ 1. 104xBOO = 3030.2 lbs
ii). P,_, =l: TrFxo = 1.19x940 = 1118.6 lbs
iii). PT'P
=:t TtTcpo = 1.114x19,788.8+1.499x2200
= 25,342.5 lbs-ft
iv). PTcf =:ETtTfO = 1.106x35,300+1.525x13,450
= 59,553.05 lbs-ft
91
Table 6-2-2
Embedment Coefficients for Spring Constant (Ref. 41)
Mode of
Vibration
ro for Rectangular
Foundation
Vertical
vBLj;
Horizontal
vBLj;
Rocking
~BL 3 /B1r
Coefficient
17• = 1
'lojo
{/BL(B~ + L 2 )/67r
Torsional
1
'h =
=
+ 0.6 (1-v)(h/ro)
+ 0.55(2-v)(h/ro)
1 + 1.2(1-v)(h/ro)
+ 0.2(2-v)(h/r
0)
3
None available
Notes: h is the depth of foundation embedment below grade;
L is horizontal dimension perpendicular to axis of rocking; B is remaining horizontal dimension.
1.5
,---.,.....
,..
1lflJ
z
I
L
1
~~-·~
01"'
!!o
><O
-<a;
l
T
1
0
0.1
--
p.
V"
... I
v
L---
I
1---
I.I
II
1.0
/
-1-- .-..
.l--- v
~
/
7
{J.
..I- 1-
7p.
r
0.5
.
0.4
0.6
1.0
UB
4
•
•
0
10
Fig. 6-2-2, Coefficient ~Z' ~X and ~~for rectangular
footings.
92
Table 6-2-3
Effect of Depth of Embedment on Damping Ratio (Ref. 41)
Damping Ratio Embedment Factor .
Mode of Vibration
1
1 + 1.9(1- •)ro
a,-----
Vertical
1
1+1.9(2-•)ro
_ __
a,~..;___
Horizontal
""·
1 + 0.7(1- •Hh/ro) + 0.6(2- •Hh/r. )"
Rocking
Table 6-2-4
Values of n '/' for Various Value of
Bt
.6
3
2
••
1.079
1.110
1.143
1.219
(3cp (Ref.
5)
0.8
0.5
0.2
1.251
1.378
1.600
E. Check of Design Criteria - Static Conditions:
1). Static bearing capacity:
Proportion footing area for 50% of allowable soil
pressure. As calculated before, it is 871 < 1,750
psf.
O. K.
2). Static settlement must be uniform.
e.G.
of footing
and machine loads should be within 5% of each linear
dimension. The center of gravity of machine loads
and footing coincide.
93
3). Bearing capacity:
Static plus
dynamic loads.
The magnification factor
should preferably be less than 1.5. The sum of static
and modified
dynamic loads should not create bearing
pressures in
excess of
75% of
the
allowable
soil
pressure for the static load conditions.
Transmitted dynamic vertical force, P
Transmitted moments
PT~
=
=
3030.2 lbs
25,342.5+3030.2(2.75)+
1118.6(8.167)
= 42,811.16 lbs. ft
PTtf
P bd n
y
3030.2
-
= 59,553.05 lbs. ft
42,811.16(6)
= ---------- -t -------------(16)(28)
= 6.76
+
20.48
(16)(28)
-
+
59,553.05(6)
28(16)
+ 49.85
= -63.57, 77.09 psf.
Total static plus dynamic bearing pressure
=
871+77.09
also
~
948.09 psf < 0.75(3500) is O.K.
871-63.57 = 807.43 psf (no effect) is O.K. too.
4). Settlement:
Static plus repeated dynamic loads. The combined
of the
dynamic loads
e.G.
and the static loads should be
within six inches of the footing
e.G.
94
For rocking and pitching motion , the axis of rocking
and pitching
should coincide with the principal axis
of the footing.
F. Limiting Dynamic Condition:
1). Vibration
amplitude
at
operating
frequency.
The
maximum amplitude of motion for the foundation should
lie in
zone A
or B
of Fig.
4-7-1 for
the
giving
acting frequency. The maximum vibration amplitude are
0.00068 in.
and 0.000364
horizontal directions
fall in
zone
A
of
in. in
the
vertical
respectively. These
Fig.
4-7-1
at
the
and
amplitude
operating
frequency of 595 rpm and are therefore acceptable.
2). Velocity is equal to 2 f (cps) x displacement
amplitude as
determined in
paragraph 1) above. This
velocity should be compared to the limiting values of
Table 4-7-2
and Fig.
condition. Velocity
in.jsec. This
4-7-1 for
at least the "good"
= 21T (595/60) (0.00068) = 0.424
velocity falls in "good" range of Fig.
4-7-1 and is therefore acceptable.
2
3). Acceleration is equal to 4~ f
2
x displacement
amplitude as determined in paragraph 1). above. This
check is only necessary if conditions in paragraph 1).
95
or 2). are not satisfied. The acceleration should fall
in zone A or B in Fig. 4-7-1.
4). Magnification
this
1.5
factors
case, the
for
both
operating
should
be
magnification
primary
less than 1.5. In
factors are less than
operating
and
secondary
frequencies.
5). Resonance:
The acting
least
a
frequencies of the machine should have at
difference
frequencies, that
of
is f
+20
>
with
1.2
fm·
the
resonance
In this example,
resonance cannot occur in the vertical and horizontal
modes (due
to large
amounts
of
damping
in
these
directions).
In
the rocking mode, 0.8 fm
rpm and
1.2fm
=1.2(2115.2)
=
0.8(2115.2)
=
1692.16
= 2538.24 rpm. Since the
primary and secondary machine frequencies are 595 and
1200 rpm,
no resonance
will occur
in
the
rocking
mode; therefore the design is deemed acceptable.
In the
rpm and
pitching mode,
1.2fm
primary and
outside of
= 1.2
= 0.8(2059.8) = 1647.84
(2059.8) = 2471.76 rpm. The
0.8fm
secondary machine
frequencies also fall
these ranges and, therefore, no resonance
conditions are possible.
96
6). Transmissibility factor:
The transmissibility
than 1.
than
In this
1
factors should normally be less
example, the
indicating
that
the
t
value are greater
dynamic
forces
are
amplified.
7). Possible vibration modes:
a). Vertical and horizontal oscillations are possible
since the force may act in those conditions.
b). Rocking oscillation is also a possibility since
the horizontal force is above the C.G. of the
foundation.
c). pitching oscillation must also be considered in
this example since unbalanced moments are likely
as indicated by the machine manufacturer.
d). Twisting or yawing oscillation is not considered
since the horizontal forces do not form a couple
in the horizontal plane.
e). The horizontal translation and rocking modes need
not be coupled if:
2
2
Jf 0 x + f 11 cp I (fnx x
fn'l')
< 2/(3f)
using values from the previous calculations
fnx
= 1056.6,
frequency)
expression
=
fn«p
= 1800.69
and
f(primary
595 rpm, substituting into above
97
J( 1056.6 )2 + ( 1800.69 )2 I (1056. 6x1800. 69)
= 1.1x10-3
and 2/(3x595) = 1.1x10-3 , which is within 1%.
Hence, uncoupled mode analysis is O.K.
G. Environmental Demands:
1). Physiological effects on persons:
For the
machine located in a building Figures 4-7-3,
4-7-4 and
of the
4-7-6 have to be used to test the adequacy
installation. In
indicates vibrations
persons" at
to
this
be
example,
"easily
Fig.
4-7-3
noticeable
to
the operating frequency of 595 cpm for a
maximum vibration amplitude of 0.00068 in.
2). Psychological effects on persons:
Use the same procedures stated in the previous
paragraph.
3). Damage to structure:
Use the limit given in Fig. 4-7-3 with 595 cpm
operating frequency for a 0.00068 in. of vibration
amplitude shows no damage to the structure.
98
4). Resonance of structure components for the
superstructure above the footing:
Resonance should
be avoided
by keeping the ratio of
operating frequency to nature frequency less than 0.5
or greater than 1.5.
H. Conclusions:
The foundation
is adequately
acceptable manner
as
designed to
indicated
numerical verifications.
through
perform in an
the
preceding
99
6.3 Design of a Foundation Block for a Centrifugal Machine:
Fig. 6-3-1, Cutaway view of a centrifugal air compressor
A. Machine Parameters:
Compressor:
Weight (We)
- 43,000 lbs
Rotor weight (Wt)
=
Operating speed (f)
= 5725 rpm
oJ = ~-tt(f)
= 600 radjsec
=
-~11'-(5725)
60
60
Critical speed (fc)
2560 lbs.
=
1st
=
2700 rpm
=
2nd
=
8500 rpm
Eccentricity of unbalanced mass
(e) = 0.0015 in. (provided by
manufacturer for static
condition)
100
Dynamic eccentricity at operating speed:
e =
e
0.0015
= -----------------[1-(5725/2700)2]
=0.000429
Centrifugal force:
2560
--------- X
0.000429
X
2
600
32.2x12
= 1023.2 lbs.
Turbine:
Weight (Wt)
=
Rotor weight (Wt)
= 665 lbs.
Operating speed (f)
= 5725 rpm
(£0)
19,500 lbs.
= 600 radfsec
Critical speed (fc) = 1st = 2200 rpm
2nd = 8520 rpm
Eccentricity of unbalanced masse= 0.0015 in.(given
by manufacturer)
Dynamic eccentricity at operating speed:
e
=
101
Centrifugal force:
2
Fo = -ltit-ell.) 2 = ---~§,5___ (0.00026)(600)
B
= 161.1 lbs.
S2.2XI2
Total centrifugal force= 1023.2 + 161.1
Base plate weight (We)
total machine weight
=
= 1184.3
lbs.
60000 lbs.
=
=
We + WT + We
=
68,500 lbs.
43,000 + 195,000 + 6000
B. Soil and Foundation Parameters (From Soil Report):
Soil is Medium Firm Clay
Soil density
r = 130
Shear modulus
G = 6500 psi
Poisson's Ratio
\1 = 0.45
D tp;
Soil Internal Damping Ratio
Static Allowable Bearing Capacity
=
Sdtl =
pcf
0. 05
1.5 ksf
Settlement at allowable pressure is negligible.
C. Selection of a Foundation Configuration:
A shallow
and wide
combined center
footing is
of gravity
foundation coincides
chosen
of the
nearly with
such
that
the
machines and of the
the centroid
of
the
area of the foundation.
A foundation
block configuration
requirement is shown in Fig. 6-3-2.
which satisfies
this
102
Wm (MACHINE WT.)
,
68500 LBS
C.G.OF MACHINE _ ___
AND fOOT lNG
y
z
Fig. 6-3-2 Foundation layout for centrifugal machine
103
a). Weight of rigid block foundation for a centrifugal
machine should be two or three times of supported
machine. Footing weight 9:137 1 000 to 205 1 500 lbs.
b). The top of the block is usually kept 1 f t above
the finished floor or pavement elevation to prevent
damage from surface water runoff.
c). (i)
(ii)
Thickness should not be less than 2 ft.
Thickness should be more than 1/5 of the least
dimension.
(iii) Thickness should be more than 1/10 of the
largest dimension.
d).
(iv)
Thickness is 3'-3" in this example.
(i)
Foundation should be wide enough to increase
damping in the rocking mode.
(ii)
The width should be at least 1 to 1.5 times
the vertical distance from the base to the
machine
center line.
(iii) Width= {((3'-3")+(3'-3")] X 1.5}+2 = 11'-9"
use 13'
,,
'
104
e). After the thickness and width is chosen from item c.
and d. above, the length can be determined from item
a. plus 1 ft clearance from the edge of the machine
base to the edge of block for maintenance purpose.
{[(Lx13'x3.25')]+[Lx(2'+2')x2']}150 = 133,088 lbs.
===> L ~ 21'
f). The length and width of the foundation are adjusted,
so that the C.G. of machine plus equipment coincides
with the e.G. of the foundation. The combined e.G.
should coincide with the center of resistance of the
soil.
g). If the dynamic analysis predicts resonance, the mass
of the foundation is increased or decreased so that
the modified structure is overtuned or undertuned
for centrifugal machines.
h). Concrete Footing Trial Outline:
Weight of the footing (Wf)
=
(21x13x3.25)150
= 133,087.5 lbs.
Weight of the machine (WM)
Total static weight
(W)
=
=
68,500 lbs.
201,587.5 lbs.
105
Ratio of footing weight to machine weight
=
133,087.5
1
=
68,5oo
1.94 ~ 2
=
Actual soil pressure
= 7 3 8 pc f
=
< 0 . 5 ( S ~~~ )
201,587.5/(21x13)
7 50 pc f
0 . K.
D. Dynamic Analysis:
The axis
of rotation
of the
shaft is located 6 1/2 ft
above the
bottom of
the foundation.
acting at
the axis of the shaft
The dynamic force
is of the form F
=
mee~
sinwt which will excite the structure in three different
modes, a).
Since the
vertical b).
horizontal
machine operates
steady-state condition,
sinusoidal force
=
c).
rocking.
constant speed
the formulas
of constant
the dynamic analysis (F
at a
and
in
associated with a
amplitude F
are used in
F0 sinwt).
1) . Equivalent radius:
Jir
b) . Horizontal excitation
r z. = 3
= 9.32 ft
r X = 9.32 ft
c). Rocking oscillation
rtp
a) . Vertical excitation
=
a
J~~
=
10.63 ft
106
2). Mass and mass moment of inertia:
W_
201587-5
= :r
= ---32:2-=
6260.5 lbs. sec 21 ft
a) •
m
b) .
m = 6260.5 lbs. sec 2 1ft
c) • Iw=
T
!
t
2
(JnL ( a iz. + b 1
f2
)
+ m; k i2 ]
= (68,500132.2) (3.25+3.25) = 22,740
I 'f found. = ( 13 3 , o8 1 • 513 2 . 2 ) ( 13 + 3 . 2 5 112 ) +
I'f'ma.chine
1. 625
=
Itp
72,760.7
= I'll Machine+
I
=95,230.7 lblsec2 jft
'f fOUnd.
= _!.L=n.:.4Sl -r.~Ql~e1~- = 0.26
4
I 30X.Cf.32wo
= _1=-tf.(f>;-\~ _zgL~8J,_'i_ = 0.37
3:ZC ,-u.~5) ( 4.04){10.63)5
= .?..CJ=i.:.'1:~t
= 0.036
4). Geometric damping ratio and Internal damping:
Because embedment depth h
hjr 0 =O
a).
====>
~
=
= _Q~~f?..f!_o( = 0.473
JB;
1 is small, therefore
1
Dz = _Q.•.§~.J!J..- rJ. = o. 83
AB z
'1.
b). DX
=
X
Internal damping are
negligible for both
vertical and
horizontal.
107
c). D = -----D~15~f------ = --------D~L5-------(If0.036)) 0.036
c1-t n'f Scp~ t1t Bt
=0.76
D~j
Internal damping
DIP=
= 0.05
0.76 + 0.05 = 0.81
5). Spring coefficient:
From Fig. 4-7-5, we found
a) .
~z= 2.15
When L/B = 13/21 = 0.62
b) .
f3x= 0.95
when L/B = 0.62
c) .
~'I'~' 0.46
when L/B = 0.62
6). Equivalent spring constant:
a). kz =
-r~v-f3zJBL7Jz= -6f.!B!J~A-J_(2.15))13x21
X
= 60.46 x 10 6 lbs.jft
b) • k X
=
=
2 ( 1 + Y) G
l3xJi3L 7'Jx
2(1+0.45)6500(144)(0.95)xA13x21 x 1
= 42.61 x 106 lbs./ft
c). kcp =
-,'!p-
~'fBL2 ~cp= -~~.Q£_£~41(0.46) (21) (13)
x1
= 2778.29 x 106 lbs. ftjrad
7) . Natural frequency: fn
= -~- ~
a) . fnz =
-~~-J-6~i~~- =
b) . fnx =
_60-~~-JV~J~~ZTT
zso.
c) . fntp =
-~"-if
Z.lf 1
=
_6Q_
ZTf
=
-~-
W
938.43 rpm
= 787.81 rpm
Z1113..~Z:.9XJJ
95230.7
= 1631 07 rpm
.
1
lOB
8). Resonance frequency
a).
fmr =
J__ f.IL __
fm~=
= ____Cf3.e..:.B-..a __
1-2 0 1
~I -2.(0.83) 2
====> Imaginary root not possible
= ---~.a-z...aL5!" = 1059 92 rpm
"'- (0-413
.
= --l~JaL~Q7__ ===> Imaginary root
J 1-2 co.al)z
b). f
c).
f
impossible
9). Magnification factor, M:
1
where
D
=
r
-~-n tJ0 =~
= cjc 0 ===> damping
ratio
a) • r
=
-Wr
wn
Mz =
b) • r
c) •
=
= ]R~ = __Cf8.2.7
§~~-- = 6.11
-fQ-6~ii)i_:f_zrJ.a-5x6~11)-;--- =
,-Rtm-
= 7.27
Mx = 7c1=7.2.-f:z)i=t-i.cd.473x7-:i.7)i"- =
r = -~~-~- = 3.51 D = 0.81
Mrp =
0 · 027
-lcl=3~sli)2 =i"z-fo~e;;-x-351}a-
0 019
·
= o·
083
10). Dynamic force:
a).
v0 =
c).
To= HoKo = 1023.2 (65) = 6650.8 lbs.-ft
1023.2 lbs.
b). H0
=
1023.2 lbs
11). (i). Vibration amplitude:
Y=M(m;e)/m
a).
ZI =
= M(m;ew 2jmwi =
Mz(V0 /kz)
M(Fo/k)
=
0.027 x
-i~~xfoEr
=
0.457
10-6
X
ft
~
109
b) .
c) .
x1 = Mx(H 0 /kx) =
=
~z=
M'f(To/kcp)
=
=
0.019
X
0.456
X
- ~.D~.J!'-ti-
4
.61)( 106
10-f> ft
0.083 ( ---' 65-~·~l-8-)
2!118.2. X 0
6 rad
0.199 x 10-
(ii). Component of rocking oscillation:
a). At edge of footing =~h
6
=
0.199 X 10-
=
1.294 X 10-6 ft
b) . At center of bearing
= '/J Ry
= 0.199
= 1.294
X
(13/2)
-6
10
(3.25+3.25)
6 ft
X 10-
(iii). Resultant vibration amplitude:
11 (i) + 11 (ii)
a). Zt
b).
= 90.457+1.294)10- 6 = 1.751x10- 6 ft
xt =
(0.456+1.294)10- 6
=
=
=
0.000021 in.
6
1.750x10ft
0.000021 in.
12). Transmissibility factor Tr and force transmitted P 0
.
a). TrCZ)
Pv
=
0.027J1+(2x0.83x6.11)2-
=
= 0.275(1023.2) = 281.56 lbs.
0.275
'
"
110
b). Tt-CXJ = 0.019J1+(2x0.473x7.27)2- 0.132
PH=
0.132(1023.2) = 135.1 lbs.
c). Tt-CI/IJ
= 0.083J1+(2x0.81x3.51)2 = 0.479
PM= 0.479x6650.8 = 3187.05 lbs.-ft
E. Check of Design Criteria:
1). Static conditions:
a). Static bearing capacity:
Proportion of footing area for 50% of allowable
soil pressure. 738 < 750 psf
is O.K.
b). Static settlement must be uniform:
e.G. of footing and machine loads should be
within
5% of each linear dimension. The e.G. of
machine loads and foundation coincide and is
O.K.
c). Bearing capacity:
Static plus
dynamic
factors are
all less
The sum
should be
of static
within 6
loads.
The
magnification
than 1.5 in this example.
and modified
in of
dynamic
loads
the footing C.G. For
75% of the allowable soil pressure given for the
static load condition ====>
'
111
= 738
+
--~ii~5~(I 3)(ZI)
= 738
+
1.03
=
+
+ __ JiLS1~Qp
__ _
2
CZI)CI3)
jtS
5.39
731.58 and 744.42 < (0.75)(1500)
=
1125 psf
O.K.
====>
d). Settlement for static plus repeated dynamic
loads for rocking motion:
The axis
of rocking
principal axis
should coincide
with
the
of the footing. In this example,
dynamic forces are rather small in comparison to
the
static
caused
loads;
consequently
settlements
by dynamic loads are negligible.
2). Limiting Dynamic Conditions:
a). Vibration amplitude at operating frequency. The
maximum amplitude of motor for the foundation
system should lie in zone A or B of Fig. 4-7-1
for the acting frequency. In this case, the
vertical vibration amplitude is
Zt =
0.000021
in. at 5725 rpm. Referring to Fig. 4-7-1, this
amplitude is within the allowable limits.
Horizontal vibration amplitude at the center of
the bearing area
Xt =
0.000021 in. at 5725 rpm
112
falls in zone A in Fig. 4-7-1, so they are both
acceptable.
b). The velocity equals 2nf (cps) x displacement
amplitude as calculate in paragraph a. above.
Compared with the limiting values in Table 4-7-2
and Fig. 4-7-1 at least for "good" condition.
Velocity equals 600(0.000021)
=
0.0126 in.jsec.
From table 4-7-2 this velocity falls in the
"smooth operation" range and is, therefore,
acceptable.
c). Acceleration
equal
4~f 2 x (displacement
amplitude. This check is not necessary
conditions
in
paragraph
a
and
b
because
both
are
satisfied.
d). Magnification factor (applicable to machines
generating unbalanced forces):
The calculated values of M and
M~
should be less
than 1.5 at resonance frequency. In this
example, all modes are less than 1.5.
e). Resonance:
The acting
have at
frequencies of
least a
the
machine
difference of ±20%
resonance frequency
equations as
should
with
indicated
the
in
113
Table 4-7-1
(0.8fmt
proved there
vertical
is no
mode.
1.2x1059.92
<
> f >
1.2fm~)·
In here, we
resonance frequency
In
5725,
the
in the
mode
horizontal
therefore,
there
no
is
resonance frequency in the rocking mode too.
f). Transmissibility factor (usually applied only to
high-frequency spring-mounted machines):
The value of transmissibility calculated in this
example by equations of Table 4-7-1 is less than
1
indicating
that
dynamic
forces
are
not
amplified.
3). Possible Vibration Modes:
a). Vertical oscillation and horizontal translation
are possible because the forces act in either
direction.
b). Rocking oscillation is also possible since the
point of the horizontal force application is
above the
e.G.
of the foundation mass.
c). No torsional oscillations are possible because
horizontal forces do not form a couple in the
horizontal plane.
114
d). Fatigue failures:
i). Machine components:
Referring to Table 4-7-2, it was found that
with the velocity of 0.0126 in./sec, the
machine is under smooth operation.
Therefore, there is no danger to the machine
components.
ii). Supporting structure:
In this example the amplitude of dynamic
forces are not appreciable to produce any
significant stress increases in excess of
those caused only by static loads.
e). This trial design is acceptable and may be used
to support the machine.
APPENDIX A
DYNAMIC SOIL PROPERTIES
A-1 Shear Modulus
The shear strains in the supporting soil of vibrating
machinery produced by its unbalanced loads are usually much
smaller in
magnitude than
the strains
produced by static
loading. The mechanism governing the stress-strain behavior
of soils
at small
strains
deformation characteristics
involves
of the
mainly
the
stress-
soil particle contacts
and is not controlled by the relative slippage of particles
associated with
large strains.
As the
consequence ,
the
stress-deformation behavior of soil is much stiffer at very
small strain levels than at usual static strain levels.
115
116
T
(SHEAR STRESS)
Fig. A-1-1, Hypothetical shear stress-strain
curve for soil. (Ref. 42)
It is inappropriate to obtain a shear modulus
directly
from a static stress-deformation test, such as a laboratory
triaxial compression
test or
a field
plate bearing test,
unless the stresses and strains in the soil can be measured
for very
strain,
small values
the
of strain.
stress-strain
At very small values of
relationship
is
nonlinear.
Therefore, it becomes expedient to define the shear modulus
as an
equivalent linear
modulus having
the slope
of the
line joining the extremities of a closed loop stress-strain
curve (Figure
A-1-1). It is obvious that the shear modulus
so defined is strain dependent and that in order to conduct
an equivalent
linear analysis
it is necessary to know the
approximate strain amplitude in the soil. For conditions of
controlled applied strain, the ordinates of shear stress in
@
117
Figure A-1-1, and therefore G, will vary (usually decrease)
with number
of cycles
reached. Henceforth,
value of
applied until a stable condition is
only
shear modulus
termed ''the
the
stable
will be
shear modulus",
equivalent
cinsidered
since
the
and
response
linear
will
be
of
the
structure after at least several thousand cycles of load is
the condition
of
principle
interest
to
the
structural
designer.
There seven
variables selected
by Richart (Ref. 42)
that always influence the shear modulus of soil:
1. Grain characteristics and structure of the soil
2. Stress history and void ratio
3. Amplitude of dynamic strain
4. Mean effective stress and length of time since the
stress was applied
5. Temperature
6. Frequency of vibrations
7. Degree of saturation
For variable 3, the amplitude of strain due only to the
dynamic component
evaluating
the
of loading
shear
should
modulus
in
be
considered
normal
practice,
when
as
superimposed static
strain levels
effect on
the static strains are of a very large
G unless
have a relatively minor
magnitude not usually present in a foundation for vibrating
machinery. The first four variables are considered directly
•
118
or indirectly
6-8 are
in the
generally of
procedures for obtaining G. Variable
secondary importance,
temperatu.re of
the soil
freezing point
of the
determined by
although
the
may alter G considerably near the
soil. Soil
field measurement,
shear
modulus
laboratory
may
be
measurement,
and use of published correlations that relate shear modulus
to other more easily measured properties.
lo-•
111
10-•
10-•
Shear Strain (Percent)
If grovel, multiply by 1.5- 2.0
Dr= relative density
Fig. A-1-2. Relationship of K? to shear strain
amplitude and relative densi~y. (Ref. 9)
1800~~10')
13 00
IOOOI-__o:~=~~~~~~~-f---+--J
_Q_.
300f-------r<-
Su
Shear Strain in Percent
Fig. A-1-3. In situ shear modulus for
saturated clay (Ref. 9).
119
Because the shear modulus is a function of shear strain
magnitude, it
is necessary
appropriate value
of shear
calculations
soil
of
to obtain
strain
spring
an estimate
magnitude
constants.
to
The
of the
use
in
following
approximate procedure are recommended by Arya, O'Neill, and
Pincus (Ref.
9)
for
vertical loading
based on an analogy
with static conditions:
1. Select a shear strain amplitude in the range of the
crosshatched
area of Figures A-1-2 and A-1-3, and
compute G.
2. Conduct the structure-soil interaction analysis and
determine the transmissibility factor Tr from Table
'i-7-1 for the forcing frequency desired.
3. Multiply the unbalanced vertical force by the
transmissibility factor and divide the result by the
contact area of the footing to obtain the dynamic
bearing stress qd.
4. The approximate average shear strain
a
in a central
block of soil below the footing of dimensions
2r0 x 2r0 (horizontal) x 4r0 (vertical) is given by
Eq. (A-1).
- - - - - - - - -(A-1-1)
120
Equation (A-1-1)
the soils
. Of
presumes qd to induce the same strains in
as it would if it were acting as a static stress
course, the
emanate
as
strain due
waves,
making
Nonetheless, Equation
strain levels
that
to a
Equation
A-1-1 will
are
dynamic bearing stress
A-1-1
nonrigorous.
yield order-of-magnitude
sufficiently
accurate
for
most
analyses. Therefore this equation can be used to verify the
assumed value of G (Step 1).
If
the
assumed
significantly,
and
these
computed
four
steps
iteratively, using the value of
trial to
shear
a
strains
should
differ
be
repeated
computed on the preceding
obtain G for the present trial, until the strains
close to within an acceptable difference. If a particularly
precise analysis
above should
is warranted
the
approach
described
be abandoned in favor of a more comprehensive
technique, such
as the
complete modeling
finite element
method,
in
which
of a relevant volume of the soil and its
constitutive relationships is considered.
The shear
strain magnitude
beneath a
footing
is
always
taken as that produced only by the dynamic component of the
footing load. The static shear strain is neglected since it
in effect
the dynamic
only provides a nonzero strain level about which
strain
modulus relative
is
to that
cycled.
The
small
nonzero reference
strain
is
shear
generally
121
about the
same as
the small
strain modulus relative to a
reference level of zero strain. Typical ranges of values of
low-strain-amplitude shear
given
in
Table
information.
for
A-1-2
However,
modulus for
it
purpose
the
should
several soils
never
general
of
be
are
used
as
a
substitute for a rational determination of modulus values.
Table A-1-2
Typical Values for Low-Strain-Amplitude Shear Modulus
Shear Modulus (psi)
Soil Type
3,00Q- 5,000
lO,OOQ--20,000
Soft Clay
Stiff Clay
Very Stiff to Hard Clay
Medium Dense Sand*
Dense Sand*
Medium Dense Gravel*
Dense Gravel*
~20,000
5,00Q-15,000
lO,OOQ--20,000
15,00Q-25,000
20,D00-40,000
*For shallow depths.
A-2 Poisson's Ratio
Soil-foundation
insensitive to
Poisson's ratio
soil type
a value
interaction
the values
can be
problems
chosen for
V
selected based
are
and
relatively
p.
Generally,
on the predominant
using Table A-2-1. It is also possible to obtain
for Poisson's ratio by measuring independently the
shear modulus (G) in a laboratory torsional resonant column
test and
Young's modulus
(E) in a laboratory longitudinal
resonant column test. Assuming isotropy,
122
\1 = (E/2G) - 1
- - - - - - - - -(A-2-1)
Table A-2-1
Typical Values for Poisson's Ratio (Ref. 9)
.
Soil Type
0.45--0.50
0.35--0.45
0.4 -D.S
0.3 -{).4
0.3 -D.4
Saturated Clay
Partially Saturated Clay
Dense Sand or Gravel
Medium Dense Sand or Gravel
Silt
A-3 Damping Ratio
The damping
ratios as
ratio can be computed from the mass or inertia
indicated in
Table 4-7-3.
The weights W in the
mass ratio expressions are the total weights of the footing
plus the
load supported
by the footing vibrating in phase
with it,
including machinery. The moment of inertia
If
and
1
are the rocking and the twisting mass moment of inertia
8
about the axis of rotation, which is an axis in the plane
of the base of the foundation perpendicular to the plane of
rocking for rocking motion and an axis perpendicular to the
foundation and
motion. The
to the
factors
rA
plane
of
are damping
twisting
for
torsional
ratio coefficients
to
account for the increased geometric damping that occurs due
to effective
embedment. Equations
for evaluation of
A are
123
given in
Table 6-2-3.
is an inertia ratio
The factor
correction factor for rocking, giving in Table 6-2-4.
Table 6-2-1
inertia of
gives typical
simple
expressions for mass moments of
volumes.
Generally,
mass
moments
of
inertia for most machines and foundations can be adequately
represented by
these expressions.
primary objective
geometric damping,
that the
foundation
with increasing
design
obvious
is
to
that
a
maximize
consistent with economy. It is observed
damping ratios
increase with
It can
of
It is
for
a
halfspace
increasing foundation size (r 0
weight (W)
be concluded,
(Table
)
4-7-3)
and decrease
or mass moment of inertia (I).
therefore, that ideally, foundations
should be as wide and shallow as is practicable.
,, .
References
1. Vierck, Robert K., "vibration Analysis", New York,
N.Y.: Harper & Row, 1979.
2. Meirovitch, Leonard., "Elements of Vibrations
Analysis", New York, N.Y.: Mcgraw-Hill, 1975.
3. Biggs, John., "Introduction to Structural Dynamics",
New York, N.Y.: McGraw-Hill, 1964.
4. Norris, Charles.,
"Structural Design for Dynamic
Loads", New York, N.Y.: McGraw-Hill, 1959.
5. Richart, F. E., Jr., Hall, J. R., Jr., and Wood, R. D.,
"Vibrations of Soil and Foundations", Englewood Cliffs,
NJ: Prentice-Hall, 1970.
6. Barkan, D. D., "Dynamics of Bases and Foundations", New
York, N.Y.: McGraw-Hill, 1962.
7. Richart, F. E., Jr., "Foundation Vibrations",
Transactions ASCE, Vol. 127, Part 1, pp. 863-898, 1962.
1?4
125
B. Kruglov, N.
v.,
"Turbomachine Vibrations Standards",
Teploenery, Vol. 8, No 85, 1959.
9. Arya, Suresh C., O'Neill , Michael W., and Pincus,
George., "Design of Structures and Foundations for
Vibrating Machines", Houston, TX: Gulf Publishing,
1981.
10. Logcher, Robert D., ICES STRUDL-II, Engineering User's
Manual, Vol 2, Second Edition, School of Engineering,
M.I.T., Cambridge, MA: June 1971.
11. Crockett, J. N. A., and Hammond, R. E. R.,
"The Dynamic
Principles of Machine Foundations and Ground", Proc.
Institution of Mechanical Engineers, Vol. 160, No. 4,
pp 512-523, London, England: 1949.
12. Seed, H. B., Discussion on "Soil Properties and Their
Measurement" , Proc. 4th ICSMFE, Vol. 3, pp 86-87,
1957.
13. Thomson, W. T., and Kobori, T. "Dynamic Compliance of
Rectangular Foundations on Elastic Half-Space", J.
Appl. Mech., Trans. ASME, pp 579-584, Dec. 1963.
J
14. Taylor, D. W., and Whitman, R. V., "The Behavior of
Soil Under Dynamic Loadings-3, Final Report on
126
Laboratory Studies'', Aug. M. I. T. , Dept. of Civil and
Sanitary
Engineering Soil Mechanics Lab., Cambridge, MA: 1954.
15. Arya, Suresh c., Drewyer, Roland P., and Pincus, G.,
"Foundation Design for Vibrating Machines", Hydrocarbon
Processing, Vol. 54, No. 11, Nov. 1975.
16. Winterkorn, Hans F., and Fang, Hsia-Yang, "Foundation
Engineering Handbook", New York, N.Y.: Van Nostrand
Reinhold, 1975.
17. Newcomb,
w.
K., "Principles of Foundation Design for
Engines and Compressors", Transactions, ASME, Vol. 73,
pp 307-318, 1951.
18. Westerguard, H. M., "Theory of Elasticity and
Plasticity", pp 176, Cambridge, MA: Harvard University
Press
and New York, N.Y: John Wiley and Sons, Inc.
1952.
19. Terzaghi, K., "Theoretical Soil Mechanics", New York:
John Wiley and Sons, Inc. 1943.
20. Maxwell, A. A., Panel Discussion on Session IV,
"Dynamically Loaded Foundations", Proc. of Symposium,
Bearing Capacity and Settlement of Foundations, Duke
127
University, pp 136-139, 1965.
21. Sowers, George., "Introductory Soil Mechanics and
Foundations", New York: Macmillan Publishing co. 1961.
22. Prakash, Shamsher., "Soil Dynamics", New York, N.Y.:
McGraw-Hill, 1981.
23. Parcher, J.
v.,
and Means, R. E., "Soil Mechanics and
Foundations" Columbus, Ohio: Charles e. Merrill
Publishing, 1968.
24. Arya, Suresh C., Drewer Roland p., and Pincus, G.,
Foundation Design for Reciprocating Compressors",
Hydrocarbon Processing, Voi. 56, No. S,May 1977.
25. Sung, T. Y., "Vibrations in Semi-Infinite Solids due to
Periodic Surface Loadings", Symposium on Dynami Testing
of soils, ASTM-STP No. 156, pp 35-64, 1953.
26. Lamb, H., "On the Propagation of Tremors over the
Surface of an Elastic Solid", Philosophical
Transactions of the Royal Society, London, England:
Ser, A, Vol. 203, pp 1-42, 1904.
27. Sung, T. Y., "Vibration in Semi-Infinite Solids due to
Periodic Surface Loading", S. D. Thesis, Harvard
128
University, May 1953.
28. Clough, Ray
w.,
and Penzien Joseph., "Dynamics of
Structures", New York, N.Y.: McGraw-Hill, 1975.
29. Bycroft, G. N., "Forced Vibration of a Rigid Circular
Plate on a Semi-Infinite Elastic Space and on an
Elastic Stratum", Philosophical Trans., Royal Society,
London, Ser. A, Vol. 248, pp 327-368, 1956.
30. Reissner, E., And Sagoci, H. F., "Forced Torsional
Oscillations of an Elastic Half-Space", J. of Appl.
Phys., Vol. 15, pp 652-662, 1944.
31. Hall, J. R., Jr.,
"Coupled Rocking and Sliding
Oscillation of Rigid Circular Footing", Proc.
International Symposium on Wave Propagation and Dynamic
Properties of Earth Materials, Albuquerque, N.M., Aug.
1967.
32. Lysmer, J., and Richart, F. E., Jr.,
"Dynamic Response
of Footings to Vertical Loading", Soil Mech. and
foundation Div., Proc. ASCE, Vol. 92, No. SM1, Jan., pp
65-91, 1966.
33. Terzaghi, K., "Evaluation of Coefficients of Subgrade
Reaction", Gee-Technique, Vol. 5, pp 297-326, 1955.
129
34. Weissmann, G. F., and Hart, R. R., "Damping Capacity of
Some Granular Soils", ASTM STP No. 305, pp 45-54, 1961.
35. Hall, J. R., Jr., and Richart, F. E., Jr., "Dissipation
of Elastic Wave Energy in Granular Soils", J. Soil
Mech.
and Foundation Div., Pro. ASCE Vol. 89, No. SM6,
Nov. pp 25-56, 1963.
36. Whitman, R.
v.,
and Lawrence, F.
v.,
Jr., Discussion of
"Wave velocities in Granular Soils", J. of Soil Mech.
And Foundation Div. Pro. ASCE Vol. 89, No. SM5, pp 112118, 1963.
37. Stevens, H. W., "Measurement of the Complex Moduli and
Damping of Soils under Dynamic Loads",
u.s.
Army Cold
Regions Research and Eng. Lab. Hanover, N.H. Tech. Rep.
No. 173, 1966.
38. Hardin, B. 0., "The Nature of Damping in Sands", J. of
Soil Mech. and Foundation Div., Proc. ASCE, Vol. 91,
No. SM1, Jan. pp 63-97, 1965.
39. Whitman, R.
v.,
and Richart, E. F., Jr., "Design
Procedure for Dynamically Loaded Foundations", J. of
Soil Mech. and Foundation Div., Proc. ASCE, Vol. 93,
No. Sm6, Nov. pp 169-193, 1967.
130
40. Kaldjian, M. J., Discussion of "Design Procedures for
Dynamically Loaded Foundations" by Whitman and Richart,
Jr., Paper No. 5569, J.
Soil Mech. and Foundation
Div., Proc. ASCE, Vol. 95, No. SM1, Jan. pp 364-366,
1969.
41. Whitman, R.
v.,
"Analysis of Soil-Structure Interaction
--A State-of- Art Review", Soils Publication No. 300,
M.I.T. Cambridge, MA: Apr. 1972.
@
'