civappa_

nda-ing om -

ies

Appendix A. Design of Foundations for Vibrating Machinery

AbstractThis appendix discusses the design of block foundations and pile-supported foutions for vibrating machinery. Two attachments will guide the engineer in collectneeded data for use in the design: “Machinery Unbalances” lists data needed frvendors of the reciprocating or rotating machinery to be mounted on the foundation; and “Soil Data” lists data needed in order to calculate the dynamic propertof the soil on site.

Contents Page

A1.0 Introduction A-2

A2.0 Block Foundations A-2

A2.1 Basic Design Information

A2.2 Modes of Vibration

A2.3 Dynamic Parameters

A2.4 Response Parameters

A2.5 Allowable Design Values

A2.6 Symbol Notation for Section A2.0

A3.0 Pile-Supported Foundations A-11

A3.1 Introduction

A3.2 Equivalent Cantilever Lengths

A3.3 Coordinate System

A3.4 Spring Constants

A3.5 Symbol Notation For Section A3.0

A4.0 Attachments A-17

A4.1 Machinery Unbalances

A4.2 Soil Data

A4.3 Design Aids

A5.0 References A-21

Chevron Corporation A-1 December 1993

Appendix A Civil and Structural Manual

aid ring ic eter-

on dge har-

soil ed

lock

the anal-

by

-d

hich r-ors

h

ith act

A1.0 IntroductionThis appendix has been formulated as a guide to the design of foundations for vibrating machinery. Specifically, this method of analysis should be used as an to the designer in determining the maximum amplitudes of vibration and compathem with acceptable values. This is accomplished by first obtaining the dynamparameters of the foundation-soil system. These parameters are then used to dmine the natural frequencies of the system from which the amplitudes of vibratiare then calculated. This appendix assumes that the designer has some knowleof vibration theory, as well as a thorough description of the static and dynamic cacteristics of the machinery, and data describing the dynamic properties of the at the installation site. Section A4.0, Attachments, should be consulted as needduring the analysis procedure.

Methods of analysis are developed herein for block foundations on grade and bfoundations on piles. The analysis for block foundations on grade is based on Richart and Whitman [14], [21] and assumes only one engine is in operation onmat. This analysis is theoretically rigorous and produces accurate results. The ysis of pile-supported foundations is based on a procedure developed by N. C. Donovan, et al, of Dames and Moore [23] and its accuracy has been confirmedfield test data.

A2.0 Block Foundations

A2.1 Basic Design InformationBlock foundations on grade are the most common support structures for reciprocating or rotating machinery in refinery operations. Figure A-1 shows a simplifieblock foundation with the coordinate system and notation used throughout this section.

Several fundamental considerations should be given to the foundation design wwill enhance its dynamic behavior. Detailed statements of these design consideations can be found in References [1], [8], [24]. The most important design factare:

1. Use as large a foundation-soil contact area as practical.

2. Locate the centroid of the foundation-soil contact area on a vertical line witthe foundation plus machine center of mass; the eccentricity of these pointsshould not exceed .05B.

3. Distribute the foundation mass for the smallest possible moment of inertia wrespect to the principal axis through the centroid of the foundation-soil contarea (e.g., use minimum pedestal height H).

4. Foundation mats should not be joined to surrounding structures.

5. For natural soil bases, the foundation embedment depth should exceed thedepth of frost penetration.

December 1993 A-2 Chevron Corporation

Civil and Structural Manual Appendix A

t

eri-

of

6. For multiple engine foundations [3], [8], [17];

a. Use a common mat.

b. Design the mat to be rigid.

c. Compute the dynamic parameters for each pedestal as if they were noconnected by a common mat.

A2.2 Modes of VibrationDuring the operation of moving machinery, the foundation-soil system may expence one of four possible modes of vibration depending on the direction of the unbalanced forces:

1. Vertical translation in the Z-Z direction.

2. Horizontal translation in either the X-X or Y-Y direction.

3. Rocking (φ) about either the X-X or Y-Y axis. These axes lie in the plane of the mat-soil contact area and intersect at the centroid of this area.

4. Twisting (θ) about the vertical Z-Z axis. This axis passes through the centergravity of the combined foundation-machine mass.

Fig. A-1 Block Foundation Coordinate System



sti-

:

rota-

c

ion

A2.3 Dynamic ParametersThe design of a foundation for reciprocating or rotating machinery requires an emation of the dynamic parameters of the foundation-soil system. The inertia, elastic, and damping parameters can be calculated from the following formulae

1. Inertia Parameters

a. Mass, m

(Eq. A-1)

where:m = Mass of the foundation-machine system, lb-sec2-in-1

g = Acceleration of gravity, in-sec-2

wM = Dead weight of machinery, lb

wF = Dead weight of foundation block, lb

wA = Dead weight of any appurtenances on foundation, lb

b. Mass Moments of Inertia, Iθ, Iφ

The mass moments of inertia about the vertical and horizontal axes of tion can be computed by breaking the foundation into components andusing the parallel axis theorem:

(Eq. A-2)

where:Io = Mass moment of inertia of component about its own axis, lb-se2-

in

mc = Mass of component, lb-sec2-in-1

l = Distance between axis of component and rotation axis, in

Ij = Mass moment of inertia of component about axis of rotation (ϕ = θ,φ), lb-sec2-in

The moment of inertia of the entire foundation/machine system about agiven rotation axis (Iθ, Iφ) is then the sum of the moments of inertia for each component about that axis. Section A4.3 contains useful informatfor determining Io.

m1g--- WM WF WA+ +( )=

I ′j Io mcI2+=



)

m:

2. Elastic Parameters

The spring constant for the six different modes of vibration (see Figure A-1can be found directly from the following formulae and the soil data of Section A4.2.

Vertical, lb-in.-1

(Eq. A-3)

Horizontal, lb-in-1

(Eq. A-4)

Rocking, lb-in

(Eq. A-5)

Twisting, lb-in

(Eq. A-6)

where:

ßx, ßz, and ßφ are coefficients from Figure A-2 and

B = Width of foundation measured along axis of rotation for rockingor normal to direction of horizontal force for translation, in

L = Length of foundation measured in the plane of rotation for rocking or in the direction of horizontal force for translation, in. See example in Figure A-3

ν = Poisson’s Ratio of soil, dimensionless

3. Damping Parameters

The equivalent damping of the foundation-soil system can be completed fro

(Eq. A-7)

KzG

1 ν–------------βz BL=

Kx or y 2 1 ν+( )Gβx BL=

KφG

1 ν–------------βφBL2=

Kθ163------G

BL B2 L2+( )6π

--------------------------------

34---

=

γ γi γr+=



γi, the internal soil damping, is given in Figure A-13, Section A4.2 for the appropriate soil conditions; γr, the radiation damping, can be found from Figure A-4, where:

For Translation:

(Eq. A-8)

For Rocking:

(Eq. A-9)

For Twisting:

(Eq. A-10)

Fig. A-2 Rectangular Coefficients βX, βZ, and βφ Fig. A-3 Length of Foundation Measured in the Plane of Rotation for Rocking

bmρ---- BL

π--------

32---–

=

bIφρ---- BL3

3π----------

54---–

=

bIθρ---- BL B2 L2+( )

6π--------------------------------

54---–

=



be aram-

a-

A2.4 Response ParametersThe response parameters, which characterize the motion of the foundation, cancalculated from the dynamic parameters and the machinery data. The crucial peters are:

1. Natural Frequencies of Foundation-Soil System, fj, cycles per second

(Eq. A-11)

where:j =. . . . . . . . . x, y For Horizontal Translation

=. . . . . . . . . z For Vertical Translation

=. . . . . . . . . φ For Rocking

=. . . . . . . . . θ For Twisting

(Note: For rocking and twisting, replace m with Iφ and Iθ, respectively.)

2. Frequency Ratios, σj

(Eq. A-12)

where:ωj = Actual frequency of machinery unbalance during j mode of vibr

tion, cycles per second

Fig. A-4 Radiation Damping γr

f j1

2π------

Kj

m-----=

σωj

f j-----=



3. Maximum Vibration Amplitudes, Aj

a. Calculate the dimensionless dynamic magnification factor (M)

(Eq. A-13)

b. Calculate the static deflection (∆j) or rotation (φ, θ) caused by the unbal-anced forces (Fj) or unbalanced couples (Mo).

Units: in . . for translation (j = x, y, z)(Eq. A-14)

Units: radians . . for rocking(Eq. A-15)

Units: radians . . for twisting(Eq. A-16)

where:Fj = The unbalanced forces, lb

Mo = The unbalanced couples, in-lb

d = The perpendicular distance from Fj to the axis of rotation, in

K = Spring constants defined in section A2.3, lb/in

c. Calculate the maximum amplitude of vibration (Aj)

Units: in . . for translation(Eq. A-17)

M 1 σj2–( )2 4γ2σj

2+[ ]12---–

=

∆j

Fj

kj----=

φFjd Mo+

Kφ----------------------=

θFjd Mo+

Kθ----------------------=

Ax, y, or z ∆M=



at

ible d. If e. e on

Units: in . . for rocking(Eq. A-18)

Units: in . . for twisting(Eq. A-19)

where: d′ is the perpendicular distance from the axis of rotation to any point where the deflection is desired.

A2.5 Allowable Design ValuesIn order to avoid excessive vibration, the foundation should be designed such th1.4 ≤ σj ≤ 0.7 for all j.

In addition, the designer should proportion the foundation such that the permissamplitudes of vibration furnished by the machine manufacturer are not exceedethese values are not provided by the vendor, Figure A-5 may be used as a guidThe amplitude of vibration for any point on the foundation should not exceed thrange marked “Easily Noticeable to Persons.” The equation of the upper boundthis area is

(Eq. A-20)

where:A = Amplitude in inches, and

f = Machine frequency in cpm

Aφ φMd′=

Aθ θMd′=

A 1f---=



-

n,

e

of

j

ut

ut

A2.6 Symbol Notation for Section A2.0

Fig. A-5 Vibration Limits [1]

Aj The minimum amplitude of vibration corresponding to the j mode of vibration, in.

B Width of foundation base measured along axis of rotation for rocking or normal to direction of horizontal force for translation, in.

b Mass ratio of foundation, dimensionless

d The perpendicular distance from an unbalanced force to an axis of rotatioin.

d′ The perpendicular distance from an axis of rotation to any point where thdeflection is desired, in.

Fj Any unbalanced external force acting on the foundation causing a j modevibration, lbs.

fj The natural frequency of the foundation-soil system corresponding to themode of vibration, cycles per second.

G Modulus of elasticity in shear of the supporting soil, lbs-in-2

g Acceleration of gravity, 386.4 in-sec-2

Iφ The mass moment of inertia of the entire foundation-machine system abothe horizontal X or Y axis, lbs-sec2-in

Iθ The mass moment of inertia of the entire foundation-machine system abothe vertical Z axis, lbs-sec2-in

Io The mass moment of inertia of a body about its own axis, lbs-sec2-in



ns nd e ondi-

-

the

e

a

ess

cy

es

of

A3.0 Pile-Supported Foundations

A3.1 IntroductionThere are many methods by which dynamically loaded pile-supported foundatiomay be analyzed. Nearly all methods prove to be quite tedious and laborious aaddress the central problem of modeling the interaction between the soil and thpiles. This interaction is dependent on the behavior of the soil under a loaded c

I′j The mass moment of inertia of a component about an axis of rotation, lbssec2-in

Kj The spring constant for the foundation-machine system corresponding toj mode of vibration, lbs-in-1 for translation, lb-in for rotation.

L Length of foundation measured in the plane of rotation for rocking or in thdirection of horizontal force for translation, in.

l In computing I′j; the distance between the axis of a component mass and rotation axis of the foundation, in.

M Dynamic magnification factor, dimensionless

Mo Unbalanced couple from machine acting on foundation, in-lb

m The mass of the foundation, machine and appurtenances, lbs-sec2-in-1

mc Component mass, lb-sec2-in-1

WA Total dead weight of appurtenances, lbs

WF Total dead weight of foundation, lbs

WM Total dead weight of machine, lbs

ßj Rectangular foundation coefficient for the j mode of vibration, dimensionl

γ Critical damping ratio, dimensionless

γi Internal soil damping ratio, dimensionless

γr Radiation soil damping ratio, dimensionless

ρ Mass density of soil, lb-sec2-in-4

σj Frequency ratio of the unbalanced force frequency to the natural frequenfor the j mode of vibration, dimensionless

ν Poisson’s Ratio of soil, dimensionless

ωj Actual frequency of machinery unbalance during j mode of vibration, cyclper second

∆j Static deflection during j mode of vibration, in.

φ Rotation of the foundation about a horizontal axis during rocking mode ofvibration, radians

θ Rotation of the foundation about the vertical Z axis during twisting mode vibration, radians



tion

ile- soil-

their hows

ted:

is

tion. Therefore, no method of analysis is exact due to the high redundant condiexisting and only a close approximation can be achieved.

The method used herein was developed by N. C. Donovan, et al, of Dames andMoore [23]. It is a straightforward method by which the spring constants for a psupported foundation may be computed. This approach attempts to simplify thepile interaction by modeling the piled foundation as an equivalent cantilever system. The equivalent cantilever lengths in the model are calculated such thatbehavior is equal to the actual pile under the same external loads. Figure A-6 sa schematic model of a piled foundation.

A3.2 Equivalent Cantilever LengthsTwo equivalent cantilever lengths for each pile in the foundation must be compuan equivalent axial length, Lc, to resist the axial loads and an equivalent bending length, Lb, to resist the lateral loads and moments at the pilehead.

Values for equivalent axial length are as follows:

Lc =. . . . . . . . . L/2, for friction piles having no end bearing.

L/2 ≤ Lc ≤ L, . for friction piles having partial end bearing.

Lc =. . . . . . . . . L, for complete bearing piles.

where:L = the driven length of the pile

The equivalent cantilever length, Lb, for bending is dependent on whether the soil granular or cohesive. For sandy soil, Lb can be computed from the following equa-tion:

Fig. A-6 Modeling of the Piled Foundation



X

(Eq. A-21)

where:E = Modulus of elasticity of the pile, lb-in-2

I = Moment of inertia of the pile in the direction of bending, in4

N = The constant of horizontal subgrade reaction, lb-in-3

(Note: For tapered piles, an average value of I should be used.)

For a clayey soil, Lb can be computed from the following equation:

(Eq. A-22)

where:k = the coefficient of vertical subgrade reaction, lb-in-3

In the absence of a reliable soils report of the surrounding soil, values for Nand k may be taken from Figures A-7 and A-8, respectively.

A3.3 Coordinate SystemAs indicated in Figure A-9, the origin for the rectangular coordinate system is located at the center of gravity of the pile group and each pile is located by the and Y coordinates. The pile is further defined by the horizontal angle, ψ, to the direction of batter and by the angle of batter, γ, measured in a vertical plane. The ranges for ψ and γ are:

Lb 1.86EIN------

15--- , in=

Lb 1.73EIk------

14---

=

Fig. A-7 Values of N [20]

Relative Density of SandN for Dry or Moist Sand,

lb/in3N for Submerged Sand,

lb/in3

Loose 8 5

Medium 25 16

Dense 65 40

Fig. A-8 Values of k [20]

Consistency of Clay

quUnconfined Compressive

Strength, lb/in2 k, lb/in3

Stiff 13-28 90

Very Stiff 28-56 175



0

tion nsid-a-

0° ≤ ψ ≤ 360°

0° ≤ γ ≤ 90°

However, the practical upper limitation on γ is much lower than 90 degrees since 9degrees represents a horizontal pile. Also, note that ψ is always measured in a clockwise direction from the positive X axis.

A3.4 Spring ConstantsThe equations used in calculating the spring constants for all six modes of vibraare given in Figures A-10 and A-11 depending on whether the pileheads are coered fixed or pinned at the pile cap. These spring constants are used in the equ

Fig. A-9 Orientation of Pile with Respect to Origin



the

h ry,

tions given in Section A2.4 to determine the natural frequencies of vibration for pile-supported foundation.

The equivalent mass and mass moment of inertia used in the equations for botfixed and free head conditions are calculated for the foundation block, machineand significant appurtenances. Furthermore, the equivalent damping, γ, of the foun-dation-soil system can conservatively be assumed to be 0.10.

Fig. A-10 Equivalent Spring Constants, Fixed Head Condition

Vertical

(Eq. A-23)

Horizontal

(Eq. A-24)

For horizontal spring constant (Ky) in y direction, values for ψi should be changed to ( ψi – 90)

Rocking about y-axis

(Eq. A-25)

Rocking about x-axis

Kφ = Replace xi with yi in the above expression and replace ψi with ( ψi – 90)

Torsion about z-axis

(Eq. A-26)

where:

n = the total number of piles supporting the foundation

A = the cross-sectional area of the pile, in2

KZ

EiAi

Lci----------- γicos3 12

Ei l iLbi

3-------- γisin2 γicos3+

i 1=

i n=

∑=

KX γicosEiAi

Lci----------- γisin2 ψicos2 12

Ei l iLbi

3-------- γicos4 ψicos2 γicos2 ψisin2+( )+

i 1=

i n=

∑=

KφEiAi

Lci-----------xi

2 γicos3 12Ei l iLbi

3--------xi

2 γisin2 4Ei l iLbi-------- γicos 6

Ei l iLbi

2-------- γisin γicos2 ψicos+ + +

i 1=

i n=

∑=

Kθ ri2 γicos

EiAi

Lci----------- γisin2 αicos2 12

Ei l iL ik

3-------- γicos4 αicos2 γicos2 αisin2+( )+

i 1=

i n=

∑=



of

A3.5 Symbol Notation For Section A3.0

Fig. A-11 Equivalent Spring Constants, Free Head Condition

Vertical

(Eq. A-27)

Horizontal

(Eq. A-28)

For horizontal spring constant (Ky) in y direction, values for ψi should be changed to ( ψi – 90)

Rocking about y-axis

(Eq. A-29)

Rocking about x-axis

Kφ = Replace xi with yi in the above expression and replace ψi with ( ψi – 90)

Torsion about z-axis

(Eq. A-30)

where:

n = the total number of piles supporting the foundation

A = the cross-sectional area of the pile, in2

KZ

EiAi

Lci----------- γicos3 3

Ei l iLbi

3-------- γisin2 γicos3+

i 1=

i n=

∑=

KX γicosEiAi

Lci----------- γisin2 ψicos2 3

Ei l iLbi

3-------- γicos4 ψicos2 γicos2 ψisin2+( )+

i 1=

i n=

∑=

KφEiAi

Lci-----------xi

2 γicos3 3Ei l iLbi

3--------xi

2 γisin2Ei l iLbi-------- γicos 1.5

Ei l iLbi

2-------- γisin γicos2 ψicos+ + +

i 1=

i n=

∑=

Kθ ri2 γicos

EiAi

Lci----------- γisin2 αicos2 3

Ei l iLbi

3-------- γicos4 αicos2 γicos2 αisin2+( )+

i 1=

i n=

∑=

A Cross-sectional area of pile, in2

E Modulus of elasticity of the pile, lb-in-2

I Moment of inertia of the pile cross-section corresponding to the direction bending in question, in4

k Coefficient of vertical subgrade reaction, lb-in-3



prop-ntial

amic

ra-

char-nd

g

n

n

lb-

the

A4.0 Attachments

A4.1 Machinery UnbalancesThe design of foundations for vibrating machinery requires certain information which describes the dynamic characteristics of the machinery and the dynamicerties of the soil. This data, together with the normal static design data, is essein the design of an economical foundation which will not experience excessive vibrations. The characteristics of the unbalanced machinery forces and the dynproperties of the soil define the design case; the objective of the designer is to couple these systems through a foundation in such a manner that excessive vibtions will not occur.

Reciprocating and rotating machinery develop inertial forces or moments duringnormal operation from the motion of unbalanced rods, cranks, blades, etc. Theacteristics of these forces are different for these two basic types of machinery, athe foundation design will reflect this difference. Reciprocating machines can develop forces at frequencies corresponding to integer multiples of the operatin

KX Spring constant for the foundation undergoing horizontal mode of vibratioin X direction, lb-in-1

KY Spring constant for the foundation undergoing horizontal mode of vibratioin Y direction, lb-in-1

KZ Spring constant for the foundation undergoing vertical mode of vibration, in-1

Kφ Spring constant for the foundation undergoing rocking mode of vibration about either the X or Y axis, in-lb

Kθ Spring constant for the foundation undergoing torsion mode of vibration about the vertical Z axis, in-lb

L Driven length of pile, in

Lb Equivalent cantilever length of pile for bending, in

Lc Equivalent cantilever length of pile for axial compression, in

N Constant of horizontal subgrade reaction, lb-in-3

r The radial distance from the center of gravity of the pile group to a pile, in

xi The X-coordinate of a particular pile i, in

yi The Y-coordinate of a particular pile i, in

α The angle between the pile batter projection on the horizontal plane and normal to ri

γ The angle a battered pile makes with the vertical

ψ The horizontal clockwise angle from the positive X-axis to the direction ofpile batter

ν Poisson’s Ratio, dimensionless



ary

, at an s ner

prop-in, ral on-

speed (e.g., secondary moments), but rotating machines can only develop primforces or moments [3]. Rotating machines produce forces whose magnitude depends on the speed of the machine; these forces are specified by a mass moeccentricity e from the shaft centerline, and the speed at which the mass rotate[14], [21]. The manufacturer or vendor of the machinery should supply the desigwith the data contained in Figure A-12.

A4.2 Soil DataThe foundation designer should have sufficient soil data such that the dynamic erties of the soil on site can be calculated. Soil data is in general difficult to obtaand its validity is often questionable. However, the designer must produce sevesoil parameters, if the methods of this appendix are to be employed. Data from

Fig. A-12 Vendor-supplied Data, Reciprocating and Rotating Machinery

Reciprocating Machinery

Range of Operating Speed Ω = to RPM

Maximum horizontal primary forces Fx1 = lbs

Fy1 = lbs

Maximum horizontal secondary forces Fx2 = lbs

Fy2 = lbs

Maximum vertical primary force Fz1 = lbs

Maximum vertical secondary force Fz2 = lbs

Maximum horizontal primary moment Fθ1 = in-lbs

Maximum horizontal secondary moment Fθ2 = in-lbs

Maximum vertical primary moment Fφ1 = in-lbs

Maximum vertical secondary moment Fφ2 = in-lbs

Any higher order forces or moments

A reference system for forces and moments as shown in Figure A-1.

Rotating Machinery

Range of Operating Speed Ω = to RPM

Eccentric Mass mo = lbs-sec2-in-1

Eccentricity e = in

Additional information should be supplied which describes the static characteristics of the machine. The necessary drawings of the machinery and mountings should also be supplied.



of soil

il mate

d ave

site borings and tests should be used, when it is available; general summaries properties are of limited value and should be used only where on-site data is unavailable.

It is essential that the designer have a thorough qualitative description of the soand subgrade structure. With this information, the designer can at least approxithe soil properties of the installation site with the information listed below, whenquantitative test data is unavailable.

The quantitative soil properties required in this appendix are:

Since G, E, and υ are interrelated for isotropic materials through

(Eq. A-31)

E and υ can be used to approximate G, when it is unknown.

Figure A-13 is a summary of values for these properties compiled by Dames anMoore. Their values are based on their records of dynamic testing which they hperformed on a wide variety of soils over the last decade.

ρ Mass density of soil, lbs-sec2-in-4

G Elastic shear modulus, lbs-in-2

υ Poisson’s Ratio, nondimensional

γi Internal soil damping ratio, nondimensional

Fig. A-13 Soil Properties

Density (lbs/ft3)

Dry Saturated G kip/in2 υ γi

Sand

Dense w/Gravel 116-135 135-145 15-30 .25-.35 .02-.04

Dense 116-130 135-145 10-20 .25-.35 .02-.00

Med. Dense 109-116 130-135 6-10

Loose 90-99 113-124 3-6

Clay

Hard ≥110 15-30

Stiff ~ 100 6-15 .35-.5 .02-0.04

Soft ~ 90 3-6

Note The classification for clays should be based on shear strength approximately as follows:soft < 3 psistiff ≈ 10 psihard > 20 psi

G E2 1 υ+( )---------------------=



A4.3 Design AidsFigure A-14 shows mass moment of inertia for various bodies. The mass of thebody is indicated by m.

Fig. A-14 Mass Moment of Inertia, Various Bodies

Body Axis Moment of Inertia

Thin rectangular sheet, sides a and b

Through the center parallel to b

(Eq. A-32)

Thin rectangular sheet, sides a and b

Through the center perpendicular to the sheet

(Eq. A-33)

Rectangular parallelepiped, edges a, b, and c

Through center perpendicular to face ab (parallel to edge c)

(Eq. A-34)

Sphere, radius r Any diameter

(Eq. A-35)

Spherical shell, external radius, r1, internal radius, r2

Any diameter

(Eq. A-36)

Right circular cylinder of radius r, length l

The longitudinal axis of the solid

(Eq. A-37)

Right circular cylinder of radius r, length l

Transverse diameter

(Eq. A-38)

Hollow circular cylinder, length l, radii r1 and r2

The longitudinal axis of the figure

(Eq. A-39)

Hollow circular cylinder, length l, radii r1 and r2

Transverse diameter

(Eq. A-40)

Right cone, altitude h, radius of base r

Axis of the figure

(Eq. A-41)

ma2

12------

ma2 b2+

12-----------------

ma2 b2+

12-----------------

m25---r2

m25---

r15 r2

5–( )

r13 r2

3–( )--------------------

mr2

2----

m r2

4---- l2

12------+

mr12 r2

2+( )2

---------------------

mr12 r2

2+( )4

--------------------- l2

12------+

m310------r2



r

ngi-er-

A5.0 References1. ARYA, S. C., R. P. DREWYER and G. PINCUS, “Foundation Design for

Vibrating Machines,” Hydrocarbon Processing, November 1975.

2. ASCHENBRENNER, Rudolf, “Three-Dimensional Analysis of Pile Founda-tions,” Journal of the Structural Division, ASCE, Vol. 93, No. ST1, February1967, pp. 201-219.

3. BARKAN, D. D., Dynamics of Bases and Foundations, McGraw-Hill Book Company, Inc., New York, 1962.

4. BOUTWELL, G. P., Jr. and D. S. SAXENA, “Design Method: Dynamically Loaded Pile Foundations,” ASCE Annual and National Environmental Engi-neering Meeting, Houston, Texas, 1972, Meeting Preprint 1833.

5. CONVERS, F. J., “Foundations Subjected To Dynamic Forces,” Foundation Engineering, ed. G. A. Leonards, McGraw-Hill Book Company, Inc., New York, 1962, pp. 769-825.

6. GRAY, H., Discussion to Francis, (1964) “Analysis of Pile Groups with Flex-ural Resistance,” Journal of Soil Mechanics and Foundation Division, ASCE, November 1964.

7. KOCSIS, Peter, “The Equivalent Length of a Pile or Caisson in Soil,” Civil Engineer, December 1976.

8. MAJOR, Alexander, Vibration Analysis and Design of Foundations for Machines and Turbines, Collet’s Holding Limited, London, 1962.

9. NAIR, K., H. GRAY, and N. C. DONOVAN, “Analysis of Pile Group Behavior,” ASTM Special Technical Publication, No. 444, 1969.

10. POULOS, Harry G., “Behavior of Laterally Loaded Piles: I - Single Piles,” Journal of Soil Mechanics and Foundation Division, ASCE, Vol. 97, No. SM5, May 1971, pp. 711-731.

11. POULOS, Harry G., “Behavior of Laterally Loaded Piles: II - Pile Groups,” Journal of Soil Mechanics and Foundation Division, ASCE, Vol. 97, No. SM5, May 1971, pp. 733-751.

12. REESE, Lymon C. and Hudson MATLOCK, “Non-dimensional Solutions foLaterally Loaded Piles with Soil Modulus Assumed Proportional to Depth,” Proceedings Eighth Texas Conference on Soil Mechanics and Foundation Eneering, Special Publication No. 29, Bureau of Engineering Research, Univsity of Texas, 1956.

13. RICHART, F. E., Jr., J. R. HALL, Jr. and R. D. WOODS, Vibration of Soils and Foundations, Prentice-Hall, Englewood Cliffs, New Jersey, 1970.

14. RICHART, F. E., Jr. and R. V. WHITMAN, “Comparison of Footing VibrationTests with Theory,” Journal of Soil Mechanics and Foundation Division, ASCE, Vol. 93, No. SM6, 1967, pp. 143-193.



ion

R

15. RODGERS, Grover L., Dynamics of Framed Structures, John Wiley and Sons, Inc., New York, 1959.

16. SAUL, William E., “Static and Dynamic Analysis of Pile Foundations,” Journal of Structural Division, ASCE, Vol. 94, No. ST5, Proceedings paper 5936, May 1968, pp. 1077-1100.

17. Shock and Vibration Handbook, Vol. I, II, III, ed. by Cyril M. Harris and Charles E. Crede, McGraw-Hill Book Company, New York, 1961.

18. SINGHAL, A. C., J. C. WACHEL and F. R. SZENASI, “Computation of Natural Frequencies of Compressor Foundations Supported on Different Soils,” Pipeline Compressor and Research Council, Southern Gas Associat, Report No. 132, Southwest Research Institute, July 15, 1969.

19. TERZAGHI, Karl and Ralph E. PECK, Soil Mechanics in Engineering Prac-tice, John Wiley and Sons, Inc., New York, 1964.

20. TERZAGHI, K., “Evaluation of Coefficients of Subgrade Reaction,” Geotech-nique, Vol. 5, No. 4.

21. WHITMAN, R. V. and F. E. RICHART, Jr., “Design Procedures for Dynami-cally Loaded Foundations,” Journal of Soil Mechanics and Foundation Divi-sion, Proceedings ASCE, Vol. 93, No. SM6, November 1967, pp. 169-193.

22. Design Manual, “Soil Mechanics, Foundations, and Earth Structures,” NAVFAC DM-7, Department of the Navy, March 1971.

23. SINGH, J. P., N. C. DONOVAN, and A. C. JOBSIS, “Design Procedures forDynamically Loaded Pile-supported Foundations,” Journal of Geotechnical Engineering Division, ASCE, Vol. 103, No. GT8, August 1977, pp. 863-877.

24. “Final Report on Machinery Foundation Design Standards,” Report No. AE1800 EPI 71-51, prepared by ARAMCO Services Company.


Documents

civappa_