smfe chapter 22

Chapter

22 Soil Dynamics and Machine Foundations

LEARNING OBJECTIVES

After reading this chapter, you will be able to:

• Enumerate the various types of machine foundations and the conditions under which they are suitable for a given type of machine.

• Explain the various methods of dynamic analysis of machine foundations.• Enumerate and define dynamic properties of soil and their interrelationship.• Know the in situ and laboratory methods of determination of dynamic properties of soil.• Outline the procedure for dynamic analysis of a machine foundation.• Specify the general design criteria for a machine foundation.• Specify the design criteria for various types of machine foundations.• Outline the measures for vibration isolation and control.

22.1 Introduction

Machines are subjected to dynamic loads and vibrations, which are transmitted to the foundation and the sup-porting soil mass. Foundations supporting various machines such as reciprocating engines, compressors, turbines, large electric motors, generators and punch presses, are subject to vibrations caused by unbalanced machine forces as well as the static weight of the machine. Foundations may also be subjected to dynamic loads other than those caused by machine vibrations, such as those due to earthquakes, blasting, pile driving, drilling for soil exploration, and action of wind or water.

Soil dynamics is the branch of soil mechanics which deals with the study of the soil behavior subjected to dynamic loads. It involves the principles of soil mechanics and dynamics of engineering mechanics. It has basically two broad applications: firstly the design of safe and economical foundations for various types of machines, which transmit vibrations to the foundation–soil system; secondly the dynamic analysis of soils subjected to vibrations due to dynamic forces such as due to earth quakes, construction equipment etc., to ensure stability of structures supported by the soil mass in the affected region. This chapter deals exclusively with the dynamic analysis of machine foundations and for dynamic analysis of foundation soil subjected to earthquakes and other dynamic loads, reference may be made to other sources.

Vibrations of machines may cause the following effects of varying degree:

1. Damage of the machine and/or its foundation.2. Excessive soil deformation.3. Damage to structures due to excessive vibrations.4. Malfunction of sensitive equipment.5. Discomfort to people in the vicinity, due to vibrations or noise.

The vibrations cause displacement of the foundation and the soil, which is sinusoidal function of time. The dis-placement increases with time, reaches a maximum value, known as amplitude, and then decreases to zero. It then occurs in the opposite direction, reaching the maximum negative value, thereby becoming zero. Thus, vibrations

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928 • Chapter 22 Soil DynamiCS anD maChine FounDationS

of the foundation–soil system can be represented by the classical simple harmonic motion. The number of cycles of vibration per unit time is known as frequency. The machine foundation–soil system may fail under the following two conditions:

1. When the frequency of the foundation–soil system is equal to the frequency of vibration of the machine, a dan-gerous condition known as “resonance” occurs, during which the amplitude will be maximum, causing the failure of the foundation–soil system.

2. When the amplitude of vibration of the foundation–soil system is too high even in non-resonance condition, it may endanger the stability of the foundation–soil system.

The dynamic response of foundations and structures depends on:

1. The magnitude, frequency, direction, and location of the dynamic loads. 2. Geometry of the foundation–soil system. 3. Dynamic properties of the supporting soils and structures.

The dynamic analysis of the foundation–soil system is done by using a mass-spring–dashpot model, in which the foundation along with the machine is mathematically represented by a mass and the soil by spring. The energy causing vibrations decreases due to the resistance offered by the soil known as damping and represented by the dashpot in the mass-spring–dashpot model. Experience has indicated that this model gives reasonably accurate results, even though the physical system may not resemble the mathematical model.

The design of foundations for control of vibrations was often done by increasing the mass of the foundation and/or strengthening the soil beneath the foundation base by using piles. This procedure generally works; how-ever, the early designers recognized that this often resulted in considerable overdesign. By the 1950s, a few founda-tion engineers began to use vibration analyses, usually based on the theory of surface load on an elastic half-space. In the 1960s, the lumped mass approach was introduced, the elastic half-space theory was refined, and both meth-ods were validated.

22.2 Types of Machines

Following are the basic types of machines:

1. Rotary type machines.2. Reciprocating machines.3. Impact or impulsive type machines.

Rotary type machines include gas turbines, steam turbines, turbo-pumps, turbo-compressors, fans, motors, centri-fuges, etc., and are characterized by the rotating motion of impellers or rotors. Table 22.1 shows a classification of rotary type machines based on their operating speed.

In reciprocating machines, a piston slides in a cylinder through a fluid by operation of a crank mechanism driven by, or driving, a rotating crankshaft. Common reciprocating machines used are compressors and diesel engines.

Equipment, such as forging hammers and some metal-forming presses, operate with regulated impacts or shocks between different parts of the equipment. This shock loading is transmitted to the foundation–soil system. Forging presses perform a similar manufacturing function as forging hammers but they operate at low velocities and greater forces.

Table 22.1 Classification of rotary type machines

S. No. Frequency Classification Frequency (Hz)

1. Low <25

2. Medium 25–50

3. High ≥50


22.3 typeS oF maChine FounDationS • 929

22.3 Types of Machine Foundations

Following are the different types of common machine foundations:

1. Block foundation.2. Box foundation.3. Wall foundation.4. Framed foundation.

Figure 22.1(a) shows a block foundation, which consists of a massive RCC block, supported on a spread footing. Block foundations are the common type of foundations suitable for most types of machines, especially for those producing periodic and impulsive forces.

(a)

(c) (d)

(b)

(e)

Vibrationisolator

(f)

Block foundation

Steel frame

Spring

(g)

Pier

Pile cap

(h)

Figure 22.1 Types of machine foundations: (a) Block foundation, (b) box foundation, (c) wall foundation, (d) framed foundation, (e) framed foundation with vibration isolators (ACI-351), (f) spring-mounted block foundation (ACI-351), (g) flexible foundation, and (h) pile foundation (ACI-351).



A box foundation is a hollow block foundation as shown in Fig. 22.1(b) and attempts to save the mass concrete and is suitable for machines subjected to small loads causing relatively low amplitude vibrations. In wall founda-tions, shown in Fig. 22.1(c), small machines are supported on walls, which are in turn supported on RCC mat like foundations. Combined blocks are used to support closely spaced machines.

In framed foundation, the machine is supported on a framed structure consisting of top slab on which the machine is placed and beams and columns, which are in turn supported on bottom beams and pad footing. Framed foundations are generally more economical than block foundations due to large saving in mass concrete. However additional measures should be adopted to provide damping of the vibrations. The space made available by exclu-sion of mass concrete is used to accommodate ducts, piping, and ancillary items to be located below the equipment. A framed foundation is also referred to as a Table top foundation and a typical framed foundation is shown in Fig. 22.1(d).

Figure 22.1(e) shows a framed foundation with vibration isolators. Isolators (springs and dampers) are provided at the top of supporting columns to absorb the vibrations and to minimize their transfer to the foundation. The effectiveness of isolators depends on the machine speed and the natural frequency of the foundation. Machines can also be mounted on springs to absorb the machine vibrations. The springs are in turn supported on a block founda-tion. Figure 22.1(f) shows a typical spring-mounted block foundation.

A flexible foundation is more economical than other type of machine foundations. When a flexible founda-tion is provided to the machines, the vibrations are more or less completely transmitted to the foundation soil. However, the vibrations transmitted to the foundation–soil system may endanger the stability, life and opera-tion of the machine itself. A machine can be supported on a flexible foundation, in case there is no danger if the dynamic loads and vibrations are transmitted to the foundation soil. Figure 22.1(g) shows typical flexible machine foundation.

Machine foundations are to be supported on pile foundation, when the soil at shallow depth is weak and/or highly compressible. Figure 22.1(h) shows typical block foundation supported on piles.

22.4 Methods of Dynamic Analysis

The following methods are available for the analysis of machine foundation–soil system:

1. Lumped mass method.2. Elastic half-space theory.3. Elastic half-space analog method.4. Barkan’s elastic soil spring method.

These methods are discussed in the following subsections.

22.4.1 Lumped Mass Method

In this method, the dynamic analysis is done by replacing the machine-foundation–soil subjected to vibrations by a mass-spring–dashpot model. Here, the mass represents the mass of the machine and the foundation and the spring represents the soil supporting the foundation. As the distance from the foundation increases radially in a semi-infinite soil mass, the energy causing vibrations is distributed over a larger mass of soil and hence the energy decreases with increase in distance from the foundation. This effect is known as damping.

Consider a block (mass) of mass m (and weight W), suspended by a spring with spring constant k. Let the spring–mass system assume the static equilibrium position O–O, as shown in Fig. 22.2(a). Now, if a force is applied on the block causing displacement from the equilibrium position, the block oscillates about the mean static equilib-rium position. The motion of the block can be represented by a simple harmonic motion, as shown in Fig. 22.2(b), which is the simplest form of vibratory motion.

An idealized simple harmonic motion may be described by

z A t= −sin( )ω φ (22.1)

where z is the displacement, A the amplitude, w the angular velocity, t the time, and f the phase angle.


22.4 methoDS oF DynamiC analySiS • 931

The displacements described by Eq. (22.4) will continue oscillating forever. In reality, the amplitude of the motions will decay over time due to the phenomenon called damping. It is of two types:

1. Viscous damping: Viscous damping is similar to that caused by a dashpot with constant viscosity in which the amplitude decays exponentially with time. It is called as linearly viscous damping.

2. Friction damping: In friction damping the amplitude decays linearly with time similar to that caused by a con-stant coefficient of friction. It is said to be linearly hysteric damping.

Let us consider dynamic analysis with a single-degree-of-freedom (SDOF) system illustrated in Fig. 22.3(a). A mass is attached to a linear spring and a linear dashpot. If the mass m is accelerating, the force causing this acceleration (Fa) must be

F ma md zdt

mza = = =2

2�� (22.2)

The dots are used to indicate differentiation with respect to time, which simplifies writing the equations. The restoring force (Fd) exerted by linear dashpot is proportional to the velocity of motion and acts in the opposite sense. This means

F cdzdt

czd = − = − � (22.3)

Finally, there may be some force P, which is a function of time, applied directly to the mass. Adding the three forces together, setting the sum equal to mÜ, and rearranging terms give the basic equation for an SDOF system, we get

mz cz kz P�� + + = (22.4)

In most practical cases, the mass m and the stiffness k can be determined physically. It is often possible to measure them directly. On the contrary, damping is a mathematical abstraction used to represent the fact that the vibration energy does decay. It is difficult, if not impossible, to measure directly and in some cases to be discussed in the following list, it describes the effects of geometry and has nothing to do with the energy absorbing properties of the material.

O

+ z

z = 0

WW

W W

W

Static equilibriumposition

= Doubleamplitude

p

pp

p

2A

Zmax = AzO

(a)

Dis

plac

emen

t, z

Ot

p

wt = p/2

wt = 2pT = 2p/w

3p/2 2p 5p/2 3p

(b)

Figure 22.2 (a) Vibration and (b) simple harmonic motion of a spring–mass system.

(a)

U (Displacement)

P (Force)

x

K

C

(b)

t

Undamped system

Damped system

U

Figure 22.3 Single degree of freedom: (a) Mass-spring–dashpot model and (b) amplitude of vibrations.



1. Free vibrations without damping: In case of no external force and no damping, the motion of the mass will be the simple harmonic motion. The frequency will be

ωo =1

2πkm

(22.5)

2. Free vibrations with damping: If damping is not zero and the mass is simply released from an initial displace-ment zo with no external force, the motion will be as shown in Fig. 22.3(b). The frequency of the oscillations will be

ω ωe o2 2

2

2= −

cm

(22.6)

When, c = 2 √km, there will be no oscillation, but the mass will simply creep back at the rest position at infinite time. This is called critical damping, and it is written as ccr. The ratio of the actual damping to the critical damp-ing is called the critical damping ratio D. It can be represented as

Dc

c=

cr

(22.7)

If Eq. (22.4) is divided by m, it can be written as

�� z D z zPm

+ + =2 2ω ωo o (22.8)

The frequency of oscillations can be written as

ω ω= −o ( )1 2D (22.9)

In almost all practical cases, D is much less than 1. For example, a heavily damped system might have a D of 0.2 or 20%. In this case, w is 98% of wo, so little error is introduced by using the undamped frequency wo in place of the damped frequency w.

3. Forced vibrations with damping: If now the SDOF system is driven by a sinusoidally varying force, the right side of Eq. (22.4) becomes

R F t= cos( )ω (22.10)

For a very low frequency, this becomes a static load and

zFk

A= = s (22.11)

where As is the static response.In the dynamic case, after the transient portion of the response has damped out, the steady-state response becomes

z MA t= −s cos( )ω φ (22.12)

In Eq. (22.12), M is called the dynamic amplification factor or magnification factor and f is the phase angle. The magnification factor is the ratio of the amplitude of the dynamic steady-state response to the static response and describes how effectively the SDOF amplifies or de-amplifies the input. The phase angle f indicates how much the response lags the input.

Through suitable mathematical manipulation, the magnification factor is given as

MD

=− +

1

1 22 2 2 2{( / ) [ ( / )] }ω ω ω ωo o

(22.13)

and φω ωω ω

=−

−tan( / )

( / )1

2 2

21D o

o

(22.14)

The magnification factor M is plotted as a function of the frequency ratio in Fig. 22.4.



In case of rotary type machinery, the amplitude of the driving force is proportional to the frequency of the rotat-ing machinery. If e is the eccentricity of the rotating mass and me is its mass, then the amplitude of the driving force becomes

F m e= ⋅ ⋅e ω2 (22.15)

In this case, the driving force vanishes when the frequency becomes zero, so it does not make sense to talk about a static response. However, at very high frequencies, the acceleration dominates; hence, it is possible to define the high-frequency response amplitude, R and can be represented as

R me

M=

e (22.16)

As in the case of the sinusoidal loading, the equations can be solved to give an amplification ratio, which is the ratio of the amplitude of the response to the high-frequency response R. The curve is plotted in Fig. 22.5.

An important point is that the response ratio gives the amplitude of the displacement response for either case. To find the amplitude of the velocity response, the displacement response is multiplied by w (or 2pf). To find the amplitude of the acceleration response, the displacement response is multiplied by w2 (or 4p 2f 2).

22.4.2 Elastic Half-Space Theory

In this method, the machine foundation is represented by an oscillating mass causing a periodic vertical pressure distributed uniformly over a circular area on the surface of an elastic half-space. The soil is assumed to be homoge-neous, isotropic, semi-infinite elastic half-space, represented by the properties of the shear modulus, Poisson’s ratio and mass density. The energy causing the vibrations travels radially from a circular footing and decreases with the increase in distance from the footing. This decrease in the energy is called geometrical damping. The elastic half-space theory, therefore, does not consider the viscous damping as used in the lumped parameter method.

f/fn

21.81.61.41.210.8

F = Fo sin wt

0.60.40.200.1

0.2

0.3

0.40.50.60.81

2

3456

810

20

304050

0.30

D = 0.01

0.40

Am

ax

As

Dyn

amic

dis

plac

emen

t

Sta

tic d

ispl

acem

ent

=M

=

0.60

0.20

0.10

0.05

0.02

0.50

Figure 22.4 Response curves for SDOF system with viscous damping.



Elastic half-space theory for vertical vibrations was first developed by Reissner in 1936. The assumptions used in the analysis are not realistic but the method serves as a useful guide for determination of spring and damping constants that are used in the elastic half-space analog method.

22.4.3 Elastic Half-Space Analog Method

In this method, the vertical vibrations of rigid circular footings resting on soil are represented by a mass-spring–dashpot system, but the spring constants and the damping constants are determined using elastic half-space theory. Thus, the elastic half-space analog method is an intermediate method between the lumped parameter method and elastic half-space theory. While Lysmer (1965) and Lysmer and Richart (1966) developed the elastic half-space analog method for vertical vibrations, Hall (1967) extended the method for rocking and sliding modes of vibration of the rigid circular footings. Reissner (1937) and Reissner and Sagoci (1944) extended the method for torsional vibrations.

The following three dimensionless parameters are used in the analysis:

1. Frequency ratio.2. Mass ratio.3. Damping ratio.

For vertical vibrations, the frequency ratio is given by

a rg

ro o

o

s

= =ωρ ω

V (22.17)

The mass ratio is given by

bmroo

=−

⋅( )1

4 3

µρ

(22.18)

The spring constant is given by

kGr

=−

41

o

( )µ (22.19)

0.50

0.10

0.40

D = 0.01

F = me ⋅ e ⋅ w 2 sinwt

0.5

f/fn

50

20

10

5

2

1

0.5

0.2

0.10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

Res

pons

e ra

tio,

Am

ax

R

0.300.20

0.60 Transl. :

Rotal. :

R = me e

mR = me eh

I

Amax

me

Amax

e

h0.02

Figure 22.5 Response curves for SDOF system with viscous damping.



The damping constant is given by

cr

G=−

3 41

2.( )

o

µρ (22.20)

The critical damping coefficient is given by

c kmc = 2 (22.21)

The damping ratio is the ratio of damping constant and the critical damping coefficient, given by

Dcc

=c

(22.22)

The resonant frequency is given by

fk D

mm =−1

21 2

π( )

(22.23)

22.4.4 Elastic Soil Spring Analog Method

Barkan (1962) simplified the lumped parameter method using the concept of elastic sub-grade reaction. The follow-ing spring constants representing the soil are given by Barkan and are popularly known as dynamic soil properties:

1. Coefficient of elastic uniform compression (Cu).2. Coefficient of elastic non-uniform compression (Cf).3. Coefficient of elastic uniform shear (Ct).4. Coefficient of elastic non-uniform shear (Cy).

The following expressions are given by Barkan for determination of spring constants under various modes of vibration:

For vertical vibration, we get

kG

zz=

−βµ( )1

(22.24)

For horizontal vibrations, we get

k G blx x= +2 1β µ( ) (22.25)

For rocking vibrations, we get

kG bl

φφ=−

β

µ

2

1( ) (22.26)

where b is the width of the foundation along the axis of rotation, l the length of the foundation perpendicular to the axis of rotation, and bz, bx, and bf the shape factors that are functions of l/b.

The resonant frequency for the vertical mode of vibrations as per Barkan’s method is given by

fC A

mnzu=

12π

(22.27)

The resonant frequency for the horizontal mode of vibrations is given by

fC Amnx =

12π

τ (22.28)



The resonant frequency for the rocking mode of vibrations is given by

fC I

Inm

φφ=

12π

(22.29)

The resonant frequency for the torsional mode of vibrations is given by

fC I

Inp

mzψ

ψ=1

2π (22.30)

where I is the moment of inertia of the base of the foundation about the axis of rotation, Ip is the polar moment of inertia of the base of the foundation about the axis of rotation, Im is the mass moment of inertia about the axis of rotation, and Imz the mass moment of inertia about the vertical axis of rotation.

22.5 Procedure for Dynamic Analysis of Machine Foundation

The dynamic analysis of machine foundations is done using the following steps, as recommended by the US Military Handbook (MIL-HDBK, 1997):

1. Convert the machine, foundation, and soil into an SDOF system, involving a spring constant k and damping ratio D. The spring constant k and the damping ratio D are computed for the anticipated modes of vibration, using the expressions given in Table 22.2. Figure 22.6 shows examples of modes of vibration.

2. Determine the exciting force. For a constant amplitude–exciting force, the load is expressed by

F F t= o sin( )ω (22.31)

or M M t= o sin( )ω (22.32)

where w, the operating frequency (rad/s), is 2pf, f is the operating frequency (cycle/s), Fo or Mo the amplitude of exciting force or moment (constant), F or M the exciting force or moment, and t the time.

The exciting force F or moment M may depend on the frequency and the eccentric mass. In this case

F meo = ω2 (22.33)

or M me Lo = ω2 (22.34)

where me is the eccentric mass, e the eccentric radius from the center of rotation to the center of gravity, and L the moment arm.

Table 22.2 Determination of lumped mass parameters

Mode Mass Ratio Damping Coefficient Spring Constant Damping Ratio

Vertical Bm

rzo

=−( )1

4 3

µρ C

r Gz

o=−

3 42

2.( )

ρµ

kGr

zo=

−41( )µ

DCk mz

z

z

=

Horizontal Bm

rxo

=−−

( )( )

7 832 1 3

µµ ρ C

r Gx

o=−

4 62

2.( )

ρµ

kGr

xo=

−−

32 17 8( )( )

µµ

DCk mx

x

x

=

Rocking BI m

rψψ=

−3 1

4 5

( )µ

ρ oC

r GBΨΨ

=− +0 8

1 1

4.( )( )

o ρµ

kGr

ψ = −8

3 1

3o

( )µD

C

k mψ

ψ

ψ

=

Torsional BIrθθ=

ρ o5 C

B G

Bθθ

θ

=−

41 2

ρ( )

kGr

θ =16

3

3o D

Ck mθθ

θ

=


22.5 proCeDure For DynamiC analySiS oF maChine FounDation • 937

3. Compute the undamped natural frequency by

fkmn =

12π

(22.35)

or fkIn =

12π

(22.36)

where k = kz for vertical mode, kx for horizontal mode, ky for rocking mode; kq for torsional mode; m is the mass of foundation and equipment for vertical and horizontal modes; I = Iy the mass moment of inertia around the axis of rotation in rocking mode; and Iq the mass moment of inertia around the axis of rotation in torsional mode. Thus, for vertical mode, we get

fkmn

z=1

2π (22.37)

For horizontal mode, we get

fkmn

x=1

2π (22.38)

For torsional (yawing) mode, we get

fk

Inq

q

=1

2π (22.39)

For rocking mode, we get

fk

Iny

y

=1

2π (22.40)

4. Compute the mass ratio B and damping ratio D for the modes analyzed using the equations in Table 22.2. Note that the damping terms are functions of mass and geometry and not of the internal damping in the soil. This damping is called radiation damping and represents the fact that energy is transmitted away from the founda-tion toward the distant boundaries of the soil.

X

Z

Y

Px, Zx

Mx, fx

G

My, fy

Mz, fz

Pz, Zz

Py, Zy

Rocking

Yawing

Pitching

Translation (Z)

Translation (X)

Translation (Y)

Figure 22.6 Modes of vibration of a machine foundation with six DOF.



5. Calculate static displacement amplitude as

AFkso= (22.41)

or calculate the static relation as

θso=

Mk

(22.42)

6. Compute the ratio f/fn.7. Calculate the magnification factor from Figs. 22.4 or 22.5 as the case may be.8. Calculate the maximum amplitude as

A MAmax = s (22.43)

θ θmax = M s (22.44)

9. If the amplitudes are not acceptable, modify design and repeat Steps (3)–(8).

22.6 Dynamic Soil Properties

For the dynamic analysis of machine foundations, soil properties, such as Poisson’s ratio, dynamic shear modulus, and damping coefficient of soil, are generally required. In general, problems involving the dynamic properties of soils are divided into small- and large-strain amplitude responses. For machine foundations, the amplitudes of dynamic motion, and, consequently, the strains in the soil are usually low (< 0.01%). A foundation that is subjected to an earthquake or blast loading is likely to undergo large deformations and, therefore, induces large strains in the soil.

The following dynamic properties of soil are used in the analysis and design of machine foundations:

1. Coefficient of elastic uniform compression: Coefficient of elastic uniform compression (Cu) is defined as the ratio of external uniform pressure to the elastic part of the settlement in compression. Its units are kgf/cm3 or kN/m3.

2. Coefficient of elastic non-uniform compression: Coefficient of elastic non-uniform compression (Cf) is defined as the ratio of external non-uniform pressure to the elastic part of the settlement in compression. Its units are kgf/cm3 or kN/m3.

3. Coefficient of elastic uniform shear: Coefficient of elastic uniform shear (Ct) is defined as the ratio of average shear stress at the foundation contact area to the elastic part of the displacement in sliding.

4. Coefficient of elastic non-uniform shear: Coefficient of elastic non-uniform shear (Cy) is defined as the ratio of the external moment applied to the vertical axis to the product of polar moment of inertia of contact area of foundation and the angle of rotation of the foundation.

5. Poisson’s ratio: Poisson’s ratio (m) is defined as the ratio of the strain in the direction perpendicular to loading to the strain in the direction of loading. Generally, Poisson’s ratio varies from 0.25 to 0.35 for cohesionless soils and from 0.35 to 0.45 for cohesive soils. If no specific values of Poisson’s ratio are available, then, the engineer may take Poisson’s ratio as 0.33 for cohesionless soils and 0.40 for cohesive soils for design purposes.

6. Dynamic shear modulus: Dynamic shear modulus (G) is defined as the ratio of dynamic shear stress and the cor-responding shear strain. Dynamic shear modulus is the most important soil parameter influencing the dynamic behavior of the foundation–soil system. Dynamic shear modulus represents the slope of shear stress versus shear strain curve. Most soils do not respond elastically to shear strains; and the shear strain is a combination of elastic and plastic strain. Hence, shear stress versus shear strain relationship of most soils is nonlinear. The value of G varies based on the strain considered. The lower is the strain, the higher is the dynamic shear modulus.

7. Damping coefficient: Damping is a phenomenon of energy dissipation that opposes vibrations of a system. Like restoring forces, damping forces oppose the motion, but the energy dissipated through damping cannot be recovered. A characteristic feature of damping forces is that they lag displacement and are out of phase with the motion. Damping of soil includes two effects: geometric damping and material damping.


22.6 DynamiC Soil propertieS • 939

Geometric damping reflects energy dissipation through propagation of elastic waves away from the immediate vicinity of a foundation. It is also called as radiation damping. It results from the semi-infinite nature of the soil medium, and it is close to viscous in character.

Many practitioners use a material damping ratio of 0.05, or 5%, instead of an experimental determination. The material damping ratio is fairly constant for small strains but increases with strain due to the nonlinear behavior of soils. The term material or hysteretic damping implies frequency independent damping. Experiments indicate that frequency independent hysteretic damping is much more typical of soils than viscous damping because the area of the hysteretic loop does not grow in proportion to the frequency.

22.6.1 Relation between Dynamic Constants

The relationship among various dynamic properties of soil depends on elastic properties of the soil, the size, and shape of the contact area, and the flexibility or rigidity of the foundation. The key soil properties, Poisson’s ratio, and dynamic shear modulus may be significantly affected by water table variations. Equations (22.45)–(22.47) give the relationship between different dynamic properties of soil.

C C Cu to = 1 5 2. τ τ (22.45)

C Cφ = 3 46. τ (22.46)

C Cψ τ= 1 5. (22.47)

Due to variations inherent in the determination of dynamic shear modulus, it may be appropriate to perform more than one foundation analysis. One analysis should be done with the minimum possible value and another using the maximum possible value, and then additional analyses may be done with intermediate values.

In the case of very stiff soils, the value of Cu may be so high that the natural frequency of the foundation soil system may not be reached because of limitations of the vibration exciting equipment. The frequency response curves in such cases may be extrapolated to obtain the resonant frequency of the foundation soil system.

22.6.2 Determination of Dynamic Soil Properties

Soil properties to be used in dynamic analyses can be measured in the field or in the laboratory. In many important projects, a combination of field and laboratory measurements is used, as follows:

1. Field methods: Field methods for determination of dynamic soil properties – in situ methods include:• Seismic cross-hole method. • Seismic down-hole method. • Seismic up-hole method.• Steady-state vibration test. • Spectral analysis method. • Seismic-cone method. • Block vibration test. • Cyclic plate load test.• Seismic-reflection test. • Seismic-refraction test.

2. Laboratory methods: Laboratory methods for determination of dynamic soil properties include:• Resonant-column method. • Cyclic triaxial test.• Ultrasonic pulse test. • Piezoelectric bender element test.

The description of some of the important methods is given in the following subsections.

22.6.2.1 In Situ Methods for Determination of Dynamic Soil Properties

Field measurements are the most common methods for determining the dynamic shear modulus of a given soil. In situ techniques are based on the measurement of the velocity of propagation of stress waves through the soil. Following are three different types of stress waves that can be transmitted through soil or any other elastic body.



1. Compression (primary, P) waves: Compression waves are transmitted through soil by a volume change associ-ated with compressive and tensile stresses. Compression waves are the fastest of the three stress waves and they are also called as primary waves or P-waves.

2. Shear (secondary, S) waves: Shear waves are transmitted through soil by the distortion associated with shear stresses in the soil and are slower than compression waves. No volume change occurs in the soil due to shear waves and they are also called as secondary waves or S-waves.

3. Rayleigh (surface) waves: Rayleigh waves occur at the free surface of an elastic body; typically, this is the ground surface. Rayleigh waves have components that are both perpendicular to the free surface and parallel to the free surface and are slightly slower than shear waves.

Most techniques measure the S-waves because the P-waves waves are dominated by the response of the pore fluid in saturated soils. Direct measurement for soil or rock stiffness in the field has the advantage of minimal material disturbance. The modulus is measured where the soil exists. Furthermore, the measurements are not constrained by the size of a sample.

The shear modulus is determined from the velocity of the S-waves using

G V= ρ s2 (22.48)

where G is the dynamic shear modulus of the soil, (N/m2); Vs the shear wave velocity of the soil, (m/s); and r the soil mass density, (kg/m3).

Several methods are available for measuring wave velocities of the in-place soil, and these are explained as follows.

A. Seismic Cross-Hole Method

In the cross-hole method (see Fig. 22.7), two boreholes are drilled to some depth, preferably on each side of the base location so that the shear wave can be measured between the two holes and across the base zone. A signal generator is placed in one hole at some depth and a sensor is placed in the other at the same depth. An impulse signal is gener-ated in one hole, and the waves travel horizontally from the source to the receiving holes. The time taken by the shear wave to travel from the signal generator to the sensor is measured. The distance divided by the travel time yields the shear wave velocity. The cross-hole method can be used to determine the dynamic shear modulus at different depths.

As it can be difficult to establish the exact triggering time, the most accurate measurements are obtained from the difference of arrival times at two or more receiving holes rather than from the time between the triggering and the arrival at single hole.

The original cross-hole velocity measurement methods used explosives as the source of energy, and these were rich in compression energy and poor in shear energy. Since P-waves travel faster than S-waves, the sensors will already be excited by the P-waves when the S-waves arrive, which makes it difficult to pick out the arrival of the S-wave. For this reason, explosives should not be used as energy sources for cross-hole S-wave velocity measure-ments. To overcome this difficulty, it is desirable to use an energy source that is rich in the vertical shear component of motion and relatively poor in compressive motion.

B. Seismic Down-Hole Method

In the seismic down-hole method, only one vertical borehole is drilled. A signal generator is placed at the ground surface some distance away from the borehole, and a sensor is placed at various depths in the borehole. A source rich in S-waves should be used. This technique does not require as many borings as the cross-hole method, but the waves travel through several layers from the source to the sensors. Thus, the measured travel time reflects the cumulative travel through layers with different wave velocities, and interpreting the data requires sorting out the contribution of the each soil layer. The travel time divided by the distance yields the shear wave velocity.

This method can be run on different times, with the signal generator located at different distances from the borehole each time. This permits the measuring of soil properties at several locations, which can then be averaged to determine an average shear wave velocity (Fig. 22.8). The seismo-cone version of the cone penetration test is one example of the down-hole method.

C. Seismic Up-Hole Method

The principle and procedure for conducting the seismic up-hole test are the same as those for the seismic down-hole test, except for the difference that the source of energy is placed at some depth in the borehole and the sensors are above it at the ground surface.



D. Steady-State Vibration Test

The method consists of generating vibrations at a point and detecting the points that are in phase with the gener-ated vibrations. The waves that are picked up by geophones are the Rayleigh waves, which primarily cause dis-placements along the ground surface and adjacent to the vibration source. The horizontal distances between such points are equal to the wavelength of the Rayleigh wave. The use of Rayleigh waves eliminates the problem of detecting wave arrivals and measuring arrival times.

For uniform soil extending up to infinite depth, the wavelength of propagating vibrations is given by

λπ

π λ λ=

+ −4

2 1 2

S( )

(22.49)

where, l is the wavelength in centimeters, S the measured distance between geophones in centimeters, l1 the phase shift of geophones with respect to wave nearer to concrete block at the frequency of the propagating vibrations in radians, and l2 the phase shift of the other geophone at the frequency of the propagating vibrations in radians.

The velocity of shear waves Vs is given by

V fs = λ (22.50)

where f is the frequency of vibration at which the wavelength has been measured.When the test is conducted using a phase meter, the phase angle corresponding to different distances between

the geophones should be recorded and a curve should be plotted between the phase angle and the distance. From the curve, the distance S between the geophones for a phase difference of 90° should be determined. The steady-state vibration test is useful to determine the shear wave velocity near the ground surface, but it cannot easily pro-vide detailed resolution of highly variable velocity profiles. For this reason, the steady-state vibration test has been replaced by the spectral analysis of the surface wave test.

E. Spectral Analysis of Surface Waves Method

The spectral analysis of surface waves (SASW) method is a recently developed technique that does not require bor-ings. The basis of the SASW method is the dispersive characteristic of Rayleigh waves when travelling through a layered medium. The dispersion curve is a plot of velocity of Rayleigh wave as a function of wavelength. Rayleigh wave velocity is determined by the material properties of the sub-surface up to a depth of approximately 1–2 wavelengths. Longer wavelengths penetrate deeper and their velocity is affected by the material properties at greater depth, as shown in Figs. 22.9(a) and (b).

The dispersion curve, shown in Fig. 22.9(c), in this method, is obtained by the use of digital data acquisition and signal processing equipment and hence the method is known as SASW method. This method is, therefore, more effective and less time-consuming. This technique uses sensors that are spread out along a line at the surface, and the source of energy is a hammer or tamper also at the surface. The surface excitation generates surface waves, in particular Rayleigh waves. The particles move in retrograde ellipses, whose amplitudes decay from the surface. The test results are interpreted by recording the signals at each of the receiving stations (see Fig. 22.10) and by performing spectral analysis of the data. Computer programs have been developed that can determine shear wave velocities from the results of the spectral analysis.

Oscilloscope

TransducerShear wave

Trigger Capacitivecircuit

Hammer

Impulserod

Figure 22.7 Cross-hole method schematic diagram.

Recorder

Woodenhammer

Woodenplate

WeightExpander

pump

Rubberexpander

Three-componentgeophone

Back plate

Triggergeophone

Figure 22.8 Seismic down-hole method schematic diagram.



The SASW method offers significant advantages, particularly for study of dynamic soil properties near ground surface. In contrast to borehole measurements that are point estimates, SASW testing is a global measurement, that is, a much larger volume of the sub-surface is sampled. The resulting profile is representative of the sub-surface properties averaged over distances of up to several hundred metres. To increase the depth of the measurements, the energy at the source must be increased. Measurements for the few meters below the surface, which may be adequate for evaluating pavements, can be accomplished with a sledge hammer as a source of energy, but deeper measurements require track-mounted seismic “pingers.” The SASW method works best in cases where the stiffness of the soils and rocks increases with depth. If there are soft layers lying under stiff ones, the interpretation may be ambiguous. A soft layer lying between stiff ones can cause problems for the cross-hole method as well because the waves will travel faster through the stiff layers and the soft layer may be masked. The method is more or less similar to Seismic refraction method used in preliminary soil investigations for static analysis.

The cross-hole technique employs waves with horizontal particle motion, and the down-hole and up-hole meth-ods use waves whose particle motions are vertical or nearly so; and the surface waves in the SASW method have particle motions in all sensors. Therefore, a combination of these techniques can be expected to give a more reliable picture of the shear modulus than any one used alone.

F. Seismic Cone Method

Seismic cone method is similar to the seismic down-hole test, except that no borehole is required but a seismic cone is used. This method directly measures the shear wave velocity by incorporating a small velocity seismometer (an electronic pickup device) inside the cone penetrometer. The test essentially consists of pushing the seismic cone to some depth and then applying a shock at the ground surface, using a hammer and a striking plate. This device has the advantage of not requiring a borehole. It requires that the site soil be suitable for a CPT, that is, the soil should

(a) (b)

Layer 1

Layer 2

Layer 3

Air

Dep

th

R1

R2

Rayleigh wave vertical particle motion

Longerwavelength

Shortwavelength

(c)

Velocity of surface wave

Wav

elen

gth

1000

100

10

10 1000 2000 3000

Garner valleyarray 1

Figure 22.9 Principle of SASW method (Geo-vision): (a) Soil profile, (b) penetration of Rayleigh wave, and (c) dispersion curve.

Vertical dynamic source:forward configuration

Dynamic signal analyzerwith disk drive

Reverse configuration

d1 - forward d2 d1 - reverse

Figure 22.10 SASW method – schematic.



be fine-grained, with little to no gravel. The seismic cone test is more efficient than the down-hole test and hence is more commonly used.

G. Block Vibration Test

Block vibration test is an in situ model test recommended by IS:5249 – 1992 for the determination of the coefficient of elastic uniform compression and the damping coefficient of the soil using either the forced vibration test or the free vibration test. The following is the procedure for conducting the test:

1. Test pit: A test pit of suitable size depending upon the size of the block is made. For a block size of 1 m × 1 m × 1.5 m, the size of the pit may be 3 m × 6 m at the bottom and a depth preferably equal to the proposed depth of foundation. The bottom of the pit should be level and horizontal, and the sides of the pit should be at a stable slope or may be kept vertical, where possible. The test should be conducted above the groundwater table. In the case of a rock, the test may be performed on the surface of the rock bed itself.

2. Test block: A plain concrete block of grade M-15 is constructed in the test pit as shown in Fig. 22.11. The size of the block is selected depending upon the sub-soil conditions. In ordinary soils, it may be 1 m × 1 m × 1.5 m; in dense soils, it may be 0.75 m × 0.75 m × 1 m. In boulder deposits, the height may be increased suitably. The block size is so adjusted that the mass ratio, given as follows, is always more than unity:

BM

zro

=−

×( )1

4 3

µρ

(22.51)

where Bz is the mass ratio, m is the Poisson’s ratio of soil, M is the mass of the block, and rro is the equivalent radius of the base.

The concrete block is cured for at least 15 days before testing. Foundation bolts are embedded into the concrete block at the time of testing for fixing the oscillator assembly. Details of the test block are shown in Fig. 22.11(a).

3. Test setup: Vibration exciter is fixed on the concrete block, and suitable connection between the power supply and the speed control unit is made as shown in Fig. 22.11(b). Any suitable electronic instrumentation may be used to measure the frequency and amplitude of vibrations.

4. Procedure: The block vibration test may be conducted using the following two methods: • Forced vibration test: The forced vibration test conducted in vertical vibration mode is explained in this sec-

tion. Vibration pickups are fixed at the top of the block, as shown in Fig. 22.11(a), such that they sense vertical motion of the block. The vibration exciter is mounted on the block in such a way that it generates purely

(a)

Pick-upsC.C.Block

d2

d1

Meter

(b)

Motor andoscillator Speed control unit

Pick up/transducer

Block

Powersupply

OscillographAmplifier

Figure 22.11 Block vibration test: (a) Test setup and (b) schematic diagram.



vertical sinusoidal vibrations, and the line of the action of vibrating force passes through the center of gravity of the block. The exciter is operated at a constant frequency. The signal of the vibration pickups are fed into a suitable electronic circuit to measure the frequency and amplitude of vibration. The frequency of the exciter is increased in steps of small values, (l–4 cycles/s) up to a maximum frequency of the exciter and the signals measured. The same procedure is repeated, if necessary, for different excitation levels. The dynamic force should never exceed 20% of the total mass of the block and exciter assembly.

Amplitude versus frequency curve is plotted for each excitation level to obtain the natural frequency of the soil and the foundation block tested. A typical plot is shown in Fig. 22.12.(a) Determination of coefficient of elastic uniform compression of soil: The coefficient of elastic uniform compres-

sion of soil for the test block (Cut) is determined from

Cf M

Autnz

t

=4 2 2π

(22.52)

where fnz is the natural frequency; M the mass of the block, exciter, and motor; and At the contact area of the test block with the soil.

From the value of Cut obtained for the test block, the value of Cuf for the foundation may be obtained from Eq. (22.53) as

C CAAuf ut

t

f

= (22.53)

where Af is the area of the foundation at the base. This relation is valid for small variations in the base area of the foundations and may be used for an area up to 10 m2.

For actual foundation areas larger than 10 m2, the value of Cuf obtained for 10 m2 may be used.(b) Determination of damping coefficient of soil: In the case of the vertical vibration test, the value of the damping

coefficient e of soil is determined using Eq. (22.54) as

ε =−f ff

2 1

2 nz

(22.54)

AA

ε =m

2 (22.55)

where f1 and f2 are the two frequencies at which the amplitude is equal to Ae; Ae the amplitude as given by Eq. (22.55); Am the maximum amplitude, and fnz the frequency at which the amplitude is maximum (resonant frequency), as shown in Fig. 22.13.

• Free vibration test: In free vibration test, the block is excited into free vertical vibrations by the impact of a sledge hammer or any suitable device, as near to the center of the top face of the block as possible. The vibrations

fn

Frequency, cps

Peak amplitude

Am

plitu

de, m

m

00

1.0

2.0

3.0

4.0

15 20 25 30 35

Figure 22.12 Amplitude–frequency curve for vertical vibration test.

f2

Am

Am

Frequency, cps

Am

plitu

de, (

A)

fnz f1

2

Figure 22.13 Determination of damping from forced vibration test.



are recorded on a pen recorder or suitable device to measure the frequency and amplitude of vibration. The test may be repeated 3–4 times.(a) Determination of coefficient of elastic uniform compression of soil: In the case of the free vertical vibration tests,

the value of CU is obtained from the natural frequency of the free vertical vibration using Eq. (22.53). The damping coefficient may be obtained from the free vibration tests using

επ

=1

2loge

m

m+1

AA

(22.56)

where Am and Am+1 are explained in Fig. 22.14.(b) Determination of coefficient of attenuation: The test set up is the same as that for the block resonance test. The

pickup fitted on the block is removed and installed at a certain distance d1 (approximately 30 cm) from the block. The second pickup is fixed in line with this pickup and the center of the block at a distance of d2. The amplitude of vibration at these two locations is measured for different frequencies. The coefficient of attenuation is calculated from the following expression:

A Add

e d d2 1

1

2

2 1= ⋅ − −α ( ) (22.57)

where A1 is the amplitude at distance d1, A2 the amplitude at distance d2 and a the coefficient of attenua-tion. Typical values of a are given in Table 22.3.

H. Cyclic Plate Load Test

The cyclic plate load test is used to determine the coefficient of elastic uniform compression and is similar to plate load test discussed in Sec.19.7 of Chapter 19. The procedure for conducting the test consists of loading a rigid test plate at foundation level in increments and in several cycles and measuring settlement of the plate under each load increment and in each cycle.

Wd = Damped natural frequency of system

Am

plitu

de o

f vib

ratio

n (A

)

A(m−1), wdt(m−1)

A(m+1), wdt(m+1)

Am, wdtm

wdto

2p 2p

Figure 22.14 Determination of damping coefficient from free vibration test.

Table 22.3 Typical values of coefficient of attenuation, a

Soil Type a (m-l)

Saturated sand or sandy silt 0.1

Saturated silty sand 0.04

Saturated sandy silty clay 0.04–0.12



1. Equipment: Suitable arrangement for providing reaction of adequate magnitude depending upon the size of the plate employed is made. The load mechanism should have the facility to apply and remove the loads quickly. A hydraulic jack or any other suitable equipment may be used.

2. Test procedure:• The equipment for the test is assembled according to the details given in IS:1888–1982 (see sec. 19.7 of

Chapter 19). The plate should be located at a depth equal to the depth of the proposed foundation in a pit excavated.

• After the setup has been arranged, the initial readings of the dial gauges are noted, and the first increment of static load is applied to the plate. This load should be maintained constantly throughout for a period till no further settlement occurs or the rate of settlement becomes negligible. The final readings of the dial gauges are then recorded.

The entire load is then removed quickly but gradually, and the plate is allowed to rebound. When no further rebound occurs or the rate of rebound becomes negligible, the readings of the dial gauges are again noted. The load should be increased gradually till its magnitude reaches a value equal to the proposed next load increment, which should be maintained constant and the final dial gauge readings are noted as mentioned earlier. The entire load should then be reduced to zero, and the final dial gauge readings should be recorded when the rate of rebound becomes negligible.

• The cycles of loading, unloading, and reloading are continued till the estimated ultimate load has been reached, the final values of dial gauge readings being noted each time.

• The magnitude of the load increment should be such that the ultimate load is reached in 5–6 increments. The initial loading and unloading cycles up to the safe bearing capacity of the soil should be with smaller increments in load. The duration of each loading and unloading cycle depends upon the type of soil under investigation.(a) Determination of coefficient of elastic uniform compression from cyclic plate load test: From the data obtained

during cyclic plate load test, the elastic rebound of the plate corresponding to each intensity of loading is obtained as shown in Fig. 22.15. The load intensity versus elastic rebound data is plotted as shown in Fig. 22.16. The value of Cu is calculated by

CPSu

e

= (22.58)

where Cu is the coefficient of elastic uniform compression in kgf/cm3 and Se the elastic rebound in centim-eters, corresponding to the load intensity P in kgf/cm2.

Limitations of In Situ Methods

The dynamic properties measured in the field correspond to very small strains. Although some procedures for measuring moduli at large strain have been proposed, none has been found fully satisfactory. Field techniques have

Elastic rebound

Se

PLoad

Figure 22.16 Determination of Cu from cyclic plate load test.

∆1P1 P2 P3 P4 P5

∆2

∆3∆4

∆5

Load

Set

tlem

ent

Figure 22.15 Load–settlement curve for cyclic plate load test D1, D2, D3, D4, and D5, are elastic rebound at loads P1, P2, P3, P4, and P5, respectively.



also failed to prove effective in measuring material damping, since the dissipation of energy during strain, which is called material damping, requires significant strains to occur.

22.6.2.2 Laboratory Methods

Laboratory measurements of soil properties can be used to supplement or confirm the results of field measure-ments. They can also be necessary to establish values of damping and modulus at strains larger than those that can be attained in the field or to measure the properties of materials that do not now exist in the field, such as soils to be compacted. These tests can generally be classified into two groups: those that apply dynamic loads and those that apply loads that are cyclic but slow enough that inertial effects do not occur.

Laboratory tests are considered less accurate than field measurements due to the possibility of sample distur-bance. But they are used to validate field measurements when a high level of scrutiny is required; for instance, when soil properties are required for a nuclear energy facility.

A. Resonant-Column Method

Resonant-column method is the most widely used laboratory test that applies dynamic loads. The method is used for the determination of shear modulus, shear damping, Young’s modulus, and damping for solid cylindrical spec-imens of soil in undisturbed and remolded conditions by vibration using a resonant column.

A resonant column consists of a cylindrical soil specimen that has platens attached to each end as shown in Fig. 22.17. The cylindrical soil specimen is placed in a device capable of generating longitudinal or torsional forced vibrations. A sinusoidal vibration excitation device is attached to the active-end platen. The other end is the passive-end platen that may be rigidly fixed. The vibration excitation device consists of springs and dashpots con-nected to the active-end platen, where the spring constants and viscous damping coefficients are known.

The complete test apparatus includes active and passive platens for holding the specimen in the pressure cell, the vibration excitation device, transducers for measuring the response, and the control and recording instrumen-tation. Platens are constructed from a non-corrosive material with a modulus at least 10 times the modulus of the material to be tested and with a stiffness that is at least 10 times the stiffness of the specimen. Each platen has a circular cross section, and a plane surface of contact with the specimen; the surface of contact with the soil specimen may be roughened to provide for more efficient coupling with the ends of the specimen. The active-end platen may have a portion of the excitation device, transducers, springs, and dashpots connected to it.

The soil specimen should have a minimum diameter of 33 mm. The largest particle contained within the test specimen should be smaller than one-tenth of the specimen diameter except that, for specimens having a diameter of 70 mm (2.8 in.) or larger, the largest particle size should be smaller than one-sixth of the specimen diameter. The length-to-diameter ratio should not be less than 2 and not more than 7. However, when an ambient axial stress greater than the ambient lateral stress is applied to the specimen, this ratio should be between 2 and 3.

The vibration excitation device consists of an electromagnetic device capable of applying a sinusoidal lon-gitudinal vibration or torsional vibration or both to the active-end platen to which it is rigidly coupled. The

Mass

Passive-end platen

Soil specimen

Active-end platen

Torsional springTorsional dashpot

Excitation deviceLongitudinal dashpot

Longitudinal spring

Figure 22.17 Resonant-column method (schematic).



vibration-measuring devices include acceleration, velocity or displacement transducers that can be attached to and become a part of the active- and passive-end platens. One transducer is mounted on each platen to produce a cali-brated electrical output that is proportional to the longitudinal acceleration, velocity or displacement of that platen.

The vibration apparatus and the specimen may be enclosed in a triaxial chamber and subjected to an all-around pressure and axial load. Transducers are used to measure the vibration amplitudes for each type of motion at the active end and also at the passive end if it is not rigidly fixed.

The frequency is varied until resonance occurs. The dynamic soil modulus can be calculated based on the fre-quency, the length of the soil sample, the end conditions of the soil sample, and the density of the soil sample. From the frequency and amplitude at resonance, the modulus and damping of the soil can be calculated. Young’s modulus, E, is determined from the longitudinal vibration, and shear modulus, G, is determined from the torsional vibration.

The dynamic Young’s modulus is calculated as,

E lfF

=

ρ π( )2 2

2L

L

(22.59)

where, r = is the mass density of the soil specimen, fL the resonant frequency for longitudinal motion, FL the fre-quency factor, and l the length of the soil specimen.

The shear modulus is calculated as,

G lfF

=

ρ π( )2 2

2T

T

(22.60)

where, fT is the resonant frequency for torsional motion and FT the frequency factor.For longitudinal motion, the damping factor is calculated as

D D f ML A= ⋅( )2π L (22.61)

where DA is the damping coefficient of the apparatus for longitudinal motion, fL the resonant frequency for longitu-dinal motion, and M the mass of the soil specimen.

For torsional motion, the damping factor is calculated as

D D f JT AT T= ⋅( )2π (22.62)

where DAT is the damping coefficient of the apparatus for torsional motion, fT the resonant frequency for torsional motion, J the rotational inertia about axis of rotation (Md2/8), and d the diameter of the soil specimen.

There are several types of resonant-column devices that have been developed. These devices provide measure-ments of both the modulus and the damping at low-strain levels. Although the strains can sometimes be raised by a few percent, these devices remain essentially low-strain.

The modulus and damping of a given soil, as measured by the resonant-column technique, depend upon the strain amplitude of vibration, the ambient state of effective stress, the void ratio of the soil, temperature, time, etc. Since the application and control of the ambient stresses and the void ratio are not prescribed in these methods, the applicability of the results to field conditions will depend on the degree to which the application and control of the ambient stresses and the void ratio, as well as other parameters such as soil structure, simulate field conditions. For further information on the test, the ASTM D 4015: Resonant-column method may be referred.

B. Cyclic Triaxial Test

The most widely used cyclic loading laboratory test is the cyclic triaxial test. In this test, a cyclic load is applied to a column of soil over a number of cycles slowly enough that inertial effects do not occur. The response at one amplitude of the load is observed, and the test is repeated at a higher load. Figure 22.18(a) shows the typical pattern of stress and strain, expressed as shear stress and shear strain. The shear modulus is the slope of the secant line inside the loop.

The critical damping ratio, D, is given by

DAA

= l

T4π (22.63)

where, Al is the area of the loop and AT the shaded area.



Other types of cyclic loading devices also exist, including cyclic simple shear devices. Their results are inter-preted similarly. These devices load the sample to levels of strain much larger than those attainable in resonant-col-umn devices. A major problem in both resonant-column and cyclic devices is the difficulty of obtaining undisturbed samples. This is especially true for small-strain data because the effects of sample disturbance are particularly apparent at small strains.

The results of laboratory tests are often presented in a form similar to Figs. 22.18(b) and (c). In Fig. 22.18(b), the ordinate is the secant modulus divided by the modulus at small strains. In Fig. 22.18(c), the ordinate is the value of the initial damping ratio. Both are plotted against the logarithm of the cyclic strain level.

Correlation to Other Soil Properties

Correlation is another method for determining dynamic soil properties. Correlation methods should be used with caution because these are generally the least-accurate methods. The most appropriate time to consider using these methods is when deciding on the preliminary design or small non-critical applications with small dynamic loads. Correlation to other soil properties should be considered as providing a range of possible values, not providing a single exact value.

In general, relative density in sand is proportional to the void ratio. Seed and Idriss (1970) provide guidance for correlating dynamic shear modulus to relative density in sand, along with the confining pressure.

G K= 6920 2 σo (22.64)

where K2 is a parameter that depends on void ratio and strain amplitude, and Table 22.4 provides values of K2 with respect to relative density of the soil.

(a)

A1 = Area of loop

GG0

AT

t

g

D = A1

4 p AT

(c)

D

(b)

g g

G

G0

1

0.8

0.6

0.4

0.2

00.00001 0.0001 0.001 0.10.01 1

1

0.8

0.6

0.4

0.2

00.00001 0.0001 0.001 0.10.01 1

Figure 22.18 Determination of dynamic soil properties in the laboratory: (a) Variation of shear stress, (b) G/Go vs cyclic shear strain, and (c) initial damping ratio vs cyclic shear strain.



22.7 Design Criteria of Machine Foundations

The basic goal in the design of a machine foundation is to limit its motion to amplitudes which neither endanger the stability and satisfactory operation of the machine nor disturb people working in the immediate vicinity.

22.7.1 General Requirements of Machine Foundation

The following requirements should be fulfilled from the design point of view of machine foundations:

1. The foundation should be able to carry the superimposed loads including the self-weight, without causing shear failure of the underlying soil.

2. The settlement of the foundation should be within the permissible limits.3. The combined center of gravity of the machine and the foundation should be on the vertical line passing

through the center of gravity of the base plane.4. There should be no resonance, that is, the natural frequency of the foundation–soil system should be either

too large or too small compared with the operating frequency of the machines. For high-frequency machines (> 1000 rpm), it is common to “low tune” the foundation so that the foundation frequency is less than half the operating frequency. For low-frequency machines (< 300 rpm), it is common to “high tune” the foundation, so that the fundamental frequency is at least twice the operating frequency.

5. For machinery with operating speeds exceeding about 1000 rpm, a foundation should be with natural fre-quency less than one-half the operating speed, by the following methods:• Decrease the natural frequency by increasing the weight of the foundation block.• During starting and stopping, the machine will operate briefly at the resonant frequency of the foundation.

Compute probable amplitudes at both the resonant and the operating frequency, and compare them with allowable values to alter the foundation.

6. For machinery operating at a speed less than about 300 rpm, a foundation should be provided with natural frequency at least twice the operating speed, by the following methods:• For spread foundations, increase the natural frequency by increasing the base area or reducing the total static

weight.• Increase the modulus or shear rigidity of the foundation soil by compaction or other means of stabilization.• Consider the use of piles to provide the required foundation stiffness.

7. The amplitude of motion at operating frequencies should not exceed the limiting amplitude, which is generally specified by machine manufacturers. Allowable amplitudes depend on the speed, location, and criticality or function of the machine. The limiting amplitude as a function of time to achieve various levels of safety/comfort is shown in Fig. 22.19. If the computed amplitude is within tolerable limit, but is close to resonance, it is important that this situation should also be avoided.

8. The foundation should be isolated from the main building at all levels and from other foundations as far as possible.

9. Concrete members are designed to prevent cracking due to fatigue and stress reversals caused by dynamic loads, and the machine’s mounting system is designed to transmit loads from the machine into the foundation.

10. It is desirable to cast the entire foundation block in one operation. If a construction joint is unavoidable, the plane of joint should be horizontal and measures should be taken to provide a proper joint.

Table 22.4 Typical values of K2 (Courtesy : Seed and Idriss, 1970)

Relative Density (%) K2

90 70

75 52

45 43

40 40

30 34


22.7 DeSign Criteria oF maChine FounDationS • 951

11. Before placing a new layer of concrete, the previously laid surface should be roughened, thoroughly cleaned, washed by a jet of water and then covered by a 2 cm thick layer of 1:2 cement grout. Concrete should be placed within 2 h of laying the grout.

12. The groundwater level should be as low as possible, and it should be at least deeper by one-fourth of the width of the foundation below the base of the foundation. To achieve this condition, the zone of soil around the machine foundation should be made water proof. Adequate drainage measures should be adopted to divert groundwater or rain water away from the machine foundation.

13. The machine foundation should be taken to a level lower than the level of the foundations of adjacent buildings.14. Any steam or hot air pipes, embedded in the foundation must be properly isolated.15. The foundation must be protected from machine oil by means of an acid-resistant coating or suitable chemical

treatment.

22.7.2 Foundations for Reciprocating Machines

The design and construction criteria applicable to foundations of reciprocating machines, as per IS:2974 (Part II)– 1980, are as follows:

1. Design:• The natural frequency of the foundation–soil system should be higher than the highest disturbing frequency,

and the frequency ratios should not be less than 0.4. Where this is not possible, the natural frequency of the foundation–soil system should be kept lower than the lowest disturbing frequency. The frequency ratios in such cases should not be lower than 1.50.

• The permissible amplitude to avoid damage to machinery specified by the manufacturer should in no case be exceeded.

• The permissible amplitude to avoid damage to neighboring buildings is 0.2 mm at frequencies below 1200 rpm. Where the disturbing frequency exceeds 1200 rpm, a lower amplitude may be necessary for cer-tain installations. The limiting amplitude for this condition at different frequencies is specified by the line ADD’ of Fig. 22.20.

Limit for m

achines andSevere to persons

Troublesome to persons

Machine foundations

Easily noticeable to persons

Barely noticeable to persons

Non noticeable to persons

Danger to structures

Caution to structures

Frequency, cpm

2.5

1.25

0.50

0.25

0.125

0.050

0.025

0.0125

0.0050

0.0025100 200 500 1000 2000 5000 10000

Am

plitu

de o

f vib

ratio

n, m

m

Figure 22.19 Limiting amplitude for vertical vibrations.



• The mass of the foundation should be greater than that of the machine.• The eccentricity of the foundation system should not exceed 5% of the width of the corresponding side of

the contact area. In addition, the center of gravity of the machine and the foundation should be below the top of the foundation block.

• The allowable bearing pressure for dynamic loads is usually taken as 80% of that for static loads.2. Construction:

• RCC block foundations are generally provided for reciprocating machines. Cellular foundations may be used in special cases where it is necessary to maintain the rigidity of a block foundation at the same time facilitating mass saving of concrete.

• To ensure reasonable stability in the case of vertical machines, the total width of the foundation (measured at right angles to the shaft) should be at least equal to the distance from the center of the shaft to the bottom of the foundation. In the case of horizontal machines, where cylinders are arranged laterally, the width should be greater.

• The minimum reinforcement in a concrete block is 25 kgf/m3. For machine, requiring special design consid-erations of foundations, such as machines pumping explosive gases, the reinforcement should not be less than 40 kgf/m3.

• The minimum reinforcement in a block should usually consist of 12 mm bars spaced at 200–250 mm center to center, extending both vertically and horizontally near all the faces of the foundation block.

• Reinforcement should be used at all faces. If the height of the foundation block exceeds 1 m, shrinkage rein-forcement should be placed at suitable spacing in all three directions.

• Reinforcements should be provided around all pits and openings equal to 0.50%–0.75% of the cross-sectional area of the opening.

• The concrete grade should be at least M-l5. The foundation block should preferably be cast in a single con-tinuous operation.

• When it is impracticable to design a foundation consisting of a simple concrete block resting on natural soils, anti-vibration mountings should be used to reduce the transmitted vibrations to acceptable levels. The anti-vibration mounting may be used between the machinery and the foundation, and between a foun-dation block and the supporting soil.

Disturbing frequency

Line ADD' limit to avoid damage to buildings; line ACC' limit to avoid serious discomfort topersons; line ABB' limit to ensure reasonable comfort to persons.

Am

plitu

de o

f vib

ratio

ns (

mic

rom

eter

s (2

))

30.25

0.50

1.00

1.502.002.50

5.0

7.510

152025

50

75100

150200250

5 10 20 30 50 100160 (Hz)

D′

C′

B′

A B C D

Figure 22.20 Permissible amplitude for foundation of reciprocating machine.



• Pile foundations may be used in cases when (a) the soil conditions are unsuitable to support block founda-tion, (b) the natural frequency of block foundations needs to be raised when it is impossible to alter dimen-sions, or (c) when amplitudes or settlement or both need to be reduced.

• The minimum thickness of pile caps for pile foundations is 0.6 m.

22.7.3 Foundations for Impact Machines

The design and construction criteria applicable to foundations of impact machines, as per IS:2974 (Part II)–1980, are as follows:

1. Design:• The centers of gravity of the anvil and the foundation block should coincide in plan. Similarly, the resultants

of the forces in the elastic pad and the foundation support, should coincide with the line of fall of the ham-mer-tup in plan.

• The load intensity on the soil below the foundation should not be more than 80% of the allowable bearing pressure of the foundation soil.

• The maximum permissible amplitudes are as shown in Table 22.5• In case any important structure exists near the foundation, the amplitude of the foundation should be adjusted

so that the velocity of vibrations at the structure does not exceed 0.3 cm/s.• The area of the foundation block at the base should be such that the safe loading intensity of the soil is never

exceeded during the operation of the hammer.• The minimum thickness of the foundation block below the anvil should be as shown in Table 22.6.• The mass of the anvil is kept generally 20 times the mass of the tup. The mass of the foundation block (Wb)

should be at least three times that of the anvil.• The mass of the block should be four to five times the mass of the anvil for foundations resting on stiff clays

or compact sandy deposits. For moderately firm to soft clays and for medium-dense to loose-sandy deposits, the mass of the block should be five to six times the mass of the anvil.

2. Construction:• The foundation block should be made of reinforced concrete of minimum M-15 grade.• Dowels of 12–16 mm diameter at 60 mm centre to centre should be embedded to a depth of at least 30 cm on

both sides of the joint.• Reinforcement should be arranged along the three axes and also diagonally to prevent shear.• More reinforcement should be provided at the top side of the foundation block than at the other sides.

Reinforcement at the top may be provided in the form of layers of grills made of 16 mm diameter bars suitably spaced to allow easy pouring of concrete. The topmost layers of the reinforcement should be provided with a cover of at least 5 cm. The reinforcement provided should be at least 25 kg/m3 of concrete.

• The protective layer between the anvil and the foundation block should be safeguarded against water, oil scales, etc. and the material selected should withstand temperatures up to 100°C.

• Air gaps and spring elements provided for the purpose of damping vibrations should be accessible to remove scales and enable inspection of springs and their replacement, if necessary.

Table 22.5 Maximum permissible amplitude

Item Permissible Amplitude (mm)

Mass of Tup (t)

<1 1–3 >3

Foundation block 1 1.5 2

Anvil 1 2 3–4

Table 22.6 Minimum thickness of foundation block

Mass of Tup (t) Minimum Thickness of Foundation Block (m)

<1.0 1.00

1.0–2.0 1.25

2.0–4.0 1.75

4.0–6.0 2.25

>6–0 2.50



22.7.4 Foundations for Rotary Type Machines (Low frequency)

The design and construction criteria applicable to foundations of rotary type machines of low frequency (<1500 rpm), as per IS:2974 (Part IV)–1979, are given in this section. Some typical machines of this type are crushing mills, pumps, motors, generators, compressors and rolling mill stands.

1. Design:• The natural frequency of any foundation should preferably not be within 20% of the operating speed of the

machine.• Maximum permissible amplitudes of vibration of displacement should be 0.2 mm.• To avoid transmission of vibration to adjoining parts of buildings or other foundations, it is necessary to

provide a suitable isolation between the equipment foundation and the adjoining structures. This may commonly be achieved by providing sand trenches around the foundation block, the thickness and depth of which should be determined for each individual case. As a rule, the equipment foundation should not be allowed to serve as a support for other structures or for machineries not related to the particular equipment.

• Where a number of similar machines are to be installed side by side in a close spacing and soil conditions do not permit construction of an independent foundation for each machine, the foundations for all the similar machines may be combined through one common mat of sufficient thickness.

• The area of the mat should be enough so that deformations are minor as compared with the resultant ampli-tude of vibration. The mass of the foundation should be at least two-and-a-half times the mass of the whole machine.

• The foundations should be so dimensioned that the resultant force due to the mass of the machine and the mass of the foundation passes through the centers of gravity of the base contact area.

2. Construction:• The grade of concrete should generally be M-15 to M-20 for block foundation and M-20 for framed foundation.• The concrete used should have the allowable slump of 50–80 mm. The water cement ratio should not exceed

0.45.• The amount of reinforcement in each foundation element unit should not be less than 50 kgf/m3 of concrete. • The minimum diameter of the mild steel bars should be 12 mm and the maximum spacing between them

should be 200 mm to ensure there is no shrinkage in concrete.• The concrete cover for protection of the reinforcement should be 75 mm at the bottom, 50 mm on the sides

and 40 mm at the top.• The soil stress below the foundations under dead loads only should not exceed 80% of the allowable bearing

pressure for static loading.

22.7.5 Foundations for Rotary Type Machines (Medium and High Frequency)

The design criteria for foundations of rotary type machines of low frequency are also applicable for foundations of rotary type machines of medium and high frequency as well. The criteria covered under this section, as per IS:2974 (Part 3)–1992, are over and above those applicable for foundations of low frequency rotary type machines. Examples of rotary type machines of medium and high frequency include turbo-generators, turbo-compressors, boiler feed pumps etc. Figure 22.21 shows a typical framed foundation for a turbo-generator.

1. Design:• The geometric layout of the foundation, the shape of the girder cross sections and columns shall be arranged,

as far as possible, symmetrically with respect to the vertical plane passing through the longitudinal axis of the machine.

• The foundation structure shall be isolated from the main building and also from other structures in the plant. An air gap shall be provided between the foundation and adjoining structures at all levels above the base mat to avoid the transfer of vibrations to the adjoining structures.

• For rafts supported directly on soil, the thickness of the base raft should not be less than 0 07 4 3. /L , where L is the average of the two adjacent clear spans for the initial sizing of the raft.



• Damping should be assumed to be 2% of critical damping under normal operating loads. A higher damping of 5% may be used under emergency loads such as blade failure, short circuit, and bearing failure.

• The bearing pressure on soil or the load on the heaviest loaded pile should not exceed 80% of the net allowa-ble bearing pressure or the safe load capacity of the piles.

• A fatigue factor of 2 should be used for the dynamic forces caused by normal unbalance.• The following guidelines may be followed for column sizing:

(a) As far as possible, pairs of columns should be provided under each transverse girder.(b) Compressive stresses and elastic shortening should be kept uniform in all the columns as far as possible.(c) The first two natural frequencies of the column with its top and bottom ends fixed should be away from

the operating frequency of the turbo-generator by at least 20%.2. Construction:

• Minimum M-20 grade of concrete should be used. For turbo-generator foundations of capacities higher than 100 MW, the minimum grade of concrete for the end columns should be M-25.

• The minimum diameter of reinforcement bars used as the main reinforcement should be 12 mm. Minimum reinforcement should be provided as given in Table 22.7.

• The maximum spacing of the reinforcement bars shall not exceed 300 mm and the minimum spacing shall not be less than 150 mm. The minimum clear cover to reinforcement should be 50 mm for the top deck and columns and 100 mm for the base mat.

• The clear spacing between bars should be at least 5 mm more than the sum of the aggregate size and the largest bar diameter used.

Base mat

Columns

Top deck

Transverse beamsLongitudinal beam

Figure 22.21 Typical framed foundation for a turbo-generator [IS:2974 (Part-III)–1992].

Table 22.7 Minimum reinforcement

Component Description Minimum Reinforcement

Beams of top deck Top and bottom 0.25A

Sides 0.1A

Columns base mat Longitudinal reinforcement 0.8A

Top and bottom 0.12A

Intermediate layer if D > 2 m 0.06A

A = Gross cross-sectional area; D = raft thickness

SMFE_Chapter_22.indd 955 4/4/2015 10:51:40 AM


22.8 Vibration and Shock Isolation

Transmission of vibrations from outside a structure or from machinery within the structure can be annoying to the occupants and can cause damage to the structure. Vibration transmission may also interfere with the operation of sensitive instruments. See Fig. 22.19 for the effects of vibration amplitude and frequency. The tolerable vibration amplitude decreases as the frequency increases.

The vibration amplitude transmitted away from the source can be estimated using the following relationship:

A A err

r r2 1

1

2

2 1= − − α ( ) (22.65)

where A1 is the computed or measured amplitude at distance r1 from the vibration source, A2 the amplitudes at distance r2, r2 > r1 and a the coefficient of attenuation depending on soil properties and vibration frequency.

α α=

50 50

f (22.66)

a50 is the coefficient of attenuation at a frequency of 50 Hz (3000 rpm) as given in Table 22.8.

22.8.1 Types of Vibration Isolation

The following are the two types of vibration isolation based on the purpose:

1. Active isolation: If the equipment requiring isolation is the source of unwanted vibration [Fig. 22.22(a)], the purpose of isolation is to reduce the vibration transmitted from the source to its support structure; this is known active isolation. This vibration producing equipment mainly consists of machines that apply severe dynamic forces on their supporting structures.

2. Passive isolation: Conversely, if the equipment requiring isolation is the recipient of unwanted vibration [Fig. 22.22(b)], the purpose of isolation is to reduce the vibration transmitted from the support structure to the machine itself to maintain performance; this is known as passive isolation. This includes equipment such as precision machine tools and measuring machines in which vibrations must be kept within acceptable limits to achieve the desired surface finish, tolerances, or accuracy.

22.8.2 Transmissibility

The ratio of the vibration transmitted after isolation to the disturbing vibration is described as transmissibility and is expressed in its basic form as

Tr

=−

112( )f

(22.67)

Table 22.8 Value of a50

S. No. Type of Soil a50

1. Loose fine sand 0.06

2. Dense fine sand 0.02

3. Silty clay (loess) 0.06

4. Dense, dry clay 0.003

5. Weathered volcanic rock 0.002

6. Competent marble rock 0.00004


22.8 Vibration anD ShoCk iSolation • 957

rfffd

n

= (22.68)

where rf is the frequency ratio defined by Eq. (22.68), fd the disturbing frequency, and fn the natural frequency of the isolator.

When considering the property of damping, Eq. (22.67) is rewritten as Eq. (22.69)

Tr

r r=

+− +

1 21 2

2

2 2 2

( )[ ( ) ] ( )

εε

f

f f

(22.69)

εω

=c

m2 n

(22.70)

where, e is the critical damping coefficient, defined by Eq. (22.70).Typical values for damping ratio are 0.005–0.01 for steel, and 0.05–0.10 for rubber. The effectiveness of vibration

isolation is obtained in per cent as (1–T) × 100.

22.8.3 Selection of Vibration Isolators

All vibration isolators are essentially springs with an additional element of damping. In some cases, the “spring” and the “damper” are separated as in the case of a coil spring isolator used in conjunction with a viscous damper. The majority of isolator designs, however, incorporate the spring and the damper into one integral unit.

The following are the important considerations with vibration isolator selection:

1. Machine location: The machine should be as far away from sensitive areas as possible and on as rigid a founda-tion as possible (on grade is best).

2. Proper sizing of isolator units: The isolator units should be of proper size to ensure correct stiffness (specified by the static deflection, more flexibility is generally better).

3. Location of isolators: Isolators should be equally loaded, and the machine should be level.4. Stability: Sideways motion should be restrained with snubbers. The diameter of the spring should also be

greater than its compressed height. Isolator springs should occupy a wide footprint for stability.5. Adjustment: Springs should have free travel, should not be fully compressed, and should not be hitting a

mechanical stop.

F(t) = Original vibration before isolationF′(t) = Damped vibration after isolation

P(t)

F(t)

F′(t)

Isolator

Supportstructure

Machine

Foundation

X(t)

a(t)

F′(t)

Isolator

Supportstructure

Machine

Foundation

(a) (b)

Figure 22.22 Types of vibration isolation: (a) Active isolation and (b) passive isolation.



6. Eliminate vibration short circuits: Any mechanical connection between the machine and the foundation that bypasses the isolators, such as pipes, conduits, binding springs, poorly adjusted snubbers, or mechanical stops, should be eliminated.

7. Fail safe operation: If a spring breaks or becomes deflated, mechanical supports must be available to rest the machine without tipping.

22.8.4 Methods of Vibration Isolation

The classical approach to vibration isolation is to use springs and dampers. Spring resists the movement of the vibration by exerting an opposing force proportional to their own displacement. Dampers consist of a piston moving through a viscous fluid, or a conductor moving in a magnetic field, whereby the kinetic energy is converted into heat energy. However, springs are effective for frequencies greater than about 10 Hz, whereby they vibrate at their natural frequency and start functioning as amplifiers.

The methods of vibration isolation include:

1. Physical separation of the vibrating unit from the structure. 2. Interposition of an isolator between the vibrating equipment and the foundation or between the structure foun-

dation and an outside vibration source.

Vibration isolating mediums include resilient materials, such as metal springs, or pads of rubber, and cork, and felt in combination.

1. Supporting system: Properly designed spring absorbers can be used for machines which are well balanced with small dynamic forces generated in higher harmonics, having light weight including mountings, as shown in Fig. 22.23. In such an installation, the machine is mounted on a rigid frame of structural steel that in turn is placed directly on spring absorbers.

For low-frequency machines with large unbalanced forces, the weight on the absorbers may be increased suitably. For example, an additional concrete block is added to the machine mounting for such cases, which in turn is placed on the absorbers fixed to the base slab. In both the cases, the absorbers are directly supporting the machine installations hence are called supporting systems. Such absorbers are relatively inexpensive, reliable in operation and effective in decreasing the amplitudes of vibrations. For high-capacity machines, absorber units with multiple springs are used.

2. Suspended system: In suspended system, the absorbers are placed on the top edges of the foundation mass below the springs as shown in Fig. 22.24. A long anchor bolt passes through the absorbers and the foundation is connected to the lower ends of the anchor bolts using girders and cantilever projections taken out of the foundation block. The construction procedure of the suspended type absorbers is similar to the supported type of systems. Suspended systems provide easy access to the spring casings for frequent inspection and maintenance.

Figure 22.23 Vibration isolation of a six-cylinder diesel engine (Narotama, 11).


objeCtiVe QueStionS Vibration anD ShoCk iSolation • 959

Figure 22.24 Suspended-type absorber ( Narotama, 11).

Additional methods available include the installation of open or slurry-filled trenches, sheet pile walls, and concrete walls. These techniques have been applied with mixed results. Analytical results suggest that for trenches to be effective, their depth should be 0.67 L or larger, where L is the wavelength for a Rayleigh wave and is approximately equal to Vs/w, where w is the angular velocity of vibration in rad/s, and Vs is the shear wave velocity of the soil.

Summary

Foundations supporting machines are subjected to dynamic stresses due to vibration or impact caused by the machinery in addition to the stresses due to static loads. Proper design of machine foundations is essential to ensure stability of the machine, the foundations, the struc-ture housing the machine, and adjacent structures and their foundations as well as to ensure effective operation of the machine itself and to avoid excessive discomfort to the people working in the vicinity. The main principle of dynamic analysis of machine foundations is to model the machine-foundation–soil system into a mass-spring–dashpot system and to determine the natural frequency and amplitude of vibration of the machine foundation. The machine foundation may undergo vibrations under six modes about x, y, and z axes, and dynamic analysis is

done under each mode separately, whose results are then superimposed. The properties of the soil under dynamic condition are different from those under static condi-tion. Various in situ and laboratory methods are availa-ble for the determination of dynamic properties of soil. The principal design criterion for machine foundation is to avoid the possibility of resonance, where the natural frequency of vibration of the machine foundation equals the frequency of vibration of the machine resulting in excessively high amplitudes of vibration of the machine foundation. The amplitude of vibration should be limited to the maximum permissible amplitude of the machine even when there is no resonance. Vibration isolation and control is another important technique in the effective design and construction of machine foundations.

Objective Questions

1. Resonance is the condition that occurs when the natural frequency of foundation–soil system is (a) Less than operating frequency of the machine. (b) Equal to the operating frequency of the machine. (c) Greater than operating frequency of the machine.(d) All of the above.

2. In mass-spring–dashpot model of analysis of machine foundation, the foundation soil is represented by (a) Mass.(b) Spring.(c) Dashpot.(d) None of these.



3. Rotary type machines with high frequency are those in which the operating frequency is greater than(a) 1000 rpm.(b) 1500 rpm.

(c) 3000 rpm.(d) 5000 rpm.

4. Increase in the mass of the machine foundation (a) Increases the natural frequency. (b) Decreases the natural frequency.(c) Increases the operating frequency.(d) Decreases the operating frequency.

5. When representing the vibrations of the machine foundation by simple harmonic motion, the dis-placements are proportional to (a) wt.(b) cos wt.

(c) sin wt.(d) tan wt.

6. Critical damping in machine foundations will occur when the damping coefficient, c is equal to

(a) 2 km.

(b) 2km

.

(c) 2km

.

(d) 2mk

.

7. When critical damping occurs in machine founda-tion, the foundation (a) Oscillates with very high frequency.(b) Oscillates with low frequency.(c) Comes to rest in infinite time.(d) Either B or C.

8. The critical dam.ping ratio D in most of the machine foundations is (a) Much greater than 1.(b) Equal to 1.

(c) Equal to zero.(d) Much less than 1.

9. The error introduced in dynamic analysis of machine foundations in using undamped frequency (wo) in place of damped frequency w is (a) Very high.(b) High.

(c) Medium.(d) Very small.

10. The ratio of amplitude of displacement under dynamic and static conditions of a machine foun-dation is known as (a) Damping coefficient.(b) Critical damping ratio. (c) Magnification factor.(d) Impact factor.

11. The magnification factor in machine foundations (a) Increases with increase in frequency ratio.(b) Decreases with increase in frequency ratio.(c) Increases initially and then decreases with

increase in frequency ratio. (d) Decrease initially and then increases with

increase in frequency ratio.

12. Frequency ratio in machine foundations is defined as the ratio of(a) Operating frequency and natural frequency. (b) Dynamic and static displacements.(c) Natural frequency and operating frequency. (d) All of these.

13. The coefficient of elastic uniform compression for a silty clay soil is found to be 3 kgf/cm3. The coefficient of elastic uniform shear for the soil is likely to be (a) 1–1.5 kgf/cm3.(b) 1.5–2 kgf/cm3.

(c) 2–2.5 kgf/cm3.(d) 2.5–3 kgf/cm3.

14. The seismic waves which have maximum velocity are (a) P-waves.(b) S-waves.

(c) Rayleigh waves.(d) Ultra sonic waves.

15. The seimic waves which cause volume change in soils when they travel through them are (a) P-waves.(b) S-waves.

(c) Rayleigh waves.(d) Ultrasonic waves.

16. The seismic waves which travel near the ground surface are (a) P-waves.(b) S-waves.

(c) Rayleigh waves.(d) Ultra sonic waves.

17. If r is the mass density of soil and Vs is the veloc-ity of S-Waves, the dynamic shear modulus is given by (a) ρVs .

(b) ρVs2 .

(c) ρ/ .VS

(d) ρ/ .Vs2

18. Select the correct statement(s) regarding the in situ methods of determining shear modulusA. In cross-hole method, two bore holes are drilled. B. In seismic down-hole method, only one bore

hole is drilled.C. In seismic up-hole method, no bore hole is drilled. (a) A only.(b) A and B only.

(c) B only.(d) B and C only.

19. Which one of the following seismic waves are used in the cross-hole and seismic down-hole or up-hole methods of determining dynamic soil properties (a) P-waves.(b) S-waves.

(c) Rayleigh waves.(d) All of these.

20. The test method which uses Rayleigh waves for determination of dynamic soil properties of soil is (a) Cross-hole method. (b) Seismic down-hole method. (c) Seismic up-hole method.(d) Steady-state vibration test.

21. The coefficient of elastic uniform compression for a foundation, (Cuf) is obtained from the corresponding


reView QueStionS • 961

value determined from block vibration test (Cut) using the relation

(a) C CAAuf ut

t

f

= ⋅ .

(b) C CAAuf ut

t

f

= ⋅ .

(c) C CAAuf ut

f

t

= ⋅ .

(d) C CAAuf ut

f

t

= ⋅ .

Where Af is the area of foundation and At is the area of test block.

22. The most widely used laboratory test to determine the dynamic soil properties is (a) Resonant-column method.(b) Cyclic triaxial test. (c) Block vibration test.(d) Cyclic plate load test.

23. The maximum frequency ratio of high frequency reciprocating machines for low-tuning of machine foundations as per IS:2974 (Part I)–1982 is (a) 2.(b) 1.5.

(c) 0.4.(d) 0.2.

24. The minimum frequency ratio of low-frequency reciprocating machines for high-tuning of founda-tions as per IS:2974 (Part-I)–1982 should be (a) 1.5.(b) 2.0.

(c) 2.5.(d) 3.0.

25. The maximum permissible amplitude for foun-dations of reciprocating machines as per IS:2974 (Part I)–1982 should be limited to

(a) 0.1 mm.(b) 0.2 mm.

(c) 1 mm.(d) 1.2 mm.

26. The allowable bearing pressure for machine foun-dations is usually taken as (a) 50% of that for static loads.(b) 75% of that for static loads. (c) 80% of that for static loads. (d) 90% of that for static loads.

27. The minimum reinforcement for foundations of reciprocating and impact type machines as per IS:2974 (Part II)–1980 is (a) 25 kgf/m3.(b) 35 kgf/m3.

(c) 40 kgf/m3.(d) 50 kgf/m3.

28. The minimum grade of concrete that should be used for foundations of impact type machines as per IS:2974 (Part II)–1980 is (a) M-15.(b) M-20.

(c) M-25.(d) M-35.

29. The minimum grade of concrete that should be used for framed foundations of rotary type machines as per IS:2974 (Part IV)–1979 is (a) M-15.(b) M-20.

(c) M-25.(d) M-35.

30. The natural frequency of foundations for rotary type machines as per IS:2974 (Part–4)–1992 should NOT be within ______ of the operating speed of the machine. Choose the correct answer to fill in the blank. (a) 20%.(b) 25%.

(c) 30%.(d) 40%.

Review Questions

1. Draw a neat sketch of a typical arrangement of a hammer foundation with a frame.

2. What is a reciprocating machine? Give one example of such a machine and indicate its possible modes of vibration with a neat sketch.

3. Derive from first principle the expressions for dynamic responses of a block foundation subjected to coupled rocking and sliding.

4. Derive the expression of amplitude for undamped forced vibrations for a two degrees of freedom system.

5. Explain the difference between free and forced vibrations with damping for single degree of freedom.

6. What is resonance? Explain its effect on single-de-gree-of-freedom system.

7. Explain the method of determination of the natural frequency of machine foundation–soil system.

8. What is a pressure bulb? Explain Pauw’s analogy with a neat sketch and indicate the limitations.

9. Explain any two methods of determination of dynamic properties of soil.

10. What are the different types of machine founda-tions? Explain them briefly with neat sketches.

11. Briefly explain the general requirements of machine foundations.

12. Explain the elastic half-space theory for analysis of machine foundations.

13. Discuss the design principles for foundations f impact type machines as per the IS code.

14. Explain briefly about vibration isolation and control.



Answers

Objective Questions

1. (b)

2. (b)

3. (c)

4. (b)

5. (c)

6. (a)

7. (c)

8. (d)

9. (d)

10. (c)

11. (c)

12. (a)

13. (b)

14. (a)

15. (a)

16. (c)

17. (b)

18. (b)

19. (b)

20. (d)

21. (b)

22. (a)

23. (c)

24. (a)

25. (b)

26. (c)

27. (a)

28. (a)

29. (b)

30. (a)


smfe chapter 22

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