JlSC~~ American Society A tIiii of Civil Engineers

DESIGN OF FOUNDATIONS

FOR DYNAMIC LOADS

2008

Design of Foundations for Dynamic Loads

Learning Outcomes Comprehend the basics of soil dynamics as related to soilstructure interaction modeling Know the major field tests that are used for evaluating the dynamic soil properties Design shallow and pile foundations for rotary machine and seismic loads Know the major steps for determining the seismic stability of cantilever and gravity retaining walls Apply the capacity spectrum method for evaluating the seismic performance of large caissons

Assessment of Learning OutcomesStudents' achievement of the learning outcomes will be assessed through solved examples and problem-solving following each session.

UNIVERSITY OF WESTERN ONTARIO Faculty of Engineering

Design of Machine Foundations

Professor M.H. EL NAGGAR, Ph.D., P. Eng.Department of Civil and Environmental Engineering Geotechnical Research Centre

LECTURE NOTESM.H. EL NAGGAR M.NOVAK

DEPARTMENT OF CIVIL ENGINEERING THE UNIVERSITY OF WESTERN ONTARIO LONDON, ONTARIO, CANADA, N6A 5B9

Course Content

1. Basic Notions: Mathematical models , degrees of freedom , types of dynamic loads, types of foundations, excitation forces of machines. 2. Shallow Foundations: Definition of stiffness , damping and inertia , circular and non circular foundation , soil inhomogeneity, embedded footings , impedance function of a layer on half-space. 3. Pile Foundations: PHe applications, mathematical models, stiffness and damping of piles, pile groups, impedance functions of pile groups, nonlinear pile response, pile batter.

4. Dynamic Response of Machine Foundations: Response of rigid foundations in 1 OaF, effects of vibration, coupled response of rigid foundations, 6 OaF response of rigid foundations, response of structures on flexible foundations.5. Dynamic Response of Hammer Foundations: Types of hammers and hammer foundations, design criteria , stiffness and damping of different foundations, mathematical models, impact forces , response of one mass foundation, response of two mass foundation, impact eccentricity, structural design .

6. Vibration Damage and Remedial Measures Damage and disturbance, problem assessment and evaluation , remedial principles, examples from different industries, sources of error. 7. Computer Workshop - DYNA5 Types of foundations, types of soil models, types of load , types of analysis and types of output, practical considerations, computer work on DYNA5.

2

1

BASIC NOTIONS

3

BASIC NOTIONSStatics deals with forces and displacements that are invariant in time. Dynamics considers forces and displacements that vary with time at a rate that is high enough to generate inertia forces of significance. Then, the external forces, called dynamic loads or excitation forces , produce time dependent displacements of the system called dynamic response. This response is usually oscillatory but its nature depends on the character of the dynamic forces as well as on the character of the system. Thus , one system may respond in different ways depending on the type of excitation. Conversely, one type of excitation can cause various types of response depending on the kind of structure . Mathematical models. The systems considered in dynamics are the same as those met in statics, i.e. buildinqs, bridges, towers, dams, foundations, soil deposits etc. For the analysis of a system a suitable mathematical model must be chosen. There are two types of models, which differ in the way in which the mass of the structure is accounted for. In distributed mass models, the mass is considered as it actually occurs, that is, distributed along the elements of the structure . In lumped mass models the mass is concentrated (lumped or discretized) into a number of points. These lumped masses are viewed as particles whose mass but not size or shape is of importance in the analysis. There is no

rotational inertia associated with the motion of the lumped masses and translational displacements suffice to describe their position. Between the lumped masses the structural elements are considered as massless . Examples of distributed and lumped mass models are shown in Fig. 1.1. As the number of concentrated masses increases,

4

the lumped mass model converges to the distributed model. In rigid bodies (Fig. 1.1c), mass moment of inertia is considered as well as mass .

Figure 1.1

--

-~

(a) distributed models

-A...... .::IL

t(b) lumped mass models

/

.,

(c)- I.:

~

rigid bodies

Degrees of freedomThe type of the model and the directions of its possible displacements determine the number of degrees of freedom that a system possesses. The number of degrees of----~- --

.:

I,

freedom is the number of independent coordinates (components of displacements) that

---------

-

---

- -- -- -

)~

t

l, ;'

\

r:

must be specified in order to define the position of the system at any time. One lumped mass has three degrees of freedom in space corresponding to three possible

5

translations, and two degrees of freedom in a plane. If a lumped mass can move only either vertically or horizontally it has one degree of freedom. Thus, if the vertical motion of a bridge is investigated using a model with three lumped masses and axial deformations are neglected, there are three degrees of freedom (displacements). However, a rigid body such as a footing has significant mass moments of inertia and hence rotations have to be considered as well . Three possible translations and three possible rotations represent six degrees of freedom for a rigid body in space . A distributed mass model can be viewed as a lumped mass model with infinitesimal distances between adjoining masses . Such a system has an infinite number of degrees of freedom . This does not necessarily complicate the analysis however.

Types of dynamic loads.The type of response of a system depends on the nature of the loads applied. The loads and the responses resulting from them can be periodic, transient or random. :+-

--------

Periodic Loads can be produced by centrifugal forces due to unbalance in rotating andreciprocating machines, shedding of vortices from cylindrical bodies exposed to air flow

and other mechanisms. The simplest form of a periodic force is a harmonic force. Such a force may represent the components of a rotating vector of a centrifugal force in the vertical or horizontal directions.

6

Figure 1.2: Harmonic time history

_t

T

If the vector P rotates with circular frequency w, the orientation of the vector at time t is given by the angle wt (Fig. 1-2) The components of the vector P in the vertical and horizontal directions, respectively, are:

P; (t)

= Psin(wt)

I

p,,(t) = PCOS({tJt)

These forces are harmonic with amplitude P and frequency to. The period measured in seconds is T

=Znk,

which follows from the condition that in one period one complete The frequency measured in cycles per

oscillation is completed and thus wT = 2n:.

second and expressed in units called Hertz (Hz) is

f=~=~

T21fConsider now the joint effect of two harmonic forces having different amplitudes P1 and P2 , different frequencies(01

and

W2

and a phase shift . The resultant is

r .-

~

t,

r,'"'1I

;r,.(

)

". ,I

. f

)j

r

"I

"

,I 1

I( I

IJI

f

7. I

;

-,

The time history of the resultant can be generated by projecting the resulting, rotating vector R horizontally. The character of the time history depends on the ratio of the

amplitudes, the ratio of the frequencies and the phase shift. When the two frequencies are equal, the resultant force is harmonic (Fig. 1.3a). When the frequency ratioW2 /W1

is

an irrational number, the resultant force is not periodic (Fig. 1.1 b). When the ratio

0)2 /OJ1

is a rational number, the resultant force is periodic but not harmonic (Fig. 1.1 c). However, the envelope of the resultant force is always periodic. When the two frequencies and the two amplitudes do not differ very much, a phenomenon called beating occurs; the force periodically increases and diminishes, similarly to Fig. 1.3b, with frequency of the peaks beingW2 - W1

In general, a periodic function can be represented by a series of harmonic components whose amplitudes and frequencies can be established using Fourier analysis. Therefore, knowledge of the harmonic case facilitates the treatment of more complicated types of excitation.

'.1

( .

r,

-(

r('

-

/

8

Fig. 1.3: Basic types of processes composed of two harmonic components

./'

/I

---

"'

""

'

/I

P2{tr=P2 Sin (wt"'f) p, ft)=-Pr Sin wi

\\-,

.-

\.../.

. ;]R(t)=p'{t).,.~m \. J1Rftl"RslO (cat 'fll)

a)

,/

/

--t,,P, (t)-:.A sin wit ~ (f)"B sin(w2t fJr)b},/

/

.......

/

P, =P26)2e

W, . 1, 188. . "

",

-,-,

\,

//

'\

'\

\I

I

\

I\ \

\

'\J

I

/

I I\

I

--,\I

I

I

,

w{

\

,,

I

//

/

",'\\,

I

I

I

I

.....

P, (t) ~ P, sm~ (I )~f3

W, t

""'--

P'="S ~wr!,5w,

sin(w2t.,.. Jf)

c)

9

Transient Loading is characterized by a nonperiodic time history of a limited durationand may have features such as those indicated in Fig. 1.4. A smooth type of loading such as the one shown in Fig. 1.4a is produced by hammer blows, collisions, blasts, sonic booms etc. and is called an impulse . Earthquakes or crushers generate more irregular time histories, similar to that shown in Fig. 1.4b. It is presumed that such a process is determined accurately either by an analytical expression or by a set of digital data. A process so defined is called a deterministic process . Often, the duration of an impulse,~t,

is much shorter than the dominant period of

the foundation response, T (Fig. 1.5). Such loading is characteristic of impacts associated with the operation of hammers and presses . The limited duration of the impact makes it possible to base the analysis of the response on the consideration of the collision between two free bodies.

Random Loading is an irregular process that cannot be predicted mathematicallywith accuracy, even when its past history is known, because it never repeats itself exactly. Fluctuating forces produced by mills, pumps, crushers, waves and by wind or traffic flow are typical of this category (Fig. 1.6a). A random force and its effect is most meaningfully treated in statistical terms and its energy distribution with regard to frequency is described by a power spectral density (power spectrum), Fig. 1.6b. Earthquake forces can also be treated in this way. The advantage of the random

approach over the deterministic approach is that the analysis covers all events having the same statistical features rather than one specific time history .

10

Figur e 1.4: Transient loading

P(f)

p( t)

t

o

t

Figure 1.5: Impact loading

. _.

'-r

k(

I

(.,

fJt

I

r(1 'J :' V.

1J.f T - 1/0

oP(f)

tFigure 1.6: Random loading

Sp(f)

oa) Time History

Frequency, f

b) Power Spectrum

11

Types of foundations.

Machine foundations are designed as block foundations, wall foundations, mat foundations or frame foundations. Block foundations, the most common type, and wall foundations behave as rigid bodies . Mat foundations of small depth may behave as elastic slabs. Sometimes the foundation features a joint slab supporting a few rigid

blocks for individual machines. The foundations can rest directly on soil (shallow foundations) or on piles (deep foundations). response . The type of foundation may result in considerable differences in

Notations and sign conventions

The vibration of rigid foundations is characterized by three translations, u, v, w and three rotations,

S, \V, 11 - These

are expressed with regard to the three perpendicular

(Cartesian) axes X, Y, Z. The origin of this system is most conveniently placed in the joint centre of gravity (CG) of the foundation and the machine (Fig. 1.7). The orientation of the axis and the signs of all displacements and forces are governed by the right-hand rule. The translations u, v, wand the forces P, , P, positive if they follow the positive directions of the axis. The rotationsI

P, are

S, 'V' 11 and

moments Mx , My ,Mz are positive if they are seen to act in the clockwise direction when looking in te positive directions of the corresponding axis, i.e. away from the origin .

12

Figure 1.7: Notations and Sign Convention

Z,w,F;

Examples of typical machine foundations Basic types of foundations for typical machines are shown in Figures 1.8 to 1.17.

13

Figure 1.8: Block Foundation for Two-cylinder Compressor

Figure 1.9: Block Foundation with Cavities for Horizontal Compressor

14

Figure 1.10: Two Compressors on Joint Pile Supported Mat

w-I

o

~/1:::\.-~,

(-{-0)(_-'

-; '''''''' V I I:z' w~:@.f~; i"'l'" ~4, n' : ,. . ',.//:~/, H".%,~';. .~./;?"I

/'1" ........ :r::J:l' r:.... , , / _+1 ~ IY '-.! -"'",H ;,. ~ ~ . .., . :" .~': . ' ,,,I I~.

C

ru

'< (.,.l0 0

o

:J

//~

_

/. !\1 I- i rl>.,

II

8(j)

/........ ~ \.81

!:1:- ~_ ---~

uJ

L/ D"2S DO

-l ~.:. 1 ,-- s ~ -J

~!

10

"-=:! -----;-;f"... t

- 0.5 __ . .0 ,0

f""1:. - =~~t 1'[-,--- ~-0. I

0 .0

_ O. 2

[,,-

7

:

..,

'0

0.5

1.0

0 .0

0 .5

1.0

00

0

0.7

0,4

0.3

."i'eoI po r t

-- - 0.1

i ma go part

- 0 .2- 0 .3- 0 .4 - 0.5

J QJ

IE

II

1.0u

o

Quadratic excitation.

The above formulae and Fig. 4.2a were derived for an

excitation force whose amplitude P is constant. In many practical cases, the amplitude

145

of the excitation force is not constant but depends on the square of frequency. This is so with excitation stemming from centrifugal forces of unbalanced rotating masses, unbalances of reciprocating mechanisms, harmonic ground motion or vortex shedding. An unbalanced mass me rotating with an eccentricity e and circular velocity (J,) produces a centrifugal force

(4.17) The horizontal component of the centrifugal force is

(4.18) which is a harmonic force with frequency dependent amplitude. Substitution of this amplitude P =mee(J,)2 into Eq. 4.10 gives the response amplitude

V=

(4.19)

in which the reduced eccentricity p = mee I m and the dynamic amplification factor of quadratic excitation is

(4.20)

The variation in the dynamic amplification factor s' with frequency is shown in Fig. 4.2b. The response amplitude starts from zero, grows to the resonant amplitude

v=p at (J,)

1 2D

(4.21)

=(J,) 0 and then asymptotically approaches p. Thus , the dynamic amplification e'146

ranges from zero to 1/2D at resonance and finally to 1 at high frequencies. Examination of the response peaks reveals that the actual maximum of the amplification factor exceeds the value of 1/2D occurring at00

= 000 and is

1

This maximum appears at frequency

with constant amplitude excitation and at frequency

1

with quadratic excitation. Thus the true peak may appear below or above the undamped resonant frequency co

=CUo depending on the type of excitation (Fig . 4.3). For small

damping this difference becomes insignificant.

147

Figure 4.3: Response Curves for Constant Amplitude Exc'tation and Quadratic Excitation

q u adz e c i c fo rce

constant force

cirnl

w

General Relation Between Complex Amplitude and Real AmplitudeIn general, complex motion can be described by

where the complex amplitude is

v =V1 + ivzAccording to Eq. 4.3, this amplitude is also

in which the real amplitude

(4.22)

and

(4.23)

Thus

148

and the real part of this motion is vet) = v cos (cot + ) The relation between complex and real amplitude is useful in many degrees of freedom. The theory of one degree of freedom can be used for vertical translation and rotation about the vertical axis (torsion); very approximately, it is also applicable to horizontal translation of very flat footings and rocking of tall footings about base axis. In cases involving rotation, mass moment of inertia, I, replaces mass, rn, in the above formulae and displacement, v, is replaced by the pertinent rotation. The basic formulae for mass moments of inertia are given below. The formulae are for rectangular bodies and cylindrical bodies; m is the mass of the body.

Figure 4.4: Mass Moments of Inertia

Mass Moments of Inertia

o[

--- ---r-----Ix-X

-~- --- --~-- -x'

V.ZII

I

x = 12

m

( a +b

2

2

)

I Xl

=

1

x

+

mx

2

I

--

-:_-~-_.:- - I. ,

,.,.

--:- j-X__'-_ xd

Z

= m

2 d 8

=

m2

R

2

----X'

I

xXl

=

2 2 d h + m(16 12)I

h-1

I

I

=

X

+ mx

2

149

Examples of Respons e of Footing With one Degree of Freedom

The theory outlined above can be used to analyze the response of shallow foundations and pile foundations. The needed stiffness and damping constants can be evaluated using the approaches described in Chapters 2 and 3. As an example, the response amplitudes are examined for the machine foundation shown in Fig. 4.5. Using Eq. 4.8a, the vertical and torsional amplitudes are evaluated for different types of foundation and quadratic excitation described by Eq. 4.17. The response curves established are shown in Figs. 4.6 to 4.8. The amplitudes are given in a dimensionless form which actually corresponds to the dynamic amplification factor,E',

defined by Eqs. 4.20 and 4.10. For quadratic excitation , the

dimensionless amplitudes of vertical and torsional response may be defined as

in which r = the arm of the horizontal force with regard to CG. Frequency independent stiffness and damping constants were assumed . A detailed numerical example is given at the end of this chapter.

150

Figure 4.5: Machine Foundation Used with Piles and Without Piles in Examples (1ft

=O.3048m)

Figure 4.6: Vertical Response of (A) Pile Foundation, (B) Embedded Pile Foundation, (C) Shallow Foundation, and (D) Embedded Shallow Foundation (Bx

= m I p Rx3

= 5.81,

Novak, 1974a)

5

r'

~

....

I

I'"

J.

, ... I~'

,

.

.

'.1r

)

r

JI-' .

_ ......:.._--:!':---:-.t._...2060 80 1 00f R E ~ U EN C Y

J2

Wo

is quite dramatic.

Hence, for low tuned foundations, high damping is unfavourable. The ratio F/P is the same as that shown in Fig. 4.10. Figure 4.11: Normalized Transmitted Force with Quadratic Excitation

~ , "'\.~\,' j

, V\\

I D=O

0.5

/

3t-----+,&.f---';t----:-f'---"'7"----""7"-l

\1

'.

\.

ll-_*~-=-------------I

158

4.3 COUPLED RESPONSE OF RIGID FOUNDATIONS IN TWO DEGREES OFFREEDOM

Rigid foundations are constructed as rigid or hollow blocks . Their motion in space is described by three translations and three rotations and consequently they have six degrees of freedom. The six components of the motion are, in general. coupled. However, there is usually at least one plane of symmetry and this reduces the coupling between individual components of the motion. Two vertical planes of symmetry

decouple the six degrees of freedom into four independent motions: vertical translation, torsion around the vertical axis, and two coupled motions in the vertical planes of symmetry; the latter motions are composed of horizontal translation (sliding) and rotation (rocking). The coupled motion in the vertical plane represents an important case because it results from excitation by moments and horizontal forces acting in the vertical plane. The motion is treated most conveniently if the centre of gravity of the footing and the machine considered together is taken as the reference point (origin of coordinates). Then, the horizontal sliding u (t) and rocking\jI

(t) describe the coupled motion as

indicated in Fig. 4.12 in which the positive directions of the two components are indicated . For the analysis, inertia forces, stiffness constants, and damping constants must be established first.

159

Figure 4.12: Notations for Coupled Motion

~

x.o

b-..

,Y,v a

4.3.1 Mass, Stiffness and DampingInertia forces are due to the mass of the footing-machine system and its massmoment of inertia about the axis Z passing through the centre of gravity. The mass of the system is m = m1 + m2 if

rn- is the mass of the footing and m2 the mass of the

machine. No additional mass to account for soil inertia is needed because this effect is taken care of by the variation of stiffness constants with frequency and the generation of geometric (radiation) damping. occur. The mass moment of inertia is calculated in a standard way. If the footing is of a simple rectangular shape with the dimensions shown in Fig. 4-12, the mass moment of inertia of the footing-machine system is: In a massless medium, these two factors would not

(4.32)

In this formula, the mass moment of inertia of the machine about its own axis is neglected because it is usually small compared to that of the footing.

160

Figure 4.13: Generation of Stiffness Constants

I JII

r---_..

-..

_.._ / ~ __ T\Tk"ttu,, /v J,e ! 'R~

1-.. ,k1{;if;, ................ 'J

. t.

~t_

~ ~

-;I '~' I

. . . . . -t-tI

VJJ=I

/-/

J

-k ub)

jv

JJ c l

L

-...:;:

-......~R= ku Yc

a)

Stiffness constants are defined as external forces to be applied at the centre ofgravity in order to produce a unit displacement at a time with all the other displacements being zero. When the centre of gravity lies above the level of the base, two external forces (stiffnesses) are needed to produce a sole unit displacement. The unit translation calls for the horizontal force kuu and the moment requires the moment k w and the force principle,ku'V k'l'ukUlI'

(Fig . 4.13a) while the unit rotation

(Fig. 4.13b). Because of Maxwell's reciprocity

=

kljlu ,

These stiffness constants are described as the stiffness constants

for translation and rotation at the centre of the base of the footing , transformed to the new reference point , the centre of gravity, CG. If the stiffness constants referred to the centre of the base are ku and k1jl' the stiffness constants referred to CG are:kuu = k u

(4.32a) (4.32b) (4.32c)

in which Yc > 0 if CG lies above the level for which ku and k., is are defined (this is the base in the cases considered in Figs. 4.12 and 4.13). Equations 4.32 are evident from

161

the geometry indicated in Fig. 4 .13 in which the reactive forces generated in the medium under the base due to unit displacement of the CG are also shown. surface foundations the constants For

ku, k, are given by Eqs. 2.20 to 2.22 . For embedded

foundations , the resultant expressions are described by Eqs. 2.26. For pile foundations, Eqs. 3.3 to 3.5 apply. If the footing is supported by an elastic layer of cork, rubber or other material whose Young's modulus is E, shear modulus G, and thickness d. the stiffness constants of the base are

k, = GA I dkv

(4.33a) (4.33b)

=El z I d

In which A = base area and Iz = second moment of base area about the axis parallel to z. These constants are to be substituted into Eqs. 4.32 . The damping constants are evaluated in the same manner. Thus , the formulae for damping are obtained from those for stiffness by replacing constants k by c in Eqs. 4.32. The resultant expressions are given in Chapters 2 and 3.

4.3.2 Governing Equations of Coupled MotionThe coupled motion can be caused by a horizontal excitation force, P(t) and a moment in the vertical plane, M(t), P(t) = P cos rot M(t) = M cos rot where00

(4.34a) (4.34b)

= circular frequency of excitation and P = the force amplitude; the moment

amplitude , M, derives from the horizontal force and possibly from an independent excitation moment , Me, and is

162

M = pYe + Me

(4.34c)

in which Ye

=the vertical distance between the horizontal force and the centre of gravity

of the machine-footing system. With the mass, stiffness and damping constants established, the governing equations of the coupled motion, composed of the horizontal translation, u(t), and the rotation in the vertical plane, \jf(t), can be written by expressing the conditions of dynamic equilibrium of the foundation in translation and rotation. Applying Newton's

second law and recalling the basic definitions of the stiffness and damping constants, the governing equations of the coupled motion are (4.35a) (4.35b) in which the dots indicate differentiation with respect to time, kUl v = kljlu, the sake of brevity, u(t) = u and ~J(t)CUIjI

=

~u

and for

=\!J.

The governing equations, Eqs. 4.35,can be rewritten in matrix form as

[rn]{ii}+[C]{Ll}+[k]{u} = {PCt)}

(4.36a)

in which the diagonal mass matrix, the displacement vector and the force vector are

[m] =[

{u(t)} = {U(t) } {P(t)} = { P(t ) } ~ 0] 1 ' If/(t) , M(t)

(4.36b)

and the stiffness and damping matrices are

[k] =

lk

uu

k tf/l~

ku'l/ [c] = k ' '1/'1/

J

[CUllC'1/U

(4.36c)

163

4.3.3 Solution of Equations of Coupled

otion

The governing equations, Eqs. 4.35 or 4.36a,of the coupled motion can be solved using two approaches: the direct solution and modal analysis. Both methods lead to closed form formulae and are easy to use.Direct Solution

The direct solution is mathematically accurate and is suitable with stiffness and damping constants which are frequency dependent or independent. For mathematical convenience, the harmonic excitation described by Eqs. 4.34 may be complemented by imaginary components iP and iM to yield pet) = P (cos rot + isin rot) = P exp (kot) M(t) = M (cos rot + isin cot) = M exp (ioit) (4.37a) (4.37b)

With this complex excitation, the particular solutions to Eq. 4.36a are also complex and can be written as

U(t ) } ;:::: c } exp(i to t) { lY(t) lYein whichUe

{u

(4.38)

and \lie are complex displacement amplitudes. Substitution of Eqs. 4.38 into

EqsA.36a yields two algebraic equations for these complex amplitudes:

P= (kuu -

rnco' + i (U Curl ~c + (kulf + i (U CuvJtc

These equations are readily solved using Kramer's rule. Introduce the auxiliary constants

164

(4.40)

Then, the complex vibration amplitudes are from Eqs . 4 .39,

(4.41 a)

(4.41b)

Separating the real and imag inary parts

(4,42a)

IJf

'l' c

. = nr + iu/ =M '1'1 '1' 2

/3& 1 1 2 &1

+j3& . /3& -/3& 2 2 + zM 2 [ 1 2 2 2 2

+ &2

&1

+ &2

(4.42b)

165

As in

Eq. 4.22, the true (real) vibration amp litudes u and

\jf

are:

(4.43a)

(4.43b)

When the motion is excited by a moment alone, P = for the real amplitudes result:

a and special, simpler expressions

u=M

(4.44a)

If/ = M

(4.44b)

The phase shifts between the excitation forces and the response follow from Eq. 4.23 as

(4.45a)

(4.45b)

Dropping the imaginary components of the response labelled by i, the real motion of the centre of gravity is:u(t) = U cos (rot + >u)

(4.46 a) (4.46b)

'V(t) ::: 0/ cos (Ot + >u)

From Eqs. 4.43 or 4.44 the response amplitudes are readily evaluated. Beredugo and

166

Novak (1972) formulated this closed form solution. For very high frequencies it may be advantageous to divide all constants a, large numbers. As in the case of uncoupled modes, dimensionless amplitudes

p and

s in Eq. 4.40 by co or co to avoid very

2

(4.47) may be introduced to facilitate the presentation and analysis of the response to forces whose amplitudes are constant. This is the case of force amplitudes independent of frequency or force amplitudes evaluated for a certain operating frequency. With excitation due to unbalanced forces of rotating or reciprocating machines, the force and moment amplitudes are proportional to the square of frequency as described by Eq. 4.17. If the excitation is caused by an unbalanced rotating mass me acting at a height Ye above the centre of gravity, then

in which e = rotating mass eccentricity; the ratio M / P = Yeo The dimensionless vibration amplitudes are, in the case of frequency variable excitation,

m Au =u

me ' e

(4.48)

The uncoupled modes of vibration in one degree of freedom are special cases of the solution described . It may be noted that an alternative direct calculation may be formulated in which the complex amplitudes u, and\jIc

are separated into their real and imaginary parts

beforehand. This approach leads to four simultaneous equations with real coefficients;

167

however, the computation requires more time and a closed form solution woul d be inconvenient. From the motion of the centre of gravity, the horizontal and vertical

components of the motion experienced by the surface of the footing can be determined. The upper edge of the footing experiences vertical amplitude amplitude u, that are:Ve

and horizontal

Ve

= If! ~ , U e = U

+ (b - y JIJI\jf

(4.49) is neglected and a,b are the

In the last formula , the phase difference between u and dimensions of the footing (Fig. 4.12).4.3.4 Examples of Coupled Response

Examples of the coupled response calculated from Eqs. 4.43 are shown in Figs. 4.14 to 4.17. The foundation is the one shown in Fig. 4.5, the excitation is quadratic and the footing is founded either directly on soil or on piles. Frequency independent stiffness and damping constants are assumed. Figs. 4.14 and 4.15 show comparisons between pile foundations and shallow foundations. As can be seen, pile foundations provide less damping than shallow

foundations . Fig. 4 .16 shows the response of the shallow foundation calculated for different soil shear wave velocities . For this more heavily damped embedded Fig. 4.17 shows the

foundation, the second resonance region often is not marked.

effect of soil stiffness on the response of pile foundation . As can be seen , the variation of resonance amplitudes with soil stiffness follows different patterns for shallow and pile foundations . Similar parametric studies can be conducted for various foundation conditions described in Chapters 2 and 3.

168

Figure 4.14: Horizontal Component of Coupled Footing Resposne to Horizontal Load. ( (Bx

= m IpR3x = 5.81, B'V = II pR\, = 3.46; (+) = modal analysis)

rI

t

A

I

I

I

-20

. .1. ..

II

I

--- --

.....

'Wz

,

40FREQUENCY w (RAD / S )

( '0\. "t

.

';

169

Figure 4 .15: Rocking Component of Coup ed Footing Response to Horizontal Load.

~i

'1.

,

.,.>-< , Cl

I L II2 :

IV

s '" f>,

IE

?

l

L

--l n,

::E

4.315x10 51b / ft / sec

Torsional Vibration:

k~1]

= k q l1 - 2fJ c1]7]OJ o = 1.789 x l 0 9 - 0.1 x 4.39 x l 0 6 x 82.3

=1.752x109 N m / rad ;:: 3x108 Ib ft / rad

= 6.563x10 6 N m / rad / sec

>

1.124x10

61b

ft / rad / sec

186

Coupled Motion: For the first mode ro ::::W1,

for the second mode

(0 ::::

W 2

For the more important first mode

k' =k /Ill -2j3c /Iii OJ l = 3.683x10 -O.lx3.18xl0 x 41.3 /Ill:::: 3.551x108 N 1m:::: 2.43x1 0 7 Ib 1ft

R

6

c'

till

=

c + 213 k11 11

I~ U

I OJ1 = 3.18 x 106 + 0.1 x 3.683 x 10 8 -.;- 41.3

= 4 .071x1 06 N I m I sec::; 2.786x1 05 Ib / ft / sec

k'~~ =k~~ -2j3c ~~ OJ1 =1.668xl0 9 -O.l x1.15xl0 7 x41.3=1 .620>

=1.554x10 7 N m / rad / sec

2.662x1061b ft / rad / sec

k'. =k1I~

u~

-2j3c

u~

OJl

=-5.34xl0 8-O.1 x(-4.611 xl0 6)x41.3

::; -5.149x1 08N/rad::; -1.156x1 08 Ib / rad

:::: -5 .904x1 06 N I rad I sec

>

-1 .325x1 06 Ib I rad / sec

Natural frequencies and damping ratios with material damping included:

For comparison the values obtained with material damping neglected are shown in brackets .

187

V ertical Motion:

kw

=3.708x10 8 N / m

Cw

=6.306x1 06 N/ m / sec

COo

fk = 62.14 rad I sec (65.14) = V;;2 km

c o = .jk;;;

= 0.528 (0.454)

Torsional Vibration: kill!

=1.752x1 09 N m / radVIII

C ll ll

=6.563x106 N m I rad / sec

COo =

fE = 81.47 rad / sec (82.3)

D = 0.1776 (0.10)

Coupled Horizontal and Rocking Vibration:001=

40.75 rad / sec (41.3)

CO2

= 110.34 rad / sec (112.3)

a1

=2.631 (2 .61)

a2

=- 0.632 (-0.638)

0 1= 0.188 (0.135)

D2 = 0.563 (0.30)

Response to Harmonic Loading The Vertical and Torsional Amplitudes: the amplitudes follow from Eq . 4.9. With Py

=rn,e co 2 and k =molo, the amplitude is, as in Eq. 4.19,

188

Expressing the amplitudes in a dimensionless form (Eqs. 4.23)

The response curves are shown in Figs. 4.24 and 4.25 .

Coupled Motion: Using the direct analysis, the horizontal and rocking amplitudes follow from Eqs. 4.43 . These are shown in Fig. 4.26 as dimensionless horizontal amplitudes and

dimensionless rocking amplitudes defined by Eq. 4.48,

Figure 4.24: Effect of Soil Material Damping on Vertical Amplitude of Shallow Foundation : 1) soil material damping neglected and (2) soil material damping includedw0

::::JI-

o

'"

u

a:.L...

....J

{l)0

-~~-- .

5'0

i

90

f RE2 UENCi iRRD. / SC: C. I

189

Figure 4.25: Effect of Soil Material Damping on Torsional Amplitude of ShallowFoundation: (1) soil material damping neglected and (2) soil material damping

included

w "]

a

= .: ~

0

(0

a: G ......

'"N

.

o

6'C

9'0

. ... '-'

~C: '"

.ec

f tH:~Q U ==' r~c r :~ ;:;:; .

I S :.C. J

190

Figure 4.26: Horizontal and Rocking Components of Coupled Response to Horizontal Loads: (1) material damping neglected and (2) material damping included

con

:::>l

w o

CJ

~

.,J

n, :L

a:

c

. -;

'

.....

a:

-.J

~

:c:

o

a::

50

I

I

i

I

90

lZO

I SO

.ac

FR:' QUf.NCY lA RD . IS EC. )

j

~o

I

90

120

I ~O

18 0

210

i

f RE.QUE NC Y [RAD. I SE C. )

J 91

CASE II - PILE FOUNDATION

Summary of Results for Pile Foundation

1) Vertical Vibration: a) Pile-soil-pile interaction neglected: kw :;:; 15.32x10 8 N 1m

m

-0

II)

'------sa

=====leo210

so

90

.

I

I

120

lS0

FREQUENCY rARo. /SEC. )

193

Figure 4.28: Horizontal and Rocking Response of Foundation Harmonic and Horizontal Excitation

by Fig

~ .19 to

J

r

.(;. ,~

4.4 RESPONSE OF RIGID FOUNDATIONS IN 6 DOF 4.4.1 Governing Equations

When the rigid foundation is of general shape , the response is in six degrees-of freedom, three translations and three rotations, all of them, possibly, coupled . These directions are indicated and labelled in Fig. 4.29. The stiffness and damping constants of the footing are best established for the elastic centre of the base (C.B.) first and then transferred to the centre of gravity (C.G.) of the footing-machine system , analogously to the coupled constants of a 2 DOF system by Eqs. 4.32a, band c. Thus, the individual stiffness constants kij and damping constantsCij,

or the impedance functions

194

refer to the centre of gravity and strictly satisfy the basic definition according to which

K ij

is the external force to be applied in the nodal direction i when there is a sole unit vibration amplitude to occur in direction j. The positive directions for forces and displacements are indicated by arrows in Fig. 4.29 . For embedded foundations, details on the coupled impedance functions can be found in Novak and Sachs (1973). For shorter writing, describe the stiffness and damping properties in terms of impedance functions Kij. Then, the typical governing equations in the directions X and\V, being conditions of dynamic equilibrium of forces and moments in the two directions

respectively, can be written using Newton's second law as (4.64a)

where dots indicate differentiation with respect to time. Figure 4.29: Notations and Sign Convention for Rigid Footing

,;/'

Pz.

f----- I I

/'

/'

P.

2 Y,v

I ~-L.:.r-:-+-----...J[-"::'--+-~

X, u

z,w

195

Similar equations can be written in the other four directions. In Eqs. 4.64, li = mass moment of inertia in direction i and Dij = products of inertia. The effect of the latter is usually small unless the asymmetry of the footing is very large. displacements and rotations in a vector, Listing all the

{u} = [u v{P(t)} = lpx(t)

W

S

ljI

SY/v(.(t)

(4.65a)

and the excitation forces in the loading vector,

Py(t)

Pz(t)

M y(t)

Al z(t)J (4.65b)

the impedance matrix can be readily written and the governing equations expressed in the standard matrix form, i.e.

[Tn J{U} + [KJ{u } =

{pCt) }

(4.66)

The mass matrix is diagonal when D jj l = O.

4.4.2 Free VibrationUndamped free vibretion In the case of free vibration, {P(t)} = 0 in Eq. 4.66 . When damping is neglected,Cij

= 0 and [K] = [k] . Assume that the stiffness matrix is frequency independent. Then, the particular solution for all displacement in Eq. 4.65a is {u(t)} = (u) sin cot, where (u) lists displacement amplitudes and co is the unknown natural frequency. Substituting the particular solution into Eq. 4 .66 yields

([k] -

0) 2

[mJXu } = {a}

(4.67)

Eq. 4.67 represents the classical eigenvalue problem whose solution yields six natural

]96

frequencies and vibration modes.

These are best determined using a suitable

subroutine such as IINROOT" in the IBM Scientific Subroutine Package or the subroutine "GVCSP" in the IMSL Mathematical Subroutines Package. If the stiffness and damping matrices are taken as frequency dependent, the eigenvalue problem becomes a nonlinear one and its solution is more difficult. The natural frequencies can be more easily identified from the response curves of the undamped or lightly damped system to harmonic excitation .

Damped free vibretionIf damping is considered and the impedance matrix is constant (frequency independent), the free vibration analysis leads to a nonclassical eigenvalue problem . Its solution, carried out in terms of complex eigenvalues and modes, yields six damped natural frequencies and associated modes, which feature phase shifts between individual motion components, and six modal damping ratios. The analysis can be carried out using a suitable subroutine such as 'IRGG" in the EISPACK package or "GVCCG" in the IMSL package. Details on the complex eigenvalue problem can be

found in Novak and EI Hifnawy (1983) or elsewhere.

4.4.3 Response to Harmonic LoadsIf the excitation forces are harmonic with frequencyCD,

they can be written as(4.68)

{P(t)}

= {P]}e iM = {P} (cos CO t + i sin (0 t)

The particular solution to Eq. 4.66 is

{u(t)} = {u}e;(lj{197

where {u}

=vector of complex amplitudes.[[K]-m 2[m]]{u} = p

Substituting into Eq . 4.66 yields (4.69)

This is a system of linear algebraic equations for the complex amplitudes that can be solved using the IMSL package or any other. Alternatively, both the complex impedance functions and the amplitudes can be split into their real and imaginary parts, i.e.

[K] = [k ] + i OJ [ C]

,{u} =

{u] } + i {u2 }

(4.70)

Substituting Eq. 4.70 into Eq. 4.69 and realizing that both the real part and the imaginary part of the latter equation must vanish, two coupled equations for the real (ul) and (U2 ) are obtained, which can be written as

(4.71)

The dimension of the problem is doubled to 12 x 12 but all is real. Consequently, the solution is easily obtained by Gaussian elimination or any of the basic subroutines available such as "SIMQII" in the IBM package. In either situation , after {U1} and {uz} have been established, the real amplitudes and phase shifts follow analogously to Eqs. 4.22 and 4.23. An example of the coupled response in 6 OOF is shown in Fig. 4.30 . The figure shows the vertical response and rotation about the horizontal axis X for an irregularly shaped, large compressor foundation exposed to harmonic unbalanced forces and moments. Notice that all six possible resonances need not be discernible. (The vertical response is shown for the edge of the footing.)

198

Figure 4.30: Response of an Irregular Compressor Foundation in 6 OOF in Vertical Translation (Z) and Rotation about Horizontal Axis X. (30m layer of clay overlying bedrock)

U

LlJ(f)

(

o .......

f f

....,

0 ~

...

L:.L:

.....: axI0 co

:1

----.--/ o -1----, . - . I, ---,..'-'-----.;:=--.- I' I I '=> J 5 10 IS 20 2 $ 30

o

)r" 35 'r--" , ~ "I 10 isI I i l l

50

55

66

6S

]C

Fr:EOUEl-IC )'

R:A.O . ISEC .

199

o

CI

o ...,

,.....1:

0 0CD

.

z;

~

00

Q

~ ,g

X..."

,::,

....-e, ;

CJ ..III

':~

40

a ... -=:::

~" ~(;)

\-0

001

III

E

:tL

..0..J ..J

C

III

... z: z:A.~

g

.4..J

JI

0.001

0 N It%

t! Z

-=~

A

0

0.0001 100

III

1000'REQUENCY

10,000

CP.

E D C BA

Dangerous, shut it down immediately Failure is ncar, Correct very quickly. Faulty, correct quickly. . ' i ' \ . Minor faults. No faults, typical of new equipment.

(

(,

'C'r:

209

Figure 4.35 Response spectra for allowable vibration at facility

Frequency.

epa

210

Figure 4.36 Vibration standards of high-speed machines

6O ...........Q...--I----I--.........- - - t - - - - t

4000

SPEED, RPM\

,-'(

.

211

Figure 4.37 Turbomachinery bearing vibration limits

2560 0 0

...'28

.-;

0.630

After Baxter and Bernhard (1967)

REFERENCESAboul -Ella, F. and Novak, M. (1980) - "Dynamic Response of Pile Supported Frame Foundations," Journal of Engineering Mechanics, Vol. 106, No. EM6, December, pp. 1215-1232. Arya , S.C., O'Neill , M.W. and Pincus, G. (1979) - "Design of Structures and Foundations for Vibrating Machines," Gulf Publishing Company, Book Division, Houston, Texas, p. 191. Baxter, R. L. and Bernhard, D. L. (1967) . "Vibration Tolerances for Industry", ASME Paper 67-PEM-14, Plant Engineering and Maintenance Conference, Detroit , MI, April.

213

Beredugo, Y.a. and Novak, N. (1972) - "Coupled Horizontal and Rocking Vibration of Embedded Footings," Canadian Geotechnical Journal, Vol. 9, No.4, pp. 477-97. Novak, M. (1974a) - "Dynamic Stiffness and Damping of Piles," Canadian Geotechnical Journal, Vol, II, pp. 574-598 . Novak , M. (1974b) - "Effect of SoH on Structural Response to Wind and Earthquake," International Journal of Earthquake Engineering and Structural Dynamics, Vol. 3, No.1 , pp.7996. Novak, M. and Beredugo, Y.O. (1972) - "Vertical Vibration of Embedded Footings," Journal of the Soil Mechanics and Foundations Division, ASCE , SM12, December, pp. 1291-1310. Novak, M. and EI Hifnawy, L. (1983) - "Effect of Soil-Structure Interaction on Damping of Structures," Journal of Earthquake Engineering and Structural Dynamics , Vol. 11, pp . 595-621. Novak, M. and EI Hifnawy, L. (1984) - "Effect of Foundation Flexibility on Dynamic Behaviour of Buildings," Proc. 8th World Conference on Earthquake Engineering , Vol, 111, San Francisco, pp. 721-728. Novak, M. and Sachs, K. (1973) - "Torsional and Coupled Vibrations of Embedded Footings," International Journal of Earthquake Engineering and Structural Dynamics, Vol. 2, No. 11, 33. Novak, M., EI Naggar, M. H., Sheta, M., EI-Hifnawy, L., El-Marsafawi, H., and Ramadan , 0 ., 1999. DYNA5 a computer program for calculation of foundation response to dynamic loads. Geotechnical Research Centre, The University of Western Ontario, London , Ontario. Richart, F.E., Hall, J.R. and Woods, R.D. (1970) - "Vibrations of Soils and Foundations," Prentice-Hall, lnc., Englewood Cliffs, U.S.A. Urlich, C.M. and Kuhlemeyer, R.L. (1973) - "Coupled Rocking and Lateral Vibrations of Embedded Footings," Canadian Geotechnical Journal, 10, pp. 145-160.

214

5

FOUNDAnONSFORSHOCKPRODUaNG MACHINES

5 FOUNDATIONS FOR SHOCK-PRODUCING MACHINES

Shock producing machines generate dynamic effects which essentially differ from those of rotating and reciprocating machines and the design of their foundations, therefore, requires special consideration.

5.1 Introduction

Many types of machines produce transient dynamic forces that are quite short in duration and can be characterized as pulses or shocks. Typical machines producing this type of load are forging hammers, presses, crushers and mills. The forces generated by the operation of these machines are often very powerful and can result in many undesirable effects such as large settlement of the foundation, cracking of the foundation , local crushing of concrete and vibration. Excessive vibration may impair the operation of the facility and the health of the workers, cause damage to the frame of the machine and expose the vicinity to unacceptable shaking transmitted through the ground. Some machines operate with fast repeating shocks and consequently, the effects of vibration may be aggravated by resonant amplification of amplitudes such as is the case with rotating or reciprocating machines. The objective of the foundation design is to alleviate these hazards and secure optimum operation of the facility. Hammers are most typical of the shock-producing machines and therefore this report is limited to them. This is not a serious limitation, however, because the design and analysis of the other shock producing machines follow criteria that are in many respects similar to those applied to hammers.

213

5.2 Ty pes of Hammers and Hammer Foundations

There are many types of hammers. According to their function, they can be divided into forging hammers (proper) and hammers for die stamping. Forging hammers work free material into the desired shape while die stamping hammers shape the material using a mould or matrix. According to their mode of operation , hammers can be classified as drop, steam and pneumatic, although other systems are also used. More details on the various types can be found in Major (1962) Because of the powerful blows generated, hammers are mounted on block foundations of reinforced concrete separated from the floor and other foundations . The basic elements of the hammer foundation system are the frame, head (tup), anvil and foundation block (Fig.5.1). The frame of forging hammers is separated from the anvil. In die stamping hammers, the frame is usually connected to the anvil to give the system rigidity and precision of blows.

Figure 5.1: Schematic of Forging Hammer and its Foundation

214

The forging action of hammers is generated by the impact of the falling head against the anvil , which is a massive steel block. The head is allowed to fall freely or in order to obtain greater forging power, its velocity is enhanced using steam or compressed air. The size of the hammer can be judged by the weight of the head, which ranges from a few hundred pounds to several tons . The intensity (energy) of the blows can be expressed as a product of the head weight and the height of the drop or the equivalent height of the drop. Only a part of the impact energy is dissipated through plastic deformation of the material being forged and conversion into heat. The remaining energy must be dissipated in the foundation and soil. this end. In small hammers, the anvil is sometimes mounted directly on the foundation (Fig.5.2a). This is done for the sake of simplicity and hard shocks . The main drawback of this arrangement is that the concrete under the anvil suffers from the shocks and, depending on the hammer type , also from high temperature. Repairs often may be necessary. To reduce the stress in the concrete and shock transmission into the frame, viscoelastic suspension of the anvil is usually provided (Fig. 5.2b). This may have the form of a pad of hard industrial felt, a layer of hardwood or, with very powerful hammers, a set of special isolation elements such as coil springs and dampers. Such a Different foundation arrangements are used to

suspension reduces the impact of the anvil on the foundation by prolonging the path of the anvil and by energy dissipation through hysteresis and plastic deformation .

2J5

J:igure 5.2: Types of Foundation Arrangement

/

r

GA P

\

/

SPR lNGS -TROUGH

(c)

( d)

The foundation block is most often cast directly on soil as indicated in Figs - 5.2a and b. When the bearing capacity of soil is not sufficient or undesirable settlement is anticipated, the block may be installed on piles. When the transmission of vibration and shock forces in the vicinity and adjoining facilities is of concern, a softer mounting for the foundation may be desirable. This can be achieved by supporting the block on a pad of viscoelastic material such as cork or rubber (Fig-5.2c) or on vibration isolating elements such as rubber blocks or steel springs possibly combined with dampers (Fig. 5.2d) . A trough, which adds to the cost, is needed to protect these elements. The material of the pads must be able to resist fatigue as well as moist environment due to condensation and must have a long lifetime. Rubber pads should

216

have grooves or holes to allow lateral expansion because the Poisson's ratio of rubber is 0.5. Slabs of solid rubber are quite incompressible. The gap around the footing, which rests on a pad (Fig-5.2c), may be filled with a suitable soft material which allows the block to vibrate freely but prevents blockage of the gap by debris.

Figure 5.3: Suspended Footing Blocks

.' .D..

~

~

4II"

-

~

"

..# ...

d.

-

..

4.

. .... ..

_

..

-. -.

~.

_

...

a)

b)

With springs, the space around the footing must be wide enough to provide access for installation, inspection and replacement. This is necessary because springs sometimes crack. However, access space is not always available. For reasons of easy access and convenience, the springs are sometimes positioned higher up and the footing is suspended on hangers or cantilevers (Figs. 5.3a,b). Such a design is more complicated and costly. Careful reinforcement of the cantilevers for shear is necessary. The soft suspension of the footing block on pads or isolators is particularly efficient on stiff soils. It increases the vibration amplitude of the block but reduces the force transmitted into the soil. Additional damping, if provided, is very useful because it reduces the vibration amplitude. The inertial block is sometimes deleted and the anvil

217

suspen ded directly on isolators (GERB). More complicated foundations are sometimes designed to protect the frame of the hammer from shocks which can hinder the operation and cause fatigue cracks . To interrupt the flow of shock waves into the frame, additional joints with viscoelastic pads or elements (Fig. 5.4) separate the upper part of the block from the rest. Klein and Crockett (1953) describe the example shown in Fig. 5.4b .

Figure 5.4: Schematic of Foundation with Additional Joints to Protect Hammer Frame: (a) outline and (b) prototype

, - . !' , -, -. .~~ . .. I' - -- . -. -.- -. . ~ . - , . , . .- .. . , . . - . - . / i '' r'x x . -. - - ,. .- -: i( :-. . ,. ~> . c;'"".".

..........

".

.-

,""' ,\

r

". '

~

/ ~ 7 ~ ~.

/"

'~ '

a) 5.3 Des ign Criteria

b)

The hammer foundation must be designed so as to facilitate efficient operation of the hammer without failure and cause minimum disturbance to the environment. This general objective may be achieved if the vibration amplitude, settlement, physiological effects, and all stresses remain within acceptable limits. In addition, resonance should be avoided with high-speed hammers.

218

5.3.1 Vibration AmplitudesThere are no unique limits on the vibration amplitude unless specified by the manufacturer or codes. In the absence of such specifications, the allowable level of vibration amplitude is estimated on the basis of experience and physiological effects with some accommodation for the fact that larger hammers produce greater shocks and usually larger amplitudes. A few values of maximum allowable amplitudes are suggested for guidance in Table 5.1.

Table 5 .1: Maximum Allowable Amplitudes for Hammer Foundations

For foundations built on soils susceptible to settlement, such as saturated sands, smaller amplitudes are desirable. On the other hand, larger amplitudes may sometimes be admitted for large hammers provided they satisfy the criteria for physiological effects and settlement. Amplitudes larger than about 0.16 in (4 mm) can impair the operation of the hammer, however.

5.3.2 Physiological Effects of VibrationPhysiological effects depend on vibration velocity and acceleration rattler than displacement, but vary with the type of vibration and the sensitivity of individuals. The

2]9

velocity may be considered a criterion in the moderate frequency range typical of hammers. The amplitude of vibration velocity can be calculated approximately as (5.1) In whichVrn =::

the maximal (peak) displacement and roo ::: the natural circular frequency

of the foundation. Various authors (see Richart et. al. 1970) have collected many data on human perceptibility. The data given by the German Code DIN4025 and shown in Table 2 are useful in that they provide an indication of perceptibility of vibration as well as the effect of vibration on work. The data are shown as a function of the physiological factor K calculated as

K =O.80vmfor vertical vibration and

r

-'

>

oJ'!(I

r.A. "

~'.

,OJ I

(5.2a)

(5.2b) for horizontal vibration; V m is peak vibration velocity in mm/s (1 inch 25.4 mm) . For

>

machines operating intermittently, the effect on work may be one category lower than that given by the calculated value of K.

l"(\I(

;' l (.'

.,(

"

'.r 1

rI

10---

\

-

_I

. "tI

- ,

[~\

II

rjr}

,

"

\ !-

~/ l

r0

30 mm Standard Sampler Sampler without liners

CR

0.95 1.0 1.0

Sampling Method

Cs

1.0

1.2

1.7.2 Cone Penetration Test (CPT) In a CPT test, the cone penetrometer (Figure 1-18) is pushed into the ground at a standard velocity of 2 cm/s (0.8 in/s) and data is recorded at regular intervals (typically 2 or 5 ern) during penetration . The standard cone penetrometer has a conical tip of 10 cm 2 (1 .55 in 2 ) area, which is located below a cylindrical friction sleeve of 150 cm 2 (23.3 in2 ) surface area. The tip as well as the friction sleeve are connected to load cells to record the tip resistance qc and the sleeve friction resistance fs during penetration, and the friction ratio FR defined as FR =fJ qc(Figure 1-19). The initial tangent shear modulus Gmax is related to the penetrometer tip stress using the following empirical formulas (Kramer 1996). For sand:

22

G max 1634(q c )0.250 (a rv )0.375 For clay:

(1-19)

G max: 406(q c )0.695 e -1.1 30 G max , qc, and a~, in equations 1-19 and 1-20 are calculated in kPa.

(1-20)

1.7.3 The Cross-hole Seismic SurveyThis method determines the variation with depth of in-situ low-strain shear wave velocity V smax As shown in Figure 1-20, the cross-hole method is based on generating shear waves in a borehole and measures their arrival times at the same elevation in neighboring boreholes. The wave velocity is calculated from the travel times and the spacing between the boreholes. The initial tangent shear modulus is calculated from the measured low-strain shear wave velocity Vsmax using equation 1-3 as: (1-21 ) in which, p is the mass density of the soil and g is the acceleration due to gravity. For successful results of a cross-hole test , there should be at least two boreholes, which are spaced about 3 to 5m (10 to 15 ft) apart. Also, the source must be rich in shear wave generation. The SPT can offer a good inexpensive solution . Moreover, the receivers must be in good contact with the surrounding soil.

1.7.4 The Seismic Down-hole SurveyThis method offers an economical alternative to the cross-hole test as it needs as shown in Figure 1-21 one borehole, inside which the receiver can be moved to different depths, while the source remains at the surface, 2 to 5 m (6 to 15 ft) away. Alternatively, the test can be done by fixing an array of multiple receivers at predetermined depths against the walls of the borehole. Travel times of body waves (S or P) between the source at surface and the receivers are recorded for various depths and a plot of travel time versus depth can be generated, from which V smax or V pmax are then computed at the same depths as the slope of the travel time curve at that depth .

1.7.5 The Seismic Cone Penetration TestThis test as shown in Figure 1-22 is a combination of the down-hole and cone penetration tests. The cone penetrometer is modified by mounting a velocity seismometer inside it, just above the friction sleeve. At different stages during penetration of the cone penetrometer, penetration is paused to generate impulses at the ground surface. Travel time-depth curves can be generated and interpreted the same way as the down-hole test..

23

s7

sJ

...!

I

~

I iI I

I. ' -': ... ...

1

:s

1I

i I~

. ,..

.~

;;It

----* 1 Conir.:lll pl"lil'1T (10 CIIl')~ l.noo cell J S tJ.~iJ ) ga . ~~~ 4" Friction sleeve ( 151) .;.oi )

35.6

rnm

I

5 AdjuRtment r ing I'J W~te tp~ lXJ r bushi ng 7 Cable S Connection I'r'ilh rods

Figure 1-18. CPT penetrometer

Friction resistance (tons/ft)

Bearing resistance (tons/ft)

Friction ratio (%)

6 4 2 0

100 200 300 400 500. .

o

2 468

o

I!

:::: .cQ)

-a. a

10

1 ......;..,.....J, ... . .

. ~=====~.!C; . . . . .. . . ' . ..

20 30

T ~

:

.:

..

. . r: .. .e O ; ..

40

_ __ _ ;

. o w w

.:. . . . .

. ~ . ':' '' A '' :

=

5060

. ' 0 ... .

. ... :

.

- . - ~ ----_

... . .

Figure 1-19. Results of cone penetration sounding

24

A significant advantage of this method is that with a single sounding test , one can obtain information for the stratigraphy of the site, the initial tangent shear modulus of different layers, as well as static strength parameters. A limitation of this method is that it may not be adequate for some types of soils containing coarse gravels

1-8 The Design SpectrumThe design spectrum should satisfy certain requirements because it is intended for the design of new structures or seismic assessment of existing structures to withstand potential earthquakes . Therefore, it should in general sense be representative of ground motions recorded at the site during past earthquakes or at other sites under similar conditions . The design response spectrum is usually based on statistical analysis of the response spectra for the ensemble of ground motions for a specific site. Different codes have developed procedures to construct such design spectra from ground motion parameters. One such procedure of FEMA-356 and the LRFD Guidelines for Seismic Design of Highway Bridges is outlined herein as an example 1. From the U.S. Geological Survey web site (http://earthquake.usgs.gov/) determine the 0.2-second and 1-second spectral accelerations Ss and S1 . These values can be obtained by submitting the latitude and longitude, or the zip code of the site of interest. These are spectral accelerations on rock outcrop (class B). 2. Classify the site according to the average shear wave velocity in the upper 30 m. Site class A is defined as hard rock with average shear wave velocity greater than 1500 m/sec (5000 ftlsec) . Site class B is defined as rock with average shear wave velocity that ranges from 750 to 1500 m/sec (2500 to 5000 ftlsec) . Site class C is defined as very dense soil and soft rock with average shear velocity that ranges from 360 to 760 m/sec (1200 to 2500 ftlsec). Site class D is defined as stiff soil with average shear wave velocity that ranges from 180 to 360 m/sec (600 to 1200 ftlsec) . Site class E is defined as soft clay with average shear wave velocity that is less than 180 m/sec (600 ftlsec) 3. Determine the site coefficients Fa and Fv for the short period spectral acceleration and the 1-second period spectral acceleration respectively. These values are displayed as function of the site class as shown in Tables 1-2 and 1-3. 4. Calculate the design earthquake response spectral acceleration at short periods, SOs :: Fa Ss, and at 1 second period, S01 :: Fv S,. 5. Determine the periods Ts and To required for plotting the design response spectrum, where Ts :: S01/S0s , and To :: 0.2 Ts .

25

-- -/"m (13fl)

Lm (l3ftl

[mpQct

+TfQnsduc~

r

- --I'"~ 1II

I

...

I _J'

"J I~~ ~

....

,

.)

~

-

Figure 1-20. Seismic cross-hole test

26

._-----~ ----------_.~.~--

._-- _ ..__ .._- .._

_-_ ..

,.

~-

Figure 1-21. The Seismic down-hole test

V. (m/s)

! felay eerueru =5';;; ) @ Ma . aI NoQ

FINESCOl\'TENT:?; 5%

"')0 .

~~o

U quefoaion l.iq:lOn l.KEfac1,00

AdjllStmenl

Pan A.merian data

Recomrreeded Jap;m=~ra By Workshop Chinese data10

A

g

0A

o

20

30

40

CorrectedBlow Count, (Nl)6Q

Figure 1-28. Curve Recommended for Determining CRR from SPT Data (Youd and Idriss, 1997)4j

4

3.53

angeof recommen I MSFfromNC Workshop

I

25

1.S

.... .~

o5.0 6.0 7.0 8.0 9.0

Earthquake Magnitude. MwFigure 1-29 Magnitude scaling factors for the SPT data (youd and Idriss, 1997)

38

0.6Volumetric strain (%)

0.5

1054 3

2

1

0.5

II

0.4~

a:

0.3

0.20.1

.' .. .',.~ ~

:0.2 .. . . .. .. .. t.: . . .0.1 . . . .... . .. . . . . . '. . . . . ....... . . .... . . . .. ... ..... . . . .. ...... . . .. .. .. ... # # #

.9

."

o

o

10

20

30

40

50

(N1)SOFigure 1-30. Estimation of post-liquefaction volumetric strain from SPT data and cyclic stress ratio for saturated clean sands and rnaqnitude > 7.5 (Tokimatsu and Seed, 1987)

39

Figure 1-31. lateral spread following the 1995 Kobe, Japan earthquake in Tempoyama Park Osaka (courtesy of EERC, Univ. of California)

Figure 1-32. Lateral spreading in Granada Hills at Rinaldi St. (Granada Hills, California) following the 1994 Northridge earthquake (courtesy of EERC, Univ. of California)

40

For free-face conditions: Log OH = - 16.3658 + 1.1782 M - O.9275Log R - 0.0133 R + 0.6572 Log W + 0.3483Log T15 + 4.5270 Log (100 - F15) -0.9224 050 1 5 (1-26) For ground slope conditions: Log OH = - 15.7870 + 1.1782 M - 0.9275 Log R - 0.0133 R + 0.4293 Log S + 0.3483 Log T 1 5 + 4.5270 Log (100 - F15) - 0.9224 050 15 (1-27) where, OH is the estimated lateral ground displacement in meters; M is the moment magnitude of the earthquake; R is the horizontal distance from the seismic energy source, in kilometers; W is the ratio of the height (H) of the free face to the distance (L) from the base of the free face to the point in question, in percent; T15 is the cumulative thickness of saturated granular layers with corrected blow counts,(N1)60, less than is, in meters; F15 is the average fines content (fraction of sediment sample passing a No. 200 sieve) for granular [ayers included in T 15 in percent; 050,5 is the average mean grain size in granular layers included in T 1 5 , in mm; and S is the ground slope , in percent. The LRFO guidelines for the seismic design of highway bridges (2004) recommends using this approach only for screening of the potential for lateral spreading , as the uncertainty associated with this method is generally assumed to be too large. Alternatively, more rigorous methods such as the Newmark sliding block analysis can be used to assess the potential of post-liquefaction lateral spreading at a site.I

1.9.3 Post-liquefaction Flow FailuresFlow failures are the most catastrophic form of ground failure that may take place when liquefaction occurs in areas of significant ground slope. Flow failure may be triggered when farge zones of soil become liquefied or blocks of unliquefied soils flow over a layer of liquefied soils. Flow slides can develop where the slopes are generally greater than six percent.

1.9.4 Mitigation of liquefaction HazardMitigation of liquefaction potential can be established either by site modification methods or by structural design methods. Site modification methods include but not limited to: Excavation of the site and replacement of liquefiable soils, which is applicable only to small projects due to the expenses of excavation and soil replacements . Oensification of in-situ soils through advanced compaction methods such as vibroflotation. This principle involves lowering a machine into the ground to compact loose soils by simultaneous vibration and saturation. As the machine vibrates, water is pumped in faster than it can be absorbed by the soil. Combined action of vibration and water saturation

41

rearranges loose sand grains into a more compact state. After the machine reaches the required depth of compaction, granular material, usually sand, is added from the ground surface to fill the void space created by the vibrator. A compacted radial zone of granular material is created. In-situ improvements of soils by using additives such as the stone column technique. The stone column technique, also know as vibro-replacement, is a ground improvement process where vertical columns of compacted aggregate are formed through the soils to be improved. These columns result in considerable vertical load carrying capacity and improved shear resistance in the soil mass. Stone columns are installed with specialized vibratory machines. The vibrator first penetrates to the required depth by vibration and air or water jetting or by vibration alone. Gravel is then added at the tip of the vibrator and progressive rising and repenetration of the vibrator results in the gravel being pushed into the surrounding soil. The soil-column matrix results in an overall mass having a high shear strength and a low compressibility Grouting or chemical stabilization. These methods can improve the shear resistance of the soils by injection of chemicals into the voids. Common applications are jet grouting and deep soil mixing. Designing for liquefaction may be accomplished by the use of deep foundations which are usually supported by the soil or rock below the potentially liquefiable soil layers. These designs would need to account for additional forces that would develop because of potential settlement of the upper soils that could occur due to Iiquefaction.

1.10 ReferencesArias, A (1970) " A measure of earthquake intensity, ~ Seismic Design for Nuclear Power Plants, MIT Press, Cambridge, Massachusetts , pp. 438-483 Bolt, B.A. (1969) "Duration of strong motion, "Proceedings of the 4th World Conference on Earthquake Engineering, Santiago, Chile, pp. 1304-1315. Bartlett, S.F. and Youd, T.L. (1992). "Empirical analysis of horizontal ground displacements generated by liquefaction-induced lateral spread, "Technical Report NCEER-92-0021 , National Center for Earthquake Engineering Research, Buffalo, New York . FEMA (2000). "Prestandard and commentary for the seismic rehabilitation of buildings", FEMA-356 , Federal Emergency Management. Washington, D.C. ldriss, I.M. and Sun, J.I. (1992). "SHAKE91: a computer program for conducting equivalent linear seismic response analyses of horizontally layered soil deposits , "User's Guide, University of California, Davis, 13pp.

42

Kramer, S.L. (1996), "Geotechnical Earthquake Engineering," Prentice-Hen, Inc ., Upper Saddle River, New Jersey, 653 pp. MCEERJATC (2003) "Recommended LRFD guidelines for seismic design of highway bridges", MCEERIATC 49, Applied Technology Council and Multidisciplinary Center for Earthquake Engineering Research. Schnabel, P.R, Lysmer, J., and Seed, H.B. (1972). "SHAKE: computer program for conducting equivalent linear seismic response analyses of horizontally layered sites," Report EERC 72-12, Earthquake Engineering Research Center, University of California Berkeley. Seed, H.B. and Idriss, I.M. (1970)."Soil moduli and damping factors for dynamic response analyses," Report EERC 70-10, Earthquake Engineering Research Center, University of California Berkeley. Seed, H.B. and Idriss, I.M. (1971). "Simplified procedure for evaluating soil liquefaction potential," Journal of Soil Mechanics and Foundations Division, ASCE, Vo1.107, NO.SM9, pp.1249-1274. Seed, H.B., Tokimatsu, K., Harder, L.F., and Chung, RM. (1985). "Influence of SPT procedures in soil liquefaction resistance evaluations," Journal of Geotechnical Engineering, ASCE, Vol. 112, No.11, pp.1016-1032. Seed, R.B. and Harder, L.F. (1990). "SPT-based analysis of cyclic pore pressure generation and undrained residual strength," Proceedings. H.B. Bolton Seed Memorial Symposium, University of California Berkley, Vol. 2, pp.351-376 . Tokimatsu , K. and Seed, H.B. (1987) Evaluation of settlements in sand due to earthquake shaking," Journal of Geotechnical Engineering , ASCE, Vol. 113, No.8, pp.861-878. Youd, T.L. and Idriss I.M (1997) Proceedings of the NCEER Workshop on evaluation of liquefaction resistance of soils. Report NCEER 97-22, National Center for Earthquake Engineering Research, Buffalo, New York.

43

2

SEISMIC DESIGN OF SHALLOW FOUNDATIONS

45

2-SEISMIC DESIGN OF SHALLOW FOUNDATIONS2.1 GeneralShallow foundations are usually suitable for sites of rock and firm soils. The stability of these foundations under seismic loads can be evaluated using a pseudo-static bearing capacity procedure. The applied loads for this analysis can be taken directly from the results of a global dynamic response analysis of the structure with the soil-foundation-interaction SFI effects represented in the structural model.

2.2 SFI Representation in Global Structural ModelsSFSI effects can be incorporated into global structural models by means of two methods, the foundation dynamic impedance function method, and the Winkler spring model method. The dynamic impedance function method is adequate if the seismic foundation loads are not expected to exceed twice the ultimate foundation capacities (FEMA 2000). The Winkler spring model approach is more applicable for life safety performance-based design, where it is essential to represent the nonlinear force displacement relationships of the soil-foundation system. As illustrated in Figure 2-1 , the dynamic impedance model is an uncoupled single node model that represents the foundation element. On the other hand, the Winkler approach can capture more accurately the theoretical plastic capacity of the soil-foundation system. The non-linear spring constant for this approach are usually established through non-linear static pushover analyses of local models of the soil-foundation system using general-purpose finite element programs such as ABAQUS or ADINA, or by using a Geotechnical soil structure interaction programs such as FLAC OR SASSI. An upper and lower bound approach to evaluating the foundation stiffness is often used because of the uncertainties in the soil properties and the static loads on the foundations. As a general rule of thumb, a factor of 4 rs taken between the upper and lower bound (ATC-1996). The procedure is to make a best estimate of foundation stiffness and multiply and divide by 2 to establish the upper and lower bounds, respectively.

22. 1 The Dynamic Impedance ApproachThis approach is based on earlier studies on machine foundation vibrations, in which , it is assumed that the response of rigid foundations excited by harmonic external forces can be characterized by the impedance or dynamic stiffness matrix for the foundation. The impedance matrix depends on the frequency of excitation, the geometry of foundation and the properties of the underlying soil deposit. The evaluation of the impedance functions for a foundation with an arbitrary shape has been solved mathematically using a mixed boundary-value problem approach or discrete variational problem approach. Both approaches are mathematically rigorous methods. In another approach the problem has been approximately solved by defining an equivalent circular base. The impedance function of a foundation is a frequency dependent complex expression, where its

46

real part represents the elastic stiffness (spring constant) of the soil-foundation system and its imaginary part represents both the material and radiation damping in the soil-foundation system. At small strain levels typically material damping ratio ~ associated with foundation response is on the order of 2% to 5%. Radiation damping is close to viscous damping behavior. and is frequency dependent. Considering the range of frequencies and amplitUdes in earthquake ground motions compared to machine foundations, it is reasonably to ignore the frequency dependence of the stiffness as well as the damping parameters. There are two methods for evaluating the dynamic impedance functions for a shallow foundation that are commonly used in practice. The first is based on the approximate solution for a circular footing rigidly connected to the surface of isotropic homogeneous elastic half-space. adopted by FHWA (FHWA-1995). which provides the static stiffness constants for each degree of freedom. The second approach is based on the more rigorous mathematical approaches. Evaluation of the stiffness coefficients using the equivalent circular footing is carried out in four steps as follows: Step 1: Determine the equivalent radius for each degree of freedom, which is the radius of a circular footing with the same area as the rectangular footing as shown in Figure 2-2: ro

=Rv =~BLl1t ro =Rh =~BL/1C-R _[16(B)(L)3]1/4-

(2-1-a) (2-1-b) (2-1-c)

r0

r1 -

---=--z....:.....:.-

31t

_R

ro

-

r2 -

_ [16(B)3(L) 31t

]1/4

(2-1-d)

_[16(B +L r0 -R t - ---=---'61t

2 2>]1I4

(2-1-e)

Step 2: Calculate the stiffness coefficients for the transformed circular footing for vertical translation ksv , horizontal (sliding) translations ksh1 and ksh2 , rocking, kr , and torsion, k t

k sv

-

_ 4GR v1-u 2-u

(2-2-a)(2-2-b)

8GRh k sh1 = k Sh2 =

47

p

Foundation forces

Uncoupled stiffnesses

Winkler spring model

Figure 2-1. Analysis models for shallow foundations

x

/ : RECTANGULARFOOTlNG

I-

-~

I

IJ

II

I~------if--"""'y

I I I I

coN

2l

--t--EQU1VALENT

CIRCULAR FOOTING

Figure 2-2. Calculation of equivalent radius of rectangular footing

48

(2-2-c)

where, G and v are the dynamic shear modulus and Poisson's ratio for the soil foundation system. Step 3: Multiply each of the stiffness coefficients values obtained in step 2 by the appropriate shape correction factor C1 from figure 2-3 (Lam and Martin 1986). This figure provides the shape factors for different aspect ratios LIB for the foundation . Step 4: Multiply the values obtained from step 3 by the embedment factor ~ using Figure 2-4 for values of D/R :5:0.5, and Figure 2-5 for D/R > 0.5 (Lam and Martin 1986). D in these figures is the footing thickness . The second approach for calculating the impedance functions for the soil foundation system is based on the results of rigorous formulations (Gazetas 1991). This approach is adopted by FEMA-356. Using Figures 2-6 and 2-7, a two-step calculation process is required. First, the stiffness terms are calculated for a foundation at the surface. Then, an embedment correction factor is calculated for each stiffness term. The stiffness of the embedded foundation is the product of these two terms. According to Gazetas , the height of effective sidewall contact, d, in Figure 2-7 should be taken as the average height of the sidewall that is in good contact with the surrounding soil.

2.3 Dynamic Bearing Capacity of Shallow FoundationsThe general vertical soil bearing stress capacity of a shallow footing is:qull

= en, Sc + yDNqSq +2yBNySy

1

(2-3)

In this expression : C = cohesion property of the soil bearing capacity factor depending on angle of internal friction, ' Nc , Nq ,N.( and evaluated as:

=

N = eittan

eoefl. of Varia on of SoH Ree.ction Modulus with Depth, f {lblln S } T [TTTTIT I T 1 111111

~

1=80 1=60'"

,=

1=20 r- f= 10 ~ 1= 5t-.. f = 1 r-..

f~40

1=0.5 f = 0..1

10' 1

1012

BendingStiffness, EI (lb-ln2 )

II/

(

I1

!--......IiI

,

J t I

I

Figure 3-3 , Cross-coupling pile-head stiffness (Lam and Martin 1986).

82

c

8as

~ 10 6 "d cv

L:::

--:::. r;::; v.---: __

I:;:::~

V-::::---

"'"-'"

......-

...----: v....v

\:12( \f\V'\ '\

"C(I)

J: I

V10 5

~

-:: ,.,/'

vVi--'

.......VV

,......

...,)t'\\ \

\l\\\f\f\ f=200 1\ f= 150 f=100 I' f= 80 1\ f= 60 1 \ f;;;;4O

,I-'

I--"

It:,gG)

...tZ

$

>v

~

V

as ...(I)

~ en

V4 10

a;

~~~;ijg~~~*mt~~~mJ~l\~\.\ :....V

V

V V I--'

I--'

---v

v

I\.

v

,1\

---VI--"

~

l\\

f;;;;20 f= 101\ f= 5

1\\\.

-I

~-+--H--H++++--t-+-+-H-+l+il---+-+-H-t+tH--"n\ 1\

f;;;; 1 f r::s 0.5

-+---I---t-+++1I-+tt---f--I--H-+-Hf+---L---J.......I..-JL....I..Lu.J----'1 \

f = 0.1 Coeff. of Variation of Soil Reaction

Modulus with Depth. (lbJin 3 ) 3 10 -+--+--+-++++trH---+--+-i-+++t+t--r--r-T"T'TTTr'lr--"""T"""""T""T""TT'T'TT1 I I I 111111 I I I I I III 13 11 12 109 10 10 10 1010

Bending Stiffness, EI (Ib-in 2) Free head Pile Stiffness/

J

II

I I

r

1 1

I

Figure 3-4. Lateral pile-head stiffness for free-head condition (Lam and Martin 1986).

83

10 7 : EmbedmentI I I I I

I I

Co

t: ~ .0~

--.

-

'C

ee

-::

-

- - - - - 51

a

-

- 10'

- 10 6 ~

.""

"...

...

...I ....

".

V

coCD

I

-

f=100_~

-

~ 10ct) tI)

CD .~

~

--.,

...........

5

...

,," ...

...

-' ......~

~

~

--'" "...

....

...

... ....... ..... ""

.

+:

c:

CD

CiS~

m c 10 40

~CI)

-v.....~

v

_.....

".

e- :_

I

1-/ v~

...

/

......

....

=

~

F

~10 3

~--'"

....

>

......1

......Coeff. of Variation 01 Soil reaction MOdulus with D9eth, f llb/in 3 )II I I I1III

..

.....

I

-

""1010

J I I 1III

10

11

10

12

Bending Stiffness, EI (Ib-in 2 )

PIle H9w:l at Gradll Lswl

Embedded pae Head

Figure 3-5. Lateral embedded pile-head stiffness for fixed-head cond ition(Lam and Martin 1986)

84

I

I

I

I I III

= :

Embedment 0'

"'C

a:I

.~

c:

-

- - -

- - 5' ----- 10'IA

~

l.t?

~

/?

~

m 1010A.

Vv:

V .1,,1

,.

i-'..~~

.0

,.,

C :.;:;..~Q)

P/accncrtl,

0 0.00'5

0

'I

21/

D5'5./0'5

01!/81'5

2'5'5(0 )X

9W

cor ::=

:= cor

50'515~~80'5

981IU'5 1'5!/8 1819 1819 }104

y ;:= cor

< /}

950920

R'iqid pile ~olufion if:> obfained

by xf1f1inq fhe din [ricilon and end bearlnq

capocliie af each axial dif:>place:f'7enf.

0 0 ,00'5,0 '5'5

0211

')

f,1581'5

./0'5 ,2'5'5x := cor

960 984124-'5 1'558 1819 1819 )

(0)(1)

cor :=

50'5150'5 5.80'5

y:= cor

95f,9 20

2000

Riqid Pile: 5olufion

1'500~ ~~

j

_Y_l000

~

~

'500

Oi------..L.---......L.----L---~

o

10

20

xDi:?pJocet?CfJ1 (f'lf'i)

105

Flexible pileXJ!ulion /~ ochieYcd by:!Vf'7I'?/rq ibepilehead di~placer?Cnf of eachloadleYd 10 lheriqid pile::dulion Area of pileeedion0pi! := (0 Z11 6i5 8/'5

9GO 984 12+'5 /'558

/8i9

/819 ) iN

106

/'dl.loJ pile=!ufion b oMaiocd by o/eroqln; lhe fJaiMe: and tiqld pile!X>!u/io~

o0.'5/61::;2'5

o '\

211

61;;8/'5

1,61

Ac ;=v;:=

2072.16

960 984/21-'51'5;;8/819

k(O) (I)

;;66z:~

I1c

6.112.9

2;;.5'5 /819 )

2OO0r------r -----.- ---.---~-__,

I1dl.ld :301u/ion

1'500'"'

8-.l

~I:l

--/000z

y v

~'500

6raphlcd ~iffnc~ :>oIu/ion i~ oblaincd by dderrJlninq a vdoe for lhe pile: xcad diffrt=" (or lhe odod =/ufion al10X of fheullinak oxjdpile: capacity

Di:>placenad cotcopondirq10 107. of lhe vllimle: copociiv

L\ := 2.62

rJrJ

/Uiolpile: ::Nfocx>f1

LRFD :>o/ufion(e:qualion ;;-8)

107

Pile group ~/jf[rx;~~

n :=G

Dldance 10 kax/~ of roidion DI:;-fance fa Yraxi of rofalion

50/

:=

0:1 (7

"XX6 := n"xx

KXX6

= 10'507G.I (7=

xN

r. '('(6

kN 1'5/GI4-Y5 ("J

('1

KXeY6 :: 8540'5.8G

xl\!

KYeX6 = 221'521.4-4- kN

KeX6 = 18829'50.GI

kN('1.

rad

KeY6 = 4078908.1'5

('1.

kN rad

r. eZ6 = 2'5/4-:5G,4-'5

kN('1.

rod

108

:5 focz fhe bridqc i~ fo be buill oaoso a krqe flood plaIn corrlr/buflon of fk f'OX'~ re!>idance of Ik pile cop10 Ik lafad ~iff~ CQflrof befakenido = rr.:iderafion

Tbado: Ik ::J'rffne= nalrixcan be o:pr~d o

10"5076.1

01'516/4'.1'5

0 0~0/2-66~,12

02 2./'521.#

-1 882.9'50.6 1

0

1

0

0

00

K . '

0

022.1'52./,4-4

0/882C;X;.6/

004018908.1'5

0- /882.9 '50.6 1

0 0 0

0 0

00

0

0

0

0

2.'514-%.4-'5 )

3.4 ReferencesBrown, D., Reese , L. and O'Niell, M.(1987), "Cyclic lateral loading of a large scale pile," Journal of Geotechnical Engineering, ASCE , Vol. 113, No.11. Gadre, A. (1997), "lateral responseof pile-cap foundation systems and seat-type bridge abutments in dry sand, Ph.d. Dissertation, Rensselaer Polytechnic Institute. Hoit, M.L and McVay , M.C., (1996), FLPIER User's Manual, University of Florida, Gainsville. Florida . Lam, I.P. and Martin , G .R. (1986), "Seismic Design of Highway Bridge Foundations," Report No. FHWAIRD-86-102, U.S. Department of Transportation, Federal Highway Administration, McLean, Virginia, 167 p. Lam, LP., Kapuskar, M., and Chaudhuri,D . (1998) "Modeling of pile footings and drilled shafts for seismic design" Technical Report MCEER-98-0018, Multidisciplinary Center for Earthquake Engineering Research, Buffalo , New York. Lam , I.P. Martin , GR. , and Imbsen, R. (1991) , "Modeling bridge foundations for seismic design and retrofitting," Transportation Research Record 1290. LPILE (1995), "Program lPILE Plus, Versiuon 2.0" Ensoft Inc., Austin , Texas,:fId

Matlock, H.(1970), "Correlations for design of laterally loaded piles in soft clay", Offshore Technology Conference, Vol. 1, Houston , pp.579-594.

McVay, M.C., O'Brien, M., Townsend, F.C., Bloomquist, D.G., AND Caliendo, J.A. (1998), "Numerical analysis of vertically loaded pile qrcups ," ASCE Foundation Engineering Congress, Northwestern University, Illinois, pp.675-690.

109

McVay, M.C., Casper, R. and Shang, T.(1995),"Lateral response of three-row groups in loose to dense sands at 3D and 50 Pile Spacing," Journal of Geotechnical Engineering, ASCE, Vol. 121, NO.5. NAVFAC, (1986), "Foundations & Earth Structures, " Naval Facilities Engineering Command, Design Manual 7.02. O'Neill , M.W. and Murchison, J.M . (1983), "An evaluation of p-y relationships in sands", Report No. PRAC 82-41-1 to THE American Petroleum Institute Terzaghi, K. (1955), "Evaluation of coefficients Geotechnique, vol. 5, No.4, pp.297-326. of subgrade reaction",

Vesic, A.S. (1977}, " Design of pile foundations", Transportation Research Board, National Research Council, Washington, D.C.

110

- --

4-RETAINING WALLS UNDER SEISMIC LOADS4.1 General During the past earthquakes, gravity earth retaining walls have suffered considerable damage which ranged from negligibly small deformations to disastrous collapses. Performance of retaining walls during past earthquakes has revealed the fact that the damage is much more pronounced jf the wall is extending below the water level. According to Seed and Whitman (1970), failures in walls extending below water level may have resulted from a combination of increased lateral pressure behind the walls, a reduction in water pressure on the outside of the wall and a loss of strength due to liquefaction. As an example, extensive failure of quay walls during the 1960 Chilean earthquake and the 1964 Niigata earthquake in Japan have been attributed to backfill liquefaction. Fewer cases were reported for walls constructed above the water level. Few cases of minor movements of bridge abutments were reported during both the San Fernando and Alaska earthquakes. This section will focus on two of the most commonly retaining walls used in construc