Design of Foundations for Dynamic Loads

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 Outcomes Students' 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 NOTES M.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 NOTIONS Statics 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 freedom The 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,

)~

l, ;'

t

\

r:

­

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

---------

-

---

- -- -- -

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 and reciprocating 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)

I

= Psin(wt)

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

oscillation is completed and thus wT = 2n:.

The frequency measured in cycles per

second and expressed in units called Hertz (Hz) is

f=~=~

T

21f

Consider 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 .-

.

r j

f

I

I (

I

,I

I

,I 1

,'"'1

. f

" "

-

"

)

)

r

t,

~

7

I

J I

;

-,

.

I

;r,. (

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 ratio

W2 /W1

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

is

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 being

W2 - 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

""

'

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

/ I

\ \ -,

-­

.-­

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

\.../.

a)

,/

/

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

,/

/

.......

/

P, =P2 6)2

e

W, . 1, 188. . "

b}

",

-, -,

/

\,

'\

/

'\

\

I

\

\

I I

\ \

I I \

\

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

W, t

sin(w2t.,.. Jf)

,,

I

/ /

.....

/

I

/

'\J

I

--,\ I

",'\

I

\,

I

I

I

I

,

w{

I

""'--­

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

c)

9

Transient Loading is characterized by a nonperiodic time history of a limited duration and 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 mathematically with 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( t)

P(f)

t

o

t

Figure 1.5: Impact loading

. _.

'-r

k

I

(

fJt

o

(.,­

I

r(1 'J :' V.

1J.f« T - 1/0

t Figure 1.6: Random loading

P(f)

Sp(f)

o a) 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).

The type of foundation may result in considerable differences in

response .

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 rotations

I

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

~/1:::\.-~·,

o

w

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



1

.)~

-. ... _-­ ---

... _----- ...

--

---- -

-- - -

J~

~/, ~~~~i'~~~~r~&{1,/~0;1

r---­

/I'lI4

(#,... . , ~-

19

e xWe .....• (I

(V

L

.l....t. "

c'

Excitation Forces of Machines

= In rotating machines the excitation forces stem from centrifugal forces associate

~o

"-'1..

':;"'/-with

residual unbalances. Their magnitude can be estimated on the basis of balancing experiments or experience. The centrifugal force is represented by vertical and horizontal forces, the amplitude P for rigid rotors is usually defined as:

)

'" \ \.,\ •.1

:

• rl ('~ '

ftf, e.

Q. . \

~ ,

I." J

7

(,...\ " \ J t."

'I .

,F

t ' l ( ".. )

and co = circular frequency of rotation. The magnitude of e is typically a fraction of a " .tr . C; , .( millimeter such a s,I Q1-0~rnrn.] I !

{ it

,

In reciprocating machines the excitation forces stem from inertial forces and centrifugal forces associated with the motion of the pistons, the fly wheel and the crank mechanism . Many of these forces can be balanced by counterweights but often, higher harmonic components and couples remain unbalanced.

In design situations the

excitation forces should be provided by the manufacturer of the machine.

DESIGN OBJECTIVES The design of foundations for vibrating equipment is always governed by displacement considerations. The displacement of foundations subjected to dynamic loads depends on i) the type and geometry of the foundation ; ii) the flexibility of the supporting ground; and iii) the type of the dynamic loading. The main objective of the design is to limit the response amplitudes of the foundation in all vibration modes to the specified tolerance. Usually the tolerance is set by the machine manufacturer to ensure a satisfactory performance of the machine and minimum disturbance for people working in its

r

I

ru

"",J <

immediate vicinity.

Another objective which could be extremely importan t in some

cases is to limit vibration propagating from the footing into the surroundings.

DESIGN CRITERIA

\ .. I

} ij~ ~ .

'

" '.I' ) , ' -

"

Factors that may be included in the design requirements .

( ,

I ....

' :

_.

.

1,_ static requirements for bearing capacity and settlement. .

~ ot"

•

\

I

C'

{

2. Dynamic behaviour , '

,1 -. ft

• limiting vibration amplitude • limiting velocity

,I

• limiting acceleration • maximum dynamic magnification factor • maximum transmissibility factor • resonance conditions

t

0­

3. Possible modes of vibration

vertical; horizontal; torsional; rocking; pitching and possibility of coupled modes.

4. Possible fatigue failures in the machine, in the structure, or in connections.

5. Environmental considerations

I

,

• physical and physiological effects on people • effects on nearby sensitive equipment

I

.-- '

\

,

• possible resonance of structural components

I

I '

• consideration of foundation isolation 6. Economy

,J

J•

1./ ''rI

~ 1'.'

• initial cost L

2]

I"

i.,

+.

Uv..l,'. ' "

~

('"

.

Ir1

-

!

\

• maintenance costs • down time costs

~. . r: ,-.

I

\ \

I . , (1

\

..\

• replacement costs . ~1 ­ , r

"

\J'

1

I

, '( .1

)

Ijt't-· ,~ DESIGN PROCEDURE

','r,'

\

{ (,' I

I':' I

~

(

r

o E

- I­

~

~ o

4 .877m ( 16 f t )

77

3. The Soil : Unit weight (y)

100 Ib/ft (15.714x10N/m)

Unit mass (p)

3.105 slug/ft (1602 kg/m)

Shear wave velocity (Vs )

492.1 fUsec (150 m/sec)

Material damping (tanc)

0.1

Poisson's ratio (v)

0.25

4. Masses: from 1 and 2

6583 slug (9.60x10 4 kg)

Total mass of the system, m

Solution

The stiffness and damping constants will be calculated for the following vibration modes:

1. Vertical mode 2. Coupled horizontal and rocking vibration in X-V plane

3. Torsional mode.

Equivalent radii: from formulae (2.19)

Translation :

Ru , R" =

Rocking:

R'I'

lab

V-; == 2.174 m == 7.132 ft

A~

= v3"; =1.96 m =6.43 ft

78

Torsion:

RTI:::

4

ab (a 2 + b 2 ) 6Jr

:::

2 .26 m ::: 7.414 ft

Shear modulus of soil :

G = p V} :::

1602 X (150)2::: 3.6045x107 N/m ::: 7.519x10s Ibltf

CASE (1) - SHALLOW FOUNDATION Shallow foundation overlying a deep homogeneous soil layer (halfspace) with no embedment.

a. Soil Material Damping Neglected: -

-

-

Use formulae (2 .24 - 2 .27). Setting all constants SV1, 5 "2' S'11, 5"/2 S",1 , 5 'f/ 2' SU1, and I

Su 2

equal to zero (Le. no embedment cont ribution) and reading the values of the other

constants CV1, C"2'" from Table 2.2 for granular soil the stiffness and damping constan ts are:

Vertical Motion : = 3.604x10 7 x 2.174 x 5.2

=4.074x108 N/m

= 2.788x10 7 Iblft = (2 .174)2 x (1602 x 3.60 x 10 7)1/2 x 5 6

::: 5.68x10 Nrrn/sec

>

s 3 .887x10 Iblftlsec

79

Coupled Motion: 8

=3.604x10 7 x 2.174 x 4.7 = 3.683x10 N/m 7

= 2.520x10 Ib/tt ;::: (2.174)

X

2.403x10 5 x 2.8

= 3.18x10 6 N/m/sec;::: 2.176x10 5Ib/fUsec

=1.668x10 9 N.m/rad ;::: 2.857x10 8 Ib.fUrad

4

4

1.45"1 2.17

= 2.403x10 5 [(1.96) x 0.5 +(2.174} x (- ) - x 2.8] ;::: 1.15x10 7 Nrn/rad/sec

>

1.97x106Ib .ftJrad/sec

7

= -3.604x10 x 2.174 x 1.45 x 4.7

= -5.34x10 8 N/rad

r:-;::;

CUl{/

2

=- '\j P G Ru

­

YcCu 2

=-1.199x10 8Ib/rad

= -2.403x1 05 x (2.174)2 x 2.8 x 1.45

= - 4.611x10 6 N/rad/sec = -1.035x10 6Ib/rad/sec

80

Torsion: k'7'1 :::;;

GR~ C'11

=3.604x10 7 X (2.26)3 X 4.3 =1.789x10 9 N.m/rad =3.065x10 8Ib.fUrad

=(2.26t X 2.403x10 5 x 0.7 = 4.39x10 6 Nrn/rad/sec = 7.52Ix10 5Ib.ftIrad/sec

b.Soii Material Damping included (tano

=2~ =0.1)

Use Eqs, 2.18. For hysteretic material damping frequency w is needed. If the footing response is to be evaluated for a given operating frequency this operating frequency is substituted for w. If whole response curves are to be calculated, the frequency is better taken as equal to the natural frequencies of the footing, l.e.

(0

=

WI.

The natural frequencies are calculated in Chapter 4, in which the effect of material damping on the stiffness and damping constants of the footing is accounted for.

81

3

STIFFNESS AND DAMPING

OF PILE FOUNDATIONS

) . .,

Ilrrl'U:."~

'"'I......... _. _

Examples of pile supported structures are shown in Fig. 3.0. Figure 3.0: Examples of Pile Supported Structures a) offshore towers

b) nuclear reactors

; i

.IU '-1.\

-

_.

~~

J r

0 .2

•

~

0 .4

0 .6 0 .6 FRECUENCY b,

1.0

l.z

For slender piles in average soils, dynamic stiffness can be considered to be practically independent of frequency as indicated by curves 2 and 3. The troughs visible on curves 2 and 3 are caused by soil layer resonances but they completely disappear for higher values of soil material damping , 0 = tano. The imaginary part of stiffness (pile damping) grows almost linearly with frequency and therefore can be represented by constants of equivalent viscous damping

Cj

which are also almost frequency

independent. Only below the fundamental natural frequencies of the soil layer given by Eqs. 2.29 does geometric damping vanish and material damping remains the principal source of energy dissipation; then the soil damping can be evaluated using Eq. 2.30b.

88

The disappearance of geometric damping may be expected with low frequencies, shallow layers and/or stiff soil.

Apart from these situations , frequency independent

viscous damping constants and functions f 2 which define them, are sufficient for practical applications. The mass ratio PP is another factor whose effect is limited to extreme cases. Pile stiffness and damping changes significantly with the mass ratio only for very heavy piles (Novak and AbouJ-Ella, 1978b). The Poisson's ratio effect is very weak for vertical vibration, absent for torsion and not very strong for the other modes of vibration unless the Poisson 's ratio approaches 0.5 and frequencies are high. The effect of Poisson's ratio on parameters f 1,2 can be further reduced if the ratio E/Ep rather than G/Ep is used to define the stiffness ratio.

Figure 3.4: Comparison of Vertical Stiffness and Damping Parameters of Floating Piles with End Bearing Piles (Novak, 1977, ao

=0.3)

D Ob ,..

I

\

- - fl OA""O PIl[

\

-

- - [NO

\

a~AFl'I'j(; PIL£.

\ I ,? l OA....PIN al

,

\ \

" ....

-­ '\.----­

__~'"'r-- -

I

40 Pll£.

&0

Sl C N O

VERTi CAL L/ D"2S · DO

!:1:- ~_--·-·~

~ - o_o,

uJ

~ c: =""

~

\

' - STAT IC ( POU LO S J974)

,

, --~----,----""",,-----,-----'------r-'

0-

_.-.--_ .--., -

30 ·

20 ­

1 O.

' --" -' -' -' -' -. -. -' j

+0·

s id

Figure 3.11 : Group Efficiency of Vertical Stiffness and Damping of Two Floating

Piles for Different Separations and Weakened Zones Around Piles (rO

~o

t / :l

c.'a

~ t

-1 0

r l. O;' ''

i

1. ,

~r: .

30.

=R)

1 ·~ C.

.0,_

s Id

101

Another remarkable feature of dynamic behaviour of pile groups is the oscillatory variation of stiffness and damping with frequency (Fig. 3.12) . Curves numbered 4 and 5 were calculated including pile-soli-pile interaction while for the other curves this interaction was neglected.

Different soil profiles were considered as well as a

composite soil medium that incorporates the weakened zone (curves 5).

This zone

reduces the sharp peaks observed in the homogeneous medium (curve 4) but does not eliminate them. Obviously, dynamic group-effects are quite complex and there is no simple way of alleviating these complexities. The use of suitable computer programs appears necessary to describe the dynamic group stiffness and damping over a broad frequency range.

However, a thorough experimental verification of the phenomena

indicated by theoretical analysis is still lacking. The only simplifications available are the approximate approach due to Dobry and Gazetas (1988) in which the interaction problem is reduced to the consideration of cylindrical wave propagation . The replacement of the group by an equivalent pier, considering the dynamic interaction as equal to static interaction or using dynamic interaction factors. The equivalent pier cannot yield the peaks shown in Fig. 3.12, may be applicable only for very closely spaced piles and may overestimate the damping (Novak and Sheta, 1982).

The static interaction may be sufficiently accurate for

dynamic analysis if the frequencies of interest are low and especially if these frequencies are lower than the natural frequencies of the soil deposit given by Eqs . 2.29 .

102

Figure 3.12: Variation of (a) Stiffness and (b) Damping of Group of Four Piles with Frequency and Soil Profile (Sheta and Novak, 1982)

...

ir­ '--

~

"2 .

~

\0 ;

z

I:

f';

ec

'"

,.

~



[,,-

7

»:

..,­

'0

t

~!

0.5

1.0

0 .0

0 .5

00

1.0

°0

0.7

0,4

0.3

."i'eoI po r t

-- -

i ma go part

- 0.1

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

::: j

I fi

1

-J

~ ~ ~ ~ i

-1

0

{ ' '\

II" I

.\

II

~ ~l ~--. : : J: C

1

!r ii !,~

, I '\

,~l

:< ~

I

~ ~i o

:::

~

Nl •

G

Col.¥led :-buo11, No L'1t.eractic'A'1

I

I

.1

\ 11

i t - SDJF, No rnt.e::=-action

~l

~ I~ \

J/ I

.

\

eoupled

\

/JM ~ \

I

,

3C·

!,' ~\ so

.

v.()t ion

I::1te;.ac::.ion

SrY"'.~

Tnt,:>..- cci,~ __ 5~_ ~,

'

~"

~~'- w

eo

;20

: :N

l Bli

e,

2!C

2H

137

4

RESPONSE OF FOUNDATIONS AND

STRUCTURES TO HARMONIC EXCITATION

4. RESPONSE TO HARMONIC LOADS

Harmonic loading is shown in Fig. 1.2. It represents the basic case of periodic loading and can be described as

pet) = P coset

(4.1)

where P force amplitude and co excitation frequency.

For reasons of mathematical

convenience, an auxiliary imag inary component may be added making the excitation force complex. Denoting the imaginary unit i is described as the complex harmonic force

pet)

=p ei ~)l= P (cosot +

/sincot)

(4.2)

analogous to a complex number (Fig . 4.1) z= a + b

=r (coso +isin 100 KW 1,000 RPM> 100 KW 750 RPM 600 RPM 500 RPM 6) Factory Machines Lathe, drilling machines Shaper, milling machines Grinders , precise lathe 7) Paper Machines (according to make & component)

0.20 1.2 1.0 2.0 0.30 0.025 = 1 mil 0.050 0.100 0.050 0.090 0.10 0.12 0.16 0.20 0.01 0.003 0.008 to 0.025

Table 4.2 Noise Levels of Typical Machines That Range Above 90 dB Machine Hydraulic pres s Pneumatic press Wing bar drop press Swagger Shell press Mechanical power press Header Drop hammer Automatic screw machine Circular saw Ball mill Grinder

Average dB level 130 130 128 108 - 118 98 - 112 98 - 110 101 - 105 99 -101 93 - 100 100 99 80-95

155

p Va = - &

(4.25)

k

where k is stiffness and n is the dimensionless dynamic amplification factor which depends on the natural frequency w 0 and damping ratio D according to the equation

1

c = -r===========

(4.26)

The total force transmitted into the supporting medium results from both the restoring force and the damping force and is

F(t)

=

k vet) + c v(t)

(4.27)

With the motion given by Eq. 4.24, the expression for transmitted force becomes

F(t)

=

k V COS(OJt + rjJ) - c vOJ sin(OJ t + rjJ)

This is a sum of two harmonic motions having different amplitudes, the same frequency and a phase difference of 90 degrees. Thus, the transmitted force is also harmonic and its amplitude is

(428) Upon substituting for the amplitude from Eq. 4.25 and realizing that

the amplitude of the transmitted force is obtained as

156

(4.29)

With constant excitation amplitude P, the transmitted force varies with frequency and damping as shown in Fig. 4.10 in which the dimensionless ratio FJP, called transmissibility, is plotted . Damping reduces the transmitted force in the resonance region but increases it for frequencies w>

.J2 COo.

Figure 4.10: Transmissibility

3t-------:~....I....---+-----+------==-l

2 t-------ti:.......+"'+-i---+-----+----l

1

e::----.~~-_+_--_____ll----_i

i:""' .

1

2

3

With excitation stemming from the rotation of unbalanced masses , the force amplitude

157

is proportional to the square of frequency (4.30) where me and e are the unbalanced mass and its eccentricity, respectively . P can be rewritten as

P

mee =--m()) In

in which p

=:

2 = p k( ­ ca J2

(4.31 )

())o

mse I m is equal to the amplitude v as

product pk is the restoring force as

0)

---7

00,

0)

---7

co and m

=:

k I ulo

The

This force can be used to normalize the

transmitted force. This normalized transmitted force F/(pk) is shown in Fig. 4.11. The increase in the transmitted force due to damping for co >

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

I D=O

, V

~ , "'\.~\ ,' j

0.5

/

\\

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

\1

'.

\.

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

158

4.3 COUPLED RESPONSE OF RIGID FOUNDATIONS IN TWO DEGREES OF FREEDOM

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 Damping Inertia forces are due to the mass of the footing-machine system and its mass moment 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.

In a massless medium, these two factors would not

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:

(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 J

r---_..

-..

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

II

_.._ / ~ __ T\Tk"ttu,,­ /

. t.

vJ,e !'

~t_

~~

-;I '~ '

I

R~

-...:;:

jv

-k u

I

. . . . . -t-t-

L

JJ c l

VJJ=I

J

/ -/

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

b)

a)

Stiffness constants are defined as external forces to be applied at the centre of gravity 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

=

kljlu ,

k'l'u

kUlI'

(Fig . 4.13a) while the unit rotation

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

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 d

(4.33a)

=El z I d

(4.33b)

kv

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 Motion The coupled motion can be caused by a horizontal excitation force, P(t) and a moment in the vertical plane, M(t),

where

00

P(t) = P cos rot

(4.34a)

M(t) = M cos rot

(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

(4.34c)

M = pYe + Me

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] =[

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

(4.36b)

and the stiffness and damping matrices are

[k] =

lk

uu

k tf/l~

J

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

[CUll C'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)

(4.37a)

M(t) = M (cos rot + isin cot) = M exp (ioit)

(4.37b)

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

{u

U(t ) } ;:::: c } exp(i to t) { lY(t) lYe in which

Ue

(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 CuvJ¥tc

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 =

a and special, simpler expressions

for the real amplitudes result:

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)

'V(t) ::: 0/ cos (Ot + ¢>u)

(4.46b)

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,

p and

2

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

large numbers. As in the case of uncoupled modes, dimensionless amplitudes

(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

(4.48)

me ' e

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

Ve

and horizontal

amplitude u, that are:

Ve

= If! ~ , U e = U

+ (b - y JIJI

In the last formula , the phase difference between u and

(4.49)

\jf

is neglected and a,b are the

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

foundation, the second resonance region often is not marked.

Fig. 4.17 shows the

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)

r£

t

I

A

I

I

I

-­

I

•• ••• •• ••• ••• • •••.• •• .1. •.. •

I

--- --

..... ­

I

'Wz 20

,

40 FREQUENCY w (RAD / S )

( '0\. "t

.

';

169

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

'1.­

~i

,

I L I

.,.

I

s '" f>,

IV

>-< , Cl

IE

2 :­

Ii

?­

i

l­

--l n,

I

::E l­ -l

0..

!

~

I

q~uz

l

::t I

I

j '

I t

",

C,. t·1

,I

r

0.10

':~

\-0

~" ~(;)

O

...a-= :::»

•4

•I li•.z:.. •z: -!C a •-> .....­ III

0·01

0

III

E

.. 0

III

g

:t

C

L

JI

.• 4

..J

-... •z: • •z:

0.001

..J

A.

~

..J

t! Z

0

N

It

­

-=

A

~

III

•

0

%

0.0001 100

'REQUENCY •

E D C B A

10,000

1000

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

(

CP.

'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

256 0 0 0

...

.-;



~

"1(



0

320

5

L

( \ '

\

\

r

212

Table 4.3 General machinery-vibration-severity data

Horizontal Peak Velocity

Machine Operation

(in/sec) < 0.005

Extremely smooth

0.005-0.010

Very smooth

0.010-0.020

Smooth

0.020-0.040

Very good

0.04-0.080

Good

0.080-0.160

Fair

0.160-0.315

Slightly rough

0.315-0.630

Rough

>0.630

Very rough

After Baxter and Bernhard (1967)

REFERENCES Aboul -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.

Different foundation arrangements are used to

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

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

~

~

. ­.' 4

D ..

-

a)

..

4.

•

~

-

II"

"

d.

...

_

. .... ..

..

.. #

-. -.

~.

_

...

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)

b)

5.3 Des ign Criteria

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 Amplitudes There 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 Vibration Physiological 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 which

Vrn =::

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 r

K =O.80vm

oJ'!(

>

-'

r.A. "

I

~'.

(5.2a)

,OJ I

for vertical vibration and (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 (.'

rI

10---

\

(

"

'. r 1

. "t-

-

_I

- ,

[-

II

rjr}

,

"

I

\ !-

r

~/ l

<

r

~\

1

0"

v-

,.

0 J\

\

220

Table 5.2: Physiological Effects of Vibration

D1N4025 (German Code)

K (m/s)

Classification

Work

0.1

Threshold value, vibration

Not affected

just perceptible 0.1 - 0.3

Just perceptible , scarcely

Not affected

unpleasant, easily bearable 0.3 - 1.0

Easily noticeable,

Still not affected

moderately unpleasant if lasting an hour, bearable 1.0 - 3.0

Strongly noticeable, very

Affected but possible

unpleasant if lasting over an hour, still tolerable 3.0 - 10

Unpleasant, can be

Considerably affected , still

tolerated for one hour, not

possible

tolerable for more than one hour 10- 30

Very unpleasant, not

Barely possible

tolerable more than ten minutes 30 -100 Over 100

Intolerable

Impossible Impossible

221

Another physiological effect is noise. Hammers are noisy and have a level of noise of about 100 db (decibel) but some presses are even noisier.

5.3.3 Stresses Stresses in all parts of the foundation have to remain within allowable limits. This includes compressive stresses in the pad and the concrete underlying it; bending, shear and punching shear stresses in the block, and finally, the stresses in the soil or piles as well as in the vibration absorbers , if used. Dynamic stress is repetitive and can cause fatigue . This effect can be accounted for by lowering the static allowable load or by multiplying the dynamic stress by a fatigue factor.

A fatigue factor

J.!

=3

is often

recommended for all parts of the system. Steel springs are also subject to fatigue, particularly when they have initial cracks.

To eliminate this possibility, the springs should be X-rayed before they are

installed. Temperature effects can contribute to the decay of the pad under the anvil and the underlying concrete because the temperature of the anvil can rise to as high as 1000C (2120F) with hot forging.

'- I . 5.3.4 Foundation Settlement

I

c r

t •

I

1-' I

•

"' \

Settlement can be a serious problem with hammer foundations. Where the soil is unreliable in this respect, piles or absorbers should be considered (Figs .5.2c, d and 5.3).

Piles limit the settlement by transmitting the loads to deeper strata while

absorbers reduce the settlement by reducing the forces transmitted to the soil.

222

5.3.5 Mass of the Foundation Block The adequacy of the foundation mass and dimensions is best proven by detailed analysis of stresses and amplitudes . This is particularly true for the more complex foundations.

Nevertheless, some guidelines have been suggested for the preliminary

choice of the weight of the foundation block. Assuming that the anvil weight is about twenty times the weight of the head, Go, the weight of the block. G, can be estimated using the formula (Rausch, 1950)

(5.3) in which Co

= the

maximum velocity of the head and c,

:=

reference velocity taken as

1.8.37 ft/s (5.6 m/s). Smafler masses can be used if the response is limited by special measures such as shock absorbers. As for the general layout of the foundation, it is desirable that the centreline of the anvil and the centroid of the base area lie on the vertical line passing through the centre of gravity of the footing with the hammer. Misalignment and eccentric blows can result in tilting of the foundation and differential settlement

5.3.6 Vibration Effects on Environment Vibration propagates from the footing into the surroundings in the form of various types of waves. At greater distances, surface waves (Rayleigh waves) usually prevail. The vertical amplitude of the ground motion, vr , at a distance x from the vertical axis of the foundation can be evaluated approximately as

223

'1 GJ

Y

t'.

6 I l,l

' ,'

,I

Vr

- .ff

r:

-va -e -a(r-roJ r

~ \

in which v» = the footing amplitude, ro

",I

fc-~ ,

\

.

iJ """.J. {~ _

"I

/

"'--"

~

.l

30 mm

1.0

Standard Sampler Sampler without liners

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

(1-19)

For clay:

G max: -- 406(q c )0.695 e -1.1 30

(1-20)

G max , qc, and a~, in equations 1-19 and 1-20 are calculated in kPa.

1.7.3 The Cross-hole Seismic Survey This 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 Survey This 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 Test This 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

s

...

s

I

7

Ii

~

I

!

:s

1

J

1 I

I

. ' -'

I

: .­

. ,..

.. ...

i I

.~

;;It

----*­

35.6

rnm

• I

~

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

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

6 4 2 0

o 10

---

20

a. a

30

::::­ .c Q)

40

50

Friction ratio (%)

Bearing resistance (tons/ft)

Friction

resistance

(tons/ft)

I !

100 200 300 400 500 .

o

2 468

.

··1···......;..,.....J,• .•.. . .

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

··

.

· · ·····T ··· · · · ·~ · ·· · ··

_ __ • _ • • ;

. o w • w• • •

.:. . . . .

:

:

·

. ' 0 ... .

•

..

. r:. .···. .

.

. ~ . ':' '' A '' •• · :

.

. ...:

e O ;

..

=

• ••

- . - ~ ----_

... . .

60

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 Spectrum The 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

-- -Lm (l3ftl

/"m (13fl)

[mpQct

+ TfQnsduc~

I

...

"J I

I _

~ J'

- --­

r

~

~

I'" ~ 1

I

....

,

.

~

)­

-

I

Figure 1-20. Seismic cross-hole test

26

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

._-- _.•.__ .._- .._

_-_..

,.

~-

Figure 1-21. The Seismic down-hole test

V. (m/s)

o

z

~

lQl

I5ll 100 1'iO



._ - - - - -_."._ - -_ .._._..

_ .~_._--

Figure 1-22. The Seismic cone penetration test

27

.5v .: '7-

,

. cD

~

..

"

.(.,

l

.s,­ \

\. .f

6. For periods less than or equal to To, the design spectral acceleration, Sa, shall be defined by: Sos To

Sa = 0.60-T + 0.4080s

(1-22)

Note that for T ;;; 0 seconds, Sa shall be equal to the effective peak ground acceleration. 7. For periods greater than or equal to To and less than or equal to Ts. the design spectral acceleration shall be defined by: Sa =Sos

(1-23)

8. For periods greater than Ts , the design response spectra! acceleration, Sa, shall be defines by:

S = 8 01 a

T

(1-24)

The steps involved in the development of the design spectrum are displayed in Figure 1-23.

1.9 Site Response Analysis The local soil profile at a project site can have a pronounced effect on the earthquake ground motions, and subsequently on the response of the structure to the earthquake. The nonlinear behavior of soils under strong earthquake loading is a highly complicated problem. Generally soil is a nonlinear, anisotropic material. Despite this complex behavior, isotropic elastic models either linear or nonlinear have been used in the past for practical considerations. Methods of evaluating the effect of local soil conditions on ground response during earthquakes are based on the assumption that the main responses in a soil deposit are caused by the upward propagation of shear waves from the underlying rock formation. The most commonly used model to represent the soil behavior in seismic analysis is the equivalent linear model (Seed and Idriss , 1970). According to this model, the appropriate equivalent linear shear modulus G as shown in Figure (1­ 24) is the secant modulus, which is less than the initial tangent shear modulus Gmax.

28

Table 1-2. Values of Fa for different values of spectral acceleration (LRFD guidelines for seismic design of highway bridges 2004) Spectral Acceleration at Short Periods Site Class Ss::;; 0.25 g

S5

=0.50 g s, =0.75 9 s, =100 9

Ss ~ 1.25 g

A

0.8

0.8

0.8

0.8

0.8

B

1.0

1.0

1.0

1.0

1.0

C

1.2

1.2

1.1

1.0

1.0

D

1.6

1.4

1.2

1.1

1.0

E

2.5

1.7

1.2

0.9

0.9

Table 1-3. Values of Fv for different values of spectral acceleration (LRFD guidelines for seismic design of highway bridges 2004) Spectral Acceleration at Short Periods Site Class 51::;; 0.10 g

51

= 0.20 9

51

=0.30 9

51 = 0.40 9

51 ~ 0.50 9

A

0.8

0.8

0.8

0.8

0.8

B

1.0

1.0

1.0

1.0

1.0

C

1.7

1.6

1.5

1.4

1.3

0

2.4

2.0

1.8

1.6

1.5

E

3.5

3.2

2.8

2.4

2.4

29

I

I I I " " ' -~- -- ~ - - ._- -

SOS == FaS s

S01 == FvS 1 TO

==O.2Ts

s . . S01

T

SOS

PGA

~

I

:

I

:

!! ,' '

-

._- - j-

---­

. \ \

I \

:

I So1 - ~ -I -··· T

I_l: I

"

~,, ;

-. i ._ _

-- - -- -_. _ . . . .~:=--_

TO 02 Ts

1.0

Figure 1-23. Example of a Design Spectrum

30

Meanwhile, the area of hysteresis loop has expanded, indicating an increased dissipation of energy resulting from sliding at particle contacts, hence, the equivalent linear hysteretic damping ratio ~ is larger than~o . Therefore, the bigger the cyclic shear strain, the smaller the equivalent shear modulus G and the larger the equivalent damping ratio ~ as illustrated in Figure (1-25). Two different levels of site-specific seismic site response analysis are available. In levell, the simplified methods recommended by codes are usually followed. As an example , FEMA-356 established six classes of sites for seismic depending on their shear wave velocities. According to this classification sites range from hard rock (class A) to peats and organic clays (class F). Table 1-1 illustrates the recommended values for the effective shear modulus to account for the non­ linear behavior of soils for different sites and peak ground accelerations. The recommended LRFD guidelines for the seismic design of highway bridges (2004) also recommends for regions of low-to-moderate seismicity (PGA 0.5g), a value of G = 0.25 Gmax is recommended. Both FEMA-356 and LRFD guidelines request level II, which is a dynamic site response analysis , for organic soils (class F). Dynamic site response analysis is also requested for major projects and critical facilities when global time history analysis is mandated to establish the ground motions at the foundation levels. I

Typically, a one dimensional soil column that extends from the ground surface to bedrock is used to model the soil profile. Two dimensional soil profiles may also be used in special cases such as basins. The soil layers in a one dimensional model are characterized by their unit weighs, maximum shear wave velocities and by relationships defining the nonlinear shear stress-strain relationships of the soils such as the one shown in Figure 1-24. The computer program SHAKE originally developed by Schnabel et al. (1972) and updated by ldriss and Sun (1992)under the name SHAKE91, is the most commonly used computer program for one dimensional equivalent-linear seismic site response analysis . The program requires a set of properties (shear modulus, damping and total unit weight) to be assigned to each sublayer of the soil deposit. The analysis is conducted using these properties and the shear strains induced in each sublayer is calculated. The shear modulus and damping values for each sublayer are then modified according to the relationship relating these two properties to shear strain. The analysis is conducted iteratively until strain-compatible modulus and damping values are reached.

1.9 Liquefaction of Soils Liquefaction occurs during earthquakes due to loss of strength of soil, which may occur in sandy soils as a result of an increase in pore pressure. This phenomenon can take place in loose and saturated sands.

31

G-eao-GIIIO.Il Mol'c.ooic toadin9 Q,jry.

Figure 1-24. Equivalent linear representation of the soil hysteretic cyclic stress­ strain behavior

O.S l---- --

x:

0.6

_i__

-~-+---.

.. f---+--

--I-- .- - ­

--+-----+-~.----------,P__----+-----l 1 5 i!­ t

o E

~ C)

~

1 0.41--- - - 1..- - -

c: .&

-j- ---;f--'\'-.;-------'f------ - ­

O.2t-----~+_--++_---+__'"

0001

0 .01 Shear SJroin -

- -

-:--­

10

eo

0

5

O.i %

Figure 1-25 Typical shear modulus and damping relationships used in equivalent linear soil

32

Table 1-4. Effective shear Modulus Ratio (G/G max) after FEMA-356

Effective Peak Acceleration 0.40 50S SITE CLAS5 0.40 50S = 0

0.40 50S = 0.1 0.40 50S = 0.4 0.40 5DS

A

1.00

1.00

1.00

1.00

B

1.00

1.00

0.95

0.90

C

1.00

0.95

0.75

0.60

0

1.00

0.90

0.50

0.10

E

1.00

0.60

0.05

*

F

*

*

*

*

*Site-specific geotechnical investigation analyses shan be performed

and

dynamic

=0.8

site response

33

The increase in pore pressure causes a reduction in the shear strength, which in certain cases may be totally lost. In such cases soil will behave like a viscous fluid, and in some earthquakes liquefaction appeared in the form of sand fountains . Structures founded on liquefiable sands may settle, tilt, or even overturn during earthquakes due to loss of bearing capacity as a consequence of soil liquefaction. Examples of failures of structures due to liquefaction during the 1964 Niigata Japan earthquake and the 1999 Izmit Turkey earthquake are illustrated in Figures 1-26 and 1-27. 1.9.1 How Liquefaction Build-up during Earthquakes

Seismic shear waves during its passage through different soil layers will tend to compact loose saturated sand deposits, and thus decrease its volume . If these deposits cannot drain rapidly, there will be a gradual increase of pore water pressure and decrease of the effective stress of soil with increasing the ground shaking. Since the shear strength of cohesionless soils is directly proportional to the effective stress, liquefaction will occur at the point when the pore water pressure becomes equal to the total overburden pressure . At this stage there will not be any shear strength for the soil, and it will tend to boil like a fluid. 1.9.2 Liquefaction Potential Evaluation

The LRFD guidelines for seismic design of highway bridges (2004) recommend that no evaluation of liquefaction hazard potential at a site be done if the following conditions occur: • The distance from the ground surface to existing or potential ground water level is more than 15 m. • Bedrock underlies the site. • The soil is clayey with particle size < 0.005 mm greater than 15% A simplified procedure for the evaluation of liquefaction potential was originally developed by Seed and Idriss (1971, 1982) with subsequent refinements by Seed et al. (1985, and 1990). The procedure compares earthquake cyclic stress ratio (CSR) at a certain depth of a cohesionless stratum to the cyclic resistance ratio (CRR), which is defined as the cyclic stress ratio required inducing liquefaction for that given depth. This evaluation procedure uses correlation between the liquefaction characteristics of soils and field tests such as the Standard Penetration Test. The procedure involves the following basic steps: • Determine the cyclic stress ratio (CSR). During an earthquake, the soils will be subject to cyclic shear stresses induced by the ground shaking. The average cyclic stress ratio (CSR) may be estimated by the following formula: CSR==

'r ~v ==0 .65 (amax)(cr~Jrd Go

g

(1-25)

Go

34

Figure 1-26. Failure of the Kawagishi-cho apartment buildings following the 1964

Niigata earthquake due to soil liquefaction (courtesy of EERC, Univ. of California)

Figure 1-26. Failure of a building following the 1999 Izmit Turkey earthquake (courtesy of EERC, Univ. of California)

35

o0

0.1

02 03 0.4 05

0.6 0.7 0.8

0.9 1.0

3 (0) 6(20)

-­

::= '-' E

'" ..r:: -..

0..

Average values Mean values of rd

9(30)

12(40) 15(50)

. . .. . .

tU

0

18(60)

21 (70)

24(80)

27(90) 30 (l 00)

.. . ..... . . . .

. . . ..

...

. . '

l--...;.....L...---J.~.a...;...~:.......;....L_...o...-..;...a...----JI.-.-""""'--..I

Figure 1-27. Stress reduction coefficient rd versus depth curves (youd and ldrlss , 1997)

36

Where amax is the maximum acceleration at the ground surface; ;l,",~

1.1'0! felay eerueru =5';;; ) @

"')0 .

Ma . aI

No

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

•

Pan • A.merian data

AdjllStmenl

•

Recomrreeded Jap;m=~ra By Workshop Chinese data

30

0 A

A

20

10

Q

g

40

CorrectedBlow Count, (Nl)6Q

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

I

4

angeof recommen I MSFfromNC Workshop

3.5 3

25

1.S

­

.... . ~

o 5.0

6.0

7.0

8.0

9.0

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

38

0.6 Volumetric strain (%)

0.5

1054 3

2

~

0.5

· · ·:0.2 .... .. .. ..• • 0.1 t.: ....... . . ........ . .....'......... .... . . . .. ... ..»...» . . . .. ...... . . .. ...... ·•···• •

II

0.4

a:

1

.

•

0.3

#

#

0.2

.9

#

•

•

."

.' .. .'

0.1

o

•

•

,.~ ~

o

10

20

30

40

50

(N1)SO Figure 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 15 (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 15 + 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 15 , 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 Failures Flow 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 Hazard Mitigation 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 References Arias, 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 FOUNDATIONS 2.1 General Shallow 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 Models SFSI 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 Approach This 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:

=Rv =~BLl1t ro =Rh =~BL/1C

(2-1-a)

ro

r0

-R _[16(B)(L)3]1/4 r1 -

-

_R

ro

(2-1-b)

-

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

31t

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

]1/4

r2 -

2 2>]1I4

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

(2-1-c)

(2-1-d)

(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 v

-

(2-2-a)

1-u

8GR k sh1 = k Sh2 = ­h 2-u

(2-2-b)

47

p

Foundation forces

Uncoupled stiffnesses

Winkler spring model

Figure 2-1. Analysis models for shallow foundations

x

/ : RECTANGULAR FOOTlNG

I-­

-~

I

I

I

J

I

I

I I I I

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

2l

co N

--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 Foundations The general vertical soil bearing stress capacity of a shallow footing is: 1

qull

= en, Sc + yDNqSq +2yBNySy

(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

-: -:

V

.,,­ A ... t =200 I;::: 150 100

,=



G)

a;

....-

-'"

/'

i--'

I--'

V V I--'

V 4 10

V

\:12( \f\

v

v

.......-

V

V

~

V'\

,......

'\

...,)t'\\ \

\l\\\f\

I--"

V

v

tZ

as...

...----: v

......-

"'"

V

,g

~ en

V

---v

v

----

~

l\\

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

1\\

1\ f= 5

I\.

v

, 1\

V-

I--"

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

....-

\.

f;;;;20 f= 10

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

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

f = 0.1 Coeff. of Variation of Soil Reaction

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

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

I I

I

r

1

I 1

I

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

83

10 7 : Embedment I

--

--.

t: ~ .0

-

I I

I I

a

-­

ee

I

- - - - - 51

-

Co

I

-

.­

- ·10'

~

- 10 6 ~

::

CD

-

--

I

--­­

~

CD .~

~ 10

5

CD

c:

.,

CiS

-­

+:

0

~

...

,," ...

""

v

v

_.....

~

".

I­

~

....

­ ......

10 3

»>

.

...

e-­ :_

~

=

....

... 1­

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

...­

/

...­

~--'"

F

....

1-/ v~

~

CI)

~

~

--'" "...

.....

~

V

-'......

...­

...........

ct) tI)

m c 10 4

".

... I ....

f=100_~

'C

co

...­

"...­

-

......

-

.....

..­

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

""

I

10

10

10

I I I I1III

11

10

I

-

J I I 1III

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:

.~

- - 5' ----- 10'

/?

"'C

-

m 1010

IA

a:

V

~~

C :.;:;..

k?

v?ft< ~

::> of fop of second

layer

r Z := r, + Yl:vb .nz r z = 12~.4-;;

kFa

Effedive ~rex> of pile lip

f'1 := f' Z f', =

+ y Z~b

1~/.o2

-tJ;J Pt =157 I..n,,; - --

---'

kFa

r oin! Rc:>i~af)Ce

op

= 1;;/~,4- kN

GirOJrJ. lire:a// rt

Dala for cdcddionof ('1(Zi.irvn di::>p/acerlCrJf::>: crlbedded depfh of pile

enpatca poin! rc>i~ooxcodficial

enpcricaJ::>kin ltkiion codliciat

102

Ullirnfe: poin! ~re:!!>!}

Ullrrnfe: dl!!>p1accncrl cor{c>pandlrq 10u/Ilrde: paIn! t=blarcc

UI/if'lafe: d/~p1ace:f'lCfIf

corrc~pandirq 10:>Kin Iridian rc~idaocc:

Zd = 02

rJ/'I

P oin!(e:~bl(jr= OJNe:

I

o co ""= 0 p .[.-!...-'I~ Z cp)

coo "­

x := coo

o 'I

0

z :=00 .l rr: .. lo.rJ/'I

.00'5

101

0'5'5

2.51

./0'5

2-'74

2'5'5

:571

XJ'5

4-20

1!/0'5

682­

(0)

:;130'5 '71:;

«)

'7.%'7

y :=coo

20

1'500 r ­-

-

-,­-

-

-

J:;I'5 J:;I'5 )

, - - - - - - ,-

-

--,

1000

O L-­

o

---'---­

-...L-­

- --'-­

- -'

20

103

5kfn [tidion asve

ZI ;:= 0 ,0'5 om 0.2 of'7/')

coo

x

::= coa

y

;:= coa

0

0

00CY'5

164­

O~'

4!/0

./0'5

'521

2'5'5

%4­

50'5

%4­

tzo»

564­

!/BO'5

564­

9509

%4­

'I

::=

(0 )

(I )

20

%4- )

5kin Frfdfon!? c~;:;fafKX C UNe I

t

I

-

100

-

I­

obfaincd by!:JJmirK} fhe: din (ridion anderJ bearfrK} copoalte» aI oxh axial di!>P/accncrtl,

Riqid pile: XJ!vlion b

(0 ) X

:= cor

y ;:= cor

obfained

by xf1f1inq fhe din [ricilon and end bearlnq

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

x := cor

cor :=

(0)

y:= cor

(1)

')

0

0

0 ,00'5

211

,0 '5'5

f,15

./0'5

81'5

,2'5'5

960

50'5

984­

150'5

124-'5

5.80'5

1'558

95f,9

1819

20

1819 )

Riqid Pile: 5olufion

2000

1'500 ~

~ ~

j

_Y_l000

~

~

'500

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

o

10

20

x Di:?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 pileeedion 0pi! := (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~

Ac ;= v;:=

z

:~

o '\

o

k(O) (I)

I1c

0.'5/6

211

1::;2'5

61;;

1,61­

8/'5

207

960

2.16

984­

;;66

/21-'5

6.1

1'5;;8

12.9

/819

2;;.5'5 /819 )

I1dl.ld :301u/ion

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

1'500 '"'

~

8 -.l

I:l

y v

--/000 z

~ '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

50/

Dldance 10 kax/~ of roidion

:=

0:1 ·(7

DI:;-fance fa Yraxi» of rofalion

"XX6 := n·"xx

xN

KXX6

= 10'507G.I (7

r. '('(6

=

kN 1'5/GI4-Y5 ­ ("J

('1

KXeY6 :: 8540'5.8G

xl\!

KYeX6 = 221'521.4-4- kN

KeX6 = 18829'50.GI

KeY6 = 4078908.1'5

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

kN ('1.­

rad

kN rad

('1.­

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 befaken ido = rr.:iderafion

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

K .­ '­

- 1882.9'50.6 1

0

0

0

0

0

0

0

/882C;X;.6/

0

0

0

0

0

4018908.1'5

0

0

0

0

0

10"5076.1

0

0

0

1'516/4'.1'5

0

0

0

~0/2-66~,12

0

22.1'52./,4-4­

- /882.9 '50.6 1

0

0 2 2./'521.#

1

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

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

:fId

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 LOADS 4.1 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 construction, gravity retaining walls and cantilever retaining walls. Under static conditions, these walls will sustain body forces related to the mass of the wall, soil pressures, and any external forces. Equilibrium of these forces is mandated for a proper design of the retaining wall. During an earthquake. however, inertial forces and changes in soil strength may breach equilibrium and cause unfavorable deformation of the wall. Failure in sliding, tilting, or bending mode may occur when excessive permanent deformations take place. Gravity walls usually fail by rigid~body mechanisms such as sliding, which occur when the lateral pressures on the back of the wall produce a thrust that exceeds the available sliding resistance on the base of the wall. Cantilever walls are SUbject to the same failure modes as gravity walls and also to flexural failure modes. If the bending moments required for equilibrium exceeds the flexural strength of the wall, flexural failure may occur. The seismic performance of retaining walls is usually evaluated using pseudo~ static methods, where the transient dynamic pressures induced by the earthquake are added to the initial gravitational static forces exerted on the wall before the occurrence of the earthquake. Hence, a brief overview of the static earth pressure is presented in the following section.

4.2 Static Pressures on Retaining Walls Static earth pressures on retaining walls are strongly affected by the movements of both the wall and soil. Minimum active earth pressures are mobilized when the wall moves away from the soil behind it, and this movement is sufficient enough to activate the soil strength behind the wall. On the other hand, the maximum passive earth pressures develop as a result of movement of the wall towards the soil. A number of simplified approaches are available for the computation of static

112

loads on retaining walls. The most commonly methods used in practice are outlined below.

4.2.1 Rankine Theory According to this theory, the pressure at a point on the back of a retaining wall can be expressed as: PA =

KAO'~ -2C~KA

(4-1)

where O'~ is the vertical effective stress at the point of interest, c is the cohesive strength of the soil, and KA is the coefficient of minimum active earth pressure evaluated as;

K

A

= 1-sincP = tan 2(4S-

1+sincj)

~)

2

(4-2)

For the case of backfills inclined at angle ~ to the horizontal, the following equation can be used to compute 1TAH (45+~f2.)

OA :1'7 1I1N2.(45~M)-2C'DiN(~-~> PA : (~)TAH2(4S-.I2H()oIWIl(.45"'4li: +2C2/y

Ptt-$$IVE PfU$l)flES ,.........2C

J-­ ,2

< ) l(

""

Kp:DH2(4$"~)

a'p'KpYZ

ppfl(p1'tt2/z

.... "V

~·1'Z.2C

Pp'

TYH2 +ZCH

O"p:;.rz TAN! (45+#Zh2CTAN(45~) pp.;( ~I TAN2(4!S+~)+2CH'aN

(45+eW2)

Figure 4-1. Rankine active and Passive earth Pressure Distributions For Backfills with Various Combinations of Frictional and Cohesive Strength (NAVFAC 1982).

115

~

F

(a)

(b)

Figure 4-2. Wedge of Soil and assumed Failure Surface for Calculation of

Coulomb Active Earth Pressure Coefficient

116

Table 4-1. Typical Interface Friction Angles (NAVFAC 1982) ,f'llc codflclerl

Kv:"'O.o

Irlerface anejle bdween !>OIl and wall

5 := 21- ·dt!:tJ (fab/e4--f).

f?;odfll! Irdlnaflon

Ii

:= O·deej

124

!1ononobe ~ Okabe t1e:fhod

'II

KItE

:=

kh "\ alan ( I-k ) v

'II

= ~.111

(co~ (.

;==

co::> (V

(e:quafion 4--12)

deq 9

).(co~ (9))2 .co~ (li + 9 + 'J1 ).[1 +

KItE

== 0.4-~8

F/IE

:== 0.'5

'II ))2

~in(a + ej))·~/n(, - p - V) ]~ (co~(li +9 +'II)·co::>(P -9)) (equafJon 4--11J

PItE

.KItE·y·t1/·(/-kv)

r

= 14-.2-64- -

(e:quafion4--I:?)

f?

Prakadrt1e:fhod 11~5U/'1e

n:=OO

fir!?! c~ of no len~ion crack

5e1e:d failure: 5Urface:~ 50 thai their inc/ina/ion anqle:!> ranqe: approxlnafely frof? to (45++°)

.0

a := 10 -dec; .1'5 -Je:q ..GO .Je:q B (a) := a + 9 +. + a

[Cn + ,~).(lanC a) + Ian (9)) + n2. .fan (9)].(CO~ (a + +) + kh'~in( a + 4

NolB ,a):= - - - - - - - - - - - - - - - - - - - - - - ­ f>in(B( a))

(equafJon 4--/1aJ

Nac(B ,a)

co~( a

+ 9 + ej))·xcC9) + co~( +)·xc( a) (equa/ion4--11c)

:=

. (

~m

2

(

B a.

F/tE(a):=y·t1, .Na/B,a)

))

(equalion 4-~/G)

125

P/rC(a)::::

r

9.4 10.9

f1

12.1 13.1 13.8

F/t£(u)

1~

= 10°

".~

f/)

0.6 W (,) 0.5

\~

0.4

\

0

\2d'

20

~\

Z

\

~

\ \ 30°

0.3

'\

02

1=2

'40°

\30°

\

\.400

0.1

00' '0.0 0.2 0.4 0.6 0.8 1.0 t

•

'

,

,

0.0

~.2

0.4 0.6 0.8 1.0

0.0 0..2 0.4 0.6 0.8 1.0

kh (~)

NcE/Ncs

131

Phi

Phi = 30 degree

00' '0.0

, 0.2

• 0.4

,,"--','- I 0.6 0.8 1.0

=35 degree

0.0 0.2

Phi = 40 degree

0.0

0.2

0.4

0.6

0.8

1.0

(c) NYe/Nys

Figure 4-8 (continued). Ratio of seismic to static bearing capacity factors (Shi 1993).

132

displayed in Figure 4-8. The ratio for NcelN cs is presented in terms of the friction factor f F/Nkh n tan(~)/~, instead of n.

=

=

The effective stress must be used to compute q in the second term and the submerged unit weight must be used in the third term of equation 4-28, if the foundation is submerged above the base of the footing. If the foundation is submerged below the base of the footing an equivalent unit weight must be used in the third term of equation 4-28 as:

'Yeq

= 'Ysub(1-Z/B)+ y(Z/B)

(4-29)

where Z is the depth to the ground water surface below the base of the footing and B is the width of the footing. If Z is greater or equal to B, then 'Yeq 'Y.

I

=

The seismic bearing capacity is evaluated by comparing the seismic vertical force resultant at the base of the retaining wall to the seismic bearing capacity of the foundation soils computed with equ~tion 4-28 with eccentricity e computed as:

B Mnet N

e="2-

(4-30)

in which, N is the vertical force resultant determined using equation 4-26, and Moat is the net moment of forces about the toe of the wall (point C in Figure 4-17) calculated as MR-Mo1 where Mo is the overturning moment computed with reference to Figure 4-17 as: Mo = khWYc +PAE cos(5 w +8)h+PpE sin(ow -8 2 )(0/3)(tan82 )

(4-29)

and the resisting moment can be computed as: ~ =

WX c +PAE sin(Bw +8)(B-htan8)+PpE cos(Bw -82 )D/3

(4-30)

4.5 Seismic Stability of Retaining Walls The safety factor against seismic induced bearing capacity failure as: F:'C = PdB' (4-31) N The wall is considered stable under seismic induced loss of bearing capacity if the computed factor of safety F:'c for the peak acceleration is equal to or greater than one.

133

The safety factor against seismic overturning instability is quantified as: O.T _ MR

F5

(4-32)

--

MO

The wall is considered stable with respect to overturning if the computed factor of safety is equal to or greater than one. 4.6 Seismic Displacements of Retaining Walls Estimation of the permanent displacement of retaining walls is necessary for performance based seismic design. Richards and Elms (1979) method is used for the estimation' of this allowable permanent displacement. According to this method, the fevel of acceleration that is just large enough to cause the wall to slide on its base is defined as the yield acceleration defined as:

[t

.I.

ay = an'l'b -

PAECOS(O+9)-PAESin(o+9] W 9

(4-33)

where, ~b is the angle of internal friction of the soil beneath the wall's base. This method works with the M-O method for calculation of PAE. Hence, the solution of equation 4-33 must be obtained iteratively because the M-O method requires that ay be known. According to this method, the permanent displacement is quantified as: 2

dperm

3

=0.087 vmax ~max

(4-34)

ay

where,

Vmax is

the peak ground velocity. amax is the peak ground acceleration.

Example 2: Check the seismic stability of the reinforced concrete cantilever wall shown in Figure 4-9 for the maximum considered earthquake (2500 years return period). The wall is located near Memphis, Tennessee (350 3' latitude, _90 0 O' longitude). The site consists mainly of coarse sand of unit weight 18 kN/m3. The average initial shear modulus up to a depth of 100 meters is 180 MPa.

5olufion For a5i1e localion: ~O ~' Ialifude, 9000' IOl7t:Jifude, and a 2'500 year!>!??, fhe 5horl period 5pedral acceleralion of bedrock (rorl fhe U,5, 6eoloqical ?UNey !>ife (hHp:// earlhc,uake,u!>tJ!>.eJov/)

51> := 0,6/ '4

134

--

O.6m

-7!!!&/l1I§:,····· =30°

r =18.00 kN/m3 7.5m

p

!

121

,

I 3'

B=6m

Figure 4~9. Geometry of the retaining wall Irveraqe ~ear f'?Odulu~

6 rlax := 180 ·I1Pa

kN

y:=18·­

Un!f wek;hf of 50/1

~

f?

yc = 2;;,%;;

Un/f weicjhf of concrde

­

kN ~

f?

Irveraqe *ar wave vdocily

For f>fiff 5011 wlfh (:xci/on 1-8)

V,,:=

J 6~.q

rI ~

180 ro/f> in 8 w +' .!>in (, + ~ - 'I'

.

CO!> (

'I' ).(co!> (9))2

'co~ (8 w - 9 + '1').

1-

{co!> (8 _ 0 + 'I' ).co!> w

)] _

(~ _ OJ)

/(PE = Z~28

(eC/ualion4--I/)

PPt: := 0,'5 ·f(Pt:·y .11/ .(1 -kv) PPt: = 188,'581 IN f'J

(e:qualion4--I!'J)

CalQJlallon ofwdqhf!>

WI

:=

(tt- T,).(t? -t?,- T).y

WZ

:= (11- T,).T·yc

W.'

:=

khW1

l~

t?·T,·yc

W :=WI +WZ +W"'

(t? - t?,- T)

AI:= +t?/+T Z ItZ

:=

It)

n

.'5·T + t?,

~.

M

~'

:=.'5.t?

f;! :=

t?Z

¢

(I1-T,) Z

+ T,

t?/

T,

:=-

f;Z

Z

=

4 f'J

= 02'5 f?

Didance of re:xllartl ofverlical force5 frof'Jloe

X := WI ·1'11 + WZ. t1Z +

w"' .~

X =!'J,Z4-'5 f?

Didance of rt:Xllanf ofhorlzonfal force> frof? foe

Y:= ft?I·(WI + WZ) + t?2·W",]

W

Y = .'(:/52 f?

137

Applicaf/on poinf I!J( := /(At: -/(II

11 Kit";; +AJ(·06·11

h ---.- · - - /(IJE

h = ;;,414

IVJ

Check for dJdlng along fhe box But?rKJIion of drivlnej force!>

Fd

:= PIlE

F d = '568Bf7 kN

+kh'W

IVJ

8uf?f1aflon of rt:~/dinc;force:!>

IN

N:=W

N = 16/.8'54- ­

f'I

IN

F r := PrE + N·lan (ar)

F r = 628.444 ­

I'?

Fr

F ador of :xidy OCJaln~f ~/idinc;

F8 ~ := F d

F8:>

::=

1./0'5

O/(

Check for overlurn1trJ 81abilizinej f'IOf'leri

11/

HI( := PrE'""3" + W·x

HI( = 2.661

x

10~ kN

.(1

f'I

Overlurniinc; f'IOf?enf

110 := FAt:.h+kh·W,Y Ho

=

~

f'I

2.004 x 10 kN·­ f?

F ador of t>afely aqain!!i overlurnlnc;

F8 o

:=

HI(

Ho

F8 0

=

1:!J28

nK 138

-14TJ

.

:= ""./an ( +) ·fan (

NqC

:=

(by inferpo/afion fiqure 4--8a)

r

~ .d,,9' + i

Nqr·Nq:>

N

:>

'1

NqC

Nye./Ny~

N yr :=

Ny:>

:= 2.(Nq:> + I).fan( ep)

NyC

:=

Ol),?'5 Ny:>

= 18,"f01

=

(e;quafion:2 - f

0)

4,6

(by inferpo/afionfiqure4-8c)

=

22.402

(equafion

2 - rb)

NyC =2J28

Nyr·N y:>

fJ I1K- 110 ecc := -Z - --N-­ ecc = ZJ~ f?

eccenfrlcify

fJo

:=

f;o = /,726

fJ - 2- ·ecc

f?

/

Pd := y.t1,.Nqc + 2 ·y·f;o·NyC

5ei~jc lif?iI fo bearinq pre~t:XJre

Pd = 28/.47'5 kN f?2.

139

l~

_11 _ _.L!

~1~

_~

. : _ _ L.- _ _ ,....._

J

(JI

.. ~

....,---r-r

~

_.1#

F aclor of 5afdy acjain5f Xi5f?ic induced bearinej capacify failure

F 5~ ._ Pd·f?D vC .-

F 5f?C = 0,6:58

COf?pu!ed fador of Mdy for xi!>f?ic beadnej capacily i5 le5!> fhan one. rherefore lo!>!> of beadncj capacify and filfinq of fhe wall i!> o.peded dUrinej fhe 2500 year earfh4uake. ttencel xi!>f?ic rdrofif i!> required. /(drofil ?fraf!!ZJY,' Place a lieback fhrouc;h fhe wal15fef? fo reduce drivinq I'1OMenf!> and Increax Xi5f?ic beadnej capacify.

I

1.0 m

tD'U'-1~~---'-"

- - _.._ _.... ­ F,.cos 15'

~

T

r _______________ F.. sin15°

II

w

~

I

vJ /

1.'' ':' ': ' ' ~ · '0

1,

1

i

{

"-""

f

L-__.. ,...

I

PAE

""."".,,,,F""""""""

I! ,

!

16.5m

h

j

J

.

N

n:= 0.'5

a5!>Urte !>hear frander codficienl of fhe rdrofiHed wall

(3 := 20 ·deq

inclinafion anqle of lie-back rod

PItE + kh' W -PPE -n-W-fan(+)

Force in fiC'back rod

Flie:

:= - - - - - - - - - - ­ n'5ln ((3)·fan (cr.) + co!> ((3)

Flie:

=

1"54.4-4-1 kN f?

140

=:>ei!>f'7ic beariQ:} capacity fador1> \

Nere/Nq~

N qr

:= 0,40

(by inferpolalion fit/ure 4-8a)

NqE := Nq(.Nq~

N q£: = 1.!J6

Nye/Ny~

N yr

:=

02-0

Ny£: := Nyr·Ny~

(by inferpolalion fiqure 4--8c)

NyE =4.48

11Kr := 11R + Flit: ·co!> (rJ)·(I1-I.f'/) + Flie·!>in(rJ)·!?,

l10r := NO

N

N := W + Flie·~in(rJ)

= 8/4,616 kN

f'7

eccenfricily

I1Rr- N OT

2 N

!?

ecc :=

ecc = 0,99> r?

!?o :=!? -2·ecc =:>ei!>f'1ic lif'1if 10 beariQ:} pre!>!>Ure

!?O == 4.021

f'7

1

Pd :== 1 .I1,.NqE + 2. '1 ·!?D·Ny£: Pd = J1j'59,591 kN

2­

f'7

F ador of ~afdy acjain!:i ::>e:i!>r?ic induced beariQ:} capacify failure

Pd·!3Jo F5fX,:==

N

F5e;c

=

2,162

141

5-SEISMIC PERFORMANCE OF CAISSONS

5.1 General Caissons are very large concrete boxes that are excavated or sunk to a predetermined depth. They are used usually for the construction of bridge piers or other heavy waterfront structures, and they often become advantageous where water depths exceed 10 to 12 m. Caissons are divided into three major types: (1) open caissons, (2) box caissons -(or closed caissons), and (3) pneumatic caissons. Open caissons are concrete shafts with the top and bottom open during construction. This type is provided at the bottom as shown in Figure 5-1 with a cutting edge. After the caisson is sunk into place, soil from the inside of the shaft is removed through a number of openings by grab buckets until the bearing level is reached. Once the bearing stratum is reached, concrete is poured into the shaft, under water, to form a seal at the bottom. After the concrete has matured, the caisson is pumped dry and filled with concrete. This method does not guarantee thorough cleaning and inspection of the bottom. Box caissons as shown in Figure 5-2 are cast on land with spaces open for buoyancy. They are then transported to the construction site and gradually sunk by filling the inside with sand, ballast, or concrete. Pneumatic caissons are closed at top and open at bottom. Overburden materials are excavated by hand or machine from a working chamber while compressed air is used to keep water from entering the chamber. Penetration depth below water is limited to about 40 m (130 ft) as higher pressures are beyond human endurance. Despite of their higher cost as compared to the other two methods, this method of construction yields proper bearing stratum and concrete will be of adequate quality. A common feature of caissons produced by the three methods is that they are massive structures that respond to seismic loads in a primarily rocking mode about the base plus some translations.

143

lJ:I~r

Section at A-A

A

A

•L

~

i

_

..

'I

.,.

"

,"

·t ~

~

. !

_---.J

Water Level _ -=

~

....... , ....

, j

/)707/;:;:""

": 11/; Ihe !>ife i!> cla:?!?ified a!? ealeqory D, rhe !?ile eoefficienl:;:, 1'5 de/emined frof? r able!? (1'2-) and (I,!J) at>:

Fa"" /,0/6 Fv"" /''58 Calcu/ale Ihe de!?iqn earlhquakere!?ponx t>pedra/ acce/eralion al t>horl period 5D:::- a nd al'-xcond period, 501 501>~· /,0/6X 501 ""

1.2-lq ~ 1,25 q

1/58 X0:1% "" 0,6Gq

Delerf?ine Ihe: period!? r!? and r 0

r ~ 066/ /,25 ~ 0,'54 !? r 0 ~ O,2X 0,'54 - 0./08 :;:, d

Con!?lrud Ihe ~% danpinq de:5iqn !?pedfUf1 u!>inq equalion!> 1'2-2Ihrouqh /,24-, rhe de::.5iqn re!?pon!?C !?pedrun I!? depided in Fiqure '5'4-.

152

IA

I

I

1.2

~

v ~

0.8

l

0.6

~

l

~ O.~ 0.2

a 2-

I

0

~

4­

Period (::ec)

FiCJurc ~~4-. 'X Darlpif7CJ De5iCJn R c!:>po~ :5pcdrurl for T ocorJa WO!>hlnc;fon 5fep 2: E::>faUidl fhe Pudrover Cu/V.e:

Follow fhe !:>fep::> in ::edion 5.2.210 develop fhe capacily cUNe:

rJAfA: f oundatiGfl width

A = 24.?B4m

Fcul1dati(ifJ lenqth

B=

~.624m

P01550115 ratio

v :=

a,??

5hear wave velOCIb1

V5:=

?o?~ 5

Y

Unlt wt of soil

= 19,6?6 kN

:?

m MalC.imUm shear

modulus

/'.

. Y

tAmax'= -.

v52

Umax. = 1.86:; x I08 pa

Cj

153

--(

~ffective

moment arm

Halfspace transverse stiffness

HO :=

J(4.H

Z

Z +A )

HO = 4?J,89Bm

12

(17 ,0,67 + 0.4·-[? + 0,8J

CtA [

kXX := - . ?A· 2-v A)

A

kxX = 6,129 x 106 kN m

Halfspace rocKinCl 5tlffne55

Ct.A'?.[ OA· key := -.I-v

([7'- O.J]. A)

+

I

rad

kOY = 1.246 x I09kN.~ rad

Coefficient of vertical 5ubgrade reaction

k. := 12.

Kay

A'? ·18

k = 2,604 x 104 kN '? m

The pU!5hover CUNe I!:> depided In Fiqure 5-5, II? dlown in fhe fiqure fhe re!5pon!5e fran 0 fo It ;!:> linear wilh un/lom dlf:>fribulion of file pre!:>?Ure on fhe cai!5!50d!:> box. 5lidlnc; !:>/arf!:> 10 occur of point It and con/lIVe!:> up 10 polnf ~J whree if I!:> occonpanled by rockinq frof'7 ~ fa C, J

154

BOJ,CX::O

.'"'"-'-'

. -:-1' -~.-,-~-

__ ._....• _ _._ - _---,-_

__ . "--,,.....•.. ,

.

-· ..­

1

C

i

!

600,CX:O f

\ t'

:::;;z;» .........

r

3

v

~

4On. 5fep 5: E!:Jfab/if:>h fh~.._...r;;.qJ?Cl{;!.fy... 2f?t;d[(!.tL0u.Ne and Check Perfomonce the capacily !:Jpedrun CUNe if:> dif:>played in FiCJure ~-6, II i~ ~hONn fhaf in fhb reCJion fhe caif:>=>on f1ay experience ~one f:>/idinq under a pofenfial earfhquake. Neverfh/e~~J rocki!1CJ ('1oy nof occur. IA

!

!

;

,

1.2

11

\

-a­

..

\

0.8 ;

v

-{'

0,6

o

V -----

....... k-,""'

Q-r

0.2

~:

,.-

I

/

'1/ ,._-_ ...

:

I

:

I

'"

·ww_·

I

-IJ~

1

~... ~~

,I

--- ;

- I

-

1

-----'r---..

,

1;

I

-S:-Q.

rq

("') ~~

Jb

r'\-+('}",e (' c~

I

_.._.---!

i

-

---=

,

;

..-1-._........_...................

)

.-_ ............................... ....• "

Fic) 12 Cvvt. Qh.

J

k ~ ~~ 10-- ~0oAkr kJ P(Z etA. I ('vI I 9'~ ,:;>.\

IAJ ~ XA~

v', S"lA.

155

, ~

5.4 References Gazetas, G. (1991). Foundation vibrations, Chapter 15 in Foundation Engineering Handbook, 2nd edition, H.-Y. Fang, ed., Van Nostrand Reinhold, New York, pp.553-593. Mahaney, J.A., Paret, T.F., Kehoe,B.E., and Freeman, S.A. (1993) "The capacity spectrum method for evaluating structural response during the Lorna Prieta earthquake, "National Earthquake Conference, Memphis.

156