EARTHQUAKE ENGINEERING RESEARCH CENTER...EARTHQUAKE ENGINEERING RESEARCH CENTER SHAKE A COMPUTER PROGRAM FOR EARTHQUAKE RESPONSE ANALYSIS OF HORIZONTALLY LAYERED SITES by Per B. Schnabel

EARTHQUAKE ENGINEERING RESEARCH CENTER

SHAKE

A COMPUTER PROGRAM FOR EARTHQUAKE RESPONSE ANALYSIS

OF HORIZONTALLY LAYERED SITES

by

Per B. Schnabel John Lysmer

H. Bolton Seed

A Computer program distributed by NISEE / Computer Applications

Report No. EERC 72 - 12 December 1972

College of Engineering University of California

Berkeley, California

TABLE OF CONTENTS

1. Introduction

2. Theory

2.1 Propagation of harmonic shear waves in a onedimensional system.

2.2 Difference between rock outcrop and base rock motions.

2.3 Transient motions.

3. Description of Prog:ram SHAKE

4. System and Operation Documentation

4.1 Computer eq~ipment.

4.2 Storage requirements.

4.3 Runtime.

5. Required Input Data

5.1 Organization of input data.

5.2 Initialization card.

5.3 Run option card.

5.4 Data cards and explanatory notes for the various options.

6 • Example Run

6.1 Selection of soil system and input motion.

6.2 Input data for the analysis.

6.3 Computer output from the analysis.

ii

Page No.

1

2

3

7

9

10

12

12

13

13

14

14

15

15

16

38

38

39

43

iii

7. Program Identification and Abstract

7,1 Program Identification.

7. 2 Abstract,

$. Scu~ce Listing for Program SHAKE2

Acknowledgements

Ref er-enc es

?age N;,.;.

54

54

55

87

8f

l

1. INTRODUCTION

Several methods for evaluating the effect of local soil conditions on

ground response during earthquakes are presently available. Most of these

methods 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. Analytical procedures based on this concept in

corporating nonlinear soil behavior, have been shown to give results in

good agreement with field observations in a number of cases. Accordingly

they are finding increasing use in earthquake engineering for predicting

responses within soil deposits and the characteristics of ground surface

motions.

The analytical procedure generally involves the following steps:

1. Determine the characteristics of the motions likely to develop in

the rock formation underlying the site, and select an accelero

gram with these characteristics for use in the analysis.

The maximum acceleration, predominant period, and effective

duration are the most important parameters of an earthquake

motion. Empirical relationships between these parameters and

the distance from the causative fault to the site have been

established for different magnitude earthquakes (Gutenberg and

Richter,1956, Seed et al~ 1969, Schnabel and Seed, 1972). A

design motion with the desired characteristics can be selected

from the strong motion accelerograms that have been recorded

during previous earthquakes (Seed and Idriss, 1969) or from

artificially generated accelerograms (Housner and Jennings, 1964).

2

2. Determine the dynamic properties of the soil deposit.

Average relationships between the dynamic shear moduli and

damping ratios of soils, as functions of shear strain and static

properties, have been established for various soil types (Hardin

and Drnevich, 1970, Seed and Idriss, 1970). Th.us a relatively

simple testing program to obtain the static properties for use

in these relationships will often serve to establish the dynamic

properties with a sufficient degre~ of accuracy. However more

elaborate dynamic testing procedures are required for special

problems and for cases involving soil types for which empirical

relationahips with static properties have not been established.

3. Compute the response of the soil deposit to the base-rock motions.

A one-dimensional method of analysis can be used if the soil

structure is essentially horizontal. Programs developed for

performing this analysis are in general based on either the

solution to the wave equation (Kanai, 1951; Matthiesen et al.,

1964; Roesset and Whitman, 1969; Lysmer et al,, 1971) or on a

lumped mass simulation (Idriss and Seed, 1968), More irregular

soil deposits may require a finite element analysis.

In the following sections the theory and use of a computer program

based on the one-dimensional wave propagation method are described. The

program can compute the responses for a design motion given anywhere in the

system. Thus accelerograms obtained from instruments on soil deposits can

be used to generate new rock motions which, in turn, can be used as design

motion for other soil deposits, see Fig. 1 (Schnabel et al., 1971). The

program also incorporates nonlinear soil behavior, the effect of the elas

ticity of the base rock and systems with variable damping.

3

2. THEORY

The theory considers the responses associated with vertical propagation

of shear waves through the linear viscoelastic system shown in Fig. 2. The

system consists of N horizontal layers which extend to infinity in the

horizontal direction and has a halfspace as the bottom layer. Each layer

is homogeneous and isotropic and is characterized by the thickness, h, mass

density, p, shear modulus, G, and damping factor, e. 2.1 Propagation of harmonic shear waves in a one-dimensional system.

Vertical propagation of shear waves through the system shown in Fig. 2

will cause only horizontal displacements:

u • u(x,t) (1)

which must satisfy the wave equation:

(2)

Harmonic displacements with frequency w can be written in the form:

u(x,t) • U(x) • eiwt (3)

Substituting Eq, 3 into Eq. 2 results in an ordinary differential

equation:

(G + iwn) (4)

which has the general solution

u(x) • Eeikx + Fe-ikx (5)

in which

Pw2 Pw2

k2 - ---- --G +iwn G* (6)

Recorded Ground Motion

Al

Bose Rock Motion

Rock Modified Outcrop Rock Outcrop

Motion Possible change in Motion amplitude cl rock outcrop motion depending on distance from zone of energy release

Modified Ground Motion

tB

Modified Bose Rock

Motion

Fig. I SCHEMATIC REPRESENTATION OF PROCEDURE FOR COMPUTING EFFECTS OF LOCAL SOIL CONDITIONS ON GROUND MOTIONS

w I»

Layer No .

• • •

m

m+I

• •

N

Coordinate System

Xm

Um+I

l ~ X N+I

Propagation Direction

Properties

•- -- -~Particle motion

Incident wave

Reflected wave

Fig. 2 ONE-DIMENSIONAL SYSTEM

3b

4

where k is the complex wave number and G* is the comp:ex shear modulus,

The critical damping ratio, B, is related to the viscosity n by

WTl • 2GB

Experiments on many soil msterisls indicate that G and Bare nearly constant

over the frequency range which is of main interest in the analysis, It is

therefore convenient to express the complex shear modulus in terms of the

critical damping ratio instead of the viscosity:

G* • G + iwn • G(l+2iB) (7)

where G* can be assumed to be independent of frequency.

Equations 3 and 5 give the solution to the wave equation for a harmonic

motion of frequency w:

u(x,t) • Eei(kx+wt) + Fe-i(kx-wt) (8)

where the first term represents the incident wave travelling in the nega

tive x-directioa (upwards) and the second term represents the reflected

wave travelling in the positive x-direction (downwards).

Equation 8 is valid for each of the layers in Fig. 2. Introducing a

local coordinate system X for each layer, the displacements at the top and

bottom of layer mare:

u (X• 0) • (E + F )eiwt (9) m m m

ikmh -ikmhm iwt u (X•h ) • (E • e m+ F e ) • e (10) m m m m

The shear stress on a horizontal plane is:

£!!. + n a2u • G* au T(x,t) • G • ax oxat ax (11)

or by Eq. 8:

ikx -ikx iwt 1(x,t) = ikG*(Ee - Fe )e (12)

and the shear stresses at the top and bottom of layer mare respectively:

T (X=O) = ik G* (E -F )eiwt m mm m m

-ik h 1 (X=h ) = m m

ik h ik G* (Ee mm

mm F mm) iwt - e e m

(13)

(14)

5

Stresses and displacements must be continuoue at all interfaces. Hence, by

Eq. 9, 10, 13 and 14:

ik h -ik h mm+Fe mm

m

(E e m

ik h -ik h m m_F e m m)

m

(15)

(16)

Subtraction and addition of Eqs. 15 and 16 yield the following recursion

formulas for the amplitudes, Em+l and Fm+-l' of the incident and reflected

wave in layer m+l, expressed in terms of the amplitudes in layer m:

ik h -ik h m m+k (1-a )e mm

2 m m

ik h -ik h m m+ !p (l+a )e m m

2 m m

where a is the complex impedlltlce ratio m

k G* mm

which again is independent of frequency.

1/2

)

(17)

(18)

(19)

6

At the free surface, the shear stresses must be zero. In addition,

Eq. 12 with , 1 and x1 equa.1 to zero gives E1 = F1 --i.e., the amplitudes

0f the incident and reflected waves are always equal at a free surface.

Beginning with the surface layer, repeated use of the recursion formulas

Eqs.17 and 18 leads to the following relationships between the amplitudes

in layer m and those in the surface layer:

(20)

(21)

The transfer functions e and f are simply the amplitudes for the case m m

E1 = F1

= 1, and can be determined by substituting this condition into

the above recursion formulas.

Other transfer functions are easily obtained from thee and f m m

functions. The transfer function A between the displacements at level n n,m

and mis defined by

A (w) • u /u n,m m n

and by substituting Eqs. 9, 20 and 21:

e (w) + f (w) m m

An , m ( w) • -e"'n-,.( w""'),--+--,f'"n""(-w'"") (22)

Based on these equations the transfer function A(w) can be found between

any two layers in the system. Hence, if the motion is known in any one

layer in the system, the motion can be computed in any other layer.

The amplitudes, E and F can thus be computed for all layers in the

system, and the strains and accelerations can be derived from the displace

ment function. Accelerations are expressed by the equation:

u(x,t) _ w2(Eei(kx+wt) + Fe-i(kx-wt)) (23)

and strains by:

(24)

2.2 Ratio between rock outcrop motions and base rock motions.

If the amplitudes of the incident and reflected wave components,~

and FN, in the elastic halfspace, Fig. 3a, are known, the motions in the

halfspace with the soil system removed, Fig. 3c, are easily computed. The

shear stresses are zero at any free surface; thus FN =~·and the

7

incident wave is completely reflected with a resulting amplitude 2~ at the

free surface of the halfspace. The amplitude of the incident wave in the

halfspace is independent of the properties of the system above it since

the reflected wave is completely absorbed in the halfspace and does not

contribute to the incident wave. The incident wave component, EN, is

therefore equal in all systems shown in Fig. 3.

The ratio between the base motion,~· and the motion, "l'.i• at

the free surface may be computed from the transfer function:

(25)

The transfer function between the motion at the surface of the deposit, u1 ,

and the motion at the free surface of the halfspace is:

(26)

If the halfspace is the rock formation underlying a soil deposit, Eq, 25

shows the ratio between the motion in the base rock and in the out

cropping rock, The ratio between the amplitudes of the base rock motion

• •

m

• •

N

a)

u1

= 2E,e I.Ill

h, lE, l E,

um= (Em+ Fm)e iilt

Ihm ! Fm !Em

int UN=(EN+FN)e

~ )•, System analyzed

- -I I iilt

Um= 2Eme •

k 1E~ 1E~ m

•

N \., h. / int

UN= 2EN e •

!EN N Ioo n.,

b) Loyer m outcropping

c) Holfspoce outcropping

Fig. 3 ONE- DIMENSIONAL SYSTEM WITH OUTCROPPING LAYERS

..... O>

8

and the outcropping rock motion is always less than 1, with minimum values

at the resonance frequencies of the deposit. Transfer functions for the

deposit used in the example, (Sect. 6), are shown in Fig. 4. The amplitude

of the base reek motion is only 65% of the amplitude of the rock outcrop

motion at the fundamental frequency of the deposit. This difference is a

function of the impedance ratio between the deposit and the rock and of the

damping in the deposit.

The differences in the computed responses resulting from the use of a rigid

base, relative to the use of an elastic base, depend also on which frequencies

are dominant in the rock motion. Rock motions with frequency dominance near the

resonant frequencies of the deposit will be considerably more affected than

motions with frequency dominance between the resonance frequencies, see Fig. 4.

The effect of the elasticity of the base rock is, therefore, not only a function

of the impedance ratio between deposit and rock and of the damping in the

deposit, but also of the frequency distribution of the energy in the rock

motion relative to the resonance frequencies of the deposit.

An approximation for the free surface motion for one of the layers in

the system, Fig. 3b, may be obtained in the same way as for the halfspace,

provided the incident wave component in the outcropping layer and in the

layer within the system are equal--i.e. E • E'. This is approximately the m m

case when the properties of layer m and all layers below are equal in the

two systems and when the impedance, p V, is of the same order of magnim m

tude as for the halfspace. This is the case for example, in sedimentary

reek layers overlying a crystalline rock base. For a more accuate solution,

the motion in outcropping layers must be computed in a separate system from

the motion in the halfspace.

C 0 -0 0 -0.

E <(

10

5

00

/

§lase rock to soil surface, A1, N

I \YVI) IV 'iPY' YUIV: 1 YI/

\ \ \1

A' I ?UI IU\tC, 1, N

Rock outcrop to roe k base, A~

••• •••• :·· ........................ n •••••• ......., •••• -- • ··w-··· .... ......... ·····-............ -~··· .... --.- ....•...... ;·;.··----~. •·······. ,i

2 3 4 5 6 7

Frequency c/sec.

Fig. 4 TRANSFER FUNCTIONS

8

0, I!)

2.3 Transient motions

The expressions developed above are valid for steady state harmonic

motions. The theory can be extended to transient motions through the use

of Fourier transformation.

9

A digitized seismogram with n equidistant acceleration values, iij(j•6t),

j = 0, .... ,n-1, can be represented by a finite sum of harmonic motions:

n/2 ii(t) - L

s=O (a e

s

iw t s +be

s

-iw t s )

where ws' s•O, •••• ,n/2 are the equidistant frequencies:

211 ws • n=-2it • s

as and bs designates the complex Fourier coefficients:

n-1 a =l L'.

s n j•O

-fol t "( ) s u t e

n-1 iw t 1 ~ ··c > s b •- L.. u t e

s n j•O

(.-27)

(28)

(29)

and each term in Eq, 27 is a harmonic motion oscillating with frequency

If the series in Eq. 27 represent the motion in a layer m, a new

series representing the motion in any other layer n, is obtained by

applying the appropriate amplification factor from Eq. 22 to each term in

the series:

A (w ) • m,n s

iw t -iw t (a e s + b e s )

m,s m,s (30)

The representation of a discrete motion with its Fourier transform

gives an exact representation of the motion at the discrete points t • j•6t,

j • O, •••. ,n-1. Cyclic repetition of the motion with the period T • n•6t

10

is implied in the solution. The solution applies, therefore, to an infinite

train of ident1cal accelerograms rather than the given singl~ accelerogram.

For systems with damping this is not of any significant consequence since

the individual accelerograms can be separated by a quiet zone of zeros causing

the responses from one cycle to damp out before the beginning of the next

cycle.

The Fourier Transformation can be performed in several ways. The SHAKE

program utilizes the Fast Fourier Transform algorithm developed by Cooley

and Tukey (1965), which is faster by a factor n/logn over the conventional

method. This technique computes all values in the series simultaneously.

The method requires that the number of terms in the series be some power of

2. A typical analysis using an acceleration record of 800 terms with time

step 6t = .02 sec. will use 1024 values in the Fast Fourier Transform, with

all values between 800 and 1024 set equal to 0. This will satisfy both the

requirements of a quiet zone after the acceleration record and that the total

number of terms must be a power of two.

J. !cESCRIPTION OF PROGRAM SHAKE

Program SHAKE computes the responses in a system of homogeneous, visco

elastic layers of infinite horizontal extent subjected to vertically

travelling shear waves. The system is shown in Fig. 2. Tha program is based

on the continuous solution to the wave-equation (Kanai, 1951) adapted for

use with transient motions through the Fast Fourier Transform algorithm

(Cooley and Tukey, 1965). The nonlinearity of the shear modulus and damping

is accounted for by the use of equivalent linear soil properties (Idriss and

Se2d, 1968, Seed and Idriss, 1970) using an iterative procedure to obtain

values for modulus and damping compatible with the effective strains in each

layer.

The following assumptions are implied in the analysis:

1. The soil system extends infinitely in the horizontal direction,

2. Each layer in the system is completely defined by its value of

shear modulus, critical damping ratio, density, and thickness.

These values are independent of frequency.

3. The responses in the system are caused by the upward propagation

of shear waves from the underlying rock formation.

11

4, The shear waves are given as acceleration values of equally spaced

time intervals. Cyclic repetition of the acceleration time

history is implied in the solution.

5. The strain dependence of modulus and damping is accounted for by

an equivalent linear procedure based on an average effective

strain level computed for each layer.

The program is able to handle systems with variation in both moduli

and damping and takes into account the effect of the elastic base. The

motion used ass basis for the analysis, the object motion, can be given in

any one layer in the system and new motions can be computed in any other

layer.

The following set of operations can be performed by the program:

1, Read the input motion, find the maximum acceleration, scale the

values up or down, and compute the predominant period.

2. Read data for the soil deposit and compute the fundamental period

of the deposit,

3. Compute the maximum stresses and strains in the middle of each sub

layer and obtain new values for modulus and damping compatible with

a specified percentage of the maximum strain,

12

4. Compute new motions at the top of any sublayer inside the system

or outcropping from the system.

5. Print, plot and punch the motions developed at the top of any

sublayer.

6. Plot Fourier Spectra for the motions.

7. Compute, print and plot response spectra for motions.

8. Compute print and plot the amplification function between any two

sublayers.

9 .. Increase or decrease the time interval without changing the pre

dominant period or duration of the record.

10. Set a computed motion as a new object motion. Change the accelera

tion level and predominant period of the object motion.

11. Compute, print and plot the stress or strain time-history in the

middle of any sublayer.

These operations are performed by exercising the various available options

in the program. A list of these options is given in Section 5, Required

1nput Data.

4. SYSTEM AND OPERATION DOCUMENTATION

4.1 Computer equipment

The program has been developed on a CDC 6400 computer using FORTRAN IV

language. The CDC 6400 has a 131 k core memory and uses a 60 bit words.

The program has been run without modifications on CDC 6600, 7600 and

UNIVAC 1108 computers, and with minor modifications on IBM 360 and 370

compll t ers ~

4.2 Storage requirements

The program requires approximately 50,000 octal words of storage

excluding the blank common X. The additional s~orage is a function of the

maximum number of terms used in the Fourier Transform as shown in

Table 1.

Table 1. Storage Requirements

Number of Length of Field length terms array X octal

0 0 50,000

512 3220 57,000

1024 6420 65,000

2048 12820 102,000

4096 25620 134,000

8192 51220 220,000

4.3 Runtime

13

The runtime is a function of the number of terms, n, used in the

Fourier Transformation and of the number of sublayers in the deposit. The

time involved in the Fast Fourier Transformation is proportional to

n·logn; all other operations are approximately proportional ton. In the

computation of strain compatible soil properties, the time will also

increase in proportion to the number of sublayers.

For the example run, Sect. 6, the approximate run times on the CDC 6400

are sqown in Table 2.

14

Table 2. Runtimes.

Number of Time Run time terms interval, sec. sec .

512 . 04 45

1024 .02 80

2048 .01 170

5. R>QUIRED INPUT DATA

5.1 Organization of input data.

Following is a description of the operations performed by the different

options, the required format for the input data, and explanations of some

of the input parameters. The input format is also described at the

beginning of the main program, see listing sect. 8.3.

The various options can be executed and repeated in any logical sequence.

The operations in an option will be performed on the data given or computed

in the program when the option is called, and the data may be changed at any

time during the execution by repeating the option with new data.

For example, in order to compute new m:itions in a soil deposit, (Option 5) object

ootion (Option 1), soil profile data (Option 2), specification of location

of object motion (Option 3), dynamic soil property-strain relation (Option 8),

a:1d strain iterations (Option 4--if strain compatible properties are desired),

must precede Option S. Soil responses for a new (additional) soil deposit

may be obtained by repeating Options 2, 3, 4, and 5. The last-read soil

deposit may be subjected to a new earthquake by repeating Options 1, 4, and 5.

15

5.2 Initialization card (15,FlO.O)

Cols. 1-5 MAMAX Maximum number of terms to be used in the Fourier Transformation in any of the problems to be run. Must be a power of 2 such as 512, 1024, 2048, etc.

6-15 SKO Coefficient of earth pressure at rest for sand layers. If blank the value is set equal to 0.45. May be left blank if all layers are clay.

After the initialization card follows one run option card.

5.3 Run option card (IS)

Cols. 1-5 KK Run of option

0 - stop, no more data 1 - read input .motion, and set as object motion 2 - read soil profile data 3 - assign the object motion to a specified

sublayer 4 - iterate to obtain strain~compatible soil

properties 5 - compute new motions at the top of specified

sublayers, print maximum accelerations and punch acceleration time history

6 - print or punch acceleration time history of object motion or any specified computed motion

7 - modify object motion or set the motion in any specified sublayer as new object motion

8 - read relations between dynamic soil properties and strain

9 - compute response spectra for any specified motion

10 - increase time interval in motions 11 - decrease time interval in motions 12 - plot Fourier Spectrum of object motion 13 - compute and plot Fourier Spectrum of motion

in any specified sublayer 14 - plot acceleration time history of object

motion or any specified computed motion 15 - compute and plot amplification function

between any two specified sublayers 16 - compute and plot stress or strain history in

the middle of any specified sublayer.

After the run option card follows the data set for the selected

option:

16

5.4 Dat3 cards and explanatory notes for the various options

Optlon 1. Read Input Motion.

Operations performed

(1) Acceleration values are read from cards,

(2) The sequence of the cards is checked.

(3) The maximum acceleration value in the record is found.

(4) The acceleration values may be scaled either by a specified factor or to a specified maximum acceleration.

(5) Trailing zeros are added to the record to obtain sufficient length on the quiet zone (a) and a total number of values which are a power of 2.

(6) The higher frequencies in the record are removed and the maximum acceleration in the modified record is found--optional.

(7) The motion is set as the new object motion.

1st Card

Cols.

Data Cards

(2I5, FlO, 0, 5A6)

1-5 NV Number of acceleration values to be read from cards.

6-10 MA(a) Number of values to be used in Fourier transform. Must be a power of 2.

11-20 DT(b) Time interval between acceleration values (sec.)

22-50 TITLE(I) Identification for earthquake.

2nd Card (3F10.0)

Cols. 1-10 XF Multiplication factor for acceleration values, Used only if XMAX is 0, left blank otherwise.

11-20 XMAX Maximum acceleration value to be used. The acceleration values in the record will be scaled to give maximum acceleration= XMAX, unless XF is left blank.

21-30 FMAX(c) Maximum frequency to be used in the calculations. Acceleration amplitudes at all frequencies greater than FMAX are set equal to O.

3rd and consecutive cards. Acceleration record. (8F9,6,I7)

Cols, 1-72 X(I)

73-79 K

8 acceleration values. (g's)

Card number. Warning will he given for cards not in sequence.

17

Explanatory notes for Option 1.

(a) The acceleration values between NV and MA are set equal to 0. in the

program. Cyclic repetition of the motion <.s implied in the Fourier

transform and a quiet zone of O.'s or low values are necessary to avoid

interference between the cycles. For most problems a quiet zone of 2-4

seconds is adequate with longer time required for profiles deeper than

about 250 ft andior damping values less than about 5 percent.

(b) The predominant period of the earthquake record can be changed by

altering the time interval ~t from that originally assigned to the

acceleration record. If the original record has time interval ~t, and

corresponding predominant period T1 , a new predominant period T2 is

obtained by changing the time interval to

(c) Frequencies above 10-15 c/sec carry a relatively small amount of the

energy in earthquake motions, and the amplitudes of these frequencies

can often be set equal to O without causing any significant change in

the responses within a soil system. Table 3 shows the maximum accelera

tions and strains in the soil system used in the example run, sect. 6,

computed for the Pasadena motion with time interval 0.02 sec and a

maximum frequency of 25 c/sec. Results are also shown for the same

motion with all amplitudes above 5 c/sec set equal to O. The difference

in maximum accelerations was less than 6.5% and in maximum strains less

than 0.7% in the two cases. The difference in response spectral values

was less than 1% for periods above 0.2 sec and less than 10% for periods

from.Oto 0,2 sec,

Depth

0

7

20

30

42

62

80

100

120

Table 3. Effect of the Higher Frequencies on the Maximum Accelerations and Strains.

Maximum acceleratlon, g's Difference Maximum strain. % f = 25 c/sec 5 c/sec % f = 25 c/sec 5 c/sec max max

.0971 .0962 .9 .00725 .00724

.0958 .0949 .3 .1292 .1283

.0600 .0599 ,1 .0391 .0390

.0553 .0556 .6 .0287 .0287

.0508 .0507 .2 .00982 .00989

.0470 .0469 .2 .0505 .0504

.0319 .0299 6.3 .0349 .0348

.0239 .0235 1. 7 .0320 .0319

.0178 .0189 6.2

Difference %

,1

.7

.3

-. 7

.2

.3

.3

.... 0,

19

In the computation of responses in deep soil systems from a motion given

near the surface of the deposit, errors in the higher frequencies will be

amplified and may cause erroneous results. To avoid this source of error,

the amplitudes of all frequencies above 10-20 c/sec. may be set equal to O.,

since these frequencies generally are of little interest and do not affect

the response. Several runs should be performed with different amounts of

the higher frequencies removed to investigate the effect on the response and

to ensure a stable solution.

Removal of the higher frequencies in a motion has a smoothening effect

on the acceleration time history as shown in Fig. 5 for a segment of the

Pasadena motion. In this case the maximum acceleration for the modified and

original motion were approximately equal, but the maximum accelerations may

decrease or increase with the removal of the higher frequencies depending on

the shape of the acceleration curve near the maxi111um value.

Option 2. Read Data for Soil Deposit.


(1) The properties of the soil deposit are read from cards.

(2) The sequence of the layer cards is checked.

(3) The layers are subdivided into sublayers--optional.

(4) Effective pressures in the middle of each sublayer are computed,

(5) The fundamental period of the deposit is computed.

1st Card (3I5,6A6)

Cols. 1-5 MSOIL 6-10 ML(a)

Data Cards

Soil deposit number. Can be left blank.

Number of layer cards to be read including card for halfspace. There is one card for each layer whose properties are individually specified.(b)

6.5

fmax = 25 C/sec

f max= 5 c/sec

\!

7.0 7.5 Time in seconds

I/

I I

'l

8.0

Fig. 5 EFFECT OF THE HIGHER FREQUENCIES ON THE ACCELERATION TIME - HISTORY

>-a

"' "

Cols. 11-,15 MWL

17-51 IDNT(l)

Number of first submerged sublayer (b). llf no ground water table present, put groundwater table at top of halfspace.]

Identificaticn for soil profile.

20

2nd and consecutive cardla. One card for each layer including halfspace. (3I5,6Fl0.0,F5.0)

Cols. 1-5 K

10

11-15 NLN(a,b)

16-25 HL (a)

26-35 GMOD(d)

36-45 B (d)

46-55 W

56-65 vs<d)

66-75 FACTOR(c)

76-80 BFAC

Layer number. The layer cards must be in sequence with the surface layer as layer l. Note that the number of layers may be_!:. the number of sublayers(b).

Soil type l - clay 2 - sand 3 - rock

Number of sublayers in layer K. The Kth layer will be divided into NLN sublayers of thickness• HL/NLN.*

Layer thickness (ft.)

Initial estimate of shear modulus (kips/sq.ft.) Not necessary if VS is given.

Initial estimate of critical damping ratio (decimal).

Unit weight (kips/cu. ft.).

Initial estimate of shear wave velocity (ft/sec). Not necessary if GMOD is given.

Factor for shear modulus Clay - F • undrained shear strength

c (kips/sq. ft.) Sand - Fs • factor modifying the average

curve read in under Option 8. Set Fs • 1. for no change.

Rock - FR• Shear wave velocity for low strain values~in thousands of ft./sec.

Factor modifying the standard damping curve read in under Option 8. For example, a factor of 1.2 increases each and every value by 20 percent.

For the elastic half space, soil layer card number ML, it is sufficient

to give values for K, GMOD or VS and W.

*Maximum total number of sublayers including the base is 20.

21


(a) With th~ wave propagation method the responses can be computed in a

homogeneous layer of any thickness. A soil deposit wiil, however, have

v~rying properties not only due to the variation in the soil itself but

also due to the differences in the strain-level induced during shaking.

Since the soil deposit must be represented by a set of homogeneous

layers, each with a constant value of modulus and damping, the thickness

of each layer must be limited based on the variation in the soil

properties. For a fairly uniform deposit, a sublayer thickness

increasing from about S' at the surface to 50-200' below 100' depth

should give sufficient accuracy. Accuracy may be checked by making a

trial run and comparing results with a subsequent run where more layers

and/or sublayers are used.

(b) The division of a layer into sublayers is for convenience to avoid

punching of several cards with the same properties, and all sublayers

are treated as separate layers in the following computations. The

sublayers are numbered consecutively starting at the top of the soil

deposit and the halfspace is counted as the last layer and the last sub

layer in the deposit.

(c) Computations of shear moduli for the different soil types are based on

the following expressions:

Clay G K Fe C C

Sand G = K 1000 . (o')l/2 . F s s m s

Rock G • KR • p . (1000 · FR) 2 /2000. R

where

K • strain function given in Option 8,

F = factor given as input (FACTOR)

p = mass density in kips/cu. ft,

o' = mean effective pressure (psf), m

The strain function for clays, K, gives the average relationship C

22

between G/S and strain for saturated clays. While the undrained shear u

strength of the clay, S, is normally used in this modulus-strain u

relation, the factor for clay, F, should be given a value which gives C

the correct modulus-strain relation; thus Fe is not neceasarily equal

to S • If the modulus of the clay is found from seismic investigations, u

the value of F should be set to G /K where K is the value for 10-• C C C C

percent strain in the curve given in Option 8.

(d) The modulus and damping are in general used as initial values on the

first iteration for the computation of strain-compatible properties,

but they can also be used directly to compute the responses for the

values given, by omitting Option 4. Typical values of the modulus for

strong shaking are of the order of 500 kips/sq. ft. near the surface

increasing to 3000 kips/sq, ft. at 100-200' depth for sand, 500-

2000 kips/sq.ft, for clay with values as low as 50-100 kips/sq. ft, for

soft clay. Usually 3-5 iterations are sufficient to obtain strain

compatible values within a 5-10% error limit.

The results ere not highly sensitive to errors in the damping ratio

and values selected between 0.05 to 0.15 will usually give strain

compatible values with 2 to 3 iterations.

23

Option 3. Assign Object Motion to a Specified Sublayer.


The object motion is assigned to the top of one sublayer in the soil

deposit.

1st Card (2I5)

Cols. 1-5 IN

6-10 INT

Data Cards

Number of sublayer where object motion is assigned.

Type of sublayer 0 - Outcropping(a) sublayer 1 - sublayer within profile

Explanatory notes to Option 3. (a)

See Section 2,2,

Option 4. Obtain Strain Compatible Soil Properties.


(1) Parameters for the iterations are read from card,

(2) ~.aximum strains, stresses and times for the maxima are computed in the middle of each sublayer,

(3) Effective strains are obtained from the maximum strains and used to compute new soil properties.

(4) The operation is repeated until strain-compatible soil properties are obtained within a given error limit or until a specified ~aximum number of iterations is reached.

(5) The fundamental period of the deposit is computed after the final iteration.

(6) A set of soil data cards with the new strain compatible properties is punched--optional.

1st Card (215,2Fl0.0)

Cols. 1-5 KS(a)

6-10 ITMAX(b)

11-20 ERR(b)

21-30 PRMUL (c)

24

Data Cards

Set equal to 1 for punched set of soil data cards with the soil properties after final iteration. Leave blank if punched cards are not wanted.

Maximum number of iterations.

Maximum acceptable difference between the last-used modulus and damping values and the strain-compatible values (percent),

Ratio between effective strain and maximum st=ain (decimal).


(a) The most time consuming part of the computations is to obtain strain

compatible soil properties. A set of soil data cards with strain

compatible properties may save computer or punching time if

additional computer runs are to be made subsequently.

(b) The iterations stop when the specified maximum number of iterations

(ITMAX) is reached or when the difference between the modulus and

damping used and the strain-compatible modulus and damping values

is less than the acceptable difference (ERR). Usually 3-5 itera

tions are sufficient to obtain an error of less than 5-10%. The

values given as "new values" in the final iteration are used in all

computations following Option 4, and the actual error is less than

the error values given in the final iteration.

(c) The effective strain is used to compute new soil properties. The

ratio between the effective and the maximum strain has been

empirically found to be between 0.5 and 0.7. The responses, however,

are not highly sensitive to this value and an estimate between 0.55

to 0.65 is usually adequate, with the higher value appropriate for

giving more uniform strain histories.

25

Option 5, Compute Motion in Specified Layers.


(1) The acceleration time history is computed at the top of specified sublayers.

(2)

(3)

(4)

The maximum acceleration and times for maxima are printed for the computed motions.

The computed acceleration time histories may be punched--optional.

The acceleration time histories may also be printed or plotted (Option 6, 7 and 14)(a).

Data Cards

1st Card (1515)

Cols. 1-75 LL5(1)

2nd Card (1515)

Cols. 1-75 LT5(I)

3rd Card (1515)

Array showing the numbers of the sublayers at the top of which the motion is to be computed, Maximum of 15 locations,

Array specifying typ@s of above sublayers. 0 - outcropping lb) sublayer 1 - sublayer within profile

Cols. 1-75 LP5(I)(a) Array with mode of output for the computed motions.

0 - max. acceleration value only printed, 1 - punched carda giving acceleration time

history in addition to the printed maximum acceleration value,

Explanatory notes for Option 5

(a) The acceleration time histories can be printed or plotted through

the use of Option 7 where a specified motion is set as the new

object motion. Subsequent use of Options 6 and 14 give respectively

a printed and a plotted output of the acceleration time history of

the motion.

(b) See section 2,2,

Option 6. Print or Punch Object Motion.


(1) Maximum acceleration and time at which maximum occurs are found.

(2) The object motion is printed--optional.

(3) The object motion is punched on cards--optional.

Data Cards

1st Card (IS)

Col. 5 K2 Selects mode of output.

K2 = 0 1 2

Max. acc. only Punched output Printed and punched output,

Option 7. Change Object Motion.


26

(1) A motion at the top of a specified sublayer can be set as the new object motion and printed or punched (Option 6) or plotted (Option or used for subsequent computations--optional.

(2) The time step in the object motion can be changed--optional.

(3) The acceleration level in the object motion can be changed-optional.

Data Cards

1st Card (2IS,2Fl0.0)

Cols, 1-5 111

6-10 1Tl

Number of sublayer. Use O if object motion originally assigned is to be retained(a),

Type of above sublayer 0 - outcropping Cc) sublayer 1 - sublayer within profile

11-20 XF Multiplication factor for acceleration values--1. for no change.

21-30 DTNEW New timestep (b).


(a) The acceleration level and timestep can be changed either on the

motion originally set as the object motion, or on the computed

27

motion which is set as the new object motion through Option 7,

(b) A change in time interval will change the predominant period of

the motion. If the time interval and predominant period of the

original motion are 6t1

and T1 , respectively, a new predominant

period T2

is obtained by changing the time interval to

(c) See section 2,2,

tion 8. Read the relation between the Effective Strain an the Dynamic Properties


(1) Effective strain values with corresponding values for damping and moduli are read from cards.

(2) Parameters are computed for interpolation of modulus and damping values using a linear semilogarithmic relation between the given values.

(3) The relationship between the dynamic properties and the strain is plotted--optional.

1st Card

Cols.

(3I5,Fl0.0,l0A6)

1-5 NSOILT

10 NPL(b)

11-15 NN(b)

16-25 2&-80

SC

Data Cards

Number of different soil or rock types to be read. Maximum 4,(a)

Set equal to 1 for plot of curves.

Number of strain-values in each logarithmic unit to be plotted.

Maximum value of the ordinate in the plotting. Title or identification data.

Next follows two sets of cards for each soil or rock type. The first

set gives the relationship between the shear modulus parameters (C) and

the effective strains; the second set give the relation between the

critical damping ratios and the effective strains.

shown on page 40.

Typical data is

First Set:

1st Card

Cols.

(IS,FS.0,11.A6)

1-5 NV(L)

6-10 FPL(L) (b)

12-76 ID(L, I)

Number of strain values to be read. Maximum 20.

Multiplication factor for shear-modqlµs parameter. Used for plotting only.lb)

Identification for first data set. Used for plotting only.

28

2nd and consecutive cards (8Fl0.0)

Cols. 1-80 X(L,I) Effective strain values in percent beginning with the lowest value. 8 values per card with maximum of 20 values.

Consecutive cards (8Fl0.0)

Cols. 1-80 Y(L,I)

Second Set:

Values of the corresponding above. Eight maximum of 20

(c) shear modulus parameter to the strain values given values per card with values.

The input format for the second set is identical to that for the first

set with values of critical damping ratios in percent instead of the

values for the shear modulus parameter.

Explanatory notes- for Option 8.

(a) Three different soil or rock types can be used in the program as

described in Option 2. The relationships between effective strains

and the dynamic properties must be read in the same sequence as the

soil type using the notation:

1 - Clay

2 - Sand

3 - Rock

(b) The values for the shear modulus parameter and the damping can be

plotted against the effective strains. If plotting is specified

(NPL • 1), values for the shear modulus parameter and damping are

29

computed for a specified number of effective strains (NN) in each

log&rithmic unit. The computed values should be scaled (FPL(L)) to

obtain good representation of all curves on the same plot. The

scaled values and the corresponding effective strains are also

printed.

(c) The values are used to compute the shear modulus for the different

soil types. The relationship for sand and clay used in the program

is based on the expressions given by Seed and Idriss (1970):

G (~ Clay K (y) = _c.c--_

c Su

Sand

The relationship used for rock is the scaled ratio between the shear

modulus at low effective strain (10-• percent) and the shear modulus

at a specified effective strain:

Rock (y). G(y) • 2000

~ G(ylO-')

Crtion 9. Compute Response Spectra


(1) The motion is computed at the top of a specified sublayer.

(2) Times for maxima in the acceleration, velocity and displacement spectra are computed and printed.

(3) Acceleration and velocity spectra may be plotted and/or punched on cards--optional.

lst Card (2IS)

Cols. 1-5

10

LLl

LTl

Data Cards

Sublayer nUlllber. Use O if the response spectra sre to be computed fo~ the object motion,

Type of sublayer. 0 - outcropping sublayer 1 - sublayer within profile,

The response spectra are computed for the motion at the top of the sublayer, May be left blank if LLl is O.

2nd Card (5I5)

Col. 5 ND

3rd Card

10 KP

15 KAV

20 KPL

25 KPER

(6Fl0.0)

Total number of damping values to be used. Maximum 6 values.

Set equal to 1 for punched output.

Select plot and punch option: 0 - plot and/or punch velocity spectrum 1 - plot and/or punch acceleration spectrum 2 - plot and/or punch acceleration and velocity

spectrum.

30

Set equal to 1 for plot of spectra according to KAV. All spectra computed since last plotting will be plotted together.

Select periods to be used in the computations:

KPER = 0 9 steps from 0.1 sec to 1. sec 5 steps from 1. sec to 2. sec 4 steps from 2. sec to 4. sec

KPER = 1 18 steps from 0.1 sec to 1. sec 10 steps from 1. sec to 2. sec

8 steps from 2. sec to 4. sec

KPER - 2 38 steps from 0.05 sec to 1. sec 20 steps from 1. sec to 2. sec 30 steps from 2. sec to 5. sec

KPER • 3 Logarithmic increments with 10 steps in each log. unit from 0.1 to 5.

KPER = 4 Logarithmic increment with 25 steps in each log. unit from 0.05 to 10.

Cols. 1-60 ZLD(I) Values of critical damping ratios in decimal to be used in the spectral analysis. number of values must be given.

Option 10. Increase the Time Interval


The time interval is increased.

ND

31

Data Cards

1st Card (IS)

Cols. 1-5 IFR(a) Factor for increasing time interval. Must be a power of 2.


(a) The Fourier Transformation of a given acceleration time history

consists of a series of harmonic motions

n/2 iw t -iw t

ii(t) * L (a8

e s + bse 8

)

s•O

With the harmonic motions given, acceleration values can be computed

for any value of the time, t, and a new acceleration time history

can be generated with a time interval different from the original.

Suppose, for example an acceleration record is given with 2048

values and a timestep ~t = 0.01 sec. Through Option 10 with IFR • 2

a new record with 1024 values and timestep 0.02 sec is generated.

The acceleration values in the two records are identical at all

times n • .02 sec., n • 1,2 ••• 1024. The new record has a maximum

frequency of 25 c/sec. compared to SO c/sec. in the original

records, and frequencies from 25 c/sec. to SO c/sec. are lost in the

operation.

Increasing the time interval reduces the computer time as shown

under sect. 4.3. For computation of maximum accelerations a time

interval of 0.02 sec. will generally give adequate accuracy while

a time interval of 0.04 sec. may be sufficient for the computation

of the stresses and strains in a deposit.

The difference in maximum accelerations and strains resulting

from the use of different time intervals are shown in Tables 4

and 5 for the example run, The effect may be somewhat higher

for earthquakes with lower predominant periods and for stiffer

soil systems.

Option 11. Decrease the Time Interval


The time interval is decreased.

Data Cards

1st Card (IS)

32

Col. 1-5 IFR (a) Factor for decreasing the time interval; must be a power of 2.


(a) See explanation to Option 10. Through Option 11 a new time history is

generated with the time interval reduced by a power of 2. Compared

with the usual linear interpolation, this method has the advantage

of not introducing additional frequencies to the motion.

Option 12. Plot Fourier Spectrum of Object Motion


(1) The Fourier Spectrum of the object motion is plotted.

(2) The spectrum may be smoothed--optional.

Data Cards

1st Card (3IS)

Cols, 5 Kl Select for plotting: 0 - Store spectrum for later plotting, Max, of

2 spectra can be plotted together. l - Plot all spectra stored since last plotting.

6-10 NSW(a) Number of times the spectrum is to be smoothed.

11-15 N Number of values to be plotted--maximum of 2049.

33

Table 4. Effect of Time Interval on Maximum Strain.

,...

Depth Computed Maximum Strain %

lit - .01 lit• .02 lit • .04

3.5 .00727 .00725 .00725

13.5 .129 .129 ,127

25. .0392 .0391 ,0390

36 .0287 .0287 .0285

52 .00982 .00982 .00981

71 .0505 .0505 .0505

90 .0350 .0349 .0348

110 .0320 .0320 .0316

34

Table 5. Effect of Time Interval on Maximum Acceleration.

Depth Maximum Acceleration

At• .01 At• .02 At• .04

0 .0971 .0971 .0967

7 .0960 .0958 .0954

20 .0598 .0600 .0590

30 .0554 .0553 .0548

42 .0508 .0508 .0498

62 .0471 .0470 .0462

80 .0317 .0319 .0318

100 .0238 .0239 .0242

120 .0181 .0178 .0178

35

Explanatory notes to Option 12.

(a) The expression used to smooth the spectrum is:

where A1

is the acceleration amplitude for the 1th frequency.

Option 13, Plot Fourier Spectrum (c) of Computed Motions


(1) The motions at the tops of the specified sublayers are computed.

(2) The Fourier Spectra for the computed motions are plotted and printed.

(3) The spectrum may be smoothed--optional.

1st Card (SIS)

Cols. 1-5

10

LL(l)

LT(l)

Data Cards

Sublayer number.

Type of sublayer: O - OutcroppingCb) sublayer 1 - Sublayer within profile.

15 LP(l) Select for plotting: 0 - Store spectrum for later plotting;

max. of 2 spectra can be plotted together

1 - Plot all spectra stored since last plotting.

16-20 LNSW(l)(a) Number of times the spectrum is to be smoothed.

21-25 LLL(l) Number of values to be plotted. Max. of 2049.

2nd Card (515)

As for Card 1 for a second motion. A blank card must be used if only

one spectrum is to be computed.


(a) See Option 12.

(b) See section 2.2.

(c) See section 2.3.

Option 14. Plot Time History of Object Motion(a),


The time history of the object motion is plotted.

1st Card (215)

Cols. 1-5 NSKIP

6-10 NN

Data Cards

Number of values skipped in the plotting, 0 - every value is plotted 1 - every second value is plotted etc.

Number of values to be plotted. Max. of 2049 values.


(a) The time history of a computed motion can be plotted by setting

this motion as the object motion through Option 7.

Option 15, Compute Amplification Spectrum.


(1) The amplification spectrum between any two sublayers in a given soil system is computed.

36

(2) The maximum amplification and the corresponding period are printed.

(3) The amplification spectrum may be plotted and printed--optional.

1st Card

Cols.

(5I5,

1-5

6

F5.0,8H6) LIN(a)

LINT

11-15 LOUI(a)

20 LOTP

25 KP

Data Cards

Number of first sublayer.

Type of first subzg,er 0 - outcropping sublayer 1 - sublayer within profile

Number of second sublaver.

Type of second sublayer 0 - outcropping sublayer 1 - sublayer within profile.

Select for plotting: 0 - Store spectrum for later plo~ting.

Maximum of 8 spectra can be stored. 1 - Plot all spectra stored since last

plotting.

37

26-30 DFA Frequency steps. TI!.e amplification factor is computed for the first 200 frequencies with interval DFA c/sec. beginning at O.

32-78 IDAMP(I) Identi:'. 'ion.


(a) The amplification factors are computed from the first sublayer to the second.

(b) See section 2.2.

Ootion 16. Compute Stress or Strain History in the Middle of Specified Sublayers.


(1) The stress and/or strain time history in the middle of any two specified sublayers are computed.

(2) The computed time histories may be plotted or punched on cards.

1st Card (5I5,Fl0.0,5A6)

Cols. 1-5 LLL(l)

10 LLGS (1)

15 LLPCH(l)

20 LLPL(l)

21-25 LNV(l)

26-35 SK(l)

37-65 ID(l,)

Data Cards

Sublayer number. The stress or strain history is computed on the middle of the sublayer.

Select type of response: 0 - strain 1 - stress

Set equal to 1 for punched output.

Set equal to 1 for plotting.

Number of values to be plotted; maximum of 2049.

Scale for plotting--i.e. maximum value of ordinate. If blank, the largest value in the response is set as the maximum value of the ordinate.

Identification.

2nd Card. As for Card 1 for second sublayer. Use blank card if only one response is to be computed.

38

6. EXAMPLE RUN

6.1 Selection of soil system and input motion.

An example problem is shown in Fig. 6. Maximum accelerations, stresses

and strains in the soil deposit and response spectra for the surface accelera

tions are wanted for a magnitude 7.4 earthquake occurirg 100 miles from the

site.

Based on the relations given by Seed and Idriss (1970), the soil system

shown on Fig. 7 was selected for analysis. The factors used for clay are

equal to the undrained shear strength in kips/sq. ft. The factors for sand

are estimated from relative densities and content of gravel.

The motion in rock for a magnitude 7,4 earthquake 100 miles from the

causative fault is estimated to have maximum acceleration of ,02g and a

predominant period of 0.65 sec (Schnabel and Seed, 1972; Seed et al., 1969).

Among the available strong motion records, the Pasadena record from the 1952

Kern County earthquake seems to have characteristics most similar to those

desired. The magnitude of the earthquake was 7.7, the record was obtained

some 75 miles from the fault, the maximum acceleration was 0.057g and the

predominant period was 0.65 sec. Modification of this record to give a maximum

acceleration 0.02g gives the desired characteristics for the motion in an

outcropping rock formation near the example site.

38a

Depth Soil Deposit

0 -Sand Dr = 45 °10

Clay Su= 250psf

Clay Su= 750 psf

~ Clay Su= 1150 psf

. . . . 50 -

. . . . ~·.: Sand .. . . with . . . .

.• 0 some grovel

... ·C • '•

Cloy Su= 1400 psf

~ Distance from causative

Cloy Su= 2000 psf fault: 100 miles

100 -

~ Earthquake magnitude: 7.4

Cloy Su= 2250 psf W.L.

_..,L Ill-$: Ill

Rock Vs= 8000 fYsec

Fig. 6 EXAMPLE PROBLEM

Depth

o-

50-

100-

Soil type Factor

2 0.7

0.25

0.75

1.15

2 1.25

1.4

2.0

2.25

Holfspoce Vs= 8000 ff/sec

38h

Motion in outcropping rock:

Posodena record from the 1952 Kern County earthquake scaled to 0.02 g maximum acceleration.

Fig. 7 SYSTEM USED IN THE ANALYSIS OF THE EXAMPLE PROBLEM

39

6.2 Input data for the analysis,

i:· ; • ~ ") : 3 ' •1

O• ~, 3 ' ~ 3

V i C • • V

• ~ 0

0

• • ~ • ~ • •

0 • • u

• s • • ~ ~ • • ;

_ I',/_ ..... ,t'I ,0 ... _, 0- I:' - "'"'-# >I' ,0 ,,__.,:,, 0 - "' fP' ,# ,r, ,(I.,._ "' & C - N ... # ,,., ,0 ,._ ,r- & C' - N..,"' ,r ,£ ... C' t'" 0 - N "'

. . :

0 • . . . . .; '! . . . :a . 0

'! . :a

~

!a

. . .

I l I

1~

----------"'"'"'"'"'"'"'"'"'"'"' ...... ~~ ... ~~ ... ~ .. 4 ................... ,.,,.,

•

r

. . I:

~ 1 l I •

ii I: . ' . ,. . . ~

. 0 ~ . . . l: . . . :, . . . i

OI

! 0

~ s ~ . I

. - 0 0 N

N • • !

N • !

~

': •

~

•

I !I . ~ • N~ N • • ! • N

' ·-1 co .; • • •

-;

• ~

I: _ .... " ... l'--0 ...... ... : =1· i . fl '! .. C,,I( ..

-o •

""!!·1-~ ' • • • • > • "' • ......... .., ...... Co C: • <•• . ........ :

40

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., . •l••,••••••••••••ooH•••••••••••••u•••••u•••n•••••••••••oououo••••••••••••n•••••o,uo••••••••• . . . . . . . . 11'!•100 I"' src.

,.,.ot ... itSl tl.l"l"t.( 11yr

'~" ' -0•11 ,,;;e, , lfOi ,1u• , t 19' ,)JU

51

,)SOC ,•OH .. ,.,, ,500¥ ,HU ,6C00 ,69,U .uoo .noo ,UOt ,l!IOC .,uo .•soo

l,OUO 1.u~• I, ?oOO I ,JOU 1.,ou l•'-OOf 1,000 1,TtU 1,11000 i.tooo l,eooo l!,UGO l,5Ut ,,no• lo,000 l,1100 J,5000 J,?5011 ••otOf

,OOH , tot• , 1soo ,,oh• .1100 ,IOU .J!>RG

.. ,uoo l ,•SOI • .sooo J ,!i.500 .- ,&OU • ,11-seo ~ .,no l! ,1'500 II ,HOO

.u,o ,OOCI .,,,,

1,0000 1, UOO l ,f000 l.JUO l ,4000 1.s,00 !,MU

11 t,?UG .I 1.11000 I l,tOU

l,OOOG l,lUO l,9,0tO l,1sno ,.ou S,;>Sto 3,9,0U J,?!00 ••••••

.,111 ,1t!>t

-1••• •lH9 .,.,t ,H~1 ,lHl iJlll ,)US

:;:H .•. ,, ,Jltt ,1•u ,Jh6 •O~H ,011111 ,o?oJ ,o!iU .,.u .,ou •OlOJ ,oJt, •tlb~ ,,i.a •1236 .,1,1 ,ot•t ,ottt •DOil !tfU ., . .,

.. .. .. •• .. .. .. .. " IH l'lt• co.t . . . . . . . . . , ..................................................................................................... . ' ,, ' I

I

I

I I

' I

' ' I

• I ' I • ...................................................................................................... . . . . . . . .

c~u: I • 11'11.U'I)( ... ltH «~su, '

~·fll& .,,u .!f09 ,\~U ,Hit ,Kto

C-rt: I

••••• ,toll ,,,,. ,t3U .,,3s

52

,)10 ,l'; ,, . un .• ,,o .,oat .110• •6000 ,&!iU ,,Ht • ,~oa .,coo .~sco ,,no .'15~0

J,oou 1.1,ao 1.;ao l•JUG l ,4000 \,SOU J.!,,n i .ueo :.,,u 1,•u• 1.0~,, l,lSU 1.,00 2,rsoe 1.~ou )o1'Hj

l,!OU J,7:5,01

··~···

•11•1 • ! ,. • •lUl ollll ••Gil ,s•a ,1Ul

·'"'" t •lilt l ,lS1l l•th• t,1Sl'I l,tll!i t, ,,., 1,.ttl 1,~u• ,rut .u.s •Tl!l ,,sn ,,a1• ,,1,a •1)14 ,'JUT ,sou iu11 ·•l•t

:;~n .uu .. ,., ... .,. .,,.,.

53

7. PROGRAM IDENTIFICATION AND ABSTRACT

7.1 Program Identification

1. Program title: Vertical propagation of shear waves through a horizontally layered soil/rock system.

2. Program name: SHAKE.

3. Writers: Per B. Schnabel, Research Assistant John Lysmer, Associate Professor of Civil Engineering.

4. Organization: Geotechnical Engineering Department of Civil Engineering University of California Berkeley, California 94720

5. Date: December, 1972.

6. Version: 2

7. Source language: FOR:l'.RAN IV

7.2 Abstract

54

The program computes the response in a horizontally layered soil

rock system subjected to transient, vertical travelling shear waves. The

method is based on Kanai's solution to the wave equation and the Fast Fourier

Transform algorithm, The motion used as basis for the analysis can be

applied to any layer in the system. Systems with elastic base and with

variable damping in each layer can be analyzed. Equivalent linear soil

properties are used with an iterative procedure to obtain soil properties

compatible with the strains developed in each layer. A varied set of

operations of interest in earthquake response analysis can be performed.

55

8. SOURCE LISTING FOR PROGRAM SHAKE,

t P•Dc•tfll St-t.ttEUINP\.f, O\ll'Ut_c 19~CH1

C • • • • • • • • • • • • • • • • • • • • • • • • o • • •"' • • • • • • C •.• • • • • • • • • • • • • • • • • • • • • • • o • • • •• -. • • • • C t C C C C C C C C C C • C C C C C • C C C C C C C C C C C C C C C C C C C C C t C t C C

• C t C C C C t C t C t

THIS ,.CGIA" CO~P~TlS AESNJNSI IN A HOftlSONTAllT LAY!R!O S!Nl'"'IN(TI SYSTf"' SLIJfCTtO TO VfATICAlLY TlAVElllNG SHEAl WAY($ THE RET•OO IS ltSfO CN THf CO~TlNCUS SOlUTtCN TO THE ~~~l WAVI f®Al!Ut

'llCGlAllll'fC BY Pfa I SCHNAl!L RESEARCH ASSISTANT JOt.N LYSNU .USCC1ATI '1l0FESS0ft

,ace-.- vEASlCN SHAKU SftAKU

GIOTICHNICM. ENGINEERING 114IV(ftS1TY Of CALIFORNIA lll!IUI.Ell!Y

JANUAftY nn .Vft.lL HU JUHi: t•n OlAtreGI - OPTIOflf? c.DRletTID

- SU81'1JUTIHU STIPC AfQ FFT ADAPTED TO UfflVAC COMl'Ult!RS

lltcltt8!1l 19Tl CHANGE - SIJ1ROUTIN£S DRCTSP. Af!D STIIAIN

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

OEFl~ITIC~S (ft,UT MOTION CIJECT J«JTICN

CC:•l'llTEO NOT ICN

CPllCN

C\IIET ZCIN£

CL TC POP

CLTt•OP lllOTIO

• JIOTICN Reao IN FftOM CARDS • l«JTICN USED AS &ASIS FCR COMPUTING NEY

l'OTICNS IN A SOil P~OFILE • ,rQTICN COMPUTl:O ANYWHERE IN A CtV!N SOIL

SOIL PAOFILE FRON A GIVEN OIJECT NOTION • OP!RATIOH TO BE PERFOlt~EO ON GIVEN DATA,

INJTIALllATICN OR CHA~GE OF CATA • ZO~E CONTAINING TH! TRAll1NC ZEROES IN

THt t"PUT ~OTION • l'lACE WHER~ a SOIL SUBLAYER OR THE BASE

•ocK REACHES A f .. EE SURFACE • •OTICN AT 4N OUTCROP OF ROCK OR SOIL SUBLAYkA

••••••••••••••••••••••••••••••••••• l'I•ST C,._C FOlilMATI 11,flO .. O)

COL. 1- ' IUlf. Nl.M8Eft OF TERMS TO SE USEO IN fHE ,au•tEl TRAHSFOAMATIO... IN ANT OF THE PROBLEMS TO 9E Jl:UN

CJl. 6-1' EHTH l'AE$SURE AT •esr FOR UNO. USED ONl...l' FOR CQ~,UTING SHEA~ ~ODUlUS FOR SAND. IF BLAH- Ti,.£ VALUE IS SET TO .41

C C C .t. C C C C C (_ C C C C C C C C C C • -{ ( C C C C C C C

AIIT~ ttits ,un CAll:O FQUQIIIS SWITCH FOlt OPTION TO ae ll'EII.FiJANEO. THE AVllLAlll OPTIONS NI.E LISTEO IELOW. THE OPTIONS CAN BE !XECUT£0 A~ R!PFAT!O IN ANY LCGICAL O~OfR.

S!(OkU ta•o lNO CAIO PRECffOING EVERY OPTIOh, FOR•AT(l5) COL. 1- ' SWITCH

SJilllCH • • l

• • • • • • • • •• ll

u u •• .. ••

sro,, NO l'tORE DATA RrAO INPUT l'IOTION, TRANSFfR TO fRfQUEltCY 00~.&lN ANO SET AS NEW 08JECT ~OTION READ N,w SOIL PROFILE R!AO wtERE IN fHE SOIL PROFILE THE OBJECT IICTJCh IS GIVEN OBTAIN STRAIN CONPATl9l£ SCtL PROl'ERflES CONPUTf NEW MOTl~S P~INT OR PU/'tCH RESULTS PRINT ANO/QA PUNCH TIME HISTORY OF OIJfCT tlOTJCN C~ANG! P.tAAM!TERS Of OBJECT MOTION Ol Sl:f A COMPUTED "'1TlON •S O~JECl 1'0TtON RIAQ SOil PROPERTY/STRAllf-R!LATIO,U CDMP~T! RESPONSE SPECTRA Ike.RIAS£ TIMESTEP tl!C~!AS! Tl•ISlEP PLOT IIOURIER SPECTRU" OF OIJECT MOTICll PLOT ,ouRJER SPECTRUI' ~F CO•PuTlO IOOflO~ PlOT TIME HISTORY OF OBJECT l"OTION US£ Cl'TICttil l 'IRST FOR PLOT Cf CO .. UTEO NOTICN COMPUTI A~PltPtCAflON PACTQRS CDNPLTI STRESS OR STRAIN HISTORY

C . C • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • C C C C C C C C C C C C C C C C C C C C C C C C

DATA CAACS ,OR T•t OJFFER!Nf Cl'TIONS

RUD I

It.PUT JtOTICH CUD UIS, f'lO.o. 5AU

COL.. 1- , ftlUIIIIIIElt OF YALUU TO IIE llUO COL. 6-10 HIJJ1118flt OF VALUES to ee USED IN fOURifR

TIIAHSFOll:Jlt NUST IIE A ,0 .. 1::it o, 2 COi.. 11-20 TU41!STEP UTMEEN E•CH YAlOE. uec., CC\.. u-,o IDl!HTtPICATlO"f Po.: EAATHi:uue

2 OltO IIFU.ot . COL. 1•10 "1JlTIPLICATJmt ,lCTO~ flCII: ACC.. Ulltfi

USIO O~lY IF COi.. ll-20 ISO. sncHIEO NAlf. ACCEL!"!I.ArtON COl. u ... zo

COl. U•lO MAX. ,ilEQUENCY TO SE USED Ht Cn!J;\ 1.'.:I\ TUNSP:Cll:14 IC/SEC.I

l AND CONSEC. CiROS 18F1.0• 171 COl. l•ll t ACCELEll:11flON VILVES (Ol,. lJ-l1 (AllC NUN8fll:. CAR0S MUST IE IN SEQUEriCf

C t RUO CATA P:OR h!W SOIL PlOFILE c.uo 1115, ..... , C l

C COL. 1- 5 C COL, 6-10 t

SOIL ,•cFILf ~u~eEA, TOTAL hl.NBfll: Of SOtL TO I[ Rf:AO 11'fCWOII\C

),T!Of,AL L•~EA Ptl'C'lotUION Cf,111:QS CA~O FUA NALFs,.,, V,

°'

C

' C

-f C C C C

---~-< (

C C

--·-·-1 C

' ' C L C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C

COL. 11•15 JNl'll'IE~ Of- FIA.ST SU!Pt.-H,~0 SUl'll.11'fEA t.OL. U-'1 SOU. P-MOflLf lOENTif (CATION

l_UU:1 CCli!Sl!C. CARCS - SOil LAYO: INfCl.14ATION CUDS tNE C.,.AO f(lft EACH LAYEil INCLUOING HALFSPACE IJ15, 4Flo.o. F5.0)

COl. l• I LAVfR NL.M&Ell lATlRS NJST 81 IN SEQUENCE ~lfM SURFACE LAYER AS NUM.flflll: l

CDl. 10 $OIL 1"Pl: l - CLAY G • K•IUNOR. SHEAR STMENGTHI

-- J - SAND G • K•INEAN Eff. STA.ESSl .. l/2 ) - AOCK G • ll•tSHEAll ~OD. AT LOM STRAINI 4 • SANO '• K•INAX. EFF. STAESSJ••L/l

WNfll:E K I$ GlV!N THACVGH OPTION I COL• ll-U NJMBER OF SUBUYERS, THE LAYER ~ILL 8E

DIWIOtO INTO SU8LAYE1lS OF EQUAL THICKNESS llt.lXlNUN TOTAL MJNl'IER OF SU8LAYERS FOil ~HOLE HOfllE IS 19

Cot.. 16-Z'S THICOESS Of LAVER IFT.J cot.. a.-,, SHE,lR JtCOULUS lklPS/SQ.FT,I

COl. COL. COL •

, ... ,., ,.•-n .....

NOT IC'ECCESSARY U' SHEAR VEL, lS Gl'i'Eli OtlTIC.ll OAMPlP+G IUHO UNIT KEIGT~ IKIPS/CU8.FT.t SH!AR ~ELOCITY lfT.ISEC.I NOT NECCESSARY If SHEAR MOO. IS GIVE"!

COL. tt-15 •ACTCR fOR SHEAR MODULUS CU,Y - UNOA. SHEAR STRE"IGTH !KIPS/SQ.FT.I SAMO - fACTDR ON AVERAG! (URVE AOCA .. IIIA¥1 VELOCITY IN lOOOFT/SfC.

COL. 16-10 fAClOR #OR OAMPING

FOR ~ALFSPAC! ONLY CCL. l- 5, 26-15 Ol 16-0S, AND ~•-55 AA& NfCfSSARV

J SUILAVII -~ERE oeJECT NOTICh IS GllfEN CUD UUt

,Ol. I• ' SUUAYUI NUltltEII OUTCAOPPltfG ROCK OR OUTCftOPPING SOIL SUBLAYER

I - IASE ROCK OR SOIL SUBLAYER

COl. 10 TYPI Of l'!OtlOM O -

WITHIN Pfl.Oflll!

4 08TAlh SOIL PRCPEATIES COMPATIBLE WITH EFFECTIVE STRAINS l CARO 121S, 7FlQ.OI

CCIL. S SET EQ 1'0 l FOR P\JNCHEC OUTPUT ciF 'lei., SU 0, SOIL DATA c;.A~CS

till • .-10 IIUJMUM NU,.lliER OF ITEII.ATIONS COL. 11-zo MU flUIOft (It PRCNT • ITElUTION $TOPS WHE/'4

f~ROR IS LESS OIi wttE~ MAl. NUM8!R OF tTf•AtlCNS AR! REACHEO

tOL. Zl-JO RATJO lfFECTIVE STAAl't/lU.X. STl'tAJN

S CC~P~TE NE• NOTlONS l (ARO USUI

CtlL. 1 .. 1'

l CARO l11U1

ARAAY klTH NUJIIS(• Of SUBLAYEAS WHfRE MOTION IS TO IE CO~PUTEO. JIOJION IS COMPUTED AT TK! TOP Qfi fHE SUBUTt-R

C C

' C C C C t C C C C C C C C C C C C C C C C C C C C t C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C

C!JL.. 1-n AR~AY •Int 1"1'P! OF Al'IO~i: SUBlOEIII.S

, (ari:ocf,n1 CO\.. 1-1,

• PRthT Ahll/OR PLNCH l CARO UUI

CCL. -S

G • (lJTC~PPIHG SUt:ILl'fElt 1 - $UILAYER WITtll~ PROflLI

Aftft&Y WITH MOO! OF OUTPUT G • TAllE OF Mai. ACCEllAATIONS ONLY 1 - '1.JNCHED TINE HISTORY IN &OOtl'ION 10 TAil£

Tl114E HIST°'' ·o, oa·JECT "6TlON

0 - MAJ. ACCELERATION ONLY l - fU~CHED OUTPUT OF Tl~E HISTOAY Z - PU~CfO ANO PAINTED .OUTPUT OF Tlfl!E M1STOltY

T C~A~GE CBJECT PCTICN 1 0110 un, ·F10.01

Ctl. 1- S sue.LAYER NUMBEA THE AC,CELEAATICNS AT THE TOP OF THE 1iV'lll&YER Ult.I. U SET AS NE\e 06-JECT MOTICH SIT• t If OAlGJNAt. OSJECT MOT(ON IS a:E.PT

COL. 6-10 TYPE OF l'IOTJON O - CUtclltOPPING SU9UTU, l - MlTHIN P~OF(LE

CCI.. ll-20 JIULTIPLICATION FACTOR fQI l'lOTICN COL. 21-30 Nfllt TIMESTEP tt,c.,

I atat aELATlCN lfTWEEN SOIL PROPERTIES ANO STRAIN l (ARO nu, HO.DI

C<ll.. l- S llftJ,itBU OF CIFFlfl;E.NT SOIL OR ROCA TffU TO 8£ REAO 11.u. 4

to&.. 10 SU IQ. JO l FOR PLOT CF PROPERTY SRI.IN 11.(lAflCN

COi.. ll-1'5 NUPIEIII CF VALUES IN EACH LCGlO POR PLCTTl~G Of PRO,ER1Y/STRAIN MELATIDtll SN Sflllll,OG PLCT

cc1. .. u-z, SUL! Fca '4.0TTING IMAX. OAOINATE VC..IJU

UltT fCLLC1111 TWO sns OF CAR.CS FOllt EACH SO(l/111.CClt rY t FIRST SH - Rt;LATICN STMltt-SHUR JtOOULUS 1 (AIIIO 115,F,.O, lUCJI

COL, 1- S ~UJP9ER OF STRAIN VALUES FOR Wt!IC.11 ~AA MODULUS PAAA~ETEAS ARE Cl~EN. MAi.zt

C.<Jl.., 6-10 JftJLTIPlltATION FACTOR FOR StiEA!I: MOD;A.US USED FCR PLOfflkG ONlY

CCL. ll-76 nne: FCA SOIL TYPE

Z CARO AhO CCNSCC~TIVE CAIIIOS ftFl0,01 STRAIN VALUES. I VALUES PER CAkO lPfACENTI

CCNSK;UTIVE CARDS tlFl0.01 VALUlS OF MODULUS PAMAMETEIII C0RR£SP~~OlNC ~a THE 1TRA(N YALUlS g(Y£N, I ~ALUES P(R CARO.

SECClitll SET - AELAflON STR~ltt-OAMPING CU.CS AS FOR '(AST SET BUT lilTH VALUES 01' ClllrU.,u .. DA~PING RATIO lNSTfAC OF ~OOULUS ,,aaMf.TfM VALUf.5

t RE!PCNSE: SPECU't..N

IJ1 ._..

t t t t t C C C

' C t C C C t C C C C C C C C C C C C C C C t t C C t t t C t C t t C C t C C C t C C C C C C C C C C C C

cuo 115) CCL. l- '

CCL. 10

l CUIO 141151 CCL, 1 CCL. 10 CCL. U

COL. ZO

COL. 25

SUeLAV[ll NUJ!t&'elt SET • 0 IF O!JECT TYP( Of SUIUYEII.

Jtl'JTI ON 0 • C:UTCIIIOPPING l - WITHI~

TOTAL ~li't6EA OF DAMPING VALUES ~AX.• S£T !Q. TO l FOR PUNCHEO OUTPUT swJTC~ 0 - PLOT ANO/OR PU~CH VEL. SPECTRUM

l - PLOT AN0/0111 PUNCH ACC. $PECTRUl!t 2 - PLCT ANO/OJI. PUNC"' ACC. ~ll vn .. SPCCTRA

SWITCH fOR PLOTTINO 0 - STORE SPECTRA FOA LATEII PLOTTING

N.U, ' 1 - PLOT AlL SPECTAA COfllPUTEO ANO

STORED ~INCE LAST PLOTTING SWlTCH R)R PERIODS TO Bf useo. F(IIIST PERIOD IS .OCl SEC. 0 - '9 STEPS FROl'I ,l TO l.

' !YEPS fROl'I lo TO l, " HEPS Fil~ z. TO 4.

1 -11 STEPS FRCot ,l TO l. 10 STl!PS fRCIII lo TO z.

I STl!fl'S FROM 2, TO 4, Z -31 STEPS fllGM .05 TO lo

lO HEPS fROllt l. lO z. )0 STEPS FA.at lo TO 5 ..

) -LCGjRfTMIC INCREME~TS 10 STEPS IN fACH LOGo ON(T Flt.01'4 ,l TO 5o

4 '"'LOGllllf"!IC lNCltEl'IE"ITS - -2, !TfflS Jlril EACH t..OG. U"IIT FltO" .OJ TO 10.

011:0 UflO .. Ot VALUES CF CMITICIL tAMP1"4G 11.lTlO lltU. 6 VM..UES

10 t~C•fJS! Tl•tSTfP l COO I 151

COL. 1- !t FACTOlt FOR INCREASl"IG T1"EST!:P MUST 8( A POWEil OF l

ll cec•tASE Tl"ES1EP l (Alt:) 1151

CIJl. 1- S FACTCR FOR DECREASING Tl"1ESTEP JltUSf 8£ A POWER OF 2

12 'LCT FOUllElt s,(CTllU" Of OBJECT "OTION l CUD IJUJ

COL. ' S•ITCH FOIi; PLOTTING 0 - 5TOI.E SPECfll.U/11 FOIi. LA.TEA. PLOTTllltG

!'IA•. 2 SPECTRA PLOTTEO TOGETHER t - PlOT All. SPEC.TIU ST0'1.!:0 SINCE

UST PlOTTING COL. &-10 ~uMeE• o, Tl~Es SPE<TRU"' IS TO ee SW1~THE"4EO

CJL. ll-l5 NUll!l(lt Cf FREQUENCIES TO 8E PLOTU:O MU. 20\9

lJ PLCT fCUltllR ~PfCTRA Cf CC"'PUTEC MOTION I CARO 151SI

COL. I• ' SU~L•YER "'U~8ER FOR FIRST CO•Pureo Pl'JTION

C • • I a • < C C • $ • C C

• C • ,--c • C •

a~ • • C C • C • • • C C C C C C C < C C < C C C C • t C C C C C C C C C C t C t ,.

till. .. TYP! OF ABOVE SUBLAYER 0 - OJTCltOP

COL • .. SWITCH FOIi. PlOTTl~G 1 - Ml THIN

I-'! STORE SPfCTII.U>t FOR LATER PLOTTf~ "1U. z SPe<:TU PLCTlEO ro,eTH.EM.

1 - Pl.OT Al.\. SPECTRA STOREO SINCE UST •LOfTJ'1G co,. u-zo fllt.J,0:ISl:Jt CF TIMES SPECTRA .U;f TO 6E- SMOOTHE/iEO ..... 21-zs N&JI06U. CF FREQUENCIES TO BE PLOTTED 14.U .. 2~9

J d110 nn,

H ,LcT

•

AS FfAST CAAO FOA S!CCHO NOTION. 8LAMK IF CMLY ON6 MOTIO"

TIIIIE- HI STCRY ~ . .uo 1151

coi. 1- ,

toL. .-10

OF OIJ!C1 "OTlOM

STl!PS- Iii'" PLOTTING 1 - PLOT EACH VALUE Z • PLCT EYEAV SECONO VALUE ANO SO CN NUM8EA OF VALUE$ TO If PLOTTEO, NAX. JF ILA~K.WliOlE IIECOAO tS PLOTTED

2049

tt CCNP~TE A'""llflCATlO~ FU~TICN 1 CAAD 1,t,,F,.O, IA61

(.CL. l- 5 SU!LAYO NI.MBEA. P:llOM WHICH Al'PLIFICATIOR FACTORS IS COMPUTED

CCt.. 10 TYPE Of UOVE SUBLAYER 0 - CUTCll:iJP

COi.. 11-l!

cc1.. cm..

•• ••

CCL. 26-lO

CCL. Jl-ft

SUflAV!A NUMBER TO WHICH FACTCAS rs COMPUTED TYPE OF ABCYE SUBLAYER

1 .. WITl-l[N Al'Pl,IFfCUION

0 - CUTC•OP l - W(TH(lli

IF O - Pllf~T MAX A~P. FACfCR ANC FRfQUf"ICY STOAI! FU,,,CttON FOlt UTER Pl.JfUNG

. "AJ:lfltUM a CURVB CAN SE STO~ED 1 • ~LOT IJIIPLIFICATION FU,C:TION TOCETHER

iffTH ALL FU~TIONS STCRFO PREQUIEHC'f' SUPS IN CO~PUffNG AltPUFICATION FUHCTtCN. THE FIRST 200 fllfQ. ARE CD"'-PUTEO IDlHTIJJER

H CCIIPU1E STIUIH Cfl ST•ESS Tll'f l"'ISTCRY 1 CARO (41S,Fl0.0,5A6J

COL. 1- 5 HU118ER CF SU8lUER WI-If RE ltESll")hS~ '. !, f;J 8f

CO<•

CCL • COL. CCL• tOl.

co•PvTEO AT •lOOl£ OF LAYER lO RESPChSE TYPE

.. •• Z I-Z5

2fi-H

0 • STII.AIN 11ISTCIIY l - HRESS HISTORY

n, fQl..ll TO 1 If l'UP\CtH,C cu,,1.1r l'i W.l'ITfO SET fQU.ll TO 1 FOlt PLOTTfliG kU•!ER CF IIUUES ro 6E PLOTTEO, >to;. ZOft<J S.CALE FCR PlllTT!W~ - I.E. ll(U 'till. 'Jf r•otHATt If o. llJ!.GfST YALU£ ti SET •1 MU:. ~o- UL\Jl

CCL. Jl-65 IOENTlffCATlnN

.I c.a•o U CUD t ,:OJI HEW RESPIJNSf, ILAllll'i If OPH.Y CNE ltfSPONSf IS WANTED. l•U l iu;SPI

••••••••••••••••••••••••••••••••••• u, 0,

• • • • • ' • • • • • • • • • • • • • • 0

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59

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l PRl~T 10<>4 • Pltll'IIT 100~

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1000 l'QRIUTIJ3Hl PLUT )F A'l?UFICATIC'f SPECTRA //I 1001 FORH4T IZ31t ,..U,l14J'I ,\ICPLIFIC,t,TIOl\l • F6.ZI

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1000 f(li(!"ATlllH -- fkll; Ji( l'i l"i.:JRIEI. TllANSFOi<'4 0 IFEU•)

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82

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83

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84

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86

ACKNOWLEDGEMENTS

The development of the computer program SHAKE was conducted as part

nf a study of "Soil and Foundation Response Du,ing Earthquakes" sponsored

by the Nati~nal Science FoundatiJn. The authors are most grateful for

this support and for valuable suggestions from Professor W. N. Houston

i.n reviewing the manuscript.

87

88

References

Cooley, J. W. and Tukey, J. W. (1965) "An Algorithm for the Machine Calculation of Complex Fourier Series, 11 Mathematics of Computation, Vol. 19, No. 90, pp. 297-301.

Gutenberg, B. and Richter, C. F. (1956) "Earthquake Magnitude, Intensity, Energy and Acceleration (Second Paper)" Bulletin of the Seismological Society of America, Vol. 46, No. 2.

Hardin, B. 0. and Drnevich, V. P. (1970) "Shear Modulus and Damping in Soils: I. Measurements and Parameter Effects, II. Design Equations and Curves," Technical Reports UKY 27-70-CE 2 and 3, College of Engineering, University of Kentucky, Lexington, Kentucky.

Housner, G. W. and Jennings, P. C. (1964) "Generation of Artificial Earthquakes," Journal of Engineering Mechanics Division, ASCE, No. 90, February, pp. 113-150.

, Idriss, I. M. and Seed, H. B. (1968) "Seismic Response of Horizontal Soil Layers," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 94, No. SM4, July, pp. 1003-1031.

Kanai, K. (1951) "Relation Between the Nature of Surface Layer and the Amplitude of Earthquake Motions," Bulletin Tokyo Earthquake Research Institute.

Lysmer, J., Seed, H. B. and Schnabel, P. B. (1971) "Influence of Base-Rock Characteristics on Ground Response," Bulletin of the Seismological Society of America, Vol. 61, No. 5, October, pp. 1213-1232.

Matthiesen, R. B., Duke, C. M., Leeds, D. J. and Fraser, J. C. (1964) "Site Characteristics of Southern California Strong-Motion Earthquake Stations, Part Two:• Report No. 64-15, Dept. of Engineering, University of California, Loa Angeles, August.

Roesset, J. M. and Whitman, R. V. (1969) "Theoretical Background for Amplification Studies," Research Report No. R69-15, Soils Publication No. 231, Massachusetts Institute of Technology, Cambridge.

Schnabel, P. B. and Seed, H. B. (1972) "Accelerations in Rock for Earthquakes in the Western United States," Report No. EERC 72-2, University of California, Berkeley, July.

Schnabel, P. B., Seed, H. B., Lysmer, J. (1971) "Modification of Seismograph Records for the Effect of Local Soil Conditions," Report No. EERC 71-8, University of California, Berkeley, December.

Seed, H. B. and Idriss, I. M. (1970) "Soil Moduli and Damping Factors for Dynamic Response Analysis," Report No. EERC 70-10, University of California, Berkeley, December.

Seed, H. B., Idriss, I. M. and Kiefer, F. W. (1969) "Characteristics of Rock Motions During Earthquakes," Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 95, No. SMS, September.

SUPPLEMENT TO COMPUTER PROGRAM SHAKE*

by

T. Udaka and J. Lysmer

September 1973

Suggested corrections are shown framed on the attached segments of

subroutines SHAKIT, EARTHQ, CXSOIL, MOTION, STRT, UTPR, RESP, STRAIN,

REDUCE, FIT, RFIT and RFSN of program SHAKE.

The purpose of these changes are:

1. To decrease the execution time by up to 50% depending on the

type of problem to be solved.

2. To redefine the complex modulus from G* = G(l + 2ii3) to

c* = G(l - 213 2 + i 213~ ) •

(This change only influences subroutine CXSOIL.)

Input and output formats are unchanged by these corrections and

response values will differ only slightly from those in the published*

test example, see page 16.

*"SHAKE A Computer Program for Earthquake Response Analysis of Horizontally Layered Sites," by P. B. Schnabel, J. Lysmer, and H. B. Seed, Report No. EERC 72-12, Earthquake Engineering Research Center, University of California, Berkeley, December 1972.

S-2

C PR03RAM SHA-<E?(INPUT, our~JT, PUNCH)

C

C * " C

.. ., " .. ...... ...... tt * * * * * ~ ~ * * * * * * * * 0 * * O * O O O O * 0 0

0 D tt * 0 * * 0 * * 0 0 ~ 0 * * * * 0 0 ~ + * 0 0 0 0 0

C C C C C C C C C C C C C C C C C C C ,: C C C C C C C C C C C C C C C C C C

THIS PROGR:M CO~PUTES RESPO•ISE IN Q H0RI50NTA~LY LAYERED SEMIJ~FlNITE SYSTEM SJAJECTEU TJ VERTICAL•Y TRaVELLING SHEAR ~AVES THE METM0D IS BASED 0~ THE :O~TINOUS SOLUTION TO THE SHEAR ~AVE

PROGRAM"IED 3Y

PR03RA~ '/Er>Sl"JN

~ERB SCHNIHEL RESEARCH ASSISTANT JOHN LYSMtA aSSJC!ATE PROFESSOR

sH.:.«t.1 Sr1A>\t.?

GEOTECH~ICAL EN3INEERING U~IVERSITY JF CALIFOP~IA RER~E .. EY

JAl~UA:,Y 1972 4PR IL l ~72 J,ME 1972 CHA~G~ - JPTJON 7 CORRECTED

- su3RJUTI~ES STEPG AND FFT A~APTEJ TO IJNJVAC COMPUTFRS

DECE'l9E~ 1972 CH~~GE - SU~R~UTI~ES DRCTSP A~O STRAIN ::,1:_,- It. "It,:.-< I '<73 CHA~GE - SU3ROUTl~ES SHAKIToEARTHD,

c•sorL,'lOTIQN,STRT,STRAI~ RE)UCE,UTPR,RESP,FFT,RffT ANO '<FSN, EX~ruTION TIME REDUCED ~y UP TU SO CERCENT, COYP~EX SrlEAR ~OUULUS C~oNGEO FM'.)M G(J,+!<>2,*RETA) TU G(l,-~•*RETA**2•I*2,<>qETA*

SQRT(l,-RETA**2)l

C ~ .:~ * * 0 * 0 0 * * * * 0 * 0 * ~ ~ 0 * * * 0 0 * ~ ~ 0 0 ~ 0 0 ~ 0 0

0 * * 0 0 0 * 0 0 0 0 * * * 0 * * 0 0 O O * * 0 O O O O O O O O 0 C * * C C C C C C

N:F I rJ IT! U'JS

C C C

!'•"UT MOTI'.lr, JHJECT MOT I,l\i

::O~l-'UTEO ,aJT!ON

: YOTIJN MEAJ 1~ F~O~ CARDS = ~uTIJN USEJ A~ BASIS FOR COMPUTING NEW

~OTIJNS IN A SJlL PROFILE = YOTIJN COMPUTED ANYWHERE I~ A GIVEN SOIL

sor_ PRUF!_E FROM A GIVEN OBJECT MOTION

C C C <>

C C C

C

s-3

~UB~OLJT!NE SHAKJT(X,AX,AA,S,INV)

THIS QOJfl•1E CAL.LS THE DlFFEQENT SEJUENCES OF OPERATION,15 DIFFERENT OPEkATlDIS CAN 3E PERFOQMED 45 LISTED 9ELOW,

INTEGE"l TlTL.F,TP CDM"'LE/\ X, AX COMPLEX G• V, PLUS, Ml~US DIME"S10', A35IS (lO), AdSP=< rlOl ,AHSC:.. 1(18) DI ME •'JS I D i'J LL C 3 l , LT ( 3 ) • L NS w ( 3 ) 11I1•lE,N>;ILlN LLL(?.l, LLGS(?l•LLPCH(::>),i.LPLC2),SK(2l,LNV(2) ) l ME •\i SI O ,,, q 3 0 Q l , AX ( 3, 2 7 O l , 4 A ( 2, r:; 5 O ) , S ( 7 O l , INV ( 7 O l ,!"1EMS[D,, LL5(15l• LTs<1s1, LPS(1Sl•!,.P(3) UlMENSIO~ 1DAMP(9,lli ii I :,1 ::· hi S l CH·1 M "1 'v\ ( 3 ) ~-.-. _) c.Jf\v.(h h".J7 '-'"0Ln,MA2,TITL.E(5),')T, Mt,., MMA• DF,M)( cOtHP\i /,O[LAI rn1H(6),9L(20l,"'-(2Ql,FACT(201,H(20),R(20l,AF(20) COMV,OJ ;<;~!LR/ •AC(20), WL(i?OJ, T"12o>, DEPTH(20), fiEIG1iT(20) (UM~O~ /SJILC/ MS0Ii.,MWL CUM•G, 1,surLI 5(20), V(201, PLJS(20), "1I\JUS(20) CUM~O'J ;r~G/ IUc9,ll~, T'.2049) cb,·1~6·,;F ,·:,H 1 ;,c,n, ~ t.Ro I iJAT4 ldL~NK /brl / OAT~ (A8~IS1ll•I=l•l0i/6~ TI~E ,6HI\J SEC t6HONDS t7*6H I

7J DO 74 I = l ,'1•01.D 74 X(J) : X(I)<>X,

!JEW : 111! 73 I'll : Nf:,W

P'il \JT Iuno, ( l'J,'t..l) (",

DO 77 Il:J,MF~LO AX(!,IIl:X(II)

77 co,~TI'JUf 7h

l.) T = f) T \J ~ l'J

UF : 1,/("IA<>()T) Gv T(l l nJ

C C

C C

l ,) PRI\JT 1010,KK READ 1 on n, IF'< C~L'- -< E. [) ! , 2 l I I F >, , ~ , A X , c. L )

' ( 1 ) = ~ ~ "1"'1'21=n ,11.,Ji\1(31:n CALL FF T I X , '1 M'-1 , Ii'J V , 5 , 0 , I FE 'I "I I

. , l

" " " " .. " " * * 1 l PRI"T 11~1.><~

REA'.) 1000• IF R CAL ,~(C(!FR,X,AX,Lll

( l I = ¥ 'IMM(2l=r ;~MM ( 1l =0 CAL'- FFT X '1M'1, !NV s,n,IFERR) GO TO l (J l

.. * .. .. * .. * * "

14 PRI"T 14'4,KK REA) l006• IJSKIP, NN, NS• 1'JP i "if t 1

I F ( '" · J • LE , C l i·i \J = '1 M,:. /\JS < F' IF (,~\i,GT,2,14-,) rm : 2049 NN : w,.,,SKI" N : n 00 136 [:l• N\J, \JSK[P 'l:,+] T(Nl = FLOAT(I-1J<>UT

131'> C01;fl\lJE N : (' M = :'.hi/;;,

>JO 1.1 r I : 1 , .., N:''J•1 AA(\JP,N) : RfAL(X(!)l N = >f + 1 AA(NP,N) : AI..,AGp(I))

l,n CONfTJUc IF ( ;SK!D,Efl,1) 1.iO TO 135 ('~ = ') DU 134 I: 1,IJN ,,,S~IP \,:~J+1

AA(~P,N) : A~(,\JP,I) 01 ;d

DO 131 I = l , ::i i3J !O(NP,lJ : TITLE(!)

DO 132 I = b,ll ID(;jP,Il = HHT(I-51 IF ,,,51,,E],C;) rn,NP,J) : lBLA"K

132 CONTI JUE IF (,S.,,F'..l,Jl (.j'.) TU lll CAL c. ,, L Cl T (,,JP , N, T , "i\, ,13 5 IS, IO• 0 , , 2 , N > i\Jp • n. GO ro 1~1

s- 4

SUBROUTINE EAPTHQ(X,AX,AA,S,!NV) s- S' C c~~~**ft********=*~******~*********************************************** C C C C

Tf--'.S :;_,UTJ~E Cf A OS THE MOTION IN THE TIME OOMA;f'<, t.ODS TRAILING ZfcH-S, SCALFS THE VALUES, FJND 'IAX;\.\UM VALUE ANO VAR !CUS PARAMETERS ~' D TL ~i,SFE' THt 'l'.JTJON INTO THE FRFQUfNCY DOMAIN.

l'lTfGfc TITLE COMPLEX X, t,.X D1'1ENS!ON XR(B) D!ME"JSJ,•'\ XUOOl,AX(3,270l,AA(2,550},S(701,INV(70) C"M1"C\I /EQ/ MfJLD,MA2,TITLE!51,DT,_ "IA_t MMA, OF_. )•IX

.,_ ____ 5:1"MOl;/FRCUT/ NCUT,NlERC J C

C C C

PI2 = 6.28 PEAD 1001, NV, MA, OT, TITLE

30 X{!l = Xlll*XF x,•~x = XM*XF T ~nx = Fl C ~T (1,X"~X-1 l*DT c-r,r 7.014,X'4,T'l;..X,XF,X'IAX

X(l) = O.

C "f't 'JV': FO,t:QU 1:Y Cl ES A'l'.JVE FMAX A'JO FI NO "1A X. ACC. Of C

fCi'(;: Q.

~xx= o. Sf X = 0, IC\!T=O

,. rtJT=t-'CUT +l X(I)=).J

34 ('.'•P H.UF Xt = CWS(X(!l) SXX= S)(X + Xf.''XA

TCI 14

SFX = SFX + FFEO*XA*XA tX(l,ll = XIII

FREQ= FRfD + OF 33 cnNTINUE

SFX = SFX/SXX i r<<T='1Fr'LD-~'CUT r-.; Z f O n-NCLJT + 1 PFP 1T 2005,SFX T F ( F'-'f,.X.CT .F•t'Ol Rf TUR" L rt.LL ,-Fs"><X.i,Mx, P,v,s 1 IFF•C.1-21 [ t LL XMX( X, 1rt, XM, NXMAX) ·- -Dr' 72 I = 1, "IFOLD

7 2 X (II = AX ( l, I I PPl~'T 2001,XM,FMAX

s- 6

SUBROUTINE CXSOILINll C C*********************************************************************** C C C C C C C C C C C C C C C

Tl-'!S pnuT!NE CALCULATES THE COMPLEX SOIL PROPERTIES AND TRA"ISFER FUNCTI"NS FGR THF LAYFPS

1\11 PL GL R G V PLUS M !'JUS

= '!UMBER CF SOTL LAYERS = RATIO OF (QITICIL DAMPING = SHEAR MODULUS = DENS ITV = C'lMPLEX = CO '~P LEX = COMPLEX := COMPLEX

SHEAR 'IODULUS SHEAR WAVE VELCCITY TRANSFER FUNCTION TRANSFER FUNCT!ON

CODED 3V PE~ P SCHNABEL OCT 1971

(**************************~******************************************** C

C

COMPLEX G, V, PLUS, MINUS, MU C'.JMMON /S"1ILA/ I0NH6l ,BL!20l,Gll20l,FACT(20l,H{201,R(201, BF(ZOI CO'IMO~ /CSOIL/ G(20l, V(20l, PLUS(20l, MINUS(201

~· = Nl + l = N

G IM I! G= 2. *B LI I l *GL I I l *SQRT( l. -Bl ( I l *BL {Ill GR F ~ l = GL ( I ) * ( l • -2 • *Bl( I ) *Bl I I I ) G!Il=CMPLX!GPEAL,GIMAG)

--'ffl",.,..~ffl~~fflfflf"T------·-~ l CC''H!MUE

O.J 2 l = 1,11' l J = I + l MU= CSQRT(?(TI/RIJl*G(Il/G(Jll FLUS(!)= 11. + MlJl/2. MPIUS{U= (l. - MUl/2.

2 CO~'TI'WE F ETUR '~ E'-10

C C

SUBQOUTINE MOTION(Nl,IN,INT,LL,LT, X,AXI

C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C C C C C C C (

C C C C C C C C C

THIS PUUTI~E CILCULATES THE MOTION IN ANY TWO SOIL LAYERS OR IN ROCK FROM MOTION GIVEN IN ANY LAYER OR IN ROCK

! 'J I'H

LU I

LT ( )

X (l

t, X C )

= NUMBER OF SOIL LAYERS EXCLUDING ROCK = NUMBER _OF LAY.ER WHERE OBJECT MOTION IS GIV_EN = MJ TI ON TYPE

IF EQ O OUTCROPPING LAYER = NUMBER OF LAYERS WHERE OUTPUT MOTION IS WANTED

"!AX 3 LA YEPS = MOTION TYPE

O_ - OUTCROPPI.NG LAY.ER------------1 - LAYER W!TIN PROFILE

= OBJECT MOTION = JU TP UT MO TI ON

C CCDED. BY PER fl SCHNABEL ___ OC_T 1970 ________ . __ _ C " * C

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

! C

I NT f GER LL I 3 1, LT 13 l IN TEGEP TITLE COMPLEX AAC31 C0'-1PLEX _Xt AX_ COMPLEX G, V, PLUS, MINUS COMPLEX E, F, EE, FF, A, EX, AIN, IPI2 DIMENSION X(300l,AXl3,2701, S(70),INVl70) CJMMO~ /EQ/ MFOLD,MA2,TITLE(5l,DT, MA , MMA, OF,MX C'1"1P'ON /SOILA/ IDNTl6),BLl20l,GLl20l, FACT(201,Hl20l,R1201,BF(201 ;:QMMQN 1:sm L/ G(2Ql, V(20l, PJ.Jl.Sl2Q1_._MI"IUS_120_l ________ ·_·-· _ ---('1"1MON/FRCUT/ r~CUT,NZERO I

IPI2 = CMPLXIO., 6.281 DO 20 L = 1,3 !F (Ll(L) .GT. OJ AX(L,ll,. Xlll lf(NCUT.EQ.MFOLDl GO TO 20 00 30 I=NZERO,MFOLD AX (L, I l=CMPLX!O., O.l

30 CO"ITINUE 20 CONT!\IUE

FREQ - o. D 'l 1 9 I = 2, NC UT F : l. FF = 1. FREQ= F?EQ + OF A = FPEO*IPI2 DCJ 191 K = 1,Nl I F I K .NE. IN I GO -TO -192

---- ----------------·

SUBROUT!'ff ST~T( tT,Nl,DGMAX,PRllolUL,X,'-X,AA,SF,INVI C C

s- 8

C * • * * * • * • * * * * * * * * * * * * * * * * * * * * * * * * * * * C C C C

HiIS O('UT!'IE ciLCULt\TES STRJ\P4 TN THE "1IOOLE !:F EACH LAVER ANO F!NfJ NEW SCIL PROPERTIES COMPATIBLE k!TH THE STRAINS

COMMON /SOILOG/ 519,201, ASl9,20l, 8S(9,201, NVl91 :OM~ON /CSO!l/ G(20>t V(201, PLUS1201, MINUS(201 c __ .,,[.,"-· " ........ :c .. , ... J / .. E ... s .. c .. 1 .. 1 ... 1 ... t .... ·y..,C.,.lJa..T_. • ..,N .. l_.E_R,..R-· ....,l Dl·"F'IS!ON TMAX(20l, EMAX(201,STRl20) O!'~l=NS!J'J X( 681, AX(3, 641, AA12,128),SFl10l, IIW(lOI

C

C C C

r C

43

1

D [l 43 ! = 1, MF :JLD tA(l,Il = Rb'IL(X(lll AA 12,I) = A I 'IA G ( X ( II I or 1 I = 1,MFOLO ,x12,r1 = AX ( l, I J / 2. AX{3,I) = .i\X(2,Il PI2=6.283 IPI2=(~DLX(O.,P!2) GT = 32.2 0:1 2 K = 1,r-,~1 fCEQ = o. X(l)=O. FF= GT/1IP!2*V(K)l FF= H1KJ;2."'!PI2/V(Kl DO 20 I= __ NCUT 1 FREQ = FREQ + DF

EX= CEXPIFFEQ*EE) X(Il = (AX(2,!l*EX - AX(3,I1/EX)*FF/FREQ EX= FX*EX E = AX(2,Il''FX F = AX!3,!)/EX IX!2,I1= PLUS(Kl*E + M!NUS(Kl*F !X(3,T I= 0 L1 1S(K)*F + ''lMJSIKl*F

20 CGNTPIUF E'1AX(Kl - O. !FU1CUT.FQ.,VFnlo) GO TO 22 nr 122 ![=~1ZERJ,MFOLO X (I! l=C"PLX( O., O. I

122 CJNT!"UE >? Cf''IJTT\ll!IC:

O~TERMl~E MIX. STPAIN BY TNVEQTING FOURIER TRINSFOR"l OF STRAIN L'iTO THE TI,.E O'.'MAIN

CtLL RFSN(X,MX,'NV,SF,IFERR,-21 1 CALL XMX IX,MA, XMAX,NXMAXI

C C

SUBPOUTI 'IE UTPR IKK,DPTH,LS, K2,LH, LT,X,.&X,4A,S, P<l/1

S-9

(*********************~*********************************~****.********** C

C THIS CCJ'JT!\JE TRLNSFEDES THE VALUES IN AX(LH,) INT( lt,E TIME DCJMAYN C H' X( 1, PRINTS MW PUNCHES OUT THE RESULTS. C

C FREQ = O. S FX = 0. sxx = o.

C TCtNSFOR~ Vtl~ES IN XOR IN AX INTO THE T[MEDOMA!N O'l 24 I = 1,MF[)LD IF (LS.EJ.Ol GO TfJ 241 SAVE=X(!) X I I l = AX ( LS , T l AX(LS,IJ = SAVE

241 XA = CABS(X(I)) SXX= SXX + XA*XA SF X = SF X + XA *F of Q"' XlF PE Q =FREQ+ OF

24 CC1NTINUE

27

SFX = SFX/SXX

PUNCH 2009,( XR ( J) ,J=l, 81 t I !F (K2 .EQ. 21 PRINT 2019, (XD(J), J: 1,81, 1 NN = 4 + NN N = N + 4

ct OfFT(x 1 MXzINv,s 1!FEP 0 121 IF (LS .EQ.O l PE TUPN DCl 27 I= 1,MFOLD SlV<= = AXILS,! l AX(LS,Il = X(!) XI II = SA VE RE TU'<N

SUBF'OUTINE PESP ILL,L"l,LS,NN, X,AX,A, S, INVI C

C "'************ * * * * * * * * * * * * * * * * frp RESPONSE SPECTRUM ANALYS!S C TH!S P'l.OGP6-M READS Dl,TA

C Nf(ESSARY SUBP!lUTINFS DRCTSP, CMPMAX, PLOT

NN NO I< AV

KP L

= F ESP-'JS I: SPECTRUM NU'IBEP = PIUMRER OF DAMPING VALUES = SWITCH

EQ O ACCELERATIO"I SPECTRUM FQ 1 VFLOC I TY SPECTRUM EQ 2 ACC. AND VEL. SPECTRUM

= SWJ TCH

s- 10

******

C C C C C C C C C C C C C C C C C C

FQ l PLOT FfSPIWSE: SPECTPA ACCORDING TO KAV KP

X

ow NM T

= SWITCH FQ O ~IC PUN(HED rJUTDUf ~~ 0 PUNCHED OUTPUT ACCORDING TO KAV

= FOURIER TA~NSFCRM Of OBJECT MOTION = Ff'UR IEP TRANSFORM OF COMPUTED MOT! ONS = PEF<lf'O STEPS = NUMBER OF EACH STEP = PEQ.IQDS WHERE PESPONSE IS TO Bf COMPUTED

l O l Tl 11 = • 00 l C s""v E VALUES I~! X !N AA

C

l

D!l 11 I= 1,MF!llO l(l,Il = PEfl(X(lll A(2,Il = ~!MAG(X(lll IF !LS.fQ.Ol GO TIJ ll X ( ! l = AX ( LS, I )

l l C l"JNT! NUE

TFANSFQDM VALUES •~ X oq AX INTO THE TIME DOMAIN tELL oFsNI x,~zlN't.!S...!..l_FsflP,-21] D fJ 13 L = l , ND JF (NN.GE.51 NN= 0 \; l'l = 'IN + l D'l 131 I = 1,5

131 !D(NN,II = TITLEl!l fl" 132 T = 6,11 I fl ! Pi , I ) = T D'J T C I -5 I !F ILS.EQ.01 ID(NN,I I = ItlLANK

132 CONTINUE

s -11

SUBFOJTINE STRAIN( LL, LGS, LPCH, LPL,LNV,SK,X,AX,AA,Nl,S,INVI C C * C

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * • * * • * C C C C

C

C C

I

THIS SU'lROUTI NE CO'IPUTES STRAIN AND/OR STRESS TIME-HISTORY AT THE HP CF AN\' LAY F~ FOP t.CCELE<IA T!ON HISTORY KNOWN IN A.NY LAYER TWC RESPONSE hISTOR!ES ARE COMPUTED IN ONE RUN

PHEGFR. TrTLE,TP CGMPLEX X, /:.X COMPLEX G, V, PLUS, MINUS COl-<PLEX E,F,EE, A,AH,IPI2, AE,AF,EX,AI DIMENSION AB SI S(lOJ D!MENS!:'\I tE(2), AF(21 DIME:1,SlflN X<l/, AX(3tl.l, AA!2,ll,S(l/,INV(l) 0IMENS!GN LU21, LGS(2l, LPCH(2l, LPLl2l,SK(21,LNV(2) CC1'1"'10N /SC1ILA/ IDNT(6),BL{20l,GLl20l,FACT(20l,H(20),R{20),BFl20l CC"'1M0'i ;snILB/ FAf(20l, Wl(20), TP(20l, OEPTH(20l, WEIGHTl20l UlMMCl" /CSO!L/ G(20l, Vl201, PLUS(20l, MINUS120l C r MMON / F Q/ MF O LO, MA 2, T IT l E ( 5 1, OT, MA , MM A, D F, MX CC' M •'1f''l IC~ GI I D I Q , 111 , T( 2 04 9) C'.J'1MQ(>/FRCIJT/ NCUT,NZERO I IPI2: C'IPLX(0.,6.2831 GT : 32.2 AX(2,ll = O. AX ( 3, 1 l = 0. FREQ= O. __ DATA (A3SISlll, I=l,10)/6H TIME ,6HIN SEC, 8*6H / tI = GT/IPI2

STARTING AT THE SURFACE THE STRAIN JS COMPUTED SUCCESSIVELY OCWNWARD FfR EiCH FREQUENCY

DO 1 I=2,NCUT E = AX <l, I l/2.

0'1 2 l = 1,MFC1LD 2 f,X(l,ll: X(II

D'.' 3 L = 1, 2 'F (LLIU. EQ.01 GO TO 3

= IF(NCUT.EQ.~fnLD) G" TO 33 DO 34 I!=Nlf~G,MFOLO XI I I ) =CM PL XI O. , O. l

34 CO'JTI',UE , 3 C O'./T! 'IUE

C~Ll RfS'J(X 'IX HlV,S,!FEPR,-2) on 32 I =1,~FOLO t t I L , 2"' I - 11 =PE AL ( XI I I ) * l 00.

32 l!~(L,2*II "' AJMAG(X( II 1*100. 3 CCNT[NUE

C C

SUBROUTINE PEDUCE(IFR,X,AX,LLI

s- 12

C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C THIS ROUTINE INCREASES TIME INTERVAL ANO REDUCES NUMBER OF VALUES C C C C C C C C C C C C C C C C

! F'<

OT OF p, L

X AX

=

= = = = =

DIV!O!NG FACTOR ON LENGTH OF FRECORD MULTIPLICATION FACTOR ON TIMESTEP MUST BE A POWER OF 2. T I'1ESTEP IN SEC. FREQUENCY STEP INC/SEC. NUMBER OF POINTS USED IN FOURIER TRANSFORM FOURIER TRANSFOR"t OF OBJECT MOTIOI\! FOURIER TRANSFORM OF COMPUTED MOT!CNS

crOED BY PED B. SCHNABEL DEC. 1970. MGDIFIED SEPT. 1971

C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * C

C

Ir-.:TfGER TITLE CC!M"'O\i /E / MF'JLO MA2 TITLE{5l,OT, MA, MMA, DF,MX CcMMC\i /FQC UT/ \iCUT NZE_f:.f __ _ C 'JMPLEX X, AX DI'IENSFJN X( 68), AX(3, 641, lll3) Fl = .5/0T FR = FLOAT ( I FR I OT = DT*FP MA= "lA/lFR 1-'"lA = M"IA /!FR M A2 = 'IA + 2 MFOLD = "tA2/2 N = MFOLD + l D'.: 12 I = "IFOLD,N X { I I = O.

0012l=l,3 TF (LULI.LE.Ol GO TC 12 AXCL,Il = O.

l 2 C nrin 'fUE ~FOLD= "lFCLD + 1 F2 = .5/0T P'<H'T 1000,Fl,F2,DT, ,_,A F "l ti. = FLO A Tl MA I MX = (ALOG10(FMAI/ALOG1012.ll-l. lF CM~.LT.2"*1MX+lll 'IX= MX+l !fCNCUT.LE.MFOLDI GO TO 15 NC UT="IFOLD

15 (C,NT!'IUE FETUP'I

SUBPOUHNE FFT IA,M,INV,S,TFSET,IFEPP) OIMENSICN A(ll,!NV(ll,S(ll,Nl31,Ml3),NP(3l,Wl2),W212l,W3(21 EQUTVALF'\Ct (Nl Nill), (N2,N(2ll, (N3,Nl3l)

10 610

620

630

30

1000 40 C C

Ml=l'(ll M2="1(2) 1'3=MI 31 MTT=Ml-2 MT=MAX0(2, '1TT)

~ '4-..« T

IF (PRS(JFSET)-11 610,610,20 MT="AXO( '1( ll ,1"121,M(3l l-2 MT=MAXO ( 2, MT) IF (MT-201 630,630,620 IFEPR=l GO TO 600 IFERR=O

IFERP=l PR INT 1000 STOP Fl•RMA T( 31H !FEDR=O I' l=M( 11 !'2="12) M =M N l= 2*"'M l N2=2**'12 ,, 3=2** M3

--- ERPOR IN FOURIER TRANSFORM

IF ( IFSET I 50, 50, 70 50 NX=,ll*N2*N3

FN=NX OC 60 I=l,NX A!2*I-ll=A!2*1-ll/FN

60 A(2*I1=-Al2*I)/FN 70 NP(ll=Nl*2

NP ( 2 I = NP { ll * N2 ~· P ( 3 I= NP ( 2 l *N3 D'l 330 !0=1,3 Il,.NP( 3l-NP( ID) lll•IL+l

)

s-13

LI 21 =O L ( 3 )=O ~: TOT=2** M '< TOT 2= 2*'HOT F N=NF!T nr 10 I=2,MT~T2,2

10 t(ll=-A(I} DO 20 ! = 1, MT OT 2

20 A(Il=.tlll/H' Ct LL FFT ! A,L, !'JV, S, IFSET, I FERR l

C

s- 14

C ~rvE L\ST ~fLF OF A(JIS D~WN GNE SLOT AND ADD A(N) AT BCTTCM TO C GIVE Al<R.AV FOR AlPRIME AND A2PRIME CALCULATION r

L

l 'V ll,S(ll L(ll='-' L ( 2 l =O Ll3)=Q f TCT= .:'""'' !F--Sf t~ l "E' T 2= 'J L T +r. TD T r-.r -~,r,,12+2 t I '1'-'+2) =t ( N•:) t ('J'Hl )=t (Nr-1)

= .Ti;T N TO T3=~,T~ T 2+4 DC 70 !=3,NTOT2,2 t(Il=0.5" qT)

7 0 t I ! + 1 l ~ • 5 *i ( l + 11 D r 6 0 T = 1 , \JT ')T , 2 K 8=t: Tf' T 2+ 2-l A(KAJ= A(K8-2)

60 t.(K8+ll=~{K8-ll f; rr~'" T'.' T; z f T =' r•+ l flEL=3. 14l 5Q265/FN SS= S!'!(DELl SC= C'1S(DEU SJ=n. ( 0 = l .o DC 50 !=1,NT

s- 15

s- 16

COMPARISON BETWEEN ORIGINAL AND NEW RESULTS

The redefinition of the complex modulus G* from

G"' • G(l + Zill)

where ll is the fraction of critical damping to the improved value

c"' = G(l - 213 2 + Zill /1- 13 2 )

slightly changes the response values computed by program SHAKE. The

following table shows the influence on maximum accelerations through the

profile, see page 50 of the original report*.

Depth Original Max. Acc. New Max. Acc.

0.0 0.09377 0.09878 7. 0 0.09259 0.09758

20.0 0.05934 0.05942 30.0 0.05487 0.05540

42.0 0.05042 0.05037 62.0 0.04666 0.04667

80.0 0.03195 0.03140

100.0 0.02423 0.02364 120.0 0.01793 0.01763

all other differences observed were of similar or smaller relative

magnitude.

Documents

EARTHQUAKE ENGINEERING RESEARCH CENTER...EARTHQUAKE ENGINEERING RESEARCH CENTER SHAKE A COMPUTER PROGRAM FOR EARTHQUAKE RESPONSE ANALYSIS OF HORIZONTALLY LAYERED SITES by Per B. Schnabel