Analogue computer techniques in a seismic design

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Analogue computer techniques in a seismic design

Bycroft, G. N.

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Ser

THl

N21r2

no. 137

c.

2 NATIONAL

RESEARCH

COUNCIL

C A N A D A

DIVISION O F B U I L D I N G R E S E A R C H

ANALOGUE COMPUTER TECHNIQUES IN ASElSMlC DESIGN

BY

G. N. BYCROFT

R E P R I N T E D F R O M P R O C E E D I N G S OF T H E S E C O N D W O R L D C O N F E R E N C E O N E A R T H Q U A K E E N G I N E E R I N G . VOL. 11. 1 9 6 0 , P. 669

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_{R E S E A R C H P A P E R N O . 137} OF THE

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DIVISION O F B U I L D I N G R E S E A R C H

N 4 T 1 0 h d r -:iAricr! C L ( ! ( , C , C O T T A W A S E P T E M B E R 1961 P R I C E 25 C E N T S /-

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T h i s publication i s being d i s t r i b u t e d by the D i v i s i o n

s f

Building R e s e a r c h of the National R e s e a r c h Council. I t

should not be r e p r o d u c e d in whole o r i n p a r t , without p e r m i s

-

s i s n of t h e o r i g i n a l p u b l i s h e r . T h e Division would be glad to

be of a s s i s t a n c e i n obtaining s u c h p e r m i s s i o n .

P u b l i c a t i o n s of the Division of Building R e s e a r c h m a y

be obtained by m a i l i n g the a p p r o p r i a t e r e m i t t a n c e , ( a B a n k ,

E x p r e s s , o r P o s t Office Maney O r d e r o r a cheque m a d e p a y -

able a t p a r i n Ottawa, to the R e c e i v e r G e n e r a l of C a n a d a ,

c r e d i t National R e s e a r c h Council) to the National R e s e a r c h

Council, Ottawa. S t a m p s a r e not acceptable.

A

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a b l e i n denominations of 5, 25 and 50 c e n t s , and m a y be o b -

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T h e s e

coupons m a y b e u s e d f o r the p u r c h a s e of all National R e s e a r c h

Council publications including specifications of the Canadian

Government Specifications B o a r d .

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Reprinted from Prgceedings of the Second W o r l d Conference

ANAL^^^^

o n Earthquake Engineering

ANALOGUE COMPUTER TECHNIQUES IN ASEISM.IC DESIGN

9 G. N. Bycroft

Aseismic design i s handicapped by a lack of statistical knowledge of earthquakes and the difficulty of analyzing the response of a complex frame to these i r r e g u l a r forcing functions. Structures subject to earthquake motion present a complex picture of linear and non-iinear vibration. An

explicit solution in t e r m s of the participation factors of the modes and spectrum of the earthquake i s useful when considering linear vibrations of the structure but the introduction of non-linearities such a s friction and plastic yielding make this approach untenable. The general complexity of both linear and non-linear c a s e s has led to the simulation of structures by various mechanical and electrical models. These models a r e highly idealized and, in general, simulate only the major motions of the structure such a s shearing, bending, and rocking of the structure on its base. Me- chanical modellhg(1) produces satisfactory results but suffers f r o m complexity and lack of versatility. Further, it i s difficult t o apply to a mechanical model, which i s necessarily not small, a prescribed forcing function a s complicated a s that of an earthquake. On the other hand this method has the advantage that the structure may be m o r e iaithfully modelled than by other means. Electrical models of various types(2,3,4,5,6) have been made and have the advantage that their parameters may be readily changed over l a r g e ranges and measurements recorded by convenient instruments.

(a) Earthquake Representation

The obvious and dire-ct method of earthquake representation is to reproduce the actual accelerograrns of a number of l a r g e earthquakes, for example those given by Housner e t al(4). The usual technique is t o scan the accelerograms with some f o r m of optical-photocell arxangement, thus obtaining an electrical equivalent of the original disturbance. This rcay be used a s a source of excitation for an electrical o r mechanical simulation of a building structure. As the relevant frequencies of the original disturbance a r e too low for suitable electrical operation i t i s necessary to decrease the time scale of the disturbance by a large factor and increase the characteristic frequencies of the simulated structure by a corresponding factor. De- pending on the system used, i t may be convenient to study a family of struc- t u r e s by varying either of these factors.

*

Research Officer, Division of Building Research, National Research Council, Ottawa, Canada

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G. N. B y c r o f t

Any one earthquake i s c h a r a c t e r i z e d by i t s p a r t i c u l a r shape and a l s o i t s " s i z e " and duration. Earthquakes of the s a m e energy content but dif- f e r e n t configurations may have completely different effects o n different s t r u c t u r e s . A s each earthquake b e a r s l i t t l e relation to i t s p r e d e c e s s o r s one i s f a c e d with the question of deciding how t o r e p r e s e n t f u t u r e e a r t h - quakes. If enough r e c o r d s of s t r o n g motion earthquakes w e r e available f o r each locality the situation could be given a sound s t a t i s t i c a l b a s i s . Unfortunately, t h i s i s not the c a s e . H o u s n e r ( 7 , 8 ) f i r s t conceived the i d e a of r e p r e s e n t i n g s t r o n g motion earthquakes by a random a s s e m b l a g e of a c c e l e r a t i o n i m p u l s e s . T h i s was a m o s t i m p o r t a n t s t e p i n t h e conception of the t r u e .nature of a l a r g e earthquake. But the complete description of a random p r o c e s s i s v e r y difficult, if not impossible, and r e q u i r e s f a r m o r e

s t a t i s t i c a l knowledge of s t r o n g motton earthquakes than i s available. If o n e r e g a r d s earthquake a c c e l o g r a m s of equal "size11 a s being finite b u r s t s of a

s t a t i o n a r y .gaussian random p r o c e s s then i t is n e c e s s a r y t o d e t e r m i n e t h e power s p e c t r a l density of s u c h a s o u r c e i n o r d e r t o make i t c o r r e s p o n d t o what is known of the r e s p o n s e s p e c t r a of typical earthquakes. In a r e c e n t paper(9) the author u s e d a n analogue computer t o construct the conventional damped velocity s p e c t r a of b u r s t s of e l e c t r i c a l noise having a power s p e c t r a l density that w a s constant o v e r t h e relevant frequency range, i. e. "band- l i m i t e d white noise". F i g u r e 1 shows the m e a n of the maximum 4 e l o c i t i e s o c c u r r i n g i n a damped o s c i l l a t o r during s u c c e s s i v e b u r s t s of a c c e l e r a t i o n of duration twenty-five seconds a n d s p e c t r a l density 0.78 ft2/ sec4/cycle, f o r v a r i o u s periods and damping coefficients

,%

.

A l s o shown a r e H o u s n e r l s ( l O ) a v e r a g e d earthquake s p e c t r a with t h e s c a l e chosen to c o r r e s p o n d t o that of E l C e n t r o 1940. The c o r r e s p o n d e n c e is reasonable. A s f a r a s m e a n maximum r e s p o n s e s a r e concerned b u r s t s of random noise a p p e a r to be a suitable r e p r e s e n t a t i o n of typical l a r g e earthquakes. On t h i s baais the l l s i z e l l of the earthquake i s determined by the s p e c t r a l density and duration. Thus s u c c e s s i v e b u r s t s of equal duration give equal earthquakes although they differ somewhat i n configuration a n d hence i n their effect on any p a r t i c u l a r s t r u c t u r e . T h e objective h e r e i s to provide a r e p r e s e n t a t i o n of a p a r t i c u l a r "size" of earthquake, viz E l C e n t r o 1940. T h e distribution of r e l a t i v e l ' s i z e s l l , i n any one locality, i s another question. Depending on t h e band width of the noise u s e d this r e p r e - sentation p e r m i t s higher peak a c c e l e r a t i o n s corresponding to higher frequencies than i s o b s e r v e d on a c c e l o g r a m s . Where the band w i d t k s h o u l d be l i m i t e d could only be d e t e r m i n e d by s e i s m o g r a p h instrumentation having a higher frequency r e s p o n s e together with a c o m p a r i s o n of corresponding s p e c t r a l a n a l y s e s .

A s !'white noise" g e n e r a t o r s a r e readily available t h i s r e p r e s e n t a t i o n is p a r t i c u l a r l y suitable f o r the excitation of e l e c t r i c a l m o d e l s .

(b) Simulation of S t r u c t u r e s by P a s s i v e Networks

T h e construction of e l e c t r i c a l c i r c u i t s obeying the s a m e laws a s t h o s e of a mechanical s y s t e m i s a technique u s e d often i n engineering analysis and

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Analogue Computer Techniques in Aseismic Design

also used on many occasions i n aseismic design of structures. Within c e r t a i n Limitations electric networks have reciprocals and there i s the possibility of two analogous networks for the one mechanical system. The mechanical equations a r e the summation of forces and these may have their analogy in the summation of voltages i n one c i r c u i t or the sumxnation of loop c u r r e n t s i n the reciprocal circuit. In the one, force i s represented by voltage and i n the other by currents.

As an example of the voltage-force analogy one may consider a damped single degree of freedom osciUator having a n idealized elasto-plastic stiffness characteristic. This i s a fundamental unit of a structure vibrating into the plastic region. The restoring force i s assumed proportional to displacement up t o the yield point and then stays constant until the velocity changes sign. The f o r c e displacement characteristic now d e c r e a s e s along a line parallel t o the initial linear portion until the r e v e r s e yield point i s reached.

Let

m = m a s s of the oscillator,

4

= the linear stiffness of the oscillator,

w

= angular frequency of the system, = viscous damping coefficient,

X

= displacement of the m a s s relative to the base,

%

= displacement a t which the system yields, =

f

= yield f o r c e ,

d

@)

= ground acceleration,

then the equation of motion of the oscillator i s

( 1 )

- -

F

sgn

2

,

(x-Y)

a

X o

where

y

i s the algebraic s u m of all yield displacements up to the t i m e i n question.

Consider the circuit shown i n Fig. 2. The condenser diode branch behaves a s a condenser a s long a s the applied voltage i s l e s s than t h e biasing voltage

E

.

If

a n attempt i s made t o r a i s e the voltage above the biasing voltage c u r r e n t flows through one o r other of the diodes. A loop analysis of the circuit gives,

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G . N . B y c r o f t

where

9

and

Q

a r e the charges corresponding to the loop c u r r e n t s

i

,

I

and LE i s the algebraic sum of the charges which have flowed in the diode

branch of the circuit up to the time considered. Equations (1) and (2) a r e analogous and if

L

i s made proportional to

m

,

R

t o

d

,

fi

to

A ,

t o

f

and Q to a( @)then the charge

2

will be proportional t o the displacement

X

.

The requirement that Q be proportional to

d@)

i s that the c u r r e n t input

I

be proportional to the velocity of the ground motion.

(c) Analog Computer Simulation

Another method of simulating the mechanical equations i s to assemble a circuit of integrators and s u m m e r s i n such a fashion that the voltage relations a r e analogous to the relations of some particular variable in the m e - chanical system. The advantage of an analogue computer i s that the neces- s a r y components a r e already organized in a fashion which enables any particular circuit t o be readily a s sembled. F u r t h e r , non-linear components, scanning devices, noise generators, and plotting facilities a r e generally provided. The main disadvantage i s that one i s limited, in general, t o one dimension in space. This technique will be illustrated by a r a t h e r s p c i a l problem in aseismic design.

(d) Nuclear P i l e Subject to Earthquake Motion

The design of nuclear power stations in seismic a r e a s introduces some rather special problems t o earthquake engineering. One of t h e s e problems concerns a graphite moderated nuclear pile subject to earthquake ground motion.

Basically, the situation i s that of a cylindrical stack of graphite blocks with control rods inserted vertically. The special feature i s that the r e l a - tive horizontal movements of the blocks during a n earthquake must not be permitted to become l a r g e enough to jam the control rods. It has been suggested that the blocks be restrained by a n encircling m e t a l cylinder. It i s of i n t e r e s t therefore to estimate the s i z e of these relative displacements for various thicknesses of this cylinder when subjected t o a l a r g e earthquake of thel'size"of E l Centro 1940.

As the height and diameter of the cylinder a r e approximately equal it i s assumed that only horizontal shear of the surrounding cylinder i s signifi- cant. Lengthening and shortening of the cylinder vertically resulting in bending i s neglected. It i s further assumed that the blocks a r e stiff enough horizontally to prevent any appreciable change in the c i r c u l a r shape of

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Analogue Computer Techniques in Aseismic Design

horizontal sections of the pile. Layers of the blocks now move together horizontally a s a single flat disk. The idealized mechanical s y s t e m i s thus a pile of c i r c u l a r disks resting on top of each other. Between any two disks there i s a friction f o r c e obeying the simple friction laws and a l s o a horizontal spring force proportional t o the relative displacements of these two disks. The top i s assumed unrestrained and the bottom of the cylinder i s rigidly fixed t o the ground. The c a s e where the top i s a l s o fixed is immediately deducible.

F o r convenience on the computer a system of five disks and springs was s e t up. The disks a r e of equal height and weight and the connecting points of the s h e a r springs a r e a t the mid-points of t h e i r heights. The spring stiffnesses a r e a l l

,k

except a t the b a s e where the stiffness i s

21. Lf

i s the coefficient of friction, i s the displacement of the h* d i s k relative t o the base and

b@)

i s the representative $'white" ground acceleration of

xo

seconds duration, then

-

=

w&)

-

a&

+&-%)A

"-9

-

5m7x

s s n

+ 4m3A

ssn(k-,$i),

(3)

f o r the bottom disk,

f o r the middle disks, and,

for the top disk.

F o r the lower harmonics,

,

a d ,

)

will always have s a m e sign and we may simplify Equation (4) to,

where

u 2 =

'/,

,

i - e . , the angular frequency of one of the m a s s e s on one of the springs.

These equations w e r e suitably scaled and simulated on the computer. Figure 3 shows the circuit for the

n+h

cell.

Y,

r e f e r s to the scaled computer v a r i a b l e corresponding t o

yn

.

The c i r c u i t i s shown i n the

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G . N. B y c r o f t

normal diagrammatical fashion where c i r c l e s a r e potentiometers, triangles a r e s u m m e r s , and double-based triangles a r e integrators. The relative displacements

(y,

-

Yn-,)

w e r e recorded simultaneously on a recorder. F o r each value of W and

,

ten runs of 25 seconds duration were made; the mean of the maximum relative displacements of each of these ten runs i s shown in Figs. 4,5. These figures show the actual relative displacement in inches of

(jf,,-%-,)

.

It must be remembered that

$&

is a t a point one-tenth of the height of the pile f r o m the base but that the other points differ by a distance one-fifth of the pile height. This i s because

Yfi

i s the displacement a t the mid-point of the disk. There will be many m o r e than five blocks in a n actual pile height but a simple interpolation of these r e s u l t s should give a reasonable figure. Figure 6

shows the l a r g e s t displacement of the top relative t o the base that could occur if a i l the blocks moved i n the same direction. This i s merely the summation of a l l the maxima of the relative displacements.

The r e s u l t s show the t r e n d s that a r e t o be expected. Naturally, increasing the stiffness of the system d e c r e a s e s the displacements a s does a higher coefficient of friction. In a few c a s e s one hundred runs were made in o r d e r to find the statistical distribution of the results. The d i s - persion was found to be comparatively s m a l l and in the many runs 'made the l a r g e s t of the maxima was never greater than twice the mean.

The value of

W

in t e r m s of the pile dimensions may be simply derived.

Let

f

= radius of the cylinder (ft) = height of the cylinder (ft)

f

= density of the graphite ( l b / c u it)

= modulus of rigidity of the cylinder ( l b / s q ft)

,f

= thickness of cylinder (it) It may be shown that for five c e l l s ,

.

The s h e a r s t r e s s

S

i n theicylinder i s given by

(10)

then,

A s

,f-

r a n g e s f r o m 0.01 to 0 . 2 5 , bJ r a n g e s approximately f r o m 50 to 250. B e c a u s e of the n a t u r e of t h e non-linearity i n t h i s example i t will be m o r e s e n s i t i v e to higher frequency components i n the n o i s e than would a l i n e a r situation. A s a wide band of n o i s e w a s u s e d the resulting d i s p l a c e - m e n t s m u s t b e qualified accordingly.

BIBLIOGRAPHY

( 1 ) Jacobsen, Lydik S. a n d R o b e r t S. A y r e . "Experimentally D e t e r m i n e d Dynamic S h e a r s i n a S i x t e e n - s t o r y Model", Bull. S e i s m . Soc. A m . , 28: 269-311, 1938.

-

(2) Housner, G. W. and G. D. McCann. "The Analysis of Strong Motion Earthquake R e c o r d s with the E l e c t r i c Analog Computer", Bull. S e i s m . Soc. A m . ,

2:

47-56 (1949).

(3) Murphy, M. J . , G. N. B y c r o f t and L . W. H a r r i s o n . " E l e c t r i c a l Analog f o r E a r t h q u a k e S h e a r S t r e s s e s i n a Multi-Story Building1', F i r s t World Conference on Earthquake Engineering, B e r k e l e y , California, J u n e 1956, P a p e r No. 9 .

( 4 ) H o u s n e r , G. W.

,

R. R. M a r t e l , a n d J. L . Alford. llSpectrum A n a l y s i s of Strong-Motion E a r t h q u a k e s " , Bull. S e i s m . Soc. A m . , 43, p. 97-119,

_--

1953.

( 5 ) M e r r i t t , R. G. and G. W. Housner

.

"Effect of Foundation Compliance on Earthquake S t r e s s e s i n Multistory Buildings", Bull. S e i s m . Soc. A m . ,

-

44: 551-569 (1954).

( 6 ) Bycroft, G.N., M. J. Murphy a n d K. J . Brown. " E l e c t r i c a l Analog f o r E a r t h q u a k e Yield Spectra", Engineering Mechanics Division P r o c . A. S. C. E.

,

85: 43-64 (1959).

(7) H o u s n e r , G. W. " C h a r a c t e r i s t i c s of S t r o n g Motion Earthquakes", Bull. S e i s m . Soc. A m . ,

37: 19-31 (1947).

(8) H o u s n e r , G. W. I 1 P r o p e r t i e s of Strong Motion Earthquakes", B u l l . S e i s m . Soc. A m . , 45, p. 197-218, 1955.

-

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G . N. B y c r o f t

(9) Bycroft, G. N. "White Noise R e p r e s e n t a t i o n of Earthquakes", t o b e published s h o r t l y in t h e Engineering Mechanics Division, P r o c . A. S.C. E.

(10) Housner, G. W. "Limit Design of S t r u c t u r e s " , First World Conference on E a r t h q u a k e Engineering, Berkeley, California, J u n e 1956.

(12)

Analogu: Computer Techniques in Aseismic Design

PERIOD T SECONDS

EARTHQUAKE AND NOISE SPECTRA FIGURE I

ELECTRICAL ANALOGUE O F YIELDING O S C I L L A T O R FIGURE 2

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G. N. Bycroft + Y n t ~

/1

- Y n + l

U

RECORDER .c (t) -Yn - -Yn-1 + Y n SWITCH - Y n - I

COMPUTER DIAGRAM FOR ANALOGUE CELL N FIGURE 3

RELATIVE DISPLACEMENTS FOR X = 0.2

FIGURE

4

(14)

RELATIVE DISPLACEMENTS FOR X = 0 - 4 FIGURE

5

DISPLACEMENT OF TOP RELATIVE TO THE BASE FIGURE 6