Earthquake response analysis of reactor structures

Paper K1/1"

NUCLEAR ENGINEERING AND DESIGN 20 (1972) 303-322. NORTH-HOLLAND PUBLISHING COMPANY

EARTHQUAKE RESPONSE ANALYSIS OF REACTOR STRUCTURES

Berhn 20-24 September19TI

N.M. NEWMARK Department of Ctvll Engtneering, Untverstty of Ilhnots, Urbana, Illinois 61801, USA

Received 8 November 1971 The general nature of the principles upon which earthquake resistant design ISbased is descnbed with particular reference to components and elements of nuclear reactor facilities. Special attenUon is paid to the response and design criteria of items of equipment or of components that are mounted on or attached to responding elements, and basic procedures are developed to bound the dynamic response of such items. Consideration is given to vertical as well as horizontal excitation, and to the combination of the effects of the vanous excitations. Suitable approximations are developed for inelastic response estimates. One section of the paper is devoted to relative motions of points some distance apart, and to bounds for such relative motions. Recommendations are made for the general criteria govermng the design of nuclear facdlties, includmg the basic parameters governing response characteristics and energy absorption.

1. Introduction When a structure or a component piece of equipment or instrumentation is subjected to earthquake motions, its base or support tends to move with the ground on which it is supported or with the element on which it rests. Since this motion is relatively rapid, it causes stresses and deformations throughout the component considered. If this component is rigid, it moves with the motion of its base, and the dynamic forces acting on it are very nearly equal to those associated with the base accelerations. However, if the component is quite flexible, large relative motions or strains can be induced in the component because o f the differential motions between the masses of the component and its base. In order to survive the dynamic motions, the element must be strong enough as well as ductile enough to resist the forces and deformations imposed on it. The required strength and ductility are functions of stiffness or flexibility, among other things. Unfortunately, the earthquake hazard for which an element or component should be designed is subject to a high degree of uncertainty. In only a few areas of the world are there relatively long periods of observations of strong earthquake motions. The effects on a structure, component, or element, depend not only on

the earthquake motion to which it is subjected, but on the properties of the element itself. Among these properties, the most important are the energy absorption within it or at interfaces between the element and its support, either due to damping or inelastic behavaor, its period of vibration, and its strength or r~sistance. It is the purpose of this paper to describe the general nature of the principles upon which earthquakeresistant design is based, with particular reference to components of nuclear reactor facilities, and to consider the development of design spectra for the design of major parts of such facilities or components. Additional attention is paid to the response of items of equipment or instrumentation that are mounted on parts of responding elements or components themselves, where the response of the supported equipment or component is influenced to a major extent by the nature and type of the response of the primary element on which it :s supported. Much of the introductory part of this paper is taken from Newmark [1, 3, 7], with some additions and clarification based on more recent work.

304

N.M Newmark, Earthquake response ana(vsls of reactor structures

2. Loading and environmental criteria In determining the seismic motion for which a facility should be designed, one must proceed by correlating available records and observations with qualitative reports of the effects of earthquakes, and by making comparisons with observations in samdar geological regions. One must often make inferences from the geology and tectonic structure of the region to estimate the possible intensity of an earthquake which might occur in the future. In some regions, maximum or extreme earthquakes might not have occured in the historic past. In order to specify adequately the earthquake intensity for the design earthquake, one must do more than determine the maximum probable acceleration of the ground. The character of the earthquake motions must also be described in a way that is representatlve of the geologic conditions, taking into account the local soil conditions including overburden depths and characteristics, presence of water, depth to basement rock, and the like. A better measure of the freefield earthquake motions is a description which mcludes not only the maximum ground acceleration but also the maximum ground velocity and displacement, with some measure of the number of pulses or the duration of strong motions that should be considered. These quantities are dependent on the geologic and foundation material characteristics and on the interaction between the soil or rock and the structure supported on it. Tilting or tipping of the foundation materials under the structural foundation must often be considered.

3. Principles of seismic resistance The permissible level of response of a structure, element, or component, must be associated with the loading criteria. The response criteria should properly be dependent on the type of structure, the relative cost of repairs for minor damage, and the hazard in terms of possible loss of life should the item fail or reach extreme deformation hmits. The seismic resistance of an element is a function primarily of its natural frequency of vibration, its damping and energy absorption in the elastic range, and its ductility and energy absorption capacity an the range before failure Occurs.

Natural frequencies of vibration can be computed from the mass and stiffness &strlbutaons of the element or component, but are affected to a large degree by the Interaction with the foundations or with other supporting elements on which the component in question is based. The energy absorption m the structure from damplng depends on the nature of the structure itself, the type of joints or connections within it, and the level of stress or deformation which it undergoes during the dynamic motion. The damping involved is also a function of the mechanisms at the interfaces between the component and its supports or foundations. The Importance of damping is indicated by the fact that dynamic response of a structure in an earthquake may be affected to a greater degree by damping than by almost any other parameter. This is especially true in those instances where long-sustained nearly harmomc motions are involved. In the design of a structure, the designer can choose to resist the motions in various ways. He may elect to use a flexible, energy-absorbing structure which can comply with the base motions readily; or he can use a rigid structure to hmat the relative deformation withan the structure itself. In the former case, the stratus m the structure are determined primarily by the maxamum transient base displacement, and an the latter case they are determined primarily by the maximum transient base acceleration. In tl~e intermediate range of stiffness, the energy absorbing capacity within the structure is of the greatest importance, and involves both the strength and ductdity in a balanced fashion. A trade-off between ductility and strength as available, in general, for the intermediate range of stiffness, although only deformability is involved for very flexible elements, and only strength is involved for very rigid elements. The basis for these observations will become clearer in the summmary presented later in this paper concernlng the response of elastic and inelastic simple dynamic systems.

4. Special design considerations A number of points are often overlooked in the design of structures or components to resist dynamic motions. A summary of some of the more important

305

N.M. Newmark, Earthquake response analysis o f reactor structures

factors, but by no means a complete listing of all of them, is contained hereto. The motions due to an earthquake occur in both horizontal and vertical directions in a complex manner. It xs necessary to consider the interactions between the responses in the various directions, and especially important to consider the interaction between the vertical and the maximum horizontal response. VerUcal loads, and eccentricities of the vertical loads caused by horizontal displacements, must often be taken into account with especially heavy structures that carry large masses at or near the points which may deflect a great deal. Some of the resisting capacity for horizontal motions may be used up by the secondary effects of the eccentricities of the gravity loads. Quite often, the vertical motions may produce vertical stresses in the structure or element that exceed by a large amount those stresses due to the inertial forces corresponding to the vertical acceleration multiplied by the mass of the element. This is true when the frequencies of vxbration in the vert:cal direction of the element or component are in the range where major amplification of response can occur. One must also consider the combination of the stresses that arise from the horizontal and vertical excitations occurring more or less simultaneously. One of the factors that is commonly overlooked ~s the matter of relative motions between the parts or elements of a system having supports at different points, because the support motions may not occur simultaneously. Hence, there may be transient relative motions which produce strains in the structure, in addition to the strains produced by the dynamic effects of the overall motion. This is especially important in piping, electric wiring, or other elements connecting parts of a facility. Finally, there are a group of items which do not lend themselves readily to analytical consideration. These concern the details and material properties of the element or component, and the inspection and control of quality in the construction procedure. The details of connections of the structure to its support or foundations, as well as of the various elements or items within the structure or component, are of major importance. Failures often occur at the connections and joints because of inadequacy of these to carry the forces to which they are subjected under dynamic conditions. Inadequacy in properties of the materials

can often be encountered, leading to brittle fractures where sufficient energy can not be absorbed even though such energy absorption may have been counted on in the design. In order to insure that the intent of the designer is achieved, control of construction procedures and appropriate inspection practices are necessary. It is important that the practical aspects of seismic design be emphasized and that both designers and constructors be fully aware of their importance.

5. Elastic response to seismic loading A detailed description of the response of simple elastic systems, or more complex structures and elements, subjected to dynamic loading and especially to seismic loading, is given by Newmark [1 ]. In general, it can be shown as indicated in [ 1], that the response of a simple damped oscillator to a dynamic motion of its base can be represented graphically in a simple fashion by a logarithmic plot as shown m fig. 1. In this figure, there are shown on the plot, using four logarithmic scales, the following three quantities: D = maximum relative displacement between the mass of the oscillator and its base, V = maximum pseudo relatwe velocity = coD, A = maximum pseudo acceleration of the mass of the oscillator = w2D. In these relations, co is the circular natural frequency of the oscillator.

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The effective maximum ground motions for the earthquake disturbance for which fig. 1 is drawn are maximum ground displacement d m = 10 In, maximum ground velocity v m = 15 in per sec, maximum ground acceleration a m = 0.3 g, where g is the acceleration of gravity. The curve shown is a smooth curve rather than the actual jagged curve that one obtains from a precise calculation. The symbols 1, 2 and 3 on the curve represent oscillators, item 1 having a frequency of 20 cps, item 2 of 2 5 cps, and item 3 of 0.25 cps. It can be seen that for item l the maximum relative displacement is extremely small, but for item 3 it is quite large. On the other hand, the pseudo acceleration for Item 3 is relatively small compared with that for item 2. The pseudo relative velocities for Items 2 and 3 are substantlally larger than that for 1tern 1. The advantage of using the tripartite logarithmic plot, with frequency plotted also logarithmically, is that one curve can be drawn to represent the three quantmes D, V, and A. The pseudo relative velocity IS nearly the same as the maximum relative velocity for higher frequencies, but differs substantially for very low frequencies. It is, however, a measure of the energy absorbed in the spring. The maximum energy in the spring, neglecting that involved In the damper of the oscdlator, is ½ M V 2, where M is the mass of the oscillator.

The pseudo acceleration is practically the same as the maxunum acceleration, and the quantity M A IS precisely the maximum force in the spring Theretbre, the pseudo acceleration is exactly the same as the maximum acceleration when there Is no damping. In the discussion and figures which follow, we shall use the term "velocity" for V, and "acceleration" for A, without the explanatory words maximum, pseudo, relative or absolute. There are many strong motion earthquake records available. One of the strongest that has been measured is that for the E1Centro earthquake of May 18, 1840. The response spectra computed for that earthquake for several different amounts of damping are shown in fig 2.

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The oscillatory nature of the response spectra, especially for low amounts of damping, is typical of the nature of response spectra for earthquake motions in general. A replot of fig. 2 is given in fig. 3 in a dimensionless form where the scales are given in terms of the maximum ground motion components, in this figure, the ground displacement is given by the symbol y, and the subscript m designates a maximum value. Dots over the y indicate differentiation with respect to time. It can be seen from fig. 3 that for relatively low frequencies, below something of the order of about 0.05 cps, the maximum displacement response D is practically equal to the maximum ground displacement. For intermediate frequencies, however, greater than about 0.1 cps, up to about 0.3 cps, there is an amplified displacement reponse, with amplificatmn factors running up to about three or more for low values of the damping factor/L For high frequencies, over about 20 to 30 cps or so, the maximum acceleration is practically equal to the maximum ground acceleration. However, for frequencies below about 6 cps, ranging down to about 2 cps, there is nearly a constant amphfication of acceleration, with the higher amplification corresponding to the lower values of damping. In the intermediate range between about 0.3 to 2 cps, there is nearly a constant velocity response, with an amplification over the maximum ground velocity. The amplifications also are greater for the smaller values of the damping factor. The results shown in fig. 3 are typical for other inputs, either for other earthquake motions or for simple types of dynamic motion in general. The data from which fig. 3 was drawn, as well as other similar figures, are taken from Newmark [l ], and Veletsos and Newmark [2].

6. Inelastic response to seismic loading Let us now consider the situation m which the simple oscillator has a spring which can deform inelastically during the response. The simple resistance-displacement relationship for this spring is shown by the light line in fig. 4, where the yield point is indicated, with a curved relationship showing a rise to a maximum resistance and then a decay to a point of maximum

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usetul limit or failure at a displacement um. An effective elasto-plastic resistance curve is shown by the heavy line in the figure, rising on a straight line to a point where the yield displacement is Uy and the resistance ry, and then extending without appreciable increase in resistance to the maximum displacement u m. The effective resistance curve is d, awn so as to have the same area between the origin and Uy as the actual curve, and again the same area to the maximum displacement point. The ductility factor/a is defined as the ratio between the maximum permissible or useful displacement to the yield displacement, for the effective curve. It is convenient to use an elasto-plast~c resistancedisplacement relation because one can draw response spectra for such a relation in generally the same way as the spectra were drawn for elastic conditions in figs. 2 and 3. In fig. 5 there are shown acceleration spectra for elasto-plastic systems having 2% of critical damping for the E1 Centro 1940 earthquake. Here, the symbol Dy represents the elastic component of the response displacement, but is not the total displacement. Hence, the curves also give the elastic component of maximum displacement as well as the maximum acceleration, A, but they do not gwe the proper value of maximum velocity. This is designated by the use of the symbol V' for the pseudo velocity drawn in the figure. The figure is drawn for ductility factors ranging from 1 to 10. It is typical of other acceleration spectra for elasto-plastic systems, as indicated by the acceleration spectra shown in fig. 6 for the step displacement pulse sketched in the figure. Fig. 6 is drawn for a step displacement pulse corresponding to the two triangular pulses of acceleration

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changed to 15 cps, etc. The general nature of the similarity between figs. 5 and 6 is important. One can also draw a response spectrum for total displacement, as shown in fig. 7. This is drawn for the same condmons as fig. 5, and is obtained from fig. 5 by multiplying each curve's ordinates by the value of ductility factor/x shown on that curve. It can be seen that the maximum total displacement is virtually the

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0,5 2 I0 20 30 UndampedNaturalFrequency,f, cps Fig. 7. Total displacement response spectra, elastoplastic systems, two percent critical damping, E1Centro 1940 earthquake. same for all ductility factors, actually perhaps decreasing even slightly for the larger ductility factors in the low frequency region, for frequencies below about 2 cps. Moreover, it appears from fig. 5 that the maximum acceleration is very nearly the same for frequencies greater than about 20 or 30 cps for all ductility factors. In between, there is a transition. These remarks are applicable to the spectra for other earthquakes also. One can generalize about them in the following way for general nonlinear relations between resistance and displacement. For low frequencies, corresponding to something of the order of about 0.3 cps as an upper limit, displacements are preserved. As a matter of fact, the inelastic systems have perhaps even a smaller displacement than elastic systems. For frequencies between about 0.3 to about 2 cps, the displacements are very nearly the same for all ductility factors. For frequencies between about 2 up to about 6 cps, the best relationship appears to be to equate the energy m the various curves, or to say that energy is preserved, with a corresponding relationship between deflections and accelerations or forces. There is a transition region between 6 and 30 cps. Above 30 cps, the force or acceleration is nearly the same for all ductility ratios. For convenience, one might modify these relationships slightly, as discussed subsequently.

7. Design spectra and design criteria In the light of the preceding discussion, we can now develop a basis for design of structures,, elements, or components, where these are subjected directly to the ground or base motion for which we have maximum values of displacement, velocity, and acceleration. We first proceed with selection of values of damping. Table 1 is reproduced from Newmark [3] and Newmark and Hall [4], and gives the percentage of critical damping for various types and conditions of structures or elements, as a function of stress level. It represents the best information available at the present time, but certainly involves a great deal of judgment and interpretation. Amplification factors for the various ranges in the response spectrum were considered in [3] and [4]. The values determined therein for a number of earthquakes, with some smoothing and reduction of peaks to present a reasonably consistent prohability of failure or survival, are given in table 2. The amplification factors given in that table are used in connection with fig. 8a, as explained below. In general, for any given area or site, estimates might be made of the maximum ground acceleration, maximum ground velocity, and maximum ground displacement. The lines representing these values can be drawn

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Table l Damping values. Stress level

Type and condition of structure

Percentage of critacal damping

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on the tripartite logarithmic chart o f w h i c h fig. 8a is an example. The lines showing the ground m o t i o n m a x i m a m fig. 8a are drawn for a m a x i m u m ground acceleration o f 1.0 g, velocity o f 48 m / s e e and displacem e n t o f 36 m. These data represent m o t i o n s m o r e mtense than those o f any k n o w n or postulated earthquake. T h e y are, however, a p p r o x i m a t e l y in correct p r o p o r t i o n for a n u m b e r o f areas o f the world, where earthquakes occur either on firm ground, soft rock, or c o m p e t e n t sediments o f various kinds. F o r relatively

soft sediments, the velocities and displacements might require increases above the values corresponding to the given acceleration as scaled f r o m fig. 8a. However, it is n o t likely that m a x i m u m ground velocities in excess o f 3 to 4 ft/sec are obtainable under any circumstances. F o r each o f the a m o u n t s o f damping shown in the figure, or tabulated in table 2, the amplified displacements are shown on the left, the amplified velocities at the top, and the amplified accelerations in that part

N.M. Newmark, Earthquake response analysis of reactor structures

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same scale factor relative to the maximum ground acceleration compared with 1 g. The amplification factors given in table 2 and shown m fig. 8a differ somewhat, especially for low values of damping, from those gwen in [5] for a number of earthquakes. However, for damping values from 1.0 to about 5%, they are in reasonably good agreement with those given m [5]. The general shape of the spectra m fig. 8a is in reasonable agreement also with those computed in [5]. Table 2 Relatwe values of spectrum amplification factors.

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of the right-hand side of the figure for which the lines are parallel to the maximum ground acceleration line, but lie above it. We shall identify these portions of the line as the amplified displacement region, the amplified velocity region, and the amplified acceleration region, respectively. At a frequency of about 6 cps, the amplified acceleration region line intersects a line sloping down toward the maximum ground acceleration value, and intersecting that line at various frequencies, depending on the damping. The intersection is at a frequency of about 30 cps for 2% damping, and the other lines are parallel to the line for 2% damping. These lines are designated as the acceleration transition region of the spectra. Finally, beyond the intersection with the maximum ground acceleration line, the response spectrum continues with the maximum ground acceleration value for higher frequencies. The spectra so determined can be used as design spectra for elastic responses. The spectra are completely described when the maximum ground motion values are given for the three components of ground motion, and the damping is known. When only the maximum ground acceleration is given, the values used for maximum ground velocity and displacement are taken as proportional to those in the figure, or as scaled by the

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To use the spectra for inelastic behavior, the following suggestions are made. The amplified displacement region of the spectra, the left-hand side, remains unchanged for total displacement, and is dixnded by the ductility factor to obtain yield displacement or acceleration. The upper right-hand portion sloping down at 45 ° , or the amplified acceleration region of the spectrum, is relocated for an elasto-plastic resistance curve by choosing it at a level which corresponds to the same energy absorption for the elasto-plastic curve as for an elastic curve shown for the same period of vibration. The extreme right-hand portion of the spectrum, where the response is governed by the maximum ground acceleration, remains at the same acceleraUon level as for the elastic case, and therefore at a corresponding increased total displacement level. The amplified velocity portion of the spectrum then is obtained by drawing a straight line transition in the newly located elastoplastic spectrum, between the amplified displacement and amplified acceleration regions. The frequencies at the corners are kept at the same values as in the elastic spectrum. Similarly, the acceleration transition region

3l2

N.M. Newmark, Earthquake response analv~ts o f reactor structures

of the response spectrum IS now drawn also as straight line transition from the newly located amphfied acceleration line, using the same frequency points of intersection as in the elastic response spectrum. In all cases the "inelastic maximum acceleranon" spectrum and the "inelastic maximum displacement" spectrum differ by the factor/~ at the same frequencies. An earlier procedure for the defimtion of inelastic response spectra for design was presented by Newmark

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[1 ] in the form shown in fig 8b. In thin figure the displacement bound, the velocity bound, and the acceleration bound are determined, respectively, by keeping the displacement constant, the energy constant, and the force in the spring constant, and drawing the corresponding maximum response displacement limxts. However, In no case can the acceleration rise above the level corresponding to the velocity limit, and in no case can either the acceleration or the velocity rise above the level corresponding to the displacement bound. The revised procedure presented In this report was from Newmark [7]. A sketch of the resulting desxgn spectrum is shown in fig. 8c for 2% dampmg, for an elasto-plastic system with a ductility factor of 5. Both the maximum displacement and maximum acceleration bounds are shown, for comparison with the elastic response spectrum. The dashed lines DD, VV, and AA are bounds above which the spectra cannot rise, for the damping level assumed. A lower bound for AA is/aA, which corresponds to a lower bound for AA//~ of.4. In other words, the amplified inelastic maximum acceleratmn response cannot be less than the maximum effective ground acceleratmn. In using these design rules for nuclear power plant facilities, one should give attention to a number of special topics, some of which were outlined previously

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N.M. Newmark, Earthquake response analysts of reactor structures

in this paper, and others of which are discussed in some detail by Newmark and Hall [6]. Of course, the elasto-plastic response spectra can be used only as an approximation for multi-degree-of-freedom systems. In the development of a design spectrum one may choose to use an "effective" value of maximum ground acceleration rather than an actual value, particularly in cases where the higher spikes of acceleration are associated with very short durations and correspond to velocity changes much smaller than the maximum ground velocity, or where the duration of the earthquake motion is extremely short and the influence on failure or inelastic behavior is thereby lessened.

the author's colleague, Dr. M. Amin, for the elastic response spectra of several earthquakes. The results for the three components of motion for the E1 Centro, California, earthquake of May 18, 1940 and the Taft earthquake of July 21, 1952 are shown in figs. 9 and 10, respectively, for 2% of critical damping. There are several interesting features of these response spectra. For example, the frequencies of the spikes are not the same except m a few instances; the responses for the two horizontal directions show crossovers and significant differences in some ranges of frequency; and the vertical response is equal to or greater than the maximum horizontal response in the high frequency region, but is somewhat to a great deal less in the intermediate and low frequency regions. It is suggested that until further information becomes available the following design criteria be used: 1. The design spectrum for vertical response be considered equal to that for horizontal response for frequencies in the amplified acceleration range or higher frequencies. 2. The design spectrum for vertical response be considered equal to two-thirds that for horizontal response for frequencies in the amplified velocity or displacement ranges. These rules may be considered conservative for the E1 Centro 1940 earthquake, but they are definitely

8. Vertical and horizontal excitation Since the ground moves in all three directions in an earthquake, and even tilts and rotates, consideration of the combined effects of all these motions must be included in the design of important structures. When the responses in the various directions may be considered to be uncoupled, then consideration can be given separately to the various components of base motion, and individual response spectra can be determined for each component or direction of transient base displacement. Calculations have been made by

5o

% , - . ,.,,.v.- v

=\../.~_~/

/

Vert,col

"0.2,

!\./P',../

"

'~

. . . . .

,

'

00I

0.05

O, I

0.2

\

IN o~.o. A \ / \

~_

I/

\ I/~

/\~A",'~,/

I ^~v.. / ~

/~

o,"

I

7',,

x, Z

X~J

"k-"",z. --~

/

0,02

,.i

\/

"~ o-A"

Ns

- - -

",

7.,, ./'., ,'Cb 7,, I

,%/

313

o"

0,5 I 2 Frequency, cps

5

I0

?0

50

I00

Fig. 9. Elastic response spectra for three components of El Centro earthquake of 18 May 1940.

314

,V.M Newmark, Earthquake response

analysts o f reactor structures

\

/\

/~/

¢

!)5 A' 'j

o

O. / / ~ ,~% Domp,ng,\~ ~OO W o' o,

- - -

OI

002

,,9,

0,05

/.~xOo~//~\ o" \ / \ / ,./

,./

<,

..oo:

O,I

02

05

I

2

5

IO

20

50

I00

Frequency, cps

Fig. 10. Elastic response spectra for three components of Taft earthquake of 21 July 1952.

not overconservatlve for the Taft earthquake of 1952. Since the responses for motions m the various directions may not occur at the same time, it is considered reasonable to combine the effects of the several components of motion In a probabllistlc manner, by taking the maximum stress, deflection, or other specific response as the square root of the corresponding responses to the individual components of motion. The effects of transient tipping, ultmg, and rotation of the ground during an earthquake have not been studied extensively. An elementary treatment of some aspects of these movements has been given m section 7.7 of [8], and the effects of rotation of the ground about a vertical axis on the accidental torsxon in symmetrical buildings, for example, is given in sectmn 15.6 of the same reference When the responses of the structure or component are coupled, the analysis becomes much more complex and a three-dimensional [or at least two-dimensional] response analysis must be considered. However, data regarding the simultaneous input motions must be used in such an analysis, and little guidance is available on this topic.

9. Response of light equipment and attachments Various procedures have been suggested to slmptify the design of light eqmpment mounted on a respondmg structure subjected to earthquake or other dynamic motions. In one of the earhest procedures such as that given by Newmark [9], attention was called to the fact that the maximum response, even when the light equipment was tuned to a frequency of the system on which it was supported, could not exceed the square root of the ratio of the mass of the primary system to that of the secondary system. A more detailed study by Newmark, Walker, Veletsos and Mosborg [10] has been made. From that reference, figs. 11 and 12 are taken and presented herein. Fig. 11 shows the notation of a single-mass-spring primary system supporting a singlemass-spring secondary system. The frequency ratio of the secondary system to the primary system was consldered to have a range of values, and the mass ratio 7 had a wide range of values. The spring &stortlon ratio was computed for three response spectra, one a constant displacement bound, the second a constant velocity bound, and the third a constant acceleration bound,

315

N.M. Newmark, Earthquake response analysts of reactor structures

Primary System

Secondary System

vvvw~

Nvww~9

~'u I

~ ),=0,05

-I=," u 2

-

v/p

~,/

S = U2--LI I

PZ=K/M,

P2 = k / m ,

7" = m / M

Fig. 11. Notation, two-degree-of-freedomsystem.

~

5

0 05

OI

02

05 I 2 Frequency Roho, p/P

5

Fig. 12b. Spring distortion bounds, secondary system, step

velocity.

7=0'05

s.s_ D

il 0,1

0,2

3

!,

s

A/p 2 0,5

I 2 Frequency Retina, p/P

5

I0

2

Fig. 12a. Spring distortion bounds, secondary system, step displacement. 0 O,i

corresponding, respectively, to a step displacement, a step velocity, and a step acceleration pulse. The curves shown in figs. 12a, b and c are for these three conditions, respectively. The systems considered have no damping. However, it is seen that even for tuning of the secondary system to the same frequencies as that of the primary system, the maximum response is again x / 1 / 7 or the square root of the ratio of the primary mass to that of the secondary mass. The spring distortions for the two modes of the combine,t systems were considered to be added in numerical value, and consequently the curves represent an upper bound to the spring distortion of the secondary system. The upper bound to the response of the secondary system can then be obtained directly from the basic response spectrum for the primary system. The upper bound to the acceleration that can be used with the secondary mass is merely p2 times the spring distortion

0 2

0.5

I

2

5

lO

Frequency Ratio, piP

Fig. 12 c. Spring distortion bounds, secondary system, step acceleration.

bound of the secondary system, where p2 lS the circular frequency of the secondary system. Of course, if the secondary system has damping in it, the maximum response is limited also by the quantity 1/(2/3). Hence, as soon as the ratio of the masses becomes small enough so that the damping m the second ary system governs the maximum response, then a further decrease in the mass of the secondary system has no effect on the spring distortion bound.

10. Maximum response of "tuned" light equipment The studies reported m the preceding section are

316

N.M. N e w m a r k , f~arthquake response analysts o~ reactor structures

for very simple primary and secondary systems. A more recent study made by the author considers a much more complex primary system with a simple secondary system, as shown in fig. 13(a), for example. The primary system can be other than a linear spring mass arrangement, however. The only limitation involves a single spring and mass for the secondary system which is connected to one of the masses in the primary system. Fig. 13 (b) shows the way in which the frequencies in the combined system differ from those of the primary system. The heavy dots con show the frequencies of the nth mode of the primary system, and the cross X shows the frequency p of the secondary system. When the frequency of the secondary system falls between two of the frequencies of the primary system, then the frequencies of the combined system are shifted away from p as shown by the direction of the arrows in fig. 13 (b). When the frequency of the secondary system falls on one of the frequencies of the primary system, the frequencies of all of the modes of the combined system are shifted away, in the same fashion, but there are two frequencies at or near p such that one of them is shghtly above and one is slightly below p. Both are very close to p when the mass that IS added is small. We shall consider this situation only in the following discussion. The frequency of the secondary mass is given by eq. (I): p2 = k/m

(1)

.

The mode shapes for the nth mode of the primary system are given by Un" When these are normalized so k

m

) Secondory System

that the participation factor Is unity, according to eq. (2), 1 "M'6 c

17

n

, --

T

u ?/

"M'u

(2)

n

then one can define the normalized displacements of the nth mode by u n , where Un = C n ~ n .

(3)

From the fact that the participation factor is unity for the modal displacements u n , it follows that I'M

• u n = uT n . M . Un

(4)

It also follows from the orthogonality of the modal displacements u n and u i for mode n and mode i, that Ut7T "M ' u t = 0.

(5)

Because the added mass is very small, it has little if any influence on the primary system. Let us consider, that the modal shapes of the combined system are the same as those for the primary system, with the addition of the modal deflection of the added secondary mass, which is represented by the symbol ~o. If the displacement o f the mass of the primary system to which the secondary system is attached is represented by Un', then the value of~Pn is given by eq. (6): U ~

~on -----

n

(6)

2/p2 1 - co n

M3 "•:•::'%%"

O

~ Pomory S y s t e m /

........

o ) Ltghf SecondorySystem Added To Pr=mory System

However, for the added mode which has a frequency p, the displacement of the mass m in that mode is given by eq. (7) ~op ~-- 1 - Z n ~ O n .

0 •

~n' Frequency Of n'th Mode, Prlmory System

X

p, Frequency Of Secondary System

{ b ) Frequencies Of Combined System

Fig. 13. Simple secondary system with complex primary system.

(7)

This equation follows because of the fact that the algebraic sum of all of the normalized modal displacements is unity if the system is subjected to motion only at its base. It IS noted that eq. (6) gives an infinite value for ~0n when p = con. This obviously not the case, and therefore the situation needs further investigation when the secondary system is tuned to one of the

N.M. Newmark, Earthquake response analysis o f reactor structures

frequencies of the primary system. In this case, let us consider the added frequency p being very nearly the same as 9n, with ~0n and ---¢p representing the displacements of the added mass in the tuned nearly identical frequencies. Let us define also the displacements for mode n and mode p of the masses of the primary system as precisely one-half those of the displacements of the nth mode of the primary system. Then we obtain the following results: ½ (1" M " u n) + m~Pn = 1 (UTn . M " u n) + m~on 2, (8) T (1 "M" U n ) - m ~ o p _--~1 [u , n "M" Un)+ m~op2 , (9)

These results are obtained by considering the combined system with the same general procedure as was used to obtain eq. (4). Let us now choose to make the modes n and p orthogonal to one another if we can do so. We shall try to do so by using eq. (10). m~On~Op = 1

(uTn" M ' u n )

"

(10)

We shall also choose to make the algebraic sum Ofgn and 9p unity in order that we have the proper relationship for the participation factors. This Is exemplified by eq. (11). ~on - %

= 1.

(11)

Then eq. (8) can be put into the form: ~on 2 _ ~on = J / 4 m ,

(12)

J

(13)

where =

uTn "M" u n .

Similarly, eq. (9) can be put into the form: ~Op2 + tpp = J / a m .

(14)

317

Now since the nth and pth modes have very nearly the same frequencies, they are additive directly, and we shall designate the sum of the absolute values of ~0n and ~Opby the symbol ~0. We can then obtain the relation: ~o =

l~Onl+ kOpl"~X/-J/m.

(17)

This equation is essentially the same as the relation described in the preceding section for a primary system having only a smgle mass and spring. Of course, if the primary system has only one mass and spring, then the normalized modal deflection u n is unity, and J becomes M. Calculations were made for several simple cases to verify the accuracy of eqs. (6), (7), and (17) for zero damping. These equations are quite accurate even for relatively large added masses. When the added mass is even of the same order of magnitude as the primary effective mass J, the relationship given by eq. (17) is within 10 to 20% of the results obtained by precise numerical analysis and eqs. (6) and (7) are in error by about the same amount. When the secondary system is a more complex system having a number of springs and masses arranged in an arbitrary fashion with several points of support on the primary system, then the treatment required becomes more complex. In this case, the modes of the secondary system must be defined for base motions at the points of support corresponding to the motions of the primary system in the mode for which the frequencies of the primary system and the secondary system are equal. Under these circumstances, the appropriate definition of the participation factor uses, instead of unity in eq. (2), the proper value of the acceleration vector to be applied to the secondary system to make it move without distortion with the same displacements of the points of support on the primary system as that of the primary system. Then one can derive eqs. (18) and (19), which have a reasonable form and are fairly simple to use.

The solution of eqs. (12) and (14) leads to the following:

¢ ~ X[~,

(18)

where 'Pn = ½ + ½ X / J I m + 1 ,

(15)

- g p =½ - ½ X / 7 / m + I .

(16)

T

] = otn "m • o~n .

(19)

318

N M. Newmark, Earthquake response analvsts o2 reactor struetures

In eq. (19) the quantity a n is the mode shape for the secondary system that has the same frequency as the nth mode of the primary system When the secondary system IS supported on only two points of support, the relationships required to obtain the normalized mode shapes for it are relatively simple since these involve substituting a strmght hne having the appropriate Intercepts, corresponding to the motions of the primary system at the two points of support, instead of a unit displacement everywhere. Where the number of supports of the secondary system IS greater than two, then there are relative stratus introduced m the secondary system owing to the displacements of the primary system, in addition to those caused by the dynamic responses of the secondary system interacting with the primary system. This problem is inherently more complicated, and for the time being, eqs. (18) and (19) should be considered merely as heuristic relationships that lead to a better understanding and a reasonable approximation to the behavior of a complex system. In all cases, however, one must take account of the possibilities that for multi-degree-of-freedom primary and secondary systems, more than one mode of each may have the same frequencies. This leads to even more than the usual amount of difficulty As an example of the results of calculations to verify the accuracy of the foregoing relations, tables 3 and 4 were prepared for a main system of 2 masses to which a light third mass and spring were added. For table 3 the main system has circular natural frequencies of 1 and x/3-hz. The effective mass m mode l IS 3.0 and In mode 2, 1.0 When the added mass is 0.03 and is tuned to a frequency of 1 0, the responses, as Indicated in the table, are quite accurately given by the approximate relatmns, and the absolute sum of the modal strains for spring 3 IS about 105 for a uniform velocity spectrum of 10 in/sec, in accord with eq. (17). The excess over 100 comes from the third mode which is not considered in eq. (17), but can be computed from eq. (6). Table 4 IS for a system having real parameters and for a more typical response spectrum, similar to the right-hand part of fig. 8. The amplification for acceleration of the added mass is, of course, also just over 100 The amplification factors at resonance are affected both by the damping factor/3 and by the effective mass

ratio 7. Their effects are almost, but not quite, mteJchangeable. The net combined effect IS fairly well bounded by the relation amplification factor ~ -

1 -

2/3+v5

(20)

This relation is applicable directly when/3 is very small or very large compared with X/'Y. However, when /3 is of the same order of magnitude as x/y, a more accurate but less conservative value is l 4/3 + V/2/_

(21)

At resonance or near it, the effectwe value of damping/3 is very nearly the average of the values for the main system and the secondary system. This effective value, however, should be used only in eqs. (20) or (21), and not for the response of the main system, which is still governed by its own value of damping factor.

11. Relative m o t i o n s

It is often of Importance to have information about the relative motions of two points either close together or some moderate distance apart, in connection with the design of structures supported on or in soil or rock Information concerning relative motions, as determined from measurements, is often difficult to interpret or to assess. However, where it is fairly clear that a wave motion IS propagated in one direction without interference with other waves in other directions, and where the change in shape of the wave from point to point is relatively small, one can make inferences about the relatwe motions between nearby points In a fairly simple manner, as outlined here. For example, consider two points, point 1 and point 2, at a distance b apart, as shown in fig. 14. Consider a displacement O at point 1 and p plus an increment at point 2, as Indicated. Now let us consider a situation where a wave is propagated from point 1 towards point 2, with a displacement in the form given by a = f (x - c t ) ,

(2 2)

in which c is the velocity of tills particular wave propagatlon and t is the time. Then the various derivatives

319

N.M. Newmark, Earthquake response analysis o f reactor structures

Table 3 Parameters Masses or weights: 3 1 Spring consts: 6 1.5 Scale factor for weights, masses and springs = ****

0.03 0.03

Mode

Frequency

Circ. ~equency

Circ. ~equency squ~ed

1 2 3

0.1475451 0.1713524 0.2761886

0.9270535 1.076639 1.735344

0.8594282 1.159152 3.01142

Normahzed modal values Mode

Mass no.

Displacement

Strmn

Acceleration

1 1 1

1 2 3

0.2318585 0.760761 5.411904

0.2318585 0.5289025 4.651143

0.1992657 0.6538195 4.651143

2 2 2

1 2 3

0.2766675 0.7419382 -4.661827

0.2766675 0.4652707 -5.403765

0.3206996 0.860019 -5.403765

3 3 3

1 2 3

0.491474 -0.5026992 0.2499226

0.491474 -0.9941732 0.7526218

1.480035 -1.513839 0.7526218

Spectral parameters Spectral bounds: Displacement = 1000 in Velocity = 10 in/sec Acceleration = 100 g Response spectrum combinations Combination

Mass no.

Dlsplacement[ml

Strain [in]

Acceleration [g]

ABS sum

1 2 3

7.9029 17.9943 103.1175

7.9029 15.75568 104.6993

3.538065E-2 6.156534E-2 0.2712418

1

SRSS

2 3

4.569429 11.1058 72.69716

of the displacement p w i t h respect to x and t are given by the following relations developed by N e w m a r k [ 11 ].

x = f' (x - ct) ,

(23)

4.569429 9.167651 71.09927

a2= -c£'

0t

020- c2f"

0.0240574 3.567694E-2 0.184195

(25)

(x - ct)

(x

-

ct)

.

at 2

a2P - f " Ox 2

(x - c t ) ,

(24)

F r o m eqs. (23) and (25) one derives the following result:

(26)

320

N.M. Newmark, Earthquake response analysts oJ reactor structures

Table 4 Parameters Masses or weights. 3 1 Spring consts 5400 1350 Scale factor for weights, masses and springs = 10

0 03 27

Mode

Frequency

Circ. frequency

Circ. frequency squared

1 2 3

4.426354 5.140573 8.285659

27.81161 32.29917 52.06033

773.4854 1043.237 2710.278

lb sec2/m lb/ln

Normahzed modal values Mode

Mass no.

Displacement

Strmn

Acceleration

1 1 1

1 2 3

0.2318585 0.760761 5.411904

0.2318585 0.5289025 4.651143

179.3392 588.4375 4186,029

2 2 2

1 2 3

0 2766675 0.7419382 -4.661827

0.2766675 0.4652707 -5.403765

288.6297 774.0171 -4863.389

3 3 3

1 2 3

0.491474 -0.5026992 0.2499226

0.491474 -0.9941732 0.7526218

1332.031 -1362.455 677.3596

Spectral parameters Spectral bounds Displacement Velocity Acceleration Freq. at drop-off and at base Base acceleration value

= 12 m = 30 m/sec = 0.75 g = 6,24 Hz = 0.3 g Response spectrum combinations

Combination

Mass no.

Dlsplacement[m I

Strain [in]

Acceleration [g]

ABS sum

1 2 3

0.2120531 0.5402318 3.343894

0.2120531 0.4251735 3.31465

0.7219165 1.475324 7.728459

1

0.1256076 0.3548612 2 403559

0.1256076 0.2558937 2.29884

0.4350296 0.8697725 5.359989

SRSS

2 3

3p _ ax

1 op c 3t

(27)

strain e is o b t a i n e d f r o m eq. ( 2 7 ) , a n d the m a x i m u m strain at p o i n t 1 is t h e r e f o r e

and similarly, f r o m eqs. ( 2 4 ) and ( 2 6 ) o n e o b t a i n s o z j = 1, 32P 3x 2

(28)

c 2 3t 2

In the case w h e r e O is m the d i r e c t i o n o f x , t h e n the

e m = --Om/C ,

(29)

in w h i c h o m is the m a x l m u m v e l o c i t y at p o i n t 1. In the case w h e r e p is p e r p e n d i c u l a r t o x , e i t h e r h o r i z o n t a l l y or vertically, the m a x i m u m c u r v a t u r e at

N.M. Newmark, Earthquake response analysts of reactor structures

to it from that corresponding to the strains determined from the equations. Other relationships are of importance in the case where the motions are caused by more general disturbances than a wave of nearly constant shape transmitted in one direction. For example, it is apparent that the maximum change m the d:stance between points 1 and 2, ~b21, is related to the maximum displacements at points 2 and 1 in the following way.

,o + b 0.-~p

P

321

-2

Fig. 14. Relatwe displacements.

point 1 is obtained from eq. (28), and is as follows: 6b21 1> Umax, 2 - Umax,1 • curvature = am~C2 ,

(33)

(30)

where a m is the maximum acceleration at point 1. These relations, m eqs. (29) and (30), are often of use in determining the maximum strain that must be experienced by an element extending over some distance, or the maximum curvature in an element, as for example a tunnel, pipeline, or other structure. When the displacements in the region considered are those associated with horizontal shearing displacements occurring without longitudinal or extensional strain, then the displacement p is perpendicular to the wave front. For this case there is also an extensional deformation of an embedded element such as a cable or pipeline or tunnel, but the relations governing it are slightly different from eq. (29). Consider the case where 0 is normal to x in fig. 14, but the element considered makes an angle 0 with x. The maximum shearing distort:on 7 :n the element :s = ( o / c ) cos 2 0

In many instances, this relaUon may be trivial because the maximum displacements may be nearly equal, but since they do not occur at the same time, it is obvious that the maximum transient change in length must be greater than the difference m the maximum displacements. It is, however, true that the maximum change in length is less than the difference between the maximum displacement at either point 1 or point 2, less the minimum displacement, or the displacement in the opposite direction, at the other point. The minimum displacement would of course be zero, if the displacements do not reverse in direction. This relation is expressed as follows: 6b21 ~< [ Umax, l,2 - Umin,2,1 [ •

(34)

Similarly, the maximum change m length between points 2 and 1 must be less than the maximum strata anywhere along the line connecting the two points, multlphed by the length, as given in the following relation:

with maximum value 6b21 ~< emaxb. 7m = o m / c .

(31)

as in eq. (29). However, for the maximum longitudinal strain in the element e = (o/c) sin 0 cos 0 = ½ (v/c) sin20 and the m a x n a u m value is e m = Om[2C.

(35)

For the special case where the maximum strata is related to the maximum velocity by eq. (29), corresponding to a wave transmission situation, then one can derive from the preceding equation the following result: 6b21 ~< [(b/c) Omax,2,1 ] .

(32)

For either eq. (29) or (32), slippage of the soil against the element may reduce the force transmitted

(36)

For the special case where the deflection transverse to the line is given by an arc of a sine curve, as in the relation Y = Ym sin lrx/b (37)

N M Newmark, Earthquake response analysts o/ reactor structures

322

then the curvature IS obtained by the second derwatwe of the relation as follows curvature max

_ ~2y I = _(Tr2/b2)y m ~ Y m / C 2 ~x2 } ma~ (38)

F r o m this relation and eq. (30) one derives the following result.

Ym ~ - ( b 2 /¢r2c 2) a m

(39)

This Is apphcable, however, only for the special case considered.

References [ 1 ] N.M. Newmark, Current Trends in the Seismic Analysis and Design of High Rise Structures, Chapter 16, m Earthquake Engineering, Prentice-Hall, Inc., Englewood Chffs, N.J., (1970) 403. [2] A.S. Veletsos and N.M. Newmark, Design Procedures for Shock Isolatxon Systems of Underground Protectwe Structures, Vol III, Response Spectra of Single-Degreeof-Freedom Elastic and Inelastic Systems, Report RTD TDR-63-3096, Vol. III, Contract AF 29(601)-4565, Newmark, Hansen and Associates for Air Force Weapons Laboratory, June 1964 [3] N.M. Newmark, Design Criteria for Nuclear Reactors Subjected to Earthquake Hazards, Proceedings of IAEA Panel on Aselsm~c Design and Testing of Nuclear Facthties, Japan Earthquake Engineering Promotion Society, Tokyo, (1969) 90.

[4] N.M. Newmark and W.J. Hall, Selsmxc Design Criteria for Nuclear Reactor Facilities, Proceedings Fourth World Conference on Earthquake Engineering, Santiago, Chile, II (1969) B4-37. [5 ] Freddy Garcla and Jos~ M. Roesset, influence of Damping on Response Spectra, Research Report R70-4, InterAmencan Program, Department of Cwfl Engineering, Massachusetts Institute of Technology, January 1970. [6] N.M. Newmark and W.J. Hall, Special Topics for Consideration m Design of Nuclear Power Plants Subjected to Seismic Motion, Proceedings IAEA Panel on Aselsmlc Design and Testing of Nuclear Facilities, Japan Earthquake Engineering Promotion Socxety, Tokyo, (1969) 114. [7] N.M. Newmark, Seismic Response of Reactor Facflaty Components, in' Seismic Analysis of Pressure Vessel and Pxpmg Components, D.H. Pai,' editor, ASME (1971). [8] N.M. Newmark and E Rosenblueth, Fundamentals of Earthquake Engineering, Prentice-Hall, Inc., Englewood Chffs, N.J. (1971). [9] N.M Newmark, Notes on Shock Isolation Concepts, m: Vibration m Cwfl Engineering, Butterworths, London (1966) 71. [10] N.M. Newmark, W.H Walker, A.S. Veletsos, and R.J. Mosborg, Design Procedures for Shock Isolation Systems of Underground Protective Structures, Vol. IV, Response of Two-Degree-of-Freedom Elastic and Inelastic Systems, Report RTD TDR-63-3096, Vol. IV, Contract AF 29(601) - 6253, Newmark, Hansen and Associates for Air Force Weapons Laboratory, December 1965. [ 11 ] N.M. Newmark, Problems m Wave Propagation m Soil and Rock, Proc. International Symposmm on Wave Propagation and Dynamic Properties of Earth Materials, Univ. of New Mexxco Press (1967) 7.