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Chapter 8: Generation of Floor Response Spectra and Multiple Support Excitation

G. R. Reddy & R. K. Verma

8.1 Introduction

Industrial structures support systems and components (SCs) at different elevations. These

systems and components are designed using Floor Time History (FTH) or Floor

Response Spectra (FRS). The FTH is obtained at various floor levels and at the locations

where SCs are supported, from structural analysis. The FRS is generated for damping of

the SCs using time history analysis, stochastic analysis or direct simplified methods.

8.2 FRS generation

Generally, time history methods are used for generating FRS from FTH because of its

simplicity and realistic. For conservative design of SCs, direct method which is simple

and less time consuming can be adopted.

8.2.1 Time History Analysis

The various steps involved in the time history analysis are given below:

1. Generate design basis ground motion called design basis time history.

2. Generate mathematical model of the structure. The model could be beam model

or 3D FE Model as shown in Fig. 1(a) and 1(b).

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Raft

Foundation

Outer

Containment

Inner

ContainmentTail

Pipe

Tower

Calandria

Fig. 1(a): Typical Reactor Building and 3D FE model

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3. Generate floor time histories from the structural analysis using design basis time

history.

4. Generate FRS using floor time histories. While generating FRS, the spectrum

ordinates shall be computed at sufficiently small frequency intervals to produce

accurate response spectra, including significant peaks normally expected at the

natural frequencies of the structure. One acceptable frequency intervals to

compute FRS is, at frequencies listed in Table 1 [1]. Fig. 2 shows the FRS at

various levels of a typical reactor building.

Table 1: Frequency steps for FRS generation

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Outer

Containment

Inner

Containment

Tail Pipe

Tower

Calandria

Fig. 1(b): Typical Reactor Building and Beam model

Frequency Range

(Hz)

Increment (Hz)

0.5-3.03.0-3.63.6-5.0

5.0-8.0

8.0-15.015.0-18.0

18.0-22.0

22.0-34.0

0.100.150.20

0.25

0.501.0

2.0

3.0

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8.2.2 Stochastic Analysis

The various steps involved in stochastic method are given below:

1. Generate design basis ground motion called design basis Power Spectral Density

Function (PSDF).

2. Generate mathematical model of the structure. The model could be beam modelor 3D FE Model.

3. Generate floor Power Spectral Density Function from the structural analysis using

design basis PSDF.

4. Generate FRS using floor PSDF. The frequency intervals shall be chosen as

explained above.

15

0.00 10.00 20.00 30.000.00

0.51

1.02

1.53

2.04

Floor Response Spectra at Bottom of

Calandria (Node 2) in Z - Direction

=1%

=2%

=5%

Sa

/g

Frequency (Hz)

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.000.00

2.00

4.00

6.00

8.00

10.00

12.00Floor Response Spectra at Steam Drum Level

EL. 123.00 m (Node 14) in Z - Direction

=1%

=2%

=5%

Sa

/g

Frequency (Hz)

0.00 10.00 20.00 30.000.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

Floor Response Spectra at Top of Calandria

EL. 95.00 m (Node 5) in Z - Direction

=1%

=2%=5%

Sa

/g

Frequency (Hz)

Fig. 2: Floor Response Spectra at various floor levels of atypical reactor building

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8.2.3 Simplified Analysis

The various steps involved in the simplified analysis are given below:

1. Generate design basis ground motion called design basis response spectrum as

shown in Fig. 3(b) for given structural damping.

2. Generate mathematical model of the structure as shown in Fig. 3(a). The model

could be beam model or 3D FE Model.

3. Generate FRS based on the procedure out lined below [2].

222

2222

),()},(){(

)()(4})(1{

1AABiBi

Bi

A

Bi

ABiA

Bi

A

EihShS

hh

S

+

++

=

2( )E i Ei

i

S U S=

SE- Floor response spectrum taking into account every evaluated mode of the

structure

Ui- The i-th mode excitation function value of the floor. It is the product of

modal participation factor and the floor mode shape in i-th mode

SEi- The maximum value of absolute acceleration response of the systems and

components under i-th mode acceleration of structure

hA- Damping factor of systems and components

TA-Natural period of systems and components

hBi- Damping factor of the structure

TBi- Natural period of the structure

S (TBi, hBi)- The standard design ground spectrum corresponding to TBi, hBi of the

structure

S (TA, hA)-The standard design ground spectrum corresponding to TA, hA of the

systems and components.

Notes-

(1) The mass mA of the systems and components needs to be sufficiently smaller than

the mass mBi of the structure.

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(2) The floor response spectra, obtained from the above method, need to be

broadened by at least 10% to account the uncertainty of frequency analysis of

SSCs.

Using the above procedure, FRS at the top of the building is generated and

compared with time history analysis and shown in Fig. 3(d). It can be seen that the

spectra generated using simplified method is conservative compared to the one generated

using TH analysis.

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Fig. 3(c): Time History compatible to thegiven spectra

Fig. 3(b): Typical DesignResponse Spectrum

Y

X

Fig. 3(a): Beam model of a

typical cantilever structure

Fig. 3(d): FRS at top of the building

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8.3 Analysis of Systems subjected to Multiple Support Excitation

Consider the following simple spring mass system, for this system equation of motion

can be written as follows.

Fig. 4: Spring mass system

[ ]{ } [ ]{ } [ ]{ } [ ] { }1gxMxKxCxM =++ 1

where M is mass matrix (lumped/consistent), C is damping co-efficient matrix and K isstiffness matrix. In the absence of external damping and neglecting the structural

damping, the free vibration of the system is expressed by

[ ]{ } [ ]{ } 0xKxM =+ or

{ } [ ] [ ]{ } 0xKMx -1 =+

Let

{ } { } { } { } { } 22 -t)sin(-xsoandt)sin(x ===

Free vibration characteristic of the system becomes

{ } [ ] [ ]{ } 0KM- -12 =+

[ ] [ ]{ } { }KM 2-1 =or

[ ] [ ] andA,asKMTaking 2-1 =

[ ]{ } { } A =

The above eigen value problem can be solved for eigen values called naturalfrequencies (n) of the system and the eigen vectors called mode shapes ( n) of the

system. The number of eigen values and eigen vectors will be equal to the number of

degrees of freedom of the system. However, the number of modes that are important for

response evaluation is fixed based on the procedure explained earlier.

{ } { }XxLet = 218

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{ } ntdisplacemedgeneralizetheisXandshapeetheisWhere mod

Substituting Eq. 2 into 1,

[ ]{ } [ ]{ } [ ]{ } [ ] { }1gxMXKXCXMgetwe =++ 3

Multiplying Eq. 3 with { }T

on both sides we get

{ } [ ]{ } { } [ ]{ } { } [ ]{ } { } [ ] { }1gTTTT

xMXKXCXM =++ 4

Considering excitations from ns supports [1,3,4], Eq. 4 can also be written as

{ } [ ]{ } { } [ ]{ } { } [ ]{ } { } [ ] { }sgT

ns

s

TTTUxMXKXCXM

1=

=++ 5

where Us is the influence vector. It has unity values along each degree of freedom for

uniform support excitation. For the case of multiple support excitations, it is the

displacement vector of the structural system when the support s undergoes a unit

displacement in the direction of motion of the support while the other supports remain

fixed.

Equation 5 can be solved using direct time history method. To solve using modal

superposition technique, Eq. 5 can be simplified as follows.

Using orthogonal properties of mode shapes, Eq. 5 can be written as

[ ] [ ] 21

1 2nsp

n n n n ns g sn

X X X x =

+ + =

&& & && 6

where ns is the participation factor for support s, mode n. The equation for evaluation of

residual rigid response due to the missing mass is changed to the following.

[ ] { } [ ] { } { }{ }1 1

nsp n

r ns si i g s i

K x M U x= =

= && 7

Case Study: 1

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Let us consider a piping running from the reactor to the steam drum as shown in Fig.

5(a). This piping called tail pipe will be subjected to the support motion at top of the

reactor and at the steam drum location. The spectra at two locations are shown in the

Figs. 6(a) and (b) respectively. The interaction between two piping can be neglected

because they are connected to large size components. For explaining the method of

analysis of systems subjected to multiple support excitations, the piping is idealized as

shown in Fig. 5(b). For simplicity, consider a two degree of freedom system as shown in

Fig. 7, similar to the system shown in Fig. 5(b), the frequencies and mode shapes of the

system are evaluated using the similar procedure explained in chapter 4 and are given

below.

Frequencies are 17.6 rad/sec (2.8 Hz) and 28.3 rad/sec (4.5 Hz) in first and second

mode respectively. The ortho-normalized mode shapes are given below.

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=

=

185.0

657.0

465.0

261.0

23

22

13

12

and

In the above vectors, the first suffix indicates mode number and the second suffix

indicates the node number.

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The combined stiffness and mass matrix for the full system can be written as

follows

[ ]

1000 1000

1000 1500 500

500 1500 1000

1000 1000

K

=

[ ]

0

2

4

0

M

=

Applying unit displacement at support 2 (Node 1) and at support 1 (node 4), the

influence vectors respectively can be obtained as follows.

{ } { }

=

=

4

3

4

1

4

1

4

3

12UU

The suffix of the influence vector indicates the support number. Now using Eqs. 5

and 6, the participation factors can be obtained as follows.

The participation factor for support 1 excitation in mode 1 is obtained as follows.

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Fig.7: Idealized model for the

piping along vertical direction

Support 2

Support 1

K3= 1000

K2= 500

K1= 1000

m1

=2

m2=4

Node 1

Node 2

Node 3

Node 4

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[ ] [ ] { }11 1 T

M U =

[ ] )382.2(526.1

4

34

1

4

2465.0261.011 =

= T

Similarly the participation factor for support 2 excitation in mode 1 is

[ ] [ ] { }12 2T

M U =

[ ] )382.2(8565.0

4

14

3

4

2465.0261.012 =

= T

The values in the bracket are the participation factors for uniform support excitation.

For evaluating the acceleration response, the modal acceleration at this frequency

can be obtained from the Fig. 6 as 2g and 1g at support 2 and support 1 respectively. The

damping of piping considered is 2%. Now the acceleration response at node 2 and 3 can

be obtained using Eq. 6 as follows.

1 1

nspn

j s ij si s

x X= =

= &&

Considering one mode, the response

)243.1(845.0261.028565.0261.01526.1..

2 ggggx =+=

)576.2(506.1465.028565.0465.01526.1..

3 ggggx =+=

The values in the bracket are evaluated considering uniform excitation which is

basically coming from support 2 and corresponding spectral acceleration is 2g. It is also

sometimes called envelope response spectrum analysis. It can be clearly seen that

envelope response spectrum analysis gives conservative results compared to the response

obtained in multiple support excitation analysis.

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Case Study: 2

In general, dynamic analysis of a piping system subjected to earthquake excitation is

performed using single-point response spectrum method. In a single-point response

spectrum method, one response spectrum curve is specified at all supports. This method

is acceptable as long as the piping system is single or multiple supported system,

subjected to uniform translational excitations at all its support points. This is true for a

piping system attached to one floor of a building or structure via some passive devices

such as anchors. In case of a piping system supported at different floor levels or in

different building structures, each support point (or group) would experience a different

excitation. The current practice is to assume a set of envelope response spectra which

encompasses all the input excitations. This practice gives very conservative results as we

can see in previous case. But in case of overhang in the piping systems or equipments,

envelope response spectra may underestimate the response of the system [5].

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Analysis of a piping system with overhang as shown in Fig. 8 has been

performed. Analysis has been performed for the two cases, one using envelope response

spectrum and other using multiple support excitations. The spectra at two support points

are shown in Fig. 9. Responses at various nodes of piping are given in Table 2.

Table 2: Responses (Displacement) at various nodes of the piping

Node Envelope ResponseSpectrum

Multiple SupportExcitations

41 9.37621 14.0785

40 8.31053 12.5642

39 7.24857 11.0525

38 6.19611 9.54808

37 5.16189 8.05796

It is clear from the Table 2 that Envelope Response Spectrum analysis may

significantly underestimate the response of piping systems or equipments with overhang.

References:

1. ASCE 4-98, Seismic Analysis of Safety-Related Nuclear Structures and

commentary, Published by the American Society of Civil Engineers, 1801

Alexander Bell Drive, Reston, Virginia 20191-4400.

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Fig. 8: Piping System

Fig. 9: Response Spectra

All DOFs are

fixed

TranslationalDOFs are fixed

Node 41

Node 40

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2. IAEA-TECDOC-1347, Consideration of external events in the design of nuclear

facilities other than nuclear power plants, with emphasis on earthquakes, March

2003.

3. C. W. Lin and E. Loceff, A new Approach to compute System Response withMultiple Support Response Spectra Input, Nuclear Engineering and Design, 60,

pp 345-352, 1980.

4. J. K. Biswas, Seismic Analysis of Equipment Supported at Multiple Levels,

Proceedings of ASME Pressure Vessel and Piping Conference, PVP-Vol.65,

Oriando, Florida, 1982.

5. A. Neelwarne, H. S. Kushwaha, A. Kakodker, Seismic Qualification of Nuclear

Equipment under Multiple Support Excitations, SMiRT 11 Transactions Vol. K,

August 1991.

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