Spectral Estimation Method









 Equation B-2

where SA is the scaled spectra that accounts for foundation incoherence. In the above

equation, SA and are the estimated floor spectra that would result from the elastic, or unreduced UHS ground spectrum and the incoherence reduced ground spectrum, respectively.

The ratio within the brackets of Equation (B-2), may be identified as the reduction function associated with the incoherence reduction.

INni

eEni SAeIN_ni

B.3.2 Spectral Estimation Method

Instead of redoing the complete structural response analysis for the incoherence reduced UHS, a simple method of estimating the modal components of floor oscillator response, given a modal representation of the structure (Γ, φ) and the ground motion spectrum, SA_g{f,ζ , for the } damping value, ζ, used in the building analysis may be done. The first use of a simplified

method of direct (i.e., avoiding time-history analysis) floor spectrum generation was proposed by Biggs (Reference B2). This method, based on empirical amplification factors, used only the structure mode shape factors, φ_nij, and the participation factors, Γ , defined for mode _jr j, node n, response direction i, and input direction r. Vanmarke (Reference B3) showed that Bigg’s empirical amplification factors are similar to those as obtained from random vibration theory.

The amplification factors presented in these studies compared the response of an uncoupled Single-Degree-of-Freedom (SDOF) oscillator mounted on the structure to the same oscillator mounted on the ground. Subsequent work (Igusa and Der Kiureghian, Reference B4) has emphasized the random vibration formulation in the development of direct structure response generation methods, including the effect of equipment mass coupling. Traditional generation of floor spectra, using time-history methods, assume that the floor motion is uncoupled from the equipment response. This assumption also can result in response conservatism, however, the consideration of this effect is somewhat application specific (knowledge of the equipment mass

Development of In-Structure Response Spectra for Seismic Margin or Seismic PRA Evaluation by Scaling

and local floor mass is required), therefore, in this study the scaling procedure did not include mass coupling effects.

It should be noted that the accuracy of the spectral values obtained using an estimation method is not that important since only the ratio of the spectral estimates is used for scaling of the original spectra. The method used in this study for the estimation of floor spectra is based on the random vibration results (Crandall and Mark, Reference B5) for a cascaded set of SDOF systems

(i.e., uncoupled oscillators) with white noise base motion input. Direct generation computer codes could also be used and are the preferred choice. For purposes of illustration, the Crandal and Mack random vibrations model using existing eigensolution values are utilized. This

procedure can be carried out on a spreadsheet using the modes of interest. Modes that contribute little to response can be eliminated with very little loss in accuracy.

Basically, the floor response spectrum ordinate at each frequency, f_k, is the sum of the

contribution of each structure mode, j, at that frequency. Each mode contribution was considered as the response of two cascaded SDOF systems for which the first stage output was the structure modal acceleration response component Y_nj″ =Γ_jφnjSA_gj

for base motion input where SA is the ground response spectrum ordinate at f. The output of the first stage is then used as input to the second stage which is the response spectrum oscillator (on the structure) tuned to frequency f . The output of the second stage is the modal component of the floor response spectrum ordinate.

These relationships are illustrated in Figure B-5. In the following development, it will be understood that the modal response of the oscillator is associated with input direction r.

gj j

k

SAgj SA

fj ωj = 2πfj

f f fj

N

n

1

ωk = 2πfk

Ynj’’

fk

SA

SAnk = Σj(SAnkj)

Xnj’’

Xg’’

Ground Response Spectrum ζ^j

Floor Response Spectrum ζ^k

ZPA = |Xg’’|max

Zzpa’’ = |Yn’’|max

Znkj’’

Figure B-5

Response Spectra Relationships

The ground response spectrum is a plot of the spectral acceleration for a set of SDOF oscillators attached to the ground, with damping ζ as a function of oscillator frequency f, for a given ground motion characterized by an acceleration time-history, ″

Xg . If we denote ″

Xj as the absolute

Development of In-Structure Response Spectra for Seismic Margin or Seismic PRA Evaluation by Scaling

acceleration response of an ground mounted oscillator with frequency and damping f_j ζ , then _j the spectral acceleration is given by:

m

We assume that sufficient number of modes are included in the modal representation of the structure such that:

(

)

Σ

δ_m- _{j j}r _nj Equation B-4

where |ε| < 0.1 and where = 1, if the response at node n is in input direction r, and δ = 0, if the response direction is cross-axis to input direction r.

δm

Then the absolute acceleration response for structure node n, ″

Yn , may be obtained using modal

The floor response spectrum is a plot of the spectral acceleration for a set of SDOF oscillators attached to the structure at node n, with damping ζ_k, as a function of oscillator frequency, f, for a given floor motion characterized by an acceleration time-history, Y_n . If we denote the absolute acceleration response of the floor mounted oscillator with frequency as f_k Z , then the spectral acceleration is given by

= _nk″

Y , then each mode component may be

considered as an independent input to the floor oscillator, and the contribution of each mode component to the floor oscillator response can be considered as the response of two cascaded SDOF systems. Using the notation of Crandall and Mark (Reference B5), 1

..

X is the absolute acceleration response of the first stage with frequency and damping f_j and X

..

2 is the absolute acceleration response of the second stage frequency f_k and damping ζ . Now, given that the base input motion for the first stage is characterized as White Noise (WN), the

Root-Mean-Square (RMS) response of the first and second stage may be obtained from the WN results presented in Crandall & Mark for a two-SDOF cascade. Denoting the first stage response

Development of In-Structure Response Spectra for Seismic Margin or Seismic PRA Evaluation by Scaling

as X _{RM S} ..

1 , and the second stage response as X _{RM S} ..

2 , the functional relations presented by Crandall and Mark may be utilized to obtain an amplification factor which compares the uncoupled response of the floor oscillator to the structure response at the point of contact. We denote this amplification function as



where P and P are Peak Factors introduced by Vanmarke (Reference B3). Using

Equations (B-7), (B-8), and (B-9), the modal floor response component may be expressed in terms of the modal structure response as:

nk gj

Vanmarke showed that the peak factors, P, corresponding to a given exceedance level (such as 84%) may be considered, in general, as approximately constant for a damped oscillator over the frequency range 5-50 Hz. Vanmarke also showed that the ratio P was approximately 0.8.

We note that the order of this approximation is not relevant to the scaling procedure, since we are considering ratios of estimated

g k /P

Znk for the original and reduced ground motion spectra.

Thus, considering the input motion to be in direction, r, and the response at node, n, in direction, i, we use the notation

for the response component of the floor oscillator with frequency, , due to the structure mode with frequency, , and modal response factor,

fk

fj Γ_rjφ_nij.

Then, the total response may be estimated using the SRSS modal summation,

Development of In-Structure Response Spectra for Seismic Margin or Seismic PRA Evaluation by Scaling

where the summation is over all of the modes used in the modal analysis of the structure, and the SRSS spatial summation gives the final spectral acceleration estimate,

2

where the summation is over the three translational input directions.