Structural Dynamic Magnification factors of Storey Accelerations

The values of ) ( 1 90 T S A a r th

are provided in Table 5-12, which are the dynamic magnification

ratios of maximum storey accelerations when compared with their structural spectral acceleration.

These values can be transformed to

PGA A_r90th

by multiplying them by the ratio of the suite’s median

spectral accelerations at Sa(T1=2s) and PGA,

PGA T S_a( ₁)

= 0.833. Table 5-12 manifests values of Ar90th

normalised by Sa(T1=2s) and PGA that were markedly high; generally > 5×Sa(T1), and > 4×PGA.

They exhibited maximum values at elastic behaviour and gradually decreased with damage-

induced structural softening up to LS (≈ 0.3g) performance level. Afterwards, they increased again as structural collapse was inevitable. These magnifying factors can be utilised in calculating the

90th percentile value of maximum storey acceleration by multiplying them by the structural

Table 5-13 lists dynamic magnification ratios of maximum storey accelerations for the far-

field study. Comparison of

PGA A_r90th

values show lower magnitudes for the near-field study up to LS

level. At CP and higher levels, near-field values markedly exceed those of the far-field study

following the trend of θr90th. This suggests that collapse onset occurs at the level/location of

maximum θr90th.

Table 5-16 presents median values of peak storey accelerations at different damage levels

for non-structural components, Kircher et al (1997A). These values incorporated factors for

appropriate fixation and anchorage. On comparing values of Table 5-16 with Ar90th in Table 5-

12; one can establish that at DBE (0.88g) a near extensive damage of acceleration-sensitive

contents and non-structural components is expected, which is better performance than that of the

far-field study. Likewise and at MCE and up to CP near-field structural performance is better

than that of the far-field. For levels higher than CP, near-field damage exponentially increases

with severe adverse consequences.

5.4.3 STOREYS’MAXIMUM RESPONSE PROFILES

Seeking completeness of depicting structural response, the results of storeys’ maxima are

presented for the IM levels of MCE and 3DBE. For the complete set of θ, δ and A figures, refer to Appendix A2 – Figures A2-1 to -15 for radial, X- and Y-axes plots of θ, δ and A for Tx/Ty =

{1.0, 1.5, 2.0, 2.5, 3.0} at Elastic behaviour, DBE, MCE and 3DBE. Likewise, Figures A2-16 to

-30 provide the IM normalised ones. As with Chapter 4 and on inspecting these plots of θ, δ and A, MCE was found to be a credible representative of behaviour IM levels of elastic, DBE and

MCE. Consequently, figures for MCE (response) and 3DBE (collapse onset) are only displayed

in this chapter.

Figure 5-10 displays the medians and 90th percentiles of θr/IM, θx /IM and θy/IM at IM =

Sa(T1=2s) = Sa(MCE). Figure 5-10 (a, c and e) show typical interstorey median responses for

upper mid-section where typically the bisymmetrical structure (Tx/Ty =1) exhibit maximum

values for radial and Y-directions. Figure 5-10 (c) depicts the responses of the weak X-axis

where they all nearly coincide as X-axis maintained its capacity properties for the set of

considered structures. Figure 5-10 (b and d) and (f) [for the bisymmetrical structure] display the

90th percentile code-maximum behaviour when the structure’s mid-section exhibit degrading

strength. Subsequently interstorey maxima shift upwards where stabilising normal forces are

somewhat lower, as seen when comparing Figures 5-11 (a) [median demands] and (b) [higher

90th percentile demands]. Nevertheless, Figure 5-10 (f) show that in the strong direction the

structure retained its undegraded character of maximum interstorey drift values at mid-height,

except for the more flexible bisymmetrical structure. In essence, maximum interstorey drifts

occur in the weak direction of the structure in its upper storeys that subsequently affect the radial

(vectorial) total. Consequently, interstorey drifts of the strong direction slightly shifts upwards

due to degradation of columns in the orthogonal (weak) axis but stay within the mid-height

section of the structure. In contrast with the far-field study, the imminent collapse, due to the

loss of lateral structural strength, occurs in the mid-height section (particularly about the mid-

height) of the considered structure except for the bisymmetrical case; as confirmed by Figures 5-

11 (b, d and f) at 3DBE (collapse onset) level. The figures suggest that the collapse is shown by

a large flat increase in a brittle manner. This can be ascribed to the local failure of columns due

to higher mode (vibration) effects that rapidly degrade the already degraded elements to a

“complete” collapse. On the other hand, the more flexible bisymmetrical structure follows the

far-field trend the onset of collapse starts in the upper storeys. Furthermore, Figure 5-11

reported markedly high standard deviation values, particularly in the mid-height storeys, thus

confirming the collapse is taking place in the upper storeys.

Figures 5-12 and 5-13 exhibit the medians and 90th percentile values of radial, X- and Y-

axes lateral storey drifts at MCE and 3DBE levels respectively. Both figures display the same

structures approached incipient collapse. Once again, the bisymmetrical structure expectedly

showed the highest magnitudes. Predominantly, the values of lateral storey drifts were, more or

less, linear with maxima occurring at roof level, except for the unisymmetrical structure (Tx/Ty

=3) at collapse onset. Thus, values of lateral drifts to each storey can be linearly assigned,

starting with zero at ground level and ending with the maximum at the roof level. This can be

achieved via implementation of equations (4.6) [of Chapter 4] and either of (5.4) or (5.5) to

evaluate the maximum lateral storey drift of the studied structure. In essence, the results suggest

marked resemblance between the near- and far-field response albeit of different magnitudes.

Figures 5-14 and 5-15 show the medians and 90th percentiles of radial, X- and Y-axes

storey accelerations at MCE and 3DBE levels respectively. For MCE or less, Figure 5-14 shows

that maximum values exist at the top level, while storey accelerations increase linearly from

level 1 up to about mid-height then almost maintain the same value for the remaining storeys up

to the storey below the roof. The median of this maintained value for radial storey acceleration

was 64% of the roof acceleration, 65% for X- (weak) and 64% for Y-accelerations (strong)

respectively, as listed in Table 5-17. On studying (conservatively) the curves and their data, it is

suggested to use the following:

Aroof = A90th = a Sabγbiaxial,Aε (5.6) Aiu = 0.69 Aroof (5.7) Ail = Aroof m i 69 . 0 94 . 0 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ (5.8) A0 = PGA (5.9)

where A0 is the ground storey acceleration, Ail is the lower storey accelerations up to the mid-height

of building, Aiu is the median value of storey accelerations for the remaining upper storeys except

roof, i is the storey number, m = integer(n/2) – 1 , n = number of building storeys, Aroof is the roof

acceleration. Again, structures with deep foundations or those that have multi-storey deep

unreliable. This resembles the behaviour reported in the far-field study albeit of lesser magnitudes.

On the other hand and when inspecting Figures 5-15 (b) and (d), storey accelerations show a very

large flat increase in the mid-height section of the considered structure following the trend of

interstorey drifts at collapse, except for the more flexible bisymmetrical structure. This is a stark

difference with the far-field pattern showing maximum accelerations only occur at the roof level

even at collapse.

Again, the reported overall median values of γ-percent, previously investigated in Chapter 3, for the 90th percentile code demands of θ, δ and A were generally above 35%. Moreover, individual Tx/Ty median values of γ-percent for θ, δ and A demands peaked to as high as 39.8%, thus

vindicating the suggested use of 40-percent rule in combining the bidirectional X and Y demands.

5.5 IMPLEMENTATION INTO DCFD FORMAT AND FRAGILITY CURVES

In keeping with Chapter 4, capacity and dispersions values for the far-field are compared

with the near-field results, except for βDR that are calculated from the near-field demand values

plus values of b and γbiaxial,D developed in this chapter. This decision is made to facilitate

appropriate comparison between near- and far-field demands without being affected by other

factors. Consequently value of βor is taken as in the far-field study βor = 0, also relying on the

previously-used assumption that the stronger components applied to (weak) X-axis represent the

fault-normal ones thus eliminating the need for βor. Values of βDR were calculated from values

of the median and 90th percentile demands and are listed in Table 5-14. When comparing Tables

5-14 and 5-15 (far-field), values of βDR of the near-field study are markedly less than those of the

far-field one, ascribed to relatively lower βEQ of the near-field.

Consequently and as in Chapter 4, values of the correction factors of demand (γ) and capacity (φ) are evaluated for θr, δr and Ar, and listed respectively in Tables 5-18, 5-19 and 5-20.

Accordingly, values factored demands (γD) and factored capacities (φC) are computed and used to calculate the confidence factor λ=γD / φC. Subsequently values of λ are used to compute the

Gaussian confidence variate KX = KX =½k βUT – ln(λ)/bβUT that is used to obtain the confidence

level of non-exceedance CL% = Φ (KX). These values are listed in Tables 5-18, 5-19 and 5-20.

On comparing values of CL% of the near- and far-field studies, the considered structures

under the near-field suite achieved higher values of CL% inferring higher confidence in

achieving the studied performance levels of IO, LS and CP, particularly in roof accelerations.

This finding can be attributed to the relatively lower magnitudes of θr, δr and Ar for the near-field

study when compared with the far-field study. This outcome can be also seen in Figure 5-9

where it displays lower θr values for the near-field study up to CP performance level.

Endeavouring completeness of this comparative study and as in Chapter 4, fragility

curves for the near-field study were generated and plotted (thick solid lines) in Figure 5-17

among the far-field fragilities (thin dashed lines) to facilitate comparison. Figure 5-17 show

(visually) less fragility values for the near-field study than those of the far-field. For example,

Figure 5-17 shows that the hazard intensity of Sa(T1) = 0.3g has 19% and 81% probability of

(failure) exceedance of expected θr demands respectively at LS and CP performance levels which

are significantly better than the fragility values of 28% and 87% at LS and CP for the far-field

study. In other words, this indicates that the structural performance under the near-field suite is

generally better than under the far-field suite.

5.6 CONCLUSIONS AND RECOMMENDATIONS

This chapter has investigated the response and behaviour of three-dimensional 2-second-

period 12-storey structures by means of an IDA method under the previously-developed near-

field earthquake suite. A comparative study was conducted and the drawn conclusions can be

summarised as follows:

1. Values of the log-log regression coefficients a and b for the near-field were not that

different from the far-field suite. This suggests universal applicability of the expression

2. There was a marked drop in values of the amplification factors of the biaxial interaction

for structure with Tx/Ty≤ 1.5 for the interstorey and storey drifts. Almost no change in

values of storey accelerations was recorded for these structures when compared with the

overall median of all considered structures.

3. Overall the structural response and demands were less than for this near-field study than

those of the far-field study up to collapse prevention (CP) performance level. This

reduction was particularly large in storey accelerations. Consequently, the dynamic

magnification factors for storey accelerations dropped significantly to 5×Sa(T1), and >

4×PGA, about half those of the far-field suite. However and upon exceeding the CP

level, the response and demands of the near-field study increase in a brittle explosive-like

fashion, particularly for interstorey drifts and storey accelerations.

4. As with the far-field study, the structure degrades around its mid-height section and

subsequently the degradation shifts slightly upwards. However, and with the exception

of the bisymmetrical structure, the considered structures exhibit a blow-out increase in

interstorey drifts in the degraded mid-height section that causes a sudden explosive-like

failure accompanied with extremely high magnitudes in storey accelerations.

5. Values of storey accelerations for the upper half of the considered building would result

in moderate damage to contents and non-structural components at DBE levels and

extensive damage is anticipated at MCE, which results in better performance than the far-

field response.

6. Collapse displayed by 90th percentile demands at 3DBE suggests high mode effects

failure that can be attributed to severe degrading vibrations of the structural members

with reduced strength in the mid-height section of the considered structures. One can

postulate that this damage starts at the onset of strength degradation markedly increases

when plastic mechanisms are formed, thus allowing for larger vibration amplitudes. This

explosive-like nature. Nevertheless, this higher mode induced damage only progresses to

soft-storey failure when the seismic intensity exceeds the MCE level for seismically-

designed structures.

5.7 REFERENCES

Benjamin, J. R., and Cornell, C. A. (1970). Probability, statistics, and decision for civil

engineers, 640p, McGraw-Hill, New York.

Dutta A., and Mander J. B. (1999). “Seismic fragility analysis of highway bridges.” Proceedings

of the Center-to-Center Project Workshop on Earthquake Engineering in Transportation

Systems, Tokyo, Japan.

Federal Emergency Management Agency (FEMA) (2000). “FEMA 350 – Recommended

Seismic Design Criteria for New Steel Moment-Frame Buildings.” Report No. FEMA-

350, Washington DC.

Federal Emergency Management Agency (FEMA) (2000). “FEMA 356 – Seismic rehabilitation

pre-standard.” Report No. FEMA-356, Washington DC.

Jalayer, F. and Cornell, C. A. (2003). A Technical Framework for Probability-Based Demand

and Capacity Factor Design (DCFD) Seismic Formats, PEER Report 2003/08, Pacific

Earthquake Engineering Research Center (PEER), University of California at Berkeley,

Nov 2003.

Kircher, C. A., Nassar, A. A., Kustu, O. and Holmes, W. T. (1997A). "Development of Building

Damage Functions for Earthquake Loss Estimation", Earthquake Spectra, Vol. 13, No. 4,

pp. 663-682, Earthquake Engineering Research Institute, Oakland, California.

Mander, J. B., and Basoz, N. (1999). “Seismic Fragility Theory for Highway Bridges,”

Optimizing Post-Earthquake Lifeline System Reliability, Proceedings of the Fifth U.S.

Conference on Lifeline Earthquake Engineering, Seattle WA, Technical Council on

McDonough), ASCE, pp 31-40.

Otani, S. (1974). “SAKE, A Computer Program for Inelastic Response of R/C Frames to

Earthquakes”, Report UILU-Eng-74-2029, Civil Engineering Studies, Univ. of Illinois at

Urbana-Champaign, Nov. 1974.

Ramamoorthy, S., Gardoni, P. and Bracci, J. (2006). “Probabilistic demand models and fragility

curves for reinforced concrete structures”, Journal of Structural Engineering, ASCE, Vol.

132, No. 10, pp. 1563–1572.

Shome, N., and Cornell, C. A. (1999). Probabilistic seismic demand analysis of nonlinear

structures, Report No. RMS-35, RMS Program, Stanford University, Stanford, CA.

http://www.stanford.edu/group/rms/Thesis/NileshShome.pdf. (Last time accessed: 05 Jan 2007).

Stirling, M. W., McVerry, G. H. and Berryman, K. R. (2002). “A New Seismic Hazard Model

for New Zealand”, Bulletin of the Seismological Society of America, Vol. 92. No. 5, June

2002, pp. 1878-1903.

Vamvatsikos D. and Cornell C. A. (2002). “Incremental Dynamic Analysis”, Earthquake

Engineering and Structural Dynamics, Vol. 31, pp. 491–514, Earthquake Engineering

Research Institute, Oakland, California.

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Earthquake Spectra, Earthquake Engineering Research Institute (EERI), Vol. 20, No. 2,

pp. 523–553, Earthquake Engineering Research Institute, Oakland, California.

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simple systems to earthquake motions", Proceedings of the 2nd World Conference on

Table 5-1. Comparative values of Spectral accelerations (Sa) for Near- and Far-Field Suites.

Period T Near-Field Suite Sa (g) Far-Field Suite Sa (g) (sec) Medians Maxima βEQ Medians Maxima βEQ

0.00 (PGA) 0.551 1.226 0.363 0.382 0.883 0.618 0.25 1.007 2.123 0.414 0.939 2.285 0.673 0.50 1.025 2.455 0.439 0.822 1.761 0.705 0.60 1.000 1.932 0.387 0.725 1.239 0.691 1.00 0.873 2.269 0.449 0.483 1.008 0.774 1.50 0.598 1.278 0.392 0.306 0.511 0.659 2.00 (T1) 0.459 0.840 0.408 0.206 0.380 0.646 2.50 0.347 0.755 0.431 0.164 0.250 0.674 3.00 0.258 0.570 0.486 0.126 0.337 0.718 Overall Median for T= 0–2.5s 0.690 1.381 0.415 0.382 0.612 0.681 Sa(2s)/PGA 0.833 0.539

Table 5-2. Comparative values of Spectral accelerations (Sa) for Near- and Far-Field Suites

Normalised by Sa(T1=2s).

Period T Near-Field Suite Sa (g) Far-Field Suite Sa (g) (sec) Medians Maxima βEQ Medians Maxima βEQ

0.00 (PGA) 1.202 2.673 0.363 1.849 4.279 0.618 0.25 2.195 4.628 0.414 4.549 11.070 0.673 0.50 2.234 5.352 0.439 3.984 8.532 0.705 0.60 2.239 6.695 0.436 2.969 5.198 0.762 1.00 1.903 4.947 0.449 2.339 4.884 0.774 1.50 1.304 2.786 0.392 1.483 2.474 0.659 2.00 (T1) 1.000 1.831 0.408 1.000 1.842 0.646 2.50 0.757 1.647 0.431 0.794 1.213 0.674 3.00 0.563 1.242 0.486 0.609 1.634 0.718 Overall Median for T= 0–2.5s 1.504 3.011 0.415 1.852 2.965 0.681

Table 5-3. Increases in Median Responses and 90th percentile Demands of aθ = θr/IM due to

Biaxial interaction for Near- and Far-Field Suites. Median Responses

aθr = θr/IM

90th percentile Demands

aθr90th = θr90th/IM

In document Behaviour of three dimensional concrete structures under concurrent orthogonal seismic excitations (Page 184-194)