S TOREYS ’ M AXIMUM R ESPONSE P ROFILES

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

presented for the IM levels of MCE and 3DBE. Figure 4-9 presents a selected storey median

response and 90th percentile demands of θr at T /Tx y = {1, 2 and 3}. Figure 4-9 shows that θr at

IM levels of elastic, DBE and MCE; MCE was found to be a credible representative of this

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 A1-17 to -31 provide the IM normalised

ones. As with θr 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 4-10 displays the medians and 90th percentiles of θr/IM, θx /IM and θy/IM at IM =

Sa(T1=2s) = Sa(MCE). Figure 4-10 (a, c and e) show typical interstorey median responses for

radial, X- (weak) and Y-axes (strong). They display maximum values around the structure’s

mid-height where typically the bisymmetrical structure (T /Tx y =1) exhibit maximum values for

radial and Y-directions. Figure 4-10 (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 4-10 (b and d) and (f) [for the bisymmetric structure] display the 90th percentile code-

maximum behaviour when the structure’s mid-section exhibit strength degradation.

Subsequently interstorey maxima shift upwards where stabilising normal forces are significantly

low. Nevertheless, Figure 4-10 (f) show that the strong direction the structure retained its

undegraded character of maximum interstorey drift values at mid-height, except for the more

flexible bisymmetric 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.

The imminent Collapse, due to the loss of lateral structural strength, is likely to start in the upper

storeys; as confirmed by Figures 4-11 (b, d and f) at 3DBE (collapse onset) level. Furthermore,

Figure 4-11 reported markedly high standard deviation values, particularly in the top storeys,

thus confirming the collapse is taking place in the upper storeys.

Figures 4-12 and 4-13 exhibit the medians and 90th percentiles of radial, X- and Y-axes

behaviour, albeit the 3DBE showed higher magnitudes and larger standard deviation values as

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. 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) and (4.7) to evaluate the

maximum lateral storey drift of the studied structure.

Figures 4-14 and 4-15 show the medians and 90th percentiles of radial, X- and Y-axes

storey accelerations at MCE and 3DBE levels respectively. Maximum values exist at the top

level, while storey accelerations increase linearly from level 1 up to 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 75% of the roof acceleration, 73 for

X- (weak) and 69% for Y-accelerations (strong) respectively, as listed in Table 4-13. Kircher et

al (1997A) suggested the use of one value for upper storeys and PGA for lower ones. On

studying the curves and their data, it is suggested to use the following:

Aroof = A90th = a S abγbiaxial,Aε (4.9) A = iu 0.75 Aroof (4.10) roof A m i 75 . 0 85 . 0 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Ail = (4.11) A = PGA 0 (4.12)

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), n = number of building storeys, Aroof is the roof

acceleration. Note structures with deep foundation or have multi-storey deep basement are not

within the present scope; where even values of equation (4.12) become unreliable. The proposed

values are 0-5% conservative and successfully implemented for radial and strong Y-direction.

However, it exaggerates values for the weak X-direction due to its long fundamental period and

subsequent reduced values of storey accelerations. Nevertheless, this approximation works well

for radial storey accelerations with substantial independence of Tx/Ty values. Consequently;

equations (4.9), (4.10), (4.11 and (4.12) suffice to provide a 5% conservative or near exact

approximation of median maximum storey accelerations that facilitate their implementation in fast

calculation of expected damage.

Finally, note that 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 T /Tx y median values of γ-percent for θ, δ and A demands peaked to as high

as 40.5%, thus vindicating the use of 40-percent rule in combining the bidirectional X and Y

demands. Note that FEMA 350 (2000) - section 4.4.5.3.4 suggests a 30% combinational rule for

bidirectional responses in contrast with the γ-percent of 40% suggested by this study.

4.5 IMPLEMENTATION INTO DCFD FORMAT

The Demand and Capacity Factor Design (DCFD) format pertains pivotally to finding

representative values for the demand (D) and capacity (C) of the considered structure and their

respective factors γ and φ that are utilised in obtaining the associated confidence level (λ). λ is the factored-demand-to-capacity ratio from which the level of confidence in satisfying the designated

performance level (PL) by the considered structure is inferred. The general expression for the

confidence parameter (λ), according to Jalayer and Cornell (2003), is as follows:

C D ϕ γ (4.13) λ =

where γ is the demand factor φ is the capacity reduction factor; γ is the combined factor of variability (aleatory) γR and uncertainty (epistemic) γU and likewise φ. On recalling previous sections of the study, demands should be further factored to account for biaxial interaction for

Tx/Ty≤ 1.5. This biaxial interaction factor, γbiaxial, for θ and δ can be evaluated respectively from

equations (4.5) and (4.7), and to be taken as 0.85 for A. Because lognormal distributions are

implicitly assumed in the DCFD formulation, the parameters γ and φ are defined by exponential functions as follows:

2

γ =γRγUγbiaxial = γbiaxial exp(½ k/b βDT) (4.14a)

φ = φRφU = exp(-½ k/b β2CT) (4.14b)

where k and b are respectively the power coefficients of the log-log linear regression of seismic

input hazards and demands, see equations (2.1) and (4.2). Note that k is taken as the absolute

value of k in previous sections to avoid inversing the correction factors. The ratio k/b acts as a sensitivity parameter relating an x change in demand to x1/b change in seismic input Sa that

implies xk/b change in the probability of exceedance of demands due to the seismic input hazard.

Finally, βDT and βCT are respectively the total dispersion of demand and capacity that can be

defined, in light of equation (4.14), as follows:

2 2 2

βDT = βDR + βDU (4.15a)

2

βCT = β2CR + β2CU (4.15b)

where βDR and βCR are the aleatory (randomness) dispersions respectively for the considered

demands and capacities. Likewise βDU and βCU are the epistemic (modelling) uncertainty

dispersions for the considered demands and capacities.