** INCREMENTAL DYNAMIC ANALYSIS TO ASSESS THE PERFORMANCE, CAPACITY AND DAMAGE**

**4.3 METHODOLOGY EMPLOYED IN THE IDA STUDY**

**4.4.1 S TRUCTURAL M AXIMUM D EMANDS**

**4.4.1.1 Radial Interstorey Drifts**

Maximum values of global radial interstorey angular drifts for the IDA of the two sets of

the five prototype structures were computed and are presented in Figures 4-2 and 4-3. Tables 4-

6 and 4-7 list their overall medians for both sets of structures where they show strong

resemblance between both sets of results. Figure 4-2 is for the structures with elastically

behaving columns, Figure 4-3 presents results for the structures with columns allowed to behave

inelastically. The presented results had been further manipulated by setting a cut-off when

values of interstorey drifts exceed 10%, effectively implying structural collapse; see FEMA 350

(2000) Table 4-10. Figures 4-2 (a) and 4-3 (a) exhibit the median values of radial interstorey

drifts and Figures 4-2 (b) and 4-3 (b) display the 90th percentile values. Figures 4-2 (c) and 4-3

(c) show the median values of radial interstorey drifts normalised by their intensity measures

The rationale for normalising the maximum values of global radial interstorey angular

drifts by their relevant IM values, S (Ta 1=2s), can be found in Cornell et al (2002) and Jalayer and

Cornell (2003). Jalayer and Cornell (2003) provided the following expression for interstorey

drift demand as a power function of spectral acceleration and lognormal random variable ε:

*b _{ε}*

* (4.2) *

*θ = a Sa*

where *a* and *b* are least-squares regression coefficients (in log-log space) for displacement-based

demands, such as interstorey drifts (*θ*). For medium- to long-period structures the well-known
principle of equal displacement (Veletsos and Newmark, 1960), renders *b = 1*. The lognormal

random variable *ε* is a random variable with unit median that encompasses both epistemic and
aleatory uncertainties, which at this stage can be taken as one for this deterministic study.

Consequently, the expression is simplified to:

* (4.3) *

*a = θ/ Sa*

Therefore without randomness, the term *θ/Sa*, or *θ/IM*, reduces to a simple single

constant, *a*. However, variation in *a* can be expected due to the randomness of the input and the

output (response). Therefore, *θ/IM* in this case represents the magnitude of interstorey drifts at
the same universal value of IM = Sa(T1=2s) = 1g, thus facilitating the comparison between

different levels of IM and for all structures utilised in this study.

On inspecting figures 4-2 (a to d) and 4-3 (a to d) along with their numerical data, both

results were identical up to MCE values of Sa(T1=2s) = 0.25g. This verifies that structural

performance followed the inherent strong-column/weak-beam principle with no plastic hinges

occurring in columns except at ground and roof levels. Beyond MCE, overall structural

instability started to occur where it can be seen strongly evident at 3DBE values at Sa(T1=2s) ≥

0.45g, incipient collapse is evident. At this intensity level, interstorey drifts in excess of 8% may

be expected which is well beyond their stable capacities (4%). Such instabilities and erratic

behaviours were marked in the analyses with columns allowed inelastic behaviour due to the

On further inspection of Figures 4-2 (c) and (d) together with 4-3 (c) and (d), one can realise a

noticeable increase in the values θr/IM for the bisymmetrical structure (Tx = Ty) followed by that of

Tx/Ty = 1.5 when compared with other structures of Tx/Ty = {2.0, 2.5, 3.0}, specifically marked in the

90th percentile figures. Table 4-3 lists the increases in both median and 90th percentile values for Tx/Ty

= 1.0 and 1.5 when compared with their overall medians. Such increased response can be attributed to

the rather more flexible nature of bisymmetrical structures and those with close bidirectional

fundamental periods in comparison with those with Tx/Ty≥ 2, as well as the apparent synergic coupling

of the two principal bidirectional responses. This finding emphasises the importance of considering the

bidirectional fundamental period ratio (Tx/Ty) when studying displacement-based demands. In essence

one can postulate that for structures with close bidirectional fundamental periods, the expected drift

values would be higher by 3-15% than those with periods well apart with Tx/Ty > 1.5.

In keeping with Demand and Capacity Factor Design (DCFD), Jalayer and Cornell

(2003), and on inspecting Figures 4-2 (c) and (d) together with 4-3 (c) and (d) as well as their

data up to MCE one can substantiate that interstorey drifts are candidates of high level of

confidence as the reported standard deviation and coefficient of variations (COV) are markedly

low. Experience, and many authors such as Shome and Cornell (1999), suggests that interstorey

drifts are distributed lognormally about their medians with their standard deviations of their

natural logarithms, *βθ| Sa*. Note that, *βθ| Sa* = *σlnθ*|*Sa* is the standard deviation of the natural

logarithms of *θ*, herein referred to as the dispersion and given by the symbol *β*. According to
Benjamin and Cornell (1970), COV values are approximately numerically equal to the lognormal

standard deviation (*σlnx*) for moderate levels of dispersion less than 0.3. Consequently these

COV values are only considered as a measure of confidence in values of the considered results

when their values COV < 0.3 showing low variability in results. On studying COV*θrp*|*Sa* of

median-to-median of θr median responses (*p*=50%) and 90th percentile demands (*p*=50%), the
median of COV values was 0.065 for medians and 0.07 for 90th percentiles, Table 4-3. Further,

structural design capacity, i.e., Sa(T1=2s) ≤ Sa(MCE). This finding strongly suggests a high level

of confidence in the probabilistic demand representation of maximum radial interstorey drifts

due to their substantial independence of structural configuration, albeit noticeably dependant on

the biaxial interaction for T /Tx y ≤ 1.5. Furthermore, θr/IM substantial independence of levels of

IM can be realised from Figures 4-2 (c) and (d) along with 4-3 (c) and (d), where the values of

θr/IM largely maintain same values (almost a vertical line in the figures) for different structural
configurations and IM values up to MCE. Consequently, θr is linearly proportional to IM,
Sa(T1=2s) ≤ Sa(MCE), validating the equal displacement principle, thus *b≈ 1*.

th

Regression analysis was conducted on the results of median and 90 percentile values of

θr to evaluate the coefficients of power equation *θ = a Sab*. Table 4-8 lists the median values of *a*

and *b* for both the median response and 90th percentile code-maximum demands at Sa(T1=2s) ≤

Sa(MCE) and > Sa(MCE). Median values of *b* for median response and code-maximum demands

were respectively 1.127 and 1.055 for Sa(T =2s) 1 ≤ Sa(MCE), very close to one (equal

displacement). While for Sa(T =2s) > S1 a(MCE), values of *b* were respectively 1.4 and 1.96 (i.e.

close to parabolic). Table 4-9 shows COV values of *a* and *b*, where COV(*b*) were 0.057 << 0.3

for Sa(T =2s) 1 ≤ Sa(MCE) and 0.24 < 0.3 for Sa(T =2s) > S1 a(MCE), confirming their credible

values. Value of *a* code-maximum demands at Sa(T =2s) 1 ≤ Sa(MCE) was 0.086 with COV(*a*) =

0.144 < 0.3, renders it credible for implementation. However, its value of 0.264 for ≤ Sa(MCE)

has COV(*a*) = 0.632 >> 0.3, indicating high variability due to erratic behaviour for motions

larger than MCE. In essence, one can postulate that θr are predominately dependent on Sa(T1)

and differential interstorey capacities and marginally on differential biaxial capacities.

Accordingly, one can implement the power equation of:

*b _{γ}*

*θ = a Sa* *biaxial,θε (4.4) *

where values of *a* and *b* are obtained from Table 4-8 and *γbiaxial,θ* is the θr biaxial factor equals

{1.03–1.15} Tx/Ty≤ 1.5 for 90th percentile maxima and one for Tx/Ty > 1.5. The value of *γbiaxial,θ*

*γbiaxial,θ= 1.15 – 0.24 (Tx/Ty – 1) * *(4.5) *

Equation (4.4) is suitable for medium to long period seismic-code designed structures that

follows the equal displacement principle. In addition, it can be credibly implemented in rapid

calculation of expected maximum values of θr and in the DCFD framework.
**4.4.1.2** **Structural Maximum Response of Lateral Storey Drifts **

As in section 4.4.1.1, Figures 4-4 and 4-5 exhibit the demands of global maxima of lateral

radial storey drifts (δr) per unit IM and Tables 4-6 and 4-7 list their overall medians values. Both

figures and tables show strong resemblance as with the case of θr and thus drawing the same above- mentioned conclusion that the considered structures with columns allowed to behave inelastically

acted similar to those with elastic columns up to MCE. Consequently, the structures with realistic

(inelastic) columns followed the inherent Capacity Design principles and their columns behaved

essentially elastically up to MCE. Beyond MCE, the 90th percentile maximum values of δr

erratically increase exhibiting apparent collapse behaviour. Note the maximum lateral storey drifts

in all cases occur at the roof level, as later discussed in the following section 4.4.2.

Figures 4-4 (c) and (d) together with 4-5 (c) and (d) for Sa(T1=2s) ≤ 0.10g (elastic

behaviour) show that the values of δr/IM were almost identical, i.e., *b ≈ 1*. However, δr/IM
values for Sa(T1=2s) ≥ 0.15g (DBE) and ≤ 0.25g (MCE) show reducing magnitudes of lateral

sways due to structural softening. As the structures underwent inelastic deformation their

fundamental periods were extended reducing their structural Sa intake, and consequently a

decrease in the values of δr resulted. This indicates that the coefficient *b* is less than one. A
similar change in behaviour is reported by Ramamoorthy et al (2006) that shows discontinuities

at the onset of first yield. Nonetheless, the values of δr/IM maintain the same ratios with their

overall medians, particularly up to MCE.

As with θr/IM, on further inspecting Figures 4-4 (c) and (d) together with 4-5 (c) and (d), one can realise a noticeable increase in values of δr/IM for the bisymmetrical structure (Tx = Ty) followed

by that of Tx/Ty = 1.5 when compared with other structures of Tx/Ty = {2.0, 2.5, 3.0}. This increase

is specifically marked in the 90th percentile Figures 4-4 (d) and 4-5 (d). Table 4-4 lists the increases

in both median and 90th percentile values for Tx/Ty = 1.0 and 1.5 when compared with their overall

medians. As with θr/IM, the same conclusion was drawn which emphasises the importance of considering the bidirectional fundamental period ratio (Tx/Ty) when studying displacement-based

demands. In essence for structures with close bidirectional fundamental periods, the expected values

would be higher by 1-12% than those with periods well apart with Tx/Ty≥ 2.

Finally, on inspecting Figures 4-4 (c) and (d) together with 4-5 (c) and (d) as well as their

data up to MCE one can substantiate that maximum radial storey sways are candidates of high

level of confidence as the reported standard deviation and coefficient of variations (COV) are

markedly low. On studying COV*δp*|*Sa* median-to-median of the median δr responses (*p*=50%)

and 90th percentile δr demands (*p*=90%) in Table 4-4, the median of COV values was 0.056 for
medians and 0.042 for 90th percentiles. This finding strongly suggests a high level of confidence

in the probabilistic demand representation of storey lateral drifts due to their substantial

independence structural configuration, albeit noticeably dependant on the biaxial interaction for

T /Tx y ≤ 1.5. The substantial independence of δr can be realised from Figures 4-4 (c) and (d)

along with 4-5 (c) and (d) where the values of δr/IM largely maintain the same ratios with their overall medians for different structural configurations up to MCE. Values of δr/IM are, linearly proportional to IM, Sa(T =2s), for S1 a(T1) ≥ DBE.

th

Regression analysis was conducted on the results of median and 90 percentile values of

δ *b*

r to evaluate the coefficients of power equation *δ = a Sa* , to test its applicability. Table 4-8 lists

the median values of *a* and *b* for both the median response and 90th percentile code-maximum

demands at Sa(T =2s) 1 ≤ Sa(MCE) and > Sa(MCE). Median value of *b* for median response and

code-maximum demands were respectively 0.845 and 0.825 < 1, due to structural softening, for

Sa(T =2s) 1 ≤ Sa(MCE). While for Sa(T =2s) 1 ≤ Sa(MCE), *b* values were respectively 1.314 and

Sa(T =2s) 1 ≤ Sa(MCE) and 0.169 < 0.3 for Sa(T =2s) > S1 a(MCE), assuring their credible values.

Value of *a* code-maximum demands at Sa(T =2s) 1 ≤ Sa(MCE) was 1.196 with COV(*a*) = 0.123 <

0.3, renders it credible for implementation. However, its value of 3.044 for ≤ Sa(MCE) has

COV(*a*) = 0.489 > 0.3, indicating its variability due to erratic behaviour for motion larger than

MCE. In essence, one can postulate that δr are predominately dependent on Sa(T1) and storey

capacities and marginally on differential biaxial capacities. Accordingly, one can implement the

power equation of:

*b _{γ}*

_{ε}* (4.6) *

*δ = a Sa* *biaxial,δ*

where values of *a* and *b*are obtained from Table 4-8 and *γbiaxial,δ* is the δr biaxial factor equals

{1.02–1.11} Tx/Ty≤ 1.5 for 90th percentile maxima and one for Tx/Ty > 1.5. The value of *γbiaxial,δ*

can be interpolated for 1.0 ≤ T /Tx y≤ 1.5 by the following expression:

*γbiaxial δ= 1.11 – 0.18 (Tx/Ty – 1) * *(4.7) *

Equation (4.6) is suitable for medium to long period seismic-code designed structures that

follows the equal displacement principle. Again, it can be credibly implemented in rapid

calculation of expected maximum values of δr and in DCFD framework.
**4.4.1.3** **Structural Maximum Response of Storey Accelerations **

This section follows the same framework implemented in studying both θr and δr, in order

to assess the applicability of the power function *A = a Sab* and its credibility in representing

global maximum maxima of radial storey accelerations (Ar). Figures 4-6 and 4-7 display the

demands of Ar per unit IM and Tables 4-6 and 4-7 list their overall medians that show strong

resemblance indicating that the considered structures with columns allowed inelastic behaviour

acted as the ones with elastic columns up to MCE. Consequently, the structures with realistic

columns followed the inherent Capacity Design principles and their columns behaved elastically

up to MCE. Beyond MCE; the 90th percentile (code-maximum) values of Ar erratically increase

Figures 4-6 (c) and (d) together with 4-7 (c) and (d) for Sa(T1=2s) ≤ 0.10g (elastic

behaviour) show that the values of Ar/IM were almost identical, i.e., *b ≈ 1*. However, Ar/IM

values for Sa(T1=2s) ≥ 0.15g (DBE) and ≤ 0.25g (MCE) show slight reduction in magnitudes of

radial storey accelerations due to structural softening. As the structures underwent inelastic

deformation their fundamental periods were extended reducing their structural Sa intake, and

consequently; decreasing the values of Ar. This indicates that the coefficient *b* is less than one.

Nonetheless, the values of Ar/IM maintain the same ratios with their overall medians,

particularly up to MCE.

As with θr/IM and δr/IM, on further inspecting Figures 4-6 (c) and (d) together with 4-7 (c) and (d), one can realise a noticeable reduction in values of Ar/IM for the bisymmetrical

structure (T = Tx y) followed by that of T /Tx y = 1.5 when compared with other structures of Tx/Ty

= {2.0, 2.5, 3.0}. This reduction is specifically marked in the 90th percentile figures. Table 4-5

lists the decreases in both median and 90th percentile values for T /Tx y = 1.0 and 1.5 when

compared with their overall medians. For the 90th percentile demands, the reduction was about

15% for T /Tx y = 1.0 and 1.5 that is attributed to their relative higher flexibility. Note that Figure

4-7 (d) and data show the median response lie about T /Tx y = 2.5. Consequently one can

postulate that for structures with close bidirectional fundamental periods, Tx/Ty ≤ 1.5, the

expected values would be lower by 15% than those with periods well apart with T /T x y≥ 2. Finally, on inspecting Figures 4-6 (c) and (d) together with 4-7 (c) and (d) as well as their

data up to MCE; one can substantiate that maximum radial storey accelerations are candidates of

high level of confidence as the reported standard deviation and coefficient of variations (COV)

are markedly low. On studying COV*Ap*|*Sa* median-to-median of the median Ar responses

(*p*=50%) and 90th percentile Ar demands (*p*=90%) in Table 4-5, the median of COV values was

0.036 for medians and 0.082 for 90th percentiles. This finding strongly suggests a high level of

confidence in the probabilistic demand representation of storey accelerations due to their

interaction for T /Tx y≤ 1.5. The substantial independence of Ar can be realised from Figures 4-6

(c) and (d) along with 4-7 (c) and (d) where the values of Ar/IM largely maintain same ratios

with their overall medians for different structural configurations up to MCE. Values of Ar/IM

are linearly proportional to IM, Sa(T =2s), for S1 a(T1) ≥ DBE.

th

Regression analysis was conducted on the results of median and 90 percentile values of

Ar to evaluate the coefficients of power equation *A = a Sab*, to test its applicability. Table 4-8

lists the median values of *a* and *b* for both the median response and 90th percentile code-

maximum demands at Sa(T =2s) 1 ≤ Sa(MCE) and > Sa(MCE). Median values of *b* for median

response and code-maximum demands were respectively 0.982 ≈ 1 and 0.795 < 1 (due to
structural softening) for Sa(T =2s) 1 ≤ Sa(MCE). While for Sa(T =2s) > S1 a(MCE), values of *b*

were respectively 0.797 and 1.332. Table 4-9 shows COV values of *a* and *b*, where COV(*b*)

were 0.163 < 0.3 for Sa(T =2s) 1 ≤ Sa(MCE) and 0.2 < 0.3 for Sa(T =2s) > S1 a(MCE), assuring their

credible values. Value of *a* code-maximum demands at Sa(T1=2s) ≤ Sa(MCE) was 7.434 with

COV(*a*) = 0.385 > 0.3, indicating its high variability. However, its value of 13.939 for ≤
Sa(MCE) has COV(*a*) = 0.215 > 0.3, renders it credible for implementation. In essence, one can

postulate that Ar is predominately dependent on Sa(T1) and storey strength capacities and

marginally on differential biaxial capacities. Accordingly, one can implement the power

equation of:

*b _{γ}*

*A = a Sa* *biaxial,Aε (4.8) *

where values of *a* and *b* are obtained from Table 4-8 and *γbiaxial,A* is the Ar biaxial factor equals

0.85 for Tx/Ty ≤ 1.5 for 90th percentile maxima and one for T /Tx y > 1.5. Equation (4.8) is

suitable for medium to long period seismic-code designed structures that follow the equal

displacement principle. Furthermore, it can be credibly implemented in fast calculation of