Civil Engineering Reference
In-Depth Information
Table 3.14
Constants for trilinear behaviour factors spectra.
μ
= 2
μ
= 3
μ
= 4
μ
= 6
T
1
T
2
q
1
q
2
T
1
T
2
q
1
q
2
T
1
T
2
q
1
q
2
T
1
T
2
q
1
q
2
EPP
0.20
0.79
2.06
2.20
0.21
0.78
2.89
3.31
0.22
0.87
3.59
4.34
0.25
0.99
4.81
6.13
K
3
= 0
0.20
0.56
2.20
2.51
0.25
1.67
3.10
4.09
0.27
1.55
3.76
5.45
0.29
1.26
4.78
7.79
K
3
= 10%
K
y
0.21
0.54
2.04
2.33
0.27
1.80
2.78
3.62
0.29
1.64
3.25
4.56
0.33
1.54
3.93
6.10
K
3
= − 20%
K
y
0.26
0.26
2.43
2.43
0.24
1.76
2.83
3.93
0.25
1.69
3.25
5.12
—
—
—
—
K
3
= − 30%
K
y
0.26
0.26
2.42
2.42
0.24
1.85
2.76
3.81
—
—
—
—
—
—
—
—
Key
: EPP = elastic - perfectly plastic model; “
= ductility
K
y
and K
3
are the secant and post-yield stiffness of the primary curve of the hysteretic hardening-softening model
μ
observed that the infl uence of input motion parameters on elastic and inelastic acceleration spectra is
similar and signifi cant. However, the effect cancels out for their ratio. Ductility is the most signifi cant
parameter, infl uencing the response modifi cation factor. Consequently, analyses to defi ne period-
dependent behaviour factor functions for all the ductility levels and all structural models were under-
taken. The average values and the standard deviations were calculated considering various combinations
of input motion parameters. The period- dependent
R
-factor functions calculated were further approxi-
mated with a trilinear spectral shape. The
R
-factor is equal to 1.0 at zero period and increases linearly
up to a period
T
1
, which is defi ned as the period at which the force reduction factor reaches the value
q
1
. A second linear branch is assumed between
T
1
and
T
2
. The value of the reduction coeffi cient corre-
sponding to
T
2
is denoted herein
q
2
. For periods longer than
T
2
, the behaviour factor maintains a constant
value equal to
q
2
:
T
T
qq
=−
(
1
)
+
1
hen
TT
≤
(3.26.1)
1
1
1
qq q q
TT
TT
−
−
1
(
)
=+ −
when
TTT
<≤
(3.26.2)
1
2
1
1
2
2
1
qq
(3.26.3)
=
when
TT
>
2
2
The values
q
1
,
q
2
,
T
1
and
T
2
that defi ne approximate spectra for all ductility levels and hysteretic
parameters are summarized in Table 3.14, as they are obtained by a piece-wise linear regression, for
the sample EPP and HHS models.
To demonstrate the reasonable fi t of the trilinear representation to the regression force reduction
factors spectra, the standard deviation
between the approximate and the original spectral
values was studied. The standard deviation was calculated for all branches of the approximate spectra
and across the whole period range. These values are provided in Table 3.15 .
It is observed that the dispersion of
σ
of the ratio
γ
is close to the global standard deviation. This has an important
consequence, from a practical point of view, as the
R
-factor spectra proposed herein correspond to
constant seismic design reliability over the whole period range, a feature not previously achieved.
Finally, the coordinates of the points that allow the defi nition of the approximate spectra were expressed
as a function of ductility and given as:
γ
T
T
1
(3.27.1)
=
1
Ta
=
μ
+
b
(3.27.2)
2
T
2
T
2
