r - The Hong Kong Polytechnic University

provided in DBELA, a base rotation can be mechanics-derived for both beam- and column-sway frames and the
displacement at the center of seismic force is calculated by multiplying this rotation θ by an effective height H e
as is shown in Figure.2.
The displacement capacity can be expressed as follow:
Δ = θ ⋅ H e
(1)
where
H
e
=
n
∑( miΔi
Hi
) ∑
i=
1
n
i=
1
( m Δ )
(2a)
and m i is the mass at the height H i associated with displacement Δ i . For regular beam-sway frames, the effective
height coefficient ef h defined as the ratio of the height of the SDOF substitute structure (H e ) to the total height of
the original building (H T ), expressed as a function of the number of storey n, suggested by Priestley[1997]. Then
H e can also be computed as Equitation (2b)
H = ef ⋅ H
(2b)
e
h
ef h = 0.64
n ≤ 4
ef h = 0.64
− 0.0125( n − 4)
4 < n < 20
(3)
ef h = 0.44
n ≥ 20
Researches in deformational behavior of RC members show that the yield curvature of beams and columns are
independent of reinforce ratio and strength, thus storey yield drift of frames may be represented by the material
and geometrical properties of the buildings only. For a concrete frame, ignoring strain-hardening and using
empirical coefficients to account for shear and joint deformation, the yield base rotation of a beam-sway frames
is shown in Equation (4).
lb
θ y = 0. 5ε y
(4)
hb
where h b is the height of the beam section, l b is the length of the beam and ε y is the yield strain of the
reinforcement steel.
Combining the equations above, the yield displacement capacity (Δ y ) formula to define capacity displacement in
the yield limit state for beam-sway frames presented in Equation (4) is altered as following.
lb
Δ y = 0.5ε yefhHT
(5)
hb
From the definition of displacement ductility (Equation 6), displacement capacity of other limit states can easily
be derived when the ductility levels are known.
Δ LSi
μ LSi =
(6)
Δ
Displacement demand
Displacement response spectra, which present the displacement demand to SDOF oscillators with a range of
periods of vibration, are used in DBELA to represent the input from the earthquake to the building class under
consideration. Normally response spectra provide information on the peak elastic response for a specified elastic
damping ratio (typically 5%), and are plotted against the elastic period.
T
The concept viscous damping is generally used to account for the energy dissipated by structures in the elastic
range and this viscous damping is often taken as 5% for reinforced concrete structures an 2% for steel structures.
In order to account for non-linear behavior, the hysteretic damping included into the viscous damping term
leading to so-called equivalent viscous damping ξ eq . The equivalent viscous damping is related to the ductility (μ)
of the structural system in many studies and can be expressed in Equation (7) for concrete frame building.
⎛ μ −1⎞
ξeq = 0.05 + 0.565
⎜
⎟
(7)
⎝ μπ ⎠
The equivalent viscous damping values obtained through Equation (7), for different ductility levels, can then be
combined with Equation (8), proposed by Bommer et al. [2000] and currently implemented in EC8 [CEN, 2003],
to compute a reduction factor η to be applied to the 5% damped spectra at periods from the beginning of
y
i
i
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