Civil Engineering Reference
In-Depth Information
5.00
5.00
Rock
Soft site
4.00
4.00
3.00
3.00
Long-period exponent
2.00
2.00
Long-period exponent
1.00
1.00
T
1
T
2
T
1
T
2
0.00
0.00
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.00
0.50
1.00
1 50
2.00
2 50
3.00
3.50
4.00
Period,
T
(s)
Period,
T
(s)
Figure 4.39
Standard smoothed spectra: rock site (
left
) and soft site (
right
)
The fundamental period of vibration
T
of a structure is essential to compute the base shear
V
B
. The
importance of this dynamic parameter is twofold: the site-structure resonance and the design spectrum
ordinate. Codes attempt to supply simplifi ed, semi-empirical expressions for period estimation as a
function of height, material, system and number of storeys. These expressions are also calibrated using
regression analyses of data derived from system identifi cation procedures (e.g. Goel and Chopra, 1997 ,
1998 ).
A reasonable evaluation of the fundamental period for a multi-storey structure requires calculations
involving the mechanical properties of the members of the lateral resisting system. Clearly, for new
structures these calculations cannot be carried out until the system is designed. It is customary, however,
to check the period determined empirically through code-based formulae by using Rayleigh's method,
which provides the following expression:
N
N
⎡
⎢
⎤
⎥
⎡
⎢
⎤
⎥
∑
∑
2
T
=
2
π
W
δ
g
F
δ
(4.36)
i
i
i
i
i
=
1
i
=
1
where
W
i
is the storey weight,
F
i
the force applied at the
i
th storey and
δ
i
the corresponding lateral
displacement. Equation (4.36) is a simple application of the ' self - weight method ' .
The distribution of seismic loads along the building height depends mainly on mass and stiffness
distributions and the building confi guration in plan and elevation as also discussed in Section 2.3.1.2
and Appendix A. The contribution of higher modes in the dynamic response of the structure also affects
the load distribution. Codes attempt to supply a simplifi ed method for load distributions based only on
mass distribution and storey heights. A common expression for the seismic lateral force
F
i
at the
i
th
storey of a building structure is given as:
WH
i
i
FV
=
(4.37)
i
B
N
∑
WH
j
j
j
1
where
N
is the total number of storeys,
W
i
and
W
j
are the seismic weight of the
i
- th and
j
- th storeys,
respectively; they can be computed using equation (4.1) . Similarly,
H
i
and
H
j
are the heights from
ground level to the
i
th and
j
th level, respectively. Equation (4.37) provides a triangular distribution
over the height for uniform mass and stiffness and is thus suitable for low-rise regular structures (
T
≤
0.5 second), for which the fundamental mode of vibration departs little from a straight line. For long-
period structures, the infl uence of higher modes can be signifi cant. In high-rise regular structures, the
