Page 178 - Dynamic Loading and Design of Structures

Page 153

The displacements for the nth mode can be calculated from

(4.34)

2
where is the circular frequency of the nth mode. Recall that S /ω equals S , the
pa
d
spectral displacement (Section 4.3.2).
Since the response of a structure results from the contribution of all modes, and since
modal maxima generally do not occur simultaneously, it is customary to combine the action
effects Si from the individual modes in a statistical way. The most commonly adopted
procedure is the Square Root of the Sum of Squares (SRSS) combination that is:

(4.35)

where S , is the probable maximum value of the action effect (force or displacement), and
i max
the subscripts 1, 2,3,... refer to the first, second, third ... mode; a sufficient number of modes
should be considered in estimating S , (see code criteria in the next subsections). Note also
i max
that action effects due to earthquake should always be taken with alternate sign (i.e. both as
positive and negative). Equation (4.35) gives reasonable values in many practical cases, but is
generally unconservative when two or more modes are closely spaced (i.e. their periods are
close to each other); this is often the case in three-dimensional structures susceptible to
torsional effects. In these cases more refined combination rules, such as the Complete
Quadratic Combination (CQC) (Wilson et al., 1981) are appropriate.
A significant shortcoming of modal response spectrum analysis is that it is not possible to
define exactly the simultaneous values of forces, for instance the axial loading corresponding
to the maximum moment in a column section, and vice versa. Therefore, in addition to the
approximation of modal combination (eqn 4.35), it is customary to assume that the probable
maxima of the various action effects (M, N, V) for a given earthquake action (e.g. a response
spectrum in a particular direction) occur simultaneously; this is usually, but not necessarily,
conservative, with regard to design of members. The problem of combining moments
(generally pairs of moments) and axial loads in the design of columns for (biaxial) bending
and axial force (M ,M ,N) is discussed in more detail elsewhere (Gupta, 1990; Penelis and
y
x
Kappos, 1997).
The Eurocode 8 procedure
The basis of the method is the design response spectrum discussed in Section 4.3.4; this has to
be applied along two, properly identified, perpendicular axes of the structure.