Civil Engineering Reference
In-Depth Information
Chapter 7
DETERMINATION OF
CROSS SECTIONAL FORCES
7.1 Introduction
While we in chapter 6 focused exclusively on the determination of response
displacements, we shall in this chapter deal with the determination of the corresponding
cross sectional forces, i.e. the cross sectional stress resultants defined in chapter 1.3 (see
Fig. 1.3.b). From a design point of view it is the maximum values of these quantities that
decide the actual level of safety against structural failure. For a line like type of bridge
structure the problem at hand is equivalent to that which is illustrated in Fig. 6.1, only
that the response quantities we shall now set out to calculate are the cross sectional force
components
F
(e.g. a bending moment, a torsion moment or a shear force) rather than
the displacements which were in focus in chapter 6. The assumption of a Gaussian,
stationary and homogeneous flow over the design period
T
(e.g. 10 min) is still valid, as
well as the assumptions of linearity between load and load effects and a linear elastic
structural behaviour. Thus, any cross sectional force component
F
may be described by
the sum of its mean value and a fluctuating part that is Gaussian
(
)
()
( )
F
xt
,
F x
F xt
,
=
+
(7.1)
tot
The time domain chain of events is illustrated in Fig. 7.1.a. Similar to that which was
argued for the determination of displacements, it is in the following taken for granted
that the fluctuating part of the cross sectional response forces are quantified by their
standard deviation (
), as illustrated in Fig. 7.1.b. The maximum value of a force
component at spanwise position
σ
F
x
is then given by
()
()
()
F
x
Fx
k
x
=
+
⋅
σ
(7.2)
max
r
r
p
F
r
where
k
is the peak factor (that depends on the type of response process). The chain of
events for cross sectional forces is equivalent to that which is shown for structural
