Civil Engineering Reference
In-Depth Information
are stressed in compression and the vertical members in tension, Figure 2.1 (g). This
diagram also shows the conventional representation of tension and compression forces
in members which is used in this topic. Truss analogy is explained in more detail
in
3.10
.
The cantilever could have consisted of a beam of rectangular cross section, Figure 2.2
of width
b
and height
h
. The tensile and compressive stresses caused by the bending
moment are zero at what is termed the neutral axis, which is at the centre of the
beam for a symmetrical cross section, and they are proportional to their distance from
this neutral axis. Consequently they are a maximum at the top and bottom extreme
fi bres where they are represented by the symbol
±
. As force = area × stress, the
forces of tension and compression forming the internal couple are equal to the average
stress in the top or bottom half of the beam multiplied by the cross-section area of
half the beam,
F
= ±(
σ
bh
/4. The lever arm of the internal couple
is the distance between the centroids of the tension and compression forces. As the
force diagrams are triangular, this lever arm is 2
h
/3. The stress on the top and bottom
extreme fi bres of the cross section created by the applied moment
WL
can be found by
equating the external moment with the internal couple,
WL
=
F
×2
h
/3. Substituting
for
F
,
WL
= (
σ
/2)×(
bh
/2) = ±
σ
WL/(
bh
2
/6). The term
bh
2
/6 is
known as the elastic modulus of the rectangular cross section (not to be confused with
E
, the modulus of elasticity).
For bridge decks with cross sections that are unsymmetrical about a horizontal axis,
Figure 2.3, the elastic moduli corresponding to the top and bottom extreme fi bres
are not equal. They are conventionally designated by
z
t
and
z
b
for the top and bottom
extreme fi bres respectively. Thus the stresses on the top and bottom extreme fi bres of
the bridge deck, subjected to an external bending moment
M
are
M
/
z
t
and
M
/
z
b
.
σ
bh
/4)×2
h
/3, or
WL
=
σ
bh
2
/6, or
σ =
Figure 2.2
Rectangular cross section cantilever
Figure 2.3
Section unsymmetrical about a horizontal axis
