Civil Engineering Reference
In-Depth Information
The deflections of the beam can also be determined from the principal plane
momentsbysolvingequations5.2and5.5intheusualway[1-8]fortheprincipal
plane deflections
w
and
v
, and by adding these vectorially.
Workedexamplesofthecalculationoftheelasticstressesanddeflectionsinan
angle section cantilever are given in Section 5.12.5.
5.4 Shear stresses in elastic beams
5.4.1 Solid cross-sections
Averticalshearforce
V
z
actingparalleltotheminorprincipalaxis
z
ofasectionof
abeam(seeFigure5.9a)inducesshearstresses
τ
xy
,
τ
xz
intheplaneofthesection.
Insolidsectionbeams,theseareusuallyassumedtoactparalleltotheshearforce
(i.e.
τ
xy
=
0), and to be uniformly distributed across the width of the section, as
shown in Figure 5.9a. The distribution of the vertical shear stresses
τ
xz
can be
determined by considering the horizontal equilibrium of an element of the beam
asshowninFigure5.9b.Becausethebendingnormalstresses
σ
varywith
x
,they
create an imbalance of force in the
x
direction, which can only be compensated
for by the horizontal shear stresses
τ
zx
=
τ
v
which are equal to the vertical shear
stresses
τ
xz
. It is shown in Section 5.10.1 that the stress
τ
v
at a distance
z
2
from
the centroid where the section width is
b
2
is given by
z
2
τ
v
=−
V
z
I
y
b
2
bz
d
z
.
(5.13)
z
T
v
b
δ
x
b
δ
z
-z
T
z
z
2
V
Shear stresses
v
parallel to shear
force
V
z
and constant
across width of section
z
δ
z
b
x
b
2
∂
∂
δ
x
b
δ
z
(
)
+
x
(
v
b
)
x
∂
y
v
b
δ
z
δ
x
(
+
)
∂
z
z
(a) Assumed shear stress distribution
(b) Horizontal equilibrium
Figure 5.9
Shear stresses in a solid section.
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