Civil Engineering Reference
In-Depth Information
are negligible. Thus the longitudinal strains
ε
vary linearly through the depth of
the beam, as shown in Figure 5.4, as do the longitudinal stresses
σ
. It is shown in
Section 5.8.1 that the moment resultant
M
y
of these stresses is
M
y
=−
EI
y
d
2
w
d
x
2
(5.2)
wherethesignconventionsforthemoment
M
y
andthedeflection
w
areasshown
in Figure 5.5, and that the stress at any point in the section is
σ
=
M
y
z
I
y
(5.3)
inwhichtensilestressesarepositive.Inparticular,themaximumstressesoccurat
the extreme fibres of the cross-section, and are given by
σ
max
=
M
y
z
T
I
y
=−
M
y
W
el
,
yT
,
in compression, and
(5.4)
σ
max
=
M
y
z
B
I
y
M
y
W
el
,
yB
,
=
in tension, in which
W
el
,
yT
=−
I
y
/
z
T
and
W
el
,
yB
=
I
y
/
z
B
are the elastic section
moduli for the top and bottom fibres, respectively, for bending about the
y
axis.
The corresponding equations for bending in the principal plane
xy
are
M
z
=
EI
z
d
2
v
d
x
2
(5.5)
wherethesignconventionsforthemoment
M
z
andthedeflection
v
arealsoshown
in Figure 5.5,
σ
=−
M
z
y
/
I
z
(5.6)
2
d
2
w
δ
x
d
x
M
z
d
A
=
y
A
M
y
M
y
2
-z
T
d
w
=
-
EI
C
b
y
2
d
x
y
z
z
z
B
= E
δ
x
δ
x
z
(c) Strains
(d) Stresses
(e) Moment resultant
(a) Cross-section
(b) Bending
Figure 5.4
Elastic bending of beams.
Search WWH ::
Custom Search