Civil Engineering Reference
In-Depth Information
O
d
V
y
M
y
M
q
(
x
)
x
V
(a)
(b)
(c)
Figure 4.1
(a) Simply supported beam subjected to arbitrary (negative) distributed load.
(b) Deflected beam element. (c) Sign convention for shear force and bending
moment.
h
(b)
(c)
(a)
Figure 4.2
Beam cross sections:
(a) and (b) satisfy symmetry conditions
for the simple bending theory, (c) does
not satisfy the symmetry requirement.
The ramifications of assumption 4 are illustrated in Figure 4.2, which de-
picts two cross sections that satisfy the assumption and one cross section that
does not. Both the rectangular and triangular cross sections are symmetric about
the
xy
plane and bend only in that plane. On the other hand, the L-shaped section
possesses no such symmetry and bends out of the
xy
plane, even under loading
only in that plane. With regard to the figure, assumption 2 can be roughly quan-
tified to mean that the maximum deflection of the beam is much less than di-
mension
h
. A generally applicable rule is that the maximum deflection is less
than 0.1
h
.
Considering a differential length d
x
of a beam after bending as in Figure 4.1b
(with the curvature greatly exaggerated), it is intuitive that the top surface has de-
creased in length while the bottom surface has increased in length. Hence, there
is a “layer” that must be undeformed during bending. Assuming that this layer is
located distance
from the center of curvature
O
and choosing this layer (which,
recall, is known as the
neutral surface
) to correspond to
y
=
0
, the length after
bending at any position
y
is expressed as
d
s
=
(
−
y
) d
(4.1)
