Civil Engineering Reference
In-Depth Information
movement measured at a point midway between the supports is
the amount of deflection. The action of beams subjected to bending
forces is described as follows.
If a load of 100 pounds placed in the middle of a stick that is
2 inches
2 inches and 4 feet long, supported at both ends, bends
or deflects this stick one-eighth of an inch (in the middle), then
200 pounds will bend it about
1
×
/
8
inch, and so
on, the deflection varying directly as the load. This is in accordance
with Hooke's Law that states that stress is proportional to strain.
Soon, however, a point is reached where an additional 100 pounds
adds more than
1
/
4
inch, 300 pounds,
3
/
8
inch to the deflection—the limit of elasticity has
been exceeded.
Taking another piece from the straight-grained and perfectly clear
plank of the same depth and width but 8 feet long, the load of
100 pounds will deflect it by approximately 1 inch. Doubling the
length reduces the stiffness eightfold. Stiffness, then, decreases as
the cube of the length.
If AB in Figure 4-7 is a piece of wood, and D is the deflection
produced by a weight or load, then
P
DEFLECTION
B
A
Figure 4-7 A simple beam, loaded at the middle and sup-
ported at both ends, is used to illustrate the term deflection.
Pl
3
48
EI
where the following is true:
P
=
deflection (
D
)
=
the load, concentrated at the center of the span (in pounds)
l
=
the length of the span (in feet)
E
=
the modulus of elasticity of the material
I
=
the moment of inertia (for rectangular beams
=
bd
3
12
where
b
=
width and
d
=
height).
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