Geoscience Reference
In-Depth Information
11.6.8 r
adius
oF
C
urvature
(b
eam
)
The radius of curvature provides one of several measures of the deformation of a beam. The radius
of curvature of a beam experiencing a bending moment (inches, feet, meters) can be determined by
using the following equation:
p
=
EI
/
M
(11.12)
where
p
= Radius of curvature (in., ft, m).
E
= Modulus of elasticity of the beam material (psi, KPa).
I
= Cross-sectional area moment of inertia about its centroid (m
4
, in.
4
, ft
4
).
M
= Moment or torque applied at the cross-section (ft-lb).
11.6.9 C
olumn
s
tress
Stress is defined as force per unit area and is expressed in pounds per square inch (psi). The amount
of stress is a real indicator of how severely the member is loaded.
Tensile stress
occurs when a mem-
ber is in tension (force acts to stretch it);
compressive stress
occurs when a member is in compres-
sion (force acts to shorten or flatten it); and,
shear stress
occurs when a member is in shear (force
acts to cause one part of the material to slide over another part). To determine the stress (loading) of
a column, we use the following equation:
σ =
P
/
A
(11.13)
where
σ = Stress (lb/in.
2
or psi).
P
= Loading on the column (lb).
A
= Cross-sectional area of the member (in.
2
).
■
EXAMPLE 11.13
Problem:
A column 4 in. in diameter supports a load of 6000 lb. What is the stress on it?
Solution:
A
= π
r
2
= (3.14)(2 in.)
2
= 12.56 in.
2
σ =
P
/
A
= (6000 lb)/12.56 in.
2
= 478 psi
11.6.10 b
eam
F
flexure
In mechanics, flexure (also known as
bending
) characterizes the behavior of a structural element
(beam) subjected to an external load applied perpendicular to the axis of the element. A closet rod
sagging under the weight of clothes on clothes hangers is an example of a beam experiencing flexure
(or bending). The equation used to calculate flexure is known as the
flexure formula
(see below),
which shows the relationship between the maximum bending stress (σ) and the maximum bending
moment (
M
):
σ =
MC
/
L
(11.14)
where
σ = Maximum stress (lb/in.
2
or psi).
M
= Moment (in.-lb).
C
/
L
= Inverse of the section modulus (in.
3
) (taken from an applicable table).
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