Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

246
bENdINg
In common practice, each of these equations is often called the flexure formula. The first
form is more useful for calculating the bending stress at locations other than the top or
the bottom of the cross section of the beam. Use of this form requires careful attention
to the sign conventions for M and y. The second form is more useful for calculating
maximum bending stress magnitudes. If it is important to determine whether the bending
stress is either tension or compression, then that is done by inspection, using the
sense of the internal bending moment M.
ExAmpLE 8.1
5 kN·m
z
y
5 kN·m
40 mm
10 mm
K
150 mm
A beam with an inverted teeshaped
cross section is subjected
to positive bending moments of
M z = 5 kN · m. The cross-sectional
dimensions of the beam are
shown. Determine
(a) the centroid location, the
x
H
moment of inertia about the
15 mm z axis, and the controlling
section modulus about the
120 mm
z axis.
(b) the bending stress at points H and K. State whether the normal stress is tension or
compression.
(c) the maximum bending stress produced in the cross section. State whether the stress
is tension or compression.
Plan the Solution
The normal stresses produced by the bending moment will be determined from the flexure
formula [Equation (8.7)]. Before the flexure formula is applied, however, the section
properties of the beam cross section must be calculated. The bending moment acts about
the z centroidal axis; therefore, the location of the centroid in the y direction must be determined.
Once the centroid has been located, the moment of inertia of the cross section
about the z centroidal axis will be calculated. When the centroid location and the moment
of inertia about the centroidal axis are known, the bending stresses can be readily calculated
from the flexure formula.
SolutioN
(a) The centroid location in the horizontal direction can be determined from symmetry
alone. The centroid location in the y direction must be determined for the inverted-tee
cross section. The tee shape is first subdivided into rectangular shapes (1) and (2), and
the area A i for each of these shapes is computed. For calculation purposes, a reference
axis is arbitrarily established. In this example, the reference axis will be placed at the
bottom surface of the tee shape. The distance y i in the vertical direction from the reference
axis to the centroid of each rectangular area A i is determined, and the product
y i A i (termed the first moment of area) is computed. The centroid location y measured
from the reference axis is computed as the sum of the first moments of area y i A i divided
by the sum of the areas A i . The calculation for the inverted-tee cross section is
summarized in the accompanying table.