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

ExAmpLE 8.2
The cross-sectional dimensions
of a beam are shown in the
accompanying diagram. If the
maximum allowable bending
stress is 230 MPa, determine
the magnitude of the maximum
internal bending moment M
that can be supported by the
beam. (Note: The rounded corners
of the cross section can be
neglected in performing the
section property calculations.)
Plan the Solution
The centroid location and the moment of inertia of the beam cross section must be calculated
at the outset. Once the section properties have been computed, the flexure formula
will be rearranged to determine the maximum bending moment that can be applied without
exceeding the 230 MPa allowable bending stress.
SolutioN
The centroid location in the horizontal direction can be determined from symmetry. The
cross section can be subdivided into three rectangular shapes. In accordance with the
procedure described in Example 8.1, the centroid calculation for this shape is summarized
in the following table:
A i
(mm 2 )
M
y i
(mm)
y i A i
(mm 3 )
(1) 450 37 16,650
(2) 204 17 3,468
(3) 204 17 3,468
858 23,586
z
y
Ref.
axis
M
37 mm
x
(1)
75 mm
75 mm
6 mm
6 mm
6 mm
(2)
17 mm
(3)
40 mm
40 mm
y
yA i i
= Σ ΣA
i
3
23,586 mm
= = 27.49 mm
2
858 mm
6 mm
Therefore, the z centroidal axis is located 27.49 mm above the
reference axis for this cross section.
Ans.
The calculation of the moment of inertia about this axis is
summarized in the following table:
I c
(mm 4 )
| d i |
(mm)
d i2 A i
(mm 4 )
I z
(mm 4 )
(1) 1,350 9.51 40,698.0 42,048.0
(2) 19,652 10.49 22,448.2 42,100.2
(3) 19,652 10.49 22,448.2 42,100.2
126,248.4
Ref.
axis
12.51 mm
27.49 mm
75 mm
y 6 mm
(1)
z
40 mm
(2) (3)
6 mm
249