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

SolutioN
(a) Moment of inertia About the z Centroidal Axis
The moment of inertia I ci of each rectangular shape about
its own centroid must be computed to begin the calculation.
The moment of inertia of a rectangle about its centroidal
axis is given by the general equation I c = bh 3 /12,
62 mm
where b is the dimension parallel to the axis and h is the
8 mm
perpendicular dimension.
For example, the moment of inertia of area (1) about its
horizontal centroidal axis is calculated as I c1 = bh 3 /12 =
(90 mm)(10 mm) 3 /12 = 7,500 mm 4 . Next, the perpendicular y− = 47.0 mm 42 mm
distance d i between the z centroidal axis for the entire
flanged shape and the z centroidal axis for area A i must be
determined. The term d i is squared and multiplied by A i and
Reference axis
the result is added to I ci to give the moment of inertia for
each rectangular shape about the z centroidal axis of the
entire flanged cross section. The results for all areas A i are summed to determine the
moment of inertia of the flanged cross section about its z centroidal axis. The complete
calculation procedure is summarized in the table below.
z
30 mm
(3)
y
(2)
90 mm
6 mm
(1)
18 mm
90 mm
10 mm
I ci
(mm 4 )
⏐d i ⏐
(mm)
d i 2 A i
(mm 4 )
I z
(mm 4 )
(1) 7,500 42.0 1,587,600 1,595,100
(2) 364,500 8.0 34,560 399,060
(3) 14,580 62.0 2,075,760 2,090,340
4,084,500
Thus, the moment of inertia of the flanged shape about its z centroidal axis is
I z = 4,080,000 mm 4 .
Ans.
(b) Moment of inertia About the y Centroidal Axis
As before, the moment of inertia I ci of each rectangular shape about its own centroid must
be computed to begin the calculation. However, it is the moment of inertia about the vertical
centroidal axis that is required here. For example, the moment of inertia of area (1) about its
vertical centroidal axis is calculated as I c1 = bh 3 /12 = (10 mm)(90 mm) 3 /12 = 607,500 mm 4 .
(Compared to the I z calculation, notice that different values are associated with b and h in
the standard formula bh 3 /12.) The parallel-axis theorem is not needed for this calculation
because the centroids of each rectangle lie on the y centroidal axis of the flanged shape. The
complete calculation procedure is summarized in the table below.
I ci
(mm 4 )
⏐d i ⏐
(mm)
d i 2 A i
(mm 4 )
I y
(mm 4 )
(1) 607,500 0 0 607,500
(2) 1,620 0 0 1,620
(3) 40,500 0 0 40,500
649,620
The moment of inertia of the flanged shape about its y centroidal axis is thus
I y = 650,000 mm 4 .
Ans.
797