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

ExAmpLE A.1
Determine the location of the centroid for the flanged shape
shown.
30 mm
Plan the Solution
The centroid location in the horizontal direction can be determined
from symmetry alone. To determine the vertical location of the centroid,
the shape is subdivided into three rectangular areas. Using the
lower edge of the shape as a reference axis, Equation (A.2) is then
used to compute the centroid location.
SolutioN
The centroid location in the horizontal direction can be determined
from symmetry alone; however, the centroid location in the y
direction must be calculated. The flanged shape is first subdivided
into rectangular shapes (1), (2), and (3), and the area A i for each of
these shapes is computed. For calculation purposes, a reference
axis must be established. In this example, the reference axis will
be placed at the lower edge of the 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 shape is summarized in the table below.
109 mm
90 mm
6 mm
18 mm
90 mm
10 mm
30 mm
(3)
6 mm
18 mm
90 mm
A i
(mm 2 )
y i
(mm)
y i A i
(mm 3 )
(1) 900 5 4,500
(2) 540 55 29,700
(3) 540 109 58,860
1,980 93,060
55 mm
Reference
axis 5 mm
(2)
90 mm
(1)
10 mm
30 mm
y
yA i i
= Σ ΣA
i
3
93,060 mm
= = 47.0 mm Ans.
2
1,980 mm
18 mm
Therefore, the centroid is located 47.0 mm above the lower edge
of the shape.
z
y
6 mm
90 mm
−y = 47.0 mm
Reference
axis
90 mm
10 mm
793