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

APPENDIXA
geometric Properties
of an Area
A.1 centroid of an Area
The centroid of an area refers to the point that defines the geometric center of the area. For
an arbitrary shape (Figure A.1a), the x and y coordinates of the centroid c are determined
from the formulas:
The term first moment is used to
describe x dA since x is a term
raised to the first power, as in
x 1 = x. Another geometric
property of an area, the moment
of inertia, includes the term x 2 ,
and hence, the area moment of
inertia is sometimes referred to
as the second moment of area.
x
∫
∫
xdA
ydA
A
A
= y = (A.1)
dA
dA
A
The expressions x dA and y dA are termed the first moments of area dA about the y and the
x axis, respectively (Figure A.1b). The denominators in Equation (A.1) are expressions of
the total area A of the shape.
The centroid will always lie on an axis of symmetry. In cases where an area has two
axes of symmetry, the centroid will be found at the intersection of the two axes. Centroids
for several common plane shapes are summarized in Table A.1.
∫
∫
A
composite Areas
The cross-sectional area of many common mechanical and structural components can
often be subdivided into a collection of simple shapes such as rectangles and circles.
This subdivided area is termed a composite area. By virtue of the symmetry inherent in
rectangles and circles, the centroid locations for these shapes are easily determined;
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