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

792
gEOMETRIC PROPERTIES
OF AN AREA
y
y
c
dA
y−
Area A
y
Area A
x−
(a)
x
x
(b)
x
FIGURE A.1 Centroid of an area.
hence, the centroid calculation procedure for composite areas can be arranged so that
integration is not necessary. Expressions analogous to Equation (A.1) in which the integral
terms are replaced with summation terms can be used. For a composite area composed
of i simple shapes, the centroid location can be computed with the following
expressions:
x
xA i i
= Σ ΣA
i
y
yA i i
= Σ ΣA
i
(A.2)
where x i and y i are the algebraic distances or coordinates measured from some defined
reference axes to the centroids of each of the simple shapes comprising the composite
area. The term ΣA i represents the sum of the simple areas, which add up to the total area
of the composite area. If a hole or region having no material lies within a composite area,
then that hole is treated as a negative area in the calculation procedure.
mecmovies
ExAmpLE
A.1 the Centroids Game: learning the Ropes
A game that helps to build proficiency in centroid calculations
for composite areas made up of rectangles.