properties of area elements
TRANSCRIPT
Properties of Area ElementsNotation:
C = centroidr = radiusx = x centroid locationy= y centroid locationx', y' = centroidal axesA = areaP = perimeterIx = area moment of inertia about the x axisIy = area moment of inertia about the y axisIx' = area moment of inertia about the x' axisIy ' = area moment of inertia about the y' axis
Circle
r x'
y, y'
C
x
rP
rA
ry
x
2
0
2
441
''
445
rII
rI
yx
x
Semicircular Area
r x'
y, y'
C
x
)2(
3
4
0
221
rP
rA
ry
x
42
'
481
'
72
649rI
rIII
x
yyx
Quarter circular Area
r x'
y'
C x
y
)2(
3
4
21
241
rP
rA
ryx
42
''
4161
144
649rII
rII
yx
yx
Circular Sector Area
r
x, x'
y'
C
y
)1(2
03
sin2
2
rP
rA
y
rx
9
)(sin4))2sin((
))2sin((
))2sin((
24
214
41
'
214
41
214
41
'
rrI
rI
rII
y
y
xx
Circular Segment Area
r x
C
y, y'
x'
)sin(2
)cossin(
)cossin(3
sin2
0
2
3
rP
rA
ry
x
)cossin3
cossin23(
)cossin
cossin2(
34121
'
3441
rII
rI
yy
x
Ellipse
a
x, x'
y, y'
C
b
abA
y
x
0
0
341
341
abI
baI
y
x
Rectangle
h x'
y'
C
b
y
x
bhP
bhA
hy
bx
2
21
21
3121
'
3121
'
331
331
bhI
bhI
bhI
bhI
y
x
y
x
Trapezoid
h x'
y'
C
b
a
x
y
)(
2
3
21 bahA
ba
bahy
)(36
)4(
)3(223
'
3121
ba
babahI
bahI
x
x
Triangle
h x'
y'
C
b x
y c
hbA
hy
cbx
21
31
31 )(
)(
)(
22361
'
3361
'
22121
3121
cbcbhbI
hbI
cbcbhbI
hbI
y
x
y
x
Right Triangle
h x'
y'
C
b x
y
22
21
31
31
hbhbP
hbA
hy
bx
hbI
hbI
hbI
hbI
y
x
y
x
3361
'
3361
'
3121
3121
Quarter-circle spandrel
r
x'
y'
C x
y
rrP
rrA
ry
rx
21
2412
2
)4(3
)310(
)4(3
2
4
4
163
1
16
51
rI
rI
y
x
Parabolic semisegment
2
2
1a
xby
x '
y '
C
x
y
a
b
abA
by
ax
32
52
83
3
3
15
2105
16
baI
abI
y
x
Parabolic spandrel
22x
a
by
x '
y '
C
x
y
a
b
abA
by
ax
31
103
43
5
213
3
baI
abI
y
x