439

Appendix ASecond Moments of Area

The second moment of area, I, sometimes called the area moment of inertia, is a property of a shape that describes its resistance to deformation by bend-ing. The polar second moment of area, J, often called the polar moment of inertia, describes the resistance of a shape to deformation by torsion. Since the coordinate axes used to obtain the I’s and J’s listed here run through the centroid of each shape, all moments of area cited here may be thought of as having an additional subscript c denoting that they are taken relative to the centroid.

Remember the following: Iy = ∫ z2dA.

Iz = ∫ y2dA.

J = ∫ r2dA.

b

h

2h

b/2 Area (A) Second Moment

of Area (I ) Polar Second

Moment of Area(J )

bh Ix = bh3/12

Iy = hb3/12Ixy = 0

(bh3/12) (h2+b2)

440 Introduction to Engineering Mechanics: A Continuum Approach

Area (A) Second Momentof Area (I )

Polar SecondMoment of Area

(J )

bh/2 Ix= bh3/36

Ixy = bh2(b–2d)/72b

h

h31

d

(b+d)31

r

d

Area (A) Second Momentof Area (I )

Polar SecondMoment of Area

(J )

πr2 Ix = Iy = πr4/4= πd4/64

Ixy = 0J = πr4/2= πd4/32

Area (A) Moment of Inertia(I)

PolarSecond

Moment ofArea (J )

πr2/2 Ix = 0.1098r4

Iy = πr4/8Ixy = 0

JCG = Ix + Iy

Jo = πr4/4

r3π4r

d

2d Area (A) Second Moment of

Area (I )

PolarSecond

Moment ofArea (J )

π(d2–d12)/4 Ix = Iy = π(d4–d1

4)/64Ixy = 0

π(d4–d14)/32

d1d

Appendix A: Second Moments of Area 441

b b1

d1d

2d

Area (A) Second Moment ofArea (I )

PolarSecond

Moment ofArea ( J )

bd – b1d1 Ix = (bd3–b1d13)/12

Iy = (db3–d1b13)/12

Ixy = 0J = Ix + Iy

3π4r

3π4r

Area(A)

Second Moment of Area(I)

PolarSecond

Moment ofArea (J )

πr2/4π

π

Ix = 9π

9π

416 r4 = 0.05488r4

Iy =4

16Ixy = 0

r4

–

–

π9πJ = 8

8= 0.1097r4

r4–

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