Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 374

Formulas for the mass moment of inertia of a thin plate of thickness t and a
homogeneous material of density are obtained by specializing these results. Let

the plate be thin in the x 3 direction and consider the plate to be so thin that terms of
the order t 2 are negligible relative to the others, then the formulas (A.132) for the
components of the mass moment of inertia tensor are given by

r

t ð t ð

x 2 d x 1 d x 2 ; x 1 d x 1 d x 2 ; I 11 ¼ r I 22 ¼ r

O O

t ð x 1 þ x 2 Þ

I 33 ¼ r

O ð

d x 1 d x 2 :

t ð

I 12 ¼r

O ð

x 1 x 2 Þ d x 1 d x 2 ; I 13 ¼ 0 ; I 23 ¼ 0 (A.133)

t these components of the mass moment of inertia of a thin
plate of thickness t are called the components of the area moment of inertia matrix,

When divided by r

ð ð

I 11
r

I 22
r

I Area
11

x 2 d x 1 d x 2 ; I Area

22

x 1 d x 1 d x 2 ; ¼

t ¼

¼

t ¼

O O

ð
O ð

I 33
r

I Area
33

x 1 þ x 2 Þ

¼

t ¼

d x 1 d x 2 ;

ð
O ðx 1 x 2 Þ d x 1 d x 2 ;

I 12
r

I Area
12

I Area
13

¼ 0 ; I Area
23

¼

t ¼

¼ 0 : (A.134)

Example A.9.3
Determine the area moment of inertia of a thin rectangular plate of thickness t ,
height h , and a width of base b , and a homogeneous material of density

. Specify
precisely where the origin of the coordinate system that you are using is located and
how the base vectors of that coordinate system are located relative to the sides of the
rectangular plate.
Solution: The coordinate system that makes this problem easy is one that passes
through the centroid of the rectangle and has axes that are parallel to the sides of the
rectangle. If the base b is parallel to the x 1 axis and height h is parallel to the x 2 axis
then the integrations (A.134) yield the following results:

r

bh 3
12 ;

hb 3
12 ;

bh
12 ð

I Area
11

I Area
22

I Area
33

b 2 h 2 I Area

12

I Area
13

I Area
23

¼ ¼ ¼ þ Þ; ¼ 0 ; ¼ 0 ; ¼ 0 : Æ

Example A.9.4

Determine the area moments and product of inertia of a thin right-triangular plate of
thickness t , height h , and a width of base b , and a homogeneous material of density r .