Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 375

Let the base b be along the x 1 axis and the height h be along the x 2 axis and the sloping
face of the triangle have end points at ( b , 0) and (0, h ). Determine the area moments
and product of inertia of the right-triangular plate relative to this coordinate system.
Construct the matrix of tensor components referred to this coordinate system.

Solution: The integrations (A.134) yield the following results:

ð ð

h

x 2 d x 2 ¼

x 2
h

bh 3
12 ;

hb 3
12

I Area
11

x 2 d x 1 d x 2 ¼ I Area

22

¼ b 1 ¼

O 0

ð
O ð

ð

;

h

2

b 2 h 2
24

1
2

x 2
h

I Area
12

b 2

¼ x 1 x 2 Þ d x 1 d x 2 ¼ 1 x 2 d x 2 ¼

0

thus the matrix to tensor components referred to this coordinate system is

bh
24

2 h 2 bh

I Area ¼ :

Æ

2 b 2 hb

Example A.9.5

In the special case when the triangle in Example A.9.4 is an isosceles triangle, that
is to say b h , find the eigenvalues and eigenvectors of the matrix of tensor

components referred to the coordinate system of the example. Then find the matrix
of tensor components referred to the principal, or eigenvector, coordinate system.

¼

Solution: The matrix of tensor components referred to this coordinate system is

bh
24

2 h 2 bh

I Area ¼ :

2 b 2

hb

The eigenvalues of I are h 4 /8 and h 4 /24. The eigenvector (1/ √ 2)[1, 1] is

associated with the eigenvalue h 4 /8 and the eigenvector (1/

2)[1, 1] is associated
with the eigenvalue h 4 /24. The orthogonal transformation that will transform the
matrix of tensor components referred to this coordinate system to the matrix of
tensor components referred to the principal, or eigenvector, coordinate system is
then given by

√

1
2

11

Q ¼ p :

11

Applying this transformation produced the matrix of tensor components referred
to the principal, or eigenvector, coordinate system