Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 144

require that the stress distribution T 33 satisfy the conditions that the axial load on the
beam is zero, that the bending moment about the x 1 axis is equal to M 1 and that the
bending moment about the x 2 axis is equal to 0. The respective integrals in the 1,
2 plane over the cross-sectional area are given by
Z Z Z

T 33 d A ¼ 0 ; T 33 x 2 d A ¼ M 1 ; T 33 x 1 d A ¼ 0 :

A A A

The second integral equates M 1 to the couple generated by the integral of the
moment of the stress T 33 times the area patch at a location x 2 . The positive direction
is determined by the right hand rule. Substituting T 33 ¼ c o þ c 1 x 1 þ c 2 x 2 into

these three integrals, it follows that
c o Z c 1 Z c 2 Z c o Z c 1 Z c 2 Z

x 2 d A d A þ x 1 d A þ x 2 d A ¼ 0 ; x 2 d A þ x 1 x 2 d A þ ¼ M 1

A A A A A A

c o Z c 1 Z c 2 Z x 1 d A

x 1 d A þ þ x 1 x 2 d A ¼ 0 :

A A A

1, 2,
centroid of the cross-sectional area, denoted by x i , and the components of the area
moment of inertia tensor (A134), I 11 , I 22 , I 12 :

These results are simplified by noting the cro ss -sectional area A , the x i , i ¼

Z Z Z Z Z

1
A

x 2 d A x 1 d A

A ¼ d A ; x i ¼ x i d A ; I 11 ¼ ; I 22 ¼ ; I 12 ¼ x 1 x 2 d A :

A A A A A

Since the origin of coordinates was selected at the centroid, it follows that x i are
zero, thus c o ¼ 0. Then, since the coordinate system has been chosen to be the

principal axes of the area moment of inertia, it follows that the product of inertia I 12
vanishes, thus c 2 ¼ M 1 / I 11 and c 1 ¼ 0. It follows that T 33 ¼ M 1 x 2 / I 11 .

The solution of the stresses is then T 11 ¼ T 22 ¼ T 12 ¼ T 13 ¼ T 23 ¼ 0 and

T 33 ¼ M 1 x 2 / I 11 . The solution for the strains is

E 11 ¼ n 13 M 1 x 2
E 1 I 11

E 22 ¼ n 23 M 1 x 2
E 2 I 11

M 1 x 2
E 3 I 11 ;

; ; E 33 ¼ E 12 ¼ E 13 ¼ E 23 ¼ 0 ;

and the solution for the displacements may be obtained from the solution for the
strains and integration of the strain-displacement relations (2.49). This solution
satisfies each of the 15 equations of elasticity and the boundary conditions specified
for this problem; thus by the uniqueness theorem, it is the unique solution to this
bending problem.

Example 6.3.2
For the problem considered in Example 6.3.1, determine the solution for the
displacement field u ( x , t ) from the solution for the strains,