Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 147

constant components of u ; they are the values of u at the origin. The rigid object
rotation Y may be determined from the displacement field u using (2.49), thus

M 1 x 3
I 11

1
E 1 þ

1
E 3

Y 12 ; Y 13 ; Y 23 :

Y 12 ¼ Y 13 ¼ Y 23 ¼ þ

Y 23 therefore represent the superposed rigid object rota-
tion, the values of the rigid object rotation at the origin. If the superposed rigid object
translation and rotation are zero, then the displacement field u is given by

The constants Y 12 ; Y 13 ;

E 3 n 23 x 2

x 3

E 2 þ n 13 x 1

u 1 ¼ n 13 M 1 x 2 x 1
E 1 I 11

M 1
2 I 11

M 1 x 2 x 3
E 1 I 11 ;

; u 2 ¼ ; u 3 ¼

E 1

and the rigid object rotation Y has only one nonzero component,

M 1 x 3
I 11

1
E 1 þ

1
E 3

Y 23 ¼ ;

which represents the rotation along the beam as the distance increases from the
beam end at the origin of coordinates. The total rotation between the two ends of the
bent beam of length L is then given by

M 1 L
I 11

1
E 1 þ

1
E 3

Y Total end - to - end rotation
23

¼ :

Example 6.3.3
In mechanics of materials the deflection curve for a beam is considered to be the
deflection curve for the neutral axis, the neutral axis being by definition the curve
that coincides with the centroid of the cross-section at each cross-section. Using the
results of the problem considered in Examples 6.3.1 and 6.3.2, determine the
formula for the deflection curve for a beam subject to pure bending.

Solution : The displacement of the neutral axis of a beam subjected to pure bending
may be determined from the formulas for the displacement field given in Example
6.3.2 above,

E 3 n 23 x 2

x 3

E 2 þ n 13 x 1

u 1 ¼ n 13 M 1 x 2 x 1
E 1 I 11

M 1
2 I 11

M 1 x 2 x 3
E 1 I 11 :

; u 2 ¼ ; u 3 ¼

E 1

The centroid of the beam's cross-section in Examples 6.3.1 was set at the
origin of coordinates in the planar cross-section, thus, for the centroid, x 1 and x 2
are zero and

M 1 x 3

u 1 ¼ 0 ; u 2 ¼

2 I 11 E 3 ;

u 3 ¼ 0 :