Page 49 - Mechanics Analysis Composite Materials

34 Mechanics and analysis of composilr materials

Shear stress in new coordinates is

Permutation yields expressions for and Z~J,,.

2.4. Principal stresses

The foregoing equations, Eqs. (2.8) and (2.9), demonstrate stress transformations
under rotation of a coordinate frame. As known, there exists a special position
of this frame in which shear stresses acting on coordinate planes vanish. Such
coordinate axes are called the principal axes, and normal stresses that act on the
corresponding coordinate planes are referred to as the principal stresses.
To determine the principal stresses, assume that coordinates x', y', and z' in
Fig. 2.4 are the principal coordinates. Then, according to the aforementioned
~
J
property of the principal coordinates we should take ?zlxl = T ~ = 0 and 02 = a for
the plane z' = 0. This means that px = al;~,.,pv = ol??, and pi = aZyzin Eqs. (2.7).
Introducing new notations for directional cosines of the principal axis, i.e., taking
lyx = IF, Iz.~= I,,, 12z = IF we have from Eqs. (2.7)

(2.10)

These equations were transformed with the aid of symmetry conditions for shear
stresses, Eqs. (2.6). For some specified point of the body in the vicinity of which
the principal stresses are determined in terms of stresses referred to some fixed
coordinate frame x, y, z and known, Eqs. (2.10) comprise a homogeneous system of
linear algebraic equations. Formally, this system always has the trivial solution, i.e.,
lpx= lpy = I, = 0 which we can ignore because directional cosines should satisfy
an evident condition following from Eqs. (2.1), i.e.

(2.1 1)

So, we need to find a nonzero solution of Eqs. (2.10) which can exist if the
determinant of the set is zero. This condition yields the following cubic equation
for a

~3-1,a2-40-13=0 , (2.12)