Page 27 - Aircraft Stuctures for Engineering Student

12 Basic elasticity

while aI1 is algebraically the least. Therefore, when aII is negative, i.e. compressive, it is
possible for oII to be numerically greater than oI.
The maximum shear stress at this point in the body may be determined in an
identical manner. From Eq. (1.9)
dr
-
- - (a, - cy) cos 28 + 2rxy sin 28 = 0
de
giving
(1.13)

It follows that

-(ax - ay) 2TXy
sin 20 = 4 7
c0s28
Jm' - ay) +4T?y
=
ax
(0, - a,) -2rXy
sin 2(0 + ~/2) = , COqe + +) =
JK7-X JiL7Gi
Substituting these values in Eq. (1.9) gives

Tmax,min = *t d~ax ay)2 + 4~;y (1.14)
-
Here, as in the case of principal stresses, we take the maximum value as being the
greater algebraic value.
Comparing Eq. (1.14) with Eqs (1.11) and (1.12) we see that

(1.15)
Equations (1.14) and (1.15) give the maximum shear stress at the point in the body in
the plane of the given stresses. For a three-dimensional body supporting a two-
dimensional stress system this is not necessarily the maximum shear stress at the point.
Since Eq. (1.13) is the negative reciprocal of Eq. (1.10) then the angles 28 given by
these two equations differ by 90" or, alternatively, the planes of maximum shear stress
are inclined at 45" to the principal planes.

The state of stress at a point in a deformable body may be determined graphically by
Mohr's circle of stress.
In Section 1.6 the direct and shear stresses on an inclined plane were shown to be
given by
an = a.Y cos2 8 + ay sin2 0 + rYy sin 20 (Eq. (1.8))
and

(ax - Cy) .
r= sin 20 - T~~ cos 28
2