Page 38 - Aircraft Stuctures for Engineering Student

1.14 Mohr's circle of strain 23

If we compare Eqs (1.3 1) and (1.34) with Eqs (1 .8) and (1.9) we observe that they may
be obtained from Eqs (1.8) and (1.9) by replacing a, by E,, u, by E~, uy by E~, 7.1. by
yxy/2 and T by y/2. Therefore, for each deduction made from Eqs (1.8) and (1.9)
concerning u, and 7 there is a corresponding deduction from Eqs (1.3 1) and (1.34)
regarding E, and y/2.
Thus, at a point in a deformable body, there are two mutually perpendicular planes
on which the shear strain y is zero and normal to which the direct strain is a maximum
or minimum. These strains are the principal strains at that point and are given (from
comparison with Eqs (1.1 1) and (1.12)) by

E, + Ey ++ JGzFz
E[ =- (1.35)
2
and

EII = ~ - - (1.36)
2
If the shear strain is zero on these planes it follows that the shear stress must also
be zero and we deduce, from Section 1.7, that the directions of the principal strains
and principal stresses coincide. The related planes are then determined from
Eq. (1.10) or from

tan 20 = ~ TXY (1.37)
E, - El.

In addition the maximum shear strain at the point is

(z) =;JGZj%i (1.38)
max
or

( = 2 (1.39)
E1 - Err

{cf. Eqs (1.14) and (1.15)}.

1__IN_--
1.14 Mi
We now apply the arguments of Section 1.13 to the Mohr circle of stress described in
Section 1.8. A circle of strain, analogous to that shown in Fig. 1.9(b), may be drawn
when ux, uy etc. are replaced by E,, E~ etc. as specified in Section 1.13. The horizontal
extremities of the circle represent the principal strains, the radius of the circle, half the
maximum shear strain and so on.