Page 52 - Aircraft Stuctures for Engineering Student

2.1 Two-dimensional problems 37
-_ ---.- --
I,
2.1 Two-dimensional problems
For the reasons discussed in Chapter 1 we shall confine our actual analysis to the two-
dimensional cases of plane stress and plane strain. The appropriate equilibrium
conditions for plane stress are given by Eqs (1.6), viz.
8a.x
-+-+x=o
8Txy
ax ay
aay dry,
-+-+Y=O
aY aY
and the required stress-strain relationships obtained from Eqs (1.47), namely
1
E, = E(c, - YO,)
1
Ey = -(cy - uc,)
E
2( 1 + u)
rxy = E T x y
We find that although E, exists, Eqs (1.22)-(1.26) are identically satisfied leaving
Eq. (1.21) as the required compatibility condition. Substitution in Eq. (1.21) of the
above strains gives
&,, a2 a2
2(1 + v)- = -(ay - vu,) +-(a, - vuy)
axay ax2 aY2
From Eqs (1.6)

and

Adding Eqs (2.2) and (2.3), then substituting in Eq. (2.1) for 2a2rXy/axay, we have

or

The alternative two-dimensional problem of plane strain may also be formulated in
the same manner. We have seen in Section 1.1 1 that the six equations of compatibility
reduce to the single equation (1.21) for the plane strain condition. Further, from the
third of Eqs (1.42)
a, = u(c, + cy) (since E, = 0 for plane strain)