Page 34 - Aircraft Stuctures for Engineering Student

1 .IO Compatibility equations 19

Expanding and neglecting fourth-order powers gives
2( aw/ax)SxSz + 2( du/dz)SxSz
COS A'O'C' =
2sxsz
or
+
dw au
"ixz = - -
ax az
av au
Similarly Yxy = - + - (1.20)
ax ay
+
aw av
7,; = - - J
ay dz
It must be emphasized that Eqs (1.18) and (1.20) are derived on the assumption that
the displacements involved are small. Normally these linearized equations are
adequate for most types of structural problem but in cases where deflections are
large, for example types of suspension cable etc., the full, non-linear, large deflection
equations, given in many books on elasticity, must be employed.

p- -=J--y'*y- ...~ , ,. . .-.*, .,. , . .,
1 .10 Compatibility equations
In Section 1.9 we expressed the six components of strain at a point in a deformable
body in terms of the three components of displacement at that point, u, v and w.
We have supposed that the body remains continuous during the deformation so
that no voids are formed. It follows that each component, u, v and w, must be a
continuous, single-valued function or, in quantitative terms

If voids were formed then displacements in regions of the body separated by the
voids would be expressed as different functions of x, y and z. The existence, therefore,
of just three single-valued functions for displacement is an expression of the continu-
ity or compatibility of displacement which we have presupposed.
Since the six strains are defined in terms of three displacement functions then they
must bear some relationship to each other and cannot have arbitrary values. These
relationships are found as follows. Differentiating r,, from Eqs (1.20) with respect
to x and y gives

+--
a2y,yy a2 dv a2 au
-
~ - --
axay axay ax axay ay
or since the functions of u and v are continuous

which may be written, using Eq. (1.18)
- +-
a2y,, a*&,. d2E, (1.21)
axay ax* ay2