Page 35 - Aircraft Stuctures for Engineering Student

20 Basic elasticity

In a similar manner
2
d r,, - d2E, a2E-
-
-- +L (1.22)
ayaz az2 ay2
(1.23)

If we now differentiate rx, with respect to x and z and add the result to T:.,, differ-
entiated with respect toy and x, we obtain

or

Substituting from Eqs (1.18) and (1.21) and rearranging
(1.24)

Similarly

(1.25)

and

(1.26)

Equations (I .21)-( 1.26) are the six equations of strain compatibility which must be
satisfied in the solution of three-dimensional problems in elasticity.

Although we have derived the compatibility equations and the expressions for strain
for the general three-dimensional state of strain we shall be mainly concerned with
the two-dimensional case described in Section 1.4. The corresponding state of strain,
in which it is assumed that particles of the body suffer displacements in one plane
only, is known as plane strain. We shall suppose that this plane is, as for plane stress,
the xy plane. Then E_, "/.uz and -yr_ become zero and Eqs (I. 18) and (1.20) reduce to
dU dV
E, = -, E! = - (1.27)
dX aY
and
dv du
T.yy = - + - ( 1.28)
ax ay