Page 312 - Aircraft Stuctures for Engineering Student

9.2 General stress, strain and displacement relationships 293

X

z
Fig. 9.15 Axial, tangential and normal components of displacement of a point in the beam wall.
xy plane and is positive outwards; and w is an axial displacement which has been
defined previously in Section 9.1. Immediately, from the third of Eqs (1.18), we have
dW
& =- (9.24)
az
It is possible to derive a simple expression for the direct strain E, in terms of ut, wn, s
and the curvature 1/r in the xy plane of the beam wall. However, as we do not require
E, in the subsequent analysis we shall, for brevity, merely quote the expression

& =-+- (9.25)
aV, vn
as r
The shear strain y is found in terms of the displacements w and ut by considering the
shear distortion of an element 6s x Sz of the beam wall. From Fig. 9.16 we see that the
shear strain is given by

7 = 41 + 42
or, in the limit as both 6s and Sz tend to zero

(9.26)

Distorted shape
of element due \---**-
.
to shear 1 --._
f:.
I
L. -.._
-.
4

Fig. 9.16 Determination of shear strain y in terms of tangential and axial components of displacement.