Page 65 - Aircraft Stuctures for Engineering Student
P. 65
50 Two-dimensional problems in elasticity
corresponding equilibrating shears along the vertical edges at x = 0 and 1. As a
solution to the stress analysis problem an Airy stress function q5 is proposed, where
+=-- [5(x3 - Z2x)(y + d)2(y - 2d) - 3yx(y2 - d2)2]
120d3
Show that q5 satisfies the internal compatibility conditions and obtain the distribution
of stresses within the plate. Determine also the extent to which the static boundary
conditions are satisfied.
PX
ax = - [5y(x2 - z2) - ioy3 + 6dZy]
20d3
ay = E( - 3yd2 - 2d3)
4d3
-P
Txy = - [5(3x2 - Z2)(y2 - d2) - 5y4 + 6y2d2 - $1
40d3
The boundary stress function values of T~~ do not agree with the assumed constant
equilibrating shears at x = 0 and 1.
P.2.4 A two-dimensional isotropic sheet, having a Young’s modulus E and linear
coefficient of expansion a, is heated non-uniformly, the temperature being T(x, y).
Show that the Airy stress function 4 satisfies the differential equation
V2(V2q5 + EaT) = 0
where
is the Laplace operator.
