1.A metal plate has rectangular axes Ox, Oy marked on its surface.The
point O and the direction Ox are fixed in space and the plate is
subjected to the  following uniform stresses compressive 4p parallel
to OX ,tensile 2.5P parallel to OY and shearing 3.5P on planes
parallel to OX and OY in a sense tending to decrease the angle xOy.
Determine the direction in which point P will bedisplaced. The
coordinates of the point before straining are (1.8,
3.2). Use ?=0.25.
2.Consider the uniformly loaded simply supported beam shown in the
figure.Determine the constants of the Airy stress function and the
corresponding stress distribution. Clearly indicate the boundary
conditions used for determining the contants:
?(x,y)=Ax^2+Bx^2y+ Cy^3+ Dx^4y +Fx^2y^3+Gy^5
3.A cantilever beam is loaded at one end by a vertical force P as
shown in the figure.Show that the stress distribution, as calculated
by simply beam theory, can be represented by the expression
?(x,y)=Ay^3+ By^3x+Cxy as an Airy stress function, and determine the
coefficients A, B, and C.
4.A two-dimensional isotropic sheet, having a modulus of elasticity E
and a linear coefficient of thermal expansion ?, is heated uniformly
with temperature T(x,y).Show that the Airy stress function ? satisfies
the differential equation:
(?^2/?x^2+ ?^2/?/y^2)(?^2?/?x^2+?^2?/?y^2+E?T=0