4- A thin rectangular plate of unit thickness is loaded along the edge y=+d by a linearly varying distributed load of intensity w=px with corresponding equilibrating shears along the vertical edges at x=0 and I. As a solution to the stress analysis problem, an Airy stress function varnothing is proposed as emptyset =(P)/(120d^3)[5(x^3-l^2x)(y-2d)-3xy(y^2-d^2)^2] Show that sb satisfies the internal compatibility conditions and obtain the distribution of stresses within the plate Determine also the extent to which the static are satisfied. Hint: sigma _(xx)=(partial ^2phi )/(partial y^2) etc. and biharmonic eq. (partial ^4emptyset )/(partial x^4)+2(partial ^4emptyset )/(partial x^2)partial y^(2)+(partial ^4emptyset )/(partial y^4)=0 (25 points)

To show that the Airy stress function satisfies the internal compatibility conditions, we need to verify that it satisfies the biharmonic equation: Let's start by finding the fourth partial derivatives of with respect to and .First, we find the first partial derivatives of with respect to and : Next, we find the second partial derivatives of with respect to and : Now, we find the third partial derivatives of with respect to and : Finally, we find the fourth partial derivatives of with respect to and : Now, we can substitute these derivatives into the biharmonic equation: Simplifying, we get: This shows that the Airy stress function satisfies the biharmonic equation.To obtain the distribution of stresses within the plate, we can use the stress function to find the normal and shear stresses. The normal stress is given by: Substituting the expression for , we get: Similarly, the shear stress is given by: Substituting the expression for , we get:$\tau_{xy} = \frac{P}{120d^{3}}[15(x^{2}-l^{2})-12y(y^{2}-d^{2