Suppose we have a 4 metre long non-uniform beam. The beam is simply supported on the left and elastically supported on the right.

The beam is thinner on the right end than it is on the left end. The width at any point, 𝑥, the beam is given by the function 𝑊(𝑥)=3𝑥+2. The flexural rigidity of the beam is directly proportional to the width of the beam at that point, with the flexural rigidity equal to 100,000,𝑁𝑚2 at the thickest part of the beam.

Two blocks are placed on the beam. Each box is 3 metres long, 1 metre high, and applies 21𝑁 of downward force in total. The first box is placed directly on the beam flush with the left hand side of the beam. The second box is placed on top of the first box, but is flush with the right hand side of the beam.

In this scenario:

The load for each box is evenly distributed over it’s contact surface (i.e., evenly distributed over the length at which it touches something).

The spring of the elastic support has stiffness given by the constant 𝑘0=-1.

Find the deflection function 𝑦(𝑥) for the beam and load.

Make sure to use the following boundary conditions y(0)=0, y”(0)=0, y”(a)=0, EIy”'(a)+K0y(a)=0