There is a $1$D bar element as shown in the figure.

Solve the displacement field of a beam under the influence of a uniform load using Method of Weighted Residuals. The beam has the following characteristics:

Length $=L$

Cross $-$ sectional area $=A($ constant $)$

Supports: $2$ fixed end supports

Load: The body force in addition to the uniformly distributed load, $f$ is equal to $3fl,$ which is located at a distance of $\frac{L}{2}$ from the left end of the beam which is acting at the center of cross section of the beam.

There is a traction of $2fl$ acting in the $-x$ direction at right end of the beam.

To solve the problem, the following steps must be considered:

Model the beam using linear, two$-$noded element.

Apply the appropriate boundary conditions and element stiffness equations.

Obtain and present the displacement field and stresses, including a plot of the displacement versus the position along the beam.

Compare the approximate solution with exact solution $($hint: you may consider calculation of exact solution using dirac delta function$)$