QUESTION $3(45$ pts$)$

A pin in a knuckle joint carrying a tensile load $F$ deflects somewhat on account of this loading, making the distribution of reaction and load as shown in part $($b$)$ of the figure. A common simplification is to assume uniform load distributions, as shown in part $($c$).$ To further simplify, designers may consider replacing the distributed loads with point loads, such as in the two models shown in parts $d$ and $e.$ For the present problem, assume $F=1000N$

a$)$ Write down the corresponding loading functions $($using singularity functions$)$ for the three idealizations of the loading.

b$)$ Obtain the shear force and bending moment functions for three idealizations.

c$)$ Draw the shear force and bending moment diagrams for three idealizations.

d$)$ Calculate the maximum values of the shear force and bending moment for three idealizations.