$(45$ pts$)$ For this problem, the beam is assumed to be loaded using a service $($unfactored$)$ load. The beam shown below is assumed to be made using a homogeneous linear$-$elastic material.

a$.$ For the uniformly distributed service load, calculate the stresses at the top and bottom faces for the section corresponding to maximum positive moment and maximum negative moment $($follow common bending theory assumptions from strength of materials$).$

b$.$ Calculate the internal compressive and tensile resultant forces $($force couple$)$ for each of these sections, and the distance of these force resultants taken from the neutral surface.

c$.$ Using the maximum absolute tensile stress calculated from the positive and negative moment sections, determine if this beam is expected to crack if it is made of concrete with $f_c=4000$ psi $($Ref: Materials Handout; tensile strength of concrete is determined using the modulus of rupture in that handout, $f_r.$ The calculated stress in extreme tension fiber must not exceed $f_r$ for the beam to remain uncracked$).$