A uniform square beam of sides 200mm is to be used as one side of a ladder chassis of a commercial truck. The beam is to be constructed of steel, E=205Gpa and yield stress,320Mpa. A load analysis of the chassis reveals that the beam is uniformly varying from 30kN/m to 90kN/m spreading along the whole span. The design length of the beam is 6m and it's to be bolt fastened at the ends , hence can be treated as jointed ends. The design standard requires a design with Factor of safety of at least 2.0 and central deflection not more than 40mm. a) Determine the maximum bending moment in the beam and hence the maximum stress in the beam. b) The maximum deflection that occurs in the beam c) Does the design meet both standard requirements of the chassis design? Clearly show the working.
The text is not very clear to me, particularly about "beam is uniformly varying". There are considerations: If both ends of the beam are firmly bolted, then the beam may be over-constrained (depends on how the're bolted). Either way, you'd probably want them consider a supports (remove 1 DOF), otherwise it would get more complicated. Another thing is, that there are probably some other conditions, like cross members of the ladder frame, holes, etc., which would - in reality - affect the stress. Also, I'd be surprised, if you have only one load case.
To calculate a maximum of bending moment and stress in that part of beam for a uniform load is piece of cake. The maximum is in the middle M_max = q * l^2 / 8 (where q is load, l is length). The stress to S = M /W (where W is the cross section module, which you didn't wrote in your text. You can find the formula for W for various section in engineering hanbook). Safety is then equal to S / S_allowed (320MPa). However, when you have variyng load, it gets more complicated.
I'd use a FEM software to calculate the stress with all the boundary conditions. You can use FreeCAD and it's FEM module for a start.