This problem will be useful when we start designing beams. Very often we know the required geometry of a beam, and the external loads that will be applied to the beam. But we do not know the self$-$weight of the beam. During the design process, we will check to see if various beams with different weights will be safe $($due to the combination of external loads and self$-$weight$),$ and ultimately, we will select the lightest, safe

beam for the optimal design. To do this, it is useful if we can solve for maximum shear forces and bending moments as a function of the beam weight$/$ ft $,$ w $.$ That is what we will do in this case. We will also assume that the total weight of the beam $(50$ xxw$)$ is much less than the applied load $(20$k$)$ :

$50$w $<mtext></mtext><mn>2</mn><mn>0</mn>$k$",$ therefore, $"$w $<mtext></mtext><mn>0</mn><mn>.</mn><mn>4</mn>$k$//$ft

For the beam above, draw shear and moment $($V&M$)$ diagrams, and find the maximum positive and negative $(+$&$-)$ shear forces and bending moments in the beam. All values will be a function of w$.$

THE ANSWERES ARE: Vmax$+=(12+25$w$)$ k$,$ Vmax$-=(-8-25$w$)$ k$,$ Mmax$+=(240+300$w$)$ k$,$ Mmax$-=0$ k$*$ft