As a structural engineer for this project, your task is to design the overhang beam that is used to support the swimming pool. The beam, denoted by the bold rectangular dashed lines, is hung from the nearest support column $($not seen in the above figure$)$ to support the area of the pool as shown above with thin dotted lines. Due to the overall design of the swimming pool as has just been illustrated, each beam supports more water at the hanging end of the beam and thus we can model the loading on the beam by the pool as a linear distributed load as shown below. The beam is designed using roller and hinge support, rather than a fixed support. The area supported by the beam between the roller and the hinge is actually inside the apartment $($ie$.$ the area between the nearest and the core support columns$).$

ENGR $213$: Strength of Materials $-$ Beam Deflection

Problem $7\mathrm{.}1-$ Continued

ENGR $213$: Strength of Materials $-$ Beam Deflection

Problem $7\mathrm{.}1-$ Continued

You have done the stress analysis of this beam bending case in your previous HW$,$ so now we will do the deflection analysis for this same case $($you may re$-$use the bending moment diagram and function from the earlier HW$).$

In the present HW$,$ there is another structural criteria that the deflection of the beam must be kept below $10mm$ anywhere inside the apartment where the residents may walk around.

Given the material options below, which material will you choose in order to satisfy the above requirement? Remember the max deflection that matters is the one between points $B$ and $C($inside the apartment$).$

Stainless Steel A $(E=160$GPa, Stainless Steel B $(E=220$GPa, High Strength Ceramic $(E=350$GPa $).$

You may not have the time in this design project to determine the exact max deflection with all the material options above. How could you still make sure that Stainless Steel B would be a safe choice.