Question $3$:

Aliong thin$-$wailled cylindrical pressure vessel $($length $\backslash (\backslash$tau $\backslash ),$ radius $\text{'}\backslash ($ r $\backslash )\text{'},$ wall thickness $\text{'}\backslash (\backslash$tau $\backslash )\text{'},\backslash ($ r$>>$t $\backslash ))$ just fits between two rigid end walls when there is no pressure in the tank. $\backslash ($ A $0^\{\backslash$circ$\}-45^\{\backslash$circ$\}-90^\{\backslash$circ$\}\backslash )$ rosette is applied on its outer surface to measure strains when the tank is filled with pressurised gas.

A$.$ Find the force exerted by the tank on the walls when it filled with pressure $\text{'}\backslash ($ p $\backslash )\text{'}.$

B$.$ If the rosette is aligned such that the $\backslash (0^\{\backslash$circ$\}\backslash )$ strain gauge measures the longitudinal strain, estimate the strains measured by each strain gauge in the rosette

Question $4$:

A$.$ A carpenter uses a sawmill to cut rectangular wooden beams $(\backslash ($ E$\_\{$w$\}=10\backslash$mathrm$\{$GPa$\}\backslash ))$ from large, circular tree logs$.$ If a single $\backslash (\backslash$log $($d$=50\backslash$mathrm$\{$~cm$\})\backslash )$ is used to make two beams, each with breadth $\text{'}\backslash ($ b $\backslash )\text{'}$ and height h $\text{'},$ determine the optimal dimensions $\backslash (($b$,$ h$)\backslash )$ that will maximize their strength in terms of their resistance to pure bending.

B$.$ The carpenter decides to make a sturdy beam by smartly joining two of these rectangular beams $(\backslash ($ b $\backslash ),\backslash ($ h $\backslash )),$ either with a central square beam at the top thereby making it an inverted U$-$profile beam, or with a steel plate $(\backslash (\backslash$mathrm$\{$E$\}\_\{\backslash$pm$\}=200\backslash$mathrm$\{$GPa$\}\backslash ))$ thereby making it a composite beam. Which one of the $\backslash ($ U $\backslash )-$profile or composite beams will minimise bending deflection better?

C$.$ Now that the carpenter has figured out the most optimal sturdy beam out of the two, he uses them in the construction of a roof. Find the deflection in a horizontal beam $($length $=$ Sm $)$ located in a specific section of the roof, where it acts as a cantilever to sustain a load of $1$ kN at its midpoint.

Bonus Question:

Have you seen the pillars of the new overhead metro lines in Mumbai? Have you noticed most of them have a circular cross$-$section, but there are some that have a square cross$-$section? Which columns do you think have a square cross$-$section? Why?