$5.$ Exercise on Limit State Design $-$ Ultimate Limit State Design

Figure $4$ shows a reinforced concrete beam supported at $\backslash ($ A $\backslash )$ and $\backslash ($ B $\backslash ).$ The beam has a rectangular cross section of height, $\backslash ($ h $\backslash ),$ equal to $50$ cm and width, $\backslash ($ b $\backslash ),$ equal to $25$ cm $.$ The beam supports a uniformly distributed permanent action $($excluding its self$-$weight$)$ of $\backslash (10\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}\backslash )$ and a uniformly distributed variable action of $\backslash (7\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}\backslash ).$ Assuming that the weight of the reinforced concrete is equal to $\backslash (25\backslash$mathrm$\{$kN$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash ),$ calculate the shear forces and bending moments using ultimate limit state design. Provide the shearing$-$force and bending$-$moment diagrams.

$($N$/$m

$10$kN$/$m

Figure $4.$ Reinforced concrete beam subjected to uniform actions.