Exercise $1$

The axially loaded element shown in Figure $1$ below has a cross$-$section with a diameter of $250$ mm $.$ Assuming linear elastic behaviour and given $\backslash ($ E$=200\backslash$mathrm$\{$GPa$\}\backslash )$ :

$($a$)$ Find the support reaction at A $($in terms of $\backslash ($ P $\backslash ))$

$($b$)$ Determine the internal axial force in element AB

$($c$)$ Hence, determine the average normal stress in element AB

$($d$)$ Using Hooke$$s law determine the average the normal strain in element AB $($e$)$ Based on your answer to $($d$),$ find the axial deformation in AB$.$ Compare this to the axial deformation equation $($$)$ covered in Topic $4$ on axially loaded members. $($f$)$ Finally, what is the maximum value of P in kN that can be carried by this element while maintaining linear elastic behaviour if the $2$ separate conditions $($i$)$ and $($ii$)$ below are to be considered and compared. Take L $=3\mathrm{.}5$ m$.($i$)$ the material has a yield strength of y $=275$ MPa $($ii$)$ you have been requested to limit the average normal strains to $=0\mathrm{.}0009$

$($d$)$ Using Hooke's law determine the average the normal strain in element $\backslash ($ A B $\backslash )$

$($e$)$ Based on your answer to $($d$),$ find the axial deformation in AB$.$ Compare this to the axial deformation equation $(\backslash (\backslash$delta $\backslash ))$ covered in Topic $4$ on axially loaded members.

$($f$)$ Finally, what is the maximum value of $\backslash ($ P $\backslash )$ in kN that can be carried by this element while maintaining linear elastic behaviour if the $2$ separate conditions $($i$)$ and $($ii$)$ below are to be considered and compared. Take $\backslash ($ L$=3\mathrm{.}5\backslash$mathrm$\{$~m$\}\backslash ).$

$($i$)$ the material has a yield strength of $\backslash (\backslash$sigma$\_\{$y$\}=275\backslash$mathrm$\{$MPa$\}\backslash )$

$($ii$)$ you have been requested to limit the average normal strains to $\backslash (\backslash$varepsilon$=0\mathrm{.}0009\backslash )$