$2.]$ A $20$ mm diameter hot$-$rolled carbon steel $4340$ is subjected to fluctuating $($or cyclic$)$ axial loading of $2000$ N to $5000$ N at $\backslash (900^\{\backslash$circ$\}\backslash$mathrm$\{$F$\}\backslash ).$ Using SI units,

a$.$ What is the axial fatigue strength for $1,000$ cycles of load? $[3]$

b$.$ Determine the endurance limit $($fatigue strength at infinite life$)?[7]$

c$.$ Draw neatly the S$-$N curve $($not to scale$)$ for the rod material $($list all the necessary values$/$axis$-$labels on the S$-$N curve$).[5]$

d$.$ What is the axial fatigue strength of the rod material for $500,000$ cycles of load? $[5]$

e$.$ Draw to scale the constant$-$life fatigue diagram for this problem, $[10]$

i$.$ List the scale used.

ii$.$ Properly label everything.

iii. Show the Goodman lines for $1,000$ cycles, $500,000$ cycles and infinite life.

iv$.$ From this diagram, what is the maximum mean $($static$)$ stress at $500,000$ cycles?

Note: Clearly show all the working and from where$/$how all the values were obtained.

Answer:

$-$ Fatigue strength for $1,000$ cycles, $\backslash (\backslash$boldsymbol$\{$S$\}\_\{\backslash$boldsymbol$\{$f$\}\backslash$mathbf$\{1000\}\}=\backslash )$

$-$ Endurance limit $\backslash (=\backslash )$

$-$ Fatigue strength for $500,000$ cycles, $\backslash (\backslash$boldsymbol$\{$S$\}\_\{\backslash$boldsymbol$\{$f$\}\backslash$mathbf$\{50000\}\}=\backslash )$

$-$ Max mean stress at $500,000$ cycles, $\backslash (\backslash$sigma$\_\{$m$\}=\backslash )$