A pedestrian bridge, designed for crossing a river, features a truss structure made of a steel alloy, which has a yield strength of $400$MPa, ultimate tensile strength of $550$MPa and a fracture toughness of $80$MPa $\sqrt[\phantom{2}]{m}$ at room temperature. During a safety inspection of the truss structure, a through$-$thickness edge crack was discovered on one of the truss ties. These ties have a width, $W,$ of $100mm$ and thickness, $B,$ of $20mm.$

For the analysis, the geometry and the loading on this particular section of the tie can be assumed as a plate in tension with a through thickness edge crack, as shown in the figure above. The applied stress on the tie reaches $=180$MPa during peak usage of the bridge. Additionally, it is known that the ties possess a tensile residual stress resulting from the manufacturing process, with a maximum magnitude of $120$MPa.

Determine the minimum crack length, $,$ that can lead to failure of the tie and the corresponding failure mode.

Choose the one option that is closest to the correct answer.

$24mm,$ plastic collapse

$24mm,$ brittle fracture

$7mm,$ plastic collapse

$13mm,$ brittle fracture

$7mm,$ brittle fracture

$13mm,$ plastic collapse