A pedestrian bridge, designed for crossing a river, features a truss structure made of a steel alloy, which has a yield strength of 400MPa, ultimate tensile strength of 550MPa and a fracture toughness of 80MPa 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 =180MPa 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 120MPa.

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