A structural steel welded plate section, fabricated from two $12$ in$.x\sqrt[3]{4}$ in$.$ flange plates and one $12$ in$.x\frac{1}{2}$ in$.$ web plate, all of ASTM A$572,$ Grade $42,$ high$-$strength low$-$alloy steel, is shown on the back of this page. Assume that the welding process has introduced residual stresses into both flanges, as well as into the web, of the section in the patterns indicated. $($These residual stresses may be assumed as constant through the thickness of the plates.$)$ For each part of the problem described below, evaluate the welded plate section twice: i$)$ first by neglecting residual stresses, and ii$)$ then with the magnitude of the "baseline" residual stress $(F_r)$ value set to $9$ksi. And, before even embarking on the parts of the problem described below, please first confirm that the provided residual stress patterns, when integrated over the entire cross$-$section, sum to a net zero force, and that they generally comprise tensile and compressive residual stresses at the expected locations.

a$)$ Assuming the welded plate section is loaded in uni$-$axial tension $($or compression$),$ determine and plot the average tensile stress $(f_t)$ vs$.$ average tensile strain $({}_t)($and average compressive stress $(f_c)$ vs$.$ average compressive strain $({}_c))$ as the section goes from zero load up to a tension $($or compression$)$ force corresponding to full yield of the cross$-$section.

b$)$ Assuming that the welded plate section is used as a beam $("$plate girder"$)$ bending about its strong axis, determine and plot the bending moment $(M)$ vs$.$ curvature $()$ as the beam crosssection goes from zero moment up to the fully plastic moment.