$3\mathrm{.}19$ Three steel bars with a diameter of $25$ mm and carbon contents of $0\mathrm{.}2,0\mathrm{.}5,$

and $0\mathrm{.}8\%,$ respectively. The specimens were subjected to tension until rup$-$

ture. The load versus deformation results were as shown in Table P$3\mathrm{.}19.$

If the gauge length is $50$ mm $,$ determine the following:

a$.$ The tensile stresses and strains for each specimen at each load increment.

b$.$ Plot stresses versus strains for all specimens on one graph.

TABLE P$3\mathrm{.}19$

c$.$ The proportional limit for each specimen.

d$.$ The $0\mathrm{.}2\%$ offset yield strength for each specimen.

e$.$ The modulus of elasticity for each specimen.

f$.$ The strain at rupture for each specimen.

g$.$ Comment on the effect of increasing the carbon content on the following:

i$.$ Yield strength

ii$.$ Modulus of elasticity

iii. Ductility