Three steel bars have a diameter of 25 mm and carbon contents of 0.2, 0.5, and 0.8%, respectively. Determine the tensile stresses and strains for each specimen at each load increme... Three steel bars have a diameter of 25 mm and carbon contents of 0.2, 0.5, and 0.8%, respectively. Determine the tensile stresses and strains for each specimen at each load increment and plot stresses versus strains for all specimens on one graph. Additionally, find the proportional limit, 0.2% offset yield strength, modulus of elasticity, strain at rupture for each specimen, and comment on the effect of increasing carbon content on yield strength, modulus of elasticity, and ductility. Read more

Steps to Solve

1. Calculate Tensile Stress Calculate the tensile stress ($\sigma$) for each bar using the formula: $$ \sigma = \frac{F}{A} $$ where $F$ is the load in Newtons (1 kN = 1000 N) and $A$ is the cross-sectional area given by: $$ A = \frac{\pi d^2}{4} $$ For a diameter ($d$) of 25 mm: $$ A = \frac{\pi (0.025)^2}{4} \approx 4.91 \times 10^{-4} \ m^2 $$

2. Calculate Tensile Strain Calculate the tensile strain ($\epsilon$) for each load increment using: $$ \epsilon = \frac{\Delta L}{L_0} $$ where $\Delta L$ is the change in length (deformation) and $L_0$ is the original length (50 mm or 0.050 m).

3. Calculate Strains and Stresses for each Specimen For each specimen (corresponding to the different carbon percentages), compute the tensile stress and tensile strain at each deformation increment listed in the table.

4. Create Stress-Strain Graphs Plot the calculated stress values ($\sigma$) on the y-axis and strain values ($\epsilon$) on the x-axis for all three specimens on the same graph.

5. Determine Proportional Limit Identify the proportional limit for each specimen, generally found as the highest stress value that shows a linear relationship with strain.

6. Calculate 0.2% Offset Yield Strength To find the 0.2% offset yield strength, draw a line parallel to the linear portion of the stress-strain curve, offset by 0.002 in strain. The intersection with the stress values gives the yield strength.

7. Compute Modulus of Elasticity The modulus of elasticity ($E$) can be calculated from the slope of the initial linear portion of the stress-strain curve: $$ E = \frac{\Delta \sigma}{\Delta \epsilon} $$

8. Strain at Rupture Identify the strain at rupture for each specimen from the final deformation value before rupture.

9. Discuss Effects of Carbon Content Analyze and comment on the effects of varying carbon content on yield strength, modulus of elasticity, and ductility by comparing the calculated values.