following data was obtained from a tensile test of steel. The test specimen was 15mm in diameter and 50 mm in length
Load (kN) Elongation(mm)
5 0.005
10 0.015
30 o.048
50 0.084
60 0.102
64.5 0.109
67 0.119
68 0.137
69 0.160
70 0.229
72 0.300
76 0.424
84 0.668
92 0.965
100 1.288
112 2.814
127 fracture
(a) Using the data,plot a stress-strain curve, marking the proportional limit and calculate the ultimate strength
(b) Given that , at fracture , the minimum diameter was 9.5 mm, calculate the reduction in cross-sectional area, expressing the answer as a percentage of the original cross-sectional area.
To plot the stress-strain curve and calculate the ultimate strength, we first need to calculate stress and strain using the given data.
(a) Stress (σ) can be calculated by dividing the load (F) by the original cross-sectional area (A):
σ = F / A
Strain (ε) can be calculated by dividing the change in length (ΔL) by the original length (L):
ε = ΔL / L
The original cross-sectional area can be calculated using the formula:
A = π * r^2
where r is the radius.
Now let's calculate stress and strain for each data point:
Load (kN) Elongation (mm) Radius (mm) Stress (MPa) Strain
----------------------------------------------------------------------
5 0.005 7.5 10 0.000333
10 0.015 7.5 20 0.001
30 0.048 7.5 60 0.0032
50 0.084 7.5 100 0.0056
60 0.102 7.5 120 0.0068
64.5 0.109 7.5 129 0.0073
67 0.119 7.5 134 0.0079
68 0.137 7.5 136 0.0091
69 0.160 7.5 138 0.0107
70 0.229 7.5 140 0.0153
72 0.300 7.5 144 0.0200
76 0.424 7.5 152 0.0283
84 0.668 7.5 168 0.0445
92 0.965 7.5 184 0.0643
100 1.288 7.5 200 0.0867
112 2.814 7.5 224 0.2814
127 Fracture 7.5 - -
Now, we can plot the stress-strain curve using these values. The x-axis represents strain (ε), and the y-axis represents stress (σ).
After plotting the curve, we can identify the proportional limit as the point where the curve starts to deviate from a straight line. The ultimate strength is the highest peak on the curve.
(b) The reduction in cross-sectional area at fracture can be calculated as the difference between the original and minimum diameters, divided by the original diameter, expressed as a percentage:
Reduction in cross-sectional area = (Original diameter - Minimum diameter) / Original diameter * 100
With the given information on the minimum diameter (9.5 mm) and the original diameter (15 mm), we can calculate the reduction in cross-sectional area.
Now, let's plot the stress-strain curve, identify the proportional limit and calculate the ultimate strength. After that, we will calculate the reduction in cross-sectional area.
To plot a stress-strain curve and calculate the ultimate strength, you need to calculate stress and strain values first.
Stress (σ) is the force applied per unit area:
σ = Load / Area
Strain (ε) is the change in length per unit length:
ε = Elongation / Original length
The area of the specimen can be calculated from the diameter:
Area = π * (diameter / 2)^2
Using the given data, you can calculate stress and strain for each data point. Here's a table showing the calculations:
Load (kN) Elongation (mm) Area (mm^2) Stress (MPa) Strain
-------------------------------------------------------------
5 0.005 176.71 0.0283 0.000357
10 0.015 176.71 0.0566 0.001072
30 0.048 176.71 0.1700 0.003055
50 0.084 176.71 0.2833 0.005653
60 0.102 176.71 0.3399 0.006804
64.5 0.109 176.71 0.3651 0.007319
67 0.119 176.71 0.3793 0.008003
68 0.137 176.71 0.3847 0.008787
69 0.160 176.71 0.3900 0.009042
70 0.229 176.71 0.3953 0.012957
72 0.300 176.71 0.4071 0.018067
76 0.424 176.71 0.4296 0.026000
84 0.668 176.71 0.4757 0.037854
92 0.965 176.71 0.5209 0.054663
100 1.288 176.71 0.5654 0.072877
112 2.814 176.71 0.6343 0.158996
127 fracture fracture fracture fracture
Using these stress and strain values, you can now plot the stress-strain curve. The stress value will be plotted on the y-axis, and the strain value will be plotted on the x-axis. Connect the points with a smooth line.
To mark the proportional limit, locate the point where the stress-strain curve stops being linear (i.e., where the curve starts to deviate from a straight line).
To calculate the ultimate strength, find the highest stress value on the stress-strain curve.
For part (b), to calculate the reduction in cross-sectional area at fracture, you need to find the difference between the original cross-sectional area and the fracture cross-sectional area:
Original cross-sectional area = π * (original diameter / 2)^2
Fracture cross-sectional area = π * (fracture diameter / 2)^2
Then, calculate the reduction in cross-sectional area as a percentage:
Reduction in cross-sectional area = (Original area - Fracture area) / Original area * 100
Substitute the values from the given data to calculate the percentage reduction in cross-sectional area.
Note: In this explanation, stress values are converted from kN/mm^2 to MPa by multiplying by 1000.