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A rectangular block of steel 100 𝑚𝑚 × 100 𝑚𝑚 × 200 𝑚𝑚 long is subjected to stress
110 𝑀𝑁/𝑚2
, tensile, 60 𝑀𝑁/𝑚2
, compressive, and 45 𝑀𝑁/𝑚2
, tensile across three
pairs of faces. If 𝑣 = ⅓
and 𝐸 = 213 𝐺𝑁/𝑚2
, calculate the strain and change in
lengths in all directions. Also calculate the change in volume of the bar.?
110 𝑀𝑁/𝑚2
, tensile, 60 𝑀𝑁/𝑚2
, compressive, and 45 𝑀𝑁/𝑚2
, tensile across three
pairs of faces. If 𝑣 = ⅓
and 𝐸 = 213 𝐺𝑁/𝑚2
, calculate the strain and change in
lengths in all directions. Also calculate the change in volume of the bar.?
Most Upvoted Answer
A rectangular block of steel 100 𝑚𝑚 × 100 𝑚𝑚 × 200 𝑚𝑚 long is su...
- Strain Calculation:
< b="" />Normal Strain Calculation:
The normal strain (ε) is given by the formula:
ε = σ/E
Where,
ε = Normal strain
σ = Stress
E = Young's modulus
Given:
Stress in x-direction = 110 MN/m^2
Stress in y-direction = 60 MN/m^2
Stress in z-direction = 45 MN/m^2
Young's modulus (E) = 213 GN/m^2
Calculating strain in x-direction:
ε_x = 110/213 = 0.516 × 10^-3
Calculating strain in y-direction:
ε_y = 60/213 = 0.281 × 10^-3
Calculating strain in z-direction:
ε_z = 45/213 = 0.211 × 10^-3
- Change in Lengths:
< b="" />Change in Length in x-direction:
Using the formula:
ΔL_x = ε_x × L = 0.516 × 10^-3 × 100 = 0.0516 mm
Change in Length in y-direction:
ΔL_y = ε_y × L = 0.281 × 10^-3 × 100 = 0.0281 mm
Change in Length in z-direction:
ΔL_z = ε_z × L = 0.211 × 10^-3 × 200 = 0.0422 mm
- Change in Volume:
< b="" />Change in Volume:
The change in volume can be calculated using the formula:
ΔV/V = 3ε
Since the Poisson's ratio (ν) is given as 1/3, we have:
ΔV/V = 3(0.516 × 10^-3 + 0.281 × 10^-3 + 0.211 × 10^-3) = 3 × 1.008 × 10^-3 = 3.024 × 10^-3
Therefore, the change in volume of the steel bar is 3.024 × 10^-3 or 0.3024%.
< b="" />Normal Strain Calculation:
The normal strain (ε) is given by the formula:
ε = σ/E
Where,
ε = Normal strain
σ = Stress
E = Young's modulus
Given:
Stress in x-direction = 110 MN/m^2
Stress in y-direction = 60 MN/m^2
Stress in z-direction = 45 MN/m^2
Young's modulus (E) = 213 GN/m^2
Calculating strain in x-direction:
ε_x = 110/213 = 0.516 × 10^-3
Calculating strain in y-direction:
ε_y = 60/213 = 0.281 × 10^-3
Calculating strain in z-direction:
ε_z = 45/213 = 0.211 × 10^-3
- Change in Lengths:
< b="" />Change in Length in x-direction:
Using the formula:
ΔL_x = ε_x × L = 0.516 × 10^-3 × 100 = 0.0516 mm
Change in Length in y-direction:
ΔL_y = ε_y × L = 0.281 × 10^-3 × 100 = 0.0281 mm
Change in Length in z-direction:
ΔL_z = ε_z × L = 0.211 × 10^-3 × 200 = 0.0422 mm
- Change in Volume:
< b="" />Change in Volume:
The change in volume can be calculated using the formula:
ΔV/V = 3ε
Since the Poisson's ratio (ν) is given as 1/3, we have:
ΔV/V = 3(0.516 × 10^-3 + 0.281 × 10^-3 + 0.211 × 10^-3) = 3 × 1.008 × 10^-3 = 3.024 × 10^-3
Therefore, the change in volume of the steel bar is 3.024 × 10^-3 or 0.3024%.
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A rectangular block of steel 100 𝑚𝑚 × 100 𝑚𝑚 × 200 𝑚𝑚 long is subjected to stress 110 𝑀𝑁/𝑚2 , tensile, 60 𝑀𝑁/𝑚2 , compressive, and 45 𝑀𝑁/𝑚2 , tensile across three pairs of faces. If 𝑣 = ⅓ and 𝐸 = 213 𝐺𝑁/𝑚2 , calculate the strain and change in lengths in all directions. Also calculate the change in volume of the bar.?
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A rectangular block of steel 100 𝑚𝑚 × 100 𝑚𝑚 × 200 𝑚𝑚 long is subjected to stress 110 𝑀𝑁/𝑚2 , tensile, 60 𝑀𝑁/𝑚2 , compressive, and 45 𝑀𝑁/𝑚2 , tensile across three pairs of faces. If 𝑣 = ⅓ and 𝐸 = 213 𝐺𝑁/𝑚2 , calculate the strain and change in lengths in all directions. Also calculate the change in volume of the bar.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A rectangular block of steel 100 𝑚𝑚 × 100 𝑚𝑚 × 200 𝑚𝑚 long is subjected to stress 110 𝑀𝑁/𝑚2 , tensile, 60 𝑀𝑁/𝑚2 , compressive, and 45 𝑀𝑁/𝑚2 , tensile across three pairs of faces. If 𝑣 = ⅓ and 𝐸 = 213 𝐺𝑁/𝑚2 , calculate the strain and change in lengths in all directions. Also calculate the change in volume of the bar.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rectangular block of steel 100 𝑚𝑚 × 100 𝑚𝑚 × 200 𝑚𝑚 long is subjected to stress 110 𝑀𝑁/𝑚2 , tensile, 60 𝑀𝑁/𝑚2 , compressive, and 45 𝑀𝑁/𝑚2 , tensile across three pairs of faces. If 𝑣 = ⅓ and 𝐸 = 213 𝐺𝑁/𝑚2 , calculate the strain and change in lengths in all directions. Also calculate the change in volume of the bar.?.
A rectangular block of steel 100 𝑚𝑚 × 100 𝑚𝑚 × 200 𝑚𝑚 long is subjected to stress 110 𝑀𝑁/𝑚2 , tensile, 60 𝑀𝑁/𝑚2 , compressive, and 45 𝑀𝑁/𝑚2 , tensile across three pairs of faces. If 𝑣 = ⅓ and 𝐸 = 213 𝐺𝑁/𝑚2 , calculate the strain and change in lengths in all directions. Also calculate the change in volume of the bar.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about A rectangular block of steel 100 𝑚𝑚 × 100 𝑚𝑚 × 200 𝑚𝑚 long is subjected to stress 110 𝑀𝑁/𝑚2 , tensile, 60 𝑀𝑁/𝑚2 , compressive, and 45 𝑀𝑁/𝑚2 , tensile across three pairs of faces. If 𝑣 = ⅓ and 𝐸 = 213 𝐺𝑁/𝑚2 , calculate the strain and change in lengths in all directions. Also calculate the change in volume of the bar.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rectangular block of steel 100 𝑚𝑚 × 100 𝑚𝑚 × 200 𝑚𝑚 long is subjected to stress 110 𝑀𝑁/𝑚2 , tensile, 60 𝑀𝑁/𝑚2 , compressive, and 45 𝑀𝑁/𝑚2 , tensile across three pairs of faces. If 𝑣 = ⅓ and 𝐸 = 213 𝐺𝑁/𝑚2 , calculate the strain and change in lengths in all directions. Also calculate the change in volume of the bar.?.
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