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A point in mild steel is subjected to triaxial stresses 2.5sigma(tensile),1.0sigma(compressive) and 1.5sigma(tensile).find factor of safety by various theories of failures.take sigma=50MPa,sigma_y=250Mpa,E=200Gpa and mu=0.3?
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A point in mild steel is subjected to triaxial stresses 2.5sigma(tensi...
Solution:
Given,
σ1 = 2.5σ = 2.5 × 50 = 125 MPa (tensile)
σ2 = σ = 50 MPa (compressive)
σ3 = 1.5σ = 1.5 × 50 = 75 MPa (tensile)
σy = 250 MPa
E = 200 GPa
μ = 0.3
1. Maximum Shear Stress Theory (Tresca Theory)
According to this theory, failure occurs when the maximum shear stress developed in the material is equal to the maximum shear stress developed in a tensile test.
τmax = (σ1 - σ2)/2 = (125 - 50)/2 = 37.5 MPa
Factor of Safety (FoS) = σy/τmax = 250/37.5 = 6.67
2. Maximum Distortion Energy Theory (von Mises Theory)
According to this theory, failure occurs when the distortion energy developed in the material is equal to the distortion energy developed in a tensile test.
Distortion energy per unit volume is given by,
U = (1/2) [((σ1 - σ2)^2 + (σ2 - σ3)^2 + (σ3 - σ1)^2)/(2E(1-μ^2))]
U = (1/2) [(75^2 + 25^2 + 50^2)/(2 × 200 × 10^3 × (1-0.3^2))] = 0.00184 MPa
Factor of Safety (FoS) = σy/√(3U) = 250/√(3 × 0.00184) = 26.7
3. Maximum Normal Stress Theory (Rankine Theory)
According to this theory, failure occurs when the maximum normal stress developed in the material is equal to the maximum normal stress developed in a tensile test.
σmax = σ1 = 125 MPa (tensile)
Factor of Safety (FoS) = σy/σmax = 250/125 = 2
Conclusion:
Based on the above calculations, the factor of safety is highest for the von Mises theory, followed by the Tresca theory and the Rankine theory. Therefore, the design should be based on the von Mises theory to ensure the highest level of safety.
Given,
σ1 = 2.5σ = 2.5 × 50 = 125 MPa (tensile)
σ2 = σ = 50 MPa (compressive)
σ3 = 1.5σ = 1.5 × 50 = 75 MPa (tensile)
σy = 250 MPa
E = 200 GPa
μ = 0.3
1. Maximum Shear Stress Theory (Tresca Theory)
According to this theory, failure occurs when the maximum shear stress developed in the material is equal to the maximum shear stress developed in a tensile test.
τmax = (σ1 - σ2)/2 = (125 - 50)/2 = 37.5 MPa
Factor of Safety (FoS) = σy/τmax = 250/37.5 = 6.67
2. Maximum Distortion Energy Theory (von Mises Theory)
According to this theory, failure occurs when the distortion energy developed in the material is equal to the distortion energy developed in a tensile test.
Distortion energy per unit volume is given by,
U = (1/2) [((σ1 - σ2)^2 + (σ2 - σ3)^2 + (σ3 - σ1)^2)/(2E(1-μ^2))]
U = (1/2) [(75^2 + 25^2 + 50^2)/(2 × 200 × 10^3 × (1-0.3^2))] = 0.00184 MPa
Factor of Safety (FoS) = σy/√(3U) = 250/√(3 × 0.00184) = 26.7
3. Maximum Normal Stress Theory (Rankine Theory)
According to this theory, failure occurs when the maximum normal stress developed in the material is equal to the maximum normal stress developed in a tensile test.
σmax = σ1 = 125 MPa (tensile)
Factor of Safety (FoS) = σy/σmax = 250/125 = 2
Conclusion:
Based on the above calculations, the factor of safety is highest for the von Mises theory, followed by the Tresca theory and the Rankine theory. Therefore, the design should be based on the von Mises theory to ensure the highest level of safety.
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A point in mild steel is subjected to triaxial stresses 2.5sigma(tensile),1.0sigma(compressive) and 1.5sigma(tensile).find factor of safety by various theories of failures.take sigma=50MPa,sigma_y=250Mpa,E=200Gpa and mu=0.3?
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A point in mild steel is subjected to triaxial stresses 2.5sigma(tensile),1.0sigma(compressive) and 1.5sigma(tensile).find factor of safety by various theories of failures.take sigma=50MPa,sigma_y=250Mpa,E=200Gpa and mu=0.3? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about A point in mild steel is subjected to triaxial stresses 2.5sigma(tensile),1.0sigma(compressive) and 1.5sigma(tensile).find factor of safety by various theories of failures.take sigma=50MPa,sigma_y=250Mpa,E=200Gpa and mu=0.3? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A point in mild steel is subjected to triaxial stresses 2.5sigma(tensile),1.0sigma(compressive) and 1.5sigma(tensile).find factor of safety by various theories of failures.take sigma=50MPa,sigma_y=250Mpa,E=200Gpa and mu=0.3?.
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