Answer:
Step-by-step explanation:
We can use the following equations to calculate the safety factor versus yielding for the two failure theories:
Tresca failure theory: SF_T = σ_yield / σ_max
Von Mises failure theory: SF_VM = σ_yield / σ_eq
where σ_yield is the yield strength of the steel, σ_max is the maximum principal stress, and σ_eq is the equivalent von Mises stress.
To calculate the maximum principal stress, we can use the formula:
σ_max = P_int x r_i / t
where P_int is the internal pressure, r_i is the inner radius (half the diameter), and t is the wall thickness.
a) Tresca failure theory:
First, we need to calculate the maximum principal stress:
σ_max = P_int x r_i / t = 5 MPa x 0.75 m / 0.028 m = 131.25 MPa
Now we can calculate the safety factor using the Tresca failure theory:
SF_T = σ_yield / σ_max = 250 MPa / 131.25 MPa = 1.91
Therefore, the safety factor versus yielding for the Tresca failure theory is 1.91.
b) Von Mises failure theory:
To calculate the equivalent von Mises stress, we can use the formula:
σ_eq = √(σ_1^2 - σ_1σ_2 + σ_2^2)
where σ_1 and σ_2 are the principal stresses. For a thin-walled pressure vessel, the principal stresses can be calculated as:
σ_1 = P_int x r_i / t
σ_2 = 0
Therefore, the equivalent von Mises stress is:
σ_eq = √(σ_1^2 - σ_1σ_2 + σ_2^2) = √(3/2 x σ_1^2) = σ_1 x √3
σ_eq = P_int x r_i / t x √3 = 5 MPa x 0.75 m / 0.028 m x √3 = 226.25 MPa
Now we can calculate the safety factor using the Von Mises failure theory:
SF_VM = σ_yield / σ_eq = 250 MPa / 226.25 MPa = 1.11
Therefore, the safety factor versus yielding for the Von Mises failure theory is 1.11.