x1 and y1 are two orthogonal directions on the surface of a closed thin-walled cylindrical pressure vesselwith x1 at angle anticlockwise from the longitudinal axis of the vessel. The normal strains along x1 and y1 aremeasured to be 115x10-6 and 360x10-6 , respectively. Given R=200mm, t=2mm, E=210GPa, =0.313 for thevessel, find the principal stresses, x1y1 and the angle .
Question
x1 and y1 are two orthogonal directions on the surface of a closed thin-walled cylindrical pressure vesselwith x1 at angle anticlockwise from the longitudinal axis of the vessel. The normal strains along x1 and y1 aremeasured to be 115x10-6 and 360x10-6 , respectively. Given R=200mm, t=2mm, E=210GPa, =0.313 for thevessel, find the principal stresses, x1y1 and the angle .
Solution
The problem involves finding the principal stresses, shear stress, and the angle in a cylindrical pressure vessel. Here are the steps to solve the problem:
Step 1: Convert the given strains into base units. The normal strains along x1 and y1 are 115x10^-6 and 360x10^-6, respectively.
Step 2: Convert the given modulus of elasticity (E) and Poisson's ratio (ν) into base units. E = 210 GPa = 210x10^9 Pa and ν = 0.313.
Step 3: Use the generalized Hooke's law in cylindrical coordinates to find the principal stresses. The formula is:
σ1 = E/(1-ν^2) * (ε1 + νε2) σ2 = E/(1-ν^2) * (ε2 + νε1)
where σ1 and σ2 are the principal stresses, ε1 and ε2 are the normal strains along x1 and y1, E is the modulus of elasticity, and ν is Poisson's ratio.
Step 4: Substitute the given values into the formulas to find the principal stresses.
Step 5: To find the shear stress (τx1y1), use the formula:
τx1y1 = E/(2*(1+ν)) * (ε1 - ε2)
Step 6: Substitute the given values into the formula to find the shear stress.
Step 7: To find the angle (θ), use the formula:
θ = 0.5 * arctan((2*τx1y1)/(σ1-σ2))
Step 8: Substitute the values of τx1y1, σ1, and σ2 into the formula to find the angle.
Remember to check the units of your final answers to ensure they are consistent with the units of the given values.
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