cylindrical pressure vessel, with a diameter of 40cm and a wall thickness of 10mm is made of steel (isotropic, with Young’s modulus = 200GPa ; Poisson’s ratio = 0.28 ) and is pressurized to 20MPa . The longitudinal and hoop stresses in the wall, σl and σh , respectively, are given by
Sigma l=pr/2t
Sigma h=pr/t
What is the strain in the radial direction, through the thickness of the pressure vessel?
To find the strain in the radial direction through the thickness of the pressure vessel, we can use the formula:
εr = σr / E
where εr is the strain in the radial direction, σr is the stress in the radial direction, and E is the Young's modulus.
In this case, the stress in the radial direction can be calculated using the hoop stress formula:
σr = σh
σh = P * r / t
where P is the pressure inside the vessel, r is the radius of the vessel, and t is the wall thickness.
Given that the pressure inside the vessel is 20 MPa, the radius of the vessel is half of the diameter, which is 40 cm, and the wall thickness is 10 mm, we can calculate the stress in the radial direction as follows:
σh = P * r / t
= (20 * 10^6 Pa) * (0.4 m) / (0.01 m)
= 800 * 10^6 Pa
Now, substituting this value in the strain formula:
εr = σr / E
= 800 * 10^6 Pa / (200 * 10^9 Pa)
= 0.004
Therefore, the strain in the radial direction through the thickness of the pressure vessel is 0.004 (or 0.4%).
To find the strain in the radial direction through the thickness of the pressure vessel, we first need to calculate the radial stress, σr.
The radial stress (σr) can be calculated using the formula:
σr = (σl - σh) / 2
Given:
Diameter (D) = 40 cm
Wall thickness (t) = 10 mm
Pressure (p) = 20 MPa
Converting the units:
Diameter (D) = 0.4 m
Wall thickness (t) = 0.01 m
Pressure (p) = 20 × 10^6 Pa
Now, let's calculate the longitudinal stress (σl) and hoop stress (σh) using the given formulas:
σl = p × r / (2 × t)
σh = p × r / t
To find the strain (εr), we need to calculate the ratio of the change in radial displacement to the original thickness in the radial direction. This can be expressed as:
εr = Δr / t
Since we are assuming that the change in thickness is negligible (assuming no swelling or shrinking), the strain in the radial direction, εr, is approximately equal to the strain in the circumferential direction.
Therefore, the strain in the radial direction through the thickness of the pressure vessel is:
εr = σh / E
Where E is the Young's modulus of the material. In this case, E = 200 GPa = 200 × 10^9 Pa.
Now, let's calculate the strain in the radial direction (εr).
First, calculate the longitudinal stress (σl):
σl = (p × r) / (2 × t) = (20 × 10^6 × 0.2) / (2 × 0.01) = 2 × 10^8 Pa
Next, calculate the hoop stress (σh):
σh = (p × r) / t = (20 × 10^6 × 0.2) / 0.01 = 4 × 10^8 Pa
Finally, calculate the strain in the radial direction (εr):
εr = σh / E = (4 × 10^8) / (200 × 10^9) ≈ 2 × 10^-3
Therefore, the strain in the radial direction through the thickness of the pressure vessel is approximately 2 × 10^-3.