Lectures 20 - Pressure Vessels (2)
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University of Alberta**We aren't endorsed by this school
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CIVE 270
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Mechanical Engineering
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Dec 17, 2024
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Mechanics of Deformable BodiesCIV E 270Mechanics of deformable bodies
Thin-Walled Pressure VesselsHoop StressLongitudinal StressCIV E 270 MDB - A. Mowafy2
Thin-Walled Pressure Vessels“Commonly used in industry to serve as boilers or storage tanks.”•Vessels having an inner-radius to-wall-thickness ratio ≥10 •Stress through t is insignificant compared to σ1& σ2•σ1is in the circumferential direction (hoop stress)•σ2is in the longitudinal (or axial) direction (long. stress)CIV E 270 MDB - A. Mowafy3𝑟𝑡10Copyright ©2014 Pearson Education, All Rights Reserved.
Thin-Walled Pressure Vessels – Cylindrical VesselsHoop StressLongitudinal StressCIV E 270 MDB - A. Mowafy4σ1is in the circumferential direction (hoop stress)σ2is in the longitudinal (or axial) direction (long. stress)
Thin-Walled Pressure Vessels – Cylindrical VesselsHoop Stress 𝐹௫ൌ02ሾ σଵฏௌ௧௦௦𝑡𝑑𝑦ሿ െ 𝑝2𝑟𝑑𝑦௧ௗൌ0Longitudinal Stress 𝐹௬ൌ0σଶሺ2𝜋𝑟𝑡ሻെ 𝑝ሺ𝜋𝑟ଶሻ௧ௗൌ0CIV E 270 MDB - A. Mowafy5𝒑𝒓𝝈𝟏ൌ𝒑𝒓𝒕𝒑𝒓𝝈𝟐ൌ𝒑𝒓𝟐𝒕𝑵𝒐𝒕𝒆𝝈𝟐ൌ𝝈𝟏𝟐
Thin-Walled Pressure Vessels – Spherical Vessels 𝐹ൌ0 𝐹௬ൌ0σଶሺ2𝜋𝑟𝑡ሻെ 𝑝ሺ𝜋𝑟ଶሻ௧ௗൌ0CIV E 270 MDB - A. Mowafy6𝒑𝒓𝝈𝟏ൌ𝝈𝟐ൌ𝒑𝒓𝟐𝒕The stress is similar to 𝝈𝟐in all directions due to symmetry
CIV E 270 MDB - A. Mowafy7Thin-Walled Pressure VesselsExample (1)A pressure-vessel head is fabricated by gluing the circular plate tothe end of the vessel as shown. If the vessel sustains an internalpressure of 450 kPa,determine the average shear stress in the glueand the state of stress in the wall of the vessel.Copyright ©2014 Pearson Education, All Rights Reserved.
CIV E 270 MDB - A. Mowafy8Thin-Walled Pressure VesselsExample (2)An A-36-steel hoop has an inner diameter of 599.75mm, thickness of 6 mm, and width of 25 mm. If itand the 600-mm-diameter rigid cylinder have atemperature of 20° C,determine the temperature towhich the hoop should be heated in order for it tojust slip over the cylinder. What is the pressure thehoop exerts on the cylinder, and the tensile stressin the ring when it cools back down to 20° C.(𝜶𝒕ൌ𝟏𝟐𝒙𝟏𝟎ି𝟔/°𝑪, E = 203 GPa)Copyright ©2014 Pearson Education, All Rights Reserved.
CIV E 270 MDB - A. Mowafy9Thin-Walled Pressure VesselsExample (3) – Self-study A pressure vessel spans between a pin support at A and a roller support at B. The internalpressure is raised to 2500 kPa and the temperature is increased from 20°C to 70 °C.•What are the stresses in the x and y directions at point C? Show these stresses on anelement.•What are the total strains at point C in the x direction, y direction, and through-thicknessat point C?Assume:E = 200 GPaν= 0.33𝜶𝒕ൌ𝟏𝟐𝒙𝟏𝟎ି𝟔/°𝑪
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