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Thin Cylinders & Spherical Shells

Thin Cylinders & Spherical Shells. Analysis of above under Pressures. Thin Cylinders. Shape Use of Shape (Tanks, Boilers, pipelines, Vault) Thin and Thick Remember Thin!. Thin Cylinders. Pressure Internal and External atmospheric Stress Failure Hoop type failure

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Thin Cylinders & Spherical Shells

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  1. Thin Cylinders&Spherical Shells Analysis of above under Pressures

  2. Thin Cylinders • Shape • Use of Shape (Tanks, Boilers, pipelines, Vault) • Thin and Thick • Remember Thin!

  3. Thin Cylinders • Pressure • Internal and External atmospheric • Stress • Failure • Hoop type failure • Longitudinal failure • Strain change in • diameter • length • volume

  4. Thin Cylinders Failure Hoop Type Failure Bursting Load = Pressure x Area = p x d l Hoop Resistance = Hoop Stress X Area = σc x 2 t l For Hoop Resistance = Bursting Load σc x 2 t l = p x d l σc = pd /2t  Hoop stress (N/mm2) Longitudinal Failure Bursting Load = Pressure x Area = p x ∏d2 /4 Longitudinal Resistance = Longitudinal Stress X Area = σl x ∏ d t For Resistance = Bursting Load σl x ∏ d t = p x ∏d2 /4 σl = pd /4t  Longitudinal stress (N/mm2) Dimensions l = length (mm) d = internal diameter (mm) t = thickness (mm) ** Failure p = Pressure (N/mm2) σc =Hoop stress (N/mm2) σl =Longitudinal stress (N/mm2)

  5. Thin Cylinders • Strain change (Change in Direction) • Diameter  δd= ρc d Strain in diametric direction = ρc = δd / d = (pd/4t) (1/E) (1/m) (2m-1) • Length  δl = ρl lρl Strain in Longitudinal direction = ρl = δl / d = (pd/4t) (1/E) (1/m) (m-1) • Volume  δv = V (2 ρc + ρl ) E = Modulus of Elasticity (N/mm2) ν =1/m = Poisson’s ratio

  6. Spherical Shells Failure Bursting Load = Pressure x Area = p x ∏ d2/4 Hoop Resistance = Hoop Stress X Area = σc x 2 t l For Hoop Resistance = Bursting Load σ x ∏d t = p x ∏ d2/4 σ= pd /4t  Stress (N/mm2) Efficiency of Joints ‘η’ σ= (pd /4t) (1/ η) Another Failure is Shear Failure Shear stress, τ= (σc – σl ) / 2 = [(pd /2t) - (pd /4t)] /2 = pd /8t

  7. Spherical Shells Volumetric strain only • Volume  δV = (∏ pd4 /8t) x (1/E) x (1/m) x (m-1) E = Modulus of elasticity ν =1/m = Poisson’s ratio

  8. Other References • http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-ROORKEE/strength%20of%20materials/lects%20&%20picts/image/lect15/lecture15.htm • http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-ROORKEE/strength%20of%20materials/lects%20&%20picts/image/lect16/lecture16.htm • http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/pressure_vessel.cfm • http://www.codecogs.com/reference/engineering/materials/cylinders_and_spheres/thin_walled_cylinders_and_spheres.php

  9. Tutorials • Name and draw 3 real examples of thin cylinder • Name and draw 3 real examples of spherical shell • For a thin cylinder, • Obtain the value of ρc • Obtain the value of ρl • Show, δv = V (2 ρc + ρl ) • Using elastic theory , show equations for • Change in diameter • Change in volume • Rethaliya examples (Page 92). Example questions- 1, 2, 4, 6 & 8

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