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 Longitudinal failure – Strain change in diameter length volume

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

Thin Cylinders Strain change (Change in Direction) – Diameter  δd = ρ c d Strain in diametric direction = ρ c = δ d / d = (pd/4t) (1/E) (1/m) (2m-1) – Length  δl = ρ l lρ l Strain in Longitudinal direction = ρ l = δ l / d = (pd/4t) (1/E) (1/m) (m-1) – Volume  δv = V ( 2 ρ c + ρ l ) E = Modulus of Elasticity (N/mm 2 ) ν =1/m = Poisson’s ratio

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

Spherical Shells Volumetric strain only Volume  δV = (∏ pd 4 /8t) x (1/E) x (1/m) x (m-1) E = Modulus of elasticity ν =1/m = Poisson’s ratio

Other References 1. ROORKEE/strength%20of%20materials/lects%20&%20picts/image/lect15/lecture15.htmhttp://nptel.iitm.ac.in/courses/Webcourse-contents/IIT- ROORKEE/strength%20of%20materials/lects%20&%20picts/image/lect15/lecture15.htm 2. ROORKEE/strength%20of%20materials/lects%20&%20picts/image/lect16/lecture16.htmhttp://nptel.iitm.ac.in/courses/Webcourse-contents/IIT- ROORKEE/strength%20of%20materials/lects%20&%20picts/image/lect16/lecture16.htm lled_cylinders_and_spheres.phphttp:// lled_cylinders_and_spheres.php

Tutorials 1.Name and draw 3 real examples of thin cylinder 2.Name and draw 3 real examples of spherical shell 3.For a thin cylinder, 1.Obtain the value of ρ c 2.Obtain the value of ρ l 3.Show, δv = V ( 2 ρ c + ρ l ) 4.Using elastic theory, show equations for 1.Change in diameter 2.Change in volume 5.Rethaliya examples (Page 92). Example questions- 1, 2, 4, 6 & 8