3Thin Walled

Pressure Vessels

ASEN 3112 - Structures

ASEN 3112 Lecture 3 – Slide 1

Spherical and Cylindrical Pressure Vessels

ASEN 3112 - Structures

ASEN 3112 Lecture 3 – Slide 2

Thin Wall Cylindrical Pressure Vessel:Wall Stress Assumptions

ASEN 3112 - Structures

θx xθ

Free surface: σ = 0, τ = 0, τ = 0rθrr rx Wall thickness t

xr rx

rrσ inside body neglected since it varies from −p to 0 over wall, which is << σ & << σ

Zero because of thin body and τ = τ

θr rθ

Zero becauseτ = τ Zero

becauseτ = τ

Zero because of axisymmetry and no torque

rθ rx

No tangential forcesbecause τ = 0 & τ =0

θrΑ

Βτxr

xxσ : axial stress

xx

τxθ

θxτ

θrτ

θθσ : hoop (a.k.a. circumferential) stress

θθ

C

D

2Rx

ASEN 3112 Lecture 3 – Slide 3

Thin Wall Cylindrical Pressure Vessel:Free Body Diagrams for Hoop and Axial Stress

ASEN 3112 - Structures

t σ (2πR t)xx

dx

(a) (b) (c)

p(πR )2

σ θθ

σ θθ

σ xxσ xx

θrx

~2Rσ t dxθθ

σ t dxθθ

p 2R dx

Both interior and exterior vessel radii can be taken as R, since t << R

ASEN 3112 Lecture 3 – Slide 4

Thin Wall Cylindrical Pressure Vessel:Result for Wall Stress Distribution

Stress matrix at any wall point in cylidrical coordinates (x , θ, r ):

Nonzero wall stress component in terms of the data (p,R,t)

Hoop stress is twice axial stress, and will control the design

[σxx 0 00 σ

θθ

θθ 00 0 0

]Note: this is a plane stressstate

σθθ = p R

tσxx = p R

2t= 1

2 σ

ASEN 3112 - Structures

ASEN 3112 Lecture 3 – Slide 5

σ (2πR t)

p (πR )2

t

~2R

x

y

z

rφ

θ

σ θθσ φφ

σ φφ

σ θθ

Thin Wall Spherical Pressure Vessel:Free Body Diagram for Wall Stress

ASEN 3112 - Structures

ASEN 3112 Lecture 3 – Slide 6

[σ 0 00 σ 00 0 0

]

σ = p R

2t

Thin Wall Spherical Pressure Vessel:Result for Wall Stress Distribution

Stress matrix at any wall point in spherical coordinates

Nonzero wall stress component in terms of the data (p,R,t)

ASEN 3112 - Structures

ASEN 3112 Lecture 3 – Slide 7

Why A Spherical Pressure Vessel is MoreStructurally Efficient Than A Cylindrical One

ASEN 3112 - Structures

pp

σ

σ

σ

σ

σ

(a) Spherical vessel wall (b) Cylindrical vessel wall

σ (axial)xxσxx

θθ

σ (hoop, a.k.a.circumferential)

θθ

ASEN 3112 Lecture 3 – Slide 8

Thin Wall Cylindrical Pressure Vessel:Watch Out For End Cap Effects

ASEN 3112 - Structures

0.35 mm

0.15 mm

t

t

0.20 mm

0.35 mm

(a) Deformed shape

(b) Detail A

(c) Deformation of the same cylindrical pressure vessel at a flat head

A

R = 1000 mm

ASEN 3112 Lecture 3 – Slide 9

Gap

t 45

Joining Vessel Wall Pieces By Welds

Fair (eccentricity disturbsmembrane state) but cheap

Better but more expensive

ASEN 3112 - Structures

ASEN 3112 Lecture 3 – Slide 10

N = σ A = π σ /4bolt bolt bolt bolt

N = p π Rlid2

Design of Lid-Bolted Thin-Wall Cylindrical TankExample Worked Out in Vable's Book and Lecture Text

ASEN 3112 - Structures

ASEN 3112 Lecture 3 – Slide 11