THIN SHELLS CLASS NOTES FOR MECHANICAL ENGINEERING - ME Subjects

THIN CYLINDRICAL SHELL

CLASS NOTES FOR

MECHANICAL ENGINEERING

Steam boilers
P.G. Cylinder
Air compressor
Gas storage shell in a refinery
Pressure cooker
Reaction vessels

Cylindrical
Spherical
Modified

RESISTING FORCE

From the equations, it can be easily concluded that the radial stress is negligible because of minimum d/t ratio is 20.

Hoop stress or Tangential stress or Circumferential stress(σ_h)
It acts in the tangential direction at the point of consideration.
Hoop stress or Tangential stress or Circumferential stress(σ_h)
It acts in the tangential direction at the point of consideration (say in a vertical circle)
Radial stress(σ_r)
It acts in the radial direction at the point of consideration

(a) σ_h = hoop stress

THIN CYLINDRICAL SHELL

CLASS NOTES FOR

MECHANICAL ENGINEERING

A vessel is a thin if its thickness is less as

compared to its diameter. Mathematically

it is expressed as a thin shell if D/t. A

vessel is a thin shell where stresses are

assumed to be uniform. Uniform stresses

mean stress at the inner to outer radius is

of same magnitude.

Thin sell

where D/t > 20 and where stress are uniform from the inner surface to the outer surface of the shell.

Various Stresses in a Thin Cylindrical Shell

(i) Hoop stress or Tangential stress or Circumferential stress (σh)

It acts in the tangential direction at the point of consideration

(ii) Longitudinal stress (σl)

It acts in the longitudinal direction at the point of consideration

(iii) Radial stress(σr)

It acts in the radial direction at the point of consideration

NOTE: All the three stresses are tensile or compressive and are at right angles to each other. Therefore, these are principal stresses. Hoop stress is tensile. Longitudinal stress is tensile. Radial stress is always compressive.

Examples of thin shells

Steam boilers

P.G. Cylinder

Air compressor

Gas storage shell in a refinery

Pressure cooker

Reaction vessels

Shapes of thin shells

Cylindrical

Spherical

Modified

Derivation of formulas for various stresses

(i) σh = hoop stress

It will break the cylinder length wise as shown in Fig.

DARK PORTION IN FIGURE IS THE ELEMENT

Fig. Longitudinal failure of a cylinder

Equate resisting force = breaking force.

Material resistance = σh L t + σh L t = 2 σh L t

Breaking force due to internal fluid pressure = p x projected area

Bursting force on element with dimensions rdθ and L

dF= p Sinθ rdθ L

Integrating θ from 0 to 180 OR it will be twice of when θ is taken from 0 to 90

F = 2p∫Sinθ rdθ L . limits from 0 to 90

F = 2prL = p D L

σh L t + σh L t =2 σh L t

Therefore equating the resisting force to bursting force, we get

2 σh L t = p DL

σh = p D/2t

If the efficiency of the longitudinal joint is considered then σh = p D/2t ηlong

(ii) σl = Longitudinal stress

If the cylinder fails at the circumference joint due to fluid pressure as shown in Fig.

Fig. Circumferential failure of a cylinder

Equating the resistance of the material to the breaking force.

Material resistance = σl (π/4)(Do2 –Di2)

= σl (π/4)(Di +2t)2 –Di2)= σl π Di t

Breaking force due to internal fluid pressure = p x internal area(on which pressure is acting)

= p (π/4)Di2)

Therefore σl π Di t = p(π/4)Di2)

σl = p Di/4t

If the efficiency of the circumferential joint is considered then σl = pD/4t ηcircum

(iii) Radial stress = σr = inside fluid pressure

σr = pi

Built-up shells

Shells with joints are called built-up shells.

Change in the dimensions of the shell

(a) Change in length δL

δL/L = σl/E -μ σh/E = p Di/4tE – μ p Di/2tE

δL/L = (p Di/4tE) (1-2μ) =(σh /2E)(1-2μ)

δL = (p Di L/4tE) (1-2μ)

(b) Change in diameter of the shell, δD

δD/D = Change in circumference / Original circumference

= π δD/πD

Therefore δD/D= σh/E -μ σL/E = p Di/2tE – μ p Di/4tE

δD/D = (p Di/4tE) (2-μ) =

δD = (p Di Di /4tE) (2-μ)= (pDi2 /4tE) (2-μ)

(c) Change in volume, δV

V = (π/4) D2L

Taking log on both sides

log V= log(π/4) +log D2 +log L

log V= log(π/4) + 2 log D +log L

(i) Hoop stress or Tangential stress or Circumferential stress (σ_h)

(ii) Longitudinal stress (σ_l)

(iii) Radial stress(σ_r)

(i) σ_h = hoop stress

Material resistance = σ_h L t + σ_h L t = 2 σ_h L t

σ_h L t + σ_h L t =2 σ_h L t

2 σ_h L t = p DL

σ_h = p D/2t

If the efficiency of the longitudinal joint is considered then σ_h = p D/2t η_long

(ii) σ_l = Longitudinal stress

Material resistance = σ_l (π/4)(D_o² –D_i²)

= σ_l (π/4)(D_i +2t)² –D_i²)= σ_l π D_i t

= p (π/4)D_i²)

Therefore σ_l π D_i t = p(π/4)D_i²)

σ_l = p D_i/4t

If the efficiency of the circumferential joint is considered then σ_l = pD/4t η_circum

(iii) Radial stress = σ_r= inside fluid pressure

σ_r= p_i

δL/L = σ_l/E -μ σ_h/E = p D_i/4tE – μ p D_i/2tE

δL/L = (p D_i/4tE) (1-2μ) =(σ_h /2E)(1-2μ)

δL = (p D_i L/4tE) (1-2μ)

Therefore δD/D= σ_h/E -μ σ_L/E = p D_i/2tE – μ p D_i/4tE

δD/D = (p D_i/4tE) (2-μ) =

δD = (p D_iD_i /4tE) (2-μ)= (pD_i² /4tE) (2-μ)

V = (π/4) D²L

log V= log(π/4) +log D²+log L

log V= log(π/4) + 2 log D+log L

δV/V = 2 (p D_i/4tE) (2-μ)+ (p D_i/4tE) (1-2μ)

δV/V = (p D_i/4tE) (5–4μ) =(σ_h /2E) (5–4μ)

δV = (p D_i/4tE) (5–4μ) (π/4)D²L

δV =(πpD³L/16tE) (5–4μ)

Bursting force=(π/4)D_i² p_i

Resisting force =(πD_it σ_h)

(π/4)D_i² p_i=(πD_it σ_h)

σ_h = p D/4t

Hoop stress or Tangential stress or Circumferential stress(σ_h)

Hoop stress or Tangential stress or Circumferential stress(σ_h)

Radial stress(σ_r)

(a) σ_h = hoop stress

Material resistance = σ_h πD t

= p (π/4)D_i²

Therefore σ_h πD t = p (π/4)D_i²

σ_h = pD/4t

If the efficiency of the circumferential joint is considered then σ_h = pD/2t η_circum

(b) Radial stress = σ_r= inside fluid pressure

σ_r= p_i

Therefore δD/D= σ_h/E -μ σ_h/E = p D_i/4tE – μ p D/4tE

δD/D = (p D_i/4tE) (1-μ)

δD = (p D_iD_i /4tE) (2-μ)= (pD_i² /4tE) (2-μ)

δD = (pD_i² /4tE) (1-μ)

= 3 (p D_i/4tE) (1-μ)

δV/V= (¾)(p D_i/t E) (1–μ)

V = (4π/3) R³= (π/6) D³

δV = (p D_i/4tE) (1–μ) (π/6) D³

δV =(π/6)(pD⁴/t E) (1–μ)=(πpD⁴/6tE) (1–μ)