DR. AHMED NABHAN MAHMOUD

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DR. AHMED NABHAN MAHMOUD

MINIA University

Faculty of Engineering

Production Engineering and Design Department

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❑ Thin-walled cylinders having thickness to radius ratio small than 10%.

Where, 𝑟 : mean radius t : wall thickness 𝑅_𝑖 : inner radius 𝑅_𝑜 : outer radius

𝑡

𝑟 < 0.1

𝑟 = 𝑅_𝑖 + 𝑅_𝑜 2

Dr. Ahmed Nabhan Mahmoud

Production Engineering and Design Department Faculty of Engineering - Minia University

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Thin-walled shells include all hollow cylindrical with wall thickness to radius ratio less than 10%. Under the action of internal pressure, three perpendicular stresses will be set up, which can be listed as follows:

A. Hoop (or tangential) stress (σ_t) B. Radial (diameter) stress (σ_r) C. Longitudinal (axial) stress (σ_l)

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The value of these stress types varies proportional to the shell thickness.

However, it is recommended that, design of thin-walled can be carried out under the following assumptions:

I.

The hoop stress is constant across the wall thickness of the cylinder.

II.

The magnitude of the radial stress is small compared to the value of the hoop stress and can be neglected. Therefore, the thin-walled

cylinders are subjected to hoop and longitudinal stresses.

III.The value of the longitudinal stress can be neglected for open and

long cylinders.

IV.Thin-walled cylinders cannot resist the action of external pressure.

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Hoop (or tangential) stress

Force equilibrium tends to:

F₁= P_{i *} (2R_{i *} L) F₂ = σ_{t *} (2t _* L) Therefore, F₁= F2

σ_t = ^{Pi Ri}

t

"F₁" is the generated force in the cylinder due to the influence of the internal pressure on the cylinder projected area. Also, "F₂ "is the uniform resisting force generated in the cylinder material to overcome the effect of the force "F₁".

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Longitudinal (axial) stress

Longitudinal stress can be determined from the following relations:

F₁= P_{i *} ( R_i²) F₂ = σ_{l *}(2 R_{i *} t) Therefore,

F₁= F₂ σ_{l =} ^{Pi Ri}

2t

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❑ Thick-walled cylinders include all hollow cylindrical elements with wall thickness to internal radius ratio more than 10%.

❑ Form the force equilibrium the relation can be considered (Lame’s Theory)

σ_{t, r} =A ± [ B/R²]

Where;

A, B: Constants

R_e: The external radius of the cylinder, R_i: The internal radius of the cylinder,

R: The radius for the point at which the stresses is calculated, where R_i < R< R_e.

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The boundary conditions of the cylinder under the internal pressure can be considered as follows:

At R = R_i σ_r = - P_i

R = R_o σ_r = 0

By substituting the above conditions in equation -P_i = A-B/R_i²

0 = A-B/R_o²

A = [P_i*R_i²/ (R_o²- R_i²)]

B = [P_i*R_o²_*R_i²/ (R_o²- R_i²)]

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σ_r = [P_i*R_i²/ (R_o²- R_i²)]- [P_i*R_i²_* R_o²/ (R_o²- R_i²)(R)]

σ_t = [P_i*R_i²/ (R_o²- R_i²)]+ [P_i*R_i²_* R_o²/ (R_o²- R_i²)(R)]

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The boundary conditions of the cylinder under the external pressure can be considered as follows:

At R = R_i σ_r = 0

R = R_o σ_r = - P_o

By substituting the above conditions in equation 0 = A-B/R_i²

-P_o = A-B/R_o²

A = [-P_o*R_o²/ (R_o²- R_i²)]

B = [-P_o*R_o²_*R_i²/ (R_o²- R_i²)]

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σ_r = [-P_o*R_o²/ (R_o²- R_i²)]- [P_o*R_i²_* R_o²/ (R_o²- R_i²)(R)]

σ_t = [-P_o*R_o²/ (R_o²- R_i²)]+ [P_o*R_i²_* R_o²/ (R_o²- R_i²)(R)]

❑ The general case is the cylinder which is subjected to internal and external pressure.

σ_r = [P_i*R_i²-P_o*R_o²/ (R_o²- R_i²)]- [(P_i-P_o) _*R_i²_* R_o²/ (R_o²- R_i²)(R)]

σ_t = [P_i*R_i²-P_o*R_o²/ (R_o²- R_i²)]+ [(P_i-P_o) _*R_i²_* R_o²/ (R_o²- R_i²)(R)]

❑ The longitudinal stress will only setup in the material, if the cylinder is closed. The cylinder is subjected to internal and external pressure.

P_{i *} ( R_i²) - P_{o *} ( R_o²) = σ_{l *}2 (R_i - R_o) σ_l = [P_i*R_i²-P_o*R_o²/ (R_o²- R_i²)] =

A

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❑ The graphical solution can be performed under consideration of the general equation of Lame for the thick cylinder by plotting the stress vertically versus value ¹Τ_𝑅2

❑ The hoop stress is represented by a line with slop (B) whilst the radial has a slop of (-B)

❑ Both the lines have exactly the same intersect with the vertical axis and equal to (A) = σ_l