Thin, Thick Cylinders & Spheres

Thin Cylinders

• If the wall thickness of a cylindrical shell is less than about one-tenth to one-fifteenth of its diameter, it is treated as a thin cylinder.
• It is assumed that stresses through the thickness are uniformly distributed and that the radial stress is negligible compared with the hoop and longitudinal stresses.
• Hoop (circumferential) stress: for an internally pressurised thin cylindrical shell, the circumferential or hoop stress is

• Longitudinal (axial) stress: the tensile stress along the axis of the cylinder (for closed ends) is

• Maximum shear stress: the maximum shear in a thin cylindrical shell (from the two principal stresses) is

• Hoop (circumferential) strain: in terms of elastic constants the hoop strain is

• Longitudinal (axial) strain: the corresponding axial strain is

• Volumetric strain: total volumetric (dilatation) strain for a thin shell (radial stress ≈ 0) is

Symbols used:

• p = internal pressure
• d = diameter of cylinder
• t = thickness of the cylinder
• μ = Poisson's ratio
• E = Young's modulus

If s_a is the permissible tensile stress for the shell material, then from strength considerations the major principal stress (usually the hoop stress) should satisfy s ≤ s_a.

Hence, for design the thickness must satisfy

or, rearranged,

Remarks and common closed-form relations (for reference and quick checks):

• Hoop stress σ_h = pd / (2t).
• Longitudinal (axial) stress σ_l = pd / (4t) for a closed-ended thin cylinder.
• Maximum shear = (σ_h - σ_l) / 2 = pd / (8t).
• Hoop and longitudinal strains follow from linear elasticity using plane-stress relations: ε_θ = (1/E)(σ_h - μ σ_l), ε_l = (1/E)(σ_l - μ σ_h).
• Volumetric strain = ε_θ + ε_l + ε_r with ε_r ≈ -(μ/E)(σ_h + σ_l); simplifies to ε_v = ((1 - 2μ)/E)(σ_h + σ_l).

Thin Spherical Shells

• For a thin spherical shell under internal pressure the two principal membrane stresses (meridional and circumferential) are equal.
• Membrane stress in a thin spherical shell: σ = pd / (4 t) (tensile in nature)

• Maximum shear stress: with principal stresses σ, σ and 0 (radial), the maximum shear is

• Strain in any in-plane direction: the linear strain in a spherical shell element is

• Volumetric strain: summing principal strains gives

Notes:

• For identical internal pressure and nominal radius and thickness, a spherical shell carries stress more uniformly than a cylindrical shell; the membrane stress in a sphere is half the hoop stress of an equivalent thin cylinder (for the same diameter and thickness).
• Spherical shells are therefore efficient under internal pressure and are commonly used for pressure vessels (e.g., small pressure bulbs, domes, end caps).

Cylinders with Hemispherical Ends

Consider a cylindrical shell of thickness t_c with hemispherical ends of thickness t_s subjected to internal pressure p. For reference diagrams see the image below.

• Hoop stress in the cylindrical part: (circumferential stress in cylinder)

• Hoop (circumferential) stress in the hemispherical end: (same as spherical shell stress)

• Longitudinal (axial) stress in the cylindrical part:

• Longitudinal (meridional) stress in the hemispherical part:

• Circumferential strain in the hemispherical part:

• Circumferential strain in the cylindrical part:

At the junction of the cylinder and hemisphere the circumferential strains must match for the joint not to distort. Equating the circumferential strains of cylinder and hemisphere gives the condition:

From that relation one obtains (after simplification):

This shows that the thickness of the cylindrical portion must be greater than that of the hemispherical portion for compatible strains at the junction.

• For equal maximum stress in cylinder and hemisphere: equating the relevant maximum stresses yields

Two commonly used practical design rules derived from the above relations are:

• To avoid distortion at the junction, choose t_c = t_s (2 - μ)/(1 - μ) (for elastic behaviour) - this typically gives t_c several times t_s for μ ≈ 0.3.
• To make maximum stresses equal, t_c = 2 t_s (when equating hoop stresses only).

Thick Cylindrical Shell (Thick-Walled Cylinder)

• If the thickness of a cylindrical shell is not small compared with its diameter (typically t > d/10), it is called a thick shell.
• In thick cylinders the circumferential (hoop) stress varies across the wall thickness and the radial stress is no longer negligible.
• The three stresses present in a thick cylinder are:
  • (i) radial stress σ_r (compressive on the inner surface),
  • (ii) hoop (circumferential) stress σ_θ (tensile),
  • (iii) longitudinal (axial) stress σ_z (tensile for closed ends and relatively uniform across thickness).

The radial and hoop stresses vary with radius r and are given by Lame's solution for a thick-walled cylinder:

That is, the radial stress σ_r and hoop stress σ_θ may be written in the form

Explicitly,

and the radial pressure (radial stress) is

In the Lame expressions above:

• r_o = outer radius of shell
• r_i = inner radius of shell
• A and B are constants determined from boundary conditions (internal pressure at r = r_i, external pressure at r = r_o - often external pressure is atmospheric and taken as zero gauge).

Important observations (behaviour across the thickness):

• Longitudinal stress σ_z is approximately uniform across the thickness when the ends are closed and carries the axial resultant from pressure on the ends.
• Hoop stress σ_θ varies from a maximum tensile value at the inner face to a smaller tensile value at the outer face; the variation follows a hyperbolic form given by Lame's equations.
• Radial stress σ_r varies from -p (compressive) at the inner face to the external pressure (often zero gauge) at the outer face; it therefore decreases in magnitude across the thickness.

Design and checks:

• For thick cylinders a simple thin-wall formula (σ = pd / 2t) gives unsafe results when t is not small; use Lame's formulae to evaluate maximum hoop stress (typically at the inner surface) and check against permissible material stress.
• When external pressure is significant (e.g., vacuum or submerged vessels), radial stresses and the possibility of buckling must be considered in design.

Worked Example (thin cylindrical shell - thickness required)

Problem: Determine the minimum thickness t of a thin cylindrical shell of diameter d under internal pressure p if the permissible tensile stress of the material is s_a.

Sol.
The principal (maximum) stress for a thin cylinder is the hoop stress which must not exceed s_a.
Hoop stress σ_h = pd / (2t).
Set σ_h = s_a for limiting thickness.
Rearrange to find thickness t.

Ans.
t = pd / (2s_a).

Applications and Design Notes

• Thin shells: used where wall thickness is small relative to radius - examples include storage cylinders, thin-walled pipelines, boiler shells (historically), and many pressure vessels where t/d < />
• Spherical shells: used for pressure vessels where uniform stress distribution and minimum material for given internal volume are required - examples include small pressure bulbs, hemispherical end caps, and certain high-pressure containers.
• Thick shells: required when internal pressure is high compared with allowable stress or when wall thickness is large (e.g., gun barrels, hydraulic cylinders, high-pressure reactors). Lame's formulae must be used in design and stress analysis.
• When joining cylindrical and spherical components, check both stress compatibility and strain compatibility at junctions; ensure smooth transitions and suitable thickness ratios to avoid local overstress and distortion.