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# Stress in Thick-Walled Cylinders or Tubes

When a thick-walled tube or cylinder is subjected to internal and external pressure a hoop and longitudinal stress are produced in the wall.

### Stress in Axial Direction

The stress in axial direction at a point in the tube or cylinder wall can be expressed as:

σa = (pi ri2 - po ro2 )/(ro2 - ri2)                                   (1)

where

σa = stress in axial direction (MPa, psi)

pi = internal pressure in the tube or cylinder (MPa, psi)

po = external pressure in the tube or cylinder (MPa, psi)

ri = internal radius of tube or cylinder (mm, in)

ro = external radius of tube or cylinder (mm, in)

### Stress in Circumferential Direction - Hoop Stress

The stress in circumferential direction - hoop stress - at a point in the tube or cylinder wall can be expressed as:

σc = [(pi ri2 - po ro2) / (ro2 - ri2)] - [ri2 ro2 (po - pi) / (r2 (ro2 - ri2))]                                    (2)

where

σc = stress in circumferential direction (MPa, psi)

r = radius to point in tube or cylinder wall (mm, in) (ri < r < ro)

maximum stress when  r = ri (inside pipe or cylinder)

### Resultant Stress

Combined stress in a single point in the cylinder wall cannot be described by a single vector using vector addition. Instead stress tensors (matrixes) describing the linear connection between two physical vectors quantities can be used.

The stress in radial direction at a point in the tube or cylinder wall can be expressed as:

σr= [(pi ri2 - po ro2) / (ro2 - ri2)] + [ri2 ro2 (po - pi) / (r2 (ro2 - ri2))]                                  (3)

maximum stress when  r = ro (outside pipe or cylinder)

### Example  - Stress in Thick walled Cylinder

In a cylinder with inside diameter 200 mm (radius 100 mm) and outside diameter 400 mm (radius 200 mm) there is a pressure 100 MPa relative to the outside pressure.

Stress in axial direction can be calculated as

σa = (((100 MPa) (100 mm)2 - (0 MPa) (200 mm)2) / ((200 mm)2 - (100 mm)2

= 33.3 MPa

Stress in circumferential direction - hoop stress - at the inside wall (100 mm) can be calculated as

σc = [((100 MPa) (100 mm)2 - (0 MPa) (200 mm)2) / ((200 mm)2 - (100 mm)2)] - [(200 mm)2 (100 mm)2 ((0 MPa)- (100 MPa)) / ((100 mm)2 ((200 mm)2 - (100 mm)2))]

= 167 MPa

Stress in radial direction at the inside wall (100 mm) can be calculated as

σr = [((100 MPa) (100 mm)2 - (0 MPa) (200 mm)2) / ((200 mm)2 - (100 mm)2)] + [(200 mm)2 (100 mm)2 ((0 MPa)- (100 MPa)) / ((100 mm)2 ((200 mm)2 - (100 mm)2))]

= -100 MPa

Note! - that in addition stress caused by pressure - stress can be induced in the pipe or cylinder wall by restricted temperature expansion.

### Online Thick Walled Pipe & Cylinder Calculator

The calculator below can be used to calculate the stress in thick walled pipes or cylinders with closed ends.

## Related Topics

### • Mechanics

Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

### • Statics

Loads - forces and torque, beams and columns.

## Related Documents

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### Barlow's Formula - Calculate Internal, Allowable and Bursting Pressure

Calculate pipes internal, allowable and bursting pressure.

### Fixed Pipes - Stress vs. Change in Temperature

Temperature changes introduces stress fixed pipes.

### Hooke's Law

Hooke's law - force, elongation and spring constant.

### Process Pipes - Allowable Stress vs. Temperature

Allowable wall stress in pipes according ASME M31.3.

### Restricted Thermal Expansion - Force and Stress

Stress and force when thermal expansion a pipe, beam or similar is restricted.

### Rotating Bodies - Stress

Stress in rotating disc and ring bodies.

### Shafts Torsion

The torsion of solid or hollow shafts - Polar Moment of Inertia of Area.

### Steam Boiler Shells - Stress vs. Boiler Pressure

Calculate the stress in steam boiler shells caused by steam pressure.

### Stress

Stress is force applied on cross-sectional area.

### Stress in Thin-Walled Cylinders or Tubes

Hoop and longitudinal stress thin-walled tubes or cylinders.

### Stress, Strain and Young's Modulus

Stress is force per unit area - strain is the deformation of a solid due to stress.

### Young's Modulus, Tensile Strength and Yield Strength Values for some Materials

Young's Modulus (or Tensile Modulus alt. Modulus of Elasticity) and Ultimate Tensile Strength and Yield Strength for materials like steel, glass, wood and many more.

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