ShellRadialTemperatureGradient.xls

A spreadsheet to calculate the displacements and stresses in an open ended cylindrical shell subject to a radial temperature gradient across its thickness.

Introduction
This spreadsheet calculates the deflection and stresses in an open ended cylindrical shell wall subject to a radial temperature gradient through its wall thickness, along its entire length. The calculation assumes that the mean temperature, ₸, of the shell is zero. For a non zero mean temperature the radial displacement will simply increase by αR₸ whilst the stresses remain unchanged for an unrestrained shell. The open and free end of the cylindrical shell is assumed to be remote from any restraint that might affect the radial displacements and stresses produced by the radial temperature gradient.

Shell behaviour
At the free edge of the cylindrical shell wall the radial temperature gradient introduces bending and maximum deflection, whilst at a distance remote from the free end there is no radial deflection for the assumed zero mean temperature. The maximum radial deflection at the free end induces maximum hoop membrane stresses which in combination with bending stress produces the maximum surface stress in the cylinder wall. Away from the free end of the cylinder wall stresses tend towards a constant value.

Notes
Radial stresses are assumed to be negligible along the cylindrical shell wall in comparison to induced hoop and axial stress components.

The method was obtained from Timoshenko and Woinowsy, Theory of Plates and Shells.

Named ranges are used throughout so that cell formulae can be easily discerned.

The system of units can be chosen in reference to units of length. Other units have been chosen to
be consistent with that unit of length. No conversion of units is used in the spreadsheet.

Verification
The results from the example given compare with finite element results using the Abaqus software
 

Calculation Reference

Timoshenko and Woinowsy, Theory of Plates and Shells

Thermal Stress Analysis

Pressure Vessel Design