As a result of numerous and costly failures of the stainless-steel bellows used to take up thermal expansion in the piping of petroleum and chemical processing equipment, a basis has been developed for designing these bellows to operate within reasonable stresses. A formula has been derived to show the total stress induced in the material as a result of the combined effects of pressure and movement. The validity of the approximations used in this formula has been verified by laboratory strain-gage measurements on an experimental bellows. A relationship between several variables in the design of the disks has also been determined and serves as a basis for dimensioning them to achieve the most economical proportions. When this relationship is satisfied, the maximum permissible movement per disk can be obtained. This is shown graphically. Several supplementary formulas are given for determining bellows characteristics which may affect the design of adjacent piping. The problem of designing an expansion joint for high-pressure service is discussed briefly, and some preliminary laboratory data are presented on a commercial joint of this type.