Pipe elbow stiffness coefficients including shear and bend flexibility factors for use in direct stiffness codes

The purpose of the present paper is to present a systematic derivation of the required 12x12 stiffness matrix and load vectors for a three dimensional pipe elbow element which includes the effects of transverse shear and pipe bend flexibility according to the ASME Boiler and Pressure Vessel Code, Section III. The results are presented analytically and as FORTRAN subroutines to be directly incorporated into existing direct stiffness codes. By selecting appropriate coordinate axes a diagonally partitioned form of both the force transfer and flexibility matrices is derived and efficiently used to develop the complete 12x12 stiffness matrix. Subsequent transformations are applied to obtain the final form in the same local coordinate system used for the straight members. The subroutines presented perform all the required computations including the transformations of the elbow input parameters of end point and tangent intersection point coordinates. Also discussed and often overlooked are the stiffness contributions arising from the interaction between bending moment and axial force as a result of the bend radius. It is also shown how certain stiffness coefficients may change sign depending on the values of the bend flexibility factors