I am making the attached Mathcad 15 worksheet (Ring q Loading - Sin-Tangential Resistance - Uniform Varying Load.xmcd) available to anyone that needs such a tool. This worksheet is one of several that analyzes a structural ring stiffener (or bulkhead) given a general radial load pattern resisted by sine-tangential forces in the adjacent shell. This is a common structural model fully explained in Roark's "Formulas for Stress and Strain" (any edition).
Since I am new to this PTC community forum, I am using this worksheet as a test. If this works, and/or the reports back from other members of the forum indicate an interest for other types of loading conditions, I will continue to post them.
While this problem can be modelled using Finite Element Methods, the process is labor and time intensive, and would require frequent model alterations to converge to an efficient design. Also, the proper FEM boundary conditions needed for the sine-tangential resistance are very, VERY!, tricky. As most FEM users will agree, modeling the correct boundary conditions may be the hardest part of FEM modeling.
The cases in Roark are for a very limited and specific set of loading conditions. These MC 15 worksheets deal with completely general loading conditions. A loading condition such as that presented in ASCE 7-16 for a wind load profile on a cylindrical structure (such as a partially empty tank) is very asymmetric with positive and negative radial loads. Asymmetric ring loads result in high bending stresses in the rings and shell. The MC 15 worksheet for such a loading condition solves and integrates the ring virtual energy equations and presents graphic results of the bending moment, axial load, radial shear, and radial deflection at any point on the ring. A graphic towards the beginning of the worksheet reiterates the loadings so the user can have a check on his loading construct.
Even though this worksheet presents moment, axial forces, and shear forces in the ring, they of course, are not the final answers. Stress limits, service limits, and local and general buckling conditions still need to be evaluated by the structural engineer. This will require using the worksheet results in conjunction with the governing design Codes and Specifications such as API, AISC, ASME, ASCE, applicable EN standards, etc.
Other ring worksheets I have model radial point loads and applied moments (any number), linear varying loads (positive or negative), sin or cos loadings (any wave length, that’s reasonable), and for that matter, any loading that can be mathematically defined in the worksheet load region. Piecewise loadings can be handled using the programming features in Mathcad.
Even though many engineers/customers insist that FEM methods be used, these worksheets will at least narrow down the modelling choices so FEM models can be tested more efficiently for the final design.