Incorrect Polar Moment of Inertia in Mechanica Beam Sections

[ptc-385258]

Hello All, I am sitting here scratching my head wondering if Mechanica's beam section polar moment of intertia (J) is incorrect. For example, take a 2x2 inch square solid bar. According to hand calculations J = 2.667 in^4. Modeling the 2x2 solid bar in Pro/Engineer and analyzing the crossectional properties (Analysis -> Model -> XSection Mass Properties) it also comes up with 2.667 in ^4. However, in Mechanica when you create a new beam section as square with an "a" dimension of 2 and then click "Review" the polar moment of inertia is displayed as 2.25 in^4. Am I missing something here? This looks to be wrong. I know Mechanica has serious problems in calculating the J for custom sketched sections, but this is bizzare. Any thoughts? Steve
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[AndyDeighton-di]

In the Mechanica help it shows the following: Beam Section Property Calculations Depending on the type of beam cross section, Mechanica uses the following formulations for beam section property calculations: Solid Sections — The shear center is not calculated for solid sections. It is assumed to be at the centroid of the section, coincident with the neutral axis. You can modify the location of the shear center. Torsional stiffness, the second polar moment of area J, is approximated as: J = 4 Iy Iz / (Iy + Iz) Note: This equation gives the exact value only for circular sections and can have an error as high as 20% for rectangular sections. For other shapes, the error can be even higher. Exercise caution when using the calculated value of J. You can find exact values of J for torsional stiffness in R.J. Roark and W.C. Young, Formulas for Stress and Strain, 6th edition, Table 20, pages 348–359. I'll try and find out why it doesn't use the correct calculation for J

[AndyDeighton-di]

Actually, now that I check the approximation formula this gives 2.6666 also. I also tried creating a sketched solid in Mechanica and this gives 2.6666, so it's strange that the standard square gives 2.25 - I'll try to get an explanation.

[ptc-385258]

I think I just answered my own question. The J shown on the Review screen for the beam properties is only the polar moment of inertia if the cross section is circular. For all non-circular crossections (including open sections) this value "J" is actually the torsional K factor for beams, which for non-circular sections is typically a fraction of the true polar moment of inertia J for the section. It would be nice to have the variable name be "K" not "J" for all sketched and non-circular beam cross sections. I pulled the information between the difference of J and K from Roark's 7th Edition.