Notes on MacNeal slender beam

The MacNeal slender beam was calculated under several equivalent treatments respectively with Q4 element, the results showed evident differences of about only three significant digits around each other. These equivalent treatments include equivalent stiffness method and equivalent load method for beam´s thickness, changing elasticity Modulus E and load F in proportion, changing node numbering, different solvers of square root method and improved square root method. The 2-norm condition number of stiffness matrix checked by Matlab is up to 1e5, which indicates the problem is much ill-conditioned, accordingly the abnormal differences among results were explained by disturbance of ill-conditioned equations. Further analysis concluded the main cause for this model´s ill-condition is the beam´s slenderness, not the element´s irregular shape, and in order to obtain stable result, a necessary, simple and effective method is to calculate with double precision.