# Computing forces and moments acting on the sides of a rectangular shell element from corner forces and moments

I have a FEM program for shell elements implemented through the direct stiffness method. Using this program corner displacements of shell elements due to external loading are obtained through the inversion of a global stiffness matrix. Once the displacements are known, computing forces and moments acting on the corner of each shell element is quite straightforward.

The shell element employed in this program has 4 corner nodes, and therefore it has 24 degrees of freedom in total; 6 degrees of freedom for each node (at each corner):

• 2 dofs for membrane/axial
• 1 dof for shear
• 3 dofs for bending

The issue here, I would like to compute the angles of the principal stress for each plate element. However most literature on the topic is referring to a convention where stresses are computed using forces and moments observed on the sides of the element (see picture). My question is, how to compute (or rather, estimate) forces and moments acting on the sides of a rectangular shell element when corner forces and moments are known? Can it be done through simple averaging? (I have a hunch that averaging is not the way to go) • Why not compute the principal stress directions directly from the element stresses? (And if your FEM program doesn't already do that for you, why not get one that does?) Jul 23 at 13:47
• @alephzero This is exactly where I got stuck, i have all the corner forces. But I am not so sure how to compute element stresses using corner forces.
– ar_k
Jul 23 at 13:51
• Your program should have provided reactions that are lumped in the centroid of an element. You shall work from there. Otherwise, you shall compute using all corner forces and the applied forces/pressures to get distribution under equilibrium. The use of simple average may twist the picture in the wrong way.
– r13
Jul 23 at 16:17
• You don't calculate the stresses from the nodal forces. You calculate them from the element formulation of the internal strains and stresses. If your FE program doesn't do that already, then it is barely useable for anything IMO. Jul 23 at 18:38
• Two things you should consider: 1) Use the displacements and rotations as the degrees of freedom, 2) use the definitions of forces and moments to compute them from the stresses (see en.wikipedia.org/wiki/Plate_theory#Equilibrium_equations). Jul 23 at 21:33