The stiffness of thin-walled shell structures can be approximated well, if balanced shape functions are provided for the description of the stress state inside three-dimensional finite elements in mixed formulations.
Example: Curved shell structure
The quality of the approximated solution for the radial displacement of a curved beam under bending is analyzed for mixed finite elements with linear and quadratic shape functions of the displacement field and balanced shape functions for the stresses. A weak formulation of the overall governing equations of elastic-viscoplastic structures is possible, if the kinematics are described by velocities as primary variables. Using the principle of virtual velocities, the balance of linear momentum as well as the rate of strain balance is formulated in its weak form in the space-time domain, avoiding collocation schemes for the viscoplastic strains at discrete material points.
Example: Strip footing
Next to the supposed model, the history of the pressure state related to the imposed deformation is displayed. The pressure converges against the analytic solution.
The distribution of the normal stress in vertical direction indicates the region of viscoplastic flow.