Three years ago when we first introduced the CAE Associates blog, I wrote a post about the value of a one element model. All of those original messages are still valid today, however, there are also a number of situations, where believe it or not, one element is just not enough. So, in this post, I will discuss some examples where it is necessary to double the model complexity; hence I will discuss two element models. A couple examples are provided herein, where two element test cases can be very valuable in the development stage of complex finite element analyses:
 Evaluating the impact of modeling material mismatch such as different coefficients of thermal expansion or creep material response.
 Testing connections between elements with different degrees of freedom.
A two element model can be defined with links, beams, 2D solids, 3D bricks or tetrahedrons, or any combination of these. For boundary conditions on the two element model, I like to use symmetry boundary conditions and either a nonzero displacement on a free face, a constant thermal load, or a nodal force in the case of beam models. Care should be taken to eliminate stress concentrations on the model boundaries. The use of incremental displacement or temperature body loads provides a stable solution for evaluating nonlinear material response such as creep or relaxation.
Material Mismatch Example:
Figure 1, above, illustrates the before and after stress evaluation of a two element model representing a concrete (bottom) and steel (top) interface subjected to compressive preload and concrete creep. An initial compressive stress state is induced with a displacement boundary condition and symmetry to create a constant stress state in both elements. Concrete creep is next modeled using a time dependent material law which results in a decrease in the concrete stress as it relaxes. The stress in the steel increases slightly during the same load duration due to load redistribution.
Figure 2 illustrates the time dependent stress response of the two elements, while strains are illustrated in Figure 3. This simple two element model provides rapid valuable data which can be used to compare the impact on a range of input parameters including prestress, time and the nonlinear creep material properties.
Figure 2 – Increase & Decrease of Steel and Concrete Stresses as a Result of Concrete Creep

Figure 3 – Constant Steel & Variable Concrete Elastic Strain Response Due to Stress Relaxation in the Concrete

BeamSolid Connection:
A two element model can also be used to demonstrate the connection between elements with a mismatch of active degrees of freedom. Combining beam and solid elements is an efficient modeling approach for complex structures. One can combine the goals of computing local stress concentrations, necessitating solid elements, with a global stiffness response, captured with beam elements, using an efficient connection. If the beam and solid elements were modeled by simply sharing nodes, rigid body motion will occur since the rotational deformation of the beam, with 3 translational and 3 rotational DOF per node, is not transferred to the solid, which only has 3 translational DOF per node.
Figure 4 Example Beam to Solid Connection using MPCbased Constraint Equations

Figure 4 illustrates a combined beam and solid element model where constraint equations are used to couple the rotational and translational degrees of freedom of the beam to the translational degrees of freedom of the solid element. The value of the two element problem is that it can be quickly used test the connection process. In this case, an ANSYS MPC contact interface is developed, with the contact side being the vertex of the beam and the target surface being the face of the solid element. Axial and moment load transfer is achieved via these constraint equations as shown by the displacement response illustrated in Figure 5.
Figure 5  Axial Displacement on Solid Demonstrating Moment Load Transfer

I am sure that there are many more applications of the two element model that I have not touched on and welcome your input and feedback!