A4. Interaction between the structure and its environment

Taking into account the interactions between the structure and its environment: this consideration allows us to reduce the number of discrepancies between the model and the real-life structure. The effect of the environment on the structure is determined from the nodal values such as the nodal displacements and nodal efforts. The first ones are connected to the boundary conditions and the second ones to the external loading [1].

the displacement boundary conditions (nodal) make it possible to impose displacement values to a node (zero or non-zero). The imposed displacements are often named kinematic constraints. They enable to connect the displacements of certain nodes. Initially, it is preferable to determine if the symmetries should be taken into account before imposing the boundary conditions. When symmetries are adopted the nodal displacements perpendicular to the plane of symmetry in 3D or the axis of symmetry in 2D should be blocked (figure 3) (see chapter 4).

Boundary conditions for a plate with a hole in the center, 2 axes of symmetry, submitted to tension

Afterward, it is essential to eliminate the rigid body movements. A correct finite element model has to restrain free rotations. In 3D, the 6 rigid body movements previously introduced should be avoided (in 2D, there are 3 movements, figure 4).

Possible rigid body movements in 2D: a) possible horizontal translation; b) possible vertical translation; c) rotation around a hinge; d) all rigid body displacements restrained

Once these 2 steps are accomplished, it should be verified that every rigid body movement is correctly restrained and that no rigid body movement was blocked when this mode has already been removed. In the first case, the problem has no solution and in the second case, unexpected strains may appear.

the loading corresponds to exterior efforts exerted on certain parts of the mesh. Among the loadings, forces such as gravity and inertia should be taken into account. They are in general modeled as volumetric forces represented by nodal forces applied to all the nodes in the domain.

There are also contact forces such as pressure or any other force that needs contact with the structure. They can be surface, linear, or punctual forces. Their application should also be transformed into nodal forces. Special attention should be paid to the transformation of the contact forces when creating a model respecting its validity domain. Using a point load can generate singularities such as stress concentration in the neighborhood of the node where the load was applied. Thus, to avoid this kind of singularity, it is necessary to apply the point load as a surface load in the neighborhood of the node. This technique consists of applying a surface load containing a relatively large area around the application point. Then comes the question of mesh refinement in this area and its influence on the obtained results.

Connection conditions

There are different types of finite elements, such as volumetric, plates, shells, beams, and bars. Thin plates and shells are elements, which have a thickness smaller than their other 2 dimensions. A plate works only perpendicularly to its plan (3 degrees of freedom (DOF) for a node: 1 translation and 2 rotations), whereas a shell element works along its plan and perpendicularly to it (6 DOF: 3 translations, 3 rotations). The solid elements also have 6 DOF per node. Chapter 2 presents some practical examples.