A2. The dimensionality of the model

A2. The dimensionality of the model

It is very important to simplify the full-size problem to model the interaction between the structure and its environment. To do so, there are 2 known techniques:

• the first one consists of transforming the full-size problem into a lower-dimensional space by modeling it as an axisymmetric structure or a 2D structure (plane strain or plane stress assumptions). This technique reduces the used space (see section 2.2).
  

• the second one consists of working in a 3D space while still reducing the model thanks to kinematic hypotheses. For instance, it is possible to adopt simplifications such as beam, plate, or shell theories (see section 2.1).

Those 2 techniques help reduce the computational cost. However, the user has to be very careful when adopting those simplifications. For instance, the second technique uses hypotheses that are valid only in a limited domain. It is important to make sure that the adopted simplification stays within the domain of the full-size problem to obtain relevant results.

1) Case of finite elements applied to Strength of Materials

From a finite elements method perspective, the main difference between Continuum Mechanics and Strength of Materials relies on the geometry, which is simplified through additional hypotheses: the initial 3D problem is then transformed into a 2D problem (mean surface for plates and shells) or 1D (mean fiber for beams), but still represented in a 3D space (except for 2D problems, as mentioned above). When the FE method is used to solve a Strength of Materials problem, the finite elements have specific characteristics. It is necessary to assign to those elements geometrical properties (the section, the inertia for beams, the thickness for plates and shells). Moreover, they combine the effects of tension/compression (for beams) or membrane effects (for plates and shells) to the bending effects. The traction/compression and membrane effects are treated as they are treated in Continuum Mechanics, while the bending effects are handled separately.

For the computation of bending effects, simplifications in the geometry lead to a particular definition of deformations, which results in different expressions of the differential operator D according to the situation. Another consequence is due to the definitions of the unknowns at the nodes: while in Continuum Mechanics the unknowns are the displacement components, in Strength of Materials the unknowns corresponding to the rotations are added. This happens because it is no longer possible to evaluate directly the rotations as a consequence of the simplifications of the geometry. The initial choice of treating the problem in the frame of Continuum Mechanics or Strength of Materials implies choosing between different families of finite elements. It is, in principle, not possible to mix finite elements of different families because at the interfaces between elements of different nature the rotations are not transmitted unless they are adapted to take it into account. Moreover, the calculated stresses inside the element are usually generalized stresses (or Strength of Materials efforts: axial, shear, torsion, bending). To obtain the stresses (in Continuum Mechanics) at a given point, it is necessary to provide additional information (position inside the beam’s cross-section, for example).

In this scenario, choosing a finite element brings on an additional difficulty related to the consideration or not of the shear forces (Euler-Bernoulli or Timoshenko beam theories, Love-Kirchoff, or Reissner-Mindlin plate theories). This choice is related to geometric considerations (slenderness of the cross-section or thickness of the plate). Moreover, taking into account the shear forces might lead to numerical errors (shear locking), which renders the use of some finite elements complicated.

The Euler-Bernoulli beam finite element allows to represent exactly the bending moments that vary linearly along with the mean fiber of an element (shape functions of order 3 for the bending displacements and the bending moments obtained by taking the second derivative of the displacements): it is therefore not useful to introduce several elements between 2 nodes subjected to point loads (a dense mesh should be used when the loading is distributed between these 2 points).

Finally, for plate and shell elements, the monotonic convergence is not guaranteed depending on the mesh geometry. This behavior is connected to the formulation of the element itself. This kind of behavior is illustrated in the figure below, showing the evolution of the relative error of the deflections ω and Mx as a function of the number of degrees of freedom (in the case of a rectangular plate supported by its four sides submitted to uniform loading): in green, a non-compliant element (COQ3) for which the results are much less precise than for a compliant element (DKT). Also, for a non-compliant element (COQ3) the convergence is not monotonic (for the bending moments Mx in the figure): more elements leads, paradoxically, to results that can be less precise!

Bending plate: convergence (deflection ω and bending moment Mx) as a function of the number of degrees of freedom

This domain is particularly important to civil engineering, but it presents specific difficulties that will be explored in the next chapters.

2) Two-dimensional calculations

The studied problems are three-dimensional, however, it is faster to calculate two-dimensional problems. In some cases, it is possible to transform a three-dimensional problem into a two-dimensional problem:

• if the problem admits a revolution axis (for the geometry, the loading, and the boundary conditions): it is possible to perform an axisymmetric calculation with no additional hypotheses needed. In the case where the loading is not axisymmetric, it is possible to decompose it in the Fourier series and treat the initial problem as a superposition of axisymmetric calculations.
  
  

Cylinder under axisymmetric pressure

• if one adopts the hypothesis of neglecting the stresses or the strains that are out of the plane, for cases in which the structure is either very slender or very thick, plane stress or plane strain assumptions can be made. The obtained solution is then an approximation of the three-dimensional problem. It is important to keep in mind that in-plane stress, the strains outside of the plane are equal to zero (similarly, in-plane strains, the stresses outside of the plane are equal to zero).

Dam: important thickness of plane strain assumption

Assembly: low thickness/plane stress assumption

3) Taking into account the symmetries

Some problems present symmetries (axis of symmetry, plane of symmetry…) and it can be interesting to use them to make the finite element calculations faster. Although, it is important to notice that solving symmetric problems will provide only symmetric results (especially when computing eigenmodes).

For that matter, it is important to keep in mind that to use those symmetries in the calculations, not only the geometry has to be symmetrical, but also the loading and the boundary conditions as mentioned above. The resolution of this model will not represent the solution of the whole structure unless the symmetry conditions are added to the model. In Continuum Mechanics, one should impose the displacement components perpendicular to the axis/plane of symmetry equal to zero. In Strength of Materials, it is convenient to add no-rotation conditions around the axis/plane of symmetry. Finally, taking into account symmetries might impact the loading: for instance, a punctual load should be applied with half the intensity if applied to an axis of symmetry.