As we know, finite element analysis (FEA) is a mathematical technique to solve physical and mechanical problems. This is achieved by dividing the computational domain into a finite set of partitions called elements and by calculating the solution at a finite set of locations called nodes. In this article, we will explore how to find the appropriate solution for elemental nodes.
At each node, we define fundamental variables called degrees of freedom, which are calculated during finite element analysis. Let’s understand these using the example of an aircraft considered as a single node shown in the figure below.
This aircraft can move along 3 independent orthogonal directions which correspond to three translational DOF 1, 2 and 3. Further, this aircraft can change orientation by rotating about three independent axis of rotation which correspond to three rotational DOF 4, 5 and 6. There can be other DOF depending on the kind of problem we are solving like temperature, acoustic pressure, electric potential, pressure, axial force, and warping in open section beams. In conclusion, all the unknown variables that need to be calculated in a physical problem are considered to be degrees of freedom at a node. In FEA problem, these DOF are solved only at the nodes. At all other points in the element, the DOF are approximated using interpolation functions from the nodal values.
Formulation For Elements Using Nodal DOF
Now, let’s try to understand how the nodal DOF is used while formulating shape functions for an element in FEA using a 2D element shown in the figure below.
The linear element has 4 nodes and two displacement DOF (u, v) at each node. So, in total there are 8 DOF for this element. If the element is quadratic, there will be 8 nodes for the element. The DOF at each elemental node will not change. The total number of DOF for this element is 16.
The same type of DOF at all nodes are approximated together for the formulations. For the 2D linear element,
The number of unknown coefficients in the polynomial must be equal to the number of similar kinds of DOF for the element. In this case, this is 4 (1 per node). By substituting the nodal coordinates, we get 4 equations from the above polynomial.
Now, solving for the unknown coefficients using 4 equations and rearranging, we get
Here, N1, N2, N3 and N4 are the bilinear shape functions. The shape functions for the DOF ‘v’ will be the same as the nodal coordinates and the polynomial approximation are the same.
Element Family And DOF
The elements are divided into different families based on the geometry type that each family assumes. Now, let’s consider different element families and the type and number of DOF each of the elements have.
Continuum Elements
Among all the element types, solid or continuum elements are used widely to model a variety of components. These are three dimensional elements which assume different types of shapes like hexahedral, triangular prism and tetrahedral. They can be used to build nearly any complex geometric shape components, subjected to any type of loading. Note in the table below that, because these are 3D elements, they do not require rotational degrees of freedom. This is important to note when constraining any 3D model at a small number of points!
Shell Elements
These elements are typically used to model structures with thickness very small compared to other dimensions. The stresses in the thickness direction for these structures should be negligible. These elements assume that the plane sections perpendicular to the plane normal will remain planar during analysis. Examples of shells structures are pressure vessels, boiler, suitcase, and distillation columns. Using continuum elements in this case will exponentially increase computational costs.
Beam Elements
These elements are used to model structures which have one dimension (length) significantly larger than the other two dimensions. In these elements, longitudinal stress is more important than cross-sectional stresses. These elements are formulated based on the assumption that the deformation of the structure can be determined from the variables which are functions of the position along the length of the beam. To get results within acceptable range, the general rule of thumb is that the cross-sectional dimensions should be less than 1/10 of the structure’s length. The figure below shows general beam cross sections.
Rigid Elements
These elements are not deformable and can be used in multibody dynamic simulations for defining contact. Only surfaces of the rigid bodies are represented using elements. The boundary conditions have to be attached to the whole rigid body using a reference point.
Truss Elements
These elements are used to model structural members which are long and slender that can only transmit axial forces. These elements don’t have any moment carrying capacity. They are modelled using the centerline of the section passing through its centroid and do not support any moments or perpendicular forces to the center line.
Conclusions
Hopefully, this article has briefly explained how nodal DOF are used to formulate shape functions and types of DOF available for different element families. There are a variety of elements available while solving a finite element problem. It is very important to choose a proper element based on geometry of the physical system, type of analysis and accuracy requirements. The elements are made up of nodes at which the unknown variables (DOF) in a FEA problem are calculated. These are approximated at other points in the element using interpolation functions.
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