Why the Number of Displacement Polynomials Used for an Element Depends on Degrees of Freedom:

In the field of finite element analysis (FEA) and structural engineering, displacement polynomials are mathematical functions used to represent the displacement field within finite elements. These displacement polynomials are essential for analyzing and modeling the behavior of structures subjected to various loads and boundary conditions.

The number of displacement polynomials used for an element depends primarily on the degrees of freedom (DOF) associated with that element. Here’s a detailed explanation of why degrees of freedom are the key determinant:

1. Degrees of Freedom (DOF):

Degrees of freedom represent the number of independent ways a node or point within a finite element can move. In structural analysis, DOF typically includes translational (displacement in x, y, and z directions) and rotational (rotation about x, y, and z axes) components. For example, a 3D node with three translational and three rotational DOF has a total of six DOF.

2. Element Shape and Geometry:

The number and arrangement of nodes within a finite element determine its shape and geometry. Different types of finite elements, such as beams, trusses, plates, or three-dimensional solids, have distinct node arrangements and shapes. These shapes, in turn, affect the number and types of displacement components required to accurately represent the deformation and behavior of the element.

3. Interpolation Functions:

Displacement polynomials, also known as interpolation functions or shape functions, are used to interpolate the displacements at various points within an element based on the displacements at its nodes.

The form and complexity of these interpolation functions depend on the element’s shape, node distribution, and DOF. Elements with more nodes and DOF require more complex interpolation functions to capture deformation accurately.

4. Accuracy and Precision:

To ensure accurate and precise results in finite element analysis, it is essential to use a sufficient number of displacement polynomials to capture the deformation behavior of the element. Inadequate or insufficient displacement polynomials can lead to inaccurate results and numerical instability.

Therefore, the number of displacement polynomials used is directly related to the element’s DOF to ensure the desired level of accuracy.

Why the Other Options Are Not Correct:

A. Nature of Element:

While the nature of an element (e.g., whether it is a beam, truss, shell, or solid) does influence the arrangement of nodes and the overall behavior of the element, it is the degrees of freedom associated with the element that determine the number of displacement polynomials needed to accurately represent its deformation.

Different elements with the same number of DOF will require a similar number of displacement polynomials, regardless of their nature.

B. Type of Element:

The type of element refers to its classification in structural analysis, such as one-dimensional (beams and trusses), two-dimensional (plates and shells), or three-dimensional (solids).

While the type of element does play a role in determining the arrangement of nodes and the nature of deformation, it is again the degrees of freedom that dictate the number of displacement polynomials needed to model that deformation accurately. Different types of elements can have the same number of DOF and thus require a similar number of displacement polynomials.

D. Nodes:

The number of nodes in an element does influence the arrangement of the nodal points, but it does not solely determine the number of displacement polynomials.

It is the degrees of freedom associated with these nodes that ultimately determine the number of displacement polynomials required. Elements with the same number of DOF will need the same number of displacement polynomials, even if they have a different number of nodes.

In summary, the number of displacement polynomials used for an element in finite element analysis depends primarily on the degrees of freedom associated with that element.

Degrees of freedom determine the complexity of the interpolation functions and the accuracy of the finite element analysis. Understanding this relationship is crucial for engineers and analysts when selecting appropriate finite elements and ensuring accurate structural simulations.