FEM Concepts: Material Nonlinearity, Approximation, and Elements
1. Give the example of material nonlinearity. Material nonlinearity involves the nonlinear behaviour of a material based on a current deformation, deformation history, rate of deformation, temperature, pressure, and so on. Examples of nonlinear material models are large strain (visco) elasto-plasticity and hyperelasticity (rubber and plastic materials).
Approximation Sources in FEM
2. Give two examples of approximation sources in FEM
Shape Functions: Shape functions are used to approximate the variations of the unknown field variables within each finite element. These functions define the spatial distribution of the field variables over the element domain. The choice of shape functions depends on the type of element being used (e.g., linear, quadratic, cubic elements) and their formulation is crucial in accurately capturing the behaviour of the physical system. Commonly used shape functions include Lagrange polynomials, Hermite polynomials, and serendipity functions.
Mesh Density: The accuracy of the FEM solution is influenced by the density of the mesh, i.e., the division of the domain into smaller finite elements. A finer mesh generally provides a more accurate approximation of the solution, as it allows for better representation of complex geometries and variations in the physical field. However, increasing mesh density also increases computational costs. Therefore, mesh density is a key approximation source in FEM, balancing the trade-off between computational efficiency and solution accuracy.
10. Write the requirements that must be satisfied by shape functions.
Isotropic Elastic Material Properties
3. Explain the properties of isotropic elastic material. Give an example of such material used in civil engineering.
Isotropic elastic materials exhibit specific mechanical properties that are uniform in all directions. Here are the key properties of isotropic elastic materials:
- Isotropy: Isotropic materials have the same mechanical properties in all directions. This means that the material’s response to applied forces, stresses, and strains is independent of the direction in which they are applied.
- Linear Elasticity: Isotropic materials follow Hooke’s Law, which states that the stress is proportional to the strain within the elastic limit. The relationship between stress (σ) and strain (ε) can be expressed as σ=Eε, where E is the Young’s modulus.
- Homogeneity: Isotropic materials have consistent properties throughout their volume. The material’s behaviour is the same at any point within its structure.
- Orthotropic: While isotropic materials have the same properties in all directions, they may still exhibit different properties in different directions if they are also anisotropic.
An example of an isotropic elastic material commonly used in civil engineering is concrete. While concrete is generally isotropic for small deformations within its elastic range, it can become anisotropic at larger strains or due to factors such as cracking. In civil engineering applications, concrete is widely used in structures like buildings, bridges, and dams due to its strength, durability, and versatility. When analysing the behaviour of concrete structures, especially under normal working conditions and small deformations, the assumption of isotropy is often a reasonable approximation.
Isoparametric Element Definition
8. Define the isoparametric element.
An isoparametric element is a finite element that uses the same shape functions to describe both the geometry (or spatial coordinates) and the variation of field variables within the element. In the context of finite element analysis, elements are used to discretize a complex domain into simpler subdomains for numerical analysis. Isoparametric elements have the distinctive feature of employing the same interpolation functions (shape functions) for both the geometric coordinates and the field variables.
9. Draw Lagrange shape functions for a given 1-dimensional 3 node element.
Structural vs. Continuum Elements
5. Compare structural and continuum elements.
For continual elements, m=n, for structural elements m>n
6. Sketch the nodal degrees of freedom of 3D beam element.
Stiffness Matrix Expression
4. Write the expression for stiffness matrix. What is the size of the stiffness matrix for CST element?
