Key Concepts in Finite Element Method (FEM) Analysis

Posted on Jun 11, 2025 in Real Estate Sciences

Fundamental Concepts in Finite Element Method (FEM)

Sources of Nonlinearity:

  • Constitutive relations (material)
  • Changes of geometry (strain, displacements)
  • Boundary Conditions (contact)
  • Loads (deformation)

Why FEM is Important:

  • It replaces a continuous structure with a model having a finite number of points.
  • It describes physical phenomena for which analytical solutions are not known.

Nonlinear Elastic Materials:

Present nonlinear stress-strain relationships even at infinitesimal strains.

Linear Elastic Materials:

These materials obey Hooke’s Law, meaning the relationship between stress and strain is linear.

Rigid Body Motion:

The motion a system undergoes when the inversion of the global stiffness matrix K-1 is singular. This occurs if boundary conditions are not considered during the FEM approximation of the principle of virtual displacements.

Disadvantages of CST Element:

In the area of this element, we will always get a continuous distribution of stresses. More precise analysis may be needed.

Undesired Effects of Reduced Integration:

One remedy used to minimize the locking effect is reduced integration, but this may lead to spurious zero-energy modes.

Locking Effect:

Selection of the integration rule. It is a problem of blocking results, where displacements can be very small or even zero.

Stiffness Matrix:

∫BTDB d(vol), where B is the matrix of derivatives of the shape function and D represents the constitutive equations.

Boundary Conditions:

Must be considered while aggregating the global stiffness matrix; otherwise, the matrix is singular, and the system undergoes rigid body motion.

Requirements for Shape Functions:

  • Continuity Conditions: Shape functions must ensure continuity of displacements inside an element and their compatibility on element boundaries.
  • Rigid Body Criterion: Under rigid body motion of an element, strains and stresses must not appear inside the element.
  • Constant Strain Criterion: Shape functions must contain terms responsible for constant stress/strain.

Virtual Work vs. Virtual Displacements:

The principle of virtual work is general, while the principle of virtual displacements is expressed only by displacements.