Finite element analysis starts with an approximation of the region of interest into a number of meshes (triangular elements). Each mesh is connected to associated nodes (black dots) and thus becomes a finite element.
Consider a linear spring of stiffness k . Let the displacements of its two ends, called nodes , be denoted by u i and u j , known as nodal displacements . Let the forces acting at its two ends, called nodal forces , be denoted by f i and f j .
i j f j f i Nodes y x u j u i Nodal force Nodal displacement Reference frame
The element nodal displacements are also known as element nodal degrees of freedom (DOF)
Singularity of Element Stiffness Matrix The equation ku = f cannot be solved for the nodal displacements for arbitrary f because the matrix k is singular . Physically this means that, in static equlibrium, the displacements of the endpoints of a spring cannot be determined uniquely for an arbitrary pair of forces acting at its two ends. One of the ends must be fixed or given a specified displacement; the displacement of the other end can then be determined uniquely.
Note that the global stiffness matrix is assembled from element matrices as follows
Stiffness terms from two matrices add at coinciding DOF Stiffness matrix from element Stiffness matrix from element
Singularity of Global Stiffness Matrix As in the case of individual element matrices the global stiffness matrix K is singular . (You can check this out for this small example by calculating the determinant of the matrix; the result will be zero.) Some nodes of the structure need to be constrained (i.e. fixed or given known displacements) to make it statically determinate or overconstrained. Then the remaining DOF can be determined. Constraining some nodes in the structure corresponds to applying boundary conditions.
The plot below charts the variation of displacements across the bar for the two solutions
Formulation of the Finite Element Method f B – Body forces (forces distributed over the volume of the body; gravitational, inertia, or magnetic forces) f S – surface forces (pressure of one body on another, or hydrostatic pressure) f i – Concentrated external forces
Formulation of the Finite Element Method Denote the displacements of any point (X, Y, Z) of the object from the unloaded configuration as U T The displacement U causes the strains and the corresponding stresses The goal is to calculate displacement, strains, and stresses from the given external forces.
Using the stress-strain relations for homogeneous, isotropic
plane-stress, we have
Formulation of the Finite Element Method Equilibrium condition and principle of virtual displacements The left side represents the internal virtual work done and the right side represents the external work done by the actual forces as they go through the virtual displacement. The above equation is used to generate finite element equations. And by approximating the object as an assemblage of discrete finite elements, these elements are interconnected at nodal points.
Formulation of the Finite Element Method The equilibrium equation can be expressed using matrix notations for m elements. where B (m) represents the rows of the strain displacement matrix C (m) is the elasticity matrix of element m H (m) is the displacement interpolation matrix U is a vector of the three global displacement components at all nodes F is a vector of the external concentrated forces applied to the nodes
Formulation of the Finite Element Method The previous equation can be rewritten as follows, The above equation describes the static equilibrium problem. K is the stiffness matrix.
Example, cont’d Calculating the stiffness matrix for element 2.
Example, cont’d The stiffness of the structure as a whole is obtained by combing the two matrices.
Example, cont’d The load vector R, equals R c because only concentrated loads act on the nodes where P y is the known external force and F 1x , F 1y , F 3x , and F 3y are the unknown reaction forces at the supports.
Example, cont’d The following matrix equation can be solved for nodal point displacements
Example, cont’d The solution can be obtained by applying the boundary conditions
Example, cont’d The equation can be divided into two parts, The first equation can be solved for the unknown nodal displacements, U 3 , U 4 , U 7 , and U 8 . And substituting these values into the second equation to obtain unknown reaction forces, F 1x , F 1y , F 3x , and F 3y . Once the nodal displacements have been obtained, the strains and stresses can be calculated.
Specify analysis type (static or dynamic, linear or non-linear, plane stress, etc.)
FEA Pre-Processing Mesh Development The FEA mesh is your way of communicating geometry to the solver, the accuracy of the solution is primarily dependent on the quality of the mesh. The better the mesh looks, the better it is. A good-looking mesh should have well-shaped elements, and the transition between densities should be smooth and gradual without skinny, distorted elements.
FEA Pre-Processing - Example Coarse mesh Refined mesh - better?…probably… depends on the loading and boundary conditions
FEA Pre-Processing Finite elements supported by most finite-element codes:
FEA Pre-Processing Material Properties Material properties can be specified for element regions, elements or even within elements in most large-scale FEA codes. The material properties required for an isotropic, linear static FEA are: Young’s modulus (E), Poisson’s ratio (v), and shear modulus (G). G = E / 2(1+v) Only two of the three properties can be specified independently.
FEA Pre-Processing Nonlinear Material Properties A multi-linear model requires the input of stress-strain data pairs to essentially communicate the stress-strain curve from testing to the FE model Highly deformable, low stiffness, incompressible materials, such as rubber and other synthetic elastomers require distortional and volumetric constants or a more complete set of tensile, compressive, and shear force versus stretch curve. A creep analysis requires time and temperature dependent creep properties. Plastic parts are extremely sensitive to this phenomenon
FEA Pre-Processing Boundary Conditions In FEA, the name of the game is “ boundary condition ”, that is calculating the load and constraints that each component experiences in its working environment. The results of FEA should include a complete discussion of the boundary conditions
Boundary Conditions Loads Loads are used to represent inputs to the system. They can be in the forms of forces, moments, pressures, temperature, or accelerations. Constraints Constraints are used as reactions to the applied loads. Constraints can resist translational or rotational deformation induced by applied loads.
Boundary Conditions Degrees of Freedom Spatial DOFs refer to the three translational and three rotational modes of displacement that are possible for any part in 3D space. A constraint scheme must remove all six DOFs for the analysis to run. Elemental DOFs refer to the ability of each element to transmit or react to a load. The boundary condition cannot load or constrain a DOF that is not supported by the element to which it is applied.
Post-Processing View Animated Displacements View Displacement Fringe Plot View Stress Fringe Plot View Results Specific To the Analysis Review Boundary Conditions Review Load Magnitudes and Units Review Mesh Density and Quality of Elements Does the shape of deformations make sense? Are magnitudes in line with your expectations? Is the quality and mag. Of stresses acceptable? Yes Yes No No No Yes