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ASPECT RATIO
The ratio of the longest to shortest side lengths on an element
CRITICALLY DAMPED SYSTEM CRITICAL DAMPING
The dividing line between under damped and over damped systems where the equation
of motion has a damping value that is equal to the critical damping
CRITICAL ENERGY RELEASE
This is a material property defining the minimum energy that a propagating crack must
release in order for it to propagate. Three critical energies, or modes of crack propagation,
have been identified. Mode 1 is the two surfaces of the crack moving apart. Mode 2 is
where the two surfaces slide from front to back. Mode 3 is where the to surfaces slide
sideways.
CRACK PROPAGATION (FRACTURE MECHANICS)
The process by which a crack can propagate through a structure. It is commonly assumed
that a crack initiates when a critical value of stress or strain is reached and it propagates if
it can release more than a critical amount of energy by the crack opening.
BUBBLE FUNCTIONS
Element shape functions that are zero along the edges of the element. They are non-zero
within the interior of the element.
BUCKLING (SNAP THROUGH)
The situation where the elastic stiffness of the structure is cancelled by the effects of
compressive stress within the structure. If the effect of this causes the structure to
suddenly displace a large amount in a direction normal to the load direction then it is
classical bifurcation buckling. If there is a sudden large movement in the direction of the
loading it is snap through buckling.
CENTRAL DIFFERENCE METHOD
A method for numerically integrating second order dynamic equations of motion. It is
widely used as a technique for solving non-linear dynamic problems.
FEA: The process of assembling the element matrices together to form the global matrix.
Typically element stiffness matrices are assembled to form the complete stiffness matrix of
the structure.
DEGREES OF FREEDOM
The number of equations of equilibrium for the system. In dynamics, the number of
displacement quantities which must be considered in order to represent the effects of all of
the significant inertia forces.
Degrees of freedom define the ability of a given node to move in any direction in space.
There are six types of DOF for any given node:
§ 3 possible translations (one each in the X,Y and Z directions) and
§ 3 possible rotations (one rotation about each of the X,Y, and X axes).
DOF are defined and restricted by the elements and constraints associated with each
node.
JACOBI METHOD
A method for finding eigenvalues and eigenvectors of a symmetric matrix.
JACOBIAN MATRIX
A square matrix relating derivatives of a variable in one coordinate system to the
derivatives of the same variable in a second coordinate system. It arises when the chain
rule for differentiation is written in matrix form.
DET(J) DET J
The Jacobian matrix is used to relate derivatives in the basis space to the real space. The
determinant of the Jacobian - det(j) - is a measure of the distortion of the element when
mapping from the basis to the real space
DEVIATORIC STRESS STRESS DEVIATORS
A measure of stress where the hydrostatic stress has been subtracted from the actual
stress. Material failures that are flow failures (plasticity and creep) fail independently of the
hydrostatic stress. The failure is a function of the deviatoric stress.
DISSIMILAR SHAPE FUNCTIONS INCOMPATIBLE SHAPE FUNCTIONS
If two connecting elements have different shape functions along the connection line they
are said to be incompatible. This should be avoided since convergence to the correct
solution cannot be guarantied.
DISTORTION ELEMENT DISTORTION
Elements are defined as simple shapes in the basis space, quadrilaterals are square,
triangles are isosoles triangles. If they are not this shape in the real space they are said to
be distorted. Too much distortion can lead to errors in the solution
DYNAMIC ANALYSIS
An analysis that includes the effect of the variables changing with time as well as space
STATIC ANALYSIS
Analysis of stresses and displacements in a structure when the applied loads do not vary
with time.
DYNAMIC MODELLING
A modeling process where consideration as to time effects in addition to spatial effects are
included. A dynamic model can be the same as a static model or it can differ significantly
depending upon the nature of the problem.
DYNAMIC RESPONSE
The time dependent response of a dynamic system in terms of its displacement, velocity
or acceleration at any given point of the system.
DYNAMIC STIFFNESS MATRIX
If the structure is vibrating steadily at a frequency w then the dynamic stiffness is (K+iwCw2M)
It is the inverse of the dynamic flexibility matrix.
DYNAMIC STRESSES
Stresses that vary with time and space.
ELEMENT
In the finite element method the geometry is divided up into elements, much like basic
building blocks. Each element has nodes associated with it. The behavior of the element is
defined in terms of the freedoms at the nodes.
ELEMENT ASSEMBLY
Individual element matrices have to be assembled into the complete stiffness matrix. This
is basically a process of summing the element matrices. This summation has to be of the
correct form. For the stiffness method the summation is based upon the fact that element
displacements at common nodes must be the same.
ELEMENT STRAINS ELEMENT STRESSES
Stresses and strains within elements are usually defined at the Gauss points (ideally at the
Barlow points) and the node points. The most accurate estimates are at the reduced
Gauss points (more specifically the Barlow points). Stresses and strains are usually
calculated here and extrapolated to the node points.
EXPLICIT METHODS IMPLICIT METHODS
These are methods for integrating equations of motion. Explicit methods can deal with
highly non-linear systems but need small steps. Implicit methods can deal with mildly nonlinear
problems but with large steps.
FOURIER TRANSFORM
A method for finding the frequency content of a time varying signal. If the signal is periodic
it gives the same result as the Fourier series.
FREE VIBRATION
The dynamic motion which results from specified initial conditions. The forcing function is
zero.
FREQUENCY DOMAIN
The structures forcing function and the consequent response is defined in terms of their
frequency content. The inverse Fourier transform of the frequency domain gives the
corresponding quantity in the time domain.
GLOBAL STIFFNESS MATRIX
The assembled stiffness matrix of the complete structure.
HARDENING STRUCTURE
A structure where the stiffness increases with load.
HARMONIC LOADING
A dynamic loading that is periodic and can be represented by a Fourier series.
HERMITIAN SHAPE FUNCTIONS
Shape functions that provide both variable and variable first derivative continuity
(displacement and slope continuity in structural terms) across element boundaries.
HOOKES LAW
The material property equations relating stress to strain for linear elasticity. They involve
the material properties of Young’s modulus and Poisson ratio.
HOURGLASS MODE
Zero energy modes of low order quadrilateral and brick elements that arise from using
reduced integration. These modes can propagate through the complete body.
H-REFINEMENT P-REFINEMENT
Making the mesh finer over parts or all of the body is termed h-refinement. Making the
element order higher is termed p-refinement.
ISOPARAMETRIC ELEMENT
Elements that use the same shape functions (interpolations) to define the geometry as
were used to define the displacements. If these elements satisfy the convergence
requirements of constant stress representation and strain free rigid body motions for one
geometry then it will sat isfy the conditions for any geometry
LAGRANGE INTERPOLATION LAGRANGE SHAPE FUNCTIONS
A method of interpolation over a volume by means of simple polynomials. This is the basis
of most of the shape function definitions for elements.
LAGRANGE MULTIPLIER TECHNIQUE
A method for introducing constraints into an analysis where the effects of the constraint
are represented in terms of the unknown Lagrange multiplying factors.
MATERIAL STIFFNESS MATRIX MATERIAL FLEXIBILITY MATRIX
The material stiffness matrix allows the stresses to be found from a given set of strains at
a point. The material flexibility is the inverse of this, allowing the strains to be found from a
given set of stresses. Both of these matrices must be symmetric and positive definite.
MATRIX DISPLACEMENT METHOD
A form (the standard form) of the finite element method where displacements are
assumed over the element. This gives a lower bound solution.
NATURAL FREQUENCY
The frequency at which a structure will vibrate in the absence of any external forcing. If a
model has n degrees of freedom then it has n natural frequencies. The eigenvalues of a
dynamic system are the squares of the natural frequencies.
NEWTON-RAPHSON NON-LINEAR SOLUTION
A general technique for solving non-linear equations. If the function and its derivative are
known at any point then the Newton-Raphson method is second order convergent.
NODAL VALUES
The value of variables at the node points. For a structure typical possible nodal values are
force, displacement, temperature, velocity, x, y, and z.
NODE NODES NODAL
The point at which one element connects to another or the point where an element meets
the model boundary. Nodes allow internal loads from one element to be transferred to
another element. Element behavior is defined by the response at the nodes of the
elements. Nodes are always at the corners of the element, higher order elements have
nodes at mid-edge or other edge positions and some elements have nodes on faces or
within the element volume. The behavior of the element is defined by the variables at the
node. For a stiffness matrix the variables are the structural displacement, For a heat
conduction analysis the nodal variable is the temperature. Other problems have other
nodal variables.
PATCH TEST
A test to prove that a mesh of distorted elements can represent constant stress situations
and strain free rigid body motions (i.e. the mesh convergence requirements) exactly.
PLANE STRAIN PLANE STRESS
A two dimensional analysis is plane stress if the stress in the third direction is assumed
zero. This is valid if the dimension of the body in this direction is very small, e.g. a thin
plate. A two dimensional analysis is plane strain if the strain in the third direction is
assumed zero. This is valid if the dimension of the body in this direction is very large, e.g.
a cross-sectional slice of a long body.
PLATE BENDING ELEMENTS
Two-dimensional shell elements where the in plane behavior of the element is ignored.
Only the out of plane bending is considered.
POISSONS RATIO
The material property in Hookes law relating strain in one direction arising from a stress in
a perpendicular direction to this.
POST-PROCESSING
The interrogation of the results after the analysis phase. This is usually done with a
combination of graphics and numerics.
POTENTIAL ENERGY
The energy associated with the static behavior of a system. For a structure this is the
strain energy
PRINCIPAL STRESSES
The maximum direct stress values at a point. They are the eigenvalues of the stress
tensor.
RANDOM VIBRATIONS
The applied loading is only known in terms of its statistical properties. The loading is nondeterministic
in that its value is not known exactly at any time but its mean, mean square,
variance and other statistical quantities are known.
STIFFNESS
A set of values which represent the rigidity or softness of a particular element. Stiffness is
determined by material type and geometry.
STIFFNESS MATRIX
The parameter(s) that relate the displacement(s) to the force(s). For a discrete parameter
multi degree of freedom model this is usually given as a stiffness matrix.
STRAIN
A dimensionless quantity calculated as the ratio of deformation to the original size of the
body.
STRAIN ENERGY
The energy stored in the system by the stiffness when it is displaced from its equilibrium
position.
STRESS
The intensity of internal forces in a body (force per unit area) acting on a plane within the
material of the body is called the stress on that plane.
STRESS ANALYSIS
The computation of stresses and displacements due to applied loads. The analysis may
be elastic, inelastic, time dependent or dynamic.
STRESS AVERAGING STRESS SMOOTHING
The process of filtering the raw finite element stress results to obtain the most realistic
estimates of the true state of stress.
STRESS CONCENTRATION
A local area of the structure where the stresses are significantly higher than the general
stress level. A fine mesh of elements is required in such regions if accurate estimates of
the stress concentration values are required.
THIN SHELL ELEMENT THICK SHELL ELEMENT
In a shell element the geometry is very much thinner in one direction than the other two. It
can then be assumed stresses can only vary linearly at most in the thickness direction. If
the through thickness shear strains can be taken as zero then a thin shell model is formed.
This uses the Kirchoff shell theory If the transverse shear strains are not ignored then a
thick shell model is formed. This uses the Mindlin shell theory. For the finite element
method the thick shell theory generates the most reliable form of shell elements. There are
two forms of such elements, the Mindlin shell and the Semi-Loof shell.
TIME DOMAIN
The structures forcing function and the consequent response is defined in terms of time
histories. The Fourier transform of the time domain gives the corresponding quantity in the
frequency domain.
TRACE OF THE MATRIX
The sum of the leading diagonal terms of the matrix
VON MISES STRESS
An "averaged" stress value calculated by adding the squares of the 3 component stresses
(X, Y and Z directions) and taking the square root of their sums. This value allows for a
quick method to locate probable problem areas with one plot.
VON MISES EQUIVALENT STRESS TRESCA EQUIVALENT STRESS
Equivalent stress measures to represent the maximum shear stress in a material. These
are used to characterize flow failures (e.g. plasticity and creep). From test results the Von-
Mises form seems more accurate but the Tresca form is easier to handle
YOUNGS MODULUS
The material property relating a uniaxial stress to the corresponding strain.
ZERO ENERGY MODES ZERO STIFFNESS MODES
Non-zero patterns of displacements that have no energy associated with them. No forces
are required to generate such modes, Rigid body motions are zero energy modes.
Buckling modes at their buckling loads are zero energy modes. If the elements are not
fully integrated they will have zero energy displacement modes. If a structure has one or
more zero energy modes then the matrix is singular.

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