# The NAFEMS Glossary

# Terms N-R

Natural FrequencyThe frequency at which resonance occurs, that is when the stiffness and the inertia forces cancel. See also Resonant Frequency.

Natural ModeSee Eigenvectors.

Natural StrainSee True Strain.

Navier-Stokes EquationsPartial differential equations defining the unsteady viscous flow of fluids.

Newmark Beta MethodAn implicit solution method for integrating second order equations of motion. It can be made unconditionally stable. Also referred to as the Newmark Method.

Newmark MethodAn implicit solution method for integrating second order equations of motion. It can be made unconditionally stable. Also referred to as the Newmark Beta Method

Newmark’s Time Stepping SchemesA family of time integration methods for the solution of transient dynamic problems.

Newton Cotes FormulaeA family of methods for numerically integrating a function.

Newton-Raphson MethodAn incremental-iterative non-linear procedure to solve the equilibrium equations: the tangential stiffness matrix is updated during every iteration of every increment. See also Newton-Raphson Non-Linear Solution

Newton-Raphson Non-Linear SolutionA 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. See also Tangent Stiffness Matrix.

Nodal ValuesThe value of variables at the node points. For a structure typical possible nodal values are force, displacement, temperature, velocity, x, y, and z.

NodesThe element behaviour 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 behaviour 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.

Nominal StrainSee Engineering Strain

Non-Associative PlasticityA form of plasticity in which the yield function is not identical to the plastic potential.

Non-Conforming ElementsElements that do not satisfy compatibility either within the element or across element boundaries or both. Such elements are not generally reliable although they might give very good solutions in some circumstances.

Non-Holonomic ConstraintsConstraints that can only be defined at the level of infinitesimal displacements. They cannot be integrated to give global constraints.

Non-Linear AnalysisWhen at least one of the coefficients of stiffness, mass or damping vary with displacement or time then the system is non-linear.Superposition cannot be used to solve the problem.

Non-Stationary RandomA force or response that is random and its statistical properties vary with time.

Non-Structural MassMass that is present in the system and will affect the dynamic response but it is not a part of the structural mass (e.g. the payload).

NormA scalar measure of the magnitude of a vector or a matrix.

Normality RuleA particular plastic flow rule to ensure that the plastic strain components are in a ratio such that their resultant is in a direction normal to the yield surface. See also Flow Rule.

Norton-Bailey LawA law for steady state creep with strain rate proportional to a power of stress. Also referred to as the Norton Law.

Norton LawA law for steady state creep with strain rate proportional to a power of stress. Also referred to as the Norton-Bailey Law.

Numerical IntegrationThe process of integrating the element stiffness matrix based on numerical algorithms such as Gaussian quadrature. Evaluations are made at strategic points within each element, known as Gauss Points.

Optimal PointsStrategic locations within elements where stress evaluations are especially accurate, often at the location of the Gauss points.

Optimal Sampling PointsThe minimum number of Gauss points required to integrate an element matrix. Also the Gauss points at which the stresses are most accurate, see also Reduced Integration.

OrthotropyA material where the response to load depends on the direction within the material. It is less general than anisotropy, and up to 12 independent constants are required to relate stress and strain.

Over Damped SystemA system that has an equation of motion where the damping is greater than critical. It has an exponentially decaying, non-oscillatory impulse response. See also Critical Damping.

Overstiff SolutionsLower bound solutions. These are associated with the Assumed Displacement Method.

Parametric StudiesInitial studies conducted on small simplified models to determine the important parameters in the solution of a problem. These are often used to determine the basic mesh density required. Also referred to as Pilot Studies.

Participation FactorThe fraction of the mass that is active for a given mode with a given distribution of dynamic loads. Often this is only defined for a specific load case of inertia (seismic) loads.

Patch TestA simple element type test using a patch of several elements, one of which is arbitrarily orientated with respect to the global co-ordinates. If the patch is loaded by displacements consistent with a state of constant strain and the strain inside the selected element is constant, the test is passed. See also Continuum Region Element (CRE) Method.

Penalty FunctionIn the context of contact algorithms, a constraint on stiffness behaviour usually applied via large numbers in the equations, e.g. by introducing stiff springs. Also referred to as Penalty Stiffness.

Penalty StiffnessIn the context of contact algorithms, a constraint on stiffness behaviour usually applied via large numbers in the equations, e.g. by introducing stiff springs. Also referred to as Penalty Function.

Perfect PlasticityPlastic behaviour where the yield stress remains constant for all values of plastic strain.

Periodic Response (Force)A response (force) that regularly repeats itself exactly.

Phase AngleThe ratio of the in-phase component of a signal to its out-of-phase component gives the tangent of the phase angle of the signal relative to some reference.

Phase ChangeWhen a substance subjected to temperature changes transforms from solid to liquid or gaseous state. During this phase change, latent heat is either released or absorbed.

Pilot StudiesInitial studies conducted on small simplified models to determine the important parameters in the solution of a problem. These are often used to determine the basic mesh density required. Also referred to as Parametric Studies.

Plane StrainA 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. See also Plane Stress.

Plane StressA 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. See also Plane Strain.

Plastic StrainIrrecoverable permanent strain due to time independent plasticity.

Plastic ZonesRegions in a body where a stress measure (usually the equivalent stress) lies on the yield surface and plastic strains are accruing.

Plate Bending ElementsTwo dimensional shell elements where the in plane behaviour of the element is ignored. Only the out of plane bending is considered.

Ply Lay-UpSee Lay-up.

Poissons RatioThe material property in Hookes law relating strain in one direction arising from a stress in a perpendicular direction to this.

Post Analysis ChecksChecks that can be made on the results after the analysis. For a stress analysis these could include how well stress free boundary conditions have been satisfied or how continuous stresses are across elements.

Post Yield Fracture Mechanics (PYFM)A given crack inside a loaded structure behaves in conditions of PYFM when the crack fields local to the crack tip exhibit considerable plastic behaviour. See also Elastic Plastic Fracture Mechanics (EPFM).

Post-ProcessingThe interrogation of the results after the analysis phase. This is usually done graphically.

Potential EnergyThe energy associated with the static behaviour of a system. For a structure this is the strain energy.

Potential Energy Release RateFor a hypothetically small increase in crack length or area, this is the amount of potential energy released divided by that length or area. It equals the negative of the Strain Energy Release Rate when elastic conditions predominate. It provides the basis for fracture parameters in post yield fracture mechanics and other non-linear conditions.

Potential FlowFluid flow problems where the flow can be represented by a scalar potential function.

Power MethodA method for finding the lowest or the highest eigenvalue of a system. See also Eigenvalues.

Prandtl NumberA fluid flow measure of the ratio of momentum diffusivity to thermal diffusivity.

Prandtl-Reuss EquationsThe equations relating an increment of stress to an increment of plastic strain for a metal undergoing plastic flow. See also Von Mises Yield Criterion.

Prandtl-Reuss Flow RuleIn plasticity theory, the special form of the normality rule corresponding to von Mises yield criterion (q.v.).

Predictor-Corrector SchemesThe two-phase format of a time or load stepping scheme where the predicted solution is corrected prior to advancing to the next step.

P-RefinementMaking the element order higher is termed p-refinement. See also Mesh Refinement

Primary ComponentThose parts of the structure that are of direct interest for the analysis. Other parts are Secondary Components.

Primary CreepThe initial part of a creep test where the strain rate is decreasing.

Principal CurvatureThe maximum and minimum radii of curvature at a point.

Principal PlanesThe planes on which the shear stresses are zero. Three such planes exist at every point in a stressed body.

Principal StressesStresses normal to the principal planes.

ProfileThe profile of a symmetric matrix is the sum of the number of terms in the lower (or upper) triangle of the matrix ignoring the leading zeros in each row. Embedded zeros are included in the count. It gives a measure of the work required to factorise the matrix when using the Cholesky solution. It is minimised by node renumbering. See also Cholesky Factorisation (Skyline).

Proportional DampingA damping matrix that is a linear combination of the mass and stiffness matrices. The eigenvectors of a proportionally damped system are identical to those of the undamped system.

Proportional LoadingOccurs when all the external loads are applied simultaneously, and increase in proportion to one another throughout the loading history. This clearly does not occur when one component of load is applied and then another.

Qr MethodA technique for finding Eigenvalues This is currently the most stable method for finding eigenvalues but it is restricted in the size of problem that it can solve.

RadiationA mode of heat transfer due to electromagnetic waves. Thus, the heat energy can be transferred in a vacuum. It is characterised by the Stefan-Boltzmann law.

Radiation DampingDamping that arises from energy being carried away from a vibrating body by expanding pressure waves. Sound radiation is an example of this. Such radiating energy, both to the surrounding fluid and through the supports, often forms the main damping in a vibrating structure.

Ramberg-Osgood Power LawA stress-strain relationship where the strain is proportional to a power of stress.

Random Vibration AnalysisAn analysis undertaken when the applied loading is only known in terms of its statistical properties. The loading is non-deterministic in that its value is not known exactly at any time but its mean, mean square, variance and other statistical quantities are known.

Rank DeficiencyA measure of how singular a matrix is. See Singular Matrix

RatchettingOccurs in Cyclic Loading when plastic strains keep on accumulating incrementally with each cycle, leading to eventual failure via incremental collapse.

Rayleigh DampingA model for representing the variation of damping with frequency.

Raylelgh QuotientThe ratio of stiffness times displacement squared (2 x strain energy) to mass times displacement squared. The minimum values of the Rayleigh quotient are the eigenvalues.

Reaction ForcesThe forces generated at support points when a structure is loaded.

ReceptanceThe ratio of the steady state displacement response to the value of the forcing function for a sinusoidal excitation. It is the same as the Dynamic Flexibility.

Reduced IntegrationThe process of intentionally under-integrating the element stiffness matrix to prevent problems such as Shear Locking or to improve the element’s performance.

Redundant StructureA structure where all of the unknowns can not be found from equilibrium considerations alone. The compatibility equations must also be used. In this case the structure is said to be redundant.

Residual ForcesThe forces which are equal to the applied load minus the internal resisting forces which occur during non-linear solutions: used to measure the state of equilibrium by comparison to the convergence tolerance.

Resonant FrequencyThe frequency at which resonance occurs, that is when the stiffness and the inertia forces cancel. See also Natural Frequency

Resonant ResponseThe response of a system to vibratory forces.

Response Spectrum MethodA method for characterising a dynamic transient forcing function and the associated solution technique. It is used for seismic and shock type loads.

Restarts CheckpointsThe process whereby an analysis can be stopped part way through and the analysis restarted at a later time.

Reynolds NumberA fluid flow measure of the ratio of momentum forces to viscous forces.

Rigid Body DisplacementA non-zero displacement pattern that has zero strain energy associate with it.

Rigid Body ModesIf a displaced shape does not give rise to any strain energy in the structure then this a rigid body mode. A general three dimensional unsupported structure has 6 rigid body modes, 3 translation and 3 rotation.

Rigid LinksThis is a connection between two non-coincident nodes assuming that the connection is infinitely stiff. This allows the degrees of freedom at one of the nodes (the slave node) to be deleted from the system. It is a form of Multi-Point Constraints.

Rigid OffsetsThis is a connection between two non-coincident nodes assuming that the connection is infinitely stiff. This allows the degrees of freedom at one of the nodes (the slave node) to be deleted from the system. It is a form of Multi-Point Constraints.

Roundoff ErrorComputers have a fixed wordlength and hence only hold numbers to a certain number of significant figures. If two close numbers are subtracted one from another then the result loses the first set of significant figures and hence loses accuracy. This is round off error.

Row VectorA 1xn matrix written as a horizontal string of numbers. It is the transpose of a Column Vector (Column Matrix)Column Vector (Column Matrix)

Rupture TimeThe time required for a structure to fail due to continuous creep deformation.