Finite-Element Analysis

From continuum to algebraic model

The finite-element method replaces a continuous field by interpolation over finite elements joined at nodes. Nodal degrees of freedom are the primary unknowns. Strain follows from displacement gradients, and constitutive data convert strain to stress. Thus a colored stress plot is a derived quantity, not direct evidence of strength or safety.

For small-displacement, linear-elastic statics the assembled model is \([K]\{u\}=\{F\}\). Geometry and modulus determine stiffness \([K]\); restraints prescribe selected components of \(\{u\}\); external loads populate \(\{F\}\). Material strength does not enter this equilibrium solution. It is compared later with an appropriate stress measure and failure criterion.

Approximation, error, and the quantity of interest

Discretization error arises because finite interpolation cannot exactly reproduce every curved boundary or field gradient. The \(h\)-method reduces element size; the \(p\)-method raises interpolation order. Convergence must be demonstrated for a named quantity—such as tip displacement, strain energy, or stress averaged over a specified path—not merely by obtaining a successful solve.

Point stress at a sharp re-entrant corner, point load, or ideal fixed edge may be singular and can increase without bound under refinement. That is nonconvergence of the reported quantity, not proof that the whole model is invalid. Replace the idealization with physical load spread or radius when known, or report a structural stress away from the singularity with a documented extraction rule.

Choose dimensionality from physics

Bars carry axial force; beams carry axial force, shear, bending, and often torsion; shells represent thin midsurfaces; plane-stress elements suit thin in-plane parts; plane-strain elements suit long, restrained sections; axisymmetric elements require geometry and loading symmetric about one axis. Three-dimensional solids are appropriate when through-thickness stress, complex contact, or genuinely three-dimensional geometry matters.

The simplest valid family usually gives the clearest audit trail. Slenderness, thickness-to-curvature ratio, expected stress state, connection eccentricity, and desired output govern selection. Mixing elements requires compatible degrees of freedom or a justified coupling; an indiscriminate rigid link can create artificial stiffness and local stress.

Element stiffness, assembly, and recovery

A two-node axial element has \(k=AE/L\) and local relation \(\{f_e\}=(AE/L)\begin{bmatrix}1&-1\\-1&1\end{bmatrix}\{u_e\}\). Assembly adds overlapping nodal contributions. Essential boundary conditions remove rigid-body modes, the reduced equations yield unknown displacements, and back-substitution recovers reactions. Element strain is \(\epsilon=(u_j-u_i)/L\), followed by \(\sigma=E\epsilon\) for linear elasticity.

Refine with a declared acceptance rule

Mesh density belongs where geometry, contact, loads, or response gradients demand it. Curved boundaries need enough elements to represent curvature; bending needs adequate through-thickness interpolation; contact needs compatible surface resolution. A practical relative-change measure is \(e_i=|q_i-q_{i-1}|/|q_i|\), where \(q\) is the predeclared response quantity.

Element Quality

Excessive skew, warpage, or aspect ratio can corrupt interpolation and integration. Quality limits are formulation- and solver-dependent, so use the solver manual rather than a universal invented cutoff. Review warnings, energy behavior, and contour smoothness, but never substitute a quality color map for convergence of the engineering response.

Nodal and Element Loads

Begin with a free-body diagram. Pressures, tractions, gravity, temperatures, bearing loads, and connector forces must reproduce the physical resultant and moment without creating unintended load paths. A concentrated nodal force may be adequate for remote displacement but creates unreliable local stress. Saint-Venant reasoning applies only sufficiently far from that idealized application.

Restraining the Model

Essential conditions prescribe motion; natural conditions prescribe traction. The model must suppress rigid-body motion without suppressing real compliance. Compare reactions, contact status, and deformed shape with the physical mounting. Remote constraints and rigid couplings need particular care because they may distribute load or add rotational stiffness not present in hardware.

Symmetry and Simplification

Symmetry is valid only when geometry, material, loading, restraints, and the response mode share that symmetry. On a symmetry cut, normal displacement and appropriate rotations are constrained; an antisymmetric load requires different conditions. Submodeling can resolve a local feature after the global model supplies boundary motion, but interpolation at the cut must lie outside the disturbed local region.

Coupling Heat to Stress

A sequential analysis first solves temperature, then maps that field to the structural mesh. Free thermal strain is \(\epsilon_{th}=\alpha\Delta T\); stress arises only where expansion is constrained or gradients cause incompatible strain. Properties may depend on temperature, and mapped temperatures, reference temperature, contact conductance, and restraint stiffness require verification.

Buckling as an Eigenvalue Problem

Linear eigenvalue buckling supplies ideal load multipliers and mode shapes about a preload state. It commonly overpredicts real capacity because imperfections, residual stress, plasticity, contact, and load eccentricity are omitted. Use a nonlinear imperfection-sensitive analysis and applicable code or test evidence when buckling governs a released design.

Natural Frequencies and Mode Shapes

Modal analysis solves \(([K]-\omega^2[M])\{\phi\}=0\). An eigenvector gives relative shape, not operating amplitude. Forced response additionally needs excitation, damping, and enough modes for mass participation and response bandwidth. Density and attached mass are as important as stiffness; supports must represent operating conditions rather than a convenient laboratory idealization.