FEM Calculator

Compare a model and its reference

Choose one of the six setups, enter geometry, stiffness and loads, then Calculate. Example restores the selected setup’s defaults and calculates it. Reset clears the result while keeping inputs. Editing a model input clears the old result. The dashed analytical curve can be toggled without solving again; Reference and balance and Nodal values provide numerical details.

Models, units and signs

All elements have equal length and constant properties. Bars use axial rigidity EA, with E in Pa and A in m². The left end is fixed; positive force, uniform axial load and displacement point right. End-load displacement is PL/(EA), while uniform axial load q gives qL²/(2EA). Two-node linear elements recover the nodal values for these constant-coefficient cases, but the uniform-load profile between nodes is only piecewise linear.

Beams use Euler–Bernoulli bending with rigidity EI in N·m² and I in m⁴. Positive loads and deflections point down; θ = dw/dx and positive nodal moments act in the corresponding clockwise sense. The cantilever fixes w and θ at the left and allows a tip force P plus a uniform load w. Its tip deflection is PL³/(3EI) + wL⁴/(8EI). The supported beam fixes displacement at both ends with free rotations; uniform loading gives 5wL⁴/(384EI) at L/2. The finite-element curve uses cubic Hermite interpolation of displacement and rotation, including between-node extrema. Under uniform loading the exact profile is quartic, so even exact nodal values do not make a coarse profile exact.

The spring chain has n physical springs of stiffness k in N/m, giving end displacement nP/k. Changing n changes the physical chain, not just a mesh. Length places the spring nodes in the sketch; k is already a force per displacement and is not divided by segment length again.

Steady heat conduction has no internal source. The input k·A is conductivity times area in W·m/K; the end-to-end conductance is kA/L in W/K. Temperatures are in °C. With both ends fixed, T(x) is linear and rightward heat flow is kA(T₀−T₁)/L. Alternatively, a positive prescribed power Qᵣ enters from the right, making T(L) = T₀ + QᵣL/kA and rightward heat flow −Qᵣ. Zero prescribed power is an insulated right end. Supplied boundary powers are positive into the rod and balance to zero. Values below absolute zero are rejected.

Reactions, mesh and limits

Reaction forces, moments or boundary powers are recovered from the original system as Ku−F, including when all degrees of freedom are prescribed. Equal duplicate constraints are applied once; conflicting or invalid constraints and singular systems are rejected by the kernel. The present presets provide sufficient supports. Diagonal scaling makes the solver tolerance independent of a common stiffness-unit factor. The tool supports 1–80 elements or springs.

The plot has separate horizontal and vertical scales; the upper sketch is undeformed. The marked maximum belongs to the FEM field, not necessarily the exact continuum solution. With zero displacement every point is a maximum; the leftmost is reported. Reference differences below 5·10⁻⁹ of the result/reference magnitude and balance residuals below 10⁻⁹ of the load scale display as zero to suppress floating-point noise.

These are linear static teaching models with constant properties, small strains and slender-beam assumptions. They omit shear deformation, buckling, plasticity, variable sections and structural design checks. Large calculated deformation does not establish that linear theory is physically valid.

References: TU Delft: Euler–Bernoulli elements; MIT: beam deflection.