Finite Element Method Explained: From Mesh to Bridge Forces

From governing equations to an algebraic system

A loaded bridge is governed by the equations of linear elasticity: equilibrium (∇·σ = 0), the constitutive law relating stress to strain (σ = C:ε), and the strain-displacement relation (ε = ½(∇u + (∇u)ᵀ)). Analytical solutions exist only for the simplest geometries, so FEM instead solves the equivalent weak (variational) form — integrating the equilibrium condition against a virtual displacement rather than solving the differential equation pointwise. This reformulation is what allows a computer to handle arbitrarily complex structures, from a single truss bar to a full bridge deck.

Shape functions: approximating the unknown field

Within each element, the unknown displacement field is approximated as a weighted sum of shape functions N_i(x), where the weights are the nodal values:

u(x) ≈ Σᵢ Nᵢ(x) · uᵢ   (within one element)

Key properties of any valid shape function set:
  Nᵢ(xⱼ) = δᵢⱼ   (1 at its own node, 0 at every other node)
  Σᵢ Nᵢ = 1      (partition of unity — rigid-body motion is exact)

For a truss bar the shape functions are simply linear along the member's length; general 2D and 3D continuum elements use triangular, quadrilateral, tetrahedral or brick shape functions of increasing polynomial order. Higher-order (p-refinement) shape functions improve accuracy without shrinking the mesh, and can achieve exponential convergence for smooth problems.

Element stiffness and assembly

Substituting shape functions into the weak form gives each element's stiffness matrix K_e = ∫ Bᵀ C B dV, where B contains derivatives of the shape functions and C is the elasticity matrix built from Young's modulus E and Poisson's ratio ν. For simple bar and beam elements, this integral has an exact closed form; for general continuum elements it is evaluated numerically via Gauss quadrature. Every element's K_e is then assembled into the global system K·u = F — for a truss with N nodes and 2 DOF per node, K has size 2N×2N and is sparse, since a node only connects to its neighbouring members.

Boundary conditions and solving

A bridge model needs two kinds of boundary conditions: Dirichlet (essential) conditions prescribe known displacements, such as a pinned or roller support with u = 0, applied by modifying rows and columns of the stiffness matrix; Neumann (natural) conditions prescribe applied forces, which appear directly in the load vector F. For a linear elastic structure, K·u = F is solved once using a direct method (LU or Cholesky decomposition) or an iterative method (conjugate gradient) — no Newton-Raphson iteration is needed unless the deck undergoes large deformation or the members can yield plastically.

Trusting the result: convergence and error

FEM produces an approximate solution, so accuracy is checked by refining the model: h-refinement shrinks element size (displacement error scales roughly with h², stress error with h for linear elements); p-refinement raises the polynomial order instead. Error estimators such as the Zienkiewicz-Zhu method compare a smoothed, recovered stress field against the raw, discontinuous FEM stress across element boundaries, flagging regions — like a truss joint carrying a concentrated load — where the mesh should be refined. Engineering practice typically targets an estimated error under 5-10% before trusting a computed force or displacement for design.

Frequently asked questions

Try it live

Everything above runs in your browser — open Bridge Truss Analyser and load a Pratt truss to watch a stiffness-method solver colour each member by tension or compression while the deck deflects under the applied force. Nothing is installed, nothing is uploaded.