Linear static

Analysis methodQUALIFIED_BOUNDED · recorded combinations only

Small-strain equilibrium is assembled from element stiffnesses and consistent load vectors.

01 Model and scope

Element virtual-work contributions assemble into K and f. Essential constraints are represented by u = u₀ + Tq, producing a reduced system that satisfies the constraints by construction; reactions are recovered from the unreduced equilibrium residual.

Route assumptions and calculation checks

Free-DOF residual, constrained-DOF reactions, and global force/moment balance are separate observables.
Element quality, load resultant, sign convention, material, mesh, and solver route remain part of scope.
A small displacement is not proof of mesh convergence or physical validation.

Useful foundations and prerequisites

Units, signs and error measures Interpolation, mapping and assembly

02 Essential equations and symbols

K = \sum{}e Ae^T Ke Ae

Definitions and conventions

Aₑ: Element-to-global assembly operator.
$u_{0}$: Prescribed displacement contribution.
$T$: Basis for admissible free variations.
Kᵣ, fᵣ: Reduced stiffness and load.

01Element Kₑ and fₑ

02Assemble K, f

03Reduce supports

04Reactions + balance

Linear static · method mapElement virtual work produces the global equilibrium system; supports are reduced before solving and reactions are recovered afterward.

4 detailed steps

03 Step by step derivation

Follow the assumptions, equations and discrete operations in order. The full explanation stays visible.

1 · Derive the discrete equilibrium from virtual work

For linear elasticity, the strain is B u and stress is D B u. Equating internal virtual work to external virtual work gives each element stiffness and consistent load vector. The assembly operators scatter local contributions into global coordinates while preserving shared-node force balance.

\delta W_{int}=\sum_e\delta\mathbf u_e^T\left(\int_{\Omega_e}\mathbf B_e^T\mathbf D_e\mathbf B_e\,d\Omega\right)\mathbf u_e

2 · Impose prescribed displacements exactly

Write the full displacement as a particular field satisfying the supports plus an admissible free variation. Projecting equilibrium into the admissible space yields a reduced system. This avoids solving for prohibited motions and keeps the support values exact.

\mathbf u=\mathbf u_0+\mathbf T\mathbf q,\qquad \mathbf K_r=\mathbf T^T\mathbf K\mathbf T

3 · Recover reactions from the unreduced residual

After reconstructing u, evaluate f−Ku in the original system. Free-DOF entries should be close to zero; constrained entries are support reactions under the chosen sign convention. Summed reactions must balance applied loads and moments.

\mathbf r=\mathbf f-\mathbf K\mathbf u

4 · Separate algebraic success from model verification

A small linear residual says the assembled equations were solved; it does not establish that the mesh, loads, constitutive law, or physical boundary conditions are correct. Check load resultants, symmetry, strain energy, mesh refinement, and the exact registry row independently.