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+---
+type: concept
+title: "Displacement-Based Finite Element Formulation"
+complexity: advanced
+domain: computational-mechanics
+aliases:
+  - displacement formulation
+  - displacement method
+created: 2026-05-28
+updated: 2026-05-28
+address: c-000008
+tags:
+  - concept
+  - finite-element-method
+  - structural-mechanics
+status: current
+related:
+  - "[[Finite Element Method]]"
+  - "[[Isoparametric Finite Elements]]"
+  - "[[Mixed Finite Element Formulations]]"
+  - "[[Solid Element Notes]]"
+  - "[[Solid Element Strain-Displacement Matrix]]"
+  - "[[Solid Element Stiffness Integration]]"
+  - "[[Assumed Transverse Shear Strain Interpolation]]"
+sources:
+  - "[[Finite Element Procedures]]"
+  - "[[A Continuum Mechanics Based Four-Node Shell]]"
+  - "[[Solid Element Notes]]"
+---
+
+# Displacement-Based Finite Element Formulation
+
+## Definition
+
+The displacement-based finite element formulation uses nodal displacements as the primary unknowns and derives element and global equilibrium equations from a virtual work, energy, or variational statement.
+
+## How It Works
+
+Element displacement fields are interpolated from nodal degrees of freedom. Strains are computed from displacement gradients, stresses from constitutive laws, and element stiffness matrices from the strain-displacement and material matrices. Element contributions are assembled into the global equilibrium system, commonly represented in linear static form as `K u = R`.
+
+## Why It Matters
+
+This is the main formulation path for linear solid and structural mechanics in the source. It is direct and broadly useful, but it has limits for incompressibility, locking, and constraints, which motivates [[Mixed Finite Element Formulations]].
+
+The four-node shell paper gives a concrete locking example: direct displacement interpolation can impose nonphysical transverse shear strains in thin-shell bending, motivating [[Assumed Transverse Shear Strain Interpolation]].
+
+[[Solid Element Notes]] gives the corresponding 3D continuum path: interpolate nodal translations, compute small strains with the solid `B` matrix, apply the Hooke-law `D` matrix, and integrate `B^T D B` over the element volume.
+
+## Practical Checks
+
+- Does the interpolation reproduce rigid-body motion and constant strain states where required?
+- Are displacement boundary conditions imposed consistently?
+- Are stresses recovered in a way that reflects the approximation quality?
+- Does mesh refinement improve the relevant response quantities?
+
+## Sources
+
+- [[Finite Element Procedures]]
+- [[A Continuum Mechanics Based Four-Node Shell]]
+- [[Solid Element Notes]]