---
type: concept
title: "Beam and Frame Finite Elements"
complexity: intermediate
domain: computational-mechanics
created: 2026-05-29
updated: 2026-05-29
address: c-000065
aliases:
  - beam finite element
  - frame finite element
  - plane frame element
  - grid element
tags:
  - concept
  - finite-element-method
  - structural-mechanics
  - beams
status: current
related:
  - "[[Direct Stiffness Method]]"
  - "[[Bar and Truss Finite Elements]]"
  - "[[Finite Element Load Vector Assembly]]"
  - "[[Direct Time Integration Methods]]"
  - "[[Shell Locking Phenomenon]]"
sources:
  - "[[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]"
---

# Beam and Frame Finite Elements

## Definition

Beam and frame finite elements model slender structural members whose response includes bending, shear, axial deformation, moments, and rotations.

## How They Work

The Euler-Bernoulli beam element uses transverse displacement and rotation degrees of freedom at each node. Its displacement field is cubic so that both displacement and slope can be matched at nodes. The resulting stiffness relates nodal transverse forces and bending moments to nodal deflections and rotations.

For short or deep beams, transverse shear deformation can become significant, motivating Timoshenko beam theory. Frame elements then combine axial bar behavior with beam bending behavior and use coordinate transformation matrices so arbitrarily oriented members can be assembled into plane frames, grids, and spatial frames.

## Why It Matters

Beam and frame elements sit between simple axial trusses and full continuum or shell models. They are efficient for bridges, buildings, machine frames, and grid structures when member-level idealization is appropriate.

## Connections

- [[Bar and Truss Finite Elements]] provide the axial part of a frame element.
- [[Finite Element Load Vector Assembly]] handles distributed loads and equivalent nodal forces on beams.
- [[Direct Time Integration Methods]] uses beam mass matrices for vibration and transient structural analysis.
- [[Shell Locking Phenomenon]] is conceptually related through transverse shear treatment, though shell locking is a different element pathology.

## Sources

- [[A-First-Course-in-the-Finite-Element-Method|A First Course in the Finite Element Method]]