---
title: "Essential and natural boundary conditions"
subject: "Finite Element Method"
chapter: 2
key: "semester_8/finite_element_method:essential-natural-boundary-conditions"
source: https://www.bctengineer.com/q/semester_8/finite_element_method/essential-natural-boundary-conditions
site: BCT Engineering
---

# Essential and natural boundary conditions

Finite Element Method · Chapter 2: Mathematical Background

- Asked in: year not given, 4 marks
- Video: [Boundary conditions in Finite Element Methods | Boundary conditions in Fem | Part-03](https://www.youtube.com/watch?v=0EQkiE2BR-E) — Mahesh Gadwantikar
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/finite_element_method/essential-natural-boundary-conditions
- Whole chapter: https://www.bctengineer.com/md/semester_8/finite_element_method/chapter-2

## Question

Differentiate between essential (geometric) and natural (force) boundary conditions with examples. Classify the differential equation $EI\,\dfrac{d^4 w}{dx^4} = q$ and state the boundary conditions of a cantilever beam of length $L$ in terms of $w$.

## Short answer

Essential (geometric) boundary conditions specify primary variables like displacement (e.g., w(0)=0) and involve lower-order derivatives. Natural (force) boundary conditions specify secondary variables such as force or moment (e.g., EI w'''(L)=0) and appear in the weak form. The equation EI d4w/dx4 = q is a linear fourth-order ODE. For a cantilever: w(0)=0, w'(0)=0 (essential); EI w''(L)=0, EI w'''(L)=0 (natural).

## Answer

#### Differences

| Point | Essential (geometric) BC | Natural (force) BC |
|---|---|---|
| Specifies | Value of the primary variable ($u$, $w$, $\theta$, $T$) | Value of the secondary variable (force, moment, flux) |
| Order of derivative | Lower, up to $(m-1)$ for order $2m$ | Higher, $m$ and above |
| Example, bar | $u(0) = 0$ at fixed end | $AE\,u'(L) = P$ at loaded end |
| In weak form | Must be satisfied by the trial function | Appears in the weak form and is satisfied automatically |
| In FEM | Applied by deleting or modifying rows | Added to the load vector |

#### Classification

$EI\,w'''' = q$ is a linear, fourth-order, ordinary differential equation (one independent variable $x$). For a fourth-order equation $m = 2$, so $w$ and $w'$ are primary variables (essential), while $EIw''$ (moment) and $EIw'''$ (shear) are secondary variables (natural).

#### Cantilever beam (fixed at $x=0$, free at $x=L$)

- Essential: $w(0) = 0$, $\dfrac{dw}{dx}(0) = 0$
- Natural: $EI\dfrac{d^2w}{dx^2}(L) = 0$ (zero moment), $EI\dfrac{d^3w}{dx^3}(L) = 0$ (zero shear)

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does an essential (geometric) boundary condition specify?
   A. The value of the primary variable such as u, w, theta or T
   B. The value of a force, moment or flux
   C. The load on the whole body
   D. The derivative of the highest order
   Answer: A. Essential conditions fix primary variables, for example u(0) = 0 at a fixed end. (Key point: Essential BC)

2. How is a natural boundary condition handled in FEM?
   A. By deleting rows and columns
   B. It appears in the weak form and is added to the load vector
   C. By making the trial function satisfy it
   D. It is ignored
   Answer: B. Natural conditions enter the weak form automatically and go into the load vector; essential ones are applied by deleting or modifying rows. (Key point: Natural BC treatment)

3. How is EI w'''' = q classified?
   A. Linear, second-order, partial
   B. Nonlinear, fourth-order, ordinary
   C. Linear, fourth-order, ordinary
   D. Linear, fourth-order, partial
   Answer: C. It has one independent variable x, is linear in w, and has a fourth derivative. (Key point: Classification)

4. Which are the natural boundary conditions of a cantilever fixed at x = 0 and free at x = L?
   A. w(0) = 0 and w'(0) = 0
   B. w(L) = 0 and w'(L) = 0
   C. EI w''(0) = 0 and EI w'''(0) = 0
   D. EI w''(L) = 0 and EI w'''(L) = 0
   Answer: D. At the free end the moment EIw'' and shear EIw''' are zero; these are secondary variables, hence natural. (Key point: Cantilever BCs)

## Related in Finite Element Method

- [Truss, beam and frame element differences](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Atruss-beam-frame-element-differences) — Finite Element Method, chapter 3
- [Derive beam element stiffness matrix](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Abeam-element-stiffness-derivation) — Finite Element Method, chapter 3
- [Positive definite stiffness matrix and Gauss elimination](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Amatrix-gauss-elimination-stiffness) — Finite Element Method, chapter 2

## Related in other subjects

- [Propped cantilever boundary conditions](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Apropped-cantilever-boundary-conditions) — Theory of Structures II, chapter 3
- [Boundary conditions of fixed-fixed beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Afixed-beam-boundary-conditions) — Theory of Structures II, chapter 3
- [Boundary conditions real to conjugate beam](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aconjugate-boundary-conditions) — Theory of Structures I, chapter 4
- [Differential equation of the elastic curve](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Abeam-deflection-differential-equation) — Strength of Materials, chapter 7