---
title: "Rayleigh-Ritz method and its steps"
subject: "Finite Element Method"
chapter: 4
key: "semester_8/finite_element_method:ritz-method-steps"
source: https://www.bctengineer.com/q/semester_8/finite_element_method/ritz-method-steps
site: BCT Engineering
---

# Rayleigh-Ritz method and its steps

Finite Element Method · Chapter 4: Continuum Problems

- Asked in: year not given, 5 marks
- Video: [Rayleigh-Ritz Method](https://www.youtube.com/watch?v=JWh4Ufj5Gbg) — Josh Pilipovsky
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/finite_element_method/ritz-method-steps
- Whole chapter: https://www.bctengineer.com/md/semester_8/finite_element_method/chapter-4

## Question

Write short notes on the Rayleigh-Ritz method. Explain its steps and state its limitations.

## Short answer

The Rayleigh–Ritz method makes the total potential energy Π = U − W stationary. Steps: (1) write Π; (2) choose trial function ũ = Σ aᵢφᵢ satisfying essential BCs; (3) substitute to get Π(aᵢ); (4) set ∂Π/∂aᵢ=0; (5) solve for aᵢ. Limitations: needs a functional, difficult for complex domains; FEM applies the Ritz concept element by element.

## Answer

The **Rayleigh-Ritz method** is a variational method. Instead of solving the differential equation, it finds an approximate solution that makes a functional (the total potential energy $\Pi$) stationary. For a stable elastic system, the stationary point is a minimum.

#### Steps

1. Write the functional. For elastic bodies, $\Pi = U - W$ (strain energy minus work of external loads). For a bar, $\Pi = \dfrac{1}{2}\displaystyle\int_0^L AE\left(\dfrac{du}{dx}\right)^2dx - \int_0^L f u\,dx - Pu(L)$.
2. Choose a trial function with unknown coefficients:
$$
u(x) \approx \sum_{i=1}^{n} a_i\,\phi_i(x)
$$
The functions $\phi_i$ must be continuous, independent and must satisfy the essential (geometric) boundary conditions.
3. Substitute into $\Pi$ so that it becomes a function of $a_1, a_2, \dots, a_n$.
4. Make $\Pi$ stationary: $\dfrac{\partial \Pi}{\partial a_i} = 0$ for $i = 1, \dots, n$. This gives $n$ algebraic equations $[K]\{a\} = \{F\}$.
5. Solve for $a_i$ and compute displacement, strain and stress.

#### Remarks

- Natural boundary conditions need not be satisfied by the trial function; they come out of the minimisation.
- More terms give a better answer. The computed stiffness is never less than the true stiffness (the displacement is underestimated), so the method gives a bound.
- If the trial function includes the exact solution, the result is exact.

#### Limitations

- Choosing one trial function for the whole domain is hard for complex shapes.
- It needs a functional, which exists only for self-adjoint problems.
- FEM removes this difficulty by applying the Ritz idea element by element with simple local trial functions.

## Check yourself

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

1. What does the Rayleigh-Ritz method make stationary?
   A. The total potential energy Pi = U - W
   B. The residual of the differential equation
   C. The stiffness
   D. The stress
   Answer: A. It is a variational method that makes the total potential energy stationary, a minimum for a stable system. (Key point: Functional)

2. Which boundary conditions must the trial function satisfy?
   A. Only natural conditions
   B. Essential (geometric) conditions
   C. None
   D. Both natural and force conditions
   Answer: B. The trial functions must be continuous, independent and satisfy the essential BCs; natural BCs come out of the minimisation. (Key point: Trial function)

3. How are the unknown coefficients obtained?
   A. By setting a_i = 1
   B. By making the trial function zero
   C. By setting dPi/da_i = 0 for each i
   D. By integrating once
   Answer: C. Stationarity gives n algebraic equations [K]{a} = {F}. (Key point: Equations)

4. When is the Ritz result exact?
   A. Always
   B. Never
   C. When only one term is used
   D. When the trial function includes the exact solution
   Answer: D. If the exact solution lies within the trial family, the minimisation finds it. (Key point: Accuracy)

## Related in Finite Element Method

- [Ritz solution of beam under uniform load](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Aritz-simply-supported-beam) — Finite Element Method, chapter 4
- [Ritz method for bar with distributed load](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Aritz-bar-quadratic-trial) — Finite Element Method, chapter 4

## Related in other subjects

- [Rayleigh-Ritz and finite difference methods](https://www.bctengineer.com/question.md?key=semester_7%2Ftheory_of_mechanism_and_machine_ii%3Arayleigh-ritz-finite-difference-notes) — Theory of Mechanism and Machine II, chapter 9