Rayleigh-Ritz method and its steps

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.