## Rayleigh Ritz Method

55 Continuous systems have infinite number of degrees of freedom, those are the displacements at every point within the structure. Their behavior can be described by the Euler Equation, or the partial differential equation of equilibrium. However, only the simplest problems have an exact solution which (satisfies equilibrium, and the boundary conditions).

56 An approximate method of solution is the Rayleigh-Ritz method which is based on the principle of virtual displacements. In this method we approximate the displacement field by a function

where cj denote undetermined parameters, and $ are appropriate functions of positions.

57 $ should satisfy three conditions 1. Be continuous.

J2c3i$+ti

2. Must be admissible, i.e. satisfy the essential boundary conditions (the natural boundary conditions are included already in the variational statement. However, if $ also satisfy them, then better results are achieved).

3. Must be independent and complete (which means that the exact displacement and their derivatives that appear in n can be arbitrary matched if enough terms are used. Furthermore, lowest order terms must also be included).

In general $ is a polynomial or trigonometric function.

58 We determine the parameters cj by requiring that the principle of virtual work for arbitrary variations

Scj. or n i ^n ^n ^n \ sn(ui,u2,«s) = (^rSci + dC2Sc2 + -cScfJ = 0 (13.75)

for arbitrary and independent variations of Scj, Sc2, and Sc3, thus it follows that

dn | ||||||||

-r = 0 |
i = 1, 2, •• |
•,nj = 1 ^ 3 | ||||||

Thus we obtain a total of 3n linearly independent simultaneous equations. From these displacements, we can then determine strains and stresses (or internal forces). Hence we have replaced a problem with an infinite number of d.o.f by one with a finite number. ## 59 Some general observations1. cj can either be a set of coefficients with no physical meanings, or variables associated with nodal generalized displacements (such as deflection or displacement). 2. If the coordinate functions $ satisfy the above requirements, then the solution converges to the exact one if n increases. 3. For increasing values of n, the previously computed coefficients remain unchanged. 4. Since the strains are computed from the approximate displacements, strains and stresses are generally less accurate than the displacements. 5. The equilibrium equations of the problem are satisfied only in the energy sense Sn = 0 and not in the differential equation sense (i.e. in the weak form but not in the strong one). Therefore the displacements obtained from the approximation generally do not satisfy the equations of equilibrium. 6. Since the continuous system is approximated by a finite number of coordinates (or d.o.f.), then the approximate system is stiffer than the actual one, and the displacements obtained from the Ritz method converge to the exact ones from below. ■ Example 13-3: Uniformly Loaded Simply Supported Beam; Polynomial Approximation For the uniformly loaded beam shown in Fig. 13.6 let us assume a solution given by the following infinite series: for this particular solution, let us retain only the first term: We observe that:
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