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 observations

1. 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:

\l \

1 \

1 \

1 \

I

1 \

1 \

Figure 13.6: Uniformly Loaded Simply Supported Beam Analyzed by the Rayleigh-Ritz Method

1. Contrarily to the previous example problem the geometric B.C. are immediately satisfied at both x = 0 and x = L.

2. We can keep v in terms of ai and take = 0 (If we had left v in terms of ai and a2 we should then take both -f^ = 0, and -f^ = 0 ).

3. Or we can solve for a1 in terms of vmax(@x = L) and take

Recalling that: jMr = dXV, the above simplifies to:

aiwL3

wL2 24EIZ

Having solved the displacement field in terms of a1, we now determine vmax at L:

24EIZ V L L2 a1

v wL4

96EIz

This is to be compared with the exact value of v^XaC* = 3bwrr = rwKEr which constitutes « 17% error.

Note: If two terms were retain would be equal to v^XaC*. (Why?)

Note: If two terms were retained, then we would have obtained: ai = pr & a2 = oiEEr and

24EIz "'2 24EIz nexact vmax

0 0

Responses

  • Isotta
    What is the requirement of displacement field to be satisfied in the use of Rayleigh –Ritz method?
    2 years ago

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