# How accurate should Rayleigh-Ritz approximation be?

For a simply supported beam with a UDL, with boundary conditions of X=0, V=0 & X=L, V=0. Using a cubic trial function.

$$V=a+bx+cx^2+dx^3$$

The following curve is obtained (Red Line). It is shown in comparison to the simple beam theory curve (Blue line).

Is this the level of accuracy to be expected from a Rayleigh-Ritz method for this scenario using these conditions? I don't have an intuitive sense of how accurate the approximation should be so am unsure if I have made a mistake in my calculations or if this is an expected result.

• – Solar Mike Apr 18 at 19:31
• We have no idea how you actually made the approximation, so the question isn't answerable. But IMO either you made a mistake, or you used an inappropriate method. – alephzero Apr 18 at 22:54
• You have four unknowns. You list two boundary conditions that yield one equation each, and virtual work can provide a third. What is the forth? Please show the system of four equations and explain why you chose the ones you did. Show the calculation of the coefficients. The system is overconstrained, so you are relaxing some physical constraints. – Phil Sweet Apr 19 at 3:28
• mathcs.emory.edu/~haber/math315/chap4.pdf for more insight into methods, selection of orthagonal bases, problem conditioning, and error prediction. – Phil Sweet Apr 19 at 3:46
• @phil sweet The third equation is total energy and one of the unknowns resulted in zero leaving only 3 unknowns – FEA42 Apr 19 at 7:25