A problem in Rayleigh Approximation(NBC)

[kyong]

I am hoping to hear any suggestions or comments from fellow engineers regarding following point.

NBC(Canadian Building Code) uses Rayleigh Approximation to calculate period of vibration.

T=2*pi*sqrt(SUM(Wi*di^2)/g/SUM(Wi*di))

where Wi = weight of ith section of the structure
di = deflection of the section
a section is regarded as having uniform diameter,
uniform stiffness, unformly distributed weight over
the length.
g = acceleration of gravity

I understand that deflection di in this formula is what is calculated on the basis that the uniform weight of the section acts as a uniform lateral load.
If I am right, I can raise a problem.

Let's say we are having two identical uniform towers whose lengths are 100 feet. Top deflection of tower A regarded as ONE section must be the same as sum of top deflections of arbitrarily divided multiple sections of tower B,
i.e., d of tower A = SUM(di of tower B).

Since SUM(Wi)*(SUM(di))^2/(SUM(Wi)*SUM(di)) is mathmatically smaller than SUM(Wi*di^2)/SUM(Wi*di), T of tower A will have bigger value than T of tower B. This result is obviously wrong because two towers are identical.

I actually compared and found difference was significant.

With 100 ft high, 4 ft diameter, 50000 lbs total weight, 0.75" thick circular steel tower,
One section calculation gives 1.09 seconds of T,
Two sections calculation gives 0.80 seconds of T,
Five sections calculation gives 0.52 seconds of T.
(I also compared these T's with T calculated by different formula, i.e., T=0.0000265*(H/D)^2*(w*D/t)^0.5. It gave me 0.86 seconds)

In further calculations of WIND load or SEISMIC load, T is important factor. For example, if T is less than 0.7, no top lateral force is regarded as existing. I wonder whether I can rely on Rayleigh Approximation when it gives floating values depend on how I decide to divide the tower into sections.

 

[austim]

Kyong,

Are you quite sure that you are interpreting di correctly?

While I am not familiar with your code, I would not be at all surprised if di is meant to be the deflection of the complete tower at the height where you have concentrated the mass Wi, and not just the deflection within section i.

If you try that, do you get more consistent results?

 

[kyong]

austim,

I read the code again. di is surely deflection of ith section. I still callculated T taking di as total deflection, though. Result was surprising. T remained same, 1.09 second, regardless of number of sections. How sharp you are! But, I was disappointed soon. If di were total deflection, the formula would reduce as follow.
T=2*pi*sqrt(SUM(Wi*di^2)/g/SUM(Wi*di))
T=2*pi*sqrt(di^2*SUM(Wi)/g/(di*SUM(Wi))
T=2*pi*sqrt(di^2/di/g)
T=2*pi*sqrt(di/g)
T became regardless of distribution of weight or deflection. If so, the code wouldn't have written the complex formula.

 

[austim]

kyong,

Glad to assist with your first problem. Now to your new one, which is much easier.

Basically, your algebra is incorrect. You can't take di out of the terms SUM(Wi*di) and SUM(Wi*di^2) unless di is constant for the full height of the tower. It is easier to see your error if you expand SUM(Wi*di) into all of its terms.

For example, consider just three locations. That will give SUM(Wi*di) = W1*d1+W2*d2+W3*d3.

You can't turn that into di*SUM(Wi) [which = di*(W1+W2+W3)] unless di=d1=d2=d3. This will only occur if your tower has zero deflection full height.