Scaling law for gravitational root bending moments

Wind & Water Rotor Aerodynamics
1

Hello,

I am trying to understand the scaling laws for horizontal axis wind turbines. I am reading the Sandia report on their 100-m long all-glass baseline blade. The report is attached.

In the last paragraph of page 12, it is mentioned that “strain due to gravitational loads grow linearly with scale”. I have trouble understanding this. Strain, by definition, is the ratio of change in length to the original length. I believe both the numerator and the denominator scale linearly with the scale factor, and hence their ratio should be independent of scale. Where am I going wrong?

Scale factor is defined on page 11 of the report.

Thanks in advance,
Madhukara Putty
100m_blade.pdf (1.19 MB)

2

Dear Madhukara,

In a simplified way, the maximum normal stress, sigma, in a beam-like structure subject to a normal force, F, and bending moment, M, can be written as:

sigma = F/A + M/W

where A is the cross-sectional area and W is the beam’s resistance to bending.

The weight force scales with volume or cube of the beam length. The weight moment scales with the fourth power of the length. A and W scale with the square and cube of the length, respectively. Thus, the stress from weight will scale linearly with the length.

Stress is proportional to strain, so, strain will also scale linearly with the length.

I hope that helps.

Best regards,

3

Dear Jason,

Thanks for the reply. That makes sense. However, if we begin by the definition of strain, we get a a different result, because starin by definition, is a non-dimensional quantity.

Now, let me ask my second question. Scaling laws, as far as I know, are obtained by dimensional analysis. Does it make sense to find scaling law for a non-dimensional quantity like strain, using dimensional analysis? Is it for this reason, that we find the scaling law for stress (which is dimensional), and then use it to find the corresponding law for strain?

Regards,
Madhukara

4

Dear Madhukara,

Depending on the scaling law applied, nondimensional numbers do not necessarily hold across scales. For example, in Froude scaling often applied to offshore structures, the Reynolds number even though it is nondimensional is usually not maintained between model scale and prototype scale.

Best regards,