Simplified Tapered Cantilever Beam Generalizations

[fpjeepy]

I looked in Roark's Formulas and didn't find anything. Basically, I design parts that need to bend. Mostly plywood and plastic. Mostly cantilever beams. I like to taper the beams so that the stress along the beam is more uniform. The question I have is how much do I taper. With no taper, the stress will be highest at the base of the beam. Too much taper and the stress will be highest at the tip. Can anyone give me any ideas on how to make a crude derivation for what taper angle I should use?

 

[jrmichler]

My Roark Fifth Edition has a section on tapered beams. It's not very useful for what you want, so I suggest that you NOT look for it. My old undergrad mechanics of materials book has two pages on tapered beams, but it also is not very useful for what you want.

A beam designed for constant bending stress will have the maximum flexibility for a given stress. It also has the minimum weight for a given maximum stress and for a solid prismatic beam. So here is what I recommend:

1) Design the tip to handle your load. A theoretical analysis of a simple cantilever beam will tell you that zero bending stress at the tip requires zero thickness. A slightly more sophisticated analysis will calculate a minimum thickness to handle the shear stress. A little testing will tell you how thick the tip has to be in order to stand up to the real loads without breaking out little pieces. This is a case where a few simple tests are better than 1000 calculations.

2) Assume a load and an allowable stress, then calculate the thickness at the base. Use those same numbers to calculate thickness at the 20%, 40%, 60%, and 80% (distance from base to tip) points. Connect those points with either straight lines or a smooth curve, whichever is easier. The real world difference is minimal.

3) Test it. If too flexible or weak, redesign with the same load and a lower allowable stress. Note that only the longitudinal plies in plywood contribute to strength and stiffness, while the cross plies are dead weight spacers.

Hint: Do the calculations in a spreadsheet, so that changing the allowable stress can be done by changing only one number.

 

[fpjeepy]

Most of them work with little noticeable difference. Imperfections in the plywood seemed to be a bigger determining factor.