Derivation of the curvature equation (19th June 2002)

Curvature can be defined as how quickly the slope of a curve is changing about any point along a curve. In this case we'll only look at plane curves i.e. all the points along the curve lie in the same xy plane.


From the above figure it is well known that arc length = radius times the angle, s = R theta. The circle is a special case where the curvature is constant with the value 1/R. In general curvature could vary all along the curve. So from the principles of calculus we can choose theta to be infinitesimally small (d theta) therefore ds = R d theta or can be re-written as



The angle t1 and t2 are the tangents for points P and P1 on the curve. We can let t1 = theta and t2 = theta + d theta thus the angle t2 - t1 is = d theta. So allowing d theta to become infinitesimally small we can see the arc length ds and hypotenuse P-P1 will tend to be the same length allowing us to say tan(theta) = dy/dx and cos(theta) = dx/ds



Again from the principles of calculus dy/dx is the tangent to the curve at some point P along the curve. So if we differentiate the slope with respect to the independent variable s because we already know d theta / ds = 1/R and dx/ds = cos theta.


And by using the chain rule we can perform the differentiation

and substituting in dx/ds = cos theta

Dividing through by cos theta

leaves us with

But we can manipulate the form of sec cubed theta using the laws of indices and knowing that sec squared theta = 1 + tan squared theta

So we can now see the relationship in terms of the second derivative of y with respect to x is proportional to 1/R.

If we know the slope dy/dx of the curve is small because (say) you are designing a beam to carry a person's weight where you want to minimise the beam weight and cost by reducing beam dimensions, while also haveing a limit on the amount of deflection. Or maybe, you are using a beam as a spring in a vibrating system and you need to avoid non-linear effects we can then say



That's a lot of work to get where we wanted for the general beam bending equation, but curvature is one of the most important concepts of Dynamics ( for both statics and kinetics).

Good student texts:

"Engineering Mathematics" , Stroud K, 1987 , Third ed.Macmillian Education.
Working through Stroud examples is a must for the mechanic student if you want to practically apply the above derivation.

"A course of Mathematics for Engineers and Scientists", Chirgwin BH, Plumpton C,Pergamon Press, vol 1, 2nd ed, 1970, pp 186 - 192

"History of Strength of Materiala", Timoshenko SP, Dover Publications, 1983.

And there are plenty of web sites discussing curvature as well.

Wolfram
http://mathworld.wolfram.com/Curvature.html
The History of curvature
http://www.brown.edu/Students/OHJC/hm4/k.htm
Famous curves index
http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html

Back to the Beam Bending Equation
Cantilever Analysis

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