in my case I will heat a definite size of beam in the furnace..... the temperature would be in the order of 2000C and 4000C........ then after the heated specimens will be taken out and testing for flexure in the UTM. now I want to know how to find the radius of curvature of the specimen.if someone have a valid method please help me out. thank you
Using trigonometry, you can easily find the theoretical radius. You know 3 points, I suppose: both extremities and the mid span point after deflection. Thus, you only have to use the formula in the attached picture. The picture is the representation of your theoretical model.
You can compare this result and the one using the method suggested by Mr. Biswas.
Take photograph of the beam, then mark atleast two points ( the more points the better) on the picture and use simple trignometry you can calculate the radius of curvature
We know that bending stress is M*y/I. So, you are obtaining the bending stress and displacement from flexural test. So, form experiment you can easily get the bending moment M. We also know that the bending moment is related to the radius of curvature i.e. M/I=E/(R) where, E is Young's modulus and 'R' is radius of curvature. This way you have to measure the radius of curvature.
Using trigonometry, you can easily find the theoretical radius. You know 3 points, I suppose: both extremities and the mid span point after deflection. Thus, you only have to use the formula in the attached picture. The picture is the representation of your theoretical model.
You can compare this result and the one using the method suggested by Mr. Biswas.
Then the bending moment between the force application points is constant and the curvature as well It is enough to measure the displacement in the middle and apply the relation: R=(L²/4+f²)/(2*f) L= distance between force points/ f= middle displacement.
You can use the bending equation to do so and experimentally it depends upon how accurately you have taken the photograph
Jeevan,
based on Euler-Bernoulli beam theory:
1/R = (d2W / dx2)
where R is the radius of curvature, w is the deflection, and x is the distance where you want to find the curvature.
Also,
E(d2W / dx2)= M/I
E being the young's modulus, M the bending moment and I is the second moment of inertia.
Therefore, as Pankaj also mentioned, you can use the following:
R=EI/M
In four-point bending test the bending moment is constant between two loading noses.
As far as experimental measurements, you can either measure the modulus and then use the mentioned equations, or if it supposed to be totally based on experiments, then as mentioned by others, taking a photo at any point that you need to find radius of curvature and then use trigonometry would be your best practice.
Good question… Please check references below.. hope it will be helpful for you…
Regards…
http://www.loft.optics.arizona.edu/documents/journal_articles/Zhao_Measuring_the_radius_of_curvature_of_spherical_mirror_with_interferometer_and_laser_tracker.pdf
http://www-3.unipv.it/donati/papers/95d.pdf
We can also calculate radius of curvature by Pythagoras mathematical equation.
As shown in figure.
R2 = (BC)2 = (BD)2+ (CD)2
Hence (BC)2 = (BD)2 + (CE-DE)2
BC = CE=R
i.e. R2 = (BD)2 +(R-DE)2
R2 = BD2 +R2 - 2RDE+DE2
2RDE = BD2 + DE2
Hence R = (BD2+DE2) /2DE
Where R is (Radius of curve), C is Centre of the circle
Calculate value of radius of curvature R easily.
R2= (AB2+4f2)/8f
(AB) is the bending beam distance and (f) is the distance between peak of the arc and the horizontal axis.
Sorry to mention but what you suggest was already suggested several years ago.
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