Strength of Materials - Beam Bending Using Superposition Equations

[LFRIII]

I am working with superposition equations to solve a beam bending application. I am not a structural engineer and did not work with this for many years. I have an application which is a circular valve blade, supported by a shaft, with a bearing at each end. This system is subject to a UDL on the surface of the circular blade. I used the superposition equations for a beam with freely supported ends and a concentrated load at any point to calculate the angle of deflection at each end with a circular load profile. The load profile is (r^2-(r-x))^2)^(1/2). Theta at each end of the beam is - Theta 1 = Pb(L^2-b^2)/(6LEI) and Theta 2 Pab(2L-b)/(6LEI). I integrated these two equations and then evaluated them from x=0 to x=L/2 and added them together to get the angle at either end of the beam. This worked perfectly and the result agreed with a beam analysis done on SkyCiv. So my problem is solved and I am able to calculate what I need for my applications.

As a purely academic exercise, I tried to duplicate my calculation using a cantilever beam with a concentrated load at any point. The equation for Theta is Pa^2/(2EI). Using the same load profile and integrating, I get an answer for Theta that is 1/3 of the answer I get from the simple beam calculation. Multiplying my cantilever equation by 3 gives me the exact same answer as the simple beam and SkyCiv. The question is what aren't the results the same?

Any assistance anyone can offer would be appreciated.

 

[JStephen]

Attached sketch is how I'm visualizing your problem. Can't say about the 1/3, if done right, it should be the same.
If you assume cantilever fixed at the original end and free at the middle, results will vary. If you omit the end reaction, results will vary.
Could be math error somewhere, too.

 

[BAretired]

I am working with superposition equations to solve a beam bending application. I am not a structural engineer and did not work with this for many years. I have an application which is a circular valve blade, supported by a shaft, with a bearing at each end. This system is subject to a UDL on the surface of the circular blade. I used the superposition equations for a beam with freely supported ends and a concentrated load at any point to calculate the angle of deflection at each end with a circular load profile. The load profile is (r^2-(r-x))^2)^(1/2). Theta at each end of the beam is - Theta 1 = Pb(L^2-b^2)/(6LEI) and Theta 2 Pab(2L-b)/(6LEI). I integrated these two equations and then evaluated them from x=0 to x=L/2 and added them together to get the angle at either end of the beam. This worked perfectly and the result agreed with a beam analysis done on SkyCiv. So my problem is solved and I am able to calculate what I need for my applications.

As a purely academic exercise, I tried to duplicate my calculation using a cantilever beam with a concentrated load at any point. The equation for Theta is Pa^2/(2EI). Using the same load profile and integrating, I get an answer for Theta that is 1/3 of the answer I get from the simple beam calculation. Multiplying my cantilever equation by 3 gives me the exact same answer as the simple beam and SkyCiv. The question is what aren't the results the same?

Any assistance anyone can offer would be appreciated.

The above sketch shows deflection at point of load to be Pb^3/(3EI). It does not show slope because most people are interested in the deflection of a beam, not the slope.

Why are you comparing the slope of a cantilever to that of a simple beam? In a simple beam, the slope at the supports is large and zero at some point in the span. In a cantilever, the slope at the support is zero and maximum from load point to the free end.

A simple span with point load has positive bending moment throughout the span.
A cantilever with point load has negative bending moment throughout the span.

 

[BAretired]

If you want to find the slope of a cantilever with any load, consider the M/EI diagram as an imaginary load on an imaginary beam. At any point x in the span, the slope is the shear of the imaginary beam; the deflection is the moment of the imaginary beam.

In the case of a single load at b from support, slope is 0 at support, Pa^2/2EI at the load point (and the free end).