Statically Indeterminant Beam Problem 7

[LFRIII]

I am trying to solve the following problem which is in the method of superposition section of my Strength of Materials textbook. I am stuck and am wondering if anyone could point me in the right direction.

A beam of length 3L is supported by four supports, A, B, C, and D from left to right. The distance between any two supports is L. Between the two middle supports, B and C is a uniformly varying load with maximum intensity of w. Find reactions Ra and Rd.

I started by eliminating the two redundant reactions at A and D and used Case 10 Simple Beam with Uniformly Varying Load (see attached) to find the slope at points B and C. Then used Case 9 Simply Supported Beam with Couple Moment at End to find the moments a points B and C. From there I have tried many things without success.

The book gives the answer: Ra = wl/45; Rb = wl/36.

If anyone can help with this, I would appreciate it.

[URL unfurl="true"]https://res.cloudinary.com/engineering-com/image/upload/v1726255246/tips/Beam_Deflection_Formulae_iqr6vc.pdf[/url]

 

[BAretired]

Thanks Stress Eng,

If the rotation at B and C could be resisted by pure moments applied at each end instead of Ra and Rd acting over a 6" lever arm, the reactions at B, C could be substantially reduced, but that may not be possible with a valve shaft which is required to rotate about its axis.

A photo of the valve shaft might suggest other possibilities.

 

[Stress_Eng]

If interested, attached is the method based on M/EI. Just to mention, all the integration was done symbolically using Mathcad. It hasn't been checked but it looks a though it's working!

 

[BAretired]

Thanks again Stress Eng. That is a beautiful output.

 

[BAretired]

Vertical reactions Ra and Rd are eliminated using horizontal forces using a strut and tie as shown below. Rb and Rc reduce to 1634#. Minor adjustments in shaft deflection could be made by tightening the tie.