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.

 

[LFRIII]

Gentlemen:

First of all I want to thank you all for your help with solving my problem above. I was not understanding how to apply the beam deflection formulas and you have taught me how for which I am grateful. I now have the calculations worked out and we are applying them to our product offerings.

I have another problem that I am going to tackle and would like your help again. This challenge is to develop beam deflection formulas for a semicircular load applied to a simply supported beam. The formulas would include slope at each end of the beam, deflection at any point x along the beam and maximum deflection of the beam.

The load profile would be x^2+y^2=r^2. Since I would like the origin to be at the left hand support of the beam I assume I would modify this to be y=(r^2-(r-x)^2)^.5. Right now I am working on the integration of this load equation and it is cumbersome. If any of you have a suggestion of an easier way or have a solution already, I would appreciate the help.

Thanks.

 

[rb1957]

@human909 ... this is clearly a textbook problem. Solving it using an online program isn't, in my mind learning, anything; particularly anything applicable to the real world where we aren't (usually) presented with such simple problems. I personally dislike online beam calculators, since the work to solve these problems is pretty trivial, maybe a morning (4 hrs) at best; so using the online program is in my mind "lazy". Now someone (perhaps like yourself) who has a thorough grounding in the work, then maybe ok; but someone new to the work would just be "plugging in numbers into a black box".

 

[LFRIII]

Rb1957:

I am not sure what you are referring to above. I am trying to solve this by hand, Any direction you could give me would be appreciated.

 

[3DDave]

LFRIII, you just replied to rb1957 about a comment that was not made to you and that base comment was made before you changed the subject so it doesn't include the change of subject. ​

​

It appears you have added a different question without starting a separate thread which is different than from the original and appears to be a problem with the mathematics of integration. ​

Searching math answers I found this:

Consider a right triangle with one side given by (b2 - x2)1/2. This should let you determine the other side and the hypotenuse, thanks to the pythagorean theorem. Now rewrite your integral in terms of trigonometric functions and one of the triangle's interior angles. For example, you could orient the triangle so that (b2 - x2)1/2 = b*cos(a), where a is the adjacent angle. Then you can find x in terms of that same angle, compute dx in terms of da, and your integrand becomes b2cos2(a)da. Integrate that, then substitute back in as appropriate to get a result that depends on x.

Assuming you do all of that correctly and take limits of 0 and b on your integral, it should evaluate to πb2/4, so that when you multiply by 4 you get the area of the circle.

A longer explanation is at https://math.stackexchange.com/ques...find-the-area-under-a-portion-of-a-semicircle

 

[BAretired]

The change to a circular load should require a separate thread. Tacking it onto this thread is confusing.

The OP:
I have another problem that I am going to tackle and would like your help again. This challenge is to develop beam deflection formulas for a semicircular load applied to a simply supported beam. The formulas would include slope at each end of the beam, deflection at any point x along the beam and maximum deflection of the beam.

A circular load could be analyzed by applying the basic principles, but it is an unusual load for a beam, which suggests to me that it may not be worth the effort. Perhaps it would be more useful to use an approximate method such as Newmark's numerical procedure.

 

[GregLocock]

The semicircular load thing is easily solved using Macaulay, second year structures. Or if you don't like that use superposition of a point load at any position of the beam to whatever resolution you like, first year structures. Is this stuff homework?

 

[LFRIII]

Greg:

Good idea. I have read about McCaulay but have not worked with it at all so I would have to do some work with it. I had thought of using multiple point loads but thought if I could integrate the semicircle it would be easier to model. I will probable work on the point load idea.

This is not homework stuff. I have real world applications for this with a product line we manufacture. But I have had to go back to my Strength of Materials textbook to learn how to do all of this in more depth. Also it has been 40 years since I was in school. The homework has really helped me understand the nuances of the calculations that I never had the time to fully understand when I was in school.

Thanks for your help.

 

[Stress_Eng]

Another interesting SS beam problem. Out of interest, I looked into the distributed loading and BM. Attached is my investigation into deriving the equations by integration. In situations like this, I would usually get MathCAD to carry out the integration rather than working out the equations. Examples are the integrations conducted for the checks. Hope it gives you some ideas!