Statics-beam total deflection 5

[ntw88]

I have forgotten my statics class.



Ultimately I am trying to determine Critical Speed of a shaft supported by two bearings with loads on each end. I am struggling with correctly calculating the total deflection in the beam. We have several calculators and they all give different results so I am tasked with building a new one that yields correct results that we can agree on and stand by. The situation as shown in the picture has a round shaft of length (A+B+C+D+E=L) supported by two bearings. Bearing G is fixed, and bearing H is an expansion (can move left and right slightly). The shaft has a uniform load w (its weight). It also has two forces acting on it, Fb and Fa. if it helps it is a steel shaft, but we use a variety of steels and various temperatures so the modulus may change.



if someone can solve this for total deflection and show their work so I can follow along and apply it to our applications I would greatly appreciate it. I have been mulling over this for days and cant seem to remember how to correctly do this.

 

[robyengIT]

I should do this way (according to the attached schemes) :

scheme 8 + scheme 8 (specular) + scheme 10 + scheme 10 (specular)

Note : specular = mirror copy (sorry for non properly correct terms)

 

[ntw88]

Thank you for the reply! I hadn't found these equations in any of my searches.

I am working through the equations. On scheme 10, if i'm reading it right, Max deflection (f) along segment AB (GF in my drawing)is equal to K*(Q*l^3)/J? If I plug in some numbers (Q=3.653, J=11.8, c=8, l=20; c/l=8/20=.4, so K= 2750) I end up with 2750*3.653*20^3/11.8 = 6,810,677in, on a 28 inch shaft? I am making a mistake somewhere.

 

[FEA way]

K should be 2.750 not 2750. And be very careful with units.

 

[robyengIT]

Units are metric. Since we don't know the source of the table (really I don't know if the coeff are dimensionless), you should use metric. FEA way is correct : comma is equivalent to your dot

 

[Tmoose]

Are the loads forces, or weights/masses?
Forces change static deflections but do not change the ωn. Masses do change ωn.

I believe the relationship of static deflection and ωn is valid only for the simplest of systems.

Guitars sound about the same whether played like this -

https://dtkp6g0samjql.cloudfront.ne...gallery_hero_il_fullxfull.1207619302_bab1.jpg

or like this -

https://i.ytimg.com/vi/HJ7iLbqVgqo/hqdefault.jpg

Structures still have resonant frequencies in space, where gravity = zero and static deflection = zero.

 

[rb1957]

why can't we solve problems like this (very simple problems) from 1st principles ?

maximum deflection is either at the free end or at the CL (depending on the relationship between B and C/2), but maybe always at the free end.

another day in paradise, or is paradise one day closer ?

 

[ntw88]

Tmoose, One end (Fb) is a force (belt drive), the other (Fa) is a weight/mass (fan wheel).

I have been using

https://mechanicalc.com/calculators/beam-analysis/

to check against my results, but it doesn't show the calculations that go on behind the scenes. Also since you cant change material properties without subscribing I cant use it for general purposes.

 

[ESPcomposites]

You can also try the 1D Elements FEA program here:

https://www.structuralfea.com/

I included a sample file for the model here. You will have to adjust the geometric values, material properties, loads, and pay careful attention to the density and gravity values to ensure you get that right. You may want to create a simple cantilever beam to make sure the gravity entry is as expected.

https://www.structuralfea.com/eng_tips_model.1DL

Brian

http://www.espcomposites.com

 

[3DDave]

Doesn't the superposition method work for this case?

 

[BEMPE16524]

https://skyciv.com/free-beam-calculator/

https://mechanicalc.com/calculators/beam-analysis/

R.Efendy

 

[GregLocock]

3Ddave, superposition is definitely applicable. I am surprised to see first year structures lectures using the beam equations and Macaulay's method is such a struggle.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

http://eng-tips.com/market.cfm?

 

[ntw88]

rb,
For your information I did only take one statics class, I am degreed as an industrial/manufacturing engineer, not mechanical, so the fact that I even took one is more than many of my peers took. Since I graduated almost a decade ago those books are no longer accessible to me, and I have spent the better part of a week trying to solve it on my own before I reached out on here. The equations posted by Robyeng was extremely helpful and I am still unable to find them in google searches even knowing what the equation now is. the calculators posted all require subscriptions to get the capabilities and answers that I would need to use it for every situation, and I would much rather point to an established engineering equation than some website calculator to justify why a catastrophic failure wasn't due to an undersized shaft.

Sorry if, by your estimate, I came to the wrong place for engineering help, but after a week of digging through the annals of the internet without success I needed help from the outside.


To everyone else,
Thank you so much for your help. it is greatly appreciated and has been of great use.

 

[rb1957]

"class" is singular; "classes" is plural.

a google of "beam deflections" would've found what you wanted. "after a week of digging through the annals(sic) of the internet without success" ? good grief.
This is a first or second year problem, dealt with in just about every strength of materials text.

This is a forum for professionals asking professional questions.

another day in paradise, or is paradise one day closer ?

 

[ntw88]

I am a professional asking professional questions. I work as a mechanical engineer in a place where no one else has the solution to that. I googled beam deflections, many times in many different variants. I have read through the literature we have available to us which was not helpful. Google "beam deflections" yourself and show me the example I have shown (free ends, loads outside supports, uniform loads, etc.), show me the equation RobyengIT sent me.

Stop being an ass-hat and try being helpful like EVERYONE else in here was.

 

[ntw88]

The opinion that seemed to be shared by others was, "someone is requesting help, I will help them!" and that has been greatly appreciated.

The opinion you hold, which does not seem to be shared by others is, "I am perfect and the fact that someone is requesting help on a website called "Eng-Tips" means they are lesser than me so I will publicly mock them." And I'm glad others do not share that sentiment because it is not a community I would want to be a part of.

Was it a question that was easy for some to answer? Yes, but not for me, and I appreciate their help.

 

[GregLocock]

So look up Macaulay's method on wiki.

Or if you want to do it from first principles:

solve your FBD

Draw a shear force diagram.

Draw a bending moment diagram

Linear elastic beam theory says

M/I=E/R

R being the radius of curvature at that point along the beam.

so d[sup]2[/sup]y/dx[sup]2[/sup]=M/EI

Integrate that piecewise along your beam. The power of doing it this way is that steps in I can also be accounted for




Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

http://eng-tips.com/market.cfm?

 

[STrctPono]

Well if it makes you feel any better, you wouldn't have learned this until your solid mechanics course (typically 2nd year). Ha! So get your hands on any mechanics of materials book and there will surely be a chapter on this topic. Most likely for a simply supported beam without overhangs but once you understand the method, you can apply it to your situation quite simply.

Either way, there are several ways to solve this. I would prefer to use direct integration (1st principles)and superposition. The benefit of this is that if you are trying to write a program, this will allow you to calculate the deflection at any point along the beam. The beam deflection can be described via a 4th order differential equation. Remember that Shear, Moment, Curvature, Deflection are all related as they are the integral of the former.

Separate your concentrated loads (masses converted to loads) from your uniformly distributed load and draw your shear and moment diagram for each. Each diagram can be represented via a distinct linear equation. You will need to set up three equations for each beam since your equation is discontinuous across the supports.

GregLocock provided the constitutive equation d2y/dx2=M/EI

You can then combine your results of the different load cases together via superposition.

https://www.youtube.com/watch?v=kE7EXMf0kjs

 

[GregLocock]

No, that was first year stuff. We didn't do underwater basket weaving or other such fripperies. Mind you the electrical engineers did a common first year with the mechies, I laugh when I think of them doing the thermodynamics of steam engines, as I quietly cursed Power Electrical.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

http://eng-tips.com/market.cfm?

 

[Tmoose]

"Ultimately I am trying to determine Critical Speed of a shaft supported by two bearings with loads on each end."
"..............the other (Fa) is a weight/mass (fan wheel)."

Just checking in to confirm the ultimate goal is the (lateral) critical speeds of this over hung fan assembly.

thanks,

Dan T