Simple Strength of Material question 10

[BFstr]

Hi
We are in debate in office on the correct answer to this problem:

A uniform straight rod AB (A if fixed, B is free end)with total length of 3 meter is under some axial loads that causes strain of Ɛx = 0.01x^2
What is the axial displacement of the end B in terms of centimeter?

Interesting question. Looks simple !!!
Appreciate all steps reaching your answer

Thanks

 

[Struct123ure]

3m = 300cm
300cm*Ɛx=0.01x/x (strain is cm/cm not sure why you put x^2?)
=3cm.
Axial displacement of the end B is 3cm.

 

[BFstr]

Struct123ure (Structural)
Thank you for your response. However that is the strain equation given to us and it is 0.01 times X to the power of two.
I am not sure what value you got for epsilon?

However 3 cm is not a correct answer !!!

 

[BFstr]

Also X in the strain equation represent all point along length of the bar.

 

[ChorasDen]

Can you explain the mechanism of how the strain is increased quadratically along the member length due to axial load at the end of the straight rod?

 

[BFstr]

ChorasDen (Structural)
Assume due to an unknow loading (however axial for sure)has caused this strain equation governs for the bar. How can you get the displacement of point B in this rod having this strain equation?

 

[ChorasDen]

How can you apply the load? Curious what value there is in the question?

Without thinking about it much, this sounds to me to be very simple calculus problem, but I'm curious why the arbitrary question?

 

[rb1957]

if you had uniform shear flow into the rod, the strain would be linear with x (as the load in the rod increases)
then displacement would be the integral of the strain, so x^2 ?

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[ChorasDen]

if you had uniform shear flow into the rod, the strain would be linear with x (as the load in the rod increases)

Would this not induce eccentricity, therefore complicating the displacement check, as it would occur both axially and laterally (with additional 2nd order effects)?

 

[rb1957]

sure, probably, but the fasteners (doing the shear transfer) could be on the neutral axis of the rod (and on CL) ... as a thought exercise you can imagine almost anything !?

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[BFstr]

No sir, this is a simple straight rod only under axial loads that has caused governing axial strain equation as introduced above. We are seeking the end B ( X= 3 m) axial deformation.

 

[rb1957]

can you share the derivation of this equation ? "surely" whoever derived it has solved it for your geometry ??

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[HTURKAK]

ΔL=∫εdx = ∫ 0.01x^2dx=( 0.01 X^3)/3 =0.01*27/3= 9 cm.




Use it up, wear it out;
Make it do, or do without.

NEW ENGLAND MAXIM

 

[rb1957]

but why, under such simple conditions, is strain proportional to x^2 ?

The simple loading is described as uniform (not even linearly increasing), the rod is described as "uniform" (so constant area ?), so strain should be constant, no?

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[ChorasDen]

but OP already said!

Assume due to an unknow loading (however axial for sure)

 

[Denial]

If you rotate a uniform bar about one of its ends at a constant angular velocity, the bar's axial strain will vary QUADRATICALLY, but it will not be proportional to the square of the distance along the bar (regardless of which end you measure the distance from).

[sub][ ]—————————————————————————————————[/sub]
[sup]Engineering mathematician / analyst.[ ] See my profile for more details.[/sup]

 

[rb1957]

"If you rotate a uniform bar about one of its ends at a constant angular velocity," ... would you describe this as "under some axial loads" or "under axial loads".

and you mean spinning the rod about an axis normal to the rod (not spinning on it's CL) ?

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.

 

[BFstr]

HTURKAK
Thank you for your response but I got a question for you.
Sorry I don't have symbols to write mathematically but the arguee is the delta L you found shouldn't be multiplied by the original length of 300 CM ?
I arguee that the 9 CM is your total strain. And strain = delta L / L
So you got the strain by integration and is 9 so, 9 = delta L / (otiginal L = 300 CM)
Therefore the displacemt at B (delta L)is 9 times 300cm = 2700 CM
Is this wrong analogy?

Second dumb question is you used 3 meter in your integration how them the result comes 9 centimeter and not interms of meter????


Thank you

 

[Denial]

Attempting to answer rb1957's questions in reverse order...
»[ ] Yes. (Spinning about an axis normal to the rod.)
»[ ] I am unsure what distinction you are trying to draw here.[ ] The rotation will result in centrifugal force directed outwards from the centre of rotation.[ ] I don't think I would chose to use either of your adjectival phrases.

[sub][ ]—————————————————————————————————[/sub]
[sup]Engineering mathematician / analyst.[ ] See my profile for more details.[/sup]

 

[rb1957]

@BF ... ok, HTURKAK messed the units. But do you understand the difference between displacement and strain ? Not sure I understand "total strain" means the same as displacement. yes, strain is non-dimensional, so strain * length = displacement (length).

but then your expression for strain ... strain(x) = 0.01*x^2 presumably x is defined along the length of the rod, 0<x<300cm ? has dimensions and the integral isn't dimensionally correct.

can you share how your strain function was derived ? Is Denial's loading correct (spinning about 1 end) ??

"Hoffen wir mal, dass alles gut geht !"
General Paulus, Nov 1942, outside Stalingrad after the launch of Operation Uranus.