Can We Calculate Deflection of a Rod Placed on a Tall Building?

[Eagle9]

Here we have got tall building standing on the ground and long rod. Rod’s one end is placed on the ground and the second one-on the building.

Surely, the rod will be deflected due to its own weight and Earth’s gravity. Can we calculate the value of deflection if we know rod’s end, its material’s property (Young’s modulus), length, radius and etc?

 

[pongo38]

If deflection is small in relation to the length, then there is a reasonably simple approximation. You can add additional complications, allowing for the shear force, for example. However, there is a large deflection theory that will take account of the modified bending moments from the actual deflected shape. In any case, you would have to specify the end conditions for the rod, and make a list of all the things you are neglecting, for the sake of getting AN answer. In principle, all the theories are wrong in some respect. All you can do is get an approximation. You need to decide what sort of tolerance you are looking for. Whatever, the deflection due to bending will predominate over the effects of shear or axial load. If the wind blows, that could affect the result! A fairly neat solution is to approximate it to a catenary, and this can be further simplified to be a parabola with little loss of accuracy. Maybe this is the best approach. Assume a parabolic shape. You should find that requires the simplest maths. A first class student would, without asking, do it three ways and compare results. Are you up for that?

 

[Eagle9]

pongo38
Of course I understand that any theory can describe the real events approximately and for receiving exact answer for that rod many factors should be taken into consideration, but for now I would like to run the simple calculation and get approximate result

Assume a parabolic shape. You should find that requires the simplest maths. A first class student would, without asking, do it three ways and compare results.

Great so, which simple formulas should I use?

 

[pongo38]

Well, you could use the standard deflection formula for a udl on a simply supported beam, in order to get an initial idea (the formula with 5/384 as its coefficient, but it needs careful interpretation...

 

[AlephZero]

This way will give you a rough idea, if the deflections are small.

If the answers look crazy, then this simple linear model is no use and you would probably need to make a computer model to solve the problem.

 

[Eagle9]

pongo38

Well, you could use the standard deflection formula

And where is it?

AlephZero

This way will give you a rough idea, if the deflections are small.

If the answers look crazy, then this simple linear model is no use and you would probably need to make a computer model to solve the problem.

This is just image, but it does not indicate me the way how should I calculate……..

 

[pongo38]

For small deflections using [5*w*L^4]/[384*E*I], you would need w, the weight per unit length, and then its component perpendicular to the rod, and take L to be be the actual length of the rod between supports. (You can also use the actual weight per unit length, and the projected length of L on to the x-axis - you get the same result) If the deflection is say less than span/200, then that is a small deflection and credible. If it is more than that, then further work is needed on the deflected shape.

 

[Eagle9]

pongo38

For small deflections using [5*w*L^4]/[384*E*I], you would need w, the weight per unit length, and then its component perpendicular to the rod, and take L to be the actual length of the rod between supports. (You can also use the actual weight per unit length, and the projected length of L on to the x-axis - you get the same result) If the deflection is say less than span/200, then that is a small deflection and credible. If it is more than that, then further work is needed on the deflected shape.

w-“ you would need w, the weight per unit length”, I do not quite understand should I take rod’s total weight/mass and divide it to its length?
E-is Modulus of Elasticity?
I-is Moment of Inertia?

 

[pongo38]

pongo38


w-“ you would need w, the weight per unit length”, I do not quite understand should I take rod’s total weight/mass and divide it to its length?
E-is Modulus of Elasticity?
I-is Moment of Inertia?

yes yes and yes

 

[Eagle9]

yes yes and yes

I will try to run some simple calculations and I hope that it will be correct

Let’s assume that this rod 10 000 meters length and the radius of its circular cross-section is equal to 5 meters. The rod is made of Carbon nanotubes and the value of Young’s modulus (as I know in engineering this is much frequently used than Modulus of Elasticity) for this material is 1 000 GPa (taken from here: http://en.wikipedia.org/wiki/Young’s_modulus) As I know, Moment of Inertia (I) is equal to pi.r^4/4, in our case it is equal to 490.625.

As for W. The area of rod’s cross-section is equal to 85 square meters and with multiplying this to rod’s length (10 000 meters) we will get volume of that rod-850 000 cubic meters. The density of Carbon nanotubes varies 0.037-1.34 g/cm^3 (http://en.wikipedia.org/wiki/Specific_strength), let’s take some mean density-0.1 g/cm^3, so the mass of that rod will be 85 000 000 kg, divide this to rod’s length (10 000 meters) we will receive value of W-8 500.
So:

The value of Deflection will be 2255 meters?

 

[Eagle9]

People! My calculation is correct?

 

[JolileChat]

I don't think so.
You will need to find out the equation of the elastic line of your beam. You will need to consider the inclination of the rod.
A good startup is: http://www.efunda.com/formulae/solid_mechanics/beams/theory.cfm

 

[pongo38]

The theory mentioned by JolileChat, I think leads to the formula used by Eagle9, but I may be wrong.

 

[Eagle9]

JolileChat

You will need to consider the inclination of the rod

Let’s assume that angle of inclination is 45 degrees

You will need to find out the equation of the elastic line of your beam
A good startup is: http://www.efunda.com/formulae/solid...ams/theory.cfm

There are three options under title Choose a Boundary Condition and Calculate!:
1.Cantilevers
2.Mixed (Fixed-Simple)
3.Simply Supported
I think that the third one is the most suitable for me, so chose that one and then several other options appeared:
1.Center Load
2.Intermediate Load
3.Two Equidistant Loads
4.Uniform Load
I think that the last one is most suitable in spite of the fact that there is no load-just uniform rod, so I chose that one and the calculator appeared there (http://www.efunda.com/formulae/solid_mechanics/beams/casestudy_display.cfm?case=simple_uniformload), I put some values there:
Length of beam, L-Let it be 1 000 meters.
Line pressure load on beam, p: as I know in my case it will simply be the weight per meter length. So, if my rod’s radius is 1 meter then the volume of rod’s this part would be 3.14 m^3. As for its mass, if I use Carbon nanotubes their density varies between these values: 0.037-1.34 g/cm^3 (http://en.wikipedia.org/wiki/Specific_strength). Let’s take some medium value-0.1 g/cm^3, so the mass of that part will be 314 kg. But how can I transform this value to Pa-m?
Young's Modulus, E: for Carbon nanotubes is equal to 1 000 GPa http://en.wikipedia.org/wiki/Young's_modulus
Distance from neutral axis
to extreme fibers-as I know this will be the radius of the rod-1 meter
Moment of Inertia, I: it is equal to (pi*r^4)/4, so it is equal to 0.785

But my rod is inclined by 45 degrees, what should I do with this circumstance? :shy:

 

[Eagle9]

So, what should I do now?