Best Tunnel Shape for X,Y,Z Coordinates

[caters]

Homework Statement

X,Y,Z(coordinates)
What function corresponds to the best tunnel shape?
g = 9.8 m/s^2(earth gravity)

Homework Equations

F(x)=Y
G(x)=X^2 in the xy plane
G(z)= sin(X) in the xz plane
H(x)= parabolic sinusoid(X^2 and sin(X) both in the xy plane)

The Attempt at a Solution

I have thought about the shape of a tunnel and as far as I understand, the less the change of Y, the better since if Y changes, U changes(i am using U for upward force), namely as Y gets more negative, U gets more positive. I think I need calculus to solve this and maybe none of those functions are the best.

However, the further negative Y gets, the more the velocity increases and the more positive Y gets, the more velocity decreases. From this point of view the parabolic sinusoid is best. And the higher the velocity the higher U is.

Is there a way I can solve this problem without calculus? If so how? X and Z don't really play a part into the upward force that is required for a tunnel not to collapse from gravity. I can probably find the average density of soil to determine how much mass of soil there is above the tunnel and thus the maximum downward force it can withstand with nothing but the soil itself providing upward force and thus determine how much more upward force is needed at some depth Y.

 

[Simon Bridge]

Is that the way the problem was presented to you? (There's lots of explanation missing.)
You have used upper and lower case or x y z ... for what purpose?
What do the coordinates represent?
Is +y "upwards"? Some people use +z for that and use the x-y plane for a reference height.
Is the tunnel in the +x direction? Is it a straight tunnel etc etc?

 

[caters]

Coordinates represent position. +Y is upwards. The tunnel is in the x direction. I am trying to determine the best shape of the tunnel, that is the unknown I am trying to find.

 

[Simon Bridge]

Is post #1 the way the problem was presented to you? ie. did you copy it down verbatim?
(There is no context in the problem statement so, as written, the problem cannot be solved.)

I do not understand your notation.
What does "G(x)=Y^2 in the xy plane" end up meaning? Is that different from G(x)=y^2?

Whatever, always start from the criteria for judging the shape quality - without that there is no way to determine which is "best".
It may be the best is the one that is cheapest to build using modern bore equipment.

Basically - if you will not answer question then nobody can help you.

 

[caters]

I did however sort of set the context of it in the part where I tried to solve it. This is for an underground city and there is no currency so money isn't involved. So in this problem I am trying to find out which shape of tunnel requires the least upward force to sustain it under any given mass. I think the horizontal linear one is best in this context. If very high mass above the tunnel is expected there are 2 things that would be used in combination.

Those would be more support in more places and lower depth(the more soil there is above the tunnel, the better for supporting high mass since it can provide more upward force.

 

[Simon Bridge]

I did however sort of set the context of it in the part where I tried to solve it.

There was no way to assess your attempt without the context - the attempt is not context, since you may be doing it wrong.

This is for an underground city and there is no currency so money isn't involved.

OK - so the shape that minimises resources - as in materials, labour, administration time, that kind of thing that normally gets measured in terms of money.
So this is science fiction?

So in this problem I am trying to find out which shape of tunnel requires the least upward force to sustain it under any given mass.

All shapes require the same upwards force for a given mass.

I think the horizontal linear one is best in this context.

Which one is that? You have yet to answer the questions about your notation.
So far what you have written makes no sense. If you will not answer questions, nobody can help you.

If very high mass above the tunnel is expected there are 2 things that would be used in combination.

Those would be more support in more places and lower depth(the more soil there is above the tunnel, the better for supporting high mass since it can provide more upward force.

This makes no sense.

Guessing about what you mean:
... The shape which best supports itself and a distributed load, all other things remaining equal, is usually an arch.
That would be an inverted parabola ... the profile would be $0 < y < h(2x+w)(2x-w)/w^2$ for a tunnel w wide and h high.

Since we have reinforced concrete, and it's fairly cheap, it is more common to build tunnels as cylinders. This is usually fairly cheap to build - tunnels would be bored out.
You should probably look at real-life tunnels, particularly the ones that have to go quite deep or where materials are tricky.

 

[caters]

That F(x) = Y is horizontal linear. Lower case x,y, and z are variables. Upper case X,Y, and Z are coordinates.

So that first function means no matter what x is, the depth is a constant y value. Z is 0 in this case

That second function with 2 variables means that in the x direction, the depth is X^2 and the Z coordinate is based on the sin function.

That third function means that in the x direction, it is based on X^2 but Y periodically goes up and down from X^2 by the same amount. Z is 0 here as well.

 

[Simon Bridge]

G(x)=X^2 in the xy plane

So in G(x)=X^2 ... how do you square a coordinate?
What do you even mean by "coordinate"?
If this is in the x-y plane - then are you saying the tunnel has a quadratic profile in its travel direction (so the tunnel travels down and then up)?
basically you are still not making sense: I think you will be better just describing the shape using words.

Here's how math descriptions usually work:
Lets take the y-axis as "up" and the tunnel heads off, horizontally, in the x direction (so the tunnel does not go up and down) - treat the x-z plane as the floor level of the tunnel, and we will put z=0 in the middle of the floor. Let +z be "to the right" when we are standing on the floor, and looking down the tunnel. That will put the +x axis pointing away from us. You got that?

Then we only care about the shape in the y-z plane, and we can characterize the shape by showing y as a function of z thus: y(z).

For example: a flat-roofed tunnel width w with vertical sides height h would have function: $y(z) = h[\text{u}(z+w/2)-\text{u}(z-w/2)]$ where u is the "unit step function"
That is: u(z)=1 when z>0, but 0 elswhere.

If the height of the tunnel varies along it's length, then y is a function of both z and x like this: y(x,z).
Here I have been using the x-z plane as the floor level.

See how this looks nothing like what you wrote.

I can modify that for different profiles by letting the shape part of y(z) be an arbitrary function f(z): $y(z)=[\text{u}(z+w/2)-\text{u}(z-w/2)]f(z)$
... the effect is to take a width w bite out of f(z), and that is the tunnel shape.

For instance: put $f(z)=h\cos(\pi z/w)$ and you have a sinusoidal arch width w and height h.
put: $f(z) = -4h(z-w/2)(z+w/2)/w^2$ then you have a parabolic arch width w and height h.
... see the utility?