Fluid Mechanics: Calculating Force Required

[Sheezey]

Homework Statement

Gate AB in Fig. P2.62 is 15 ft long and 8 ft wide into the paper and is hinged at B with a stop at A. The water is at 20°C. The gate is 1-in-thick steel, SG =7.85. Compute the water level h for which the gate will start to fail.

Homework Equations

F = δh_cgA

Y_cp = I_xxsinθ/h_cgA

I_xx = bd³/12

Moment = F*D

The Attempt at a Solution

F = (62.3lb/ft³)(h/2)(120ft²)

I am getting stuck as far as how to put together the equation to find the hydrostatic force. Mainly, what height value to use for h_cg and what value to use for area.

According to a similar example in the book, h_cg represents the center of gravity for h so we would have h/2 but I'm getting confused as to what the area is for this equation. Is it the area of the submerged portion of the plate? If so, I am not sure how to compute that. Any help is greatly appreciated.

 

[billy_joule]

According to a similar example in the book, h_cg represents the center of gravity for h so we would have h/2 but I'm getting confused as to what the area is for this equation. Is it the area of the submerged portion of the plate? If so, I am not sure how to compute that. Any help is greatly appreciated.

Yes it's the submerged area, that is what you'll need to find. h_{water depth} is related to A, can you show this relationship?

Start with the easy part, what is the torque about B due to the self weight of the gate and the hanging mass? This is the torque the water must apply to open the gate.

 

[Sheezey]

Hi billy_joule,
Thank you for responding. The torque about B due to the self weight would be

10,000lb* 15ft = 150,000lbft

 

[billy_joule]

Hi billy_joule,
Thank you for responding. The torque about B due to the self weight would be

10,000lb* 15ft = 150,000lbft

That's the clockwise torque due to the hanging mass, you need to find the anticlockwise torque from the self weight of the gate itself too.
That's why you're given the density of the steel.

 

[Sheezey]

Right. You must mean the weight force of the steel plate which would act along the center of the plate.

So we would have...
(7.85)(62.3lb/ft³)(15ft)(8ft)(1/12ft)(7.5cos(60)) = 18339.6lb

Does that look right?

 

[Sheezey]

Correction
(7.85)(62.3lb/ft3)(15ft)(8ft)(1/12ft)(7.5ftcos(60)) = 18339.6lbft

 

[Sheezey]

Thinking more about it there would be a force due to the water acting on to the steel plate. I believe this force would also act in the center of the plate. I think this is what I am looking for to insert for the area in my original equation.

 

[SteamKing]

Thinking more about it there would be a force due to the water acting on to the steel plate. I believe this force would also act in the center of the plate. I think this is what I am looking for to insert for the area in my original equation.

Does the hydrostatic force act at the center of the plate? Remember, water pressure varies with depth.

 

[Sheezey]

No it wouldn't. It does vary and will be become greater the deeper you go.

 

[billy_joule]

h_{water depth} is related to A, can you show this relationship?

You'll also need to express other quantities in terms of h_{water depth} too. e.g. the length of the lever arm that the hydrostatic force acts at.

You won't actually find the submerged area directly, though you could find it after you've found h_{water depth}, at which point there's no need.

Ultimately you want an expression:
M=Fd
where the LHS is the moment about B due to the gate weight and hanging weight (which you've almost found) and the only unknown on the RHS is h_{water depth}.
(The knowns on the RHS will be 60^o, water density and 8ft..)

 

[Chestermiller]

If z is the vertical elevation above point B and h is the total depth of water, what is the water depth at elevation z? What is the pressure at elevation z? What is the direction of the resultant force per unit area (pressure force) acting on the gate at elevations between elevation z and elevation z + dz? What is the differential area of the gate between elevations z and z + dz? What is the moment arm of this force about line B?

Chet

 

[Sheezey]

You'll also need to express other quantities in terms of h_{water depth} too. e.g. the length of the lever arm that the hydrostatic force acts at.

You won't actually find the submerged area directly, though you could find it after you've found h_{water depth}, at which point there's no need.

Ultimately you want an expression:
M=Fd
where the LHS is the moment about B due to the gate weight and hanging weight (which you've almost found) and the only unknown on the RHS is h_{water depth}.
(The knowns on the RHS will be 60^o, water density and 8ft..)

Ok so the submerged area of the plate is going to be (8ft)*(the submerged length of A to B)...

If I create a triangle on the drawing and call the unknown submerged length 'x' I get x=h/sin(60).

Now I can plug that into the original equation I was asking about, hydrostatic force. So I now have F = (62.3lb/ft3)(h/2)(8*h*csc(60)) = 287.8h²

So looking at my moments I have my weight force, my 10,000 force and my hydrostatic force F acting on the plate. I should be able to sum up the moments to find h.

I have

∑_M = -(150,000lbft) + (18339.6lbft) + 287.8h²(some distance from b) = 0

Does that look like the right direction so far?

 

[Sheezey]

Ok got it! The "some distance from b" in my moment equation is = to Y_cp. Which requires me to find the moment of inertia.

I_xx = (1/12)(8)(hcsc(60))³

Y_cp = (I_xx(8)(hcsc(60))³sin(60))/((h/2)(8hcsc(60))

∑_M = -150000+18339.6+287.8h²(Y_cp) = 0 solve for h

∴ h = 10.59 ft

Thanks for your help guys!

 

[Chestermiller]

I really hate the idea of using memorized equations, like those for Ycp and Ixx. Using the simple method I outlined in post #11, you get a moment about axis B of $$M=\frac{ρgwh^3}{6}\csc^2(60)$$
where w = 8 ft. This leads to the same value for h that you obtained.

Chet

 

[Sheezey]

Interesting. I understand what you mean. That is a great way to look at it. Good advice. Thank you for your help.