Resultant force and its point of action

In summary, the pressure at the centroid of the inclined surface is 432621 Pa and the resultant force is 77.82 kN.
  • #1
MCTachyon
51
3

Homework Statement


A storage tank has the cross-sectional shape shown attached and is of 1m breadth. Calculate the resultant force acting on the inclined surface AB and its point of action.

The density of the liquid is 900kgm–3 and take g = 9.81ms–2.

hahahahahahahahahahahahah.png


Homework Equations


P = ρgh
F = ρA
Point of action = hc + (Ig Sin(θ)2 / A hc)

The Attempt at a Solution


Length of AB:

Tanθ = (O/A)
θ = Tan-1 (1.5/1)
θ = 56.31°

Cos(56.31) = 1/AB
AB = 1/Cos(56.31)
AB = 1/0.5547
AB = 1.8m

Pressure at centroid of AB:

P = ρgh
P = 900 x 9.81 x (4 + (1.8/2))
P = 900 x 9.81 x 4.9
P = 432621Pa

Resultant force at AB:

F = ρA
F = 432621 x (1.8 x 1)
F = 77818N
F = 77.82kN

This is bit when I get a bit lost.

Point of Action:

= hc + (Ig Sin(θ)2 / A hc)

Where Ig = db/12 = 1.8/12 = 0.15

= 4.9 + (0.15 x Sin(90-56.31)2 / 1.8 x 1 x 4.9)

= 4.9 + 0.00523

= 4.90523m

Therefore force of 77.82kN is acting 4.90523m from top of storage tank.
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Am I on the right track with my method? Pretty confident till working out the Point of Action.
 

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  • #2
MCTachyon said:
θ = 56.31°
It is rarely necessary to find the angle. You can go straight from one trig ratio to another. E.g. use Pythagoras.
MCTachyon said:
P = 900 x 9.81 x (4 + (1.8/2))
Pressure varies as depth. The 1.8m is not vertical.
MCTachyon said:
Where Ig = db/12 = 1.8/12 = 0.15
This doesn't seem right. Shouldn't there be a quadratic term? Certainly it seems like the resulting equation for point of action is dimensionally inconsistent...

... Or maybe I am misreading it. Please make it clearer by use of parentheses and define all the variables.
 
Last edited:
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  • #3
Thank you.

Lots of food for thought there. I will go back and retry.

Thanks again.
 
  • #4
Second attempt:

Length of AB:

Pythagorean Theorem:

A2 + B2 = C2

C = √ (12 + 1.52)
C = √3.25
C = 1.8m

Pressure at centroid of AB:

P = ρgh

P = 900 x 9.81 x (4 + (1/2))
P = 39731 Pa
P = 39.73 kPa

Resultant force at AB:

F = ρA
A = (h x b) / 2
F = 39731 x ((1 x1.5) / 2)
F = 39731 x 0.75
F = 29798 N
F = 29.80 kN

Point of Action:

Area Moment of Inertia:

Iy = IR + Ar2

Where:

IR = (d x b3) / 36

IR = (1 x 1.53) / 36
IR = 3/32 = 0.09375m4

Therefore:

Iy = 3/32 + ((1 x 1.5) / 2) x 4.52
Iy = 15.28m4

Radius of Gyration:

Ry = √(I / A)
Ry = √ (15.28 / 0.75)
Ry = √ 20.37
Ry = 4.513m

Therefore force of 29.80 kN is acting 4.513m from top of storage tank.

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Any closer to sussing out the method?

Thanks for your help with this.
 
  • #5
MCTachyon said:
F = ρA
PA not ρA.
MCTachyon said:
A = (h x b) / 2
MCTachyon said:
(d x b3) / 36
What shape do you think the area is?
 
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  • #6
haruspex said:
What shape do you think the area is?

ABC, so a right angled triangle.
 
  • #7
MCTachyon said:
ABC, so a right angled triangle.
That's not how I interpret the diagram.
C seems to be just a reference point for showing the width and height of the sloped section. The text says it is a cross-section, implying the "breadth 1m" is into the page. The area of interest is therefore a rectangle, seen side-on as AB.
 
  • #8
Right then, this is starting to make some sense now.

It is late where I am, I will come back tomorrow with some new numbers and see if this info has put me on the right track.

Thanks again for your help.
 
  • #9
With vectors you have a point of application, and a line of action.
 
  • #10
Third attempt:

Pressure at centroid of AB:

P = ρgh

P = 900 x 9.81 x (4 + (1/2))
P = 39731 Pa
P = 39.73 kPa

Resultant force at AB:

F = PA
A = (h x b)
F = 39731 x (1 x1.5)
F = 39731 x 1.5
F = 59597 N
F = 59.60 kN

Point of Action:

Area Moment of Inertia:

Iy = IR + Ar2

Where:

IR = (d x b3) / 12

IR = (1 x 1.53) / 12
IR = 9/32 = 0.28125m4

Therefore:

Iy = 9/32 + (1 x 1.5) x 4.52
Iy = 30.66m4

Radius of Gyration:

Ry = √(I / A)
Ry = √ (30.66 / 1.5)
Ry = √ 20.44
Ry = 4.52m

Therefore force of 59.60 kN is acting 4.52m from top of storage tank.

------------------------------------------------------------------------------------------------------------------

Getting closer?

Thanks again for all your help.
 

1. What is a resultant force?

A resultant force is the single force that represents the combined effect of all the forces acting on an object. It is the net force that causes an object to accelerate in a certain direction.

2. How is the resultant force calculated?

The resultant force is calculated by adding together all the individual forces acting on an object, taking into account their direction and magnitude.

3. What is the point of action of a resultant force?

The point of action of a resultant force is the point where the single force is applied on an object. It is typically represented by a vector arrow and can vary depending on the distribution of forces.

4. How does the point of action affect the motion of an object?

The point of action of a resultant force determines the direction of motion of an object. When the point of action is not in the center of mass, the object will rotate as well as move in a certain direction.

5. Can the point of action of a resultant force change?

Yes, the point of action of a resultant force can change if the distribution of forces acting on an object changes. This can result in a change in the direction and/or rotation of the object.

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