Calculating the force distribution on a triangles points

[Marni]

Hi,
I'm getting stuck on some mechanics and trig and was hoping someone could point me in the right direction. Essentially the problem is:

A triangular platform is supported at its corner points A, B, & C. A downward force F is applied to the platform at point P. Calculate the force that would be measured at points A, B, & C.

Given the corner positions A, B, & C: I'm able to work out point P and Force F, if I know the forces at the corners and their positions. I can't seem to work out how to go from knowing the force F and the points P, A, B, & C to working out the size of the forces at those points.

Any help would be great, thanks in advance!

 

[wjm11]

Hi,
I'm getting stuck on some mechanics and trig and was hoping someone could point me in the right direction. Essentially the problem is:

A triangular platform is supported at its corner points A, B, & C. A downward force F is applied to the platform at point P. Calculate the force that would be measured at points A, B, & C.

Given the corner positions A, B, & C: I'm able to work out point P and Force F, if I know the forces at the corners and their positions. I can't seem to work out how to go from knowing the force F and the points P, A, B, & C to working out the size of the forces at those points.

Any help would be great, thanks in advance!

One approach:

Write a system of equations based on rotational moments about each support point. To do this, view the triangle edge-on from three different sides.

For example, the moment about A must be zero; choose an edge-on perspective such that B and C are aligned (on top of one another), and determine the apparent distance from P to A (the distance from P to a line through A that is parallel to BC) and from BC to A. Now write a moment equation based on these distances and the force P and the sum of reactions at B and C.

Repeat this process twice more (once for B and once for C). You'll now have a system of three equations and three unknowns. Solve.

 

[DrPhil]

Hi,
I'm getting stuck on some mechanics and trig and was hoping someone could point me in the right direction. Essentially the problem is:

A triangular platform is supported at its corner points A, B, & C. A downward force F is applied to the platform at point P. Calculate the force that would be measured at points A, B, & C.

Given the corner positions A, B, & C: I'm able to work out point P and Force F, if I know the forces at the corners and their positions. I can't seem to work out how to go from knowing the force F and the points P, A, B, & C to working out the size of the forces at those points.

Any help would be great, thanks in advance!

The (vector) sum of all forces must be zero, so $\displaystyle F_A + F_B + F_C = -F$.
Since $\displaystyle F$ is downward, the other 3 are upward. This equation gives a scaling factor for forces at A,B,C but not their relative strengths.

Find the first moments at point P. The vector sum must be zero, since the triangle is not twisting. There is no moment at P due to force $\displaystyle F$ (since there is no lever arm), and the other three will have to be added as vector cross products:

$\displaystyle F_A \times \vec{PA} + F_B\times \vec{PB} + F_C\times \vec{PC} = 0$

All 3 of these axial vectors lie in the plane of the triangle, so there are two scalar equations.

 

[Marni]

Thanks for the replies.
I'm trying to work out how to get the problem in the form below:

F_A = F * (A Ratio)
F_B = F * (B Ratio)
F_C = F * (C Ratio)

Do you know how to calculate these ratios? I know its based on the positions relative to point P, and it makes sense that it uses vectors rather than just distances.
Based on wjm11's reply, would it be:

F_A = F * ((PB/AB) * (PC/AC))
F_B = F * ((AP/AB) * (PC/BC))
F_C = F * ((AP/AC) * (BP/BC))

 

[DrPhil]

Thanks for the replies.
I'm trying to work out how to get the problem in the form below:

F_A = F * (A Ratio)
F_B = F * (B Ratio)
F_C = F * (C Ratio)

Do you know how to calculate these ratios? I know its based on the positions relative to point P, and it makes sense that it uses vectors rather than just distances.
Based on wjm11's reply, would it be:

F_A = F * ((PB/AB) * (PC/AC))
F_B = F * ((AP/AB) * (PC/BC))
F_C = F * ((AP/AC) * (BP/BC))

So you know that (A Ratio) is 1 if P=A, and is 0 if either P=B or P=C, which fits with your hypothesis. HOWEVER I don't think you can get by with just the magnitudes of the distances -- I think you will need to separate out the x- and y-components, and you will end up with a system of 3 equations in the three unknowns.

I would translate the origin to be point P, and let the vertices be

$\displaystyle A = (x_1, y_1),\;\;\;\;\;\;B = (x_2, y_2),\;\;\;\;\;\;C = (x_3, y_3)$

Then by either of the two procedures, you should get this set of equations:

$\displaystyle \begin{array}{rrrcc}x_1F_A & +x_2F_B & +x_3F_C & = & 0 \\ y_1F_A & +y_2F_B & +y_3F_C & = & 0 \\ F_A & +F_B & +F_C & = & -F \end{array}$,

where the first two are the components of the moments, and the 3rd is the sum of forces.

 

[Marni]

The problem is I can't seem to figure out what to do after I've worked out those equations. I know all forces balance out as the platform is in equilibrium.

I need to get three equations in the form:

F_A = F * (Ratio_A)
F_B = F * (RatioB)
F_C = F * (RatioC)

Where: Ratio_{A +} Ratio_{B +} Ratio_C = 1

Am I right in thinking we should be able to work out the ratio from just the position of the corners and the position of the force? It seems fairly simple to go the other direction with the problem, i.e. you know the 3 forces at the corners and their positions, and you need to work out an average position of the forces.

Edit: Thanks so much for the help, silly these mental blocks you can get. I've figured it out now.

 

[DrPhil]

The problem is I can't seem to figure out what to do after I've worked out those equations. I know all forces balance out as the platform is in equilibrium.

I need to get three equations in the form:

F_A = F * (Ratio_A)
F_B = F * (RatioB)
F_C = F * (RatioC)

Where: Ratio_{A +} Ratio_{B +} Ratio_C = 1

Am I right in thinking we should be able to work out the ratio from just the position of the corners and the position of the force? It seems fairly simple to go the other direction with the problem, i.e. you know the 3 forces at the corners and their positions, and you need to work out an average position of the forces.

Edit: Thanks so much for the help, silly these mental blocks you can get. I've figured it out now.

Did you get final results like

$\displaystyle RatioA = \dfrac{|B\times C|}{|A\times B| + |B\times C| + |C\times A|}$

$\displaystyle RatioB = \dfrac{|C\times A|}{|A\times B| + |B\times C| + |C\times A|}$

$\displaystyle RatioC = \dfrac{|A\times B|}{|A\times B| + |B\times C| + |C\times A|}$ ?

That is a guess, just from looking at the matrix of coefficients in the system of equations - but I didn't try it on paper.