Theoretical Question - Planar Truss Problem

[comhedoisl]

Hi all.

This is an interesting theoretical question from a book of Feodosiev - Advanced Stress and Stability Analysis. It reads like this:

"A plane truss consisting of n>2 equal and equally spaced rods, connected in a common node, O.
The force P acts in the plane of the truss.
Show that the displacement of the node O is always directed along the force P and that the value of this displacement does not depend on the angle /alpha."

The book does provide an answer to this problem, but not with enough detail -- at least from my point of view. I'd like to have a more detailed discussion on this problem with other folks. Could you help me providing your insights?

**the blue circle is not part of the structure



Thanks.

 

[JStephen]

A quick check through, I may be missing something:
First, you have to assume "small deflections". Let x be the angle between force P and the spoke in question.
In that case, the deflection in each spoke will be cos(x) times the deflection in the direction of the force, and the component of force in each spoke, parallel to the force P, will be proportional to (cos(x))^2. Adding up the cosine-squared for N equally spaced spokes will give a constant, regardless of angle alpha. Example: Adding cosine squared for 20, 140, and 260 degrees is same as adding cosine squared for 30, 150, and 270 degrees, etc.

And, adding up the components of those same forces perpendicular to the force will give a cos(x)*sin(x) term for each spoke. And adding cosine x sine of 20, 140, 260 degrees gives zero. IE, the component of the spoke reactions normal to force P total up to zero, so deflection and force are always aligned.

 

[GregLocock]

You've drawn a bicycle wheel, so the empirical answer is yes.

 

[HTURKAK]

You can use vector algebra . Lets assume ( for simplicty ) n=4
In this case , the angles btw the spokes will be 90 degr. Then define a Cartesian coordinate system having the origin is at pt. O and , X and Y axis matching to the spikes.
Then define a force F having random angle ( α ) with X and find the components Fx and Fy.
-Fx is resisted by spikes on the X axis and one will have tension Fx/2 and the other compression Fx/2 and total displacement will be ( δx ) and for lineer elastic rods will be proportional to Fx taht is axial stiffness of the rod K= E*A and Fx= K*δx ,
-Fy is resisted by spikes on the Y axis and one will have tension Fy/2 and the other compression Fy/2 and total displacement will be ( δy) and for lineer elastic rods will be proportional to Fy that is axial stiffness of the rod K= E*A and Fy= K*δy ,
- Then draw the displacement vectors and find the resultant δ , which will have the same direction with force F .
Important point is ,number of spikes n must be at least 3. If n=2 , and the load is not parallel to spikes ,the system will be geometrically nonlinear ( although the spikes are lineer elastic )and the final displacement will not be parallel and proportional with the force .

 

[goutam_freelance]

Suggest to use strain energy method