Direct Stiffness Method for Analyzing Tripod Frame with Equally Spaced Legs

[gomerpyle]

Let's say you have part of frame contacting the ground, such as the feet on the base of the tripod. Assuming each leg is equally spaced around a center post, how would you go about analyzing this type of structure? I'm trying to teach myself how to use the stiffness method, but all the examples I find are simple 2D beam structures. Would I use polar coordinates instead of Cartesian to represent the legs being equally spaced? How would the transformation of the stiffness matrix to global coordinates work if polar coordinates were used?

 

[Divya Shyam Singh]

I think if i were to analyze this structure, i would draw Free body diagrams of all the three supporting structures, treat each of them like a beam if it is a case of 2D stress condition and calculate the tensile and shear stress in each using the basic equations of the beams. I would also take into account a uniformly acting load because of their weight.
Please do take care about the force direction. Most probably, we would have to calculate the components of the forces and solve the problems. In case there is a 3D state of stress, we can calculate the stresses by first ignoring the loads in one of the dimension(thus making it a 2D stress condition) and then calculating stress. Similarly ignoring the first load(thus again making it a 2D stress condition) and then superimpose the stresses on an element. Thus the final state of stress on an element will be 3D stresses only.

I am currently working on a similar problem. Let's discuss this further!

 

[Divya Shyam Singh]

Assuming each leg is equally spaced around a center post

The stresses in each beam will be exactly similar due to symmetry.

 

[gomerpyle]

Are you talking about using the equilibrium equations to solve for the forces? My problem is this structure is welded together and so the joints are fixed and not pinned, which in my mind would make the structure statically indeterminate, which is why I'm researching the stiffness method. Also the load in my case would be a vertical force on the center tube to which all the legs connect, so the only stresses experiences by the joints would be vertical shear, moments, and tensile forces.

 

[JBA]

Post a drawing of your structure.

 

[Divya Shyam Singh]

Assuming the joint to be rigid, the 3 beams can be approximated to be under 2 dimensional load and stress analysis can still be carried out. Whatever load is acting at the centre, can be divided equally among the beams because of symmetry. Remember that beams can only exert a reaction force along the axis of the beam, so three forces will be acting along the beam outwards. The vertical components of the beam will support the load while the horizontal component will be 120 degrees apart and will cancel out.
I strongly believe that the problem can be solved using this method.
What do you think?

 

[gomerpyle]

I probably should have included a drawing of what is in my head. This is not a homework problem of any sort, just something I thought up when learning about the direct stiffness analysis. If the tripod had crossmember reinforcements as shown (horizontal to ground) then I believe that would result in more reactions/unknowns than there are equilibrium equations. This is why I am using the stiffness method to examine this problem. I have not yet seen any examples in any text illustrating the correct approach for this type of structure.

 

[gomerpyle]

Here is the drawing

 

[JBA]

Just as a note, the analysis must include the assumed lateral restraint, i.e., the coefficient of friction, between the tripod feet and the supporting surface.

 

[gomerpyle]

Correct, but I'm not sure how applying that restraint would be worked into the stiffness method. The equation to solve for the nodal displacements is:
[f] = [K]{u}, where [K] is the master stiffness matrix comprised of the beam properties A, E, I, and L and [f] is the applied loading. Nodes are always assumed to either be fixed for a certain DOF and have a value of zero, otherwise they are left alone and solved for via {u} = [K]^-1[f].

You can solve for the reactions at the feet initially without having to go through any complicated math. The vertical reaction at each foot would have to be 1/3 of the applied load on the center post. Then the force of friction opposing the outward, 'radial' movement of the feet would be μ_s*(F/3). Knowing this, would I then use that value as an applied force in my force matrix for the above method to account for friction? That seems incorrect though as it technically is not an applied load.

 

[JBA]

gomerpyle,

What you are suggesting is straightforward an can be viewed as essentially "inverting" the problem by applying the loads at the feet and treating the top of the stand as a fixed reaction point rather than a load input point.

 

[gomerpyle]

That's what I tried initially, but what I found running ANSYS simulations was the results yielded noticeably higher stress levels using the above method than if I were to simply analyze the structure as a whole with all three legs connected to a center tube, and with a frictional contact on the feet. So somehow considering only one leg with the attachments to the tube as completely "fixed" seemed to over-constrain the problem and raise the stress values.