Find Tensile Stress in Anchor Bolts 2

[michaelb7]

I have a decoking valve that is mounted with anchor bolts. The valve is experiencing forces and torques in many directions. I need to find the tensile stress and shear stress in each anchor bolt. To me it makes sense to break up the problem into finding the tensile stress, then finding the shear stress. For now, I’d like to focus on finding the tensile stress in each bolt. Based on the shape/bulkiness of my decoking valve, I’m assuming that it is rigid compared to the bolts.

I’m not looking for a numerical answer yet, I just need help with my method. I’d like to solve a simplified version of this just to make sure I’m thinking about it correctly. Here is a simplified version:

A rigid unit cube is supported by 4 bolts at each corner. An arbitrary vertical force is applied somewhere on the cube. The bolts all have the same spring constant k. Find the reaction at each bolt.

Net forces in each direction:
ΣFx = 0; ΣFy = 0; ΣFz = 0

Net moment about arbitrary point P on x-y plane
ΣMPx= 0; ΣMPy = 0; ΣMPz = 0

I choose to do moments about an arbitrary point P rather than about the origin axes. I did this so that the forces from the reactions are not eliminated from the moment equations. (Note: I didn’t bother to expand out ΣFz = 0; ΣMPx= 0; ΣMPy = 0; into the forces, reactions and distances. I figured it could easily enough be done, but it wasn’t the part I needed help with.)

There are no forces in x or y directions and no moment about the Pz axis. We now have 3 equations.
ΣFz = 0; ΣMPx= 0; ΣMPy = 0;

And 4 unknowns (the vertical reactions at each bolt):
A, B, C, D.

I draw a free body diagram of each bolt. Since each bolts has the same spring constant:
A = kδA; B = kδB; C = kδc; D = kδD;


Now I have 7 equations:
ΣFz = 0; ΣMPx= 0; ΣMPy = 0; A = kδA; B = kδB; C = kδc; D = kδD;

And 8 unknowns:
A, B, C, D, δA, δB, δC, δD

(the spring constant of the bolts, the arbitrary force location and magnitude, and the location of P are all knowns)

Now I’ve reached the step where I’m having difficulty with this problem. I have 7 equations and 8 unknowns. I need more equations. Since the cube is a rigid body, the bolts must all lie on the same plane after they deform (there is some plane that could rest on the top of each bolt). I know the generalized equation of a plane is:

ax + by + cz + d = 0

I re-arrange the equation so that the plane is expressed as a function of z:

z = -(a/c)x-(b/c)y-d/c

My questions are:

1) It seems to me the deformation of each bolt (δA, δB, δC, δD) can be expressed as z from the equation: z = -(a/c)x-(b/c)y-d/c
Is this correct? It gives me 4 more equations:

δA = -(a/c)x-(b/c)y-d/c
δB = -(a/c)x-(b/c)y-d/c
δC = -(a/c)x-(b/c)y-d/c
δD = -(a/c)x-(b/c)y-d/c

How to I put these 4 equations into a more useful form?

2) I believe that -a/c is the slope of the plane in the x direction and -b/c is the slope in the y direction and -d/c is the offset from the z axis. I think I should either:

a) combine -a/c, -b/c and –d/c into their own variables (-a/c = L, -b/c = M, -d/c = N) so that each bolt is deforming δ = Lx + My + N

b) express –a/c, -b/c and -d/c as some trigonometric relationship based on the geometry I have.

c) do something different, what do you guys recommend?

After I get these final four equations nailed down I’m thinking I’ll have 11 equations and 11 unknowns, then I can solve them with a linear algebra solver. Once I’m convinced I’m doing the simple case correctly, I can move on to my actual calculation. Any suggestions or comments are greatly appreciated!

 

[bowlingdanish]

Honestly I think you have your time and priorities all out of whack. I'm all for some mathematical wizardry but creating yourself 11 equations for some anchor bolts seems a bit silly. If you have a dynamic system mounted on anchor bolts I think you're better off looking at concrete failure modes rather than the steel anyhow. If you post a proper description of your actual situation there's some clever people around that will sort help sort it out.

For what it's worth, thinking about it, my own opinion as a rookie would be that a rigid body would distribute load to each restraint based inversely on the distance the load is applied from that restraint. Is that what your equations say?

 

[JStephen]

For tanks, vessels, and similar objects, the assumption made to find anchor bolts loads is to treat the load distribution in the bolts as thought the bolt pattern were an equivalent beam, so you get My/I - P/A, etc., where I and A are for the total bolt pattern.

This has been refined to use a composite section to account for variations in concrete vs steel deflection, but it's doubtful in my mind if that is really an improvement in accuracy of the results, as the assumptions used in design don't seem to be any more accurate.

For an anchored cube, it would be reasonable to assume it was going to tip about an edge, calculate moments accordingly.

You can complicate this as much as desired, but if the loads are very approximate and you're going to use a very approximate analysis, you might as well use a simple approximate analysis.

 

[Ron]

As JStephen noted similarly, your valve just becomes a rigid baseplate...analyze accordingly.

 

[michaelb7]

I've attached a picture of what I'm actually trying to size the bolts for. This diagram is showing the worst case loads and moments at each flange face. There will be bolts in the holes on the baseplate.

I agree that I should probably find a simpler way to do this.

JStephen/Ron, I'm trying to wrap my head around treating the bolt Patten as an equivalent beam. Can you guys point me to a resource for a rigid baseplate analysis?

 

[Ron]

AISC has a baseplate analysis procedure that's clear and concise.

 

[michaelb7]

Thanks Ron. I'm going to have to read through that and chew on it a while.

 

[BadgerPE]

I have used the attached document for anchoring equipment before when given loading from the manufacturer.

 

[JLNJ]

First I would attempt to simply add the X and Y moment bolt tensions directly and see what kind of magnitudes I'm looking for the corner bolt.

If it can be seen that the nominal bolt sizes which go with your pre-drilled holes will work just fine, then a more precise answer is not required.

 

[michaelb7]

JLNJ,

Believe it or not, that was the first thing I attempted! It didn't seem like the proper thing to do since I figured it was a statically indeterminate problem. Do you think that method would give a conservative result?

 

[michaelb7]

I was able to solve my simplified version (this will be my last post, I just want to wrap things up). I'd like to do 3 things:

1)Show the method of my solution
2)State how I would apply this to my real life problem
3)Show weaknesses of this method


1) The key was to write the plane equation in the forum of δ = Lx + My + N. Then evaluate it at the x and y locations of the bolts.

Continuing from where I left off:

δA = Lx+My+N
x = 0, y = 0
δA = L(0)+M(0)+N
δA-N = 0

δB = Lx+My+N
x = 0, y = 1
δB = L(0)+M(1)+N
δB-M-N = 0

δC = Lx+My+N
x = 1, y = 1
δC = L(1)+M(1)+N
δC-L-M-N = 0

δD = Lx+My+N
x = 1, y = 0
δD = L(1)+M(0)+N
δD-L-N = 0

Now we have 11 equations and 11 unknowns!

The Equations:
ΣFz = 0; ΣMPx= 0; ΣMPy = 0; A-kδA = 0; B-kδB = 0; C-kδc = 0; D-kδD = 0; δA-N = 0;
δB-M-N = 0; δC-L-M-N = 0; δD-L-N = 0

The Unknowns:
A, B, C, D, δA, δB, δC, δD, L, M, N

The only thing left to would be to write out the equations and put them into this:

[11 X 11 matrix][11 X 1 matrix that contains the unknowns] = [11 X 1 matrix that contains the constants]

Now its just matter of plug and chug.


2) As far as finding the tensile stresses in bolts for the real life valve, I don't think the problem is too different. The forces in the x and y directions in the real life valve will cause some additional moments, and we have the pure moments to account for. Its just some more work with writing out all of the equations and plugging everything in.

I still think it is best to break up the bolt stresses into tensile and shear forces. I realize it could be done all at once, but the math seems more manageable by breaking it up into tensile and shear portions.

3)Weaknesses of this method.

I see several practical problems with this method. Let me list them:

a)If the bolts end up in compression, the k value of compression would be of the concrete not the bolts

b)Finding k values of bolt. If I knew the length, thickness, and modulus of the bolts I believe I could come up with an approximation of the k value. I'm not sure how being in concrete affects the bolt, and allows them to stretch.

c)Local deformation of the plate in the immediate area around the bolt. The mounting plate isn't truly rigid.


Other than the reasons mentioned above, I don't think there is anything technically wrong with what I'm doing. It’s pretty much by the book statics and mechanics of materials. I believe it would give a reasonable approximation for sizing the bolts. To me, it doesn't seem like that much work to set up the equations. Admittedly there is lots of room to make errors this way.