Determination of Center of rotation for indeterminate system

[Alexey881]

Hello fellow engineers!
I have a trouble determining axis of rotation for a few particular systems suspended on the wall, especially when multiple connections exist such that the system becomes statically in determinant. (See the file I attached). I've been looking in a few different books; Many have conflicting information. For example, ASHRAE Seismic Restraint Guide says that if the force is applied in the center of gravity of the unit then rotation happens about midpoint of the length of the unit. AISC-07 says if the bolt group is loaded by a force as shown then the center of rotation is about the center of the retaining plate (same as ASHRAE), I've also looked at Mason Industries book (example not included in the image - forgot) that if there is a steel angle attached to the unit ( the angle has 2 rows and 2 columns of anchors) then when the prying force acts, the angle will be rotating about its edge, rather than about the center of the angle (same methodology is applied to isolators with 4 or 6 holes - the rotation is always about the edge).

There is quiet a difference in maximum tension acting on a bolt if the center of rotation is in the middle or at the edge. (For example in Case 2 on my attachment, if the center of rotation is about middle bolt then only top bolt and the wall are resisting the moment thus M=R1*(2*b/4)(The 2 here is due to the fact that R1 is coupled with wall pressure on the rack)........."or could write a relationship where the wall has a triangularly distributed force acting on the rack, and the top bolt has a point force acting on the rack"............ If center of rotation is about bottom edge of the rack, then all the forces R1 R2 and R3 are oriented in the same direction and are of proportional magnitude, thus M=R1*(3*b/4)+R2*(2*b/4)+R3*(b/4)

Now, my suspended systems are often attached to plywood walls, metal studs, as well as hollow blocks. These walls are quiet weak and therefore it is extremely desirable to be able to reduce and minimize the tension.

What would you suggest? What flaws do you notice?

Also, do you think adding a cross brace would have any effect on center of rotation?

Thanks alot for the help!

 

[flash3780]

I think there is some confusion when I read the term "center of rotation". The part isn't rotating, so there is no center of rotation.

However, if I understand you correctly, you're trying to sum the forces and the moments to determine the forces on the L-shaped free body. Remember that you can choose any point in the universe to sum moments about and they should equal zero for a static, rigid body. You might choose the point strategically to remove some forces from your moment balance equations (for example, you may choose to sum moments about point 3 in the FBD below to eliminate F_3x, F_3y, and F_2y from your equation). For starters, consider creating a free body diagram for this problem... something like this:

Note that I prefer to draw all of my force vectors in the positive direction and fill in the proper relation in my equations (e.g. F_1y = -mg). This helps me to keep my signs straight (F_{sum y}= F_1y+F_2y+F_3y; M_{3 sum}=b/3*F_4x-b/3*F_2x-L/2*F_1y; etc.). For the record, I always like to assume that counterclockwise is positive when it comes to moments.

You'll have to make some further assumptions when solving this problem. You might assume, for example, that F_2y=F_3y. Note that I've also assumed that point 4 carries no vertical load in my FBD example. Once you've got enough assumptions to write an equation for each of your unknowns, you can solve for your forces. Bada-bing.

As far as your question about whether adding a brace will change the force distribution... yes and no. If you make a rigid body assumption when doing your force balance, then no, the brace will make no difference. However, if the part deflects significantly without a brace, this might not be a good assumption. Regardless, I think the rigid body assumption is a good place to start. I think that if your design is successful, the deflection of the L-shaped part will be small. Start with the forces from a rigid body analysis and then determine the deflections (see beam theory).

I hope that explanation helps a bit. Good luck.

FBD:

http://files.engineering.com/getfil...1-4d71-b7c5-e75cc4ffb268&file=FBD_example.png

 

[flash3780]

Whoops... need to correct a typo in my sum of moments about point 3 example. Should be:
M_{3 sum}=b/3*F_4x-b/3*F_2x+L/2*F_1y

Wish there was a go-back button on this site.

 

[Alexey881]

Hey, flash. I understand what you're talking about but it seems to come down back to assumptions, as you noted yourself. The whole goal is to move away from assumptions and figure some sort of systematic way. Because of the variety of the possible available assumptions it is hard to pin point what is valid and what isnt, especially if you're trying to make the most efficient design. For example, 1) I can assume triangular tensile distribution starting at the outmost edge (on the bottom of the rack) and going throughout all the bolts. Then I will be able to relate forces through similar right triangles theorem. 2) I can assume that half of the rack against the wall is in compression and the other half is in tension (being held by half the bolts) then again it is possible to relate the forces through similar triangles for the upper bolts. However if I am to compare the answeres you will find that for a given system the #2 will be about 50% more conservative. In other words maximum tension will happen in upper bolt of system#2. Now if I was to solve the same system with assumption #1 then I might get my system passing the criteria, but #2 does not. Hence the delemma, how can you just assume any distribution and be sure of its validity?

Also when I talk about center of rotation, I mean that instantaneously the system will try/tend to rotate about some specific axis just like any lever or any prying calculation would tell you (but it is true that you may count it as an assumption)

 

[flash3780]

You're right that it comes down to assumptions, any analysis does. To your point about the reaction load, my initial FBD assumed the load is concentrated at the toe of the joint. It may be more accurate to assume that the reaction load is distributed as a triangular line load if the joint is clamped.

You could imagine the joint behaving in both ways depending on whether the bolts are loose or tight... or even somewhere in the middle. So, you may solve the problem with both assumptions to bracket the problem.

It's worth noting that the bolts cannot provide a force in the negative direction, so be sure to consider that in your calculations (F_2x>0; F_3x>0). To account for this, I might run the calculations as normal, and if any of the bolt loads were to come out negative I'd set them to zero and run the calculations again.

Note that the bolt loads being solved for are external loads on the bolts only. They are not accounting for clamp loads.

Again, I don't think that a center of rotation will come into your calculations... I think the approach that you want to take is to think about the external loads on your free body, then use Newton's first law to set up force relations (i.e. set the sum of forces and sum of moments on the free body to zero). Best bet is to choose points to sum moments about which eliminate forces from your equations to make the math easier... but the points that you choose to sum moments about will not change the answer. The answer is solely dependent on the external loads on the free body.

Line Load Diagram:

http://files.engineering.com/getfil...-461c-9f6f-f51fa591936f&file=FBD_reaction.png

 

[flash3780]

Gah, another correction... the bolts can't provide a force on the free body in the positive direction, only in the negative direction as I've defined it (F_2x<0; F_3x<0).

 

[Engdoitbetter]

Dear all,

I don't know if this can be of any help to you but... As far as bolted connections undergoing bending are concerned, you might look for the neutral axis of the bolted section: compression is withstood by the plate, while tension by the bolts, so on one side of the neutral axis you have to consider only the bolt cross sections, while on the other one you have the plate with the holes.

Hope it helps,

Stefano