Analysis of Statically Indeterminate, yet very simple, Bracket 10

[nargonaut7]

Greetings everybody,

I am sorry that my question has already been presented in this forum, but the previous answers, at least those two that I have been able to find here, are not adequate, as I explain later.

My situation, as shown below, is that of the second-simplest possible structure: a two member-frame attached to a wall (this is the second-simplest structure because the two absolute simplest structures have only one member, which is either a cantilever beam or a single-standing column, with their supporting wall or floor, respectively; by the way, I am writing all this not with the purpose of giving a lecture but only to make my point as clearly as I can). So, it really bothers me that excellent books such as Roark’s “Formulas for Stress and Strain”, “Structural Engineering Handbook” by Mahamid and others, and “Structural Engineering Formulas” by Mikhelson and Hicks, these three books include formulas for much more complicated frames, when this simple bracket support I am showing here, which is used in industrial plants all over the world, is completely ignored. Neither this two-member frame is shown as an example in multiple books where I reviewed the Three-Moment Equation, the Moment Distribution Method, Castigliano Theorem, and the Slope-Deflection Method, to verify if I could apply any of them to my situation. A severe disappointment I got was with the Slope-Deflection Method, when the determinant of the 4 x 4 matrix of the coefficients of the variables, calculated with the particular dimensions and the weight “P” of my application (none of these parameters are needed to be indicated here) turned out to be zero. Another disappointment was when I used Roark’s Tables 8.8 and 8.9, to combine axial and transversal deformations using the fact that the deformations of both members at the joint “C” are the same. By trial and error, I found multiple combinations of the three internal reactions (two forces and one moment) at that joint that make those two vertical deformations equal, so that didn’t work either.

I thought I was close to calculating the reactions in this bracket, when I found the “Air Force Stress Analysis Manual”. Its 579-page PDF file can be easily downloaded for free by searching in internet with those words. The reactions and moments I calculated for a triangular frame (Table 5-5) with all three joints fixed and with an external moment applied (Case 8) satisfied all the Statics summation of forces, but this didn’t happen with an external load (Case 9), so I couldn’t combine those two cases to simulate my bracket, as was my plan. That Case 9 didn’t work even when I tried to solve the exact situation they show, which is that both members are inclined and that the load is applied between supports, not in the overhang as in my situation. So, when I couldn’t even solve the standard problem shown in the diagram of Case 9, plus my failures calculating with Roark and Slope-Deflection, as described earlier, that was when I realized that I need to ask for help. But, of course, these failures could be also because I made a numerical error.

If somebody wants to verify these formulas, or any other method, you can assume your own load, dimensions, and inertias for those two members, and check if the resulting reactions you get make all the Statics summations equal to zero.

Back to the previous answers, I have seen two of them, I want to mention this to save somebody’s time in replying. The first one is the advice to use an FEA software, which of course I will use, but I always want to see a manual calculation of forces and stresses, at least for simple cases like this bracket support, to be included in a design report together with software results. The second answer is the advice to assume all three joints as pinned. This simplification reduces the problem from statically indeterminate to determinate, and I already did that, then it becomes really simple, the maximum moment in AD is P times the distance CD, the same as if the joint “C” were a fixed support and then we only consider the overhang, I found that result surprising. This quick and simple approximation can be useful in many situations, we can always use a higher safety factor to cover us if the actual moments and stresses are higher than the calculation, but I am beyond this point, it has become a matter of honor for me to find the method or formulas to calculate all nine reactions with all three joints welded as accurately as practical. The caveat of being practical is important so, to clarify my position, the extremely long formulas of the Air Force manual and of Roark are fine with me.

Well, I will appreciate very much any help, and I hope somebody will find useful the four references I have included in this post. And sorry for writing so much.

 

[JStephen]

Anything that can be analyzed can be analyzed to a greater or lesser extent, and it is necessary to adjust the analysis to the situation.
That's where the pinned-all-around solution comes in. Otherwise, you can spend 3 weeks analyzing an item that will take a half-hour and $20 worth of material to make.
If you're just out of things to do, don't forget to analyze the semi-infinite wall itself to see what the stresses are in it. And what's the stiffness of that wall, and how much of the mass of that wall will participate when you work on the vibration problem where the wind is blowing the pipe that is supported?

 

[nargonaut7]

Well JStephen, I certainly don't appreciate the sarcastic tone of your response, starting with the "If you're just out of things to do", which evidently you can't be aware of what I need to do at my job, and then you continued your sarcasm with a long list of negligible factors that any engineer knows that their impact is extremely small. I am just looking for a formula, similar to those indicated in the references I listed. Just that those three books and the manual have formulas for frames more complicated than mine, should be proof enough to you that the formulas I am looking for are also in some book. You don't know how to calculate those reactions, but this doesn't mean that somebody has not published those formulas. But, since you have nothing useful to contribute to this discussion, it would be better for you and for the readers here if you save your energy to those discussions where you can provide something relevant. The concerning conclusion I get from your post, which is a reflection of the current social climate in the world, is that not even on something so neutral as an engineering discussion, we are able to leave aside sarcastic and demeaning answers and instead try to actually help other people. Finally, all the negligible-impact factors you mentioned, even if not practical, are very interesting to calculate, and would be evidence of powerful analysis skills from the person that would be able to calculate them. But again, I am only looking for the reactions.

 

[nargonaut7]

Oh! And I forgot to mention that of course you can't know if I need to fabricate thousands of those brackets, instead of "half-hour and $20" as you wrote, and then optimizing the solution is well worth.

 

[dik]

If you have 1000s of the little suckers, then do a simple 2D or 3D frame analysis. Easy to do. If you are still unhappy about the analysis, do a mock up and load test it.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

 

[nargonaut7]

Thank you dik for your reply. But that "simple 2D frame analysis" you mentioned is what I tried to do, but failed, with the formulas from the Air Force Manual, from Roark, and with the Slope-Deflection Method. So, I am running out of options, and I am a mechanical engineer, not civil or structural, so for example this Slope-Deflection method, which I imagine is a basic skill learned in a school, I learned it by myself in the past few weeks, but couldn't get a solution with it. I am saying this just to explain my limitations. I thought I had a good shot with the Air Force formulas, but Case 8 didn't work, and I couldn't find any error in the Excel spreadsheet I prepared to do those calcs.
And, by the way, I wrote about thousands of brackets just to show an example that I haven't given all the details of my application, I only gave the information that is needed to ask for a method of analysis. But I am curious if you would have any recommendation to make a load test, other than installing strain gages on those two steel members. I have zero experience using strain gages, but I have briefly seen them in books.

 

[BAretired]

I don't believe your response to JStephen was warranted. I would certainly design such a structure using pin joints.

If you require a thorough analysis, there is no reason why a frame analysis would fail to yield the correct answer. Same goes for the slope-deflection method, recognizing that the slope-deflection method does not consider axial deformations.

In real life, a purely fixed wall is impossible to find, so this would be an academic exercise, not the solution of a real structure.



BA

 

[Celt83]

you can get a free textbook on matrix analysis from here: [URL unfurl="true"]http://www.mastan2.com/textbook.html[/url]

 

[271828]

Here's another vote for using hinges at the member ends to make this thing statically indeterminate.

Anything beyond that is sharpening your pencil too much IMO. If you were to test the part and compare the results to your predictions, who knows how that would go. If you haven't done such a comparison before, then you'd likely be surprised at how bad the prediction would be, especially in the elastic range. Keep it simple.

 

[dik]

It shouldn't have failed. In what manner? If you check the results with statics, things should be fine. If your numbers are too high, or unreasonable look to your model. 2D or 3D should work. Due to the difference in stiffness, I'd consider the wall a rigid item. Something else, not the actual connection.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

 

[nargonaut7]

Thank you, BAretired for your reply.

My complaint to JStephen was not about his support in favor of the all-pinned approximation, but instead about the sarcasm he used, but that is fine, I have a thick skin, and can discriminate between useful and useless answers.

Well, both him and you pointed out to the value of analyzing this bracket as an all-pinned structure, and I also mentioned that this approach is useful in my original post, and also indicated its maximum moment result. If you would design this structure with this approximation, that is fine. But all this doesn't mean that a solution with all three joints fixed is useless because, as I have mentioned before, there are plenty of formulas for all-fixed joints frames in those books, the problem is that my bracket is not one of those.

And I am happy to read from your post that the slope-deflection method is applicable to this situation, because I thought that it is applicable and then learned about it in these past few weeks, so when I failed with it, is just due to my lack of experience with it. I wrote four slope-deflection equations for this bracket, and used them in four summations of moments (whole bracket, each member with their internal forces, and joint "C" with the external moment), and I was hoping to obtain the four unknowns (slope and deflection at "C", vertical reaction at "A", and horizontal reaction at "B"), but the determinant became zero, so I think that the equations are not independent, even if none of them is the multiplication or combination of others.

Now, you also wrote about a "frame analysis", but I don't know what method is that, please remember that I am a mechanical, not structural, engineer, so any reference to a book or any other indication in this regard would be appreciated. As I said, I also tried unsuccessfully to use the Moment Distribution Method.

Finally, I also agree with you that considering all three joints fixed is also an approximation, as those welded joints have certain degree of flexibility, even more that they don't have a plate welded to both members at each side of joint "C", and neither there are gussets welded to the steel columns at joints "A" and "B", but I imagine that all-fixed is a better approximation to reality that all-pinned.

 

[nargonaut7]

Thank you, Celt83, for your reply.

To get the determinants of matrices, I am simply using Excel, it has a formula that calculates them directly. Of course, studying matrix analysis as you recommended me would be interesting, but I would need to invest that time instead in studying structural analysis. As I mentioned earlier, I am a mechanical, not structural engineer.

 

[nargonaut7]

Thank you, 271828, for your reply.

And about what you wrote on "sharpening the pencil" I would look bad if I would submit the design report to our client, which is another company, with that all-pinned approximation, and then their reviewer would reply to me "oh, you didn't know that in such & such structural analysis book, they have the formula for this bracket with all-fixed joints?" As I wrote before, the Air Force Manual is "close, but not cigar", actually, very close, but they have a typo on Case 8, a formula that I mentioned in my original post, so I guess that this is why the calculation failed. And, let me also repeat here my frustration that the other books I mentioned have fixed-joints formulas for more complex frames, so I don't want to give up too easily.

 

[desertfox]

Hi nargonaut7

If you know the value of the force P ? If you do one way to analyse the bracket would be:- remove the strut and calculate the deflection of the cantilever beam and determine the deflection of the cantilever beam at the point where the strut would connect, then calculate the vertical force that would restore the beam at the said point to zero deflection, then armed with that vertical force you can resolve forces into the strut.

“Do not worry about your problems with mathematics, I assure you mine are far greater.” Albert Einstein

 

[nargonaut7]

Thank you, dik, for your second reply. Actually, you are the only one so far that has repeated.

You are right in your question, so I am sorry that I haven't indicated that, when I say that my calculation failed, is because the summation of moments at a particular point (from basic Statics that even a mechanical engineer like me knows), considering both externally applied and previously calculated, is not zero.

 

[centondollar]

You do not solve determinants when calculating end-moments (from end-rotations) and then shear force and axial force (by making free body diagrams) using the slope-deflection method. Determinants are related to buckling calculations.

Furthermore, it is not at all surprising that your case is not found in any textbook. Most real structures are not found in textbooks, since the amount of indeterminate parameters grows very quickly once the amount of joints increases. In your example, you have three equilibrium equations (SUM_FX=0 , SUM_FY=0 , SUM_Moment_about_point=0), but six unknowns (vertical and horizontal support reactions plus a moment at each connection to the wall, giving 2+1+2+1 = 6 unknowns), which means that you must solve the remaining three unknowns (two forces and a moment) by some method.

One method is the finite element method. Another method is the force method, which is explained in a structural mechanics textbook of your choice. There are a few steps to the force method (isolate statically determinate system and calculate M/N/Q, release the unknown forces and moments, apply unit forces and moments and calculate M/N/Q separately for each unit force loading the structure, use maxwell-mohr integrals to calculate flexibility coefficients, enforce compatibility, solve unknown forces and moments, superpose results), and in your case, the result would be a 3x3 flexibility matrix, 3x1 unknown force/moment vector, 3x1 known zero (to satisfy compatibility) displacement/rotation vector and quite many pages of calculations. The moral of the story: hand calculations quickly become too complicated to be of much use in practical design.

If you do not have time to study structural analysis (FEM by hand or force method by hand), you will most likely not be able to solve this problem by hand, and will have to make do with a finite element model.

PS. Keep in mind that unless your wall is very stiff, the welds are very thick and the members are stiff enough, significant bending moment will not develop at the wall-member intersections. The accuracy of calculations depends in large part on how accurately mathematical boundary conditions reflect the actual boundary conditions in the structure. Assuming pin connection at both connections to the wall leaves you with only one unknown force, the solving of which is much easier than your original problem; if you assume that the diagonal member is a bar (resists only axial tension/compression) connected in such a way that it pushes only against a perpendicular surface (some sort of slot), the problem is statically determinate and very easy to solve.

 

[nargonaut7]

Thank you, desertfox, for your reply.

I thought about doing what you wrote, but then searched and applied other methods, because I thought that the deflection I would obtain considering member AD as a cantilever beam would be much more than the actual deflection, and then the force to bring it back to zero deflection would be much higher than the real force. And, all this would be valid if the bending stresses are less than yield strength, so both members remain in the elastic range. However, as I am writing this, I am thinking that if I could have a way to estimate the actual deflection, then the force I would calculate would not be the high one to push member AD up so the deflection at point "C" is zero, but rather a lower and more realistic force that would make that deflection at "C" the real value. Even more, I just thought that I could have a curve of internal force at "C" versus deflection (considering the external force "P" which is a known value, constant) which, if I intercept this curve with another curve, would give me the solution at the point of intersection. But I can't imagine which the second curve would be, as it could not be for the strut, which has a much smaller inertia, so those two curves would never intersect. Food for thought!!

 

[rb1957]

you'd make your structure much simpler if you release the fixity at all the joints. AD is now a simple overhanging beam and BC would be a beam in compression.

To solve the structure given (which is redundant, with 3 redundancies) ... there are many methods, some of which I don't use either. It is possible to solve by hand, but very labourious.

the fixity at A and B means the beam does not rotate here, the angle to the ground remains the same (at A the tangent of the deflected beam is horizontal).
The fixity at C means the angle between the struts remains the same, but C will translate and rotate (relative to the ground).

One way to solve, iteratively. Solve AD as a cantilever, simple statics. Extract the deflection and rotation at C. Apply these to BC, which will create reactions at B and C. Now apply the reactions at C as loads onto the AD beam (reverse them), simple statics again. This will result in a different slope and deflection at C. repeat untill the differences are small

I don't have the USAF manual with me (on the road) so can't comment on that. Could you show case 8 and case 9 ?

another day in paradise, or is paradise one day closer ?

 

[BAretired]

I had a quick look at Table 5-5 in the Air Force Manual. It seems clear that their solution does not consider axial or shear deformation, because the only properties listed are the I values (moment of inertia). That means there could be a slight discrepancy between a frame analysis and the Air Force solution, likely negligible.

I think Case 8 and 9 could be combined to find the Air Force solution to your problem. I have not done that, but it should be fairly straightforward.

BA

 

[271828]

And about what you wrote on "sharpening the pencil" I would look bad if I would submit the design report to our client, which is another company, with that all-pinned approximation, and then their reviewer would reply to me "oh, you didn't know that in such & such structural analysis book, they have the formula for this bracket with all-fixed joints?" As I wrote before, the Air Force Manual is "close, but not cigar", actually, very close, but they have a typo on Case 8, a formula that I mentioned in my original post, so I guess that this is why the calculation failed. And, let me also repeat here my frustration that the other books I mentioned have fixed-joints formulas for more complex frames, so I don't want to give up too easily.

This bracket is not a typical configuration for which I would expect to find closed-form solutions in a handbook. If you must go beyond hinged connections, then the way to go forward is obvious: use a commercially available frame analysis program.

If I was a reviewer and an engineer submitted this to me with hinges and statics, I'd assume he knows what he's doing.

If he submits commercial program output, I'd assume he knows what he's doing.

If he submits slope-deflection manual calcs or matrix analysis outside of a commercial program, then I'd assume he's very inexperienced probably just coming out of school, and needs more supervision from his superiors. Sorry -- no disrespect intended.