12 Replies Latest reply on Oct 18, 2013 2:58 PM by Phil Perlich

    Beam deflection does not match hand calculations

    Phil Perlich

      I am trying to validate a beam analysis against hand calcs, but I am getting differing results. So I set up a simple simulation to compare to have a base line.

       

      Using weldments I modeled a 100" long S8 x 23 beam with a 25,000 lb point load in the center. Allowing the beams ends to rotate (simply supported) I get a very reasonable error of 3.3% on the deflection in the center, however with fixed ends the error is 21.3%. Why is the FEA computed deflection so much greater than the hand calcs?

        • Re: Beam deflection does not match hand calculations
          Anthony Botting

          Hi Phil: It might be due to the point load. Try the same beam hand calculation and FEA but use a distributed load all along the beam. If that looks good, then try the point load again, but use half the beam with a symmetry end condition (and half the point load). I have done this multiple times and found values agree similar to your 3% number. It don't really consider the percentage an 'error' though - just a difference (both hand calcs and FEA make assumptions).

            • Re: Beam deflection does not match hand calculations
              Phil Perlich

              Thanks for your help Anthony.

               

              Just to recap: 100" long S8x23 beam with 25,000 lbs applied (concentrated or evenly distributed per table below). Here are my results:

               

              results.JPG

              As you can see, the problem seems to have something to do with the fixed supports.

              Also, I did not see a symmetry option so I used reference geomtery fixture type to achieve the same effect.

               

              Any thoughts on what is going on here?

                • Re: Beam deflection does not match hand calculations
                  Jared Conway

                  just for giggles, have you tried this with solids and shells?

                   

                  also, something to check is the assumptions made by the hand calculations. they may not match the assumptions made in your analysis or by the software BCs.

                    • Re: Beam deflection does not match hand calculations
                      Phil Perlich

                      Using soild elements and symmetry the deflection for the beam with fixed ends and concentrated load is 0.0807" which is virtually the same result as the beam elements used above. The only difference was the time required to mesh and solve, as you might imagine.

                       

                      This isn't even the structure I need to analyze. This was my attempt to verify that on the simplest of simulations the two types of calculation would be comparable. The hand calculation assumes either perfectly fixed or perfect simple supports, which should match the type of restraint used in the respective simulations. I also triple checked that the material properties are consistent between the simulation and the hand calculations.

                       

                      EDIT: With shell elements the result is 0.0837" for fixed end beam with the load concentrated in the middle.

                       

                      Message was edited by: Phil Perlich

                • Re: Beam deflection does not match hand calculations
                  Phil Perlich

                  Here are the files I have been using. For the shell element study the web has a 0 offset at 0.441" thick, and the flanges have a 0.5 offset at 0.426", and you have to make sure the elements are oriented the correct direction on the flanges as i'm sure you know.

                   

                  I have been using Civil Engineering Reference Manual 6th Edition by Michael Lindeburg, P.E. page 12-31 cases 4, 5, 7, and 8. I am running SW2013 SP3.0.12-31.jpg

                    • Re: Beam deflection does not match hand calculations
                      Anthony Botting

                      Hi Phil: I tried the solid mesh half-beam w/symmetry and half load and matched your answers.

                       

                      So, that left me with an applicability question because I noticed the beam length-to-web height ratio as 100/8 = 12.5 is a bit low for B/E application.

                      So just for a check, I doubled the length of the beam to get a ratio of 25, and ran that.

                      Now, when I look at the results, it appears to match B/E theory within 4% (default mesh size).

                       

                      I researched the applicability for length-to-web height ratio limit and found that B/E theory underpredicts the deflection if that ratio is less than about 20 due to transverse shear strain, which apparently the FEA is capturing for us. This statement seems to corroborate results findings for the fixed-end conditions. I suspect the corroboration is better for this case (compared to the ss-end condition) due to the heavy shearing at the ends of the fixed restraint.

                      I believe that's the issue, but let me know what you think!

                      Tony

                        • Re: Beam deflection does not match hand calculations
                          Jared Conway

                          Anthony, thanks for doing the research on that one. That is kind of what i was figuring. I ran into a customer case with a cantilever beam case. The equation they were using was only for very long beams and their beam was very short. Once they used the right equation, the calculations matched the software. This is where writing down assumptions both on the software and calculation side helps!

                          • Re: Beam deflection does not match hand calculations
                            Phil Perlich

                            Tony,

                             

                            Do you have some references that you could post links to?

                            I don't doubt you, but I am very intrigued and would like to learn more about this. I can't believe I've never heard of this beam-length:web-height ratio limit before! I have always known that you have to know the limits of you model (formulas, assumptions, etc.) but I wouldn't have considered this beam to be too short. Thats why I was so stumped when such a seeming simple problem didn't turn out as expected. I am glad you figured this out.

                             

                            Jared, as always, thanks for your help too.

                             

                            Phil

                              • Re: Beam deflection does not match hand calculations
                                Anthony Botting

                                Hi Phil: It seems a bit elusive to find, but I do remember it from school books, and I did locate this from Wiki:

                                 

                                Euler–Bernoulli beam theory does not account for the effects of transverse shear strain. As a result it underpredicts deflections and overpredicts natural frequencies. For thin beams (beam length to thickness ratios of the order 20 or more) these effects are of minor importance. For thick beams, however, these effects can be significant. More advanced beam theories such as the Timoshenko beam theory (developed by the Russian-born scientist Stephen Timoshenko) have been developed to account for these effects.

                                 

                                The link to that article is:

                                https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory

                                  • Re: Beam deflection does not match hand calculations
                                    Phil Perlich

                                    I found a section in Roarks on the topic. It is section 8.10 in the 7th edition. Roarks claims that the deflection due to shear stress becomes neglible at s:d=24.

                                     

                                    I used Roarks to calculate the additional deflection due to shear and added it to the Euler deflection to get a deflection of 0.0854" for the fixed beam w/ concentrated load. This puts Roarks and SW Sim within 3.9% of each other. I did the same for the simply supported beam and the differnce for that case was 3.1%.

                                     

                                    EDIT: Roarks never mentions Timoshenko in the explaination or procedure, but his work is listed in the Chapter 8 references.