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Transverse Stiffeners - Torsion 6
- Thread starter ToadJones
- Start date
It may also be a gross example for the question asked for 271828, but may be useful to show the idea. Take a plate of constant thickness supporting some loads, simply supported at all edges. Imagine some party thinks it will be deflecting too much, and he decides to increase the thickness to counteract such excessive deflection; depending on from where we are starting, and to where we bring the thickness, it may well result that the overall deflection increases, and the structure in general be even worse than it initially was due to the addition of weight overcoming any gains in strength and stiffness coming from the addition of structural material.
I understand that in this case the addition wouldn't be judicious in the proposed extent, and what we have been (and somewhat are still) elucidating here is if such is the case with the addition of stiffeners, if not economically, from an structural viewpoint.
I understand that in this case the addition wouldn't be judicious in the proposed extent, and what we have been (and somewhat are still) elucidating here is if such is the case with the addition of stiffeners, if not economically, from an structural viewpoint.
I may also reconvert the example by Teguci to your satisfaction. Imagine our original cantilever is a symmetrical tee. So, instabilities apart, zero torsion and zero lateral deflection by design. Now imagine by whatever the not very reasonable reasons we decide to reinforce it by just adding a parallel plate welded to just one of the sides. Inmediately the asymmetry shows and the responses in terms of lateral deflection and torsion appear, so, augment. It is normally an unreasonable proposition, as was above to thicken so much the plate, but this is what we are thinking about at least theoretically in part of this thread, whether adding stiffeners may or not in some case make the structure to enter one of such pitfalls out of its modified setup.
Consider a WF laterally unbraced with uniform gravity load acting on it. The addition of a thin, deep vertical steel plate welded to the middle of the top flange would relieve the stresses in the beam but could cause lateral torsional buckling because the plate cannot resist the fiber stresses due to bending.
BA
I should have clarified the gauntlet. Monitor the same DOF for the before and after. Other than the variable plate, the other examples are introducing an asymmetry, so there is movement in some other DOF. The original DOF stiffness was unchanged or increased.
As for the variable plate, the stiffness for the same DOF certainly went up.
The gauntlet is worded with one DOF monitored because that is lime the problem at hand.
As for the variable plate, the stiffness for the same DOF certainly went up.
The gauntlet is worded with one DOF monitored because that is lime the problem at hand.
Well, I have made a line (frame element) model in SAP2000 and I find it less than informing for our matter, and requiring some spreadsheet or worksheet element to add to my judgement so after trying I suspend the thing for the moment in that path. It also seems that fixing the ends of the beam does not introduces restraint to warping there, to be verified.
So I return and rebuild anew entirely the Autodesk Simulation models that have no stiffener, and, essentially, I find no error in my previous trials that the new models have discovered. I can still remake as well those with stiffeners, but the likelihood is that my former conclusions of the stiffeners making the outfit show a bigger torsional response likely will stand as per AS seems to be seeing the thing. My heart is with AS in this matter, but I won't make professional statement on it till further clearance of the same. Algor was a package with all the certifications for the nuclear industry etc, and I have read that with the purchase by autodesk the buyers were at a loss to understand the pipepack module because those that made it were not already to explain. Knowledge is delicate matter to keep. This must not make in the eyes of anyone see that I am dismissing the two other programs' results nor stance in my own eyes. It is simply making a bet on who will be right.
See attachment.
You can make your own models with your own programs, I'll be happy to read of your results. I may use these two cases for benchmark with other FEM programs when I'll use them. The sooner may be the Inventor linear built-in solver, that I have used in some occasion a pair of times, or the old visualNastran if I have some PC with it still installed. For others (generic FEM packages), It will take me more time, have not used them.
Essentially, the benchmark problems are
1.
CONCENTRATED TORQUE AT MIDLENGTH
HEB 300 beam 7 m long, fixed at both ends, a pair of 1.9 tonne forces applied at midlength on flanges to create a concentrated torque
2.
UPPER FLANGE EDGE LOAD
Same HEB 300 beam 7 m long, fixed at both ends, but now with an edge distributed load at one of the edges of the upper flange of 3 tonnes on each meter.
Then I have introduced for the stiffened cases (quite proportionally thick) 15 mm complete stiffeners.
So I return and rebuild anew entirely the Autodesk Simulation models that have no stiffener, and, essentially, I find no error in my previous trials that the new models have discovered. I can still remake as well those with stiffeners, but the likelihood is that my former conclusions of the stiffeners making the outfit show a bigger torsional response likely will stand as per AS seems to be seeing the thing. My heart is with AS in this matter, but I won't make professional statement on it till further clearance of the same. Algor was a package with all the certifications for the nuclear industry etc, and I have read that with the purchase by autodesk the buyers were at a loss to understand the pipepack module because those that made it were not already to explain. Knowledge is delicate matter to keep. This must not make in the eyes of anyone see that I am dismissing the two other programs' results nor stance in my own eyes. It is simply making a bet on who will be right.
See attachment.
You can make your own models with your own programs, I'll be happy to read of your results. I may use these two cases for benchmark with other FEM programs when I'll use them. The sooner may be the Inventor linear built-in solver, that I have used in some occasion a pair of times, or the old visualNastran if I have some PC with it still installed. For others (generic FEM packages), It will take me more time, have not used them.
Essentially, the benchmark problems are
1.
CONCENTRATED TORQUE AT MIDLENGTH
HEB 300 beam 7 m long, fixed at both ends, a pair of 1.9 tonne forces applied at midlength on flanges to create a concentrated torque
2.
UPPER FLANGE EDGE LOAD
Same HEB 300 beam 7 m long, fixed at both ends, but now with an edge distributed load at one of the edges of the upper flange of 3 tonnes on each meter.
Then I have introduced for the stiffened cases (quite proportionally thick) 15 mm complete stiffeners.
ishvaaag,
Here is a reference which may provide further insight on the behavior of beams in torsion.
BA
Here is a reference which may provide further insight on the behavior of beams in torsion.
BA
- Thread starter
- #70
BA has been talking about classical Saint-Venant torsion; this assumes that the angle of twist is small, the elastic limits of the material are not exceeded, and that the shape of the cross section does not change. The section just rolls (rotates), but retains it shape, and this is generally true for most WF shapes, or distortions in shape are so small they can be neglected, thus the web stiffeners we are talking about will not appreciably improve the torsional capacity of these beams. If some of the member components are thin or tall and slender, such as some webs, plate girders, the shape of the cross section is likely to distort due to the warping torsional moment, just as SAIL3 suggests. This is called the ‘Goodier-Barton effect’ of web distortion, and in some shapes and cases might have to be considered. (See J.N. Goodier and M.V. Barton, “The Effects of Web Deformation on the Torsion of I-Beams,” Journal of Appl. Mechanics, 11, MAR1944).
St. Venant torsion assumes that the cross section shape doesn’t change, but is free to warp, this is a uniform or pure torsion, represented by St.Venant’s soap film analogy. But, since most members are not free to warp under torsion, normal stresses exist or are induced, particularly in the flanges, due to the bending of the flanges. Thus, both St. Venant (shear stresses) and warping torsional stresses (normal & shear stresses) occur together in the member. In addition, there will be the regular normal and shear stresses in the member due to bending as a beam, gravity loaded, about its shear center. All of these stresses must be combined, in appropriate fashion, to start to approach the output of a FEA. And, on a problem like this the FEA is incredibly sensitive to the way the model is put together; the types of elements used, the boundary conditions, the types of constraints employed, etc. etc. You certainly better know the limitations and idiosyncrasies of the program you are using. And, it may be quite edifying to run the same problem with several different programs, as Ishvaaag has, to start to understand what these infallible programs are doing to our thinking and understanding. You want to model this problem with small solid elements, but can’t afford to; but to model it as a beam or smaller plate elements may be missing half the detail at the important locations.
I pretty much agree with BA and 271828 that these type web stiffeners shouldn’t do much to improve the torsional strength of the WF shape, except preventing the shape from distorting in the immediate area of the stiffener. And, given Ishvaaag’s FEA they may actually cause some high, but quite localized stress points, if not detailed very carefully. I would have to see much more detail on the FEA input and output and the way it was modeled, the actual stress output, etc. I assume Ishvaaag’s model didn’t show the stiffeners clipped at the radius btwn. the web and flg. and I’ll bet that is one of the places where the max. stresses occur, as a triaxial stress condition no less. The fixity at the end reactions is suspect for high stresses too.
Some of my thoughts, or food for thought, RE: Ishvaaag’s models and results:
1. The first model with the 1.9 tonne point load applied at the center of the beam and out at the tip of the flgs.: I have two questions; are there two point loads, one at each flg. tip, one on each side of the beam, the near side, up, on the top flg. and the far side, down, on the bottom flg.; thus the torsional moment would be (1.9)(300mm)? I’m having trouble with the orientation of the load and the distorted shape of the beam, on the third page, his 24JUL11, 13:48 post, seem bass-ackwards; the visible load should be pointing upward for the distortion shown. This is actually a pretty complex problem from the Theory of Elasticity standpoint, and we should remember Saint-Venant’s principle as relates to stress conditions and our simplified methods near point loads and reactions, but there aren’t any structural detail anomalies which would cause a high von Mises stress, except at the application of the loads and the reaction points at the ends. Funny enough, Saint-Venant is also associated with BA’s soap film analogy for representing torsional shear stress. The point loads cause canti. plate type bending in the vert. direction in the flgs., but torsionally cause the flg. to bend in its strong direction in the plane of the flg. This latter bending, or the torsional shear forces associated with it, is what causes the WF section rotation, but this torsional loading is distributed over some considerable length of the WF, about the beam center, in inputting the total torsional moment. I suspect the von Mises stress and the Max. Principal stress will exist in the area of the flg. tips or flg./web connection near the load application, or at the end reaction plates and beam flg. tips.
2. In the second example with the 15mm thick stiffeners at 500mm o/c, one pair of stiffeners is at the center of the beam, right at the load application, and that torsional moment is input to the whole WF right there at length/2 from the reactions, not over some beam length, thus the rotational angle and displacements will be significantly greater in this case. I think this is about what Ishvaaag is trying to say in his 24JUL11, 18:26 post, but I think the results have more to do with the abrupt input of the torsional moment to the whole WF section at a max. distance from the reactions, thus a max. rotational angle, rather than any great change in the flg. lateral bending.
3. The third and forth examples, a distributed torsional loading over the full length of the beam, and on only one side is an apples vs. oranges comparison to the first two examples. Quite a different loading condition and a much larger total load on the beam. Despite the larger total load, it might well cause relatively smaller deformations and stresses per unit of loading given the way it is applied, and when applied over the full length of the beam. In the first two cases the moment is applied at the center of the beam, and has the full half length in which to induce the angle of rotation; in case 1 gradually through flg. bending, and in case 2 harshly and abruptly through two stiffeners. In cases 3 & 4 the distributed load induces less regular beam bending and also less rotation per unit of load since part of the load is nearer the reactions.
I am always floored by the fact that when all else fails in our ability at comprehension, we go to FEA for the ultimate solution. But, then are unable to explain how that actually works, how it should be modeled, or what the results mean, and we sight a von Mises stress and a Max. Principal stress, but can’t see where they are, the volume over which they act, or their orientation, and we take them as gospel. Not giving much of a second thought to the fact that they might be (probably are) caused by the way the structure was modeled at a few nodes or at one boundary or point of intersection of two or three plane elements. And, we can’t seem to square the FEA results with our gut feelings, or the simplified solutions which we were taught years ago and which seemed to have work without disaster for so many years. I’m all for seeing your (more exact?) solution, but not when it doesn’t square with what I know has worked for years. I expect FEA to refine my detailed solutions and assist me in refining my gross analysis, not to reinvent the Theory of Elasticity. Ishvaaag may be doing us a service here by forcing us to consider that the first FEA solution may not be better, or more correcter, than our gut feeling, our simplified solution, and we better think twice about taking all FEA results for the gospel. Thanks for the effort Ishvaaag.
St. Venant torsion assumes that the cross section shape doesn’t change, but is free to warp, this is a uniform or pure torsion, represented by St.Venant’s soap film analogy. But, since most members are not free to warp under torsion, normal stresses exist or are induced, particularly in the flanges, due to the bending of the flanges. Thus, both St. Venant (shear stresses) and warping torsional stresses (normal & shear stresses) occur together in the member. In addition, there will be the regular normal and shear stresses in the member due to bending as a beam, gravity loaded, about its shear center. All of these stresses must be combined, in appropriate fashion, to start to approach the output of a FEA. And, on a problem like this the FEA is incredibly sensitive to the way the model is put together; the types of elements used, the boundary conditions, the types of constraints employed, etc. etc. You certainly better know the limitations and idiosyncrasies of the program you are using. And, it may be quite edifying to run the same problem with several different programs, as Ishvaaag has, to start to understand what these infallible programs are doing to our thinking and understanding. You want to model this problem with small solid elements, but can’t afford to; but to model it as a beam or smaller plate elements may be missing half the detail at the important locations.
I pretty much agree with BA and 271828 that these type web stiffeners shouldn’t do much to improve the torsional strength of the WF shape, except preventing the shape from distorting in the immediate area of the stiffener. And, given Ishvaaag’s FEA they may actually cause some high, but quite localized stress points, if not detailed very carefully. I would have to see much more detail on the FEA input and output and the way it was modeled, the actual stress output, etc. I assume Ishvaaag’s model didn’t show the stiffeners clipped at the radius btwn. the web and flg. and I’ll bet that is one of the places where the max. stresses occur, as a triaxial stress condition no less. The fixity at the end reactions is suspect for high stresses too.
Some of my thoughts, or food for thought, RE: Ishvaaag’s models and results:
1. The first model with the 1.9 tonne point load applied at the center of the beam and out at the tip of the flgs.: I have two questions; are there two point loads, one at each flg. tip, one on each side of the beam, the near side, up, on the top flg. and the far side, down, on the bottom flg.; thus the torsional moment would be (1.9)(300mm)? I’m having trouble with the orientation of the load and the distorted shape of the beam, on the third page, his 24JUL11, 13:48 post, seem bass-ackwards; the visible load should be pointing upward for the distortion shown. This is actually a pretty complex problem from the Theory of Elasticity standpoint, and we should remember Saint-Venant’s principle as relates to stress conditions and our simplified methods near point loads and reactions, but there aren’t any structural detail anomalies which would cause a high von Mises stress, except at the application of the loads and the reaction points at the ends. Funny enough, Saint-Venant is also associated with BA’s soap film analogy for representing torsional shear stress. The point loads cause canti. plate type bending in the vert. direction in the flgs., but torsionally cause the flg. to bend in its strong direction in the plane of the flg. This latter bending, or the torsional shear forces associated with it, is what causes the WF section rotation, but this torsional loading is distributed over some considerable length of the WF, about the beam center, in inputting the total torsional moment. I suspect the von Mises stress and the Max. Principal stress will exist in the area of the flg. tips or flg./web connection near the load application, or at the end reaction plates and beam flg. tips.
2. In the second example with the 15mm thick stiffeners at 500mm o/c, one pair of stiffeners is at the center of the beam, right at the load application, and that torsional moment is input to the whole WF right there at length/2 from the reactions, not over some beam length, thus the rotational angle and displacements will be significantly greater in this case. I think this is about what Ishvaaag is trying to say in his 24JUL11, 18:26 post, but I think the results have more to do with the abrupt input of the torsional moment to the whole WF section at a max. distance from the reactions, thus a max. rotational angle, rather than any great change in the flg. lateral bending.
3. The third and forth examples, a distributed torsional loading over the full length of the beam, and on only one side is an apples vs. oranges comparison to the first two examples. Quite a different loading condition and a much larger total load on the beam. Despite the larger total load, it might well cause relatively smaller deformations and stresses per unit of loading given the way it is applied, and when applied over the full length of the beam. In the first two cases the moment is applied at the center of the beam, and has the full half length in which to induce the angle of rotation; in case 1 gradually through flg. bending, and in case 2 harshly and abruptly through two stiffeners. In cases 3 & 4 the distributed load induces less regular beam bending and also less rotation per unit of load since part of the load is nearer the reactions.
I am always floored by the fact that when all else fails in our ability at comprehension, we go to FEA for the ultimate solution. But, then are unable to explain how that actually works, how it should be modeled, or what the results mean, and we sight a von Mises stress and a Max. Principal stress, but can’t see where they are, the volume over which they act, or their orientation, and we take them as gospel. Not giving much of a second thought to the fact that they might be (probably are) caused by the way the structure was modeled at a few nodes or at one boundary or point of intersection of two or three plane elements. And, we can’t seem to square the FEA results with our gut feelings, or the simplified solutions which we were taught years ago and which seemed to have work without disaster for so many years. I’m all for seeing your (more exact?) solution, but not when it doesn’t square with what I know has worked for years. I expect FEA to refine my detailed solutions and assist me in refining my gross analysis, not to reinvent the Theory of Elasticity. Ishvaaag may be doing us a service here by forcing us to consider that the first FEA solution may not be better, or more correcter, than our gut feeling, our simplified solution, and we better think twice about taking all FEA results for the gospel. Thanks for the effort Ishvaaag.
slickdeals
Structural
There is some good information here
I would think you could still use AISC design guide #9. The only thing is you would have to figure out what J and Cw are. I am not sure if they are in the 13th edition manual, but a descent software program that analyzes cross-sectional shapes should be able to give these to you.
Well, dhengr, thanks. Now I will explain how I modeled the torque for the concentrated torque cases.
In AS I first introduced a small bump antymmetrical in the center of the flanges where I later put the total amount of the 1.9 tonnes as a traction in +X and -X (rounded to kN) so its average point of application was at midheight of the flanges. The ugly minor bump I put because one of the things that can be ameliorated in AS I think is introduction of the loads, it is not very clement on it for those accustomed to enter ordinary hypotheses at ease.
Then, when making the RISA-3D models I took care to specify the height of the web between the midheight of the flanges, i.e., 281 mm once you deduce two half flange thickness, that is of 19 mm. So when putting there the 1.9 tonne loads there it was at the same point, only that without the to kN rounding.
Then, for the SAP2000 models, all the geometry was OK except that the fillets were beveled to excess of material (and I accepted the end stiffeners to be like the others "to axes" against the AS models. Since I had 2 joints at the incumbent flange sections and flanges, I used two 0.95 tonnes at the atop flange, and two contrary in the bottom flange, so for the same resultant at midheight.
For the line model the torque was directly entered as a torque itself.
And the last AS models to verify the former ones, I took the liberty of using just 1 node per flange to enter the load, i.e., I put the 19 tonne or more really, the 19000 kN force at just two nodes to form the torque. Really I could have made the same than in SAP2000, since I have also 2 nodes available.
In general I have proceeded at ease not looking for entire agreement but the behaviour.
Now I will add a further example of how the modification of the structure may introduce a variation of some particular kind of response that we want controlled. It is the well known structural type of strong colum-weak beam. If you have a well tuned system to such intent, you may alter the required characteristics of the system by increasing the stiffness of the beams.
In general, as work that is, inner work is the integration of products of stresses of strains. For some given inner work, you may have more stress and less strain, or more strain and less stress; the attribution of which to every case to be got by the pertaining laws governing the case.
So it may turn that some modification of some particular aspect of one structure causes a perturbation on some kind of response, stress of strain (or its more visible or accountable correspondent solicitations and displacements) that we want controlled.
In fact we as animals use it continously: when we are to fall we extend a leg or arm to precisely modify the response.
In AS I first introduced a small bump antymmetrical in the center of the flanges where I later put the total amount of the 1.9 tonnes as a traction in +X and -X (rounded to kN) so its average point of application was at midheight of the flanges. The ugly minor bump I put because one of the things that can be ameliorated in AS I think is introduction of the loads, it is not very clement on it for those accustomed to enter ordinary hypotheses at ease.
Then, when making the RISA-3D models I took care to specify the height of the web between the midheight of the flanges, i.e., 281 mm once you deduce two half flange thickness, that is of 19 mm. So when putting there the 1.9 tonne loads there it was at the same point, only that without the to kN rounding.
Then, for the SAP2000 models, all the geometry was OK except that the fillets were beveled to excess of material (and I accepted the end stiffeners to be like the others "to axes" against the AS models. Since I had 2 joints at the incumbent flange sections and flanges, I used two 0.95 tonnes at the atop flange, and two contrary in the bottom flange, so for the same resultant at midheight.
For the line model the torque was directly entered as a torque itself.
And the last AS models to verify the former ones, I took the liberty of using just 1 node per flange to enter the load, i.e., I put the 19 tonne or more really, the 19000 kN force at just two nodes to form the torque. Really I could have made the same than in SAP2000, since I have also 2 nodes available.
In general I have proceeded at ease not looking for entire agreement but the behaviour.
Now I will add a further example of how the modification of the structure may introduce a variation of some particular kind of response that we want controlled. It is the well known structural type of strong colum-weak beam. If you have a well tuned system to such intent, you may alter the required characteristics of the system by increasing the stiffness of the beams.
In general, as work that is, inner work is the integration of products of stresses of strains. For some given inner work, you may have more stress and less strain, or more strain and less stress; the attribution of which to every case to be got by the pertaining laws governing the case.
So it may turn that some modification of some particular aspect of one structure causes a perturbation on some kind of response, stress of strain (or its more visible or accountable correspondent solicitations and displacements) that we want controlled.
In fact we as animals use it continously: when we are to fall we extend a leg or arm to precisely modify the response.
Toad,
See p. 12 of the attached for WF with Channel cap.
BA
See p. 12 of the attached for WF with Channel cap.
BA
- Thread starter
- #77
100000e's Torsion Experiment FWIW.
Objective: Determine the effect of regular transverse stiffeners on torsional stiffness and warping normal stresses for a typical W-shape beam subject to point moment at midspan.
Test Beam
• W18x35
• 20 ft long, 40 kip-in. concentrated torque at midspan
• Simply supported at ends
SAP Model Details
• Shell elements
• 1 in. mesh everywhere
• Flange shells at d-tf=17.7 in. – 0.425 in. apart, measured to centerlines.
• Lateral restraint at end nodes at each flange-web junction.
• Vertical restraint at end nodes at bottom flange-web junction.
• Longitudinal 0.1 kip/in. spring at left end, bottom, flange-web junction.
Unstiffened Beam Model (and closed-form solution verifications)
• Beam subject to strong-axis load of 20 kip at midspan. Monitor displacement at midspan.
oApply the 20 kip at each of the 17 midspan nodes, so 1.1765 kip each
o First order analysis
o Shear deflection from virtual work: 0.0207 in.
o Flexural deflection from virtual work: 0.3895 in.
o Total deflection: 0.4101 in.
o SAP2000 total deflection: 0.4158 in. Speculate that the increased deflection is due to compressive deformations near the reactions.
o SAP2000 modification: add axially rigid, otherwise flexible member at the end supports. New deflection is 0.4107 in.
• Weak-axis natural frequency of the beam (self mass only)
o Closed form solution = 6.61 Hz
o SAP2000 first mode = 6.61 Hz
• Apply 40 kip-in. concentrated torque at midspan.
o DG9 analytical solution for Case 3, Page 110.
o In SAP, apply 2.3148 kip laterally, as a couple to the nodes at the flange-web junctions.
o First-order solution in SAP.
o J=0.51 in.4, a=76.1 in., Wno=25.9 in.2, alpha=0.5
o Rotation at midspan = 10.06 deg. from the equation on Page 110; SAP2000 flange displacement = 1.657 in. which translates into 10.86 deg (8% higher than closed-form solution)
o Warping normal stress = 31.7 ksi from DG9 Eq. 4.3a. (Second derivative of theta equation using Mathcad); SAP2000 max flange stress = 33.2 ksi (5% higher than closed-form solution)
o Note that the model does not have quite as much material as the W-shape due to its lack of k-area material at the web-flange junction. The model has slightly more rotation and slightly higher stress, both of which make sense.
• Closed form solutions are close enough to the analytical solutions to conclude that the model is behaving well enough to use as a basis for comparison.
o Midspan rotation = 10.86 deg.
o Max warping normal stress = 33.2 ksi
Stiffened Beam Models (1/2 in. transverse stiffeners)
• Unstiffened: rotation=10.9 deg., stress=33.2 ksi
• One pair of stiffeners at each end: rotation=9.78 deg. (-10.3%); stress=31.5 ksi (-5.12%)
• Five pairs of stiffeners at each end, 12 in. o.c. spacing: rotation= 7.89 deg. (-27.6%); stress=27.8 ksi (-16.3%)
• Stiffeners at 12 in. the entire beam length: rotation= 7.13 deg. (-34.6%); stress= 23.6 ksi (-28.9%)
Conclusion
• For this beam, modest decreases in the rotation angle and warping normal stress may be attained by adding typical transverse stiffeners, especially if they'e closed spaced.
Experiment Mode Off / Editorial+Opinion Mode On (LOL)
I was pleasantly surprised that the stiffeners helped as much as they did. While other types of stiffeners, such as those shown in DG9, are undoubtedly far, far more efficient, these do offer a small help. For example, if torsional rotation is only slightly too much, perhaps it can be dropped 10% or so just by adding or considering (perhaps they're already there) stiffeners at the ends. In general, I think the initially visualized behavior was correct: the stiffeners bend out-of-plane near the beam ends, providing some restraint to warping, but not a lot.
Objective: Determine the effect of regular transverse stiffeners on torsional stiffness and warping normal stresses for a typical W-shape beam subject to point moment at midspan.
Test Beam
• W18x35
• 20 ft long, 40 kip-in. concentrated torque at midspan
• Simply supported at ends
SAP Model Details
• Shell elements
• 1 in. mesh everywhere
• Flange shells at d-tf=17.7 in. – 0.425 in. apart, measured to centerlines.
• Lateral restraint at end nodes at each flange-web junction.
• Vertical restraint at end nodes at bottom flange-web junction.
• Longitudinal 0.1 kip/in. spring at left end, bottom, flange-web junction.
Unstiffened Beam Model (and closed-form solution verifications)
• Beam subject to strong-axis load of 20 kip at midspan. Monitor displacement at midspan.
oApply the 20 kip at each of the 17 midspan nodes, so 1.1765 kip each
o First order analysis
o Shear deflection from virtual work: 0.0207 in.
o Flexural deflection from virtual work: 0.3895 in.
o Total deflection: 0.4101 in.
o SAP2000 total deflection: 0.4158 in. Speculate that the increased deflection is due to compressive deformations near the reactions.
o SAP2000 modification: add axially rigid, otherwise flexible member at the end supports. New deflection is 0.4107 in.
• Weak-axis natural frequency of the beam (self mass only)
o Closed form solution = 6.61 Hz
o SAP2000 first mode = 6.61 Hz
• Apply 40 kip-in. concentrated torque at midspan.
o DG9 analytical solution for Case 3, Page 110.
o In SAP, apply 2.3148 kip laterally, as a couple to the nodes at the flange-web junctions.
o First-order solution in SAP.
o J=0.51 in.4, a=76.1 in., Wno=25.9 in.2, alpha=0.5
o Rotation at midspan = 10.06 deg. from the equation on Page 110; SAP2000 flange displacement = 1.657 in. which translates into 10.86 deg (8% higher than closed-form solution)
o Warping normal stress = 31.7 ksi from DG9 Eq. 4.3a. (Second derivative of theta equation using Mathcad); SAP2000 max flange stress = 33.2 ksi (5% higher than closed-form solution)
o Note that the model does not have quite as much material as the W-shape due to its lack of k-area material at the web-flange junction. The model has slightly more rotation and slightly higher stress, both of which make sense.
• Closed form solutions are close enough to the analytical solutions to conclude that the model is behaving well enough to use as a basis for comparison.
o Midspan rotation = 10.86 deg.
o Max warping normal stress = 33.2 ksi
Stiffened Beam Models (1/2 in. transverse stiffeners)
• Unstiffened: rotation=10.9 deg., stress=33.2 ksi
• One pair of stiffeners at each end: rotation=9.78 deg. (-10.3%); stress=31.5 ksi (-5.12%)
• Five pairs of stiffeners at each end, 12 in. o.c. spacing: rotation= 7.89 deg. (-27.6%); stress=27.8 ksi (-16.3%)
• Stiffeners at 12 in. the entire beam length: rotation= 7.13 deg. (-34.6%); stress= 23.6 ksi (-28.9%)
Conclusion
• For this beam, modest decreases in the rotation angle and warping normal stress may be attained by adding typical transverse stiffeners, especially if they'e closed spaced.
Experiment Mode Off / Editorial+Opinion Mode On (LOL)
I was pleasantly surprised that the stiffeners helped as much as they did. While other types of stiffeners, such as those shown in DG9, are undoubtedly far, far more efficient, these do offer a small help. For example, if torsional rotation is only slightly too much, perhaps it can be dropped 10% or so just by adding or considering (perhaps they're already there) stiffeners at the ends. In general, I think the initially visualized behavior was correct: the stiffeners bend out-of-plane near the beam ends, providing some restraint to warping, but not a lot.
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