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**an extension to Tutorial 1**

Tutorial 3: Exploring how cross-section changes influence cross-section stability an extension to Tutorial 1 prepared by Ben Schafer, Johns Hopkins University, version 1.0

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Acknowledgments Preparation of this tutorial was funded in part through the AISC faculty fellowship program. Views and opinions expressed herein are those of the author, not AISC.

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**Target audience This tutorial is targeted at the under-graduate level.**

It is also assumed that Tutorial #1 has been completed and thus some familiarity with the use of CUFSM is assumed.

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Learning objectives Study the impact of flange width, web thickness, and flange-to-web fillet size on a W-section Learn how to change the cross-section in CUFSM Learn how to compare analysis results to study the impact of changing the cross-section

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Summary of Tutorial #1 A W36x150 beam was analyzed using the finite strip method available in CUFSM for pure compression and major axis bending. For pure compression local buckling and flexural buckling were identified as the critical buckling modes. For major axis bending local buckling and lateral-torsional buckling were identifies as the critical buckling modes.

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**W36x150 column – review of Tutorial 1**

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**web and flange local buckling is shown remember, applied**

load is a uniform compressive stress of 1.0 ksi

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Pcr,local = x 42.6 = 2007 k or fcr,local = x 1.0 ksi = ksi Pref = 42.6 k or fref = 1.0 ksi load factor for local buckling = 47.12

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**this is weak axis flexural**

buckling...

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note that for flexural buckling the cross- section elements do not distort/bend, the full cross-section translates/rotates rigidly in-plane.

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Pref = 42.6 k or fref = 1.0 ksi load factor for global flexural buckling = 7.6 at 40 ft. length Pcr = 7.6 x 42.6 k = 324 k or fcr = 7.6 x 1.0 ksi = 7.6 ksi

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**Tutorial #1: Column summary**

A W36x150 under pure compression (a column) has two important cross-section stability elastic buckling modes (1) Local buckling which occurs at a stress of 47 ksi and may repeat along the length of a member every 27 in. (it’s half-wavelength) (2) Global flexural buckling, which for a 40 ft. long member occurs at a stress of 7.6 ksi (other member lengths may be selected from the curve provided from the analysis results)

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**Modifying the cross-section**

Once we start changing the depth, width, thickness, etc. the section is no longer a W36x150 – but by playing with these variables we can learn quite a lot about how geometry influences cross-section stability. Let’s see what happens when the web thickness is set equal to the flange thickness see what happens when the flange width is reduced by 2 inches.

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**Modifying the cross-section**

Once we start changing the depth, width, thickness, etc. the section is no longer a W36x150 – but by playing with these variables we can learn quite a lot about how geometry influences cross-section stability. Let’s see what happens when the web thickness is set equal to the flange thickness see what happens when the flange width is reduced by 2 inches.

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load up the default W36x150

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change the web thickness to 0.9 in

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the model should look like this now.

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**default post-processor**

results, change the half-wavelength to the local buckling minimum

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local buckling at a stress of 84.6 ksi let’s save this file and load up the original file, so we can compare.

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**that the local buckling stress increases from 47 ksi to 85 ksi. **

now we can readily see that the local buckling stress increases from 47 ksi to 85 ksi. load the actual W36x150 (Advanced note: if one was using plate theory the prediction would be that the buckling stress should increase by (new thickness/old thickness)2 but the increase is slightly less here because the web and flange interact – something that finite strip modeling includes.)

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At longer length the section with the thicker web buckles at slightly lower stress, this reflects the increased area, with little increas in moment of inertia that results with this modification. 40’ fcr= 7.6 ksi Pcr= 324 k “W36x150” w/ tw=tf fcr=6.2 ksi Pcr=328 k

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**Modifying the cross-section**

Once we start changing the depth, width, thickness, etc. the section is no longer a W36x150 – but by playing with these variables we can learn quite a lot about how geometry influences cross-section stability. Let’s see what happens when the web thickness is set equal to the flange thickness see what happens when the flange width is reduced by 2 inches.

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**Modifying the cross-section...**

The W36x150 we have been studying in local buckling is largely dominated by the web. Do the fillets at the ends of the web help things at all? Let’s make an approximate model to look into this effect.

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Load up the W36x150 model and go to the input page.

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Let’s divide up these elements so that we can increase the thickness of the web, near the flange to approx- imate the role of the fillet.

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now divide element 5 at 0.2 of its length..

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the model should look this this now, let’s change the thickness of elements 5 and elements 10 to 2tw=2x0.6=1.2in.

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**save this result, so that we**

can load up earlier results and compare them. After hitting save above I named my file “W36x150 with approx fillet” this now shows up to the left and in the plot below. next, let’s load the original centerline model W36x150...

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After loading “W36x150” now I have two files of results and I can see both buckling curves and may select either bucking mode shape. Let’s change the axis limits below to focus more on local buckling..

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**of course global flexural**

buckling out in this range changes very little since the moment of inertia changes only a small amount when the fillet is modeled the reference stress is 1.0 ksi, the fillet increases local buckling from 47 ksi to 54 ksi, a real change in this case.

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Other modifications... Change the web depth and explore the change in the buckling properties Add a longitudinal stiffener at mid-depth of the web and explore Modify the material properties to see what happens if your W-section is made of plastic or aluminium, etc. Add a spring (to model a brace) at different points in the cross-section

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