Tutorial 1:Cross-section stability of a W36x150

Learning how to use and interpret finite strip method results for cross-section stability of hot-rolled steel members

prepared by Ben Schafer, Johns Hopkins University, version 1.0

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.

Learning objectives• Identify all the buckling modes in a W-section

– For columns explore flexural (Euler) buckling and local buckling– For beams explore lateral-torsional buckling and local buckling

• Predict the buckling stress (load or moment) for identified buckling modes

• Learn the interface of a simple program for exploring cross-section stability of any AISC section and learn finite strip method concepts such as

– half-wavelength of the bucking mode– buckling load factor associated with the applied stresses

Going further with other tutorials...• Show how changes in the cross-section

– change the buckling modes– change the buckling stress (load or moment)

• Explore the provided WT, C, L, HSS sections..• Exploring higher modes, and the interaction of buckling modes• Understand how the results relate to the AISC Specification

Start CUFSM

• The program may be downloaded from www.ce.jhu.edu/bschafer/cufsm

• Instructions for initializing the program are available online

select the input page

load a file

select W36x150(these files are available online where you down-loaded this tutorial)

questionmarks givemore info...

nodeelement

The geometry is defined by nodesand elements, youcan change these as you like, here aW36x150 is shown

Each element hasmaterial propertiesassociated with itin this example E is29000 ksi, and is 0.3. (Each element also has a thickness)

the model is evaluatedfor many different“lengths” this allowsus to explore all the buckling modes, moreon this soon.

select properties

basic properties of the cross-section,you can compare themwith the AISC manualthey will be close, buthere we use a straightline model – so theywon’t be identical.

advanced note: these properties areprovided for convenience,but the program does notactually use them to calculate the buckling behavior of the section,instead plate theory is used throughout tomodel the section.

Let’s explore one of the ways we can apply loads

enter 1 here

uncheck this box

press this button to generate stress

max referencestress

appliedreferencemoment

Generated stressdistribution

when done, goback to the inputpage

this last column ofthe node entriesreflects the appliedreference stress.

now, go backto the propertiespage

put compression of1 ksi on ths section,enter 1, uncheck Mxxgenerate stress –should get thisdistribution...

go back to the inputpage when you are done.

notestresses areall 1.0 now(+ = comp.)

analyze thesection

Finite strip analysis results – lots to takein here!

buckled shape, herewe can figure out whattype of buckling modewe are looking at, is itlocal? global? etc.

half-wave vs.load factor plothere we find thebuckling load andwe find the critical buckling lengths...

undeformed shape

buckled shape

the little red dottells you where you are

at half-wavelength = 22.6and load factor = 48.7

explaining load factor and half-wavelength

move the little red dotto the minimum on the curve with thesecontrols, then selectplot shape and youwill get this bucklingmode shape result.

Local buckling

How do you know this is local buckling?Where is flange local buckling?Where is web local buckling?

In the beginning, looking at the buckledshape in 3D can help a lot...

select (and be patient)

web and flange local buckling is shown

remember, appliedload is a uniform compressive stressof 1.0 ksi

let’s rotate this section so we cansee the bucklingfrom the end onview.

buckled shape at“midspan” of the half-wavelength, thisis the 2D buckledshape

Go back to 2D nowand see if the shapemakes more sense...

we call this local bucklingbecause the elements whichmake up the section aredistorted/bent in-plane.

Also, the half-wavelength ismuch shorter than typicalphysical member length, infact the half-wavelength is less than the largest dimensionof the section (this is typical).

At what stress orload is this elastic local buckling predicted to occur at?

our referenceload of 42.6 k

or, equivalentlyour referencestress of 1.0ksi every-where..

you also can get a quick check on the applied stress by selectingthis plot within the post-processor.

Pref = 42.6 korfref = 1.0 ksi

load factor for localbuckling = 47.12

Pcr,local = 47.12 x 42.6 = 2007 k

or

fcr,local = 47.12 x 1.0 ksi = 47.12 ksi

now let’s take a look at long half-wavelengths

change half-wavelengthto ~480” = 40ft and plotthe shape to get the result shown here.

try out the 3D shape tobetter see thismode...

this is weak axis flexuralbuckling...

note that for flexuralbuckling the cross-section elements donot distort/bend, thefull cross-sectiontranslates/rotates rigidly in-plane.

Pref = 42.6 korfref = 1.0 ksi

load factor for globalflexural buckling = 7.6at 40 ft. length

Pcr = 7.6 x 42.6 k = 324 k

or

fcr = 7.6 x 1.0 ksi = 7.6 ksi

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)

A W36x150 is really intended for beam applications more than columns, let’ssee how it behaves as a beam...

go back to thepropertiespage

enter a referencestress of 1.0 ksi

calculate

uncheck P

reference momentis 500.5 kip-in.

generate stress

check everythingout on the input page, you can even look at thestress dist.to double check..

then analyze

Results page...

move to the firstminimum to explorelocal buckling of thisbeam further

Local buckling..

Mcr,local = 231 x 500 kip-in. = 115,500 kip-in. = 9,625 kip-ft

fcr,local = 231 x 1.0 ksi = 231 ksi

compression

tension

tension helps stiffenthe bottom of the weband elevates local buckling a great deal.

local buckling half-wavelength is 25.6 in.,as shown here in the 3D plot of thebucklingmode

what about long half-wavelengths, say 40’?

Lateral-torsional buckling..

In-plane the cross-sectionremains rigid and onlyundergoes lateral translationand twist (torsion), as shownin this buckling mode shape

Lateral-torsional buckling..

Mcr = 15.8 x 500 kip-in. = 7,900 kip-in. = 660 kip-ft

fcr = 15.8 x 1.0 ksi = 15.8 ksi

also predicted by this classical formula:

Beam summary

• A W36x150 under major-axis bending (a beam) has two important cross-section stability elastic buckling modes

• (1) Local buckling which occurs at a stress of 231 ksi and may repeat along the length of a member every 26 in. (it’s half-wavelength)

• (2) Global lateral-torsional buckling, which for a 40 ft. long member occurs at a stress of 15.8 ksi (other member lengths may be selected from the curve provided from the analysis results)

Learning objectives• Identify all the buckling modes in a W-section

– For columns explore flexural (Euler) buckling and local buckling– For beams explore lateral-torsional buckling and local buckling

• Predict the buckling stress (load or moment) for identified buckling modes

• Learn the interface of a simple program for exploring cross-section stability of any AISC section and learn finite strip method concepts such as

– half-wavelength of the bucking mode– buckling load factor associated with the applied stresses

Going further with other tutorials...• Show how changes in the cross-section

– change the buckling modes– change the buckling stress (load or moment)

• Explore the provided WT, C, L, HSS sections..• Exploring higher modes, and the interaction of buckling modes• Understand how the results relate to the AISC Specification