# Tutorial: Cross-section stability of a W-section

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

element The geometry is defined by nodes and elements, you can change these as you like, here a W36x150 is shown

node Each element has material properties associated with it in this example E is 29000 ksi, and is 0.3. (Each element

also has a thickness) the model is evaluated for many different lengths this allows us to explore all the buckling modes, more on this soon.

select properties basic properties of the cross-section, you can compare them with the AISC manual they will be close, but here we use a straight line model so they

wont be identical. advanced note: these properties are provided for convenience, but the program does not actually use them to calculate the buckling behavior of the section,

instead plate theory is used throughout to model the section. Lets explore one of the ways we can apply loads enter 1 here uncheck this box press this button to generate stress

when done, go back to the input page max reference stress applied reference

moment Generated stress distribution now, go back to the properties page this last column of

the node entries reflects the applied reference stress. go back to the input page when you are done. put compression of

1 ksi on ths section, enter 1, uncheck Mxx generate stress should get this distribution... note stresses are all 1.0 now

(+ = comp.) analyze the section Finite strip analysis results lots to take in here!

buckled shape, here we can figure out what type of buckling mode we are looking at, is it local? global? etc. half-wave vs. load factor plot here we find the

buckling load and we find the critical buckling lengths... buckled shape at half-wavelength = 22.6 and load factor = 48.7 undeformed shape the little red dot

tells you where you are explaining load factor and half-wavelength move the little red dot to the minimum on the curve with these controls, then select

plot shape and you will get this buckling mode 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 buckled shape in 3D can help a lot... select (and be patient) web and flange local buckling is shown remember, applied load is a uniform

compressive stress of 1.0 ksi lets rotate this section so we can see the buckling from the end on view.

Go back to 2D now and see if the shape makes more sense... buckled shape at midspan of the half-wavelength, this is the 2D buckled shape

we call this local buckling because the elements which make up the section are distorted/bent in-plane. Also, the half-wavelength is much shorter than typical physical member length, in fact the half-wavelength is

less than the largest dimension of the section (this is typical). At what stress or load is this elastic local buckling predicted to occur at? our reference

load of 42.6 k or, equivalently our reference stress of 1.0 ksi everywhere.. you also can get a quick check on the

applied stress by selecting this plot within the postprocessor. Pref = 42.6 k or fref = 1.0 ksi load factor for local buckling = 47.12

Pcr,local = 47.12 x 42.6 = 2007 k or fcr,local = 47.12 x 1.0 ksi = 47.12 ksi now lets take a look at long half-wavelengths change half-wavelength

to ~480 = 40ft and plot the shape to get the result shown here. try out the 3D shape to better see this mode...

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

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 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. (its 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, lets see how it behaves as a beam... go back to the properties page enter a reference stress of 1.0 ksi

calculate uncheck P reference moment is 500.5 kip-in. generate stress check everything out on the input page, you can

even look at the stress dist. to double check.. then analyze Results page... move to the first minimum to explore local buckling of this

beam 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 helps stiffen the bottom of the web and elevates local buckling a great deal. tension local buckling halfwavelength is 25.6 in.,

as shown here in the 3D plot of the buckling mode what about long half-wavelengths, say 40? Lateral-torsional buckling.. In-plane the cross-section

remains rigid and only undergoes lateral translation and twist (torsion), as shown in 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. (its 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

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