Lateral-Torsional Buckling of Flange-Tapered I … Lateral-Torsional Buckling of Flange-Tapered I-Beams Michael Brett Thomas Department of Civil Engineering, BYU Master of Science

Lateral-Torsional Buckling of Flange-Tapered I-Beams

Michael Brett Thomas

A project submitted to the faculty of

Brigham Young University

in partial fulfillment of the requirements for the degree of

Master of Science

Paul W. Richards, Chair

Richard J. Balling

Michael A. Scott

Department of Civil Engineering

Brigham Young University

June 2014

Copyright © 2014 Brett Thomas

All Rights Reserved

ABSTRACT

Lateral-Torsional Buckling of Flange-Tapered I-Beams

Michael Brett Thomas

Department of Civil Engineering, BYU

Master of Science

I-beams are often used to carry bending moments because of their superior moment of

inertia to area ratio. The capacity of an I-beam is typically controlled by either the formation of a

plastic hinge or lateral-torsional buckling (LTB). LTB is affected by the un-braced length of the

compression flange and the moment gradient; this will be addressed in more depth in this paper.

If the moment gradient is constant, a uniform cross section I-beam is most efficient, but often the

moment gradient is not constant. As a result, to be most efficient, the cross sectional shape

should vary proportionally to the moment gradient. A common case is when the moment

gradient varies linearly. This particular case will be examined in this paper. Through a literature

review, it was discovered that it is impossible to determine a closed form solution for the LTB

capacity of a flange-tapered I-beam. Many papers use numerical methods to determine the

buckling capacities. The results of these studies will be used to develop graphs that will show the

capacity of a flange-tapered I-beam versus its un-braced length.

Keywords: Michael Brett Thomas, lateral-torsional buckling, flange-tapered I-beams,

v

TABLE OF CONTENTS

LIST OF FIGURES ..................................................................................................................... vi

1 Introduction ........................................................................................................................... 1

2 Literature Review ................................................................................................................. 3

3 Procedure ............................................................................................................................. 11

4 Results and Discussion ........................................................................................................ 15

4.1 Results ........................................................................................................................... 15

4.2 Possible Sources of Error .............................................................................................. 20

4.3 Further Research ........................................................................................................... 21

5 Conclusion ........................................................................................................................... 23

REFERENCES ............................................................................................................................ 25

vi

LIST OF FIGURES

Figure 1-1 Flange-tapered I-beam .........................................................................................……..2

Figure 3-1 Bifurcation of flange-tapered I-cantilever ............................................................……12

Figure 3-2 Graph used to determine the dimensionless load parameter for alpha = 0 ..........……13

Figure 4-1 Comparison of the developed graph and the AISC manual equations.................……16

Figure 4-2 Developed graph for W12x210 ............................................................................……16





Figure 4-7 Developed graph for W14x99 ..............................................................................……19

1

1 INTRODUCTION

I-beams are commonly used to carry bending moments because they are designed to have

the largest possible moment of inertia for a given amount of material. Although I-beams are

effective at carrying bending moments, there are several failure modes that need to be

considered. A couple of the more common failure modes are the formation of a plastic hinge and

LTB. This paper will discuss, in more depth, the failure mode of LTB. LTB is affected by the un-

braced length of the compression flange and the moment gradient to which the I-beam is subject.

When the moment gradient is constant, a uniform cross section I-beam is most efficient, but

often the moment gradient is not constant. For example, when the moment gradient varies

linearly, the cross section of the beam should also vary linearly. This can be achieved by tapering

the flanges as shown in Figure 1-1. Tapering the flanges allows a smaller amount of steel to carry

the same amount of load, making it more efficient. One of the issues with tapering an I-beam is

that it complicates the calculation of the possible failure modes, especially LTB. Discovering the

LTB capacity of a flange-tapered I-beam is one of the objectives of this project.

A literature review on the LTB of flange tapered I-beams was performed for this project.

During this literature review it was discovered that it is impossible to directly determine the LTB

capacity of a flange tapered I-beam. Numerical methods are used to approximate the buckling

capacity. Most of the papers that used a numerical method to solve this problem have plotted

their results on a graph that shows the dimensionless load versus the dimensionless beam

parameter. The purpose of this project was to develop more user friendly curves from the already

2

Figure 1-1 Flange-tapered I-beam

established plots. The user friendly plots will display the moment that causes elastic LTB versus

the length of the cantilevered beam. A spreadsheet was developed to produce a graph of the

moment versus length for several degrees of taper by inputting the I-beam size. This graph will

allow for easy sizing of any beam subject to a bending moment about its strong axis.

3

2 LITERATURE REVIEW

Al-Sadder, S. Z. (2004) “Exact expressions for stability functions of a general non-prismatic

beam–column member.” J. Constr. Steel Res., 60(11), 1561-1584.

Al-Sadder used power series to solve the 4th

order differential equation. That equation

was used to determine the capacity of a beam-column subject to a uniform axial load. He never

specifically addressed LTB, but he did state that he finds the elastic stability of the beam column.

It would appear that he does not consider LTB, because at no point in the calculations presented

in his paper is the torsional stiffness of the beam-column used.

There was still the possibility that the capacity of a beam-column, subject to a uniform

axial load, could have been used to get a lower bound estimate for the LTB by treating the lower

flange as a column buckling about its strong axis. This did not work either because his method

was only valid for constant axial force, while the I-beam of interest for this project had a linearly

increasing axial force in the bottom flange.

Suryoatmono, B., and Ho, D. (2002). “The moment–gradient factor in lateral–torsional buckling

on wide flange steel sections.” J. Constr. Steel Res., 58(9), 1247-1264.

Suryoatmono and Ho investigate the accuracy of the AISC equations for the Cb factor

associated with LTB. They use a finite difference approach to determine how closely the

equations predict the actual Cb factors. It was determined that the equations are relatively close

for most situations, but there are a few cases where they recommend using a different equation

that more accurately computes the Cb factor.

4

Their research was relevant to this project because the beams of interest may be laterally

braced at intermediate points along the length of the beam. In between each of these different

sections the moment gradient would be different and thus require a different Cb factor.

Bradford, M., and Cuk, P. (1988). ”Elastic Buckling of Tapered Monosymmetric I-Beams.” J.

Struct. Eng., 114(5), 977–996.

Bradford and Cuk defined a method to determine the lateral buckling of tapered I-beams.

Bradford and Cuk did not make the assumption that the shear center is parallel to the centroidal

axis. By not making that assumption, they arrived at a solution that they believe is more accurate

than the solutions proposed previously by Kitipornchai and Trahair. The method used to derive

the finite element sections is laid out. Instead of assuming uniform elements, they used tapered

elements because they feel that uniform elements lead to inefficient calculations. The rate of

convergence of the tapered element occurs much faster than the uniform element. It takes only

two tapered elements to converge to within 5% error, while it takes 6 uniform elements to

converge to the same amount of accuracy. They plotted the dimensionless load versus the

dimensionless length for an end loaded cantilevered beam. The graph produced by Bradford and

Cuk only represents the elastic LTB capacity; it does not represent any other failure mode. Their

graph had the same problem as Kitipornchai and Trahair.

Chan, S. (1990). ”Buckling Analysis of Structures Composed of Tapered Members.” J. Struct.

Eng., 116(7), 1893–1906.

Chan used the energy method to develop the equations for his tapered elements. The

calculations required to complete this form of analysis are complex and are performed by an

integration package: Macsyma. He used much the same method as Bradford and Cuk, except that

he used the energy method to come up with his finite elements. He used the method from an

5

earlier paper by Chan and Kitipornchai to validate his results. Chan believes that Bradford and

Cuk were wrong in the assumptions they made about neglecting large pre-buckling deformations

and axial loads. Chan used the minimum residual displacement method to account for pre-

buckling deformations. Most papers just found the first bifurcation value using an eigenvalue

problem. Chan plotted the dimensionless load versus the dimensionless beam parameter. The

dimensionless load parameter is ��∗��

��∗∗�∗�, the dimensionless beam parameter is � ∗ ��∗�

�∗� and Iw

is the torsional constant and has units of length to the sixth. He also applied the tapered elements

to a frame and found that the global buckling load decreased considerably.

The graphs in this project were produced using the dimensionless plots Chan produced.

Eisenberger, M. (1991). “Buckling loads for variable cross-section members with variable axial

forces.” Int. J. Solids Struct., 27(2), 135–143.

Eisenberger went through and derived exact stiffness matrices for columns of varying

cross section, varying axial load, and different boundary conditions. This method was used to

find the exact solution to the differential equation for any polynomial cross section. The

advantages of this method are that only one element is needed to exactly determine the buckling

capacity, instead of several elements needed to determine an approximate solution. The difficulty

arises in determining the exact solution for each loading and variable cross section for each case

of interest.

This paper presented the method which would best be used to determine a conservative

estimate of the LTB capacity of a flange tapered I-beam. However, the derivations presented in

Eisenberger’s paper were complex or simply too difficult to understand and thus the derivations

are not reflected in this project. Eisenberger presented a table with values for the buckling

6

capacity of the columns of interest. The problem with his table was that it gave the values for a

beam whose moment of inertia started at one value and ended at twice that value. It would help

to provide values for more than just that one particular case.

Gupta, P., Wang, S. T., and Blandford, G. E. (1996). “Lateral-torsional buckling of nonprismatic

I-beams.” J. Struct. Eng., 122(7), 748-755.

Gupta et al. investigated the LTB of non-uniform I-beams. They used a finite element

approach using both tapered elements and linear elements. They looked at linear taper of simple

supported beams and two span continuous I-beams. They do not investigate beams that narrow in

the middle, only beams that get wider in the middle. They also never looked at cantilevered

beams subject to LTB.

Because this project is interested in a cantilevered I-beam, this paper was not useful.

Kitipornchai, S., and Trahair, N. S. (1972). "Elastic stability of tapered I-beams." J. Struct. Div.,

ASCE, 98(ST3), 713-728.

Kitipornchai and Trahair discussed how to determine the capacity of tapered I-beams

subjected to different loading conditions. They used a finite element approach with uniform

sections to model the tapered beams. They demonstrated the use of this method by calculating

the buckling loads of a cantilevered and simply supported I-beam and comparing their results to

previous results from other publications.

They discovered the web tapered I-beam is adversely affected only slightly, while the

flange tapered I-beam is adversely affected by a large amount. They developed a chart displaying

the dimensionless capacity of a cantilevered I-beam subjected to a point load at the end to the

dimensionless length of that beam. Their chart only considered the LTB of the I-beam. So when

7

the length got relatively short, their capacities were no longer accurate because a different mode

of failure began to govern. They concluded that the finite element method was an efficient

method for determining the buckling load of tapered I-beams and that web tapering was more

efficient than flange tapering.

This was the first paper found during the research. Flange tapering was not considered

less efficient, because this project was more interested in the ease of connection. Their chart was

used to develop another chart that gave the capacity in kip-ft. and the length in feet. At some

point, the plastic hinge capacity of the beam began to govern, but it was not possible to

determine from their chart when there would start to be inelastic LTB.

Qiusheng, L., Hong, C., and Guiqing, L. (1995) “Stability analysis of bars with varying cross-

section.” Int. J. Solids Struct., 32(21), 3217-3228.

Quisheng et al. derived the buckling capacity of variable cross section columns with

variable distributed and concentrated loads. They also considered columns with boundary

conditions other than fixed-free. Several examples were worked through to confirm the

differential equation solutions were correct. They used Bessel functions and super geometric

series in order to determine the buckling capacities of the columns.

This project was interested in a linearly distributed load acting on a linearly varying cross

section column. The two boundary conditions of interest are the fixed-hinged and hinged-hinged

boundary conditions. The moment produces an axial force in the bottom flange and the flanges

are tapered. In order for lateral buckling of the I-beam to occur the bottom flange must buckle

about its strong axis. The buckling value presented by Quisheng et al. would have given a lower

bound for the LTB of the I-beam.

8

Rajasekaran, S. (1994). ”Instability of Tapered Thin-Walled Beams of Generic Section.” J. Eng.

Mech., 120(8), 1630–1640.

Rajasekaran used the virtual work method to formulate the finite elements. An updated

Lagrangian basis is used that accounts for the geometric nonlinearities. The finite elements are

used to determine the LTB capacity of varying types of beams and tapers. He claimed to have the

same results as Bradford and Cuk, but their results do not appear to match up.

He did not provide a graph for a flange tapered cantilever with the load applied at mid

height. The results did not seem to be reliable and they did not cover the conditions of interest.

Timoshenko, S. P., and Gere, J. M. (1961). Theory of elastic stability. 2nd

Ed. McGraw-Hill, New

York, N.Y.

Timoshenko and Gere wrote a book that goes into great detail about elastic stability of all

sorts of members and structures. There is a chapter about LTB of I-beams, but it only had the

derivation for the uniform I-beam. The problem becomes much more difficult once a taper is

introduced. This book also contains a chapter on the buckling of variable cross section columns

with variable distributed loads. Timoshenko and Gere derived the equations for basic cases and

some of the more complicated cases, but not every case. Tables were provided for many different

cross section and loading combinations, but they were only valid for a fixed-free condition. The

derivations of these tables were not presented in great depth in this book. Timoshenko and Gere

used Bessel functions in order to determine the buckling capacities of the columns.

This project was interested in a linearly distributed load acting on a linearly varying cross

section column. There was a table for coefficients of buckling for this exact case in the book, but

it is only valid for a fixed-free condition. The two conditions of interest are the fixed-hinged and

9

hinged-hinged boundary conditions. This project looked at the buckling of a column to try to get

a lower bound for the buckling values obtained from other journal papers. The moment produces

a variable axial force in the bottom flange and the flanges are tapered.

Yang, Y. and Yau, J. (1987). ”Stability of Beams with Tapered I-Sections.” J. Eng.

Mech., 113(9), 1337–1357.

Yang and Yau derived the differential equation of equilibrium for a tapered I-beam. They

also used virtual work and an updated lagrangian basis that accounted for the geometric

nonlinearity. They used the membrane theory of shells to determine the strains throughout the

beam. The displacements were determined using Vlasov’s thin-walled beam assumptions.

They present the solutions for the LTB of many loading cases of a simply supported

beam. This project was only interested in the LTB of a flange tapered cantilevered beam.

11

3 PROCEDURE

A literature review was performed on the topic of LTB of flange tapered I-beams. During

this review, graphs were discovered that gave the dimensionless load versus the dimensionless

beam parameter. The graph from Chan can be seen below in Figure 3-1. A dimensionless graph

allows a single graph to represent every possible type of I-beam. They are not user friendly when

only trying to size the I-beam. The goal was to develop user friendly graphs that show the

moment capacity versus the member length. In order to develop these graphs, Microsoft Excel

was used to transform the dimensionless load and beam parameter into the moment and the

length of the member respectively. The method used to determine the equation for the LTB was

to enlarge the graph and print it. The load values were obtained by finding the y-coordinate

associated with whole numbers along the beam parameter axis (x-axis). Those ordered pairs were

then entered into Excel. A graph was then created where the trend line function was utilized to

produce an equation for each line on the dimensionless graph.

The graph in Figure 3-1 has different lines for different amounts of taper. These amounts

of taper are represented by α. The width at the wide end is represented by a w, as shown in

Figure 1-1, and the width at the smaller end is w multiplied by α. All the graphs had a minimum

α of 0.25. The purpose of this research was to determine the LTB capacity of an I-beam that

tapers all of the way down to just the web thickness at the smaller end, which corresponds to an

12

Figure 3-1 Bifurcation of flange-tapered I-cantilever

α of essentially zero. The method used to extrapolate from the given values was similar to the

method used to determine the equations of the LTB. For each beam parameter, the value of the

load was obtained for each amount of taper and plotted in Excel. A trend line was used to create

a line through each set of four given points. An equation was then used to determine the value at

α = 0. This same process was used for each beam parameter value. This method can be seen in

Figure 3-2. The last number in each function on Figure 3-2 is the value when α = 0. For each

beam parameter, the α = 0 point was plotted to determine an equation for α = 0.

13

Figure 3-2 Graph used to determine the dimensionless load parameter for alpha = 0

Once the equations were determined, an Excel spreadsheet was used to solve for the

actual critical moment and length from the dimensionless parameters. In order to solve for the

actual values, a method was created to determine the cross sectional properties of any given W-

shape. The v-lookup function was used to find the properties in a spreadsheet from AISC that

lists all the properties of all standard steel shapes. This allowed the spreadsheet to automatically

update each of the cross sectional properties used in determining the actual moment and length

that cause LTB.

The equations produced in excel were used to determine the moment at each length. The

dimensionless graphs only gave the elastic LTB capacity. The plastic moment capacity was

determined and the lower of the two was plotted to form a graph of the capacity of the given I-

beam. Each graph produced contains five lines representing different amounts of taper for each

W shape.

14

In order to validate the reasonableness, the no taper (α = 1.0) line was compared to the

LTB capacity determined using the AISC formulas. A Cb factor of 1.67 was used because of the

expected moment distribution on the I-beam.

15

4 RESULTS AND DISCUSSION

4.1 Results

The no taper plot from the Chan paper was compared with the AISC manual equations.

As shown in Figure 4-1, the W14x22 shape matches up reasonably well with the manual

equations. A similar comparison was done with every other shape as well, but they are not

shown, because they are all similar. The comparisons to the manual equations were only done

out to a length of 50 feet because beams are not likely to have a longer un-braced length. The

maximum error for the W14x22 shape was just under 6%, and that occurred right at 50 feet. The

maximum error that was not associated with such a long un-braced length was just over 4%.

Similar values were calculated for many of the other W shapes. Since this amount of error is

acceptable, some confidence was developed in the graphs produced by Chan.

Other shapes of interest were examined and are presented here. In Figure 4-2, a

W12x210 is examined to determine the capacity of the shape versus its un-braced length. It can

be seen for zero taper, or alpha=1.0, that this shape could be approximately 80 feet long before

LTB would even be a concern. Prior to that point, a plastic hinge would form as the mode of

failure. According to the graphs developed from Chan’s work, when alpha=0.25, the un-braced

length would be reduced to 45 feet. For the case of a complete taper, or alpha=0.0, which is an

extrapolation of Chan’s work, the un-braced length would decrease to about 36 feet. There is

also a line included on the graphs that shows the moment when the extreme fibers of the member

16

Figure 4-1 Comparison of the developed graph and the AISC manual equations.

Figure 4-2 Developed graph for W12x210

would start to yield. The point where this line intersects the complete taper line is the point of

interest. An objective of this project was to have a beam with no permanent deformation. For the

W12x210 shape the maximum un-braced length would be 42.5 feet. As shown in Figure 4-2,

increasing the amount of taper will decrease the maximum un-braced length or decrease the

capacity if LTB is the governing failure mode.

17

The W14x311 has a maximum un-braced length of 55 feet, as shown in Figure 4-3. By

comparing Figure 4-4 to Figure 4-3, even though they have approximately the same amount of

steel per foot, the maximum un-braced length of the W14 shape is much greater than that of the

W33 shape. On the other hand, the W33 has a much higher capacity for shorter spans than the

W14 shape. From this, it can be seen that deep shapes are more effective in carrying a moment

for shorter un-braced lengths and shallow shapes are more effective in carrying a moment when

the un-braced lengths are long. Bracing would be the better option if bracing is relatively easy to

do. This point is also illustrated by comparing Figure 4-5 and Figure 4-6. An extreme example of

the superiority of shallow members with respect to LTB is shown by comparing Figure 4-4 and

Figure 4-7. Figure 4-4 is a W33x318 and Figure 4-7 is a W14x99. The maximum un-braced

length for each of the members is approximately 25 feet, even though the W33x318 has more

than three times the amount of steel per foot than the W14x99. Of course, if the demand was too

large for the W14x99 to handle, the fact that it would not fail due to LTB would mean absolutely

nothing.


18



19



20

It is evident that heavy shallow shapes are never really governed by LTB. LTB only

becomes an issue as beams become deeper or lighter. A beam that is both deep and light would

have a very short maximum un-braced length.

The proper method to size a beam would be to start with determining the demand. Once

the demand was determined any shape could be chosen from a drop down list of all the shapes.

Then, depending on the length of the required beam, a beam size could be chosen that has

adequate capacity, even when considering the effects of the tapered flange on the LTB capacity.

In general, if the shapes being selected for design have compact flanges, LTB will not govern.

4.2 Possible Sources of Error

The obvious source of error came from reading Chan’s graphs and coming up with

equations, by hand, which do not perfectly match the lines he had on his graph. This error should

be relatively small for the larger values of beam parameter; the length is not very sensitive to

changing beam parameter at the larger values. The lengths are very sensitive to changing beam

parameter as they approach zero. This is largely where the error is observed in the results, and

thus the most likely source of error.

Another possible source of error is that Chan did not specify the limitations for which W

shapes his dimensionless graph should be applied. It appears that it is geared toward slender

members because he never discusses any inelastic failures, but it would be possible at large

lengths to get elastic LTB even from compact members.

21

4.3 Further Research

One of the issues to be researched further is what occurs in the inelastic buckling region

of the graphs. How does the Cb factor affect the inelastic LTB? The current method proposed in

the AISC manual is to visualize the increased capacity, due to the Cb factor, as an upward

movement of the baseline capacity. Using this method, with a Cb factor of 1.67, often results in

raising the entire inelastic LTB section above the plastic moment line, thus essentially saying

that a plastic hinge will form before any inelastic LTB will occur. This may not be the best way

to think about what is actually happening. These issues with the Cb factor seem to be even more

complicated as taper is introduced into the problem.

Additional research should also be done on the limitations of the graphs obtained from

Chan. It would also be important to check other modes of failure in determining the actual

capacity of an I-beam including local flange buckling, web buckling, and shear failure.

23

5 CONCLUSION

The graphs used from Chan appear to be accurate, because they are in close agreement

with the AISC manual’s values for an I-beam with no taper. Tapering the flanges for an I-beam

will decrease the maximum un-braced length of the shape or it will decrease the capacity if LTB

is the governing failure mode. For shallow heavy shapes LTB is not an issue, even when

considering tapered flanges. As long as no permanent deformation is the objective, what occurs

in the possible inelastic LTB zone is not relevant. The intersection between the elastic LTB line

and the yielding moment line will provide the capacity of the shape in question.

25

REFERENCES

Al-Sadder, S. Z. (2004). “Exact expressions for stability functions of a general non-prismatic

beam–column member.” Journal of Constructional Steel Research, 60(11), 1561-1584.

Bradford, M., and Cuk, P. (1988). ”Elastic Buckling of Tapered Monosymmetric I-

Beams.” Journal of Structural Engineering, 114(5), 977–996.

Chan, S. (1990). ”Buckling Analysis of Structures Composed of Tapered Members.” Journal of

Structural Engineering, 116(7), 1893–1906.

Eisenberger, M. (1991). “Buckling loads for variable cross-section members with variable axial

forces.” International Journal of Solids and Structures, 27(2), 135–143.

Gupta, P., Wang, S. T., and Blandford, G. E. (1996). “Lateral-torsional buckling of nonprismatic

I-beams.” Journal of Structural Engineering, 122(7), 748-755.

Kitipornchai, S., and Trahair, N. S. (1972). "Elastic stability of tapered I-beams." Journal of the

Structural Division, ASCE, 98(3), 713-728.

Qiusheng, L., Hong, C., and Guiqing, L. (1995). “Stability analysis of bars with varying cross-

section.” International Journal of Solids and Structures, 32(21), 3217-3228.

Rajasekaran, S. (1994). ”Instability of Tapered Thin-Walled Beams of Generic Section.” Journal

of Engineering Mechanics, 120(8), 1630–1640.

Suryoatmono, B., and Ho, D. (2002). “The moment–gradient factor in lateral–torsional buckling

on wide flange steel sections.” J. Constr. Steel Res., 58(9), 1247-1264.

Timoshenko, S. P., and Gere, J. M. (1961). Theory of elastic stability. 2nd

Ed. McGraw-Hill, New

York, N.Y.

Yang, Y. and Yau, J. (1987). ”Stability of Beams with Tapered I-Sections.” Journal of

Engineering Mechanics, 113(9), 1337–1357.

Documents

Lateral-Torsional Buckling of Flange-Tapered I … Lateral-Torsional Buckling of Flange-Tapered I-Beams Michael Brett Thomas Department of Civil Engineering, BYU Master of Science