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INVESTIGATION OF SINGLE SPAN Z-SECTION PURLINS SUPPORTING STANDING SEAM ROOF SYSTEMS CONSIDERING

DISTORTIONAL BUCKLING

by

Scott D. Cortese

Thesis submitted to the faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

CIVIL ENGINEERING

APPROVED:

_________________________________ Thomas M. Murray, Committee Chairman

__________________________________ ________________________________ W. Samuel Easterling, Committee Member Raymond H. Plaut, Committee Member

May 2001

Blacksburg, Virginia

INVESTIGATION OF SINGLE SPAN Z-SECTION PURLINS SUPPORTING STANDING SEAM ROOF SYSTEMS CONSIDERING


by

Scott D. Cortese

Committee Chairman: Thomas M. Murray

Civil Engineering

(ABSTRACT)

Presently, the industry accepted method for the determination of the governing

buckling strength for cold-formed purlins supporting a standing seam metal roof system is

the 1996 AISI Specification for the Design of Cold-Formed Steel Structural Members, which

contains provisions for local and lateral buckling. Previous research has determined that the

AISI provisions for local buckling strength predictions of cold-formed purlins are highly

unconservative and that the AISI provisions for lateral buckling strength predictions of cold-

formed purlins are overly conservative. Therefore, a more accurate “hand” method is

needed to predict the buckling strengths of cold-formed purlins supporting standing seam

roof systems. The primary objective of this study is to investigate the accuracy of the

Hancock Method, which predicts distortional buckling strengths, as compared to the 1996

AISI Specification provisions for local and lateral buckling.

This study used the experimental results of 62 third point laterally braced tests and

12 laterally unbraced tests. All tests were simple span, cold-formed Z-section supported

standing seam roof systems. The local, lateral, and distortional buckling strengths were

predicted for each test using the aforementioned methods. These results were compared to

the experimentally obtained data and then to each other to determine the most accurate

strength prediction method.

Based on the results of this study, the Hancock Method for the prediction of

distortional buckling strength was the most accurate method for third point braced purlins

supporting standing seam roof systems. In addition, a resistance factor was developed to

account for the variation between the experimental and the Hancock Method’s predicted

strengths.

ACKNOWLEDGEMENTS

The author would like to express his genuine gratitude to his committee chairman,

Dr. Thomas M. Murray. Without the needed guidance and extreme patience of Dr. Thomas

M. Murray the completion of this thesis would not have been a possibility. Special thanks is

extended to committee member Dr. Raymond H. Plaut, who assisted as a technical advisor,

but is thought of as a friend. Also, appreciation is extended to Dr. W. Samuel Easterling for

serving as a committee member for this thesis. Other structures department faculty that had

key roles in providing the necessary educational background to the author include Dr.

Richard M. Barker, Dr. Siegfried M. Holzer, Professor Donald A. Garst, and Dr. Thomas E.

Cousins.

Numerous graduate students in the structural engineering program not only helped

the author with the completion of this thesis, but also with the many experiments conducted

at the Virginia Tech Structures Laboratory. These include Vincenza Italiano, Spencer Lee,

Matt Rowe, Alvin Trout, and Ron Fink. In addition, the help from Brett Farmer and Dennis

Huffman in the lab, and Ann Crate in the structural engineering office, will not go

unforgotten.

Without the support of my mother, father, and brother; this thesis would not have

been a reality. Their wisdom of life, advice of choice, and love has helped more than they

will ever know. I continue to feel grateful to have them in my life.

iii

TABLE OF CONTENTS

Page ABSTRACT……………………………...………………………………………………. ii ACKNOWLEDGEMENTS………………………...…………………………………... iii LIST OF FIGURES……………………………………………...……………………… vi LIST OF TABLES…………………………………………………………….....….…… vii CHAPTER I. INTRODUCTION AND LITERATURE REVIEW…………………… 1 1.1 Introduction………………………………………………………… 1

1.2 Literature Review…………………………………….……………....8 1.3 Need for Research…………………………………………………... 14

1.4 Scope of Research…………………………………………………... 16 1.5 Overview of Study..…………………………………………………. 18 II. DISTORTIONAL BUCKLING………………………………………… 20 2.1 Background…………………………………………………………. 20 2.2 AISI Specification Oversights……………………………………….. 21 2.3 AISI Local and Lateral Provisions………………………………….. 23 2.4 Determination of Local and Lateral Buckling Strengths..………...….. 27 2.5 Determination of Distortional Buckling Strength...……...…………... 28 2.5.1 Background………………………………………………….. 28 2.5.2 The Hancock Method for Determination of Distortional

Buckling Strength…………………………………………… 30 2.6 Determination of Section Strength.………………….……………… 40

III. EXPERIMENTAL TEST DETAILS AND RESULTS………………….41

3.1 Background and Test Details..………………………………………. 41 3.2 Experimental Results...……………………………………………… 44 3.3 AISI Specification Analysis…………...…………………………….. 49 3.4 Distortional Buckling Analysis……………………………………… 54

iv

TABLE OF CONTENTS (continued)

CHAPTER Page

IV. COMPARISON OF RESULTS……………………………………… 59

4.1 General………………..…………………………………………….. 59

4.2 Third Point Braced and Unbraced Analyses.........…..………………... 60 4.3 Prior Research………………………………………………………. 75 4.4 Possible Causes of Scatter in Data……………………………...…… 79 4.5 Resistance Factor for Design………………………………………... 85 V. EXAMPLE CALCULATIONS………………………....……………….. 88 5.1 Problem Statement for an 8 in. Deep Z-Section…………………….. 88 5.2 Calculation of Section Properties……………………………………. 89 5.3 Local and Lateral Buckling Strength Predictions..…………………… 98 5.4 Distortional Buckling Strength Prediction……………………………99 VI. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS……… 105 6.1 Summary…………………………….……………………………... 105 6.2 Conclusions………………………....…………….………………... 106 6.3 Design Recommendations.……………………………………...… 108 REFERENCES………………………………………………………………………… 110 APPENDIX A ……………………………………………………………….….……… 115 APPENDIX B…………………………………………………….……………………. 119 VITA…...….………………………………………………….………………………… 128

v

LIST OF FIGURES

FIGURE Page

1.1 Standing Seam Roof System Profile…………………………………………… 2

1.2 Point Symmetry of a Typical Z-Section………………………………………... 3

1.3 Buckling Modes of a Z-Purlin…………………………………………………. 4

1.4 Buckling Modes Subject to a C-Purlin for Major Axis Bending………………... 6

1.5 Buckling Modes of a Z-Purlin for Bending about a Horizontal Axis…………... 7

2.1 Geometric Properties Measurement Plan..…………………………………….. 27

2.2 Measurements for Section Properties………………………………………….. 31

2.3 Stiffness Restraints…………………………………………………………….. 35

2.4 Beam Web Behavior in Flexure……………………………………………...… 36

3.1 Typical Base Test Setup……………………………………………………….. 42

3.2 Clip Types…………………………………………………………...………… 43

3.3 Steel Panel Types……………………………………………………………… 44

4.1 Experimental Strengths Vs. Local Buckling for 8 in. Deep Z-Sections………… 61

4.2 Experimental Strengths Vs. Local Buckling for 10 in. Deep Z-Sections……… 61

4.3 Experimental Strengths Vs. Lateral Buckling for 8 in. Deep Z-Sections……… 63

4.4 Experimental Strengths Vs. Lateral Buckling for 10 in. Deep Z-Sections……… 63

4.5 Experimental Strengths Vs. Distortional Buckling for 8 in. Deep Z-Sections..… 65

4.6 Experimental Strengths Vs. Distortional Buckling for 10 in. Deep Z-Sections… 65

4.7 Overall Experimental Strengths Vs. Predicted Buckling Strengths……………... 66

5.1 Properties of Section 1G………………………………………………………. 89

vi

LIST OF TABLES

TABLE Page

3.1 Summary Table of Experimental Strengths and Properties of Third

Point Braced Z-Sections………………………………………………………. 45

3.2 Summary Table of Experimental Strengths and Properties of Laterally

Unbraced Z-Sections………………………………………………………….. 48

3.3 Summary Table of 1996 AISI Specification Strengths of Third

Point Braced Z-Sections………………………………………………………. 51

3.4 Summary Table of 1996 AISI Specification Strengths of Laterally

Unbraced Z-Sections….………………………………………………………. 54

3.5 Summary Table of Nominal Distortional Buckling Strengths of

Third Point Braced Z-Sections………………….……………………………... 56

3.6 Summary Table of Nominal Distortional Buckling Strengths of

Laterally Unbraced Z-Sections………………….……………………………... 58

4.1 Summary Table for Nominal Strengths of Third Point Braced

Z-Section from Murray and Trout (2000)…….……………...……………….... 69


Z-Section from Bryant et al. (1999a)………….……………...……………….... 70


Z-Section from Almoney and Murray (1998).………………..……………….... 70


Z-Section from Davis et al. (1995)…….…….……………...…………………. 71

vii

LIST OF TABLES (continued)

TABLE Page


Z-Section from Bathgate and Murray (1995).…………….….…....…….…….... 71


Z-Section from Borgsmiller et al. (1994)….…...….……..…..….…….……….... 72


Z-Section from Earls et al. (1991)………….….….….……………………….... 72


Z-Section from Brooks and Murray (1989)…..….…………...……………….... 73


Z-Section from Spangler and Murray (1989)…….…………...……………….... 73

4.10 Summary Table for Nominal Strengths of Laterally Unbraced

Z-Section from Bryant et al. (1999b)….…….….….….……...……………….... 74

4.11 Summary Table for Nominal Strengths of Laterally Unbraced

Z-Section from Bryant et al. (1999c)………..….….….……...……………….... 74

4.12 Summary Table of All Z-Purlin Strength Data………………………………… 75

4.13 Summary Table for Comparison of Results for Laterally Braced Z-Sections…....78

4.14 Summary Table of Third Point Braced Test Components……………………... 81

4.15 Summary of Effect of Clip Type for 8.0 in. Deep, 0.102 in. Thick

Third Point Braced Tests……………………………………………………… 85

5.1 Summary of Predicted and Experimental Strengths for Z-Section 1G…………102

viii

CHAPTER I

INTRODUCTION AND LITERATURE REVIEW

1.1 Introduction

Cold-formed steel products such as Z-purlins have been commonly used in the

metal building construction industry for more than 40 years. The popularity of these

products has dramatically increased in recent years due to their wide range of application,

economy, ease of fabrication, and high strength-to-weight ratios. Z-Purlins are

predominantly used in light load and medium span situations such as roof systems.

A conventional through-fastened roof system consists of C- or Z-section purlins

supporting steel deck. This steel deck is directly fastened to the purlin, usually by self-

tapping screws, and therefore provides full lateral bracing to the purlins. However, due

to the nature of steel to contract and expand with a change in temperature, the holes

through which the steel deck is attached to the purlin become enlarged. In turn, this

allows water to seep into the structure through these enlarged holes. This problem was

alleviated by the advent of the standing seam roof system.

The standing seam roof system differs from the conventional through-fastened

roof system by the introduction of a clip placed intermediately between the purlin and the

decking (see Figure 1.1). Water leakage into a structure is prevented because the clip is

embedded into the seam of the deck panels and only fastened to the purlin. However, the

1

advantage the clips provide over water leakage comes at a cost. Unlike the through-

fastened system, the introduction of the clips does not allow the steel deck to provide full

lateral bracing for the purlins. Studies have shown that this standing seam roof system

acts somewhere between a fully braced and an unbraced condition (Brooks and Murray

1990).

Steel Panel

Purlin

Fastener Clip

TopFlange

of Purlin

Figure 1.1 Standing Seam Roof System Profile

The point symmetric section (a section where the shear center and centroid of the

section coincide) properties of a typical Z-section are such that when attached to steel

decking and subject to gravity loading, it tends to twist and deflect in both the vertical

and horizontal directions (see Figure 1.2). However, this torsional force is partially

resisted by the interaction of the clip and deck to the purlin. In turn, this interaction can

increase the strength of the purlin. Other lateral bracing, such as light gage angles spaced

at third points, can further increase purlin strength.

2

X

Y

x y

X,Y - Shear center axes

x.y - Principal axes

Figure 1.2 Point Symmetry of a Typical Z-Section

Conventional design of these cold-formed standing seam roof systems has

mandated the checking of two major types of buckling: local and lateral-torsional. New

provisions, as recent as 1996, in the American Iron and Steel Institute’s Specification for

the Design of Cold-Formed Steel Structural Members (AISI 1996) (hereafter referred to

as the 1996 AISI Specification) have been unsuccessful in correctly predicting the

bending strength of most of these standing seam roof systems. This is in part due to the

AISI Specification overlooking distortional buckling as a possible cause of failure, and

the inability of the AISI Specification to take into account the partial lateral bracing

provided by the standing seam roof system to the supporting purlin. This is shown by

careful study of research completed by others where intermittently braced roof systems

have been tested to failure. These other experimental works show that distortional

buckling, not lateral or local buckling, control the strength of the system under study.

3

A cold-formed Z-section can buckle in three modes: local, lateral, and

distortional. Local buckling of a Z-section purlin is the internal buckling of the section’s

elements so that there is no relative movement of the nodes; both corners of the

compression element remain in longitudinal alignment and the adjoining lip, flange, and

web elements buckle by plate flexure at half-wavelengths comparable with the flange

width (Rogers and Schuster 1997). Lateral buckling is a rigid-body translation of the

purlin without any change in the purlin cross-sectional shape (Hancock et al. 1998). It is

important to note that when a cold-formed Z-section under flexure is unrestrained

laterally between supports, it is liable to displace laterally and twist after yielding, and the

full strength of the cross-section cannot be reached unless the section is laterally braced at

frequent intervals (Pi et al. 1997). Distortional buckling will be discussed in the

following paragraphs. Figure 1.3 shows the three main types of buckling which a typical

Z-section cold-formed purlin can experience.

Lip-Flange Distortional BucklingLocal Buckling

Tension

Compression Compression

Flexural-Torsional Buckling

(Lateral Buckling)

Tension Tension

Compression

Figure 1.3 Buckling Modes of a Z-Purlin

4

Z-section purlins subject to both flexure and torsion may go through a buckling

phenomenon where only the compression flange and lip rotate about the flange-web

junction. Therefore, the web can be said to torsionally restrain the flange-lip component.

In less common situations, the lip-flange component may rotate about the web-flange

junction, which is followed by a lateral movement of the flange-web corner. This

includes transverse bending of the web near ultimate failure. In this case, the flange-lip

component will torsionally restrain the web. Unless otherwise noted, from this point

forward any reference to distortional buckling will describe the event where the

compression lip-flange component rotates about the web-flange junction.

Distortional buckling most often occurs in purlin sections where lateral

deformations (i.e. lateral buckling) are prevented by intermittent bracing (Ellifritt et al.

1998). Therefore, lateral bracing may heighten this buckling phenomenon. As

previously mentioned, the compression lip-flange component rotates about the web-

flange junction. This rotation alleviates the stress built up in the compression lip (Ellifritt

et al. 1992). However, the presence of lateral braces and steel panels (if any) prevent this

rotation from occurring, which increases the stress on the compression lip.

Distortional buckling is a distortion of the angle between the lip-flange

component and the web of Z- and C-sections under load. This distortion results in a

reduction of the section stiffness, which in turn can cause failure. It is important to note

that this failure mode is termed distortional buckling because unlike local or lateral

buckling, the section actually distorts. This distortion is commonly caused by a rotation

of the lip-flange component about the web plate, but can also arise from a rotation of the

web plate about the lip-flange component (Davies et al. 1998). In addition, distortional

5

buckling can occur at wavelengths intermediate to local and lateral buckling and at

stresses less than local buckling, as shown in Figure 1.4. This is especially important

since purlins that are adequately designed for lateral buckling are assumed to have a

strength equal to that of local buckling.

Lateral distortional buckle (tension flange restraint)

Flexural- torsional buckle (Lateral buckle)Flange

distortional buckle

Local buckle

800

700

600

500

400

300

200

100

100 1000 10000 100000

Stre

ss in

Com

pres

sio n

Fla

nge

at B

uck l

ing

(MP a

)

Buckle Half-Wavelength (mm)

Top flange in compression, bottom flange in tension

2

1

3

4

Figure 1.4 Buckling Modes Subject to a C-Purlin for Major Axis Bending

Recreated from Design of Cold-Formed Steel Structures, 3rd Edition (Hancock 1998)

In Figure 1.4, the first minimum (Point 1) is a local buckling mode, which

involves buckling of the web, compression flange, and lip stiffener. The second

minimum (Point 2) is the flange distortional buckling mode and involves the rotation of

the compression lip-flange component about the web-flange junction. At longer

wavelengths where the purlin is unrestrained, a flexural-torsional or lateral buckling

6

mode occurs (Point 3). However, if the tension flange is torsionally restrained, then a

lateral distortional buckling mode may take place, as shown by Point 4 (Hancock 1998).

This lateral distortional buckle strength is dependent on the degree of torsional restraint

provided to the tension flange (Hancock 1998). Furthermore, these same buckling modes

may occur in a Z-section, as shown in Figure 1.5.

800

700

600

500

400

300

200

100

100 1000 10000 100000

Buc

klin

g S

tre s

s (M

Pa)

Buckle Half-Wavelength (mm)

Top flange in compression, bottom flange in tension

Vertical Lip

Sloping Lip

Local Buckle

Flange Distortional Buckle

Figure 1.5 Buckling Modes of a Z-Purlin for Bending about a Horizontal Axis

Recreated from Design of Cold-Formed Steel Structures, 3rd Edition (Hancock 1998)

As Figures 1.4 and 1.5 show, the flange distortional buckling failure mode is of

particular importance due to its tendency to be the limiting state of failure in purlins that

support standing seam cold-formed roof systems. Because of this, an accurate and

precise method of determining the strength of a Z-section for the failure mode of

distortional buckling is required. Different models have been devised by researchers that

7

account for distortional buckling in cold-formed Z-section purlins. Finite strip models

present the most accurate method to determine distortional buckling, but may not be

economically efficient for designers. Therefore, a “hand” method that uses the unified

effective width approach is needed to accurately predict distortional buckling.

Of the available hand methods that can predict distortional buckling strength, the

Modified Lau & Hancock Method (Hancock et al. 1996) appears to be one of the best

ways to determine distortional buckling. In other studies, this method has been compared

to experimental data and numerous other hand methods. The results from these studies

show that the Modified Lau & Hancock Method gives slightly conservative strength

predictions when compared to experimental data, but more precise and accurate results

when compared to the other hand methods.

1.2 Literature Review

Distortional buckling in cold-formed steel is a relatively new failure mode.

However, a large amount of research has been completed on this subject in a fairly short

amount of time. Completed research includes various methods to predict distortional

buckling. Experimental results from tests of cold formed Z- and C-section purlins used

in roof systems, storage racks, and columns analyzed the applicability of most of these

devised methods. Included herein is a summary of the more important research works

pertaining to distortional buckling.

Distortional buckling is the controlling failure mode for most Z-section purlins

with deep, slender webs. The 1996 AISI Specification tries to account for this buckling

mode through an empirical reduction of the plate buckling coefficient (k). The

8

experimental work for this (Desmond et al. 1981) concentrated on local buckling of the

flange and subsequently used back-to-back sections so the web did not buckle. This

study was completed to better study flange buckling. In turn, this severely restricted

distortional buckling from occurring. Because of this, more recent experiments on

laterally braced flexural members with edge stiffened flanges such as Hancock (1997),

Ellifritt et al. (1998), and Willis and Wallace (1990) yielded unconservative strength

predictions using the AISI Specification.

Research completed on longitudinal stiffeners for compression members provided

a method for determining the elastic buckling strengths of columns, plates, and flanges

with stiffening lips (Sharp 1966). Although this study does not directly pertain to

distortional buckling, it is the basis for the Lau & Hancock Method and the Modified Lau

& Hancock Method, as well as others. This is accomplished by the introduction of the

elastic buckling formula for a plate structure and the rotational stiffness restraint equation

(KΦ) for lipped flanges.

It was not until 1985, with the help of the finite strip method, that a detailed

design chart was devised for computing the critical stress for the distortional mode of

buckling in cold-formed sections (Hancock 1985). However, this design chart was

specific only to certain geometries of channel sections.

A hand method for determining distortional buckling stress for thin-walled, cold-

formed compression members was derived by Lau and Hancock in 1986 and published

by Lau and Hancock in 1987. The expressions were developed in part from the flexural-

torsional buckling theory of undistorted thin-walled columns developed by Timoshenko

and Gere in 1959 and from Sharp’s elastic buckling stress equation for aluminum plates.

9

This new distortional buckling equation was compared to finite strip buckling analyses to

determine its validity and range of application (Lau and Hancock 1987). The researchers

determined that the Lau & Hancock Method agreed well with the finite strip results if the

lip stiffeners satisfied the 1980 AISI Specification (AISI 1980), the ratio of web depth to

flange width was between 0.5 and 2.5, and the translational stiffness restraint was

assumed to be equal to zero. This is the basis of the Modified Lau & Hancock Method

used in this study.

While not initially concerned with the effects of distortional buckling, a study in

1990 was performed to determine if fastener location played an important role in purlin

capacity in through-fastened roof systems (Willis and Wallace 1990). The primary

finding of this study was that fastener location is vital to the torsional restraint in C-

section purlins, but had no effect in Z-section purlins. Therefore, fastener location should

affect the local, lateral, and distortional buckling strength predictions for C-sections. A

secondary result, and the most important for this study, was that the researchers found the

AISI Specification to predict unconservative purlin strengths in the local buckling mode.

The ¼-point bracing requirement by the American Iron and Steel Institute’s

Specification for the Design of Cold-Formed Steel Structural Members first appeared in

the 1956 edition and was further tested in 1992 using different experimental setups at the

University of Florida (Ellifritt et al. 1992). The primary finding was that the ¼-point

bracing was not required for cold-formed flexural members that are not attached to

decking or sheathing. Subsequently, this provision was removed in the 1996 AISI

Specification. Furthermore, this study also determined that all unbraced tests failed by

translation-rotational buckling and all braced (brace spacing closer than mid-point) tests

10

failed by distortional buckling. The significance of this is that some of the tests, which

failed by distortional buckling, failed at a load less than predicted by the lateral buckling

equations of AISI Specification Section C3.1.2.

In 1995 a draft ballot and commentary containing the Modified Lau & Hancock

Method was submitted to the American Iron and Steel Institute Specification Committee,

detailing a procedure to determine cold-formed purlin strength considering distortional

buckling for any purlin geometry (Hancock 1995). This ballot is reproduced in Appendix

A. The intent of the ballot was to alleviate problems within the original ballot containing

provisions for the determination of distortional buckling. With slight modifications, this

1995 draft ballot became a working ballot on October 2, 2000.

Research was conducted at the University of Sydney, in Australia, to further study

the effects of both types of distortional buckling (Hancock et al. 1996). As previously

discussed, the most common type is where the web torsionally restrains the compression

lip-flange component, and the other type is where the compression lip-flange component

torsionally restrains the web. The intent of this study was to experimentally validate the

Modified Lau & Hancock Method that takes into account distortional buckling through

an iterative process. This modified method compared well with experimental tests and

subsequently strengthened the draft ballot of 1995. Furthermore, the Modified Lau &

Hancock Method is the procedure used for this study.

Most distortional buckling equations conservatively predict purlin strengths. In

light of this, General Beam Theory was thought to better account for the interaction of

different buckling modes and alternative load patterns. General Beam Theory was used

in conjunction with the Modified Lau & Hancock Method to achieve better accuracy of

11

predicted buckling stresses (Davies and Jiang 1996). However, it was realized by the

authors that General Beam Theory should not be used to predict buckling strengths due to

its lack of a practical basis for a design code.

In a research study titled, “Lateral Buckling Strengths of Cold-Formed Z-Section

Beams,” a nonlinear inelastic finite element model for analyzing cold-formed Z-sections

and the effect of lateral-distortional buckling was discussed (Pi et al. 1997). This model

is unique because it takes into account the effects of web distortion, the rotation of a

yielded cross-section, pre-buckling in-plane deflections, initial imperfections, residual

stresses, material inelasticity, and the effects of a stiffening lip. This study showed that

cold-formed Z-sections need to be braced at frequent intervals to develop their full

moment capacity, and that Z-sections with web distortion have a lower strength

prediction than sections without web distortion.

In an effort to determine the best “hand” calculation method for predicting

distortional buckling, nine different approaches were compared to experimental results

and to each other (Rogers and Schuster 1997). The nine tested equations consisted of the

S136-94 Standard, AISI Specification (AISI 1996), Lau & Hancock S136-94 Standard

Method (Hancock and Lau 1990), Lau & Hancock AISI Method (Hancock and Lau

1987), Modified Lau & Hancock S136-94 Standard Method (Hancock 1994), Modified

Lau & Hancock AISI Method (Hancock and Lau 1996), Marsh Method (Marsh 1990),

and a Moreyra & Pekoz Method (Moreyra and Pekoz 1993). The Modified Lau &

Hancock Method with the S136-94 Standard for calculating the effective section modulus

was found to most precisely account for distortional buckling and therefore gave the best

results when compared to the experimental data. However, the S136-94 effective width

12

provisions are from the Canadian Standards Association and are not used in the AISI

Specification.

The Modified Lau & Hancock Method for determining elastic distortional

buckling stress and two Sharp Methods for determining elastic buckling stress were

compared to each other using the results from various section models based on a finite

strip method (Hancock 1997). The Modified Lau & Hancock Method was shown to be

more accurate than both Sharp Methods.

An experimental study of laterally braced cold-formed steel flexural members

with edge stiffened flanges determined that traditional design methods for cold-formed

steel takes into account local buckling, but not distortional buckling (Schafer and Pekoz

1998). This study was completed to determine a unified width treatment of distortional

buckling, and to achieve more accurate results than the slightly conservative Sharp

method and Modified Lau & Hancock method. A new hand design method based on the

unified effective width approach for strength prediction considering distortional buckling

was presented. In addition, this new design method used new expressions for the

prediction of local and distortional buckling and presented a new approach for the

determination of the web effective width. The developed design method was compared

to experimental data and resulted in more accurate and precise strength predictions than

the AISI Specification. However, a comparison between this method and the Modified

Lau & Hancock Method was not made.

Although the Modified Lau & Hancock Method does present a means for

determining distortional buckling, it can prove to be an over-intensive method for design.

This is due to the need for calculation of several section properties not currently given in

13

any design tables of standard shapes. Therefore, a simplified method for a fast and easy

way to determine distortional buckling was sought (Ellifritt et al. 1998). The Modified

Lau & Hancock Method was analyzed to determine which parameters had the most

influence on the results. From this, an equation was devised involving only yield

strength, section thickness, web depth, flange width, and lip depth. The output of this

simplified approach compared well to the Modified Lau & Hancock Method

(approximately 3% conservative error). However, the simplified approach was deemed

to only serve as an approximate method as stated by the authors, “If a more exact result is

desired, one can always go back to the more exact method.”

In 1998 a research paper titled, “Buckling Mode Interaction in Cold-Formed Steel

Columns and Beams” detailed how distortional, local, and lateral buckling may occur

together in conjunction with compression force and bending moment interaction (Davies

et al. 1998). The authors used General Beam Theory as a means to account for the

interactions of buckling modes and axial forces. In addition to this, the main fault of the

AISI Specification was described to be its assumption that the failure load is based on the

stress in the most highly stressed fiber in compression rather than using a stress gradient

throughout the section.

1.3 Need for Research

To date, distortional buckling experimentation has been completed without

standing seam roof systems. However, the experimental tests used for this study

incorporated standing seam roof systems. These standing seam roof systems utilize a clip

to attach the steel panels to the compression flange of a purlin. In turn, these three

14

elements can act as a diaphragm to resist lateral forces. The stiffness of this diaphragm,

which is governed by the clip type (fixed, articulating, sliding), will control the amount of

lateral resistance that can be provided, and ultimately have an effect on the load capacity

of the purlin.

Because of the interaction the clips and steel panels present, the compression

flange of the purlin is not completely laterally braced, nor completely laterally unbraced.

Hence, the steel panels and clips represent a form of torsional and lateral restraint on the

compression lip-flange component of a purlin, which increases the purlin’s strength to a

certain degree. In addition to this, the use of numerous clip types from different

manufacturers, as well as different types of steel panels and panel thicknesses (such as

the case for this study) can have widely varying effects on standing seam roof system

strengths and ultimately the purlin strength itself. Clips with the ability to move, such as

the articulating clips, provide far less lateral support than fixed clips, which transfer

almost all lateral force between purlin and steel panel. The ability of the purlin and steel

deck to work together in sharing the resistance of this lateral force results in a stronger

standing seam roof system, unlike the articulating or sliding clip system (Murray and

Trout 2000). Moreover, thicker steel panels can resist more lateral force and further

increase the strength of the system.

The ability to accurately predict the strength of these standing seam roof systems

considering only conventional lateral buckling design considerations is extremely

conservative. On the other hand, local buckling strength prediction provisions are

extremely unconservative. As previously mentioned, this is due to the fact that the tested

purlins are somewhere between fully unbraced and fully braced. The cause of this is the

15

interaction of the clip and steel panel with the supporting purlin. Since each series of

tests used different combinations of steel panel, clip type, Z-section depth and thickness,

it is nearly impossible to accurately predict the lateral and local buckling strengths using

the guidelines in the AISI Specification, Sections C3.1.1 and C3.1.2.

All experimental research conducted on the effects of distortional buckling used

laterally braced purlins without steel panels and clips (standing seam roof system). This

lateral bracing was closely spaced in order to control for lateral buckling. However, most

data used in this study consist of purlins that support standing seam roof systems.

Furthermore, the spacing of the lateral braces in this study was not originally designed to

study the effects of distortional buckling. Therefore, these tests provide a means to

obtain experimental results to test the validity of the AISI Specification provisions for

lateral buckling strength predictions as compared to the distortional buckling strength

prediction guidelines.

1.3 Scope of Research

The purpose of this study is to determine a means to accurately predict the

strength of cold-formed Z-section purlins that support standing seam roof systems by

examining predicted strengths considering the limit states of lateral buckling and

distortional buckling. This is accomplished in three major steps.

The first step is to verify that the current AISI Specification does yield

unconservative strength predictions for local buckling and conservative strength

predictions for lateral buckling for cold-formed, third point braced, Z-sections in flexure.

This will show that further study into the causes of failure in cold-formed Z-section

16

supported standing seam roof systems is needed. Moreover, if the 1996 AISI

Specification cannot accurately predict purlin section strength from either local buckling

or lateral buckling provisions, then there is the possibility of another controlling failure

mode. This first step is completed by determining section strengths using AISI

Specification Section C3.1.1 for local buckling and AISI Specification Section C3.1.2 for

lateral buckling. This process is aided by a computer program, called Cold Formed Steel

Design Software, Version 3.02, which from this point forward will be referred to as CFS

(RSG Software, Inc. 1998). AISI Specification Section C3.1.1 provides a fully braced

purlin strength prediction, which is the local buckling strength of the section, while the

AISI Specification Section C3.1.2 provides a strength prediction that takes into account

the spacing of the lateral braces (the flanges are considered to be unbraced by the steel

decking for this study). This strength prediction from AISI Specification Section C3.1.2

is the section’s lateral buckling strength. These two strength predictions will be

compared to the experimental strength results.

The second step is to define an alternate method of estimating the positive

moment strength of Z-section purlins supporting standing seam roof systems. The

method used in this research is the Modified Lau & Hancock Method (from here forth

referred to as the Hancock Method for simplicity). This method was chosen due to its

consistent and accurate results regardless of section geometries to determine distortional

buckling. In addition, this method uses the unified width approach, which allows for

easy implementation into the AISI Specification. In order to compare the Hancock

Method with the 1996 AISI Specification results, distortional buckling values were

determined using this method. All three strength values (AISI local buckling, AISI

17

lateral buckling, distortional buckling) are then compared to the experimental results,

which is the third purpose of this study.

The third step of this research is to validate the chosen distortional buckling

method, and show the need to consider another failure mode other than local or lateral

buckling. This is accomplished by comparing the Hancock Method’s results and the

AISI Specification results to compiled experimental data. To determine the accuracy of

the Hancock Method and the two AISI predictions, standard deviations and coefficients

of variation are calculated for each series of tests. Experimental data consist of different

experimental test setups (differences in clip type, span length, purlin depth and thickness,

and steel panel thickness) in order to better test the range and application of the Hancock

Method and the AISI Specification to purlins that support standing seam roof systems.

1.4 Overview of Study

Chapter II of this study presents a detailed description of the Hancock Method for

calculating distortional buckling. Included with this is a discussion of why the current

AISI Specifications do not accurately account for distortional buckling and result in

unconservative (for local buckling) and conservative (for lateral buckling) strength

predictions. In addition, a brief introduction of Cold Formed Software is included to

show where and how the AISI strengths in this research were obtained.

Details of test setups and the results for these experimental setups are found in

Chapter III. In addition, AISI Specification strength prediction results, as well as the

Hancock Method strength prediction results, are found in Chapter III. Experimental tests

consist of laterally braced and unbraced purlins, all with steel deck attached using various

18

clip types to form a standing seam roof system. The 1996 AISI Specification results

consist of strength predictions for third point laterally braced test purlins (local and lateral

buckling), and laterally unbraced test purlins (local and lateral buckling). The Hancock

Method uses the full and effective section modulus from the AISI Specification

guidelines along with other geometric properties to calculate the distortional buckling

strength of the section.

Comparisons are made between the experimental results, AISI Specification

provisions, and the Hancock Method in Chapter IV. Also discussed are the effects that

purlin orientation, clip type, angle of edge (lip) stiffener, and panel type may have on the

experimental strengths and how these can be a cause of data scatter when comparing the

experimental data to the AISI and distortional buckling results.

Chapter V presents a step-by-step procedure for determining local, lateral, and

distortional buckling strengths using the aforementioned methods. This includes example

calculations of distortional buckling, and AISI Specification example calculations of

local and lateral buckling.

Chapter VI gives a summary of the study and presents major conclusions. This

chapter also contains the development of a resistance factor for design when considering

distortional buckling, and recommendations for future research on standing seam roof

systems taking into account distortional buckling.

19

CHAPTER II


2.1 Background

A type of buckling mode, called distortional buckling, which is unlike local or

lateral buckling, may control the design of certain laterally braced, cold-formed steel

sections. If distortional buckling occurs, these sections actually distort while failing,

which is uncharacteristic of both local and lateral buckling. Since the distortion of the

section is the cause of failure, this type of buckling was appropriately named distortional

buckling and has been under study since 1962 (Yu 2000). In 1985, Dr. Hancock of the

University of Sydney, in Australia, introduced a method for the determination of

distortional buckling in cold-formed channel sections. This was later followed by a

“hand” method in 1987, which allows for the analysis of both C- and Z-cold formed

sections with the lip stiffener at any angle.

Historically, cold-formed Z-sections have been designed for two different

buckling modes. The first, local buckling, typically occurs in well laterally braced C- and

Z-purlin systems at higher stresses and shorter wavelengths than lateral buckling

(Hancock and Lau 1987). On the other hand, laterally unbraced C- and Z-purlin systems

tend to fail by deflecting normal to the load applied, to a point where the system

experiences lateral buckling. Therefore, to increase the strength of a purlin, adequate

20

bracing is required so that the wavelengths are short enough for only local buckling to

occur. However, distortional buckling has been shown to occur at wavelengths shorter

than lateral buckling and at stresses less than local buckling (Davies and Jiang 1996;

Hancock and Lau 1987). This is of particular importance because lateral buckling can be

prevented by economically efficient bracing, but distortional buckling cannot. Figure 1.4

shows the relationship between the three types of buckling, and the stresses and

wavelengths associated with each mode. To sufficiently brace against distortional

buckling would yield a standing seam roof system uneconomical due to the high number

of braces spaced at close intervals. Therefore, distortional buckling is a design

consideration.

2.2 AISI Specification Oversights

Currently, the 1996 AISI Cold-Formed Design Specification does not have

sufficient procedures for design against distortional buckling. The AISI Specification

attempts to account for distortional buckling through an empirical reduction of the plate

buckling coefficient (k) when calculating the effective design width of the compression

element (Schafer and Pekoz 1998). This effect was supposed to account for the inability

of the edge stiffener (lip) to prevent distortional buckling (Yu 2000). The experimental

work carried out for this was completed in 1981 (Desmond et al. 1981) and concentrated

on flange local buckling. The experiments used back-to-back sections so the web did not

buckle, in order to solely concentrate on local buckling of the flange. In turn, this

severely restricted distortional buckling from occurring. This resulted in the inability of

21

the current AISI Specification to effectively determine a Z-purlin’s true strength capacity

when considering distortional buckling.

Typically, purlin systems are not designed using back-to-back sections. More

recent experimental research on adequately, laterally braced flexural members with edge-

stiffened flanges that were not placed back-to-back such as Wallace and Willis (1990),

Hancock et al. (1996), and Ellifritt et al. (1998) yielded unconservative strength

predictions using the AISI Specification. This is due to the fact that distortional buckling

occurs at shorter wavelengths than lateral buckling and at lower stresses than local

buckling. Hence, the AISI Specification, which improperly accounts for distortional

buckling, gives the designer an unconservative or false sense of strength for most purlins.

In short, a designer following the current AISI Specification may unconservatively design

a purlin even though it is adequately braced for lateral buckling.

This oversight has been shown in recent experimentation. In the mid-1990’s,

research was carried out at the University of Florida to further study distortional

buckling. The study not only showed that the AISI Specification yielded unconservative

strength predictions, but also revealed that the failure mode in most well-braced tests was

distortional buckling, as compared to local or lateral buckling (Ellifritt et al. 1998). This

study further showed that as unbraced lengths became large, lateral buckling controlled,

while distortional buckling controlled for shorter brace lengths.

Another inaccuracy of the AISI Specification, Section C3.1.1, Nominal Section

Strength, deals with lateral brace spacing. The same members at longer unbraced lengths

(when these members do not fail by lateral buckling) have the same strength prediction

using the AISI Specification (Schafer and Pekoz 1998). Therefore, the AISI

22

Specification, Section 3.1.1, is not a function of brace length because local buckling

assumes a given section to be fully braced. As previously mentioned, distortional

buckling may occur at stresses lower than the local buckling strength obtained from AISI

Specification, Section C3.1.1.

2.3 AISI Local and Lateral Buckling Provisions

Equations used in this study for the prediction of local and lateral buckling

strengths are from the 1996 AISI Specification, Sections C3.1.1 for local buckling and

C3.1.2 for lateral buckling. Example calculations using Sections C3.1.1 and C3.1.2 can

be found in Sections 5.1 through 5.3 of this study.

The AISI provision used in this study for constrained bending local buckling

(AISI Eqn C3.1.1-1) is

(2.1) yen FSM =

where

Mn = Nominal flexural strength

Fy = Design yield stress

Se = Elastic section modulus of the effective section calculated

with the extreme compression fiber at Fy

The calculation of the elastic section modulus (Se) of a Z-section is typically an

iterative process with an initial guess of the location of the horizontal neutral axis (X-

axis) and with the assumption that the web is fully effective. If the web is determined to

not be fully effective, then the horizontal neutral axis has to be relocated using the

partially effective web. The elastic section modulus is calculated using AISI Section

23

B4.2 for the compression flange, Section B3.2(a) for the stiffener lip, and Section B2.3

for the web.

The AISI provision used in this study for the determination of lateral buckling

strength (AISI Eqn C3.1.2-1) is

=

f

ccn S

MSM (2.2)

where

Sc = Elastic section modulus relative to the extreme compression

fiber of the effective section calculated at a stress Mc/Sf

Sf = Elastic section modulus of the full unreduced section for the

extreme compression fiber

Mc = Critical moment

In Eqn. 2.2, the ratio of Sc/Sf is used to account for the effect of local buckling on

the lateral buckling strength of the beam (Yu 2000). The calculation of the critical

moment (Mc) involves a lengthy determination of the elastic critical moment (Me), which

is defined in AISI Section C3.1.2-1(a). However, for Z-sections bent about the centroidal

axis perpendicular to the web (X-axis), the simplified AISI equation (C3.1.2-16) for the

determination of Me can be used, where

2

2

2LdIEC

M ycbe

π= (2.3)

where

d = Depth of section

L = Unbraced length of member

24

Iyc = Moment of inertia of the compression portion of the section

about the gravity axis of the entire section parallel to the web,

using the full unreduced section

Cb = Bending coefficient dependent on the moment gradient. This

is permitted to be conservatively taken as unity for all cases.

E = Modulus of elasticity of steel (29500 ksi)

Point symmetric sections such as Z-sections will buckle at lower strengths than

doubly or singly symmetric sections (Yu 2000). Therefore, a conservative approach has

been used in the AISI Specification where Me is multiplied by 0.5 (thus the value of 2.0

in the denominator of Eqn. 2.3).

On the other hand, for members bent about the centroidal axis perpendicular to

the web, the calculation of the lateral-torsional buckling strength is not required if the

unbraced length does not exceed a certain length (Lu), which is determined for the case of

Me = 2.78My. When the unbraced length is less than or equal to length Lu, then Sc = Se

and Mc = My. For Z-sections bent about the centroidal axis perpendicular to the web (X-

axis), Lu is calculated from Part II, Section 1.3 of the 1996 AISI Specification as

5.0218.0

=

fy

ycbu SF

EdICL

π (2.4)

For flexural members, both full and effective dimensions are used to calculate

sectional properties, with the full dimensions being utilized when computing a critical

stress, and effective dimensions being used to calculate a predicted strength. In addition,

the reduction in thickness that occurs at corner bends is ignored, and the base metal

25

thickness of the flat steel, exclusive of any coatings, is used in all sectional property

calculations, per AISI Specification provisions.

The effective design width plays a key role in determining the effective section

modulus of a given section. The effective design width is a reduction of the gross width

to an effective width. This reduction method is based on an empirical correction to the

work of von Karman et al. (1932) completed by Winter (1947), and was later extended to

all member elements by the unified approach of Pekoz (1987) (Schafer and Pekoz 1998).

It is a method that takes into account the effects of local buckling and postbuckling

strength, and varies depending on the magnitude of the stress level, the distribution of

stress, and the geometric properties (w/t ratio) of the element. Furthermore, the effective

design width method is described by AISI (1996) as follows: “For plate elements it is

assumed that the total load in a plate element is carried by a fictitious effective width

subject to a uniformly distributed stress equal to the maximum edge stress in the element

while eliminating the remainder of the plate element. This concept eliminates the need to

consider the non-uniform distribution of stress over the entire width of the plate. The

non-uniform distribution of stress occurs in cold-formed steel design because of the

consideration of postbuckling strength in member elements. The use of postbuckling

strength behavior complicates member design, but does permit more efficient use of

steel” (AISI 1996). Figures B4-2 and B2.3-1 in the 1996 AISI Specification are good

representations of typical effective widths of a web, compression flange, and

compression lip for Z-sections.

26

2.4 Determination of Local and Lateral Buckling Strengths

Cold Formed Software (CFS) was utilized to determine the local and lateral

buckling strengths, as well as section properties needed in the various analyses of this

study. CFS is a Windows based analysis software package (RSG Software, Inc. 1998)

which provides designers a means to quickly and accurately determine purlin strength

using only the geometric properties and yield strength. Figure 2.1 shows the measured

geometric properties that were used as input data

R4R5

Length 3

R3

Length 4Length 5

R2

Length 2Length 1

Angle 3

Angle 1

Angle 2

Thickness

Figure 2.1 Geometric Properties Measurement Plan

A strength increase factor due to cold-forming of the steel can also be applied to

purlin sections as an option in CFS. For bending, this factor is only applied to the flat

portion of the extreme fibers (RSG Software 1998). This factor is specific to each purlin

27

and is based on a purlin’s yield strength and flat portion areas. This increase only applies

if the section being analyzed is fully effective with the extreme fiber at Fy. This strength

increase factor was allowed to occur in this study, however it is unlikely that any section

was fully effective due to the applied compression force.

It is important to note how the top (compression) flange was treated in this study.

For local buckling, the strength predictions were determined assuming the purlins were

fully laterally braced, per AISI Specification. However, for lateral buckling, the

predicted strengths were determined assuming the purlins were only braced by the lateral

braces, not the standing seam roof system. The reason for this is that in a standing seam

roof system, the steel deck only provides partial lateral bracing to the purlin (Brooks

1989). Because of this, the purlins are somewhere between being fully braced and fully

unbraced. In addition, the extent of torsional resistance that the steel decking and

fasteners provide to the compression lip-flange component is not known. Therefore, for

this study, lateral buckling strengths were calculated assuming the compression lip-flange

component was torsionally unrestrained by the standing seam roof components. This

simplification process was obtained by studying the results of other research (Hancock et

al. 1996).

2.5 Determination of Distortional Buckling Strength

2.5.1 Background

Cold-formed Z-section purlins subjected to both flexure and torsion may

experience a buckling mode in which only the compression flange and lip rotate about the

flange-web junction, as shown in Figure 1.3 (Hancock 1997). For simplicity this

28

phenomenon is called Type 1 distortional buckling. In less frequent occurrences, initial

rotation of the lip-flange component about the flange-web corner is followed by a lateral

translation of the flange-web corner, which includes transverse bending of the web near

ultimate failure (Rogers and Schuster 1997). This phenomenon is called Type 2

distortional buckling. The torsional restraint stiffness (KΦ) determines which type of

distortional buckling will occur. If the torsional restraint stiffness is positive, then Type 1

distortional buckling is more likely to occur. However, if the torsional restraint stiffness

is small or negative and the section has h/t ratios greater than 150, then Type 2

distortional buckling is favored. Furthermore, during distortional buckling failure, the

web goes through double curvature flexure at the same half wavelength as the flange

buckle, and the compression flange may translate in a direction normal to the web also at

the same half wavelength as the flange and web buckling deformations (Hancock et al.

1996).

Cold-formed Z-sections in flexure are commonly known to deflect in the direction

of the load and also move laterally and twist in such a manner as to relieve compressive

stress on the stiffening lip (Ellifritt et al. 1992). In turn, as the stiffener angle flattens out

from this twisting, the section stiffness lessens, and the purlin becomes unable to hold the

applied load. However, lateral braces restrain this twisting and lateral movement, and at

high enough loads, the section distorts. This distortion is followed by buckling of the

flange and lip (Ellifritt et al. 1998). Distortion is caused when the angle between the web

and flange changes dramatically under load (from the twisting and lateral movement).

As a result of this, distortional buckling failure most often occurs in purlin sections where

lateral deformation of the section is prevented by sufficient bracing (Ellifritt et al. 1998).

29

Distortional buckling plays an extremely important role in the design of cold-

formed sections due to its ability to be the controlling failure mode. According to the

AISI Specification, if a section is adequately braced so that lateral buckling will not

control, the section will have a strength comparable to local buckling. However, it has

been shown by numerous studies that this is not always true (Hancock et al. 1996, Ellifritt

et al. 1998, Schafer and Pekoz 1998). To support this, Figure 1.4 shows how distortional

buckling occurs at wavelengths less than lateral buckling and at stresses less than local

buckling. Therefore, to properly design a standing seam roof system, a method that can

account for the effects of distortional buckling is needed.

2.5.2 The Hancock Method for Determination of Distortional Buckling Strength

The Hancock Method is a ‘hand’ method for determining the distortional buckling

strength of a cold-formed purlin. The Hancock Method is based on Sharp’s effective

column approach for the calculation of the elastic distortional buckling strength (Sharp

1966), which in turn is based on the geometric properties of an effective column (Rogers

and Schuster 1997). Sharp presented design data on the buckling strength of plates

simply supported on all edges, and the buckling strength of flanges with lips on the free

edges (Sharp 1966). The Hancock Method consists of an adaptation of Sharp’s

expressions in order to account for post-buckling strength, and the interaction of buckling

and yielding, which commonly occurs in thin, cold-formed steel elements.

The Hancock Method requires a number of non-standard section properties.

These include the product of inertia and the moments of inertia about the centroidal X-

and Y-axes of only the compression flange and lip. The buckle wavelength (λd),

30

torsional restraint stiffness (KΦ), and several other variables are also required and have to

be calculated.

The moments of inertia about the centroidal X- and Y-axes of the compression lip

and flange are calculated by using the centroidal distances (bf and bl) of the compression

flange and lip, which neglect the radius at the flange-web junction. Figure 2.2 shows the

distances used for the distortional buckling equations. The x and y distances are

determined from the flange-web junction to the contriod of the compression flange-lip

component. The moments of inertia and product of inertia are then taken about the

centroid of the compression lip-flange component.

B L

θ

b

Dw

f blx

y

t

Figure 2.2 Measurements for Section Properties

The formulas for the X- and Y-axes moments of inertia also take into account the

angle between the compression flange and compression lip. This angle is of particular

31

importance because of the impact it has on the stiffness of the compression component

(the larger the angle, the stiffer the section). Moments of inertia about the X- and Y-axes

(Ixflg, Iyflg) and the product of inertia (Ixyflg) are

( ) ( ) ( )

++

−+

×=

22223

lg 122sin

12sin

ybtb

yb

bb

tIx ffl

ll

fθθ

(2.5)

( ) ( ) ( )

+

−++

−+

×=

12cos

2cos

212

23223

lgθθ ll

flf

ff

fb

xb

bbxb

bb

tIy (2.6)

( ) ( ) ( )

−×

×+−+

−×= ybbxbb

bxybtIxy llfl

fff 2

sin2

cos2lg

θθ (2.7)

where

t =Thickness

θ =Angle between compression flange and lip in radians

bl =Centroidal length of the lip;

−=

2tLlb

bf =Centroidal length of the flange;

−=

2tBb f

Aflg =Area of flange; ( )flf bbtA +=lg

=y Centroid location along Y-axis; ( )

=

lg

2 sin2 f

l

Abty

θ

=x Centroid location along X-axis; ( )

++

=

lg

22 cos22 f

lflf

Abbbbtx

θ

32

Several factors are needed for the calculation of the torsional restraint stiffness

(KΦ) for a purlin. These include the buckle half-wavelength (λd), St. Venant torsional

constant of the compression lip-flange component (Jflg), and the elastic distortional

buckling stress (σ′ed).

Of particular importance is the buckle half-wavelength (λd) for the calculation of

the elastic distortional buckling stress. The buckle half-wavelength is the half-length at

which distortional buckling will occur for the section under analysis. If the compression

lip-flange component of a cold-formed purlin is able to freely rotate about the web-flange

junction without restraint from any other connective element besides the web, then the

calculated half-wavelength (λd) is used. On the other hand, when the compression lip-

flange component is additionally restrained, the smaller value between the calculated

half-wavelength and the measured distance between restraints (λc) is used (Hancock et al.

1996). Studies have shown that the torsional restraint offered by the fastener location is

influential on the strength of C-sections, but does not have any effect on Z-sections

(Wallace and Willis 1990). In addition, the amount of lateral bracing supplied to the

supporting purlin from a standing seam roof system is unknown. Therefore, the torsional

restraint herein is conservatively considered to be small and equal to zero. Consequently,

the lip-flange component is not considered to be torsionally restrained by the steel

decking. This is analogous to procedures used in previous research (Hancock et al. 1996)

and therefore will be followed in this study. The torsional constant (Eq. 2.8) represents

the resistance of the section to torsion acting on the section. The St. Venant torsional

constant (Jflg) and buckle half-wavelength (λd) are given by the Hancock Method as

33

+=

33

33

lgtbtb

J lff (2.8)

25.0

3

2lg

280.4

=

tDbIx wff

dλ (2.9)

where

Dw =Depth of the web

The determination of the elastic distortional buckling stress (σ’ed) is an iterative

process with an initial assumption used to determine the torsional restraint stiffness (KΦ).

Hence, the elastic distortional buckling stress is dependent on the torsional stiffness

restraint. Therefore, for the first iteration of the elastic distortional buckling stress, the

torsional stiffness restraint is assumed negligible and taken as zero. In addition, the

translational restraint stiffness (Kx) for sections with inward facing lips is small and also

taken as zero (Hancock and Lau 1987). The α1, α2, and α3 characteristic values are

related to KΦ, λd, and the dimensions of the compression flange and lip. The result of the

first iteration of the distortional buckling stress is then used to find the actual torsional

restraint stiffness. The first iteration of the distortional buckling stress with the torsional

stiffness restraint equal to zero is

( )

−+±+

= 3

22121

lg

' 42

ααααασf

ed AE (Use smaller positive value) (2.10)

where

E =Modulus of elasticity

( )2lg

2lg

11 039.0 dfff JbIx λ

βηα +

=

34

+= lg

1lg2

2fff IxybyIy

βηα

−= 22

lg1

lg13 fff bIxyIyβηαηα

++=

lg

lglg2

1f

ff

AIyIx

xβ

2

=

dλπη

As previously mentioned, the calculation of the torsional restraint stiffness (KΦ) of the

web is an iterative process with an initial, conservative assumption of zero. This

represents the torsional stiffness supplied to the lip-flange component by the web at the

web-flange junction, which is in pure compression (see Figure 2.3). In addition to this,

for sections with inward facing lips, the lateral restraint or translational spring stiffness

(Kx) provided by the web is small (Hancock and Lau 1987). Consequently, the

translational spring stiffness (Kx), which is the resistance of lateral movement of the

section, is also assumed to be small and equal to zero throughout the calculations herein.

Therefore, the pin connection shown in Figure 2.3 can be thought of as a roller. This

manipulation of Kx is analogous to procedures contained in Hancock et al. (1996).

KΦK x

Flange-Web Junction

Figure 2.3 Stiffness Restraints

35

When considering distortional buckling for a Z-section, the cross-sectional

distortion is not important for the flange and the flange is therefore treated as a column

undergoing flexural-torsional buckling (Schafer and Pekoz 1998). For the web, this

cross-section distortion must be considered. Furthermore, if the web of a Z-section in

compression is treated as a simply supported beam in flexure, the rotational stiffness at

the end is 2EI/d, which is a result of the equal and opposite end moments, as shown in

Figure 2.4. If the web of the Z-section in Figure 2.4 is treated as a beam simply

supported at one end and fixed at the other, the rotational stiffness at an end is 4EI/d,

which is double the case for compression. Therefore, the change in end restraint between

the two cases will double the torsional restraint stiffness, (KΦ) (Davies and Jiang 1996).

M

M

M

θ

θ

θ

M = 2EI θ d

M = 4EI θ d

Figure 2.4 Beam Web Behavior in Flexure

Furthermore, the width of the buckled section of the web is substantially reduced

compared with the full web width for the web simply supported at one end and fixed at

the other (Hancock et al. 1996). Because of this, the ratio of buckle half-wavelength to

buckle width is increased since the distortional buckle half-wavelength remains relatively

36

unchanged. For certain sections with web flat width ratios (h/t) above 150 and narrow

flange elements, it is possible for the buckle width of the web to extend past the centroid

of the section, therefore reducing the torsional restraint on the lip-flange component

(Hancock et al. 1996). For this reason, the equation for determining the torsional

restraint stiffness (KΦ), assumes the web element is under a stress gradient caused by

flexure in the member and the section will tend to fail by web-flange distortional

buckling, in which a negative KΦ is obtained, which is Type 2 distortional buckling

(Rogers and Schuster 1997). Also included in the torsional restraint stiffness equation is

the plate buckling coefficient (k). The plate buckling coefficient of a web element under

pure in-plane bending varies as a function of the aspect ratio. Hence, the resulting

flange-web junction torsional restraint stiffness used in the Hancock Method uses the

plate buckling coefficients described by Timoshenko and Gere (Timoshenko and Gere

1961, Table 9-6) to include a reduction factor based on the compressive stresses in the

web (Hancock et al. 1996). This reduction factor, which takes into account the stress

gradient of the web, is a modification from the first draft ballot submitted to the AISI

Specification Committee (Hancock 1995). The resulting torsional stiffness restraint

equation used to determine the actual elastic distortional buckling stress is

( )

++

−

+

=Φ 2244

24

2

'3

39.13192.256.1211.1

106.046.5

2

wdwd

dwed

dw DDD

EtDEtK

λλλσ

λ (2.11)

The second iteration of this process is solely dependent on the torsional stiffness

restraint from the above equation. If the web torsionally restrains the lip-flange

37

component (KΦ≥0), then this value is used in the second iteration to update the α1 term

and consequently the α3 term (see below). The actual elastic distortional buckling stress

(σed) is thus calculated using the updated α1 and α3 equations.

However, if the lip-flange component torsionally restrains the web element

(KΦ<0), then KΦ is recalculated without an initially assumed elastic distortional buckling

stress. In this situation, the interaction of local and distortional buckling is ignored. This

new KΦ term is thus used to update the α1 and α3 equations to determine the actual elastic

distortional buckling stress, as shown below

If KΦ≥0 :

( ) ( )

−+±+

= 3

22121

lg

42

ααααασf


where

( )

++

= Φ

EK

JbIx dfff ηβλ

βηα

1lg

2lg

11 039.0

−= 22

lg1


If KΦ<0 :

( ) ( )

−+±+

= 3

22121

lg

42

ααααασf


38

where

( )

++

= Φ

EK

JbIx dfff ηβλ

βηα

1lg

2lg

11 039.0

( )

+

=ΦdwD

EtKλ06.046.5

2 3

−= 22

lg1


After the actual elastic distortional buckling stress has been found, the inelastic

critical stress (fc) is determined for the strength calculations. The inelastic critical stress

is a function of both the elastic distortional buckling stress and the yield stress of the

section. This inelastic critical stress calculation is significant because it allows for the

interaction of buckling and yielding, as well as post-buckling strength in the distortional

mode (Hancock et al. 1996). The procedures for the calculation of fc are

If ed 2> yf2.σ then:

yff = (2.14) c

If ed 2≤ yf2.σ then:

−

y

ed

y

edyc ff

fσσ

22.01=f (2.15)

where

fy =Yield stress of section

39

Once the inelastic critical stress is calculated, the predicted moment resistance of

a section is only dependent on the torsional restraint stiffness and is determined as

follows:

If 0≥K then Φ cfpred fSM =

If 0<K then Φ ccpred fSM =

where

Sf = Elastic section modulus of the full unreduced section for the

extreme compression fiber

Sc = Elastic section modulus of the effective section calculated at

stress fc in the extreme compression fiber with k = 4.0 for the

flange, and f = fc for the edge stiffener

Chapter 5 includes a step-by-step procedure for determining the distortional

buckling strength.

2.6 Determination of Section Strength

Once the predicted strength of a section is found from the distortional mode, it

can be compared to the local buckling and lateral buckling values. Consequently, the

smallest value is the limiting section design strength.

40

CHAPTER III

EXPERIMENTAL TEST DETAILS AND RESULTS

3.1 Background and Test Details

The experimental data used in this report were compiled from experimental tests

conducted at the Structures and Materials Laboratory at Virginia Polytechnic Institute and

State University (henceforth referred to as Virginia Tech), unless otherwise stated. The

experimental data used for comparison to the buckling strength predictions were gathered

from previous third point braced standing seam roof system tests and consist of Murray and

Trout (2000), Bryant et al. (1999a), Almoney and Murray (1998), Bathgate and Murray

(1995), Davis et al. (1995), Borgsmiller et al. (1994), Earls et al. (1991), Brooks and

Murray (1990), and Spangler and Murray (1989).

As previously mentioned, this study concentrates on simple span, Z-section purlins

that support standing seam roof systems. This was the base criterion for all data used in

this study. Since distortional buckling is being considered, most tests were intermittently

laterally braced, which helps control the effects of lateral buckling. However, several

laterally unbraced experimental test results are included to show the effects of distortional

buckling on simple spanned, laterally unbraced, Z-section purlins supporting standing seam

41

roof systems. These were also tested at Virginia Tech and include Bryant et al. (1999b)

and Bryant et al. (1999c). Figure 3.1 shows a generic test setup.

PURLINLENGTH

SPAN LENGTH

CTC PURLINSPACING

STEEL SUPPORTBEAM

ANTI-ROLLCLIPS

STANDING SEAMFOOF PANELS

EAVEANGLE

RAKEANGLE

RIDGEANGLE

RIDGE

EAVE

CHAMBERWALL

3'-6"

STANDS

7'-0"8'-0"

6-MILPOLYETHELENE

SHEET

Figure 3.1 Typical Base Test Setup

From Trout (2000)

Various combinations of roof panels (ribbed or pan) and panel thickness, clip types

(short/tall fixed, short/tall sliding), Z-section depths and thickness, and lateral braces (light

42

gage angles, Z-purlins) were used in each test series. All test setups were simple span, two

purlin-lined, third point braced or unbraced, and consisted of only Z-section purlins. Span

length, bracing configuration, clip type, purlin thickness, and panel type and thickness were

unique to each series of tests. The clip and steel panel types utilized in the tests are

illustrated in Figures 3.2 and 3.3. The combinations of these different components can

have an impact on the strength of each individual test and will be discussed in detail in

Chapter 4.

Sliding ClipFixed Clip

Figure 3.2 Clip Types

From Trout (2000)

43

Rib Type Panel

Pan Type Panel

Figure 3.3 Steel Panel Types From Trout (2000)

3.2 Experimental Results

The experimental data collected for this study consist of 82 standing seam roof

system tests in eleven different series. However, eight of these tests were considered to

have unusable data due to events such as failed lateral braces, pre-test damage to the purlin,

and limitations of CFS (such as a fifth point braced configuration, which is not in the CFS

database). The experimental buckling strengths were obtained by using the Base Test

Method (Carballo et al 1989). Table 3.1 lists the experimental results for the third point

braced tests. Table 3.2 lists the experimental results for the laterally unbraced Z-section

purlin tests. Included within both of these tables are each test’s measured yield strength,

experimental strength, and selected geometric properties of each individual test: measured

span length, purlin depth, and purlin thickness.

44

Test 18 (Z08-37) 25.0 8.00 60.1 0.102 12.47

Test 19 (Z08-39) 25.0 8.00 60.6 0.101 13.83

Test 20 (Z08-41) 25.0 8.00 60.5 0.102 11.73

Test 25 (Z08-44) 25.0 8.00 60.3 0.102 11.98

Test 26 (Z08-45) 25.0 8.00 59.6 0.102 12.19

Test 27 (Z08-47) 25.0 8.00 60.0 0.102 11.90

Test 21 (Z08-14) 25.0 8.00 63.6 0.057 6.46

Test 22 (Z08-16) 25.0 8.00 65.2 0.057 6.37

Test 23 (Z08-18) 25.0 8.00 62.9 0.056 5.39

Test 24 (Z08-20) 25.0 8.00 64.9 0.056 5.03

Test 28 (Z08-22) 25.0 8.00 63.4 0.057 6.41

Test 29 (Z08-24) 25.0 8.00 64.3 0.057 5.61

Test B (Z10-Eave) 30.0 10.063 54.2 0.103 13.14

Test C (Z10-Eave) 30.0 10.063 52.6 0.103 15.98

Test D (Z10-Eave) 30.0 10.063 52.4 0.103 18.19

Test G (Z10-Eave) 30.0 10.063 52.6 0.103 16.05

Test E (Z10-Eave) 30.0 10.063 52.5 0.076 9.02

Test F (Z10-Eave) 30.0 10.063 52.3 0.076 8.50

(Bryant et al. 1999a)

Depth (in.)

Summary Table of Experimental Strengths and Properties of Third Point Braced Z-Sections

Table 3.1

(Murray and Trout 2000)

Test No.Measured

Yield Stress (ksi)

Measured Thickness

(in.)

Experimental Strength

(k-ft)

Span (ft)

45

46

Test #9 (Eave) 25.0 10.000 60.2 0.078 13.19

Setup 2-0.105b 27.0 7.938 67.6 0.104 16.73

Setup 3-0.105a 27.0 7.938 67.7 0.104 14.30

Setup-0.105b 27.0 7.938 67.3 0.104 14.01

Setup 4-0.105 27.0 7.938 62.0 0.103 12.70

Setup 4-0.076 27.0 7.938 64.5 0.074 10.66

Setup 3-0.060a 27.0 7.875 64.1 0.059 7.10

Setup 3-0.060b 27.0 7.875 66.2 0.059 7.48

Setup 3-0.060c 27.0 7.875 65.3 0.059 6.74

1G-Eave 25.0 8.000 57.1 0.060 6.69

2G-Eave 25.0 8.000 57.1 0.060 7.91

6G-Eave 25.0 8.000 57.1 0.060 7.60

3G-Eave 25.0 10.000 58.6 0.087 13.37

4G-Eave 25.0 10.125 58.6 0.087 15.87

5G-Eave 25.0 10.000 58.6 0.087 15.01

Test #1 (Eave) 25.0 8.438 60.5 0.091 12.32

Test #2 (Eave) 25.0 8.563 60.7 0.091 12.67

Test #5 (Eave) 25.0 8.500 65.7 0.091 12.07

Test #3 (Eave) 25.0 8.500 64.5 0.060 7.51

Test #4 (Eave) 25.0 8.500 64.0 0.061 7.16

Test #6 (Eave) 25.0 10.000 65.5 0.103 22.67

Test #7 (Eave) 25.0 10.000 65.9 0.102 21.63

Test #8 (Eave) 25.0 10.000 61.3 0.078 13.11

(Almoney and Murray 1998)

(Davis et al. 1995)

(Bathgate and Murray 1995)

Measured Thickness

(in.)


(k-ft)


Table 3.1 Continued

Test No. Span (ft)

Depth (in.)

Measured Yield Stress

(ksi)

G8ZTP/S-1 (1A) 20.0 8.0 56.6 0.071 6.85

G8ZTP/S-1 (1B) 20.0 8.0 56.6 0.071 6.84

G8ZTP/S-1 (1C) 20.0 8.0 56.6 0.071 6.77

G8ZTP/S-1 (5A) 20.0 8.0 58.5 0.075 7.93

G10ZTP/S-1 (2A) 20.0 10.0 47.5 0.101 16.24

G10ZTP/S-1 (2B) 20.0 10.0 47.5 0.101 16.23

G10ZTP/S-1 (2C) 20.0 10.0 47.5 0.101 16.19

Test #1 25.0 8.000 61.4 0.058 4.97

Test #2 25.0 8.000 50.1 0.059 4.67

Test #3 25.0 8.063 62.3 0.098 16.24

Test #4 25.0 8.063 64.7 0.098 16.26

Test #5 25.0 8.063 60.1 0.086 12.58

Test #6 25.0 8.063 62.8 0.079 11.10

Test #7 25.0 8.125 67.2 0.066 8.45

Z-T-P/F-1 25.0 7.890 53.6 0.078 10.81

Z-T-P/S-1 25.0 9.480 63.7 0.074 10.54

Z-T-R/S-1 25.0 9.575 63.5 0.075 10.96

N-ZIS-12-SF-1 25.0 9.903 62.0 0.095 16.39

N-ZISO-12-SF-1 25.0 9.848 65.5 0.098 20.98

N-ZISO-12-SS-1 25.0 9.875 63.7 0.094 19.03

N-ZISO-12-TF-1 25.0 9.911 57.6 0.098 18.50Notes:1) Self weight of system not accounted for in experimental strength for Borgsmiller et al. (1994)and Brooks and Murray (1989). For these tests 8plf added for 8" deep sections, 10plf addedfor 10" deep sections which was based on an analysis of Trout and Murray (2000).


Table 3.1 Continued

Test No. Span (ft)

Depth (in.)

Measured Yield Stress (ksi)

Measured Thickness

(in.)


(k-ft)

(Borgsmiller et al. 1994)

(Earls et al. 1991)

(Brooks and Murray 1989)

(Spangler and Murray 1989)

47

(Bryant et al. 1999b)Test #1 22.75 8.000 71.9 0.062 4.48

Test #2 22.75 8.000 70.1 0.062 4.61

Test #3 22.75 8.000 69.9 0.062 4.45

Test #4 22.75 8.063 56.3 0.106 8.29

Test #5 22.75 8.063 56.0 0.106 7.61

Test #6 22.75 8.063 56.5 0.106 7.99

(Bryant et al. 1999c)

Test #1a 30.0 10.000 54.7 0.076 6.15

Test #2a 30.0 10.063 53.0 0.076 6.20

Test #3a 30.0 10.000 55.7 0.077 6.72

Test #4a 30.0 10.063 55.3 0.103 10.10

Test #5a 30.0 10.063 64.1 0.100 9.61

Test #6a 30.0 10.063 65.8 0.102 9.09

Measured Thickness

(in.)


(k-ft)

Summary Table of Experimental Strengths and Properties of Laterally Unbraced Z-Sections

Table 3.2

Test No. Span (ft)

Depth (in.)

Measured Yield Stress

(ksi)

Each completed test includes a purlin correction load added onto the experimentally

observed failure load. The purlin correction expression takes into account the effect of the

overturning moment as defined in Section D3.2.1 of the 1999 AISI Specification

Supplement No. 1 (Trout 2000). The purlin correction factor (PL) and resulting

experimental failure moment (Mts) are distinctive to each individual test and are calculated

in three steps as shown below:

48

Step One:

( )( ) ( )

( ldw

fL WW

tDb

P +

= 6.09.0

5.1

041.0 ) (3.1)

where

PL = Purlin correction factor

bf = Flange width (in.)

Dw = Web depth (in.)

t = Thickness (in.)

Wd = Weight of deck plus purlin (plf)

Wl = Applied line loading (plf)

Step Two:

( ) ( )

++=

SDPWWW w

Ldlts 2 (3.2)

where

Wts = Failure line load (plf)

Dw = Web depth (ft)

S = Purlin spacing

Step Three:

( )

=

8

2LWM tsts (3.3)

where

Wts = Failure load (klf)

L = Span length (ft)

3.3 AISI Specification Analysis

This section includes the buckling strength analyses for all experimental tests using

the 1996 AISI Specification sections C3.1.1 for local buckling predictions and C3.1.2 for

49

lateral buckling predictions. Previous studies that consider distortional buckling have

usually compared the distortional buckling strength only to the local buckling strength

obtained from AISI Specification, Section C3.1.1. Lateral buckling was not considered in

these outside studies since the purlins were adequately (sometimes excessively) braced for

lateral buckling. However, the original intent of the data collected for this study was not

initially intended to analyze the effects of distortional buckling and because of this, lateral

buckling may not have been controlled. Therefore, to investigate the effects of lateral

buckling as compared to distortional buckling, and to further study the relationship between

local buckling and distortional buckling on standing seam roof systems, an analysis was

performed using AISI Specification Sections C3.1.1 and C3.1.2 on all tests. These

analyses were conducted using CFS for ease of computation, and the results of the third

point braced Z-purlin analyses are found in Table 3.3 while the laterally unbraced Z-purlin

analyses results are listed in Table 3.4.

As discussed earlier, the extent of lateral and torsional restraint that the standing

seam roof system provides to the purlin is somewhere between the fully braced state and

the unbraced state. In the AISI Specification provisions, an “in-between” state cannot be

modeled and therefore all tests (for both local and lateral buckling analyses) were

conservatively modeled by considering the lateral and torsional restraint offered by the

standing seam roof system to be negligible and equal to zero. Hence, to be consistent with

the assumption, the only lateral support for the purlins were the lateral braces, if present.

Because of this, there is a very large difference in strength predictions using the provisions

of AISI Specification C3.1.1 and C3.1.2 for the unbraced tests, as shown by Table 3.4.

50

Section C3.1.1 Section C3.1.2

Test 18 (Z08-37) 16.18 11.31

Test 19 (Z08-39) 16.26 10.21

Test 20 (Z08-41) 16.21 10.27

Test 25 (Z08-44) 16.52 10.89

Test 26 (Z08-45) 16.31 10.58

Test 27 (Z08-47) 16.45 10.91

Test 21 (Z08-14) 8.60 5.12

Test 22 (Z08-16) 8.82 5.21

Test 23 (Z08-18) 8.22 4.73

Test 24 (Z08-20) 8.40 4.82

Test 28 (Z08-22) 8.63 5.18

Test 29 (Z08-24) 8.47 4.62

Test B (Z10-Eave) 22.00 11.01

Test C (Z10-Eave) 22.06 12.00

Test D (Z10-Eave) 21.84 11.89

Test G (Z10-Eave) 22.06 11.80

Test E (Z10-Eave) 13.91 7.95

Test F (Z10-Eave) 14.00 8.19

Setup 2-0.105b 20.35 9.84

Setup 3-0.105a 20.40 9.68

Setup 3-0.105b 20.38 9.84

Setup 4-0.105 18.51 10.98

Setup 4-0.076 12.92 8.29

Setup 3-0.060a 9.47 5.11

Setup 3-0.060b 9.58 5.26

Setup 3-0.060c 9.65 5.73

(Murray and Trout 2000)

(Bryant et al. 1999)


Table 3.3

Summary Table of 1996 AISI Specification Strengths of Third Point Braced Z-Sections

Test No.AISI Specification Nominal Strength (k-ft)

51


1G-Eave 8.77 5.76

2G-Eave 8.68 5.94

6G-Eave 8.51 5.25

3G-Eave 19.01 12.09

4G-Eave 20.37 12.47

5G-Eave 19.46 9.43

Test #1 (Eave) 16.12 8.39

Test #2 (Eave) 16.88 8.74

Test #5 (Eave) 17.08 8.21

Test #3 (Eave) 8.79 4.53

Test #4 (Eave) 9.07 4.66

Test #6 (Eave) 26.42 17.90

Test #7 (Eave) 27.41 21.04

Test #8 (Eave) 16.38 9.36

Test #9 (Eave) 15.77 8.26

G8ZTP/S-1 (1A) 10.63 7.85

G8ZTP/S-1 (1B) 10.77 8.22

G8ZTP/S-1 (1C) 10.83 8.41

G8ZTP/S-1 (5A) 11.95 9.30

G10ZTP/S-1 (2A) 20.48 16.13

G10ZTP/S-1 (2B) 20.27 15.98

G10ZTP/S-1 (2C) 20.23 16.03

(Davis et al. 1995)



Table 3.3 Continued


Test No.AISI Specification Nominal Strength

(k-ft)

52


Test #1 8.30 5.63

Test #2 7.61 5.51

Test #3 17.83 13.12

Test #4 18.34 13.39

Test #5 14.94 10.68

Test #6 13.91 9.77

Test #7 11.23 7.59

Z-T-P/F-1 11.01 7.29

Z-T-P/S-1 14.70 9.71

Z-T-R/S-1 16.64 9.69

N-ZIS-12-SF-1 24.43 15.61

N-ZISO-12-SF-1 26.56 16.57

N-ZISO-12-SS-1 24.76 16.48

N-ZISO-12-TF-1 25.43 16.68

(Earls et al. 1991)



Table 3.3 Continued


Test No.

AISI Specification Nominal Strength (k-ft)

53


Test #1 10.21 0.84

Test #2 10.50 0.86

Test #3 10.63 0.91

Test #4 18.51 1.70

Test #5 18.55 1.86

Test #6 18.53 1.80

Test #1a 15.20 1.04

Test #2a 14.14 0.94

Test #3a 15.21 1.00

Test #4a 22.86 1.48

Test #5a 24.00 1.32

Test #6a 25.32 1.39

(Bryant et al. 1999b)

(Bryant et al. 1999c)

Table 3.4

Summary Table of 1996 AISI Specification Strengths of Laterally Unbraced Z-Sections

Test No.AISI Specification Nominal Strength

(k-ft)

3.4 Distortional Buckling Analysis

A distortional buckling analysis was completed using the previously discussed

Hancock Method (Chapter 2) for both the third point laterally braced and unbraced tests.

In the same manner as the AISI analyses, the distortional buckling strength analyses

assumed the compression flange-lip component was not laterally or torsionally restrained

by the interaction of the clips and steel roof panels. Laterally braced distortional buckling

54

strengths are found in Table 3.5 and unbraced distortional buckling strengths are found in

Table 3.6.

The torsional restraint stiffness (KΦ) can be either positive or negative. This

depends on whether the web torsionally restrains the flange-lip component (positive), or the

web is torsionally restrained by the flange-lip component (negative). Coincidentally, for all

cases in this study, none resulted in the web being torsionally restrained by the flange-lip

component and thus having a negative torsional restraint stiffness (-KΦ).

As discussed earlier, the calculated distortional buckle half wavelength is used

when the compression lip-flange component is able to freely rotate about the flange-web

corner without restraint from any connective elements, such as a standing seam roof system

(Hancock et al. 1996). However, if the lip-flange component is torsionally restrained, the

smaller value of the calculated half wavelength and the measured distance between the

restraints (for standing seam roof systems this distance is the clip spacing) should be used.

Consequently, to be consistent with the assumption made in the AISI Specification

analyses for lateral buckling strength predictions, only the calculated half wavelength was

used in the distortional buckling strength predictions. This procedure has been applied in

previous research, such as Hancock et al. (1996), and has provided consistent and accurate

results.

55

Test 18 (Z08-37) 14.18 Test #1 (Eave) 13.01

Test 19 (Z08-39) 14.10 Test #2 (Eave) 13.28

Test 20 (Z08-41) 14.20 Test #5 (Eave) 13.64

Test 25 (Z08-44) 14.31 Test #3 (Eave) 7.11

Test 26 (Z08-45) 14.24 Test #4 (Eave) 7.31

Test 27 (Z08-47) 14.30 Test #6 (Eave) 21.32

Test 21 (Z08-14) 6.30 Test #7 (Eave) 21.69

Test 22 (Z08-16) 6.49 Test #8 (Eave) 13.32

Test 23 (Z08-18) 6.12 Test #9 (Eave) 12.68

Test 24 (Z08-20) 6.24

Test 28 (Z08-22) 6.33 G8ZTP/S-1 (1A) 8.18

Test 29 (Z08-24) 6.32 G8ZTP/S-1 (1B) 8.20

G8ZTP/S-1 (1C) 8.28

Test B (Z10-Eave) 17.88 G8ZTP/S-1 (5A) 9.24

Test C (Z10-Eave) 17.81 G10ZTP/S-1 (2A) 15.55

Test D (Z10-Eave) 17.79 G10ZTP/S-1 (2B) 15.45

Test G (Z10-Eave) 17.84 G10ZTP/S-1 (2C) 15.43

Test E (Z10-Eave) 11.01

Test F (Z10-Eave) 11.11

(Bryant et al 1999)

(Murray and Trout 2000) (Bathgate and Murray 1995)

Table 3.5

Summary Table of Nominal Distortional Buckling Strengths of Third Point Braced Z-Sections

Test No.Predicted Distortional

Buckling Strength (k-ft)



(Borgsmiller et al 1994)

56

Setup 2-0.105b 16.49 Test #1 5.74

Setup 3-0.105a 16.46 Test #2 5.20

Setup 3-0.105b 16.55 Test #3 14.53

Setup 4-0.105 15.68 Test #4 14.89

Setup 4-0.076 10.30 Test #5 11.48

Setup 3-0.060a 7.23 Test #6 10.30

Setup 3-0.060b 7.42 Test #7 7.80

Setup 3-0.060c 7.32

Z-T-P/F-1 9.09

1G-Eave 6.55 Z-T-P/S-1 11.29

2G-Eave 6.75 Z-T-R/S-1 12.41

6G-Eave 6.71

3G-Eave 14.91 N-ZIS-12-SF-1 19.48

4G-Eave 15.59 N-ZISO-12-SF-1 21.17

5G-Eave 14.95 N-ZISO-12-SS-1 19.85

N-ZISO-12-TF-1 19.73



(Davis et al 1995)

(Earls et al 1991)(Almoney and Murray 1998)

Table 3.5 Continued

Summary Table of Nominal Distortional Buckling Strengths of Third Point Braced Z-Sections





57

Test #1 7.81

Test #2 7.93

Test #3 7.94

Test #4 15.02

Test #5 15.02

Test #6 14.97

Test #1a 11.54

Test #2a 11.21

Test #3a 11.86

Test #4a 18.40

Test #5a 18.51

Test #6a 19.75

(Bryant et al 1999b)

(Bryant et al 1999c)

Table 3.6



Summary Table of Nominal Distortional Buckling Strengths of Laterally Unbraced Z-Sections

58

CHAPTER IV

COMPARISON OF RESULTS

4.1 General

Predicted buckling strengths from the Hancock Method and the 1996 AISI

Specification are now compared to the experimentally obtained strengths and then to each

other. More specifically, local, lateral, and distortional buckling strength predictions are

compared to experimental results to determine which method most accurately predicts the

actual buckling strength of cold-formed Z-section purlins supporting standing seam roof

systems. This comparison is comprised of two main groups: the first is for the laterally

braced configurations, and the second is for the laterally unbraced configurations.

Within each strength prediction method (AISI Specification for local and lateral

buckling, Hancock Method for distortional buckling) there are certain parameters that

cannot be accounted for in the strength predictions of Z-sections supporting standing

seam roof systems. These parameters are: clip type, purlin orientation, and roof panel

type and thickness. The extent to which each of these parameters affects the

experimental strengths is discussed in detail in Section 4.4.

In each graph in this Chapter there are three lines. The first (or middle) is a 45°

line which represents a zero percent error between the predicted buckling strength and the

59

experimentally obtained buckling strength for a given test. The other two lines are

denoted with 10% symbols and represent a ten percent confidence envelope on either side

of the zero percent error line. Any data points below the zero percent error line are

unconservative and occur when the predicted strength is larger than the experimental

strength for a given test. Data points above the zero percent error line are conservative

and occur when the predicted strength is smaller than the experimental strength. In

addition, each graph plots third point braced data (denoted with hollow black marks) and

laterally unbraced data (denoted with solid gray marks). These marks are broken down

by section thickness to show the relationship between thickness and strength.

4.2 Third Point Braced and Unbraced Analyses

Figures 4.1 and 4.2 are comparisons of the experimentally obtained buckling

results and the 1996 AISI Specification local buckling predicted strengths for 8 in. and 10

in. deep Z-sections. The local buckling strength predictions are excessively

unconservative, as expected, and thus do not provide an accurate method for determining

the strength of a standing seam roof system. The local buckling strength prediction is

based on the assumption that full lateral support is supplied to the purlin. As is evident

from these results, a standing seam roof system does not provide full lateral support. The

unconservative nature of the AISI local buckling strength prediction is further shown by

only three data points within the 10% confidence envelope for 8 in. deep sections and

none for the 10 in. deep sections.

60

Experimental Strength Vs. Local Buckling Strength of 8" Deep Stiffened Z-Sections

0.0

5.0

10.0

15.0

20.0

25.0

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Local Buckling Strength, Mdb (k-ft)

Expe

rimen

tal S

treng

th, M

exp (

k-ft)

0.055"-0.0649"0.065"-0.0749"0.075"-0.0849"0.085"-0.0949"0.095"-0.1049"0.055"-0.0649" (Unbraced)0.105"-0.1149" (Unbraced)

Unconservative

Conservative

Z-SectionThickness

10% 10%

Figure 4.1 Experimental Strengths Vs. Local Buckling for 8 in. Deep Z-Sections

Experimental Strength Vs. Local Buckling Strength of 10" Deep Stiffened Z-Sections

0.0

5.0

10.0

15.0

20.0

25.0

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Local Buckling Strength, Mdb (k-ft)

Expe

rimen

tal S

treng

th, M

exp (

k-ft)

0.065"-0.0749"0.075"-0.0849"0.085"-0.0949"0.095"-0.1049"0.075"-0.0849" (Unbraced)0.095"-0.1049" (Unbraced)Unconservative

Conservative

Z-SectionThickness

10% 10%

Figure 4.2 Experimental Strengths Vs. Local Buckling for 10 in. Deep Z-Sections

61

Figures 4.3 and 4.4 show the data comparisons of the experimentally obtained

buckling strengths and AISI Specification lateral buckling strength predictions for 8 in.

and 10 in. deep Z-sections. These plots show that the lateral buckling strength prediction

is conservative when compared to experimental data. Hence, when this method is used

for strength prediction, the full strength of the analyzed purlin will not be utilized. The

lateral buckling strength predictions were made assuming the purlin sections are only

braced at lateral brace locations. This assumption ignores any lateral restraint provided

by the standing seam roof panels. However, it is evident from the results that significant

restraint is actually present in standing seam roof systems.

Although the AISI lateral buckling strength prediction method gives conservative

results, it is more accurate when compared to the AISI local buckling provisions. This is

verified by the number of data points within the 10% confidence envelopes. The AISI

lateral buckling provisions have seven data points inside the 10% confidence envelopes

for 8 in. deep Z-sections and nine data points inside the 10% confidence envelopes for 10

in. deep Z-sections, respectively.

62

Experimental Strength Vs. Lateral Buckling Strength of 8" Deep Stiffened Z-Sections

0.0

5.0

10.0

15.0

20.0

25.0

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Lateral Buckling Strength, Mdb (k-ft)

Expe

rimen

tal S

treng

th, M

exp (

k-ft)


Unconservative

Conservative

Z-SectionThickness

10% 10%

Figure 4.3 Experimental Strengths Vs. Lateral Buckling for 8 in. Deep Z-Sections

Experimental Strength Vs. Lateral Buckling Strength of 10" Deep Stiffened Z-Sections

0.0

5.0

10.0

15.0

20.0

25.0

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Lateral Buckling Strength, Mdb (k-ft)

Expe

rimen

tal S

treng

th, M

exp (

k-ft)


Conservative

Z-SectionThickness

10% 10%

Figure 4.4 Experimental Strengths Vs. Lateral Buckling for 10 in. Deep Z-Sections

63

Illustrated in Figures 4.5 and 4.6 are the experimentally obtained strengths versus

the Hancock Method’s distortional buckling strength prediction results for 8 in. and 10 in.

Z-sections supporting standing seam roof systems. While this method yields slightly

unconservative strength predictions, it is much less unconservative than the local

buckling strength predictions obtained by the 1996 AISI Specification. It is also more

accurate when compared to the 1996 AISI lateral buckling provisions. This is shown by

the number of data points inside the 10% confidence envelopes. For the Hancock

Method’s distortional buckling strength predictions, 19 data points are inside the 10%

confidence envelopes for 8 in. deep Z-sections, and 13 data points are inside the 10%

confidence envelopes for 10 in. deep Z-sections.

The laterally unbraced test strength predictions for both 8 in. and 10 in. deep Z-

sections (represented in Figures 4.5 and 4.6 by solid gray symbols) are unconservative in

nature. The unconservative strengths predicted by the Hancock Method stem from the

calculated distortional buckle half wavelength. The Hancock Method assumes the tested

section will fail in the distortional mode at this calculated half wavelength. Furthermore,

as shown in Figure 1.4, distortional buckling occurs at wavelengths shorter than lateral

buckling and at stresses higher than lateral buckling. However, the laterally unbraced

tests can fail at longer wavelengths, and at lower stresses than the Hancock Method may

predict. If a laterally unbraced test does fail at a wavelength longer than that predicted by

the Hancock Method, per Figure 1.4, this test should also fail at a stress lower than that

predicted by the Hancock Method. Graphically this is shown as an unconservative data

point.

64

Experimental Strength Vs. Distortional Buckling Strength of 8" Deep Stiffened Z-Sections

0.0

5.0

10.0

15.0

20.0

25.0

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Distortional Buckling Strength, Mdb (k-ft)

Expe

rimen

tal S

treng

th, M

exp (

k-ft)


Unconservative

Conservative

Z-SectionThickness

10% 10%

Figure 4.5 Experimental Strengths Vs. Distortional Buckling for 8 in. Deep Z-Sections

Experimental Strength Vs. Distortional Buckling Strength of 10" Deep Stiffened Z-Sections

0.0

5.0

10.0

15.0

20.0

25.0

0.0 5.0 10.0 15.0 20.0 25.0 30.0

Distortional Buckling Strength, Mdb (k-ft)

Expe

rimen

tal S

treng

th, M

exp (

k-ft)


Conservative

Z-SectionThickness

10% 10%

Figure 4.6 Experimental Strengths Vs. Distortional Buckling for 10 in. Deep Z-Sections

65

Figure 4.7 shows the results from the three buckling strength prediction methods

(local buckling, lateral buckling, and distortional buckling) compared to the experimental

buckling results, for the third point braced tests. Data for the laterally unbraced tests

were not included. Essentially, this is a summary of Figures 4.1 through 4.6. The three

black, dashed lines are of particular importance as they represent a linear regression of

each buckling prediction method. As shown by Figure 4.7, the distortional buckling

strengths predicted by the Hancock Method more accurately reflect the experimental

strengths associated with a specific test when compared to the predicted strengths of the

1996 AISI Specification provisions for lateral and local buckling.

Strength Prediction Vs. Experimental Buckling for All Third Point Braced Tests

0.0

5.0

10.0

15.0

20.0

25.0

0.0 5.0 10.0 15.0 20.0 25.0 30.0Predicted Strength (k-ft)

Exp

erim

enta

l Str

engt

h (k

-ft)

Distortional Buckling Lateral Buckling Local Buckling

Lateral Buckling Trend Line Distortional Buckling Trend Line Local Buckling Trend Line

10%10%

Conservative

Unconservative

Figure 4.7 Overall Experimental Strengths Vs. Predicted Buckling Strengths

66

Tables 4.1 through 4.9 show the coefficients of variation and standard deviation

of the predicted buckling strength versus experimental buckling strength ratios for the

laterally braced standing seam roof systems. Tables 4.10 and 4.11 show the coefficients

of variation and standard deviation of the predicted strength versus experimental strength

ratios for the laterally unbraced standing seam roof systems. Table 4.12 is a summary of

Tables 4.1 through 4.11.

The coefficients of variation and standard deviations were determined by

statistical analysis and represent the precision of the predicted strength as compared to

the experimental strength for a given series. The lower the numbers, the more accurate

and precise the strength prediction. As shown by Table 4.12, the distortional buckling

predictions determined by the Hancock Method represent the most accurate method

(MDB/Mexp of 1.063, or 6.3% unconservative mean error) for determining the strength of a

standing seam roof system supported by a Z-section purlin. The predictions for the

lateral buckling strength yield a conservative mean value of 15.6%, or an MC3.1.2/Mexp

ratio of 0.844. On the other hand, the predictions for the local buckling strength yields an

unconservative mean value of 35.3%, or an MC3.1.1/Mexp ratio of 1.353.

Tables 4.10 and 4.11 for the laterally unbraced tests show inaccurate strength

predictions for all prediction methods and are further summarized in Table 4.12.

Furthermore, local (137.5% or MC3.1.1/Mexp of 2.375) and distortional (85.6% or

MDB/Mexp of 1.856) buckling provisions unconservatively predict the strengths of the

unbraced tests, while lateral buckling (82.0% or MC3.1.2/Mexp ratio of 0.18) provisions

conservatively predict the strengths of the unbraced tests. These results are due to the

assumption that the standing seam roof system does not provide any lateral bracing to the

67

Z-purlin. The ranges of the laterally unbraced tests are not discussed below due to the

large error associated with these strength predictions.

The range of strength prediction ratios (Mpred/Mexp) of the third point braced tests

for each buckling mode method are listed in Table 4.12. Local buckling has a range of

1.109 to 1.674. This shows that even the lowest strength predictions by this method are

unconservative. Lateral buckling has a strength prediction ratio range of 0.588 to 1.242.

This range shows that the local buckling strength predictions have the ability to be overly

conservative. The distortional buckling strength prediction ratio range is 0.841 to 1.361.

This range shows that the Hancock Method is less unconservative when compared to the

local buckling strength predictions, and less conservative when compared to the lateral

buckling strength predictions. In addition to the Hancock Method predicting strengths

more similar to the experimental strengths, it also has the smallest range. This shows a

tendency to give fewer unreasonable results.

Figure 4.7 shows that distortional buckling more accurately predicts the buckling

strengths of third point braced purlins that support standing seam roof systems, when

compared to local and lateral buckling strength predictions. However, this does not

necessarily conclude that all these previous tests failed by distortional buckling. Without

witnessing each individual test as it fails, the actual failure mode can only be

hypothesized.

68

Local Buckling (C3.1.1)

Lateral Buckling (C3.1.2)

Dist. Buckling

MnC3.1.1---

Mnexp

MnC3.1.2---

Mnexp

MnDB-----

Mnexp

Test 18 (Z08-37) 12.47 16.18 11.31 14.18 1.298 0.907 1.137

Test 19 (Z08-39) 13.83 16.26 10.21 14.10 1.176 0.738 1.020

Test 20 (Z08-41) 11.73 16.21 10.27 14.20 1.382 0.876 1.211

Test 25 (Z08-44) 11.98 16.52 10.89 14.31 1.379 0.909 1.194

Test 26 (Z08-45) 12.19 16.31 10.58 14.24 1.338 0.868 1.168

Test 27 (Z08-47) 11.90 16.45 10.91 14.30 1.382 0.917 1.202

Test 21 (Z08-14) 6.46 8.60 5.12 6.30 1.331 0.793 0.975

Test 22 (Z08-16) 6.37 8.82 5.21 6.49 1.384 0.818 1.019

Test 23 (Z08-18) 5.39 8.22 4.73 6.12 1.526 0.878 1.135

Test 24 (Z08-20) 5.03 8.40 4.82 6.24 1.670 0.959 1.241

Test 28 (Z08-22) 6.41 8.63 5.18 6.33 1.347 0.808 0.988

Test 29 (Z08-24) 5.61 8.47 4.62 6.32 1.510 0.824 1.127

Mean: 1.394 0.858 1.118

St. Dev.: 0.120 0.060 0.090

Co. of Var.: 0.086 0.070 0.080

Table 4.1Summary Table for Nominal Strengths of Third Point Braced Z-Sections from

Murray and Trout (2000)

Test No.

Experimental Buckling Strength

(k-ft)

Predicted Buckling Strength (k-ft)

69



Dist. Buckling

MnC3.1.1---

Mnexp

MnC3.1.2---

Mnexp

MnDB-----

Mnexp

Test B (Z10-Eave) 13.14 22.00 11.01 17.88 1.674 0.838 1.361

Test C (Z10-Eave) 15.98 22.06 12.00 17.81 1.381 0.751 1.115

Test D (Z10-Eave) 18.19 21.84 11.89 17.79 1.201 0.653 0.978

Test G (Z10-Eave) 16.05 22.06 11.80 17.84 1.375 0.735 1.112

Test E (Z10-Eave) 9.02 13.91 7.95 11.01 1.542 0.881 1.221

Test F (Z10-Eave) 8.50 14.00 8.19 11.11 1.647 0.964 1.307

Mean: 1.470 0.804 1.182

St. Dev.: 0.167 0.102 0.129

Co. of Var.: 0.114 0.127 0.109

Table 4.2

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Bryant et al. (1999a)

Test No.Experimental





Dist. Buckling

MnC3.1.1---

Mnexp

MnC3.1.2---

Mnexp

MnDB-----

Mnexp

Setup 2-0.105b 16.73 20.35 9.84 16.49 1.217 0.588 0.986

Setup 3-0.105a 14.30 20.40 9.68 16.46 1.426 0.677 1.151

Setup 3-0.105b 14.01 20.38 9.84 16.55 1.454 0.702 1.181

Setup 4-0.105 12.70 18.51 10.98 15.68 1.458 0.864 1.235

Setup 4-0.076 10.66 12.92 8.29 10.30 1.212 0.778 0.966

Setup 3-0.060a 7.10 9.47 5.11 7.23 1.333 0.720 1.018

Setup 3-0.060b 7.48 9.58 5.26 7.42 1.281 0.704 0.992

Setup 3-0.060c 6.74 9.65 5.73 7.32 1.432 0.849 1.086

Mean: 1.352 0.735 1.077

St. Dev.: 0.098 0.086 0.095

Co. of Var.: 0.073 0.117 0.088


Almoney and Murray (1998)




70



Dist. Buckling

MnC3.1.1---

Mnexp

MnC3.1.2---

Mnexp

MnDB-----

Mnexp

1G-Eave 6.69 8.77 5.76 6.55 1.312 0.861 0.979

2G-Eave 7.91 8.68 5.94 6.75 1.097 0.751 0.853

6G-Eave 7.60 8.51 5.25 6.71 1.120 0.691 0.883

3G-Eave 13.37 19.01 12.09 14.91 1.422 0.904 1.115

4G-Eave 15.87 20.37 12.47 15.59 1.284 0.786 0.982

5G-Eave 15.01 19.46 9.43 14.95 1.296 0.628 0.996

Mean: 1.255 0.770 0.968

St. Dev.: 0.113 0.094 0.085

Co. of Var.: 0.090 0.122 0.088

Table 4.4

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Davis et al. (1995)






Dist. Buckling

MnC3.1.1---

Mnexp

MnC3.1.2---

Mnexp

MnDB-----

Mnexp

Test #1 (Eave) 12.32 16.12 8.39 13.01 1.308 0.681 1.056

Test #2 (Eave) 12.67 16.88 8.74 13.28 1.332 0.690 1.048

Test #5 (Eave) 12.07 17.08 8.21 13.64 1.415 0.680 1.130

Test #3 (Eave) 7.51 8.79 4.53 7.11 1.170 0.604 0.947

Test #4 (Eave) 7.16 9.07 4.66 7.31 1.266 0.650 1.021

Test #6 (Eave) 22.67 26.42 17.90 21.32 1.165 0.790 0.940

Test #7 (Eave) 21.63 27.41 21.04 21.69 1.267 0.973 1.003

Test #8 (Eave) 13.11 16.38 9.36 13.32 1.250 0.714 1.016

Test #9 (Eave) 13.19 15.77 8.26 12.68 1.195 0.626 0.961

Mean: 1.263 0.712 1.014

St. Dev.: 0.077 0.100 0.057

Co. of Var.: 0.061 0.140 0.056


Bathgate and Murray (1995)




71



Dist. Buckling

MnC3.1.1---

Mnexp

MnC3.1.2---

Mnexp

MnDB-----

Mnexp

G8ZTP/S-1 (1A) 6.85 10.63 7.85 8.18 1.552 1.146 1.194

G8ZTP/S-1 (1B) 6.84 10.77 8.22 8.20 1.574 1.202 1.199

G8ZTP/S-1 (1C) 6.77 10.83 8.41 8.28 1.599 1.242 1.223

G8ZTP/S-1 (5A) 7.93 11.95 9.30 9.24 1.507 1.173 1.165

G10ZTP/S-1 (2A) 16.24 20.48 16.13 15.55 1.261 0.993 0.958

G10ZTP/S-1 (2B) 16.23 20.27 15.98 15.45 1.249 0.984 0.952

G10ZTP/S-1 (2C) 16.19 20.23 16.03 15.43 1.250 0.990 0.953

Mean: 1.427 1.104 1.092

St. Dev.: 0.153 0.103 0.120

Co. of Var.: 0.107 0.093 0.110

Table 4.6

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Borgsmiller et al. (1994)






Dist. Buckling

MnC3.1.1---

Mnexp

MnC3.1.2---

Mnexp

MnDB-----

Mnexp

Test #1 4.97 8.30 5.63 5.74 1.670 1.133 1.155

Test #2 4.67 7.61 5.51 5.20 1.629 1.180 1.113

Test #3 16.24 17.83 13.12 14.53 1.098 0.808 0.895

Test #4 16.26 18.34 13.39 14.89 1.128 0.823 0.916

Test #5 12.58 14.94 10.68 11.48 1.188 0.849 0.913

Test #6 11.10 13.91 9.77 10.30 1.253 0.880 0.928

Test #7 8.45 11.23 7.59 7.80 1.329 0.898 0.923

Mean: 1.328 0.939 0.977

St. Dev.: 0.216 0.141 0.100

Co. of Var.: 0.162 0.150 0.103


Earls et al. (1991)




72



Dist. Buckling

MnC3.1.1---

Mnexp

MnC3.1.2---

Mnexp

MnDB-----

Mnexp

Z-T-P/F-1 10.81 11.01 7.29 9.09 1.019 0.675 0.841

Z-T-P/S-1 10.54 14.70 9.71 11.29 1.395 0.921 1.071

Z-T-R/S-1 10.96 16.64 9.69 12.41 1.519 0.884 1.132

Mean: 1.311 0.827 1.015

St. Dev.: 0.213 0.109 0.125

Co. of Var.: 0.162 0.131 0.124

Table 4.8

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Brooks and Murray (1989)






Dist. Buckling

MnC3.1.1---

Mnexp

MnC3.1.2---

Mnexp

MnDB-----

Mnexp

N-ZIS-12-SF-1 16.39 24.43 15.61 19.48 1.490 0.952 1.189

N-ZISO-12-SF-1 20.98 26.56 16.57 21.17 1.266 0.790 1.009

N-ZISO-12-SS-1 19.03 24.76 16.48 19.85 1.301 0.866 1.043

N-ZISO-12-TF-1 18.50 25.43 16.68 19.73 1.375 0.902 1.066

Mean: 1.358 0.877 1.077

St. Dev.: 0.086 0.059 0.068

Co. of Var.: 0.063 0.067 0.063

Table 4.9

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Spangler and Murray (1989)




73



Dist. Buckling

MnC3.1.1---

Mnexp

MnC3.1.2---

Mnexp

MnDB-----

Mnexp

Test #1 4.48 10.21 0.84 7.81 2.279 0.188 1.743

Test #2 4.61 10.50 0.86 7.93 2.278 0.187 1.720

Test #3 4.45 10.63 0.91 7.94 2.389 0.204 1.784

Test #4 8.29 18.51 1.70 15.02 2.233 0.205 1.812

Test #5 7.61 18.55 1.86 15.02 2.438 0.244 1.974

Test #6 7.99 18.53 1.80 14.97 2.319 0.225 1.874

Mean: 2.322 0.209 1.818

St. Dev.: 0.070 0.020 0.085

Co. of Var.: 0.030 0.098 0.047

Table 4.10

Summary Table for Nominal Strengths of Laterally Unbraced Z-Sections from Bryant et al. (1999b)






Dist. Buckling

MnC3.1.1---

Mnexp

MnC3.1.2---

Mnexp

MnDB-----

Mnexp

Test #1a 6.15 15.20 1.04 11.54 2.470 0.169 1.876

Test #2a 6.20 14.14 0.94 11.21 2.281 0.152 1.809

Test #3a 6.72 15.21 1.00 11.86 2.264 0.149 1.765

Test #4a 10.10 22.86 1.48 18.40 2.263 0.147 1.822

Test #5a 9.61 24.00 1.32 18.51 2.497 0.137 1.926

Test #6a 9.09 25.32 1.39 19.75 2.786 0.153 2.173

Mean: 2.427 0.151 1.895

St. Dev.: 0.187 0.009 0.134

Co. of Var.: 0.077 0.063 0.071

Table 4.11

Summary Table for Nominal Strengths of Laterally Unbraced Z-Sections from Bryant et al. (1999c)




74

Local Buckling

Lateral Buckling

Distortional Buckling

Local Buckling

Lateral Buckling

Distortional Buckling

1.353 0.844 1.063 2.375 0.180 1.856

0.156 0.153 0.118 0.157 0.035 0.124

0.115 0.181 0.111 0.066 0.192 0.067

1.019 to 1.674

0.588 to 1.242

0.841 to 1.361

2.233 to 2.786

0.137 to 0.244

1.720 to 2.173

Range of Ratios (Mpred/Mexp)

Overall Mean (Mpred/Mexp)

Overall Standard Deviation

Overall Coefficient of

Variation

Table 4.12

Summary Table of All Z-Purlin Strength Data

Third Point Braced Tests Laterally Unbraced Tests

4.3 Prior Research

The methods used in this study for the prediction of local, lateral, and distortional

buckling need to be compared to outside studies to check for correctness. To accomplish

this, other research was found that did not use a standing seam roof system and

considered distortional buckling.

A study completed in 1997 used data from 42 laterally braced purlin sections to

test the Hancock Method as well as six other distortional buckling prediction methods

(Rogers and Schuster 1997). The Hancock Method was determined to be unconservative

by Rogers and Schuster as it had a mean value of 1.061 (Mpredicted/Mtest). In addition, this

study further determined that the Hancock Method had an average standard deviation of

0.068 and an average coefficient of variation of 0.075. These results compare well to

those determined by the distortional buckling procedure of this current study, which had a

75

mean value of 1.063 (Mpredicted/Mtest), a standard deviation of 0.118, and a coefficient of

variation of 0.111. While the mean strength predictions are similar, the increase in

standard deviation and coefficient of variation for the current study may be attributed to

the use of different standing seam roof components, whereas the study completed by

Rogers and Schuster did not use standing seam roof components.

Studies of distortional buckling by Hancock et al (1996) show the Hancock

Method to be conservative with a mean strength prediction of 0.92 (Mpredicted/Mtest).

However, most tests used in Hancock et al (1996) used purlin sections with opposed

orientation. Furthermore, only four of the 62 tests used in this study used opposed purlin

orientation. Brooks (1989) determined that opposed purlin orientation can provide up to

16% more strength when compared to “same direction” purlin orientation. This strength

increase is due to a decrease in lateral movement of the purlins in an opposed

configuration.

Research completed at the University of Florida (Ellifritt et al 1998) used third-

point and mid-point laterally braced Z-sections without a standing seam roof system. In

that research, local buckling strengths from AISI Specification C3.1.1 and distortional

buckling strengths from the Hancock Method were determined and compared to

experimental strengths. Shown in Table 4.13 is the data used by Ellifritt et al, and their

results. Also found in Table 4.13 are the results computed by the methods and

assumptions used in the present study. Ratios were determined by dividing the predicted

buckling strengths from the methods used in this study by the predicted buckling

strengths in Ellifritt et al (1998).

76

Table 4.13 clearly shows that the methods used in this study for the buckling

strength predictions compare well to the data presented in Ellifritt et al (1998). The slight

difference in local and distortional buckling strengths can be attributed to possible

rounding error, and a lack of radii information given in Ellifritt et al (1998). The radius

between the lip and flange, and between the flange and web were not given and were

assumed to be 0.15 in.

77

78

1) M

: mid

-poi

nt b

race

d, T

: thi

rd-p

oint

bra

ced

Not

es:

3) E

xp.:

E

Z14M

-8

Z16T

-1

Z14T

-1

Z14T

-1

Tes

t No.

Z16M

-4

Z14M

-5

Z14M

-6

2) L

ater

al b

uckl

ing

stre

ngth

s wer

e no

t det

erm

Dis

tort

iona

l B

uckl

ing

1.03

1.03

1.02

1.03

1.01

1.01

1.01

Loc

al

Buc

klin

g

1.00

1.02

1.03

1.03

1.00

1.00

0.99

Dis

tort

iona

l B

uckl

ing

81.1

106.

2

102.

9

107.

3

82.9

102.

4

111.

0

Loc

al

Buc

klin

g

104.

6

133.

5

130.

4

133.

8

112.

9

140.

1

154.

9

Dis

tort

iona

l B

uckl

ing

79.0

103.

2

100.

8

104.

5

82.1

101.

4

110.

3

Loc

al

Buc

klin

g

104.

6

130.

9

126.

6

129.

3

113.

2

140.

8

155.

8

Exp

.

59.3

86.4

86.2

103.

2

77.3

109.

8

85.7

xper

imen

tally

obt

aine

d bu

cklin

g st

reng

ths,

Dis

t.: D

isto

rtion

al b

uckl

ing

stre

ngth

pre

dict

ions

1 2 4

8.3

8.23

2

8.19

6

8.27

6

Sect

ion

Dep

th (i

n.)

8.26

9

8.31

8

8.36

5

2.41

6

2.30

3

2.31

4

2.23

0

Flan

ge

Wid

th

(in.)

2.32

2

2.33

0

2.34

1

0.07

2

0.06

1

0.07

2

0.07

2

Thi

ckne

ss

(in.)

0.06

1

0.07

2

0.07

2

0.91

1

0.96

3

0.96

3

0.96

0

ined

in th

e re

sear

ch o

f Elli

fritt

et a

l 199

8.

Yie

ld

Stre

ngth

(k

si)

61.4

60.6

60.6

60.6

61.4

42 40 38

Tab

le 4

.13

Sum

mar

y T

able

for

Com

pari

son

of R

esul

ts fo

r L

ater

ally

Bra

ced

Z-S

ectio

ns

Pred

icte

d St

reng

th R

atio

s

[M(2

)/M(1

)]

60.6

Buc

klin

g St

reng

ths f

rom

th

is S

tudy

(k-in

) (2)

Buc

klin

g St

reng

ths f

rom

Elli

fritt

et a

l. 19

98

(k-in

) (1)

Lip

D

epth

(in

.)

0.93

1

0.92

1

0.82

7

67.2

Lip

Ang

le

(deg

.)

35 40 40 41

4.4 Possible Causes of Scatter in Data

Standing seam roof components are of particular importance due to their ability to

provide an increase in torsional and lateral restraint supplied to the supporting purlin.

Furthermore, various component combinations acting on identical purlin sections can

affect the experimental strength of these purlins. However, the analytical prediction

methods used in this study cannot account for additional strength that these components

may provide, which is a cause of the data scatter in Figures 4.1 through 4.7. Variables

such as purlin orientation, clip type, panel type, and panel thickness affect the

experimental strength of the supporting purlin as discussed below. Table 4.14 shows the

components for all third point braced tests.

Tests were conducted at Virginia Tech to determine how purlin orientation can

affect the strength of a standing seam roof system supported by a Z-section purlin

(Brooks 1989). These tests showed that purlins oriented with their compression flanges

opposed can increase the strength of a standing seam roof system by as much as 16%

when compared to the same standing seam roof system in which the purlins have their

compression flanges facing in the same direction (Brooks 1989). This could be one

reason for the data scatter in Tables 4.1 through 4.12. However, only four of the 62

braced and none of the twelve unbraced tests utilized an opposed purlin orientation (see

Table 4.14).

Clip type plays an important role in the stiffness of a standing seam roof system

and could be another reason for data scatter in this study. The more rigid a standing seam

roof system is, the more lateral and torsional restraint it provides to the supporting purlin.

In evidence of this, Brooks (1989) reported that fixed clips can provide an approximately

79

2.8% strength increase over sliding clips. An increase is also apparent in the data

reported by Trout (2000), although percentages of the increase in strength were not

determined.

Another variable that may affect the strength of a standing seam roof system is the

type of steel roof panel used. For the tests reported in this study, two different types of

roof panels were used, ribbed type and pan type. Previous research has shown that ribbed

type panels can increase the strength of purlins supporting standing seam roof systems by

3.8% versus pan type panels (Brooks 1989). The reason for this increase is that the joint

between rib panels provides more torsional restraint to the purlin when compared to the

joint between the pan type panels (Brooks 1989).

Panel thickness may increase the stiffness of a standing seam roof system by

increasing the lateral and torsional restraint provided to the supporting purlin. However,

this is not necessarily the case. Studies have shown that panel thickness does not

significantly affect the strength of standing seam roof systems built with 10 in. deep Z-

sections (Trout 2000). On the other hand, there is a strength increase for standing seam

roof systems built using 8 in. deep Z-sections. For thin, 8 in. deep Z-purlins, a thinner

gage roof panel can decrease the strength of the purlin by as much as 15.8%, but for

thicker 8 in. deep Z-purlins, a thicker roof panel may decrease the strength of the purlin

by as much as 17.2% (Trout 2000).

80

Test No. Clip Type

Panel Type

Panel Thickness

Purlin Orientation

Test 18 (Z08-37) HS P 24 ga. S

Test 19 (Z08-39) LS P 24 ga. S

Test 20 (Z08-41) LF P 24 ga. S

Test 25 (Z08-44) HS P 24 ga. S

Test 26 (Z08-45) LS P 24 ga. S

Test 27 (Z08-47) LF P 24 ga. S

Test 21 (Z08-14) LF P 24 ga. S

Test 22 (Z08-16) LS P 24 ga. S

Test 23 (Z08-18) HS P 24 ga. S

Test 24 (Z08-20) HS P 24 ga. S

Test 28 (Z08-22) LF P 24 ga. S

Test 29 (Z08-24) LS P 24 ga. S

Test B (Z10-Eave) HS P 24 ga. S

Test C (Z10-Eave) HS P 24 ga. S

Test D (Z10-Eave) HS P 24 ga. OP

Test G (Z10-Eave) HS P 24 ga. S

Test E (Z10-Eave) HS P 24 ga. S

Test F (Z10-Eave) HS P 24 ga. S

Setup 2-0.105b HS R 26 ga. S

Setup 3-0.105a HS R 26 ga. S


Setup 4-0.105 HS R 26 ga. S

Setup 4-0.076 HS R 26 ga. S

Setup 3-0.060a HS R 26 ga. S


Setup 3-0.060c HS R 26 ga. SNotes:HS: High Sliding LF: Low Fixed P: Pan OP: OpposedLS: Low Sliding R: Ribbed S: Same


Table 4.14

Summary Table of Third Point Braced Test Components

(Trout and Murray 2000)

(Bryant et al. 1999a)

81

Test No. Clip Type

Panel Type

Panel Thickness

Purlin Orientation

1G-Eave HS P 26 ga. S






Test #1 (Eave) HS R 24 ga. S









G8ZTP/S-1 (1A) HS P 26 ga. S

G8ZTP/S-1 (1B) HS P 26 ga. S

G8ZTP/S-1 (1C) HS P 26 ga. S



G10ZTP/S-1 (2B) HS P 26 ga. S

G10ZTP/S-1 (2C) HS P 26 ga. S

Notes:HS: High Sliding LF: Low Fixed R: Ribbed OP: OpposedLS: Low Sliding HF: High Fixed S: Same P: PanF: Fixed (general) S: Sliding (general)


Table 4.14 Continued


(Davis et al. 1995)


82

Test No. Clip Type

Panel Type

Panel Thickness

Purlin Orientation

Test #1 HS R 24 ga. S







Z-T-P/F-1 F P 26 ga. S

Z-T-P/S-1 F P 26 ga. S

Z-T-R/S-1 S R 26 ga. S

N-ZIS-12-SF-1 SF R 26 ga. S

N-ZISO-12-SF-1 SF R 26 ga. OP

N-ZISO-12-SS-1 LS R 26 ga. OP

N-ZISO-12-TF-1 HF R 26 ga. OPNotes:HS: High Sliding LF: Low Fixed R: Ribbed OP: OpposedLS: Low Sliding HF: High Fixed S: Same P: PanF: Fixed (general) S: Sliding (general)


Table 4.14 Continued


(Earls et al. 1991)


Alone, some of the variables discussed may not significantly impact the

experimental strength of a Z-section purlin, but coupled together the effects may require

consideration. In spite of this, the marginal increase in the magnitudes of the coefficients

of variation and standard deviations as compared to other studies, such as Rogers and

83

Schuster (1997) and Hancock (1996), generally show that the effects of specific standing

seam roof components on the strength of a supporting purlin are not significant. As

previously mentioned, research completed by Rogers and Schuster (1997) showed that

distortional buckling strength predictions are unconservative as the average test to

predicted (MDB/Mexp) strength ratio was 1.061. For the current study, the average

strength ratio for the third point braced tests is 1.063 (MDB/Mexp), which demonstrates

that standing seam components do not drastically affect the strength of the supporting

purlins when distortional buckling is the controlling limit state.

This is further shown by an example in Table 4.15. Sections were sought that

kept all variables relatively the same except for clip type. Table 4.15 shows the effects of

clip type on experimental strength compared to local, lateral, and distortional buckling

predictions. The predicted to experimental strength ratios in Table 4.15 show little

variation amongst similar Z-sections when only considering clip type. This further shows

that standing seam roof components (in this case clip type) has little effect on the strength

of the supporting purlin.

84

Test No. Clip Type Local Buckling Lateral

BucklingDistortional

Buckling

Test 18 (Z08-37) HS 1.298 0.907 1.137

Test 20 (Z08-41) LF 1.382 0.876 1.211

Test 25 (Z08-44) HS 1.379 0.909 1.194

Test 26 (Z08-45) LS 1.338 0.868 1.168

Test 27 (Z08-47) LF 1.382 0.917 1.202

Ave: 1.356 0.895 1.182

SD: 0.037 0.022 0.030

COV: 0.028 0.024 0.025

Notes: LF: Low Fixed LS: Low SlidingHS: High SlidingData from Murray and Trout (2000)

Buckling Strength Prediction Ratios from Table 4.1

Table 4.15

Summary of Effect of Clip Type for 8.0 in. Deep, 0.102 Thick Third Point Braced Tests

4.5 Resistance Factor for Design

Although the Hancock Method does present the most accurate procedure for

predicting the buckling strength of a purlin supporting a standing seam system, it is

slightly unconservative for purlins oriented in the same direction. This is especially

important since this presents an over-prediction of section strength. In light of this, a

resistance factor (Φ) can be applied to a Hancock Method strength prediction for design

to compensate for the tendency to be slightly unconservative. This can be determined by

using Appendix F1.1 in the 1996 AISI Specification. Shown below is the calculation of a

85

resistance factor for the distortional buckling strength predictions used in this study.

Note that Mm, Fm, Pm, Vm, Vf are from AISI Specification Table F1 and described at the

end of this calculation.

nExp

DB RMM

==

063.1

.

(Mean value for all test results) (4.1)

( ) 051.12

11=

−

+

=m

mnC p (4.2)

where

Cp = Correction Factor

n = Number of tests (62 tests)

m = Degrees of freedom (m = n-1 or 61 tests)

( )( ) 118.0

11

2

=−

−=∑=

n

xxs

n

ii

(4.3)

where

s = Standard deviation

xi = Individual test results from Tables 4.1 through 4.9

x = Mean (From 6.1)

111.0=

=

np R

sV (4.4)

86

where

Vp = Coefficient of variation, must be greater than 0.065

Rn = From Equation 6.1

( ) 85.05.122225.2==Φ

+++− Qppfm VVCVV

mmm ePFM (4.5)

Where:

Mm = Mean value of the material factor, 1.10

Fm = Mean value of the fabrication factor, 1.00

Pm = Mean value of the professional factor, 1.00

Vm = Coefficient of variation of the material factor, 0.10

Vf = Coefficient of variation of the fabrication factor, 0.05

βo = Target reliability index (2.5 for structural members)

VQ = Coefficient of variation of the load effect (0.21)

87

CHAPTER V

EXAMPLE CALCULATIONS

5.1 Problem Statement for an 8 in. Deep Z-Section

The following example shows the procedures for determining the local and lateral

buckling strength predictions using the 1996 AISI Specification and the distortional

buckling strength prediction using the Hancock Method, as discussed in detail in Chapter

2. The Z-section used for this example is test 1G taken from Davis et al (1995), which

had an experimental strength of 6.69k-ft at a span of 25 feet with lateral braces located at

the third points. Figure 5.1 shows the dimensions of the Z-section used for this example.

The standing seam roof supported by the Z-section in test 1G is comprised of high sliding

clips, 26 ga. pan type roof panels, and the purlins are oriented in the same direction.

Definitions of symbols used are located at the end of these calculations along with Table

5.1. Table 5.1 is a comparison of the section strengths determined by hand and CFS

calculations to the experimental strength. Found in Appendix B are the CFS data runs for

the determination of local and lateral buckling strength predictions, and a MathCad

solution for the distortional buckling strength prediction for section 1G.

88

1.013" 2.499"

8"

2.551"

0.889"

R0.2656

R0.343848.3°

50°

0.06"

Fy=57.1ksi

Figure 5.1 Properties of Section 1G

5.2 Calculation of Section Properties

Three section modulii are needed to calculate local and lateral buckling strengths

(Se, Sc, Sf). Se is the effective section modulus calculated with the extreme compression

fiber at Fy. Sc is the effective section modulus calculated at a stress (Mc/Sf) in the extreme

compression fiber, where Mc is the inelastic critical moment. Sf is the full, unreduced

section modulus for the extreme compression fiber. Nomenclature in parentheses to the

left of an equation represents its location in the 1996 AISI Specification.

Section Modulus Se Calculations

The following calculations for Se use AISI Sections B4.2 and B2.1 for the

determination of the effective width of the compression flange, AISI Sections B3.2(a),

B2.1, and B4.2 for the determination of the effective width of the compression stiffener

lip, and AISI Sections B2.3(a), and B2.1 for the determination of the effective width of

89

the web in compression. Flat dimensions were used for components in tension. In the

first iteration, it was assumed that the horizontal neutral axis (X axis) was located at 4 in.

from the extreme compression fiber and the web was fully effective. Both assumptions

were revised in the second iteration.

AISI Section B4.2: Compression Flange

"2956.0211 =+=tRr

TT wtrtrBb ==

+++−′= "074.2

2tan

221

11γ

α

0.6056.34 <=t

wT (O.K. per Section B1.1-a-1)

(B4-1) 094.2928.1 ==fES

St

wT ≥ Therefore use Case III with 31

=n

(B4.2-11) ( ) 44 001835.05115 intSt

w

IT

a =

+

=

dtrCcT ==

+−′= "7372.0

2tan

21

1γ

α

0.14287.12 <=td (O.K. per Commentary Section B4.2)

(B4-2) ( ) 4

23

001176.012sin intdI s ==

θ

(B4.2-5) 0.1641.02 ≤=

=

a

s

II

C

"889.0==′ DC

90

"429.0=Tw

D

For simple lip stiffeners with 140 and oo 40≥≥θ 8.0≤Tw

D :

(B4.2-8) 107.30.4525.5 =≤

−=

Ta w

Dk

(B4.2-7) ( ) 738.22 =+−= auan kkkCk

(B2.1-4) 673.0967.0052.1>=

=

Ef

tw

kyTλ (Flange is not fully effective.)

(B2.1-3) 799.0

22.01=

−

=λλρ

(B2.1-2) ( ) "657.1== Twb ρ

Tension Flange and Stiffener Lip

"3738.0222 =+=tRr

"900.12

tan22

222 =

+++−′′=

γα trtrBbB

( ) "832.02

tan2

22 =

+−′′=

γα trCcB

AISI Section B3.2(a): Compression Stiffener Lip

287.12==td

tw

ksi

ww

y

rtrD

fDF

164.55cos

222 111

1 =

+−−

=γ

91

f1N.A.

0.03"+0.3738"

0.03"+0.2956"

4.0"

4.0"

57.1ksi

(B2.1-4) 673.0852.0052.1 1 >=

=

Ef

tw

kλ

where k = 0.43 per B3.2(a)

Since 673.0>λ the stiffener lip is not fully effective

(B2.1-3) 8704.0

22.01=

−

=λλρ

"642.0==′ dd s ρ

(B4.2-9) "4113.02 =′= ss dCd

AISI Section B2.3(a): Web

( ) "271.721 =++−′= trrAw

( ) ( )ksi

ww

y

rtDf

DF

45.525.05.0 1

1 =−−

=

( ) ( )ksi

www

y

rtDDf

DF

34.515.05.0 2

2 =−+−

=

where f1 is in compression and f2 is in tension

(B2.3-5) 9787.01

2 −=−

=Ψff

(B2.3-4) ( ) ( ) 452.2312124 3 =Ψ−+Ψ−+=k

(B2.1-4) 673.011.1052.1 1 >=

=

Ef

tw

kλ [Use f1 per B2.3(a)]

Since 673.0>λ the web may not be fully effective, need to check.

(B2.1-3) 722.0

22.01=

−

=λλρ

(B2.1-2) "25.5== wbe ρ

f1N.A.

0.03"+0.3738"

0.03"+0.2956"

4.0"

4.0"

57.1ksi

f2

92

(B2.3-1) ( ) "32.131 =

Ψ−= eb

b

For :236.0≤Ψ

(B2.3-2) "625.222 == eb

b

221wbb ≤+ and "64.3

2"945.321 =>=+

wbb

(Therefore web is fully effective for this iteration.)

Corners (Mean)

( )

"464.02

1 ==rUT

π

( )

"587.02

2 ==rU B

π

Compute Properties by Parts

Element Length (L)

(in.)

y From Top

Fiber (in.) (L*y) (L*y2)

I`x about own

axis (in3)

Top Flg 1.660 0.030 0.049 0.002 ---

Btm Flg 1.900 7.970 15.140 120.700 ---

Web 7.271 4.000 29.084 116.340 32.033

Top Cnr 0.464 0.137 0.064 0.010 0.004

Btm Cnr 0.587 7.835 4.600 36.100 0.010

Top Lip 0.411 0.274 0.113 0.031 0.004

Btm Lip 0.832 7.554 6.285 47.500 0.030

Total 13.126 55.335 320.683 32.081

93

( )( ) "216.4*

==∑∑

LyL

y below the top fiber

Since the assumed neutral axis (4.0 in.) does not equal the determined

neutral axis (4.216 in.), a second iteration is performed. No change occurs in

the compression flange properties because the neutral axis is below the

centerline and the maximum flexural stress (Fy) will still occur in the

compression (top) flange as assumed. The change in neutral axis location

slightly changes the stress gradient on the stiffener lip. However, this change

is minute and the resulting change in effective width is small enough to be

neglected. Furthermore, because of this change in neutral axis location, the

web is rechecked for effectiveness, as follows:

Web – 2nd Iteration

ksiy

rty

fy

F69.52

2 1

1 =

−−

=

ksi

w

y

rtyD

fy

F78.45

2 2

2 =

−−−

=

(B2.3-5) 8689.01

2 −−

=Ψff

(B2.3-4) ( ) ( ) 793.2012124 3 =Ψ−+Ψ−+=k

(B2.1-4) 673.0182.1052.1 1 >=

=

Ef

tw

kλ (Web may not be fully effective.)

(B2.1-3) 689.0

22.01=

−

=λλρ

(B2.1-2) 01.5== wbe ρ

f1N.A.

0.03"+0.3738"

0.03"+0.2956"

4.216"

3.784"

57.1ksi

f2

94

(B2.3-1) ( ) 295.131 =

Ψ−= eb

b

For Ψ :236.0−≤

(B2.3-2) "505.222 == eb

b

221wbb ≤+ and "852.3"216.0

2"80.321 =+<=+

wbb

Therefore, the web is not fully effective. The variables b1 and b2 represent

the effective compression parts of the web. When these are greater than the

compression portion of the web, the web is fully effective. However, if b1 and

b2 are less than the compression portion of the web (as shown here), the web is

not fully effective. The ineffective portion of the web is not included in the

section modulus.

Recompute Properties By Parts

The ineffective part of the web is represented as an element with a negative

length

"052.0"80.3"852.3 −=−=negb

with centroid location at

"647.122 11 =+++= negb

brty below top fiber

The horizontal neutral axis and moment of inertia about this axis are

calculated using only the effective portion of the web. The resulting neutral

axis is 4.2 in. below the extreme compression fiber, which is in good

agreement with the neutral axis location calculated in the first iteration,

therefore no further iterations are necessary. The resulting effective section

modulus (Se) computed with the extreme compression fiber at Fy is 1.75 in3

95

where .

=

yI

S xe

Section Modulus Sf Calculations

The calculation of Sf is more straightforward since it uses full, unreduced section

properties. The calculated Sf was determined to be equal to 2.05 in3, as the following

calculations show:

Horizontal Neutral Axis Location

With the assumption that all other properties are the same, the horizontal (X-

axis) neutral axis for the full, unreduced section is a ratio of the compression

lip and flange to the tension lip and flange multiplied by half of the full depth

of the web, which yields a neutral axis location of 088.4=y in. from the

extreme compression fiber. In an effort to reasonably account for differences

in geometric properties, similar elements were paired and their average was

used in the calculation of the moment of inertia about the X-axis (Ix) shown

below. For example, b is the average of the top and bottom flange unreduced

widths.

Calculation of Sf

4

223

2

23

2

22

1

23

38.7

sin2

cos212

sin

sin2

sin2

cossin

149.0637.022

0417.0

in

cracc

raur

rrauraba

I x =

−+++

++

−

++

+

++

++

=

γγγ

γγ

γγγγγα

305.2 inyI

S xf =

=

96

Section Modulus Sc Calculations

Calculation of Sc follows the same procedure as Se, except at a stress Mc/Sf for the

extreme compression fiber. Mc is determined by using AISI Eq. C3.1.2-16 for the

calculation of Me, and AISI Eq C3.1.2-5 for the calculation of My. Per AISI Eq C3.1.2-3,

Mc is 81.0 k-in and the resulting stress for the computation of Sc is 39.51 ksi. In turn, Sc is

determined to be equal to 1.84 in3. From these section properties, local buckling and

lateral buckling strength predictions are determined, as shown below.

Mc is the critical moment used in determining the stress (fSc) at which the section

modulus Sc is calculated. The coefficient of bending (Cb) is conservatively taken as 1.0

per AISI Section C3.1.2(a).

Calculation of Iyc

Iyc is the moment of inertia of the compression portion of the full,

unreduced section about the centroidal axis parallel to the web. Assuming the

Y-axis is located at mid-thickness of the web and using the same procedure as

in the above calculation, .074.0 3inI yc =

(C3.1.2-16) inkycb

e LdIEC

M −=

= 18.86

2 2

2π

(C3.1.2-5) ink

yfy FSM −== 1.117

By observation , therefore: yey MMM 56.078.2 >>

(C3.1.2-3) ( ) ink

e

yyc M

MMM −=

−

= 0.81

36100.1

910

ink

f

cSc S

Mf −=

= 51.39

97

The section modulus (Sc), calculated at stress fSc in the extreme compression fiber

is determined in the same manner as used for Se. Aside from the stress used and different

effective lengths, another difference between Se and Sc is that the web remains fully

effective for both iterations for Sc. Therefore:

"1.4=y below extreme compression fiber 355.7 inI x =

3842.1 inyI

S xc =

=

5.3 Local and Lateral Buckling Strength Predictions

The local buckling moment strength is:

(C3.1.1-1) ftkyepred FSM −== 33.8

Lateral buckling will not occur if the member is adequately braced.

Section 1G has a 25 ft. span that is third point braced, which results in an

unbraced length of 100 in. with

"100"48.5118.0 2

<=

=

fy

ycbu SF

EdICL

π

Lateral buckling needs to be checked.

The lateral buckling moment strength is

(C3.1.2-1) ftk

f

ccpred S

MSM −=

= 07.6

98

5.4 Distortional Buckling Strength Prediction

Strength prediction for distortional buckling for section 1G follows the procedures

of the Hancock Method as described in Chapter 2, Section 2.5.

Compression Flange Section Properties

( ) ( ) ( ) 4222223

lg 0061.0122

sin12

sininyb

tby

bb

btIx f

fll

lf =

++

−+

×=

θθ

( ) ( ) ( ) 423223

lg 172.012

cos2

cos212

inb

xb

bbxb

bb

tIy llfl

ff

ff =

+

−++

−+

×=

θθ

( ) ( ) ( ) 4lg 0195.0

2sin

2cos

2inybbxbb

bxybtIxy llfl

fff =

−×

×+−+

−×=

θθ

433

lg 000243.033

intbtb

J lff =

+=

Distortional Buckle Half Wavelength

int

DbIx wffd 85.24

280.4

25.0

3

2lg =

=λ

99

Formula Variables

( ) 22lg

2lg

11 0002.0039.0 inJbIx dfff =+

= λ

βηα

2lg

1lg2 0028.02 inIxybyIy fff =

+=

βηα

4722lg

1lg13 1074.3 inbIxyIy fff

−×=

−=

βηαηα

3

lg

lglg2

1 603.3 inA

IyIxx

f

ff =

++=β

22

016.0 −=

= in

dλπη

Distortional Buckling Stress Assuming KΦ=0

( ) ksi

fed A

E 045.1942 3

22121

lg

' =

−+±+

= ααααασ

Torsional Stiffness Restraint

( ) ..223.0

39.13192.256.1211.1

106.046.5

22244

24

2

'3

ininkip

DDD

EtDEtK

wdwd

dwed

dw

=

++

−

+

=Φ λλλσ

λ

100

Actual Distortional Buckling Stress

Since KΦ is positive, the procedure below is followed per Chapter 2 with

revised α1 and α3 terms:

( ) 2

1lg

2lg

11 00033.0039.0 in

EK

JbIx dfff =

++

= Φ

ηβλ

βηα

4722lg

1lg13 1034.7 inbIxyIy fff

−×=

−=

βηαηα

( ) ( ) ksi

fed A

E 35.3742 3

22121

lg

=

−+±+

= ααααασ

Predicted Strength Considering Distortional Buckling

Since yed f2.2≤σ then

ksi

y

ed

y

edyc ff

ff 963.3722.01 =

−

=

σσ

Since then 0≥ΦK ftkcfpred fSM −== 49.6

or (using Sftkcfpred fSM −== 55.6 f from CFS)

101

Strength Determination

Method

"Hand" Calculations

(k-ft)

Ratio (Mhand/Mexp)

CFS Calculations

(k-ft)

Ratio (MCFS/Mexp)

Local Buckling (C3.1.1) 8.33 1.25 8.77 1.31

Lateral Buckling (C3.1.2) 6.07 0.91 5.76 0.86

Distortional Buckling (Hancock Method) 6.49 0.97 6.55 0.98

Notes:

Table 5.1

Summary of Predicted and Experimental Strengths for Z-Section 1G

1) Experimental strength for 1G = 6.69 k-ft2) Distortional buckling calculations used S e from CFS for CFS Calculations

Nomenclature

Aflg = full cross-sectional area of the compression flange and lip

b1, b2 = Effective widths of element

b = Flange width (subscript denotes top or bottom location)

be = Effective design width of element

bf = Compression flange width

bl = Length of lip

B′ = Full flange width

C′ = Full lip length

C2, C1 = Effective width coefficients

=′sd Actual effective width of stiffener

ds = Reduced effective width of stiffener

102

Dw = Depth of web

D = Overall depth of lip

d = Depth of section

E = Modulus of elasticity

Fy = Yield stress

f = Stress in the compression element computed on the basis of the effective design width

f1, f2 = Web stresses

fsc = Stress at which Sc is calculated

Iyc = Moment of inertia of the compression portion of a section about the centroidal axis

of the entire section parallel to the web, using the full unreduced section

Ixflg = Moment of inertia of compression lip-flange component about principal axis

Iyflg = Moment of inertia of compression lip-flange component about principal axis

Ixyflg = Product of inertia of compression lip-flange component about major and minor

centroidal axes

Ix = Moment of inertia of full section about principal axis

Ia = Adequate moment of inertia of stiffener so that each component element will behave

as a stiffened element

Is = Actual moment of inertia of the full stiffener about its own centroidal axis parallel to

the element to be stiffened

Jflg = St. Venant torsion constant

k = Actual plate buckling coefficient

ka = Ideal plate buckling coefficient

KΦ = torsional stiffness restraint

Lu = Maximum unbraced length

Me = Elastic critical moment

Mc = Critical moment

My = Moment causing initial yield at the extreme compression fiber of the full section

R = Inside bend radius between web and flange (subscript 1 denotes top and subscript 2

103

denotes bottom)

r = Mean bend radius between web and flange (subscript 1 denotes top and subscript 2

denotes bottom)

Sf = Elastic section modulus of full, unreduced section for the extreme compression fiber

Se = Elastic section modulus of the effective section calculated with the extreme

compression fiber at a stress Fy

Sc = Elastic section modulus of the effective section calculated with the extreme

compression fiber at a stress Mc/Sf

t = Thickness

uT, uB = Lengths of radii

wT = Flat width of compression flange exclusive of radii

wB = Flat width of tension flange exclusive of radii

α = Parameter in determining the effective area of a stiffener

γ = Angle between lip and flange (radians) (subscript 1 denotes top and subscript 2

denotes bottom)

λ = Slenderness factor

λd = Distortional buckle half wavelength

σed = Actual distortional buckling stress

=′edσ Distortional buckling stress assuming KΦ is zero

ρ = Reduction factor

ιf = Torsional constant of the compression flange and lip

uT, uB = Length of radii

=y Location of neutral axis with respect to the extreme compression fiber

=ηβααα ,,,, 1321 parameters related to the section wavelength and geometric properties

104

CHAPTER VI

SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS

6.1 Summary

This study used the experimental results obtained from 62 third point braced and

12 laterally unbraced standing seam roof system tests conducted at Virginia Tech. All

tests were simple span and utilized cold-formed Z-sections. These experimental data

were used to determine which of three buckling methods most accurately predicted the

strength of the Z-sections. The three methods analyzed were the AISI Specification

provision for local buckling, the AISI Specification provision for lateral buckling, and the

Hancock Method for distortional buckling. The analyses consisted of predicted-to-

experimental strength ratios, and standard deviations and coefficients of variation of these

ratios to determine which method was the most accurate. After the analyses were

completed, the Hancock Method was determined to provide the most accurate overall

strength predictions for third point braced purlins supporting standing seam roof systems.

Given that the amount of lateral bracing a standing seam roof system provides to

the supporting purlin is not known, the amount of lateral bracing provided was

conservatively assumed to be zero for all tests. In addition, a resistance factor for design

was developed to account for the variation between predicted and experimental results.

105

6.2 Conclusions

Previous research has determined that the AISI provisions for local buckling

strength predictions of cold-formed purlins supporting standing seam roof systems is

highly unconservative and that the AISI provisions for lateral buckling strength

predictions of cold-formed purlins is too conservative. Although the Base Test Method

(Carballo et al 1989) does provide a means to accurately predict the strengths of purlins

supporting standing seam roof systems, it can take up valuable time and resources.

Therefore, a “hand” method is needed to quickly and accurately predict the buckling

strengths of cold-formed purlins supporting standing seam roof systems.

Based on Table 4.12, the Hancock Method represents the most accurate method

for predicting the strength of third point laterally braced Z-purlins that support standing

seam roof systems. While the Hancock Method does represent the best technique to

predict the buckling strengths of Z-section purlins supporting standing seam roof

systems, this does not necessarily mean that the tests included in this study failed solely

by distortional buckling. Without observing each individual test as it goes through its

failure event, the actual failure mode can only be hypothesized with the use of the

previously mentioned formulas and provisions.

The torsional spring stiffness (KΦ) for every test in this study was positive. This

occurs when the web torsionally restrains the compression lip-flange component.

Because of this, the conclusions discussed herein only pertain to the Hancock Method

when a positive torsional restraint (+KΦ) is used.

For the tests used in this study, the standing seam roof systems were assumed to

not provide any torsional or lateral restraint to the supporting Z-purlins. Overall, this is a

106

conservative assumption, yet necessary since the amount of lateral bracing and torsional

restraint a standing seam roof system provides to a purlin is not known. While this

assumption produces good results for the laterally braced configurations (especially the

Hancock Method), it does not produce good results for the laterally unbraced

configurations for any strength prediction method, as shown by Table 4.12.

The AISI Specification provisions for local buckling assume a test is fully

laterally braced, therefore producing unconservative strength predictions for third point

braced standing seam roof systems. The AISI Specification provisions for lateral

buckling assume a test is only supported by the lateral braces at the third points (since the

roof system is neglected), therefore producing conservative strength predictions. The

Hancock Method for distortional buckling also predicts strengths assuming the purlin is

only laterally braced (no support from the roof system), but uses a shorter buckle

wavelength and different equations to arrive at a less conservative (in fact slightly

unconservative) strength prediction. Moreover, the 1996 AISI Specification for the

prediction of the lateral buckling strength yields a conservative value (an average of

15.6%) which could result in an inefficient use of a purlin’s strength. On the other hand,

the 1996 AISI Specification for the prediction of the local buckling strength yields an

unconservative value (an average of 35.3%), which could result in an over-prediction of

purlin strength and induce a failure event.

Data for strength prediction methods for the laterally unbraced tests were not

included in Figure 4.7 due to the error associated with assuming that the standing seam

roof system does not provide any lateral bracing to the purlin. In the case of local

buckling, this assumption is meaningless because AISI provisions determine the strength

107

assuming the section is fully laterally braced, when in fact it is not. Hence, the AISI

Specification for the prediction of local buckling is unconservative. For lateral buckling

this is a conservative assumption because the AISI provisions predict strengths assuming

the purlin is not supported by the standing seam roof system (therefore the purlins are

“stand alone”). In the case of the Hancock Method, this assumption is unconservative

because the calculated failure half wavelength (λd) is used instead of the unbraced length

of the purlin. As previously mentioned, the nature of a purlin is that it will tend to fail by

lateral buckling at lower stresses if longer wavelengths can develop. Therefore, the

Hancock Method should not be used for strength prediction of laterally unbraced

configurations of standing seam roof systems supported by Z-section purlins.

6.3 Design Recommendations

To further understand the effects of distortional buckling on standing seam roof

systems, several points need to be studied. First, studies need to be conducted to

determine exactly how a purlin supporting a standing seam roof system fails (in the past

this failure has been called lateral-torsional buckling in conjunction with local buckling).

This study should include several simple span, laterally braced purlins that support

standing seam roof systems, and the failure event should be analyzed in order to correctly

determine the governing failure mode.

The amount of lateral restraint supplied by a standing seam roof system to a

supporting purlin needs to be determined. The current Hancock Method and the AISI

provisions for local and lateral buckling cannot effectively determine this. As previously

mentioned, a Z-section purlin supporting a standing seam roof system is neither fully

108

braced nor unbraced by the standing seam roof system. This partial laterally braced state,

which is based on the system’s components, will increase the strength of a supporting

purlin to a degree. For example, when identical Z-sections supporting standing seam roof

systems built with different components are tested, the resulting experimental strengths of

these purlins will be different. However, the AISI provisions for local and lateral

buckling and the Hancock Method for distortional buckling will predict the same

buckling strengths (for each respective buckle mode) for all identical purlins. Hence, the

effect of different standing seam roof system components is not accounted for by any

buckling strength prediction method. The outcome of this proposed study should provide

a correction factor for any standing seam roof system component combination.

109

REFERENCES Almoney, K., and Murray, T.M., (1998), “Gravity Loading Base Tests Using Standing Seam CFR Panels and 8 in. Deep Purlins,” Report No. CE/VPI-ST98/08, Charles Via Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University. American Iron and Steel Institute (1980). “Specification for the Design of Cold-Formed Steel Structural Members,” Cold-Formed Steel Design Manual, Washington, D.C. American Iron and Steel Institute (1989). “Specification for the Design of Cold-Formed Steel Structural Members,” Cold-Formed Steel Design Manual, Washington, D.C. American Iron and Steel Institute (1996). “Specification for the Design of Cold-Formed Steel Structural Members,” Cold-Formed Steel Design Manual, Washington, D.C. American Iron and Steel Institute (1999). “Specification for the Design of Cold-Formed Steel Structural Members with Commentary, Supplement No. 1,” Cold-Formed Steel Design Manual, Washington, D.C. Borgsmiller, J.T., Murray, T.M., and Sumner, E.A., (1994), “Gravity Loading Tests of Z-Purlin Supported SLX 264-FL Roof Covering System,” Report No. CE/VPI-ST94/01, Charles Via Department of Civil Engineering, Virginia Polytechnic Institute and State University. Bathgate, C.S., and Murray, T.M., (1995), “Gravity Loading of Z-Purlin Supported Starshield Building Roof Covering Systems,” Report No. CE/VPI-ST95/03, Charles Via Department of Civil Engineering, Virginia Polytechnic Institute and State University. Brooks, S.D., and Murray, T.M., (1990), “A Method for Determining the Strength of Z- and C-Purlin Supported Standing Seam Roof Systems,” Proceedings of the Tenth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, MO, October 23-24. Brooks, S.D., (1989), “Evaluation of the Base Test Method for Determining the Strength of Standing Seam Roof Systems Under Gravity Loading,” M.S. Thesis, Charles Via Department of Civil Engineering, Virginia Polytechnic Institute and State University. Bryant, M.R., Murray, T.M., and Sumner, E.A., (1999a), “Gravity Loading Base Tests Using LTC Standing Seam Panels Supplemental Tests,” Report No. CE/VPI-ST99/06, Charles Via Department of Civil Engineering, Virginia Polytechnic Institute and State University.

110

REFERENCES (continued) Bryant, M.R., Murray, T.M., and Sumner, E.A., (1999b), “Gravity Loading Base Tests Using LTC Standing Seam Panels and 8 in. Deep Z-Purlins,” Report No. CE/VPI-ST99/02, Charles Via Department of Civil Engineering, Virginia Polytechnic Institute and State University. Bryant, M.R., Murray, T.M., and Sumner, E.A., (1999c), “Gravity Loading Base Tests Using LTC Standing Seam Panels and 10 in. Deep Z-Purlins,” Report No. CE/VPI-ST99/03, Charles Via Department of Civil Engineering, Virginia Polytechnic Institute and State University. Carballo, M., Holzer, S.M., and Murray, T.M., (1989), “Strength of Z-Purlin Supported Standing Seam Roof Sytems Under Gravity Loading,” Research Progress Report CE/VPI-ST89/03, The Charles E. Via Department of Civil Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA. Davies, J.M. and Jiang, C., (1996), “Design of Thin-Walled Beams for Distortional Buckling,” Proceedings of the Thirteenth International Specialty Conference on the Design of Cold-Formed Steel Structures, St. Louis, MO., October 17-18.

Davies, J.M., Jiang, C., and Ungureanu, V. (1998), “Buckling Mode Interaction in Cold-Formed Steel Columns and Beams,” Proceedings of the Fourteenth International Specialty Conference on Cold-Formed Steel Structures, St. Louis MO., October 15-16. Davis, D.B., Otegui, M.A., and Murray, T.M., (1995), “Gravity Loading Base Tests,” Report No. CE/VPI-ST95/07, Charles Via Department of Civil Engineering, Virginia Polytechnic Institute and State University.

Desmond, T.P., Pekoz, T., and Winter, G., (1981), “Edge Stiffeners for Thin-Walled Members,” Journal of the Structural Division, ASCE, Vol. 107, No. ST2, pp. 329-353. Earls, C.J., Pugh, A.D., and Murray, T.M., (1991), “Base Test for Z-Purlin Under Gravity Load With SSR System,” Report No. CE/VPI-ST91/08, Charles Via Department of Civil Engineering, Virginia Polytechnic Institute and State University. Ellifritt, D. S., Glover, R.L., and Hren, J.D. (1998), “A Simplified Model for Distortional Buckling of Channels and Zees in Flexure,” Proceedings of the Fourteenth International Specialty Conference on Cold-Formed Steel Structures, St. Louis MO., October 15-16.

Ellifritt, D. S., Haynes, J., and Sputo, T., (1992), “Flexural Capacity of Discretely Braced C’s and Z’s,” Proceedings of the Eleventh International Specialty Conference on Cold-Formed Steel Structures, St. Louis, MO., October 20-21.

111

REFERENCES (continued)

Hancock, G.J., (1995), “Draft Ballot and Commentary in Combined ASD/LRFD Format,” AISI Committee on Specifications for the Design of Cold-Formed Steel Structural Members – Subcommittee 24, Ballot S95-55B.

Hancock, G.J., Merrick, J.T., and Bambach, M.R., (1998), “Distortional Buckling Formulae for Thin Walled Channel and Z-Sections with Return Lips,” Proceedings of the Fourteenth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, MO., October 15-16.

Hancock, G.J., (1985), “Distortional Buckling of Steel Storage Rack Columns,” Journal of Structural Engineering, ASCE, Vol. 111, No. 12, pp. 2770-2783.

Hancock, G.J., (1994), “Design of Cold-Formed Steel Structures (to Australian Standard AS 1538-1988) 2nd Edition,” Australian Institute of Steel Construction, Sydney, Australia, pp. 38.

Hancock, G.J., (1997), “Design for Distortional Buckling of Flexural Members,” Thin-Walled Structures, Vol. 27, No. 1, pp. 3-12. Hancock, G.J., and Lau, S.C.W., (1986), “Distortional Buckling Formula for Thin-Walled Channel Columns,” Research Report No. R-521, School of Civil and Mining Engineering, University of Sydney, Sydney, Australia. Hancock, G.J., and Lau, S.C.W., (1987), “Distortional Buckling Formulas for Channel Columns,” Journal of Structural Engineering, ASCE, Vol. 113, No. 5, pp.1063-1078.

Hancock, G.J., and Lau, S.C.W., (1990), “Inelastic Buckling of Channel Columns in the Distortional Mode,” Thin-Walled Structures, Vol. 10, pp. 59-84.

Hancock, G.J., Rogers, C.A., and Schuster, R.M., (1996), “Comparison of the Distortional Buckling Method for Flexural Members with Tests,” Proceedings of the Thirteenth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, MO, October 17-18.

Hancock, G.J., (1998), Design of Cold-Formed Steel Structures, 3rd Edition, Australian Institute of Steel Construction, North Sydney, Australia.

Marsh, C., (1990), “Influence of Lips on Local and Overall Stability of Beams and Columns,” Proceedings of the Structural Stability Research Council, Annual Technical Session, pp. 145-153.

112

REFERENCES (continued) Moreyra, M.E., and Pekoz, T., (1993), “Behavior of Cold-Formed Steel Lipped Channels Under Bending and Design of Edge Stiffened Elements,” Research Report 93-4, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY. Murray, T.M., and Trout, A.M., (2000), “Reduced Number of Base Tests,” Report No. CE/VPI-ST00/17-00, Charles Via Department of Civil Engineering, Virginia Polytechnic Institute and State University.

Pi, Y-L., Put, B.M., and Trahair, N.S., (1997), “Lateral Buckling Strengths of Cold-Formed Z-Section Beams,” Research Report No. R572, The University of Sydney, Australia, Center for Advanced Structural Engineering.

Rogers, C.A., (1995), “Local and Distortional Buckling of Cold-Formed Steel Channel and Zed Sections in Bending;” M.A.Sc. Thesis presented to the Department of Civil Engineering, University of Waterloo, Waterloo, Ontario.

Rogers, C.A., and Schuster, R.M., (1996), “Cold-Formed Steel Flat Width Ratio Limits, d/t and di/w,” Proceedings of the Thirteenth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, MO., October 17-18. Rogers, C,A., and Schuster, R.M., (1997), “Flange/Web Distortional Buckling of Cold-Formed Steel Sections in Bending,” Thin-Walled Structures, Vol. 27, No. 1, pp. 13-29. RSG Software, Inc., (1998), Cold-Formed Steel Design Software Version 3.02, Lee’s Summit, MO. S136-94, (1994), Cold-Formed Steel Structural Members, Canadian Standards Association, Rexdale (Toronto), Canada. Schafer, B.W., and Pekoz, T., (1998), “Laterally Braced Cold-Formed Steel Flexural Members with Edge Stiffened Flanges,” Proceedings of the Fourteenth International Specialty Conference on Cold-Formed Steel Structures, St. Louis, MO., October 15-16. Sharp, A.M., (1966), “Longitudinal Stiffeners for Compression Members,” Journal of the Structural Division, ASCE, Vol. 92, No. ST5, pp. 187-211. Spangler, D., and Murray, T.M., (1989), “Integration of Standing Seam Roof Systems,” Report No. CE/VPI-ST89/07, Charles Via Department of Civil Engineering, Virginia Polytechnic Institute and State University.

Timoshenko, S.P., and Gere, J.M., (1961), Theory of Elastic Stability, 2nd Edition, McGraw-Hill, New York, NY.

113

REFERENCES (continued) Trout, A.M., (2000), “Further Study of the Gravity Loading Base Test Method,” M.S.C.E., Thesis Presented to the Charles Via Department of Civil Engineering, Virginia Polytechnic Institute and State University. von Karman, T., Sechler, E.E., Donnell, L.H., (1932), “The Strength of Thin Plates in Compression,” Transactions of the ASME, 54, pp. 53-57. Willis, C.T., and Wallace, B., (1990), “Behavior of Cold-Formed Steel Purlins Under Gravity Loading,” Journal of Structural Engineering, ASCE, Vol. 116, No. 8, pp. 2061-2069. Winter, G., (1947), “Strength of Thin Steel Compression Flanges,” Transactions of the ASCE, Paper No. 2305, Trans., 112, 1. Yu, W.W., (2000), Cold-Formed Steel Design, 3rd Edition, John Wiley & Sons, Inc., New York, NY.

114

APPENDIX A

1995 Distortional Buckling draft Ballot S95-55B 2000 Distortional Buckling Working Ballot S95-55B

115

AISI Committee on Specifications For the Design of Cold-Formed Steel Structural Members

Subcommitte 24 – Flexural Members Ballot No. S95-55B

Date: December 20, 1995 (draft ballot), October 2, 2000 (working ballot)

ADDITION TO SPECIFICATION and COMMENTARY SECTION C3.1.4, COMBINED LRFD AND ASD

C3.1.4 Distortional Buckling Strength The nominal strength of C- and Z-sections subject to distortional buckling, Mn, where distortional buckling involves rotation of the compression flange and lip about the flange-web junction, shall be calculated as follows:

ccn fSM = (Eq. C3.1.4-1) 67.1=Ωb (ASD) 90.0=Φb (LRFD)

where

Sf =Elastic section modulus of the full unreduced section for the extreme compression fiber

Sc = Sf when KΦ as given by Eq. C3.1.4-12 is positive or zero

Sc =Elastic section modulus of the effective section calculated at a stress fc in the extreme compression fiber, with k = 4.0 in Eq. B2.1-4 and ignoring Section B4.2 when KΦ as given by Eq. C3.1.4-12 is negative

fc = Critical stress calculated as follows: For ed 2> yF2.σ (Eq. C3.1.4-2) Ff yc = For ed 2 yf2.≤σ (Eq. C3.1.4-3)

−

y

ed

y

edyc fF

σσ22.01 = Ff

116

where Fy = Yield point of section

σed = Elastic critical distortional buckling stress calculated as follows:

( ) ( )[ ] 32

2121 42

ααααασ −+±+=f

ed AE (Eq. C3.1.4-4)

( )E

KJbIx dfff ηβλ

βηα

1

22

11 039.0 Φ++= (Eq. C3.1.4-5)

+= fff IxybyIy

12

2β

ηα (Eq. C3.1.4-6)

−= 22

113 fff bIxyIy

βηαηα (Eq. C3.1.4-7)

++=

f

ff

AIyIx

x2

1β (Eq. C3.1.4-8)

25.0

3

2

280.4

=

tbbIx wff

dλ (Eq. C3.1.4-9)

( )

++

−+

=Φ 2244

24

2

'3

39.13192.256.1211.1

106.046.5

2

wdwd

dwed

dw bbb

EtbEtK

λλλσ

λ

(Eq. C3.1.4-10) 2

=

dλπη (Eq. C3.1.4-11)

where '

edσ is obtained from Eq. C3.1.4-4 with

( )22

11 039.0 dfff JbIx λ

βηα += (Eq. C3.1.4-12)

When KΦ is negative in Eq. C3.1.4-10, compute KΦ with σ′ed = 0. The smaller positive value of σ′ed given by Eq. C3.1.4-4 must be used. When bracing, which fully restrains rotation of the flange and lip in the distortional mode, is located at an interval less than λd computed by Eq. C3.1.4-9, use the bracing interval in place of λd in Eq. C3.1.4-10 and C3.1.4-11. Af = Full cross-sectional area of compression flange and lip bf = Compression flange width

117

bw = Web depth E = Modulus of elasticity

Ixf, Iyf = Moment of inertia of compression flange and lip about x, y axes respectively where the x,y axes are located at centroid of flange and lip with x-axis parallel with flange Ixyf = Product of inertia of compression flange and lip about x,y axes

Jf = St. Venant torsion constant of compression flange and lip

=yx, Distance from flange-web junction to centroid of compression flange and lip in x,y directions respectively

X bar bar

y

x

fY

b

Centroid of compression flange and lip

118

APPENDIX B

CFS Data Runs for Section 1G MathCad data Run for Section 1G

119

Standard Conditions

ksi 1000 lb

in2kip 1000 lb

Input Data

E 29500ksi. F y 57.1 ksi b w 8.0 in L 0.889 in θ 50.2 π

180.

S f 2.0729in3 S xe 0.0 in3 t 0.06 in b 2.551 in L b 100.0 in

Section Properties

Centroidal Lengths

lip L t2

b f b t2

Flange & Lip Area

A f t b f lip. A f 0.2028 in2=

Centroid Location of Flange & Lip

y ot

2 A f.lip2 sin θ( )..

xot

A f

b f2

2lip b f lip cos θ( )

2...

xo 1.6507 in= y o 0.0839 in=

Flange & Lip Moment of Inertia

I xf t lip3 sin θ( )2.

12lip lip sin θ( ).

2y o

2.

b f t2.

12b f y o

2.. I xf 6.101710 3. in4=

I yf tb f

3

12b f

b f2

xo

2

. lip b flip cos θ( ).

2xo

2. lip3 cos θ( )2

12.. I yf 0.172 in4=

I xyf t b f xob f2

. y o. lip b f xo lip cos θ( )2

.. lip sin θ( )2

. y o.. I xyf 0.0195 in4=

J f13

b f. t3. 13

lip. t3. J f 2.433610 4. in4=

123

Formula Variables

β 1 xo2 I xf I yf

A fβ 1 3.6033 in2=

λ 4.80I xf b f

2. b w.

2 t3.

0.25

. λ 24.8481 in=

λ d if λ L b< λ, L b, λ d 24.8481 in=

ηπ

λ d

2η 0.016 in 2=

α 1η

β 1I xf b f

2. 0.039 J f λ d2.. α 1 1.980310 4. in2=

α 2 η I yf2β 1

y o. b f. I xyf..α 2 2.786510 3. in2=

α 3 η α 1 I yf. η

β 1I xyf

2. b f2..

α 3 3.736310 7. in4=

124

Distortional Buckling Stress with Kφ=0

σ ed1'E

2 A f.α 1 α 2 α 1 α 2

2 4 α 3.. σ ed1' 19.0454 ksi=

σ ed2'E

2 A f.α 1 α 2 α 1 α 2

2 4 α 3.. σ ed2' 415.1013ksi=

σ ed' if σ ed1' 0 σ ed1' σ ed2'<. σ ed1', σ ed2', σ ed' 19.0454 ksi=

K φ2 E. t3.

5.46 b w 0.06 λ d.1

1.11σ ed'. b w4. λ d

2.

E t2. 12.56λ d4. 2.192 b w

4. 13.39λ d2. b w

2...

K φ 0.2227 kip inin.=

Used only if Kφ is greater than or equal to zero.

α 4η

β 1I xf b f

2. 0.039 J f. λ d2..

K φ

β 1 η. E.α 4 3.290910 4. in2=


β 1I xyf

2. b f2.. α 5 7.340310 7. in4=

σ ed1E

2 A f.α 4 α 2 α 4 α 2

2 4 α 5.. σ ed1 415.8621ksi=

σ ed2E

2 A f.α 4 α 2 α 4 α 2

2 4 α 5.. σ ed2 37.3484 ksi=

σ eda if σ ed1 0 σ ed1 σ ed2<. σ ed1, σ ed2, σ eda 37.3484 ksi=

125

Used only if Kφ is less than zero.

K φ '2 E. t3.

5.46 b w 0.06 λ d..K φ ' 0.2459 kip in

in.=

α 6η

β 1I xf b f

2. 0.039 J f. λ d2..

K φ 'β 1 η. E.

α 6 3.427710 4. in2=


β 1I xyf

2. b f2..

α 7 7.716510 7. in4=

σ ed3E

2 A f.α 6 α 2 α 6 α 2

2 4 α 7.. σ ed3 39.2547 ksi=

σ ed4E

2 A f.α 6 α 2 α 6 α 2

2 4 α 7.. σ ed4 415.9458ksi=

σ edb if σ ed3 0 σ ed3 σ ed4<. σ ed3, σ ed4,σ edb 39.2547 ksi=

Actual Distortional Buckling Stress and Axial Load

σ ed if K φ 0 σ eda, σ edb, σ ed 37.3484 ksi=

P 1E2

α 4 α 2 α 4 α 22 4 α 5.. P 1 7.5743 kip=

P 2E2

α 6 α 2 α 6 α 22 7 α 5.. P 2 14.3356 kip=

P cr if K φ 0 P 1, P 2, P cr 7.5743 kip=

126

Nominal Strength of Section Resulting from Distortional Buckling

M ed S f σ ed. M ed 77.4194 kip in.=

M y S f F y.M y 118.3626kip in.=

M cr M yM edM y

. 1 0.22M edM y

..M cr 78.6942 kip in.=

M c if M ed 2.2 M y.> M y, M cr, M c 78.6942 kip in.=

S c if K φ 0 S f, S xe, S c 2.0729 in3=

M n S cM cS f

. M n 78.6942 kip in.=

f c1 F y

f c1 F yσ edF y

. 1 0.22σ edF y

..F c2 F y

σ edF y

. 1 .22σ edF y

..

f c if σ ed 2.2 F y.> F y, f c1, f c if σ ed 2.2 F y> f c1, F c2,

M n1 S f f c.S c if K φ 0 S f, S xe,

M n2 S xe f c.

M n S c f c.M n if K φ 0 M n1, M n2,

M n 78.6942 kip in.=M n 78.6942 kip in.=

127

VITA Scott David Cortese was born on January 9th, 1975 in Augusta, Georgia. He

graduated from Corning East High School in Corning, New York in 1993 and subsequently

attended Bowling Green State University. Scott graduated from Bowling Green in 1997 with

a Bachelor of Science in Geology and a Bachelor of Science in Environmental Science.

Immediately following his undergraduate education, the author worked as an Environmental

Scientist/Geologist at Hull & Associates, Inc. in Toledo, Ohio. Scott’s graduate education

in structural engineering began in the fall of 1998 at Virginia Polytechnic Institute and State

University and concluded in May 2001 with this thesis. Currently, he is working as a bridge

design engineer at Kimley-Horn & Associates, located in West Palm Beach, Florida.

128

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