Post on 03-Apr-2020
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
DISTORTIONAL BUCKLING
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
4.2 Summary Table for Nominal Strengths of Third Point Braced
Z-Section from Bryant et al. (1999a)………….……………...……………….... 70
4.3 Summary Table for Nominal Strengths of Third Point Braced
Z-Section from Almoney and Murray (1998).………………..……………….... 70
4.4 Summary Table for Nominal Strengths of Third Point Braced
Z-Section from Davis et al. (1995)…….…….……………...…………………. 71
vii
LIST OF TABLES (continued)
TABLE Page
4.5 Summary Table for Nominal Strengths of Third Point Braced
Z-Section from Bathgate and Murray (1995).…………….….…....…….…….... 71
4.6 Summary Table for Nominal Strengths of Third Point Braced
Z-Section from Borgsmiller et al. (1994)….…...….……..…..….…….……….... 72
4.7 Summary Table for Nominal Strengths of Third Point Braced
Z-Section from Earls et al. (1991)………….….….….……………………….... 72
4.8 Summary Table for Nominal Strengths of Third Point Braced
Z-Section from Brooks and Murray (1989)…..….…………...……………….... 73
4.9 Summary Table for Nominal Strengths of Third Point Braced
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
DISTORTIONAL BUCKLING
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
ed AE (Use smaller positive value) (2.12)
where
( )
++
= Φ
EK
JbIx dfff ηβλ
βηα
1lg
2lg
11 039.0
−= 22
lg1
lg13 fff bIxyIyβηαηα
If KΦ<0 :
( ) ( )
−+±+
= 3
22121
lg
42
ααααασf
ed AE (Use smaller positive value) (2.13)
38
where
( )
++
= Φ
EK
JbIx dfff ηβλ
βηα
1lg
2lg
11 039.0
( )
+
=ΦdwD
EtKλ06.046.5
2 3
−= 22
lg1
lg13 fff bIxyIyβηαηα
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.)
Experimental Strength
(k-ft)
Summary Table of Experimental Strengths and Properties of Third Point Braced Z-Sections
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).
Summary Table of Experimental Strengths and Properties of Third Point Braced Z-Sections
Table 3.1 Continued
Test No. Span (ft)
Depth (in.)
Measured Yield Stress (ksi)
Measured Thickness
(in.)
Experimental Strength
(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.)
Experimental Strength
(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)
(Almoney and Murray 1998)
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
Section C3.1.1 Section C3.1.2
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)
(Bathgate and Murray 1995)
(Borgsmiller et al. 1994)
Table 3.3 Continued
Summary Table of 1996 AISI Specification Strengths of Third Point Braced Z-Sections
Test No.AISI Specification Nominal Strength
(k-ft)
52
Section C3.1.1 Section C3.1.2
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)
(Brooks and Murray 1989)
(Spangler and Murray 1989)
Table 3.3 Continued
Summary Table of 1996 AISI Specification Strengths of Third Point Braced Z-Sections
Test No.
AISI Specification Nominal Strength (k-ft)
53
Section C3.1.1 Section C3.1.2
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)
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
(Brooks and Murray 1989)
(Spangler and Murray 1989)
(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
Test No.Predicted Distortional
Buckling Strength (k-ft)
Test No.Predicted Distortional
Buckling Strength (k-ft)
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
Test No.Predicted Distortional
Buckling Strength (k-ft)
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)
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 4.3 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)
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 4.4 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)
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 4.5 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)
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 4.6 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
Local Buckling (C3.1.1)
Lateral Buckling (C3.1.2)
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
Buckling Strength (k-ft)
Predicted Buckling Strength (k-ft)
Local Buckling (C3.1.1)
Lateral Buckling (C3.1.2)
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
Table 4.3Summary Table for Nominal Strengths of Third Point Braced Z-Sections from
Almoney and Murray (1998)
Test No.Experimental
Buckling Strength (k-ft)
Predicted Buckling Strength (k-ft)
70
Local Buckling (C3.1.1)
Lateral Buckling (C3.1.2)
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)
Test No.Experimental
Buckling Strength (k-ft)
Predicted Buckling Strength (k-ft)
Local Buckling (C3.1.1)
Lateral Buckling (C3.1.2)
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
Table 4.5Summary Table for Nominal Strengths of Third Point Braced Z-Sections from
Bathgate and Murray (1995)
Test No.Experimental
Buckling Strength (k-ft)
Predicted Buckling Strength (k-ft)
71
Local Buckling (C3.1.1)
Lateral Buckling (C3.1.2)
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)
Test No.Experimental
Buckling Strength (k-ft)
Predicted Buckling Strength (k-ft)
Local Buckling (C3.1.1)
Lateral Buckling (C3.1.2)
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
Table 4.7Summary Table for Nominal Strengths of Third Point Braced Z-Sections from
Earls et al. (1991)
Test No.Experimental
Buckling Strength (k-ft)
Predicted Buckling Strength (k-ft)
72
Local Buckling (C3.1.1)
Lateral Buckling (C3.1.2)
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)
Test No.Experimental
Buckling Strength (k-ft)
Predicted Buckling Strength (k-ft)
Local Buckling (C3.1.1)
Lateral Buckling (C3.1.2)
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)
Test No.Experimental
Buckling Strength (k-ft)
Predicted Buckling Strength (k-ft)
73
Local Buckling (C3.1.1)
Lateral Buckling (C3.1.2)
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)
Test No.Experimental
Buckling Strength (k-ft)
Predicted Buckling Strength (k-ft)
Local Buckling (C3.1.1)
Lateral Buckling (C3.1.2)
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)
Test No.Experimental
Buckling Strength (k-ft)
Predicted Buckling Strength (k-ft)
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 3-0.105b 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 3-0.060b 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
(Almoney and Murray 1998)
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
2G-Eave HS P 26 ga. S
6G-Eave HS P 26 ga. S
3G-Eave HS P 26 ga. S
4G-Eave HS P 26 ga. S
5G-Eave HS P 26 ga. S
Test #1 (Eave) HS R 24 ga. S
Test #2 (Eave) HS R 24 ga. S
Test #5 (Eave) HS R 24 ga. S
Test #3 (Eave) HS R 24 ga. S
Test #4 (Eave) HS R 24 ga. S
Test #6 (Eave) HS R 24 ga. S
Test #7 (Eave) HS R 24 ga. S
Test #8 (Eave) HS R 24 ga. S
Test #9 (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
G8ZTP/S-1 (5A) HS P 26 ga. S
G10ZTP/S-1 (2A) 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)
(Borgsmiller et al. 1994)
Table 4.14 Continued
Summary Table of Third Point Braced Test Components
(Davis et al. 1995)
(Bathgate and Murray 1995)
82
Test No. Clip Type
Panel Type
Panel Thickness
Purlin Orientation
Test #1 HS R 24 ga. S
Test #2 HS R 24 ga. S
Test #3 HS R 24 ga. S
Test #4 HS R 24 ga. S
Test #5 HS R 24 ga. S
Test #6 HS R 24 ga. S
Test #7 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)
(Spangler and Murray 1989)
Table 4.14 Continued
Summary Table of Third Point Braced Test Components
(Earls et al. 1991)
(Brooks and Murray 1989)
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.
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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
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APPENDIX B
CFS Data Runs for Section 1G MathCad data Run for Section 1G
119
120
121
122
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=
α 5 η α 4 I yf. η
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=
α 7 η α 6 I yf. η
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