Read STRENGTH OF SINGLE SPAN Z-SECTION PURLINS SUPPORTING STANDING SEAM ROOF SYSTEMS CONSIDERING text version

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 coldformed 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.

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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 1.2 1.3 1.4 1.5 II. Introduction.................................................................. 1 Literature Review..............................................................8 Need for Research............................................................ 14 Scope of Research............................................................ 16 Overview of Study............................................................ 18

DISTORTIONAL BUCKLING................................................ 20 2.1 2.2 2.3 2.4 2.5 Background................................................................... 20 AISI Specification Oversights...............................................21 AISI Local and Lateral Provisions......................................... 23 Determination of Local and Lateral Buckling Strengths................... 27 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 3.2 3.3 3.4 Background and Test Details................................................ 41 Experimental Results......................................................... 44 AISI Specification Analysis.................................................. 49 Distortional Buckling Analysis............................................. 54

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TABLE OF CONTENTS (continued)

CHAPTER IV. COMPARISON OF RESULTS............................................. 4.1 4.2 4.3 4.4 4.5 V. Page 59

General......................................................................... 59 Third Point Braced and Unbraced Analyses................................... 60 Prior Research................................................................ 75 Possible Causes of Scatter in Data.......................................... 79 Resistance Factor for Design................................................ 85

EXAMPLE CALCULATIONS................................................... 88 5.1 5.2 5.3 5.4 Problem Statement for an 8 in. Deep Z-Section.......................... 88 Calculation of Section Properties...........................................89 Local and Lateral Buckling Strength Predictions.......................... 98 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

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LIST OF FIGURES

FIGURE 1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 2.4 3.1 3.2 3.3 4.1 4.2 4.3 4.4 4.5 4.6 4.7 5.1

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Standing Seam Roof System Profile................................................... 2 Point Symmetry of a Typical Z-Section................................................ 3 Buckling Modes of a Z-Purlin.......................................................... 4 Buckling Modes Subject to a C-Purlin for Major Axis Bending..................... 6 Buckling Modes of a Z-Purlin for Bending about a Horizontal Axis............... 7 Geometric Properties Measurement Plan.............................................. 27 Measurements for Section Properties.................................................. 31 Stiffness Restraints....................................................................... 35 Beam Web Behavior in Flexure......................................................... 36 Typical Base Test Setup................................................................. 42 Clip Types................................................................................. 43 Steel Panel Types........................................................................ 44 Experimental Strengths Vs. Local Buckling for 8 in. Deep Z-Sections............ 61 Experimental Strengths Vs. Local Buckling for 10 in. Deep Z-Sections......... 61 Experimental Strengths Vs. Lateral Buckling for 8 in. Deep Z-Sections......... 63 Experimental Strengths Vs. Lateral Buckling for 10 in. Deep Z-Sections......... 63 Experimental Strengths Vs. Distortional Buckling for 8 in. Deep Z-Sections..... 65 Experimental Strengths Vs. Distortional Buckling for 10 in. Deep Z-Sections... 65 Overall Experimental Strengths Vs. Predicted Buckling Strengths..................66 Properties of Section 1G................................................................89

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LIST OF TABLES

TABLE 3.1 3.2 3.3 3.4 3.5 3.6 4.1 4.2 4.3 4.4 Summary Table of Experimental Strengths and Properties of Third

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Point Braced Z-Sections................................................................ 45 Summary Table of Experimental Strengths and Properties of Laterally Unbraced Z-Sections.................................................................... 48 Summary Table of 1996 AISI Specification Strengths of Third Point Braced Z-Sections................................................................ 51 Summary Table of 1996 AISI Specification Strengths of Laterally Unbraced Z-Sections.................................................................... 54 Summary Table of Nominal Distortional Buckling Strengths of Third Point Braced Z-Sections..........................................................56 Summary Table of Nominal Distortional Buckling Strengths of Laterally Unbraced Z-Sections.......................................................... 58 Summary Table for Nominal Strengths of Third Point Braced Z-Section from Murray and Trout (2000)............................................... 69 Summary Table for Nominal Strengths of Third Point Braced Z-Section from Bryant et al. (1999a)..................................................... 70 Summary Table for Nominal Strengths of Third Point Braced Z-Section from Almoney and Murray (1998)........................................... 70 Summary Table for Nominal Strengths of Third Point Braced Z-Section from Davis et al. (1995)...................................................... 71

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LIST OF TABLES (continued)

TABLE 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 5.1 Summary Table for Nominal Strengths of Third Point Braced

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Z-Section from Bathgate and Murray (1995)............................................. 71 Summary Table for Nominal Strengths of Third Point Braced Z-Section from Borgsmiller et al. (1994)...................................................72 Summary Table for Nominal Strengths of Third Point Braced Z-Section from Earls et al. (1991)........................................................ 72 Summary Table for Nominal Strengths of Third Point Braced Z-Section from Brooks and Murray (1989).............................................. 73 Summary Table for Nominal Strengths of Third Point Braced Z-Section from Spangler and Murray (1989)............................................73 Summary Table for Nominal Strengths of Laterally Unbraced Z-Section from Bryant et al. (1999b)...................................................... 74 Summary Table for Nominal Strengths of Laterally Unbraced Z-Section from Bryant et al. (1999c)...................................................... 74 Summary Table of All Z-Purlin Strength Data....................................... 75 Summary Table for Comparison of Results for Laterally Braced Z-Sections.......78 Summary Table of Third Point Braced Test Components........................... 81 Summary of Effect of Clip Type for 8.0 in. Deep, 0.102 in. Thick Third Point Braced Tests...............................................................85 Summary of Predicted and Experimental Strengths for Z-Section 1G............102

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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 selftapping 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

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advantage the clips provide over water leakage comes at a cost. Unlike the throughfastened 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 Top Flange 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.

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X,Y - Shear center axes X

x Y y 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.

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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.

Compression

Compression

Compression

Tension

Tension

Tension

Local Buckling

Lip-Flange Distortional Buckling

Flexural-Torsional Buckling (Lateral Buckling)

Figure 1.3 Buckling Modes of a Z-Purlin

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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 webflange 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

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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.

Stress in Compression Flange at Buckling (MPa)

800 700 600 500

1

Top flange in compression, bottom flange in tension Lateral distortional buckle (tension flange restraint)

400 300 200 100

Local buckle

2

4

Flexuraltorsional buckle (Lateral buckle)

Flange distortional buckle

3

100

1000 Buckle Half-Wavelength (mm)

10000

100000

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

Buckling Stress (MPa)

Top flange in compression, bottom flange in tension

600 500 400 300 200 100

Local Buckle

Vertical Lip

Sloping Lip

Flange Distortional Buckle

100

1000 Buckle Half-Wavelength (mm)

10000

100000

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

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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

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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 (K) 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). specific only to certain geometries of channel sections. A hand method for determining distortional buckling stress for thin-walled, coldformed 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 flexuraltorsional 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. However, this design chart was

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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 Csection 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

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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

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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

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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

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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

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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

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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

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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. 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 These two strength predictions will be

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. consideration. Therefore, distortional buckling is a design

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 edgestiffened 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

M n = S e Fy

where Mn = Nominal flexural strength Fy = Design yield stress

(2.1)

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 (Xaxis) 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

M M n = Sc c S f

where

(2.2)

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

Me =

2 EC b dI yc

2L2

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

0.18C b 2 EdI yc Lu = Fy S f

0.5

(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

Length 5 Length 4

R4 R5 Angle 3 Length 3

Angle 2

Thickness Angle 1 R2 R3 Length 1 Length 2

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 2.5.1

Determination of Distortional Buckling Strength 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 (K) 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 coldformed 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 Xand Y-axes of only the compression flange and lip. The buckle wavelength (d),

30

torsional restraint stiffness (K), 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 bf x

y

L

b l

Dw

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

Ix f lg

(2.5)

Iy f lg

bl3 sin ( )2 = (t ) × 12

2 bf t2 2 bl sin ( ) + bl - y + + bf y 2 12

2 2 2 3 b 3 bf bl cos( ) bl cos( ) f = (t ) × + b f - x + bl b f + - x + 2 2 12 12

(2.6)

Ixy f lg = (t ) × b f

bf y x - 2

cos( ) sin ( ) + bl b f - x + bl × - y bl × 2 2

(2.7)

where t =Thickness

=Angle between compression flange and lip in radians

t bl =Centroidal length of the lip; bl = L - 2 t bf =Centroidal length of the flange; b f = B - 2 Aflg =Area of flange; A f lg = t (bl + b f

)

2 t b sin ( ) y = Centroid location along Y-axis; y = l 2 A f lg 2 2 t b f + 2bl b f + bl cos( ) x = Centroid location along X-axis; x = A f lg 2

32

Several factors are needed for the calculation of the torsional restraint stiffness (K) for a purlin. These include the buckle half-wavelength (d), St. Venant torsional constant of the compression lip-flange component (Jflg), and the elastic distortional buckling stress (ed). Of particular importance is the buckle half-wavelength (d) 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 (d) is used. On the other hand, when the compression lipflange component is additionally restrained, the smaller value between the calculated half-wavelength and the measured distance between restraints (c) 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 (d) are given by the Hancock Method as

33

b f t 3 bl t 3 J f lg = + 3 3

Ix f lg b 2 Dw f d = 4.80 3 2t

0.25

(2.8)

(2.9)

where Dw =Depth of the web

The determination of the elastic distortional buckling stress ('ed) is an iterative process with an initial assumption used to determine the torsional restraint stiffness (K). 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 1, 2, and 3 characteristic values are related to K, d, 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

' + 2 ± ed = 2 A 1 f lg

E

( 1 + 2 )2 - 4 3

(Use smaller positive value)

(2.10)

where E =Modulus of elasticity

1 = (Ix f lg b 2 + 0.039 J f lg 2 ) f d 1

34

2 = Iy f lg + yb f Ixy f lg 1

3 = 1 Iy f lg - Ixy 2 lg b 2 f f 1

2

1 = x +

= d

2

2

Ix f lg + Iy f lg A f lg

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

K

Kx

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, (K) (Davies and Jiang 1996).

M

M

M = 2EI d

M

M = 4EI 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 (K), 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 K 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

' 1.11 ed 2 Et 3 K = 1- 2 5.46(Dw + 0.06 d ) Et

4 Dw 2 d 12.564 + 2.192 D 4 + 13.392 D 2 d w d w

(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 (K0), then this value is used in the second iteration to update the 1 term and consequently the 3 term (see below). The actual elastic distortional buckling stress (ed) is thus calculated using the updated 1 and 3 equations. However, if the lip-flange component torsionally restrains the web element (K<0), then K is recalculated without an initially assumed elastic distortional buckling stress. In this situation, the interaction of local and distortional buckling is ignored. This new K term is thus used to update the 1 and 3 equations to determine the actual elastic distortional buckling stress, as shown below If K0 : ( + 2 ) ± ed = 2 A 1 f lg where

K 1 = (Ix f lg b 2 + 0.039 J f lg d ) + f E 1 1

E

( 1 + 2 )2 - 4 3

(Use smaller positive value)

(2.12)

3 = 1 Iy f lg - Ixy 2 lg b 2 f f 1 If K<0 : ( + 2 ) ± ed = 2 A 1 f lg E

( 1 + 2 )2 - 4 3

(Use smaller positive value)

(2.13)

38

where

K 1 = (Ix f lg b 2 + 0.039 J f lg d ) + f E 1 1 2 Et 3 K = 5.46(Dw + 0.06 d )

3 = 1 Iy f lg - Ixy 2 lg b 2 f f 1

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 > 2.2 f y then: fc = f y

(2.14)

If ed 2.2 f y then:

ed fc = f y fy 1 - 0.22 ed fy

(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 K 0 then M pred = S f f c If K < 0 then M pred = S c f c 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.

RAKE ANGLE

RIDGE EAVE ANGLE EAVE RIDGE ANGLE

STANDING SEAM FOOF PANELS

6-MIL POLYETHELENE SHEET

8'-0" 7'-0"

SPAN LENGTH

PURLIN LENGTH

CHAM BER W ALL STANDS

3'-6"

STEEL SUPPORT BEAM

ANTI-ROLL CLIPS

CTC PURLIN SPACING

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.

Fixed Clip

Sliding 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

Table 3.1 Summary Table of Experimental Strengths and Properties of Third Point Braced Z-Sections

Test No.

Span (ft)

Depth (in.)

Measured Yield Stress (ksi)

Measured Thickness (in.)

Experimental Strength (k-ft)

(Murray and Trout 2000)

Test 18 (Z08-37) Test 19 (Z08-39) Test 20 (Z08-41) Test 25 (Z08-44) Test 26 (Z08-45) Test 27 (Z08-47) Test 21 (Z08-14) Test 22 (Z08-16) Test 23 (Z08-18) Test 24 (Z08-20) Test 28 (Z08-22) Test 29 (Z08-24) 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 8.00 60.1 60.6 60.5 60.3 59.6 60.0 63.6 65.2 62.9 64.9 63.4 64.3 0.102 0.101 0.102 0.102 0.102 0.102 0.057 0.057 0.056 0.056 0.057 0.057 12.47 13.83 11.73 11.98 12.19 11.90 6.46 6.37 5.39 5.03 6.41 5.61

(Bryant et al. 1999a)

Test B (Z10-Eave) Test C (Z10-Eave) Test D (Z10-Eave) Test G (Z10-Eave) Test E (Z10-Eave) Test F (Z10-Eave) 30.0 30.0 30.0 30.0 30.0 30.0 10.063 10.063 10.063 10.063 10.063 10.063 54.2 52.6 52.4 52.6 52.5 52.3 0.103 0.103 0.103 0.103 0.076 0.076 13.14 15.98 18.19 16.05 9.02 8.50

45

Table 3.1 Continued Summary Table of Experimental Strengths and Properties of Third Point Braced Z-Sections

Test No. Span (ft) Depth (in.) Measured Yield Stress (ksi) Measured Thickness (in.) Experimental Strength (k-ft)

(Almoney and Murray 1998)

Setup 2-0.105b Setup 3-0.105a Setup-0.105b Setup 4-0.105 Setup 4-0.076 Setup 3-0.060a Setup 3-0.060b Setup 3-0.060c 27.0 27.0 27.0 27.0 27.0 27.0 27.0 27.0 7.938 7.938 7.938 7.938 7.938 7.875 7.875 7.875 67.6 67.7 67.3 62.0 64.5 64.1 66.2 65.3 0.104 0.104 0.104 0.103 0.074 0.059 0.059 0.059 16.73 14.30 14.01 12.70 10.66 7.10 7.48 6.74

(Davis et al. 1995)

1G-Eave 2G-Eave 6G-Eave 3G-Eave 4G-Eave 5G-Eave 25.0 25.0 25.0 25.0 25.0 25.0 8.000 8.000 8.000 10.000 10.125 10.000 57.1 57.1 57.1 58.6 58.6 58.6 0.060 0.060 0.060 0.087 0.087 0.087 6.69 7.91 7.60 13.37 15.87 15.01

(Bathgate and Murray 1995)

Test #1 (Eave) Test #2 (Eave) Test #5 (Eave) Test #3 (Eave) Test #4 (Eave) Test #6 (Eave) Test #7 (Eave) Test #8 (Eave) Test #9 (Eave) 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 8.438 8.563 8.500 8.500 8.500 10.000 10.000 10.000 10.000 60.5 60.7 65.7 64.5 64.0 65.5 65.9 61.3 60.2 0.091 0.091 0.091 0.060 0.061 0.103 0.102 0.078 0.078 12.32 12.67 12.07 7.51 7.16 22.67 21.63 13.11 13.19

46

Table 3.1 Continued Summary Table of Experimental Strengths and Properties of Third Point Braced Z-Sections

Test No. Span (ft) Depth (in.) Measured Yield Stress (ksi) 56.6 56.6 56.6 58.5 47.5 47.5 47.5 61.4 50.1 62.3 64.7 60.1 62.8 67.2 53.6 63.7 63.5 62.0 65.5 63.7 57.6 Measured Thickness (in.) 0.071 0.071 0.071 0.075 0.101 0.101 0.101 0.058 0.059 0.098 0.098 0.086 0.079 0.066 0.078 0.074 0.075 0.095 0.098 0.094 0.098 Experimental Strength (k-ft) 6.85 6.84 6.77 7.93 16.24 16.23 16.19 4.97 4.67 16.24 16.26 12.58 11.10 8.45 10.81 10.54 10.96 16.39 20.98 19.03 18.50

(Borgsmiller et al. 1994)

G8ZTP/S-1 (1A) G8ZTP/S-1 (1B) G8ZTP/S-1 (1C) G8ZTP/S-1 (5A) G10ZTP/S-1 (2A) G10ZTP/S-1 (2B) G10ZTP/S-1 (2C) 20.0 20.0 20.0 20.0 20.0 20.0 20.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 25.0 8.0 8.0 8.0 8.0 10.0 10.0 10.0 8.000 8.000 8.063 8.063 8.063 8.063 8.125 7.890 9.480 9.575 9.903 9.848 9.875 9.911

(Earls et al. 1991)

Test #1 Test #2 Test #3 Test #4 Test #5 Test #6 Test #7 Z-T-P/F-1 Z-T-P/S-1 Z-T-R/S-1 N-ZIS-12-SF-1 N-ZISO-12-SF-1 N-ZISO-12-SS-1 N-ZISO-12-TF-1 Notes:

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 added for 10" deep sections which was based on an analysis of Trout and Murray (2000).

(Brooks and Murray 1989)

(Spangler and Murray 1989)

47

Table 3.2 Summary Table of Experimental Strengths and Properties of Laterally Unbraced Z-Sections

Test No. Span (ft) Depth (in.) Measured Yield Stress (ksi) Measured Thickness (in.) Experimental Strength (k-ft)

(Bryant et al. 1999b)

Test #1 Test #2 Test #3 Test #4 Test #5 Test #6 22.75 22.75 22.75 22.75 22.75 22.75 8.000 8.000 8.000 8.063 8.063 8.063 71.9 70.1 69.9 56.3 56.0 56.5 0.062 0.062 0.062 0.106 0.106 0.106 4.48 4.61 4.45 8.29 7.61 7.99

(Bryant et al. 1999c)

Test #1a Test #2a Test #3a Test #4a Test #5a Test #6a 30.0 30.0 30.0 30.0 30.0 30.0 10.000 10.063 10.000 10.063 10.063 10.063 54.7 53.0 55.7 55.3 64.1 65.8 0.076 0.076 0.077 0.103 0.100 0.102 6.15 6.20 6.72 10.10 9.61 9.09

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: (b f )1.5 PL = 0.041 (Wd + Wl ) 0.9 0.6 (Dw ) (t ) 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: D Wts = (Wl + Wd ) + 2(PL ) w S where Wts = Failure line load (plf) Dw = Web depth (ft) S = Purlin spacing Step Three: (3.2) (3.1)

Wts (L )2 M ts = 8 where Wts = Failure load (klf) L = Span length (ft)

(3.3)

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

Table 3.3 Summary Table of 1996 AISI Specification Strengths of Third Point Braced Z-Sections

Test No. (Murray and Trout 2000)

Test 18 (Z08-37) Test 19 (Z08-39) Test 20 (Z08-41) Test 25 (Z08-44) Test 26 (Z08-45) Test 27 (Z08-47) Test 21 (Z08-14) Test 22 (Z08-16) Test 23 (Z08-18) Test 24 (Z08-20) Test 28 (Z08-22) Test 29 (Z08-24) 16.18 16.26 16.21 16.52 16.31 16.45 8.60 8.82 8.22 8.40 8.63 8.47 22.00 22.06 21.84 22.06 13.91 14.00 20.35 20.40 20.38 18.51 12.92 9.47 9.58 9.65 11.31 10.21 10.27 10.89 10.58 10.91 5.12 5.21 4.73 4.82 5.18 4.62 11.01 12.00 11.89 11.80 7.95 8.19 9.84 9.68 9.84 10.98 8.29 5.11 5.26 5.73 AISI Specification Nominal Strength (k-ft) Section C3.1.1 Section C3.1.2

(Bryant et al. 1999)

Test B (Z10-Eave) Test C (Z10-Eave) Test D (Z10-Eave) Test G (Z10-Eave) Test E (Z10-Eave) Test F (Z10-Eave)

(Almoney and Murray 1998)

Setup 2-0.105b Setup 3-0.105a Setup 3-0.105b Setup 4-0.105 Setup 4-0.076 Setup 3-0.060a Setup 3-0.060b Setup 3-0.060c

51

Table 3.3 Continued Summary Table of 1996 AISI Specification Strengths of Third Point Braced Z-Sections

AISI Specification Nominal Strength (k-ft) Section C3.1.1 Section C3.1.2

5.76 5.94 5.25 12.09 12.47 9.43 8.39 8.74 8.21 4.53 4.66 17.90 21.04 9.36 8.26 7.85 8.22 8.41 9.30 16.13 15.98 16.03

Test No.

(Davis et al. 1995)

1G-Eave 2G-Eave 6G-Eave 3G-Eave 4G-Eave 5G-Eave 8.77 8.68 8.51 19.01 20.37 19.46 16.12 16.88 17.08 8.79 9.07 26.42 27.41 16.38 15.77 10.63 10.77 10.83 11.95 20.48 20.27 20.23

(Bathgate and Murray 1995)

Test #1 (Eave) Test #2 (Eave) Test #5 (Eave) Test #3 (Eave) Test #4 (Eave) Test #6 (Eave) Test #7 (Eave) Test #8 (Eave) Test #9 (Eave)

(Borgsmiller et al. 1994)

G8ZTP/S-1 (1A) G8ZTP/S-1 (1B) G8ZTP/S-1 (1C) G8ZTP/S-1 (5A) G10ZTP/S-1 (2A) G10ZTP/S-1 (2B) G10ZTP/S-1 (2C)

52

Table 3.3 Continued Summary Table of 1996 AISI Specification Strengths of Third Point Braced Z-Sections

AISI Specification Nominal Strength (k-ft) Section C3.1.1 Section C3.1.2

5.63 5.51 13.12 13.39 10.68 9.77 7.59 7.29 9.71 9.69 15.61 16.57 16.48 16.68

Test No.

(Earls et al. 1991)

Test #1 Test #2 Test #3 Test #4 Test #5 Test #6 Test #7 8.30 7.61 17.83 18.34 14.94 13.91 11.23 11.01 14.70 16.64 24.43 26.56 24.76 25.43

(Brooks and Murray 1989)

Z-T-P/F-1 Z-T-P/S-1 Z-T-R/S-1

(Spangler and Murray 1989)

N-ZIS-12-SF-1 N-ZISO-12-SF-1 N-ZISO-12-SS-1 N-ZISO-12-TF-1

53

Table 3.4 Summary Table of 1996 AISI Specification Strengths of Laterally Unbraced Z-Sections

AISI Specification Nominal Strength (k-ft) Section C3.1.1 Section C3.1.2

0.84 0.86 0.91 1.70 1.86 1.80 1.04 0.94 1.00 1.48 1.32 1.39

Test No.

(Bryant et al. 1999b)

Test #1 Test #2 Test #3 Test #4 Test #5 Test #6 10.21 10.50 10.63 18.51 18.55 18.53 15.20 14.14 15.21 22.86 24.00 25.32

(Bryant et al. 1999c)

Test #1a Test #2a Test #3a Test #4a Test #5a Test #6a

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 (K) 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 (-K). 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

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)

(Murray and Trout 2000)

Test 18 (Z08-37) Test 19 (Z08-39) Test 20 (Z08-41) Test 25 (Z08-44) Test 26 (Z08-45) Test 27 (Z08-47) Test 21 (Z08-14) Test 22 (Z08-16) Test 23 (Z08-18) Test 24 (Z08-20) Test 28 (Z08-22) Test 29 (Z08-24) 14.18 14.10 14.20 14.31 14.24 14.30 6.30 6.49 6.12 6.24 6.33 6.32

(Bathgate and Murray 1995)

Test #1 (Eave) Test #2 (Eave) Test #5 (Eave) Test #3 (Eave) Test #4 (Eave) Test #6 (Eave) Test #7 (Eave) Test #8 (Eave) Test #9 (Eave) 13.01 13.28 13.64 7.11 7.31 21.32 21.69 13.32 12.68

(Borgsmiller et al 1994)

G8ZTP/S-1 (1A) G8ZTP/S-1 (1B) G8ZTP/S-1 (1C) 8.18 8.20 8.28 9.24 15.55 15.45 15.43

(Bryant et al 1999)

Test B (Z10-Eave) Test C (Z10-Eave) Test D (Z10-Eave) Test G (Z10-Eave) Test E (Z10-Eave) Test F (Z10-Eave) 17.88 17.81 17.79 17.84 11.01 11.11

G8ZTP/S-1 (5A) G10ZTP/S-1 (2A) G10ZTP/S-1 (2B) G10ZTP/S-1 (2C)

56

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)

(Almoney and Murray 1998)

Setup 2-0.105b Setup 3-0.105a Setup 3-0.105b Setup 4-0.105 Setup 4-0.076 Setup 3-0.060a Setup 3-0.060b Setup 3-0.060c 16.49 16.46 16.55 15.68 10.30 7.23 7.42 7.32

(Earls et al 1991)

Test #1 Test #2 Test #3 Test #4 Test #5 Test #6 Test #7 5.74 5.20 14.53 14.89 11.48 10.30 7.80

(Brooks and Murray 1989)

Z-T-P/F-1 9.09 11.29 12.41

(Davis et al 1995)

1G-Eave 2G-Eave 6G-Eave 3G-Eave 4G-Eave 5G-Eave 6.55 6.75 6.71 14.91 15.59 14.95

Z-T-P/S-1 Z-T-R/S-1

(Spangler and Murray 1989)

N-ZIS-12-SF-1 N-ZISO-12-SF-1 N-ZISO-12-SS-1 N-ZISO-12-TF-1 19.48 21.17 19.85 19.73

57

Table 3.6 Summary Table of Nominal Distortional Buckling Strengths of Laterally Unbraced Z-Sections

Predicted Distortional Buckling Strength (k-ft)

Test No.

(Bryant et al 1999b)

Test #1 Test #2 Test #3 Test #4 Test #5 Test #6 7.81 7.93 7.94 15.02 15.02 14.97

(Bryant et al 1999c)

Test #1a Test #2a Test #3a Test #4a Test #5a Test #6a 11.54 11.21 11.86 18.40 18.51 19.75

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

25.0

10% 10%

Experimental Strength, Mexp (k-ft)

20.0 Conservative 15.0

Z-Section Thickness

0.055"-0.0649" 0.065"-0.0749" 0.075"-0.0849" 0.085"-0.0949" 0.095"-0.1049"

10.0 Unconservative 5.0

0.055"-0.0649" (Unbraced) 0.105"-0.1149" (Unbraced)

0.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Local Buckling Strength, Mdb (k-ft)

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

25.0

10% 10%

Experimental Strength, Mexp (k-ft)

20.0 Conservative 15.0

Z-Section Thickness

0.065"-0.0749" 0.075"-0.0849" 0.085"-0.0949" 0.095"-0.1049" 0.075"-0.0849" (Unbraced)

10.0 Unconservative 5.0

0.095"-0.1049" (Unbraced)

0.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Local Buckling Strength, Mdb (k-ft)

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

25.0

10% 10%

Experimental Strength, Mexp (k-ft)

20.0

Conservative

Z-Section Thickness

0.055"-0.0649" 0.065"-0.0749" 0.075"-0.0849" 0.085"-0.0949"

15.0

10.0 Unconservative 5.0

0.095"-0.1049" 0.055"-0.0649" (Unbraced) 0.105"-0.1149" (Unbraced)

0.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Lateral Buckling Strength, Mdb (k-ft)

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

25.0

10% 10%

Experimental Strength, Mexp (k-ft)

20.0 Conservative 15.0

Z-Section Thickness

0.065"-0.0749" 0.075"-0.0849" 0.085"-0.0949" 0.095"-0.1049" 0.075"-0.0849" (Unbraced)

10.0 Unconservative 5.0

0.095"-0.1049" (Unbraced)

0.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Lateral Buckling Strength, Mdb (k-ft)

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 Zsections (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

25.0

10% 10%

Experimental Strength, Mexp (k-ft)

20.0 Conservative 15.0

Z-Section Thickness

0.055"-0.0649" 0.065"-0.0749" 0.075"-0.0849" 0.085"-0.0949" 0.095"-0.1049"

10.0 Unconservative 5.0

0.055"-0.0649" (Unbraced) 0.105"-0.1149" (Unbraced)

0.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Distortional Buckling Strength, Mdb (k-ft)

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

25.0

10% 10%

Experimental Strength, Mexp (k-ft)

20.0 Conservative 15.0

Z-Section Thickness

0.065"-0.0749" 0.075"-0.0849" 0.085"-0.0949" 0.095"-0.1049" 0.075"-0.0849" (Unbraced)

10.0 Unconservative 5.0

0.095"-0.1049" (Unbraced)

0.0 0.0 5.0 10.0 15.0 20.0 25.0 30.0 Distortional Buckling Strength, Mdb (k-ft)

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

10% 10%

25.0

20.0

Experimental Strength (k-ft)

Conservative

15.0

10.0

Unconservative

5.0

0.0 0.0 5.0 Distortional Buckling Lateral Buckling Trend Line 10.0 15.0 Predicted Strength (k-ft) Lateral Buckling Distortional Buckling Trend Line 20.0 25.0 Local Buckling Local Buckling Trend Line 30.0

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

Table 4.1

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Murray and Trout (2000)

Predicted Buckling Strength (k-ft) Local Buckling (C3.1.1)

16.18 16.26 16.21 16.52 16.31 16.45 8.60 8.82 8.22 8.40 8.63 8.47

Test No.

Experimental Buckling Strength (k-ft)

12.47 13.83 11.73 11.98 12.19 11.90 6.46 6.37 5.39 5.03 6.41 5.61

Lateral Buckling (C3.1.2)

11.31 10.21 10.27 10.89 10.58 10.91 5.12 5.21 4.73 4.82 5.18 4.62

Dist. Buckling

14.18 14.10 14.20 14.31 14.24 14.30 6.30 6.49 6.12 6.24 6.33 6.32 Mean: St. Dev.: Co. of Var.:

Mn

C3.1.1--exp

Mn

C3.1.2--exp

Mn

DB----exp

Mn

Mn

Mn

Test 18 (Z08-37) Test 19 (Z08-39) Test 20 (Z08-41) Test 25 (Z08-44) Test 26 (Z08-45) Test 27 (Z08-47) Test 21 (Z08-14) Test 22 (Z08-16) Test 23 (Z08-18) Test 24 (Z08-20) Test 28 (Z08-22) Test 29 (Z08-24)

1.298 1.176 1.382 1.379 1.338 1.382 1.331 1.384 1.526 1.670 1.347 1.510 1.394 0.120 0.086

0.907 0.738 0.876 0.909 0.868 0.917 0.793 0.818 0.878 0.959 0.808 0.824 0.858 0.060 0.070

1.137 1.020 1.211 1.194 1.168 1.202 0.975 1.019 1.135 1.241 0.988 1.127 1.118 0.090 0.080

69

Table 4.2

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Bryant et al. (1999a)

Predicted Buckling Strength (k-ft) Local Buckling (C3.1.1)

22.00 22.06 21.84 22.06 13.91 14.00

Test No.

Experimental Buckling Strength (k-ft)

Lateral Buckling (C3.1.2)

11.01 12.00 11.89 11.80 7.95 8.19

Dist. Buckling

17.88 17.81 17.79 17.84 11.01 11.11 Mean: St. Dev.: Co. of Var.:

Mn

C3.1.1--exp

Mn

C3.1.2--exp

Mn

DB----exp

Mn

Mn

Mn

Test B (Z10-Eave) Test C (Z10-Eave) Test D (Z10-Eave) Test G (Z10-Eave) Test E (Z10-Eave) Test F (Z10-Eave)

13.14 15.98 18.19 16.05 9.02 8.50

1.674 1.381 1.201 1.375 1.542 1.647 1.470 0.167 0.114

0.838 0.751 0.653 0.735 0.881 0.964 0.804 0.102 0.127

1.361 1.115 0.978 1.112 1.221 1.307 1.182 0.129 0.109

Table 4.3

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Almoney and Murray (1998)

Predicted Buckling Strength (k-ft) Local Buckling (C3.1.1)

20.35 20.40 20.38 18.51 12.92 9.47 9.58 9.65

Test No.

Experimental Buckling Strength (k-ft)

Lateral Buckling (C3.1.2)

9.84 9.68 9.84 10.98 8.29 5.11 5.26 5.73

Dist. Buckling

16.49 16.46 16.55 15.68 10.30 7.23 7.42 7.32 Mean: St. Dev.: Co. of Var.:

Mn

C3.1.1--exp

Mn

C3.1.2--exp

Mn

DB----exp

Mn

Mn

Mn

Setup 2-0.105b Setup 3-0.105a Setup 3-0.105b Setup 4-0.105 Setup 4-0.076 Setup 3-0.060a Setup 3-0.060b Setup 3-0.060c

16.73 14.30 14.01 12.70 10.66 7.10 7.48 6.74

1.217 1.426 1.454 1.458 1.212 1.333 1.281 1.432 1.352 0.098 0.073

0.588 0.677 0.702 0.864 0.778 0.720 0.704 0.849 0.735 0.086 0.117

0.986 1.151 1.181 1.235 0.966 1.018 0.992 1.086 1.077 0.095 0.088

70

Table 4.4

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Davis et al. (1995)

Predicted Buckling Strength (k-ft) Local Buckling (C3.1.1)

8.77 8.68 8.51 19.01 20.37 19.46

Test No.

Experimental Buckling Strength (k-ft)

Lateral Buckling (C3.1.2)

5.76 5.94 5.25 12.09 12.47 9.43

Dist. Buckling

6.55 6.75 6.71 14.91 15.59 14.95 Mean: St. Dev.: Co. of Var.:

Mn

C3.1.1--exp

Mn

C3.1.2--exp

Mn

DB----exp

Mn

Mn

Mn

1G-Eave 2G-Eave 6G-Eave 3G-Eave 4G-Eave 5G-Eave

6.69 7.91 7.60 13.37 15.87 15.01

1.312 1.097 1.120 1.422 1.284 1.296 1.255 0.113 0.090

0.861 0.751 0.691 0.904 0.786 0.628 0.770 0.094 0.122

0.979 0.853 0.883 1.115 0.982 0.996 0.968 0.085 0.088

Table 4.5

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Bathgate and Murray (1995)

Predicted Buckling Strength (k-ft) Local Buckling (C3.1.1)

16.12 16.88 17.08 8.79 9.07 26.42 27.41 16.38 15.77

Test No.

Experimental Buckling Strength (k-ft)

Lateral Buckling (C3.1.2)

8.39 8.74 8.21 4.53 4.66 17.90 21.04 9.36 8.26

Dist. Buckling

13.01 13.28 13.64 7.11 7.31 21.32 21.69 13.32 12.68 Mean: St. Dev.: Co. of Var.:

Mn

C3.1.1--exp

Mn

C3.1.2--exp

Mn

DB----exp

Mn

Mn

Mn

Test #1 (Eave) Test #2 (Eave) Test #5 (Eave) Test #3 (Eave) Test #4 (Eave) Test #6 (Eave) Test #7 (Eave) Test #8 (Eave) Test #9 (Eave)

12.32 12.67 12.07 7.51 7.16 22.67 21.63 13.11 13.19

1.308 1.332 1.415 1.170 1.266 1.165 1.267 1.250 1.195 1.263 0.077 0.061

0.681 0.690 0.680 0.604 0.650 0.790 0.973 0.714 0.626 0.712 0.100 0.140

1.056 1.048 1.130 0.947 1.021 0.940 1.003 1.016 0.961 1.014 0.057 0.056

71

Table 4.6

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Borgsmiller et al. (1994)

Predicted Buckling Strength (k-ft) Local Buckling (C3.1.1)

10.63 10.77 10.83 11.95 20.48 20.27 20.23

Test No.

Experimental Buckling Strength (k-ft)

Lateral Buckling (C3.1.2)

7.85 8.22 8.41 9.30 16.13 15.98 16.03

Dist. Buckling

8.18 8.20 8.28 9.24 15.55 15.45 15.43 Mean: St. Dev.: Co. of Var.:

Mn

C3.1.1--exp

Mn

C3.1.2--exp

Mn

DB----exp

Mn

Mn

Mn

G8ZTP/S-1 (1A) G8ZTP/S-1 (1B) G8ZTP/S-1 (1C) G8ZTP/S-1 (5A) G10ZTP/S-1 (2A) G10ZTP/S-1 (2B) G10ZTP/S-1 (2C)

6.85 6.84 6.77 7.93 16.24 16.23 16.19

1.552 1.574 1.599 1.507 1.261 1.249 1.250 1.427 0.153 0.107

1.146 1.202 1.242 1.173 0.993 0.984 0.990 1.104 0.103 0.093

1.194 1.199 1.223 1.165 0.958 0.952 0.953 1.092 0.120 0.110

Table 4.7

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Earls et al. (1991)

Predicted Buckling Strength (k-ft) Local Buckling (C3.1.1)

8.30 7.61 17.83 18.34 14.94 13.91 11.23

Test No.

Experimental Buckling Strength (k-ft)

Lateral Buckling (C3.1.2)

5.63 5.51 13.12 13.39 10.68 9.77 7.59

Dist. Buckling

5.74 5.20 14.53 14.89 11.48 10.30 7.80 Mean: St. Dev.: Co. of Var.:

Mn

C3.1.1--exp

Mn

C3.1.2--exp

Mn

DB----exp

Mn

Mn

Mn

Test #1 Test #2 Test #3 Test #4 Test #5 Test #6 Test #7

4.97 4.67 16.24 16.26 12.58 11.10 8.45

1.670 1.629 1.098 1.128 1.188 1.253 1.329 1.328 0.216 0.162

1.133 1.180 0.808 0.823 0.849 0.880 0.898 0.939 0.141 0.150

1.155 1.113 0.895 0.916 0.913 0.928 0.923 0.977 0.100 0.103

72

Table 4.8

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Brooks and Murray (1989)

Predicted Buckling Strength (k-ft) Local Buckling (C3.1.1)

11.01 14.70 16.64

Test No.

Experimental Buckling Strength (k-ft)

Lateral Buckling (C3.1.2)

7.29 9.71 9.69

Dist. Buckling

9.09 11.29 12.41 Mean: St. Dev.: Co. of Var.:

Mn

C3.1.1--exp

Mn

C3.1.2--exp

Mn

DB----exp

Mn

Mn

Mn

Z-T-P/F-1 Z-T-P/S-1 Z-T-R/S-1

10.81 10.54 10.96

1.019 1.395 1.519 1.311 0.213 0.162

0.675 0.921 0.884 0.827 0.109 0.131

0.841 1.071 1.132 1.015 0.125 0.124

Table 4.9

Summary Table for Nominal Strengths of Third Point Braced Z-Sections from Spangler and Murray (1989)

Predicted Buckling Strength (k-ft) Local Buckling (C3.1.1)

24.43 26.56 24.76 25.43

Test No.

Experimental Buckling Strength (k-ft)

Lateral Buckling (C3.1.2)

15.61 16.57 16.48 16.68

Dist. Buckling

19.48 21.17 19.85 19.73 Mean: St. Dev.: Co. of Var.:

Mn

C3.1.1--exp

Mn

C3.1.2--exp

Mn

DB----exp

Mn

Mn

Mn

N-ZIS-12-SF-1 N-ZISO-12-SF-1 N-ZISO-12-SS-1 N-ZISO-12-TF-1

16.39 20.98 19.03 18.50

1.490 1.266 1.301 1.375 1.358 0.086 0.063

0.952 0.790 0.866 0.902 0.877 0.059 0.067

1.189 1.009 1.043 1.066 1.077 0.068 0.063

73

Table 4.10

Summary Table for Nominal Strengths of Laterally Unbraced Z-Sections from Bryant et al. (1999b)

Predicted Buckling Strength (k-ft) Local Buckling (C3.1.1)

10.21 10.50 10.63 18.51 18.55 18.53

Test No.

Experimental Buckling Strength (k-ft)

Lateral Buckling (C3.1.2)

0.84 0.86 0.91 1.70 1.86 1.80

Dist. Buckling

7.81 7.93 7.94 15.02 15.02 14.97 Mean: St. Dev.: Co. of Var.:

Mn

C3.1.1--exp

Mn

C3.1.2--exp

Mn

DB----exp

Mn

Mn

Mn

Test #1 Test #2 Test #3 Test #4 Test #5 Test #6

4.48 4.61 4.45 8.29 7.61 7.99

2.279 2.278 2.389 2.233 2.438 2.319 2.322 0.070 0.030

0.188 0.187 0.204 0.205 0.244 0.225 0.209 0.020 0.098

1.743 1.720 1.784 1.812 1.974 1.874 1.818 0.085 0.047

Table 4.11

Summary Table for Nominal Strengths of Laterally Unbraced Z-Sections from Bryant et al. (1999c)

Predicted Buckling Strength (k-ft) Local Buckling (C3.1.1)

15.20 14.14 15.21 22.86 24.00 25.32

Test No.

Experimental Buckling Strength (k-ft)

Lateral Buckling (C3.1.2)

1.04 0.94 1.00 1.48 1.32 1.39

Dist. Buckling

11.54 11.21 11.86 18.40 18.51 19.75 Mean: St. Dev.: Co. of Var.:

Mn

C3.1.1--exp

Mn

C3.1.2--exp

Mn

DB----exp

Mn

Mn

Mn

Test #1a Test #2a Test #3a Test #4a Test #5a Test #6a

6.15 6.20 6.72 10.10 9.61 9.09

2.470 2.281 2.264 2.263 2.497 2.786 2.427 0.187 0.077

0.169 0.152 0.149 0.147 0.137 0.153 0.151 0.009 0.063

1.876 1.809 1.765 1.822 1.926 2.173 1.895 0.134 0.071

74

Table 4.12 Summary Table of All Z-Purlin Strength Data

Third Point Braced Tests

Local Buckling Lateral Buckling Distortional Buckling

Laterally Unbraced Tests

Local Buckling Lateral Buckling Distortional Buckling

Overall Mean (Mpred/Mexp) Overall Standard Deviation Overall Coefficient of Variation Range of Ratios (Mpred/Mexp)

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 1.019 to 1.674

0.181 0.588 to 1.242

0.111 0.841 to 1.361

0.066 2.233 to 2.786

0.192 0.137 to 0.244

0.067 1.720 to 2.173

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 thirdpoint 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

Table 4.13

Summary Table for Comparison of Results for Laterally Braced Z-Sections

Buckling Strengths from Ellifritt et al. 1998 (k-in) (1)

Buckling Strengths from this Study (k-in) (2)

Predicted Strength Ratios [M(2)/M(1)]

Test No. Exp.

Section Depth (in.) Lip Angle (deg.) Local Buckling Distortional Buckling Local Buckling

Flange Width (in.) Thickness (in.) Lip Depth (in.) 0.931 0.921 0.827 0.911 103.2 129.3 113.2 140.8 155.8 77.3 109.8 85.7 0.963 0.963 60.6 67.2 0.960 41 38 40 61.4 42 60.6 40 60.6 86.2 126.6 100.8 104.5 82.1 101.4 110.3 40 60.6 86.4 130.9 103.2 35 61.4 59.3 104.6 79.0 104.6 133.5 130.4 133.8 112.9 140.1 154.9 0.061 0.072 0.072 0.072 0.061 0.072 0.072

Yield Strength (ksi)

Distortional Buckling

Local Buckling

Distortional Buckling

Z16M-4

8.269

2.322

81.1 106.2 102.9 107.3 82.9 102.4 111.0

1.00 1.02 1.03 1.03 1.00 1.00 0.99

1.03 1.03 1.02 1.03 1.01 1.01 1.01

Z14M-5

8.318

2.330

78

Z14M-6

8.365

2.341

Z14M-8

8.3

2.416

Z16T-11

8.232

2.303

Z14T-12

8.196

2.314

Z14T-14

8.276

2.230

Notes:

1) M: mid-point braced, T: third-point braced

2) Lateral buckling strengths were not determined in the research of Ellifritt et al 1998.

3) Exp.: Experimentally obtained buckling strengths, Dist.: Distortional buckling strength predictions

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 Zsections (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

Table 4.14 Summary Table of Third Point Braced Test Components

Test No. Clip Type Panel Type Panel Thickness Purlin Orientation

(Trout and Murray 2000)

Test 18 (Z08-37) Test 19 (Z08-39) Test 20 (Z08-41) Test 25 (Z08-44) Test 26 (Z08-45) Test 27 (Z08-47) Test 21 (Z08-14) Test 22 (Z08-16) Test 23 (Z08-18) Test 24 (Z08-20) Test 28 (Z08-22) Test 29 (Z08-24) HS LS LF HS LS LF LF LS HS HS LF LS HS HS HS HS HS HS HS HS HS HS HS HS HS HS

LF: Low Fixed R: Ribbed

P P P P P P P P P P P P P P P P P P R R R R R R R R

24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 26 ga. 26 ga. 26 ga. 26 ga. 26 ga. 26 ga. 26 ga. 26 ga.

P: Pan S: Same

S S S S S S S S S S S S S S OP S S S S S S S S S S S

OP: Opposed

(Bryant et al. 1999a)

Test B (Z10-Eave) Test C (Z10-Eave) Test D (Z10-Eave) Test G (Z10-Eave) Test E (Z10-Eave) Test F (Z10-Eave)

(Almoney and Murray 1998)

Setup 2-0.105b Setup 3-0.105a Setup 3-0.105b Setup 4-0.105 Setup 4-0.076 Setup 3-0.060a Setup 3-0.060b Setup 3-0.060c

Notes: HS: High Sliding LS: Low Sliding

81

Table 4.14 Continued Summary Table of Third Point Braced Test Components

Test No. Clip Type Panel Type Panel Thickness Purlin Orientation

(Davis et al. 1995)

1G-Eave 2G-Eave 6G-Eave 3G-Eave 4G-Eave 5G-Eave HS HS HS HS HS HS P P P P P P 26 ga. 26 ga. 26 ga. 26 ga. 26 ga. 26 ga. S S S S S S

(Bathgate and Murray 1995)

Test #1 (Eave) Test #2 (Eave) Test #5 (Eave) Test #3 (Eave) Test #4 (Eave) Test #6 (Eave) Test #7 (Eave) Test #8 (Eave) Test #9 (Eave) HS HS HS HS HS HS HS HS HS R R R R R R R R R 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. S S S S S S S S S

(Borgsmiller et al. 1994)

G8ZTP/S-1 (1A) G8ZTP/S-1 (1B) G8ZTP/S-1 (1C) G8ZTP/S-1 (5A) G10ZTP/S-1 (2A) G10ZTP/S-1 (2B) G10ZTP/S-1 (2C)

Notes: HS: High Sliding LS: Low Sliding F: Fixed (general) LF: Low Fixed HF: High Fixed S: Sliding (general) R: Ribbed S: Same OP: Opposed P: Pan

HS HS HS HS HS HS HS

P P P P P P P

26 ga. 26 ga. 26 ga. 26 ga. 26 ga. 26 ga. 26 ga.

S S S S S S S

82

Table 4.14 Continued Summary Table of Third Point Braced Test Components

Test No. Clip Type Panel Type Panel Thickness Purlin Orientation

(Earls et al. 1991)

Test #1 Test #2 Test #3 Test #4 Test #5 Test #6 Test #7 HS HS HS HS HS HS HS R R R R R R R 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. 24 ga. S S S S S S S

(Brooks and Murray 1989)

Z-T-P/F-1 Z-T-P/S-1 Z-T-R/S-1 F F S P P R 26 ga. 26 ga. 26 ga. S S S

(Spangler and Murray 1989)

N-ZIS-12-SF-1 N-ZISO-12-SF-1 N-ZISO-12-SS-1 N-ZISO-12-TF-1

Notes: HS: High Sliding LS: Low Sliding F: Fixed (general)

SF SF LS HF

LF: Low Fixed HF: High Fixed S: Sliding (general)

R R R R

26 ga. 26 ga. 26 ga. 26 ga.

R: Ribbed S: Same

S OP OP OP

OP: Opposed P: Pan

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

Table 4.15

Summary of Effect of Clip Type for 8.0 in. Deep, 0.102 Thick Third Point Braced Tests

Buckling Strength Prediction Ratios from Table 4.1 Test No.

Test 18 (Z08-37) Test 20 (Z08-41) Test 25 (Z08-44) Test 26 (Z08-45) Test 27 (Z08-47)

Clip Type

HS LF HS LS LF Ave: SD: COV:

Local Buckling

1.298 1.382 1.379 1.338 1.382 1.356 0.037 0.028

Lateral Buckling

0.907 0.876 0.909 0.868 0.917 0.895 0.022 0.024

Distortional Buckling

1.137 1.211 1.194 1.168 1.202 1.182 0.030 0.025

Notes: LF: Low Fixed LS: Low Sliding HS: High Sliding Data from Murray and Trout (2000)

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 () 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.

M DB = 1.063 = R n (Mean value for all test results) M Exp.

1 1 + n m = 1.051 C p = (m - 2) where Cp = Correction Factor n = Number of tests (62 tests) m = Degrees of freedom (m = n-1 or 61 tests)

(4.1)

(4.2)

s=

(x

n i =1

i

-x

)

2

(n - 1)

= 0.118

(4.3)

where s = Standard deviation xi = Individual test results from Tables 4.1 through 4.9

x = Mean (From 6.1)

s V p = = 0.111 R n

(4.4)

86

where Vp = Coefficient of variation, must be greater than 0.065 Rn = From Equation 6.1

2 2 2 2 - 2.5 Vm +V f + C pV p +VQ

= 1.5(M m Fm Pm )e Where:

= 0.85

(4.5)

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

o = 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

2.551" 0.889" 50°

R0.2656 0.06" 8"

Fy=57.1ksi

R0.3438 48.3°

1.013"

2.499"

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

r1 = R1 +

t = 0.2956" 2

t t bT = B - r1 + + r1 + tan 1 = 2.074" = wT 2 2 2

wT = 34.56 < 60.0 (O.K. per Section B1.1-a-1) t

(B4-1)

S = 1.28

E = 29.094 f

wT 1 S Therefore use Case III with n = 3 t

(B4.2-11)

wT t 4 + 5 t = 0.001835in 4 I a = 115 S

( )

cT = C - r1 +

t 1 tan = 0.7372" = d 2 2

d = 12.287 < 14.0 (O.K. per Commentary Section B4.2) t

(B4-2)

Is

(d =

3

t sin 2 = 0.001176in 4 12

)

(B4.2-5)

I C 2 = s = 0.641 1.0 I a

C = D = 0.889"

90

D = 0.429" wT For simple lip stiffeners with 140 o 40 o and

D k a = 5.25 - 5 w T 4.0 = 3.107

D 0.8 : wT

(B4.2-8)

(B4.2-7)

n k = C 2 (k a - k u ) + k a = 2.738

(B2.1-4)

=

1.052 wT k t

fy = 0.967 > 0.673 (Flange is not fully effective.) E

(B2.1-3)

0.22 1 - = = 0.799

(B2.1-2)

b = ( wT ) = 1.657"

Tension Flange and Stiffener Lip

r2 = R2 +

t = 0.3738" 2

= 1.900"

t t b B = B - r2 + + r2 + tan 2 2 2 2 t c B = (C ) - r2 + tan 2 2 2

= 0.832"

0.03"+0.2956"

57.1ksi

AISI Section B3.2(a): Compression Stiffener Lip

w d = = 12.287 t t

Fy f1 = = 55.164 ksi Dw Dw t - r1 - + r1 cos 1 2 2 2

f1 N.A.

4.0"

4.0" 0.03"+0.3738"

91

(B2.1-4)

=

1.052 w k t

f1 E

= 0.852 > 0.673

where k = 0.43 per B3.2(a) Since > 0.673 the stiffener lip is not fully effective

(B2.1-3)

0.22 1 - = 0.8704 =

d s = d = 0.642"

(B4.2-9)

d s = C 2 d s = 0.4113"

AISI Section B2.3(a): Web

w = A - (r1 + r2 + t ) = 7.271"

0.03"+0.2956"

57.1ksi

f1

4.0"

f1 = = 52.45 ksi 0.5(Dw ) 0.5(Dw - t ) - r1 0.5(Dw ) Fy = f2 = 51.34 ksi Dw - 0.5(Dw + t ) - r2

Fy

N.A. f 4.0" 2 0.03"+0.3738"

where f1 is in compression and f2 is in tension

(B2.3-5)

=

- f2 = -0.9787 f1

3

(B2.3-4)

k = 4 + 2(1 - ) + 2(1 - ) = 23.452

(B2.1-4)

=

1.052 w f 1 = 1.11 > 0.673 [Use f1 per B2.3(a)] k t E

Since > 0.673 the web may not be fully effective, need to check.

(B2.1-3)

0.22 1 - = 0.722 =

(B2.1-2)

be = w = 5.25"

92

(B2.3-1)

b1 =

be = 1.32" (3 - )

For 0.236 :

(B2.3-2)

b2 =

be = 2.625" 2

b1 + b2

w w and b1 + b2 = 3.945" > = 3.64" 2 2

(Therefore web is fully effective for this iteration.)

Corners (Mean)

UT = UB =

(r1 )

2

= 0.464" = 0.587"

(r2 )

2

Compute Properties by Parts

Length (L) (in.) y From Top Fiber (in.) I`x about own axis (in3)

Element

(L*y)

(L*y2)

Top Flg Btm Flg Web Top Cnr Btm Cnr Top Lip Btm Lip

Total

1.660 1.900 7.271 0.464 0.587 0.411 0.832

13.126

0.030 7.970 4.000 0.137 7.835 0.274 7.554

0.049 15.140 29.084 0.064 4.600 0.113 6.285

55.335

0.002 120.700 116.340 0.010 36.100 0.031 47.500

320.683

----32.033 0.004 0.010 0.004 0.030

32.081

93

y=

(L * y ) = 4.216" below the top fiber (L )

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

Fy y Fy y = f1 = 52.69 ksi t y - - r1 2 f2 t Dw - y - - r2 2 = 45.78 ksi

0.03"+0.2956"

57.1ksi

f1 N.A.

4.216"

=

3.784" f

2

(B2.3-5)

- f2 = - 0.8689 f1

0.03"+0.3738"

(B2.3-4)

k = 4 + 2(1 - ) + 2(1 - ) = 20.793

3

(B2.1-4)

=

1.052 w f 1 = 1.182 > 0.673 (Web may not be fully effective.) k t E

(B2.1-3)

0.22 1 - = = 0.689

(B2.1-2)

be = w = 5.01

94

(B2.3-1)

b1 =

be = 1.295 (3 - )

For -0.236 :

(B2.3-2)

b2 =

be = 2.505" 2

b1 + b2

w w and b1 + b2 = 3.80" < + 0.216" = 3.852" 2 2

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 bneg = 3.852"-3.80" = -0.052" with centroid location at y= bneg t + r1 + b1 + = 1.647" below top fiber 2 2 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

I where S e = x y

.

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 (Xaxis) 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 y = 4.088 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

2 2 a a 3 2 0.0417 a + b + r + u1 + 0.637 r + 0.149r 2 2 2 + sin cos sin 2 3 a r sin r + u 2 + = 7.38in 4 - I x = + 2 2 3 2 2 c + c sin + a + c r cos - sin 12 2 2

I Sf = x y

= 2.05in 3

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, I yc = 0.074in 3 .

(C3.1.2-16)

2 EC b dI yc k -in Me = = 86.18 2 2L

M y = S f Fy = 117.1k -in

(C3.1.2-5)

By observation 2.78M y > M e > 0.56 M y , therefore:

(C3.1.2-3)

10 10 M y M c = (M y )1.0 - 9 36 M e M f Sc = c S f = 39.51k -in

= 81.0 k -in

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: y = 4.1" below extreme compression fiber I x = 7.55in 3

I S c = x = 1.842in 3 y

5.3

Local and Lateral Buckling Strength Predictions

The local buckling moment strength is:

(C3.1.1-1)

M pred = S e Fy = 8.33 k - ft

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 0.18C b 2 EdI yc Lu = = 51.48" < 100" Fy S f Lateral buckling needs to be checked. The lateral buckling moment strength is

(C3.1.2-1)

M M pred = S c c S f

= 6.07 k - ft

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 Ix f lg bl3 sin ( )2 = (t ) × 12

2 b2t 2 2 b sin ( ) f + bl l + b f y = 0.0061in 4 - y + 2 12

Iy f lg

2 2 2 b 3 3 b f + b f f - x + bl b f + bl cos( ) - x + bl cos( ) = (t ) × 2 2 12 12

= 0.172in 4

Ixy f lg = (t ) × b f J f lg

bf y x - 2

cos( ) sin ( ) + bl b f - x + bl × - y = 0.0195in 4 bl × 2 2

b f t 3 bl t 3 = 0.000243in 4 = + 3 3

Distortional Buckle Half Wavelength

Ix f lg b 2 Dw f d = 4.80 3 2t

0.25

= 24.85in

99

Formula Variables

1 = (Ix f lg b 2 + 0.039 J f lg 2 ) = 0.0002in 2 f d 1

2 = Iy f lg + yb f Ixy f lg = 0.0028in 2 1 3 = 1 Iy f lg -

2

2

2 2 Ixy f lg b f = 3.74 × 10 -7 in 4 1

1 = x +

= d

2

Ix f lg + Iy f lg = 3.603in 3 A f lg

= 0.016in - 2

Distortional Buckling Stress Assuming K=0

' + 2 ± ed = 2 A 1 f lg

E

( 1 + 2 )2 - 4 3 = 19.045 ksi

Torsional Stiffness Restraint

' 1.11 ed 2 Et 3 K = 1- 2 5.46(Dw + 0.06 d ) Et 4 Dw 2 d 12.564 + 2.192 D 4 + 13.392 D 2 d w d w

in. = 0.223kip in.

100

Actual Distortional Buckling Stress Since K is positive, the procedure below is followed per Chapter 2 with revised 1 and 3 terms:

K 1 = (Ix f lg b 2 + 0.039 J f lg d ) + = 0.00033in 2 f E 1 1

3 = 1 Iy f lg - Ixy 2 lg b 2 = 7.34 × 10 -7 in 4 f f 1 E

( + 2 ) ± ed = 2 A 1 f lg

( 1 + 2 )2 - 4 3 = 37.35 ksi

Predicted Strength Considering Distortional Buckling Since ed 2.2 f y then ed fc = f y fy 1 - 0.22 ed fy = 37.963 ksi

Since K 0 then M pred = S f f c = 6.49k - ft or M pred = S f f c = 6.55k - ft (using Sf from CFS)

101

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

Strength Determination Method Local Buckling (C3.1.1) Lateral Buckling (C3.1.2) Distortional Buckling (Hancock Method) "Hand" Calculations (k-ft) 8.33 Ratio (Mhand/Mexp) CFS Calculations (k-ft) 8.77 Ratio (MCFS/Mexp)

1.25

1.31

6.07

0.91

5.76

0.86

6.49

0.97

6.55

0.98

Notes: 1) Experimental strength for 1G = 6.69 k-ft 2) 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 d s = 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 K = 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

= Parameter in determining the effective area of a stiffener = Angle between lip and flange (radians) (subscript 1 denotes top and subscript 2

denotes bottom)

= Slenderness factor d = Distortional buckle half wavelength ed = Actual distortional buckling stress

ed = Distortional buckling stress assuming K is zero

= Reduction factor f = 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

1 , 2 , 3 , 1 , = 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 (K) 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 (+K) 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 (d) 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 ZPurlin 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 Zand 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/VPIST99/02, 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/VPIST99/03, 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 ColdFormed 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 ColdFormed 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," ThinWalled Structures, Vol. 27, No. 1, pp. 3-12. Hancock, G.J., and Lau, S.C.W., (1986), "Distortional Buckling Formula for ThinWalled 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 ColdFormed 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 ColdFormed Steel Structures, St. Louis, MO., October 17-18. Rogers, C,A., and Schuster, R.M., (1997), "Flange/Web Distortional Buckling of ColdFormed 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. 20612069. Winter, G., (1947), "Strength of Thin Steel Compression Flanges," Transactions of the ASCE, Paper No. 2305, Trans., 112, 1. Yu, W.W., (2000), Cold-Formed Steel Design, 3rd Edition, John Wiley & Sons, Inc., New York, NY.

114

APPENDIX A

1995 Distortional Buckling draft Ballot S95-55B 2000 Distortional Buckling Working Ballot S95-55B

115

AISI Committee on Specifications For the Design of Cold-Formed Steel Structural Members Subcommitte 24 ­ Flexural Members Ballot No. S95-55B Date: December 20, 1995 (draft ballot), October 2, 2000 (working ballot) ADDITION TO SPECIFICATION and COMMENTARY SECTION C3.1.4, COMBINED LRFD AND ASD C3.1.4 Distortional Buckling Strength

The nominal strength of C- and Z-sections subject to distortional buckling, Mn, where distortional buckling involves rotation of the compression flange and lip about the flangeweb junction, shall be calculated as follows: M n = Sc fc b = 1.67 (ASD) b = 0.90 (LRFD) where Sf =Elastic section modulus of the full unreduced section for the extreme compression fiber Sc = Sf when K 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 K as given by Eq. C3.1.4-12 is negative fc = Critical stress calculated as follows: For ed > 2.2 Fy (Eq. C3.1.4-1)

(Eq. C3.1.4-2)

f c = Fy For ed 2.2 f y f c = Fy (Eq. C3.1.4-3) 1 - 0.22

ed

Fy

ed

fy

116

where Fy = Yield point of section

ed = Elastic critical distortional buckling stress calculated as follows:

ed = 1 =

E ( 1 + 2 ) ± 2Af

{

[(

1

+ 2 ) - 4 3

2

]}

(Eq. C3.1.4-4) (Eq. C3.1.4-5) (Eq. C3.1.4-6) (Eq. C3.1.4-7) (Eq. C3.1.4-8) (Eq. C3.1.4-9)

(Ix f b 2f + 0.039 J f 2d ) + KE 1 1

2

yb f Ixy f 2 = Iy f + 1 3 = 1 Iy f - Ixy 2 b 2 f f 1

1 = x +

2

Ix f + Iy f Af

0.25

Ix f b 2 bw f d = 4.80 3 2t

K =

' 1.11 ed 2 Et 3 1- 5.46(bw + 0.06 d ) Et 2 2

4 bw 2 d 12.564 + 2.192b 4 + 13.392 b 2 d w d w (Eq. C3.1.4-10)

= d

(Eq. C3.1.4-11)

' where ed is obtained from Eq. C3.1.4-4 with

1 =

(Ix f b 2f + 0.039 J f 2d ) 1

(Eq. C3.1.4-12)

When K is negative in Eq. C3.1.4-10, compute K with ed = 0. The smaller positive value of 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 d computed by Eq. C3.1.4-9, use the bracing interval in place of d 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 x, y = Distance from flange-web junction to centroid of compression flange and lip in x,y directions respectively

bf

X

Centroid of compression flange and lip

bar

Y

bar

x

y

118

APPENDIX B

CFS Data Runs for Section 1G MathCad data Run for Section 1G

119

120

121

122

Standard Conditions

ksi 1000 lb in

2

kip

1000 lb

Input Data

E Sf 29500.ksi 2.0729in

3

Fy S xe

57.1 ksi 0.0 in

3

bw t

8.0 in 0.06 in b

L

0.889 in

Lb

50.2. 180 100.0 in

2.551 in

Section Properties

Centroidal Lengths lip L t 2 bf b t 2

Flange & Lip Area Af t. b f lip A f = 0.2028 in

2

Centroid Location of Flange & Lip xo t . bf Af 2

2

lip. b f

lip.

cos ( ) 2

yo

t 2 .A f

. lip2 .sin ( )

x o = 1.6507 in Flange & Lip Moment of Inertia

2 3 lip .sin ( )

y o = 0.0839 in

I xf

t.

12

3

lip.

lip.sin ( ) 2

2

2

yo

b f .t 12

2

b f .y o

2

. I xf = 6.101710

3

in

4

I yf

t.

bf

12

b f.

bf 2

xo

lip. b f

lip.cos ( ) 2

2

xo

3 cos ( ) lip . 12

2

I yf = 0.172 in

4

I xyf

t . b f. x o

bf 2

.y o

lip. b f

xo

lip.

sin ( ) cos ( ) . lip. 2 2

yo

I xyf = 0.0195 in

4

Jf

1 . .3 bft 3

1 . .3 lip t 3

. J f = 2.433610

4

in

4

123

Formula Variables

2

1

xo

I xf I yf Af

1 = 3.6033 in

2

4.80.

2 I xf.b f .b w 3 2 .t

0.25

= 24.8481 in

d

if < L b , , L b d

2

d = 24.8481 in

2

= 0.016 in

1

1

. I .b 2 xf f

0.039 J f . d

2

. 1 = 1.980310

4

in

2

2

. I yf

2 1

.y .b .I o f xyf

. 2 = 2.786510

3

in

2

3

. 1 .I yf

1

.I 2 .b 2 xyf f

. 3 = 3.736310

7

in

4

124

Distortional Buckling Stress with K =0

ed1' E . 1 .A 2 f 2 1 2

2

4 . 3

ed1' = 19.0454 ksi

ed2'

E . 1 .A 2 f

2

1

2

2

4 . 3

ed2' = 415.1013ksi

ed'

if ed1' 0 . ed1' < ed2' , ed1' , ed2'

ed' = 19.0454 ksi

K

2 .E.t 5.46 b w

3

0.06. d

. 1

2 4 1.11. ed' .b w . d 4 2 E.t . 12.56. d 4 2.192.b w 2 2 13.39. d .b w

K = 0.2227 kip .

in in

Used only if K is greater than or equal to zero. 4 . 2 I xf.b f 1

2 0.039.J f . d

K 1 . .E

. 4 = 3.290910

4

in

2

5

. 4 .I yf

1

.I 2 .b 2 xyf f

. 5 = 7.340310

7

in

4

ed1

E . 4 .A 2 f

2

4

2

2

4 . 5

ed1 = 415.8621ksi

ed2

E . 4 2 .A f

2

4

2

2

4 . 5

ed2 = 37.3484 ksi

eda

if ed1 0 . ed1 < ed2 , ed1 , ed2

eda = 37.3484 ksi

125

Used only if K is less than zero.

K '

2 .E.t 5.46. b w . 2 I xf.b f 1 . 6 .I yf

3

0.06. d K ' 1 . .E

K ' = 0.2459 kip.

in in

6

2 0.039.J f . d

. 6 = 3.427710

4

in

2

7

1

.I 2 .b 2 xyf f

. 7 = 7.716510

7

in

4

ed3

E . 6 .A 2 f E . 6 .A 2 f

2

6

2

2

4 . 7

ed3 = 39.2547 ksi

ed4

2

6

2

2

4 . 7

ed4 = 415.9458ksi

edb

if ed3 0 . ed3 < ed4 , ed3 , ed4

edb = 39.2547 ksi

Actual Distortional Buckling Stress and Axial Load

ed if K 0 , eda , edb ed = 37.3484 ksi

P1

E . 4 2 E . 6 2

2

4

2

2

4 . 5

P 1 = 7.5743 kip

P2

2

6

2

2

7 . 5

P 2 = 14.3356 kip

P cr

if K 0 , P 1 , P 2

P cr = 7.5743 kip

126

Nominal Strength of Section Resulting from Distortional Buckling

M ed My S f . ed S f .F y M ed My M ed My M ed = 77.4194 kip .in M y = 118.3626kip .in . 1 0.22.

M cr

M y.

M cr = 78.6942 kip.in M c = 78.6942 kip.in S c = 2.0729 in

3

Mc Sc

if M ed > 2.2.M y , M y , M cr if K 0 , S f , S xe Mc Sf

Mn

S c.

M n = 78.6942 kip .in

f c1 F y. ed Fy . 1 0.22. ed Fy F c2

Fy ed Fy ed Fy

f c1

F y.

. 1

.22.

fc

if ed > 2.2.F y , F y , f c1

fc M n1 M n2

if ed > 2.2 F y , f c1 , F c2 S f .f c S xe.f c if K 0 , M n1 , M n2

Sc

if K 0 , S f , S xe S c .f c

Mn

Mn

M n = 78.6942 kip .in

M n = 78.6942 kip .in

127

VITA

Scott David Cortese was born on January 9th, 1975 in Augusta, Georgia. He

graduated from Corning East High School in Corning, New York in 1993 and subsequently attended Bowling Green State University. Scott graduated from Bowling Green in 1997 with a Bachelor of Science in Geology and a Bachelor of Science in Environmental Science. Immediately following his undergraduate education, the author worked as an Environmental Scientist/Geologist at Hull & Associates, Inc. in Toledo, Ohio. Scott's graduate education in structural engineering began in the fall of 1998 at Virginia Polytechnic Institute and State University and concluded in May 2001 with this thesis. Currently, he is working as a bridge design engineer at Kimley-Horn & Associates, located in West Palm Beach, Florida.

128

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

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