Subject: Progress Report No. 4 AISC Faculty Fellowship Cross section Stability of Structural Steel

Transcription

1 April 2 To: Tom Schlafly AISC Committee on Research Subject: Progress Report No. 4 AISC Faculty Fellowship Cross section Stability of Structural Steel Tom, Please find enclosed the fourth progress report for the AISC Faculty Fellowship. The report summarizes research efforts to study the cross section stability of structural steel, and to extend the Direct Strength Method to hot rolled steel sections. The finite element parametric analysis reported herein focuses on local global interaction of buckling modes, and comparisons of the AISC, AISI Effective Width, and AISI Direct Strength design methods for locally slender columns and beams with variable lengths at preselected slenderness ratios. Sincerely, Mina Seif (mina.seif@jhu.edu) Graduate Research Assistant Ben Schafer (schafer@jhu.edu) Associate Professor

2 Summary of Progress The primary goal of this AISC funded research is to study and assess the cross section stability of structural steel. A timeline and brief synopsis follows. Research begins March 26 (Note, Mina Seif joined project in October 26) Progress Report # June 27 Completed work: Performed axial and major axis bending elastic cross section stability analysis on the W sections in the AISC (v3) shapes database using the finite strip elastic buckling analysis software CUFSM. Evaluated and found simple design formulas for plate buckling coefficients of W sections in local buckling that include web flange interaction. Reformulated the AISC, AISI, and DSM column design equations into a single notation so that the methods can be readily compared to one another, and so that the centrality of elastic buckling predictions for all the methods could be readily observed. Performed a finite strip elastic buckling analysis parametric study on AISC, AISI, and DSM column design equations for W sections to compare and contrast the design methods. Created educational tutorials to explore elastic cross section stability of structural steel with the finite strip method, tutorials include clear 2

3 learning objectives, step by step instructions, and complementary homework problems for students. Publications: Schafer, B.W., Seif, M., Comparison of Design Methods for Locally Slender Steel Columns SSRC Annual Stability Conference, Nashville, TN, April 28. Progress Report #2 April 28 Completed work: Performed axial, positive and negative major axis bending, and positive and negative minor axis bending finite strip elastic crosssection buckling stability analysis on all the sections in the AISC (v3) shapes database using the finite strip elastic buckling analysis software CUFSM. Evaluated and determined simple design formulas that include webflange interaction for local plate buckling coefficients of all structural steel section types. Performed ABAQUS finite element elastic buckling analyses on W sections, comparing and assessing a variety of element types and mesh densities. Initiated an ABAQUS nonlinear finite element analysis parameter study on W section stub columns, and assessed and compared results to the sections strengths predicted by AISC, AISI, and DSM column design equations. 3

4 Publications: Seif, M., Schafer, B.W., Elastic Buckling Finite Strip Analysis of the AISC Sections Database and Proposed Local Plate Buckling Coefficients Structures Congress, Austin, TX, April 29. Progress Report #3 April 29 Completed work: Studied the influence of the variation of some design parameters on the ultimate strength of W section steel stub columns; further understanding, highlighting, and quantifying the uncertainties of parameters that lead to the divergence of the columns strength than what one might typically expect. Performed an ABAQUS nonlinear finite element analysis parameter study on W section stub columns, and assessed and compared results to the sections strengths predicted by AISC, AISI, and DSM column design equations. Performed a similar nonlinear finite element analysis parameter study on W section short beams, assessing and comparing results to the strengths predicted by AISC, AISI, and DSM beam equations. Initiated a nonlinear finite element analysis parameter study for columns with variable lengths at preselected slenderness ratios, as a step towards the completion of a database that will allow extension of the Direct Strength Method to hot rolled steel sections. Publications: Seif, M., Schafer, B.W., Finite element comparison of design methods for locally slender steel beams and columns SSRC Annual Stability Conference, Phoenix, AZ, April 29. 4

5 Progress Report #4 April 29 Completed work: Studied the influence of the variation of some design parameters on the ultimate strength of W section steel long columns and beams where local and global buckling modes interact; further understanding, highlighting, and quantifying the uncertainties of parameters that lead to the divergence of the columns strength than what one might typically expect. Performed an ABAQUS nonlinear finite element analysis parameter study on W section columns with variable lengths at preselected slenderness ratios, and assessed and compared results to the sections strengths predicted by AISC, AISI, and DSM column design equations. Performed a similar nonlinear finite element analysis parameter study on W section beams with variable lengths at preselected slenderness ratios, assessing and comparing results to the strengths predicted by AISC, AISI, and DSM beam equations. Started studying the stress and strain distributions in all the sections analyzed in the nonlinear finite element analysis parameter study database as a final step towards the extension of the Direct Strength Method to hot rolled steel sections. Publications: Seif, M., Schafer, B.W. Elastic Local Buckling of Structural Steel Shapes. Journal of Constructional Steel Research (JCSR), doi:.6/j.jcsr Seif, M., Schafer, B.W., Design methods for local-global interaction of locally slender steel members SSRC Annual Stability Conference, Orlando, FL, May 2. Seif, M., Schafer, B.W., Cross-sectional Stability of Structural Steel. International Conference of Stability and Ductility of Steel Structures (SDSS) Proceedings, Rio de Janeiro, Brazil, September 2, In Press. 5

6 Table of Contents Summary of Progress...2 Introduction Finite Element Comparison of Design Methods for Local-Global Interaction of Locally Slender Steel Beams and Columns Introduction and Motivation Design Methods and Equations Parameter Study and Modeling Approach Geometric Variation: Element Local Slenderness Geometric Variation: Member Length Finite Element Modeling Mesh and element selection Material modeling Residual stresses Geometric imperfections Results Columns Beams Discussion Columns Beams Overall Summary and Conclusion Strain Distribution in Locally Slender Structural Steel Cross-Sections Introduction References...5 6

7 Appendix A : NRC Research Proposal

8 Introduction The research work presented in this progress report represents a continuing effort towards a fuller understanding of hot rolled steel cross sectional local stability. Typically, locally slender cross sections are avoided in the design of hot rolled steel structural elements, but completely avoiding local buckling ignores the beneficial post buckling reserve that exists in this mode. With the appearance of high and ultra high yield strength steels this practice may become uneconomical, as the local slenderness limits for a section to remain compact are function of the yield stress. Currently, the AISC employs the Q factor approach when slender elements exist in the cross section, but analysis in Progress Report # indicates geometric regions where the Q factor approach may be overly conservative, and other regions where it may be moderately unconservative as well. It is postulated that a more accurate accounting of web flange interaction will create a more robust method for the design of high yield stress structural steel cross sections that are locally slender. Progress Report # summarized how the locally slender W section column design equations from the AISC Q factor approach, AISI Effective Width Method, and AISI Direct Strength Method (DSM) can be reformulated and arranged into a common set of notation. This common notation highlights the central role of cross section stability in predicting member strength. 8

9 Progress Report #2, provided results of finite strip elastic cross section buckling analysis performed on all the sections in the AISC (v3) shapes database (25) under: axial, positive and negative major axis bending, and positive and negative minor axis bending. The results were used to evaluate the plate local buckling coefficients underlying the AISC cross section compactness limits (e.g., bf/2tf and h/tw limits). In addition, the finite strip results provided the basis for the creation of simple design formulas for local plate buckling that include webflange interaction, and better represent the elastic stability behavior of structural steel sections, for all different loading types. Those design formulas are essentially a proposed replacement for the AISC s Table B4. which defines the slenderness limits. Progress Report #2 also provided a comparison and assessment of the different two dimensional shell elements which are commonly used in modeling structural steel. The assessment is completed through finite element elastic buckling analysis performed on W sections using a variety of element types and mesh densities in the program ABAQUS. The concluding section of that report discussed the initiation of a finite element parameter study (performed in ABAQUS) on W section stub columns. Progress Report #3, provided a finite element reliability analysis study on hot rolled W sectioned structural steel columns. The study aimed to assess the 9

10 influence of the variation of some design parameters on the ultimate strength of such type of members; further understanding, highlighting, and quantifying the uncertainties of parameters that lead to the divergence of the columns strength than what one might typically expect. Progress Report #3 also presented and discussed a nonlinear finite element analysis parameter study (performed in ABAQUS) on W section stub columns and short beams. The study aimed to highlight the parameters that lead to the divergence of the section strength capacity predictions, provided by the different design methods: AISC, AISI, and DSM design equations. The first part of this document, Progress Report #4, discusses the extension of the parameter study presented in Progress report #3 to include longer columns and beams, thus including global buckling modes and the effect of local global mode interactions. The columns and beams in this study have variable lengths at preselected slenderness ratios. This extension leads to a further completion of a database of failure mechanisms of W-sections at different element slenderness ratios. The second part of this report describes the current ongoing work where the strain distributions observed in the failure mechanisms of the FE parameter study database are closely examined, and compared to finite strip analysis results using CUFSM, as well as to theoretical distributions. This will allow us to

11 utilize the elastic buckling information, for cross sections with large variations in element slenderness, and ultimately propose improvements to DSM so it may be applied to hot rolled structural steel with locally slender cross sections. The Appendix of this report shows a copy of a research proposal titled Multi scale Structural Stability under Realistic Fire Loading. The proposal was submitted to the National Research Council (NRC), as part of a post doctoral fellowship application, and it aims to extend this research where the effect of realistic fire loading scenarios on locally slender structural steel members will be studied.

12 2 Finite Element Comparison of Design Methods for Local-Global Interaction of Locally Slender Steel Beams and Columns 2. Introduction and Motivation With the advent of high and ultra high yield strength steels, the increased yield stress drives even standard hot rolled steel shapes from locally compact to locally slender (noncompact or slender), making it inefficient to avoid such crosssections in the design of hot rolled steel structural members (see Seif and Schafer 29a and 29b for details). Efficient and reliable strength predictions are needed for locally slender hot rolled steel cross sections. Analysis of existing AISC (25) provisions for locally slender stub columns and short beams (Seif and Schafer 29a) indicated geometric regions where AISC design may be excessively conservative, and other regions where it may be moderately unconservative. The work on the stub columns and short beams isolated and studied the effect of local buckling modes on the predicted strength. However, most failures occur do to combinations and interactions between local and global buckling modes (see Figure 2-5 and Figure 2-22). The work herein represents a direct extension of previous studies on stub columns and short beams (Seif and Schafer 29b and 29c) now to include long columns and long beams, where 2

13 the locally slender cross sections may interact with global (flexural, lateraltorsional, etc.) buckling modes. 2.2 Design Methods and Equations The design of locally slender steel cross sections may be completed by a variety of methods, three of which are examined here: () The hot rolled steel AISC method, as embodied in the 25 AISC Specification, labeled AISC herein, (2) The AISI Effective Width Method from the main body of the 27 AISI Specification for cold formed steel, labeled AISI herein, and, (3) The Direct Strength Method as given in Appendix of the 27 AISI Specification, labeled DSM herein. For each of these three design methods the expressions for strength prediction of locally slender braced columns and beams have been provided in a common notation in Seif and Schafer (28, 29b, and 29c). In those equations the centrality of elastic local buckling is made clear. For long (unbraced) columns and beams global buckling must be considered as well as local global interaction. In AISC, AISI, and DSM global column buckling is predicted using the same (single) expression. However, local global interaction is handled by the Q factor method in AISC, the unified method in AISI, and a variation of the unified method in DSM. In all cases the global strength is reduced due to local cross 3

14 section slenderness. The Q factor approach reduces the strength and increases the long column slenderness to arrive at its reduction. The unified method uses the effective area of the column at the long column buckling stress. DSM uses a similar approach, but the effective area calculation is replaced by a reduction of the full cross section (at the long column strength). Figure 2- shows the effect of how the AISC and AISI differently handle the predicted strength reduction due to global slenderness. Also it shows the column global slenderness regions that were covered in Progress Report #3 and Progress Report #4..2 Progress Report #3 Local modes Progress Report #4 Local-Global modes interaction.8 stub P n / P y.6 λ c λ c Figure 2- Effect of global column slenderness on predicted strength, and regions covered in the parametric studies of progress reports #3 and #4 4

15 AISC and AISI/DSM use different formats for the global (lateral-torsional buckling) provisions of beams. However, for no moment gradient (C b = ) the resulting expressions are actually quite similar with the exception that AISI only provides capacities up to first yield (M y ) for sections subject to lateral-torsional buckling. For AISI/DSM local-global interaction in beams is treated in the same conceptual manner as for columns; not so for AISC, which uses nothing like the Q-factor approach, and instead provides direct reductions based on the flange and web plate slenderness (also see White 28). A result of AISC s approach in not adopting one consistent philosophy for local-global interaction in beams is some unusual changes in strength as local slenderness is varied. 2.3 Parameter Study and Modeling 2.3. Approach The purpose of the nonlinear finite element (FE) analysis parameter study studied herein is the understanding and highlighting of the parameters that lead to the divergence between the capacity predictions of the different design methods under axial and bending loads. Previous FE analysis (Seif and Schafer 29b and 29c) was conducted on stub (short) members, avoiding global (i.e., flexural, or lateral-torsional) buckling modes, and focusing on local buckling modes alone. The length of the studied members was determined according to the stub column definitions of SSRC (i.e., 5

16 Galambos 998). The FE analysis herein is extended to longer members, thus including global buckling modes where the interaction between local and global modes is allowed and examined. The columns and beams in this study have variable lengths at preselected slenderness ratios. Based on the authors judgment, AISC W4 and W36 sections are selected for the study as representing common sections for columns and beams in high-rise buildings. The W4x233 section is approximately the average dimensions for the W4 group and the W36x33 for the W36 group. All sections are modeled with globally pinned, warping fixed boundary conditions, and loaded via incremental displacement or rotation for the columns and beams respectively Geometric Variation: Element Local Slenderness To examine the impact of slenderness in the local-global buckling mode interaction, and the impact of web-flange interaction in I-sections, four series of parametric studies are performed under axial and bending loading at preselected slenderness ratios: W4FI: a W4x233 section with a modified Flange thickness, that varies Independently from all other dimensions, W4FR: a W4x233 section with variable Flange thickness, but the web thickness set so that the Ratio of the flange-to-web thickness remains the same as the original W4x233, 6

17 W36FR: a W36x33 section with variable Web thickness, but the flange thickness set so that the Ratio of the flange-to-web thickness remains the same as the original W36x33, and W36WI: a W36x33 section with a variable Web thickness, that varies Independently from all other dimensions, as summarized in Table 2- and Figure 2-2. Figure 2-2 indicates that for the W4FI group, the web slenderness is held constant (compact), while the flange slenderness varies from compact to noncompact and slender. Similarly, for the W36WI group, the flange slenderness is held constant (compact) while the web slenderness is varied for compact to noncompact and slender. Finally the W4FR and W36WR groups range a whole range of slenderness combinations. Table 2- Parametric study of W-sections b f /2t f h/t w h/b f t f /t w W4x W4FI varied fixed fixed varied W4FR varied varied fixed fixed W36x W36FR varied varied fixed fixed W36WI Fixed varied fixed varied 7

18 35 f y = 5 ksi (345 MPa) 3 25 λ rf k f =.2 k w =.5 k f =.6 k w =6. 2 b f /2t f 5 λ pf 5 W4FI k f =.9 k w =2. k f =.6 k w =5. k f =.6 k w =5.6 k f =.5 k w =27 W4FR W36FR W36WI k f =.5 k w =29 k f =. k w =36 k f =.5 k w =29 k f =.5 k w =36 5 λ pw λ rw 5 h/t w Figure 2-2 Variation of parameters as a function of h/t w and b f /2t f with back-calculated elastic buckling k values, and AISC λ limts for beams shown. For the purpose of this study, element thicknesses were varied between.5 in. (.27 mm) and 3. in. (76.2 mm). While not strictly realistic, the values chosen here are for the purposes of comparing and exercising the design methods up to and through their extreme limits. Local slenderness may be understood as the square root of the ratio of the yield stress to the local buckling stress (i.e., f y /f cr ). The element local buckling stress is proportional to the square of the element thickness, thus the local slenderness is proportional to /t. Here element thickness is varied and used as a proxy for investigating local slenderness, in the future, material property variations are also needed. 8

19 2.3.3 Geometric Variation: Member Length The initial FE analysis (Seif and Schafer 29b and 29c) was conducted on stub (short) members, avoiding global (i.e., flexural, or lateral-torsional) buckling modes, and focusing on local buckling modes alone. The length of the studied members was determined according to the stub column definitions of SSRC (i.e., Galambos 998), and fixed at that length. To examine the impact of local-global buckling modes interaction on the strength of locally slender members, longer members are included in the FE parameter study taking the member s length as a variable in the parameter study. Each member s length is determined so to achieve certain preset slenderness parameter values, where the slenderness parameter, λ, is defined in terms of the member s length and cross-section dimensions. For columns, two groups of analysis were chosen to be performed at axial slenderness parameter, λ c, values fixed at.9 and.5, where λ c is defined as: f y λc = = f cr euler KL r f y 2 π E (2-) Note that varying the thicknesses (flange, web, or both at constant ratio) will vary the moment of inertia, I, and the cross-sectional area, A, and accordingly the radius of gyration, r. The member s length, L, is then backcalculated to maintain the specified λ c values. 9

20 Similarly for beams, two groups of analysis were chosen to be performed at flexural slenderness parameter, λ e, values fixed at.6 and.34, which are the AISI values defining the non-compact from the compact and slender members respectively (see, e.g. Shifferaw and Schafer 28). The λ e is defined as: M y λ e = (2-2) M cre Fixing λ e, the critical buckling moment, M cre, is calculated for each section. M cre is also defined as follows: M cre π EI ygj π EI yπ C = Cb L L w (2-3) Again, note that varying the thicknesses (flange, web, or both at constant ratio) will vary all the parameters on the right hand side of Eq. (3). Accordingly, the member s length, L, is then back-calculated to maintain the specified λ e values Finite Element Modeling Mesh and element selection ABAQUS was used to perform the analysis. Members were modeled using S4 shell elements. The S4 element has six degrees of freedom per node, adopts bilinear interpolation for the displacement and rotation fields, incorporates finite membrane strains, and its shear stiffness is yielded by full integration. 2

21 Considering computational speed and accuracy it was decided that a mesh density of five elements across each flange outstand, ten across the web, and an aspect ratio of was adequate for this study. The choice of element type and density are based on comparisons with three-dimensional solid elements as reported in Seif and Schafer (28 and 29b). It is noted that some debate exists in the literature regarding the selection of the S4 vs. S4R element (see, e.g. Dinis and Camotim 26, and Earls 2) Material modeling The material model used is similar to that of Barth, K.E. et al. (25). It follows classical metal plasticity: Von Mises yield criteria, associated flow, and isotropic hardening. The uniaxial σ-ε diagram is provided in Figure 2-3 is defined for the finite element analysis as a multi-linear stress-strain response, consisting of an elastic region, a yield plateau, and a strain hardening region. The elastic region is defined by the modulus of elasticity, E, and the yield stress, f y. The yield plateau is defined by a small slope of E ~ E/2, to help in avoiding numerical instabilities during analysis. A strain hardening modulus E st = 45 ksi which initiates at a strain of. was chosen. The curve shown in Figure 2-3 is converted to a true stress-strain curve for the analysis. 2

22 Engineering Stress (ksi) f y = 5 Slope, E =45 Slope, E =29 Slope, E st =72 f u = 65 ε y ε st =. Engineering Strain Figure 2-3 Idealized engineering stress strain curve used for analysis Residual stresses For this work, the classic and commonly used distribution of Galambos and Ketter (959), as shown in Figure 2-4, is employed. Similar to other researchers (e.g., Jung and White 26) the residual stresses are defined in the finite element analysis as initial longitudinal stresses, and given as the average value across the element at its center. (See Seif and Schafer 29b for further discussion). 22

23 - - σ c - =.3 f y + σ t = σ c b f t f + t b t f f d 2 t w ( f ) - - Figure 2-4 Residual stress distribution used for analysis as given by Galambos and Ketter (959) Geometric imperfections Geometric imperfections have an important role to play in any collapse analysis involving stability. For the previous work on short (stub) members, the imperfections were defined by scaling the local buckling eigenmode from elastic buckling analysis. Since the focus at this point is on longer members, global buckling modes are also included. Initial geometric imperfections are added through linearly superposing a scaled local and a scaled global eigenmode solution from a finite strip analysis performed on each section, using CUFSM (Schafer, B.W., Ádány, S. 26). Figure 2-5 shows a typical CUFSM curve, where the local and global buckling modes are determined. It is noted that the global buckling mode is that at a half wave-length equal to the member s unbraced length, L, while the local buckling mode is at the minima of the curve. For the purposes of this study, the local buckling is chosen as the mode closest to the 23

24 minima that can fit as a whole number of half waves within the member s unbraced length. The local buckling mode as shown in Figure 2-6, is scaled so that the maximum nodal displacement is equal to the greater of b f /5 or d/5 which is a commonly employed magnitude (see, e.g Kian and Lee 22), while the global buckling mode is scaled so that the maximum nodal displacement is equal to L/, as shown in Figure Load x Local x Global 2 3 Half wave length L Figure 2-5 Typical CUFSM curve where local and global buckling modes are determined. 24

25 d /5 (b) b f /5 (a) b f bf /5 d d /5 (c) Figure 2-6 Typical local buckling mode and initial geometrical imperfections for the analysis (a) ABAQUS 3D view, (b) ABAQUS front view, and (c) CUFSM front view, with typical scaling factors. 25

26 L / (b) (a) (a) L / (c) Figure 2-7 Typical global (flexural) buckling mode and initial geometrical imperfections for the analysis (a) ABAQUS 3D view, (b) ABAQUS front view, and (c) CUFSM front view, with typical scaling factors. 2.4 Results As discussed previously (see table 2- and Figure 2-2), the parametric study is broken into 4 groups: W4FI, W4FR, W36FR, and W36WI analyzed at different preset slenderness limits. Here the results of the parametric study are presented for each group, including comparisons to the AISC, AISI, and DSM 26

27 design methods. Analysis results are provided first for the columns, then the beams Columns ABAQUS results for the parametric study of locally slender long columns (denoted with and given for the 4 parametric studies) are reported as a function of long column slenderness (λ c ~.25,.9, and.5) in Figure 2-8. In Figure 2-8 the standard (compact) W4 and W36 cross-sections have been denoted with a *. If the long column curve is exact, the * would be in perfect agreement with the upper curve shown. As can be observed, as the local slenderness is increased the strength predictions fall further and further below the global column (upper) curve, which for compact/fully-effective sections is identical in AISC, AISI, and DSM. Also highlighted in Figure 2-8, so that a locally slender section may be observed, is the cross-sections with a back-calculated Q or A eff /A g.7, denoted with a o, and the AISC and AISI (both effective width and DSM) strength curves for Q or A eff /A g =.7. Figure 2-8 does not allow for a complete study of the impact of local slenderness as a full family of strength curves would need to be generated and each point compared to a different curve. Rather than do this, to compare all the sections in a given study the results are expressed as a function of local slenderness (at a given global slenderness, λ c ). 27

28 W4FI W4FR P n /P y W36FI.5.5 W36WI P n /P y Lambda c Lambda c Figure 2-8 ABAQUS results for the parametric study reported as a function of long column slenderness Complete comparisons of the studied columns with the AISC, AISI, and DSM methods are provided in Figure 2-9 through Figure 2-4. Figure 2-9 and Figure 2- provide the summary of results for the stub column study of Seif and Schafer (29b and 29c). In a similar manner, Figure 2- and Figure 2-3 present the results for each of the 4 parameter studies at λ c =.9 and λ c =.5 respectively. Figure 2-2 and Figure 2-4 present all 4 studies directly compared against each of the design methods, for λ c =.9 and λ c =.5 respectively. All results are plotted as a function of elastic local slenderness of the cross-section: f y /f crl, 28

29 determined by finite strip analysis. Finally, Figure 2-5 provides the deformed shapes for a W4 section at λ c =.9 and λ c =.5. The figure shows the interaction between the local and global (about the minor axis) buckling modes. W4FI W4FR P n /P y W36FR AISC AISI DSM ABAQUS 2 3 W36WI P n /P y (f y /f crl ).5 (f y /f crl ).5 Figure 2-9 Results of column parametric study for 4 study groups (stub) 29

30 P n /P y (f y /f crl ).5 P n /P y.5 AISC.5 AISI 23 DSM ABAQUS 2 3 (f y /f crl ).5 Figure 2- Results of column parametric study for 3 design methods (stub) W4FI W4FR P n /P y W36FR AISC AISI DSM ABAQUS 2 3 W36WI P n /P y (f y /f crl ).5 (f y /f crl ).5 Figure 2- Results of column parametric study for 4 study groups (λ c =.9) 3

31 P n /P y (f y /f crl ).5 P n /P y.5 AISC.5 AISI DSM 23 ABAQUS 2 3 (f y /f crl ).5 Figure 2-2 Results of column parametric study for 3 design methods (λ c =.9) W4FI W4FR P n /P y W36FR AISC AISI DSM ABAQUS 2 3 W36WI P n /P y (f y /f crl ).5 (f y /f crl ).5 Figure 2-3 Results of column parametric study for 4 study groups (λ c =.5) 3

32 P n /P y (f y /f crl ).5 P n /P y.5.5 AISC 23 AISI DSM ABAQUS 2 3 (f y /f crl ).5 Figure 2-4 Results of column parametric study for 3 design methods (λ c =.5) Figure 2-5 Deformed shapes for a W4FI section (a) λ c =.9, (b) λ c = Beams For the beams the predicted capacities from the nonlinear collapse analysis in ABAQUS are shown for each of the 4 parameter groups in Figure 2-6, Figure 2-8, and Figure 2-2; for the short specimens, intermediate length specimens at λ e =.6, and long specimens at λ e =.34 respectively. Results are also compared 32

33 against the design methods directly in Figure 2-7, Figure 2-9, and Figure 2-2 for the same three lengths (short, intermediate, long). In all the preceding plots the local slenderness f y /f cr (or equivalently M y /M cr ) is plotted against the capacity, normalized to the plastic moment, M p. Finally, Figure 2-22 provides the deformed shapes for a W36 section with a slender web at λ e =.6 and λ e =.34 (intermediate and long lengths); indicating the interaction between the local and lateral-torsional buckling mode at failure. 33

34 W4FI W4FR M n /M p W36FR AISC AISI DSM ABAQUS 2 3 W36WI M n /M p (f y /f crl ).5 (f y /f crl ).5 Figure 2-6 Results of beam parametric study for 4 study groups (short) M n /M p (f y /f crl ).5 M n /M p.5 AISC.5 AISI DSM 23 ABAQUS 2 3 (f y /f crl ).5 Figure 2-7 Results of beam parametric study for 3 design methods (short) 34

35 W4FI W4FR M n /M p W36FR AISC AISI DSM ABAQUS 2 3 W36WI M n /M p (f y /f crl ).5 (f y /f crl ).5 Figure 2-8 Results of beam parametric study for 4 study groups (λ e =.6) M n /M p (f y /f crl ).5 M n /M p.5.5 AISC AISI DSM 2 3 ABAQUS 2 3 (f y /f crl ).5 Figure 2-9 Results of beam parametric study for 3 design methods (λ e =.6) 35

36 W4FI W4FR M n /M p W36FR AISC AISI DSM ABAQUS 2 3 W36WI M n /M p (f y /f crl ).5 (f y /f crl ).5 Figure 2-2 Results of beam parametric study for 4 study groups (λ e =.34) M n /M p (f y /f crl ).5 M n /M p.5.5 AISC AISI 23 DSM ABAQUS 2 3 (f y /f crl ).5 Figure 2-2 Results of beam parametric study for 3 design methods (λ e =.34) 36

37 Figure 2-22 Deformed shapes for a W36WI section (a) λe=.6, (b) λe= Discussion The focus of the following discussion is the performance of the design methods in comparison with the capacities predicted by the nonlinear finite element analysis Columns Unlike the case of stub columns, where the AISI s implementation of the Effective Width Method provided, by far, the best prediction of the column capacity, there isn t a specific design method that outperforms the others when it comes to predicting the capacity of longer columns. (Recall all methods use the same global column curve, but reduce the strength in different manners to account for local-global interaction.) For longer columns, similar to stub columns, AISC provides reliable predictions when the flange is non-slender; however AISC is unduly conservative whenever the flanges become slender (regardless of the web). The level of conservatism is large enough to make AISC design with 37

38 slender flanges completely uneconomical. AISI works well in nearly all cases; however, when the flange is specifically varied the unified method for reducing the column capacity does not properly capture the reduction in global capacity (through loss of I). DSM s accuracy is excellent when the flange and web vary at fixed ratios, and conservative (sometimes significantly) when one element is markedly more slender than its neighbor Beams The AISC predictions are overall best characterized as conservative, often excessively so when compared with the FE predictions. The strength prediction as the web and flange move from compact, to non-compact, to slender often have abrupt transitions as the related design methods use different formulae in these different local slenderness ranges. For example, see the W36WI study at λ e =.6 of Figure 2-8. In general the expressions related to local flange slenderness provide smooth but quite conservative design predictions, while those related to local web slenderness suffer from the abrupt transitions. The study shows that the AISC expressions are essentially intended for compact, and semi-compact sections; but for locally slender sections the results are safe, but unduly conservative. An important proviso to this conclusion, particularly for long beams, is that users must take care when utilizing the approximations provided in AISC as in some cases the conservatism is derived from these approximations as opposed to the fundamentals of the design approach itself. For example, the AISC s Equation F2-4 for lateral-torsional buckling stress is: 38

39 F cr 2 2 Cbπ E Jc L b = (2-4) L Sh b x o rts rts AISC allows the approximation of the term under the square root to be taken equal to.. that approximation is reasonable for compact sections. However for very slender sections it blows up (~5. for sections in this study). Figure 2-23 provides the change in AISC s results for the W4FI (λ e =.34) depending on whether or not the approximation suggested for the lateraltorsional buckling stress (Eq. F2-4) is utilized it is clear the use of this approximation must be done with care..8 (a) AISC AISI DSM ABAQUS.8 (b) M n /M p.6 M n /M p (fy/fcrl) (fy/fcrl).5 Figure 2-23 Beam results of W4FI study group at λ e =.34; (a) AISC without Eq. F2-4 s approximation, (b) AISC with Eq. F2-4 s approximation Figure 2-24 through Figure 2-27 are re-presentations of Figure 2-8 through Figure 2-2, but with the exact computation of the lateral-torsional buckling stress (Eq. F2-4) utilized. 39

40 W4FI W4FR M n /M p W36FR AISC AISI DSM ABAQUS 2 3 W36WI M n /M p (f y /f crl ).5 (f y /f crl ).5 Figure 2-24 Results of beam parametric study for 4 study groups (λ e =.6) M n /M p (f y /f crl ).5 M n /M p.5 AISC AISI.5 DSM 23 ABAQUS 2 3 (f y /f crl ).5 Figure 2-25 Results of beam parametric study for 3 design methods (λ e =.6) 4

41 W4FI W4FR M n /M p W36FR AISC AISI DSM ABAQUS 2 3 W36WI M n /M p (f y /f crl ) (f y /f crl ).5 Figure 2-26 Results of beam parametric study for 4 study groups (λ e =.34) M n /M p (f y /f crl ).5 M n /M p.5 AISC.5 AISI 23 DSM ABAQUS 2 3 (f y /f crl ).5 Figure 2-27 Results of beam parametric study for 3 design methods (λ e =.34) 4

42 AISI s Effective Width Method is overall the best performer in comparison with the FE results. However, the method is unconservative for long beams with locally slender webs (see the W36WI study at λ e =.34 of Figure 2-8). Note, as per AISI for any section which is subject to lateral-torsional buckling (such as those studied here) the capacity is limited to M y as shown. The DSM results for beams are in excellent agreement at all lengths when the flange and web slenderness vary at a fixed ratio (the W4FR and W36FR studies). The method has smooth transitions in all ranges of local slenderness. However, when one of the elements becomes significantly more slender than its neighbor DSM assumes the entire cross-section capacity degrades and this assumption becomes excessively conservative particularly for the W36WI cases, though less so than AISC. Note, multiple curves are presented for DSM in Figures 2-5, 2-7, and 2-9 because of the normalization to M p (as opposed to M y ) and further the inelastic bending provisions allowing strengths up to M p, as proposed for DSM and currently under ballot at AISI, are utilized here Overall AISC s solutions are overly approximate for locally slender sections and deserve improvement, particularly for flanges (unstiffened elements). AISI s effective width, while the most complicated of the methods, appears to provide the most accurate solution, particularly for braced (stub) columns. The simplicity of DSM is obvious in the expressions and the curves, but the elastic web-flange interaction assumed in the method is not always realized. DSM provides a 42

43 consistently conservative, and conceptually simple prediction method that is worthy of further study. 2.6 Summary and Conclusion The design of locally slender steel cross-sections may be completed by a variety of methods. For braced (short) columns and beams, design expressions in common notation are provided for the AISC Specification, the AISI Specification (effective width method) and DSM the Direct Strength Method (as adopted in Appendix of the AISI Specification as an alternative design procedure). The key parameters, found throughout all 3 design methods, are the elastic local (element, or member) buckling stress and the material yield stress. The design expressions indicate significantly different solution methodologies to this common problem, particularly for beams. A parametric study of braced (short) columns and beams is conducted with nonlinear finite element models in ABAQUS, deformed to collapse, and compared with the AISC, AISI, and DSM design predictions. The parametric study focuses on W4 and W36 sections, where through modification of element thicknesses, the flange slenderness, and/or web slenderness are systematically varied (from compact, to noncompact, to slender in the parlance of AISC). The results indicate that AISC is overly conservative when the flange is slender, AISC s assumption of little to no post-buckling reserve in unstiffened elements is not borne out by the analysis. AISI s effective width method is a 43

44 reliable predictor, only for the beam studies does AISI provide overly conservative solutions when the web is compact but the flange slender. DSM provides reliable predictions when both the flange and web slenderness vary together, but is overly conservative when one element is significantly more slender than another. Additional work on long beams and columns with localglobal interaction is underway. 44

45 3 Strain Distribution in Locally Slender Structural Steel Cross- Sections 3. Introduction The work presented herein is part of the continuing effort towards fully understanding the local stability, including the beneficial web-flange interaction, of structural steel. Through describing and analyzing a series of finite element (FE) analysis, efforts in Progress Report #3 and this Progress Report #4 showed comparisons of three design methods for locally slender steel short beams and stub columns; (i) AISC, and two methods from cold-formed steel specifications which focus on locally slender cross-sections: (ii) AISI-Effective Width, and (iii) AISI-Direct Strength Method (DSM). It was shown that in AISC, AISI, and DSM global column buckling is predicted using the same (single) expression. However, local-global interaction is handled by the Q-factor method in AISC, the unified method in AISI, and a variation of the unified method in DSM. In all cases the global strength is reduced due to local cross-section slenderness. The Q- factor approach reduces the strength and increases the long-column slenderness to arrive at its reduction. The unified method uses the effective area of the column at the long column buckling stress. DSM uses a similar approach. The underlying mechanics of a locally unstable cross-section at failure involves a complex nonlinear stress-strain state in the cross-section. Next generation design methods should at least in part reflect this stress-strain state in 45

46 their predictions; Q-factor approach does not, while the unified method simplifies the distributions based on elements of the cross-section. We seek here a simple means to utilize knowledge of the complete cross sectional stability (including element interactions) to predict this fundamental underlying stressstrain distribution. To develop what at its heart is essentially a semi-empirical method, we have used nonlinear FE collapse analysis to generate information on the stress-strain state at collapse for locally slender steel cross sections. The nonlinear FE analysis parameter study, using ABAQUS, that was used in Progress Report #3 and this Progress Report #4 for the purpose of understanding and highlighting the parameters that lead to the divergence between the capacity predictions of the different design methods, lead to the establishment of a database of failure mechanisms of W-sections at different element slenderness ratios. The main objective of this current work is to closely examine the strain distributions observed in the failure mechanisms of the FE parameter study database, and compare them to finite strip analysis results using CUFSM, as well as to theoretical distributions. The final goal of this research is to propose improvements to DSM so it may be applied to hot-rolled structural steel with locally slender cross-sections. The stress distributions and the strain distributions of all sections are examined along the length of the member as well as through the thickness of the elements. Distributions throught the thickness of the elements are examined at different levels; at the top, mid-thickness, and bottom, as well as the average 46

47 through the through the thickness. Figure 3- shows the average stress distribution through the thickness for the four study groups at different thicknesses. Avg str at different sections for W4arstress34 Avg str at different sections for W4afstress (a) (b) thin original thick thin original thick Avg str at different sections for W36arstress34 Avg str at different sections for W36awstress (c) thin original thick (d) thin original thick Figure 3- Average stress distributions through the thickness for the four study groups at different thicknesses: (a) W4FR, (b) W4FI, (c) W36FR, and (d) W36WI. 47

48 Figure 3-2 shows the average strain distribution through the thickness for the four study groups at different thicknesses. x -3 Avg str at different sections for W4arstrain34-5 x -3 Avg str at different sections for W4afstrain34-5 (a) (b) thin original thick thin original thick x x x -3 x -3 Avg str at different sections for W36arstrain34-5 x -3 Avg str at different sections for W36awstrain34-5 (c) (d) thin original thick thin original thick x x -3 x x -3 Figure 3-2 Average strain distributions through the thickness for the four study groups at different thicknesses: (a) W4FR, (b) W4FI, (c) W36FR, and (d) W36WI. 48

49 Figure 3-3 shows stress distribution at the mid-thickness for the four study groups at different thicknesses. Avg str at different sections for W4arstress34 Avg str at different sections for W4afstress (a) (b) thin original thick thin original thick Avg str at different sections for W36arstress34 Avg str at different sections for W36awstress (c) (d) thin original thick thin original thick Figure 3-3 Stress distributions at mid-thickness for the four study groups at different thicknesses: (a) W4FR, (b) W4FI, (c) W36FR, and (d) W36WI. 49

50 Figure 3-4 shows strain distribution at the mid-thickness for the four study groups at different thicknesses Avg str at different sections for W4arstrain34 x -3 Avg str at different sections for W4afstrain34-5 (a) (b) thin original thick thin original thick x x -3 Avg str at different sections for W36arstrain34-5 x -3 Avg str at different sections for W36awstrain34-5 (c) (d) thin original thick thin original thick x x x -3 Figure 3-4 Strain distributions at mid-thickness for the four study groups at different thicknesses: (a) W4FR, (b) W4FI, (c) W36FR, and (d) W36WI. 5

51 4 References AISC (25). Specification for Structural Steel Buildings, American Institute of Steel Construction, Chicago, IL. ANSI/ASIC AISI (27). North American Specification for the Design of Cold-Formed Steel Structures, Am. Iron and Steel Inst., Washington, D.C., AISI-S. Barth, K.E. et al (25). Evaluation of web compactness limits for singly and doubly symmetric steel I-girders, Journal of Constructional Steel Research Dinis, P.B., Camotim, D. (26). On the use of shell finite element analysis to assess the local buckling and post-buckling behavior of cold-formed steel thin-walled members, III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering C.A. Mota Soares et.al. (eds.) Lisbon, Portugal, 5 8 June 26. Earls, C.J. (2). Constant moment behavior of high-performance steel I-shaped beams, Journal of Const. Steel Research 57 (2) Galambos, T.V., Ketter, R.L. (959). Columns under combined bending and thrust, Journal of Engineering, Mechanics Division, ASCE ; 3. Galambos, T.V. (998). Guide to Stability Design Criteria for Metal Structures. 5 th ed., Wiley, New York, NY, Jung, S., White, D.W. (26). Shear strength of horizontally curved steel I-girders finite element analysis studies, Journal of Constructional Steel Research 62, 26: Kim, S., Lee, D. (22). Second-order distributed plasticity analysis of space steel frames, Engineering Structures 24, 22: Schafer, B.W., Ádány, S. (26). Buckling analysis of cold-formed steel members using CUFSM: conventional and constrained finite strip methods. Proceedings of the Eighteenth International Specialty Conference on Cold-Formed Steel Structures, Orlando, FL Schafer, B.W., Seif, M. (28). Comparison of Design Methods for Locally Slender Steel Columns SSRC Annual Stability Conference, Nashville, TN, April 28. 5

52 Seif, M., Schafer, B.W. (27). Cross-section Stability of Structural Steel. American Institute of Steel Construction, Progress Report No.. AISC Faculty Fellowship, July 27. Seif, M., Schafer, B.W. (28). Cross-section Stability of Structural Steel. American Institute of Steel Construction, Progress Report No. 2. AISC Faculty Fellowship, April 28. Seif, M., Schafer, B.W. (29a). Elastic Buckling Finite Strip Analysis of the AISC Sections Database and Proposed Local Plate Buckling Coefficients Structures Congress, Austin, TX, April 29. Seif, M., Schafer, B.W.(29b). Finite element comparison of design methods for locally slender steel beams and columns SSRC Annual Stability Conference, Phoenix, AZ, April 29. Seif, M., Schafer, B.W. (29c). Cross-section Stability of Structural Steel. American Institute of Steel Construction, Progress Report No. 3. AISC Faculty Fellowship, April 29. Shifferaw, Y., and Schafer, B. W. (28). "Inelastic bending capacity in cold-formed steel members." Report to American Iron and Steel Institute Committee on Specifications, July 28. White, D.W. (28). Unified flexural resistance equations for stability design of steel I- section members: Overview. ASCE, Journal of Structural Engineering, 34 (9)

53 Appendix A : NRC Research Proposal This Appendix shows a copy of a research proposal titled Multi-scale Structural Stability under Realistic Fire Loading. The proposal was submitted to the National Research Council (NRC), as part of a post-doctoral fellowship application, and it aims to extend this research where the effect of realistic fire loading scenarios on locally slender structural steel members will be studied. Multi-scale Structural Stability under Realistic Fire Loading Summary: Stability is paramount in the performance of steel structures under fire. The work proposed here advances a multi-scale approach whereby heat transport and stability are addressed at the cross-section level, and then coupled to member-level models to enable predictions of complete building systems under fire. The work is significantly aided by recent advances in the efficient prediction of locally unstable hot- rolled steel cross-sections that formed the proposer s Ph.D. work, and will be validated with facilities uniquely available at NIST. A. Problem: The catastrophic collapse of the World Trade Center buildings (WTC Towers and 2, and Building 7) brought the nation s attention to the vulnerability of our structures to extreme loading conditions, especially fire. Fire 53