TABULATED LOCAL AND DISTORTIONAL ELASTIC BUCKLING SOLUTIONS FOR STANDARD SHAPES

Transcription

1 test TECHNICAL NOTE On Cold-Formed Steel Construction Cold-Formed Steel Engineers Institute Washington, DC $5.00 TABULATED LOCAL AND DISTORTIONAL ELASTIC BUCKLING SOLUTIONS FOR STANDARD SHAPES Summary: This note provides elastic buckling moments and forces for local and distortional buckling of typical standard sections such as those in the AISI S201 Standard for Cold-Formed Steel Framing Product Data and the Steel Stud Manufacturers Association (SSMA) Product Technical Information Catalog. These tabulated values allow designers to quickly examine and evaluate the use of the direct strength method (DSM) for design. Note that basic information on DSM is discussed in CFSEI Technical Note G102. Disclaimer: Designs cited herein are not intended to preclude the use of other materials, assemblies, structures or designs when these other designs and materials demonstrate equivalent performance for the intended use; CFSEI documents are not intended to exclude the use and implementation of any other design or construction technique. Starting in 2004 Appendix 1 of the North American Specification for Cold-Formed Steel Structural Members (AISI-S100)[1] introduced an alternative design method for strength prediction of beams and columns: the Direct Strength Method (DSM). The basic premise of DSM and the advantages of the method are discussed in CFSEI Tech Note G The objective of this Note is to provide the elastic buckling forces and moments for local and distortional buckling of standard sections. These buckling values are at the heart of the DSM approach. By providing the complete values in tabular form an engineer can more readily examine the application of DSM to standard member shapes., kips Conventional FSM solution Distortional L cr =12.65, P cr =8.72 DSM requires that the elastic buckling load or moment be known in local, distortional, and global buckling modes. Specifically, P crl or M crl (local buckling), P crd or M crd (distortional buckling), and P cre or M cre (global or Euler buckling) must be known. DSM does not specify how these elastic buckling values should be calculated since numerous methods exist as discussed in the DSM commentary found in Appendix 1 of AISI-S100 [1]. By far the most popular method is the conventional, or semianalytical, Finite Strip Method (FSM) approach, and a sample result from FSM is shown in Figure 1. FSM is a variant of the finite element method. Software is freely available for FSM calculation as discussed in CFSEI Tech Note G Provided in this Tech Note are the elastic buckling loads and moments for all standard Stud Sections [2] calculated using FSM. buckling load L cr =2.02, P cr = ,8.72 Global Figure 1: The cross-section stability signature curve from an FSM solution 5 2.0,5.72 Local half-wave length, inch Cold-Formed Steel Engineers Institute 1 TECH NOTE G103-11a June, 2011

2 Calculation of elastic buckling via FSM is generally a straightforward task; however some small challenges may still remain for the analyst. In the ideal case, as demonstrated in Figure 1, the results of the FSM analysis provide unique minima for local and distortional buckling and no judgment is required. However, if either or both minima are indistinct the FSM solution may be characterized as having non-unique minima. Studies show that the problem of non-unique minima is common for standard stud sections [3]. This is in large part due to the use of relatively small lip lengths in these shapes. Recently, the FSM approach has been extended to address this problem with the creation of the constrained finite strip method (cfsm) which allows each of the buckling modes to be analyzed separately i.e., as pure modes. Figure 2 illustrates a conventional FSM solution with an indistinct distortional mode along with separate pure mode cfsm solutions for local, distortional, and global buckling., kips-in Critical Load M cr FSM solution, rounded corner m odel cfsm solution, straight-line m odel Distortional X: X FSM@cFSM-L cr =19.31, M crd =9.24 Local X: FSM@cFSM-L cr =4.09, M crl = , Half-wave length, inch Figure 2 : Signature curve augmented with pure mode cfsm solution and illustration of the proposed cfsm L cr solution to identifying non-unique minima Two basic issues remain, hampering direct use of the cfsm pure mode solution: (1) DSM s strength expressions are calibrated to the conventional FSM minima instead of pure mode solutions from cfsm (which are generally a few percent higher), and (2) cfsm can not handle rounded corners and still provide a meaningful separation of local and distortional buckling. From a strictly practical standpoint all cross-sections have rounded (not sharp) corners. Round corners may be modeled in FSM directly as shown in Figure 3 (a), or ignored and a straight-line model utilized as shown in Figure 3 (b). Numerical studies in [3] show that although rounded corners typically have only a modest influence on the local and distortional buckling solutions (and may even be beneficial) the decrease in gross properties and hence global buckling as well as yield loads (and moments) can Figure 3: Cross-section models and dimension for use in FSM analyses H r t B D H t B D (a) Rounded corner model TECH NOTE G103-11a June, 2011 (b) Straight-line model 2 Cold-Formed Steel Engineers Institute

3 be significant. However, FSM models with rounded corners are even more likely to suffer from indistinct minima thus a general method that accounts for corners, but still generates the needed local, distortional, and global buckling solutions is needed. To address these issues a two-step procedure has been adopted for determining the elastic buckling loads and moments. Step 1: the analyst develops a rounded corner model of the section and runs a conventional FSM model. If unique minima exist, then stop, and use those values. Step 2: the analyst develops a straight-line model of the section and runs cfsm pure mode solutions for local and distortional buckling, only for the purpose of determining the half-wavelengths (L cr ) at which the modes occur. Non-structural stud sections (unpunched) The elastic buckling load (or moment) is determined from the conventional FSM with round corners (Step 1) model at the L cr identified in the Step 2 model. An abbreviation for this solution method is FSM@cFSM-L cr, which is illustrated for a 550S stud section under axial compression in Figure 2. The validity of this approach is fully explored in [3] and shown to provide consistent results. Using the FSM@cFSM-L cr approach and distortional elastic buckling load and moment (both major and minoraxis) were determined for the structural and non-structural sections in AISI S201. [2]. Results are provided in the following tables. Notes: P crl and M crl : the critical loads and moments of local buckling; P crd and M crd : the critical loads and moments of distortional buckling; The highlighted italics are those with non-unique minima and replaced by the recommended values from the suggested method. These values do not include holes/punchouts, elastic buckling and design methods that include holes/punchouts are currently under study and will be the topic of a future Tech Note. Cold-Formed Steel Engineers Institute 3 TECH NOTE G103-11a June, 2011

4 Structural stud sections (unpunched) TECH NOTE G103-11a June, Cold-Formed Steel Engineers Institute

5 The P cr and M cr values provided here along with the DSM provisions of Appendix 1 of AISI-S100 provide an efficient means to explore DSM s strength predictions for standard steel framing shapes. See Tech Note G for further discussion of DSM. In addition, the new distortional buckling provisions of the Main Specification (C3.1.4 for beams, and C4.2 for columns in AISI-S100-07) require calculation of the distortional buckling stress. M crd /S g and P crd /A g could be utilized in those sections, with M crd and P crd taken from the table above. See Tech Note G for more discussion on distortional buckling. It is worth noting that the tables include the case of distortional buckling for minor-axis bending of standard sections when the lips are in compression. This limit state is not explicitly covered in the main Specification of AISI- S [1]. Thus, inclusion of this limit state in design is a grey area. However, the AISI Design Manual has long recommended a design approach for the bending capacity of standing seam panels when the uprights of the panel are in compression a mode of buckling now known as distortional buckling, and similar to minor-axis distortional buckling of stud sections. In the current version of the AISI Design Manual [4] use of the Direct Strength Method AISI-S Appendix 1 [1] is recommended in such situations. The minor-axis values tabled here provide this calculation. As in distortional buckling about the major-axis, distortional buckling about the minor-axis may be mitigated by sheathing or other attachments. Tech Note G provides guidance on accounting for such restraint in majoraxis distortional buckling and may be readily extended for minor-axis distortional buckling. Conclusions This Tech Note provides local and distortional elastic buckling values in compression and major and minoraxis bending for common standard stud sections. The values are intended to provide cold-formed steel engineers with a means to evaluate the potential application of the Direct Strength Method to their designs. References [1] AISI S North American Specification for the Design of Cold-Formed Steel Structural Members Edition. American Iron and Steel Institute, Washington, D.C., USA, [2] SSMA (2006). Product Technical Information Catalog, 2006 International Building Code (IBC) Edition. Steel Stud Manufacturers Association, Chicago, IL. [3] Li, Z. and Schafer, B.W. Application of the finite strip method in cold-formed steel member design. Journal of Constructional Steel Research, (8-9); p [4] AISI (2008). Cold-Formed Steel Design Manual (D100-08). American Iron and Steel Institute, Washington, D.C. [5] AISI S Standard for Cold-Formed Steel Framing Product Data Edition. American Iron and Steel Institute, Washington, D.C., USA, Primary Authors of this Technical Note: Zhanjie Li Benjamin W. Schafer Cold-Formed Steel Engineers Institute 5 TECH NOTE G103-11a June, 2011