Direct Strength Design for Cold-Formed Steel Members with Perforations
Solicited proposal submitted to:
The American Iron and Steel Institute (AISI)
Principal Investigator:
Ben Schafer
Assistant Professor
Department of Civil Engineering
The JohnsHopkinsUniversity
Baltimore, Maryland
June 2004
Abstract
The recently adopted Direct Strength Method has opened the potential for the Specification to encourage the creation of highly optimized and efficient cold-formed steel members by integrating numerically determined elastic buckling loads directly into the design process. A significant current shortcoming of Direct Strength design is that it does not apply to members with perforations (holes). Perforations are commonly used in nearly all cold-formed steel applications and therefore extension of Direct Strength design to members with holes is sorely needed. A three phase (year) proposal is proposed herein to address this shortcoming. The first phase focuses on using existing data coupled with finite element analysis to efficiently provide a means to bring conventional members with holes into the Direct Strength procedures. Phases 2 and 3 focus on more in-depth theoretical challenges and experimental validation so that a rational analysis method can be provided for any cold-formed steel member with perforations, including unique configurations such as flanged holes, slotted holes, and the like. Specific budgets are provided for the three separate phases. A rational analysis method for designing any cold-formed steel member with holes stands to significantly increase the flexibility of cold-formed steel applications and allow industry and practicing engineers to take full advantage of the potentials of the Direct Strength design method.
1.Background
Perforations in cold-formed steel members come in a variety of different forms, as shown in Figure 1. The two most common types of perforations are isolated holes, and patterned holes. Isolated holes include load bearing studs with holes for services and/or bridging (Figure 1a), and isolated holes in joists, purlins, and girts (Figure 1a). Patterned holes include those in pallet rack post uprights for tabbed beam connections (Figure 1b). Other more unique situations exist for cold-formed steel members with perforations; including isolated holes that include formed-in flanges for increased stiffness around the hole (Figure 1c) and new ideas for better thermal behavior using small patterned perforations (Figure 1d).
(a) Isolated holes in studs, may also be found in joists, purlins, and girts, several holes may exist along the length, though spacing is generally far apart / (b)Patterned perforations in rack posts, and in some rack beams, in addition other specialty industries include such perforations (picture from UNARCO product catalog)(c) Isolated flanged holes in joists, purlins; holes stiffened through transverse and possibly longitudinal external stiffeners also relevant (picture from Dietrichweb site) / (d) Specialty perforations as found in slitted cross-sections used in Europefor improved thermal performance (Sections tested and analyzed in Kesti 2000, picture from Kesti 2000)
Figure 1 Examples of perforation patterns found in cold-formed steel systems
Current Specification (NAS 2001) methods for designing members with holes are limited. For isolated circular holes in lipped channel columns the experiments of Ortiz and Peköz (1981) are used to define a reduced effective width for geometries in the tested range only. Isolated holes in lipped channel beams follow Shan et al. (1994), Langan et al. (1994), Uphoff (1996), and Deshmukh (1996) and are used to define a hole size which is small enough to be ignored – design for other hole sizes conservatively require the effective width to be calculated assuming the hole creates unstiffened elements on either side, i.e., the classic solution for a long plate simply supported on three sides and free on the third (the hole) is used for finding the effective width. Alternatively, and with much greater accuracy for local buckling, stub column testing may be performed. This is common practice for members with patterned perforations only. However, testing is only done at short lengths, and distortional buckling which dominates at intermediate lengths, is problematically not considered (Hancock et al. 1994). Thorough compilations of other relevant work are provided in Kesti (2000) and Shanmugam and Dhanalakshmi (2001). Existing methods have an intentionally limited range of applicability, but may be excessively conservative in some situations and ignore important limit states in other situations. A consistent design approach is needed.
Perforations (holes) are a common need in cold-formed steel systems; whether for services in buildings, for convenient connection of other members, or other uses. The main Specification (NAS 2001) covers specific cases of members with holes, but the use is effectively limited to the testing performed. Further, in some cases the Specification methods are known to be overly conservative. The Direct Strength Method (DSM) was created to provide a means to efficiently design highly optimized cross-sections; cross-sections that could not be readily handled by the design method of the Specification. Extension of the Direct Strength Method is needed in order to provide a general method for the efficient design of members with perforations (holes).
2.Elastic buckling and the Direct Strength Method
The basic premise of the Direct Strength Method may be expressed for a column as:
Pn=f(Pcr,Pcrd,Pcre,Py)(Eq. 1)
where:
Pn=nominal column strength
Pcr=axial load at which elastic local buckling occurs
Pcrd=axial load at which elastic distortional buckling occurs
Pcre=axial load at which global column buckling (flexural, torsional, flexural-torsional occurs)
Py=axial load at which yielding occurs (squash load).
The development of functional expressions (the “f” of Eq. 1) preceded from considering the large amount of available experimental data and empirically fitting limit-state specific strength curves for global buckling, distortional buckling, and local-global buckling.
For members without holes the primary means for examining the elastic buckling loads employed the classical finite strip method (FSM), e.g., CUFSM. However, this method cannot discretely handle the presence of holes without significant modifications. Thus, when considering members with holes the more general finite element method (FEM) using shell elements is typically preferred. Recent work by Sarawit (2003) on patterned perforations for rack posts (Figure 1b) and by Kesti (2000) on small patterned perforations for a unique stud with better thermal properties (Figure 1d) has shown how the finite element method may be used for elastic buckling prediction of members with holes. For example, Figure 2 shows an open source tool developed by Sarawit for isolated plates and demonstrates the reduction in the elastic local plate buckling coefficient. Figure 3extends the analysis to full members with a general purpose FE package, ABAQUS, for elastic buckling in the local, distortional, and global buckling modes.
Figure 2 and Figure 3 demonstrate, at least visually, that members with holes have elastic buckling modes similar to members without holes. But, what these figures fail to show is (1) many more buckling modes may exist – particularly for larger holes and (2) determining which of the myriad of buckling modes should be defined as “local” or “distortional” or “global” without the aid of the half-wavelength vs. load factor plot that finite strip provides is difficult at best and impossible at worst. The finite element method is less restricted than the finite strip method, with this generality comes much complication. A methodology is needed to post-process finite element analysis (FEA), similar to that used in finite strip analysis, so that specific modes can be identified and even isolated. A consistent approach to identifying elastic buckling modes in members with holes is a key challenge in the work proposed herein.
Figure 2 FE predications of elastic plate buckling coefficients for isolated plates with rack post patterned perforations completedby Sarawit 2003
Figure 3FE predictions of the elastic buckling modes of rack post columns with patterned perforations (a) Local (b) Distortional (c) Flexural-torsional, completed by Sarawit 2003
3.Ultimate strength and the Direct Strength Method
Development of the Direct Strength Method relied on the wide body of available experimental data on members without holes. Strength curves were selected that connected elastic buckling loads to ultimate strength, for example, for a column in distortional buckling the expression is:
(Eq. 2)
Consider the rack post of Figure 3b, the question for members with holes is, if Pcrd is properly calculated to reflect the reduction due to the holes will the same strength expression (Eq. 2) be applicable to this member? It is certainly possible that this will be the case, but it is not guaranteed, as theoretically the presence of the hole will influence the elastic buckling response (represented by Pcrd) and the post-buckling response (handled through Eq. 2). While a clear means exists to account for the change in elastic buckling, and thus the reduced Pcrd due to the hole, needed modifications to the Direct Strength expressions, such as Eq. 2, remain unknown.
Use of the available experimental data on members with holes from Ortiz and Peköz (1981), Shan et al. (1994), Langan et al. (1994), Uphoff (1996), Deshmukh (1996) and others available in the literature will be crucial to initially evaluating the Direct Strength expressions. For each existing test (1) general purpose finite element models will have to be built to determine the elastic buckling loads in the presence of the holes (2) the FE elastic buckling modes will have to be mapped appropriately to local, distortional, and global buckling values and only then can we (3) determine the accuracy of existing Direct Strength expression and evaluate the necessity of potential modifications to those expressions.
4.Statement of Work and Work Plan
Objective: Development of a general design method for cold-formed steel members with perforations.
The developed design method is envisioned as an extension to the Direct Strength Method. While it is believed that completion of the entire project plan is required in order to fully implement the Direct Strength Method for members with perforations it is recognized that funding may not be available for this complete effort immediately. Therefore, the project has been specifically broken into three years, each year with separate tasks and goals so that the funding and effort levels can be matched appropriately.
Year 1: Benefiting from existing experimental data$66,056
Objective:Efficiently extend DSM to members with holes that are already covered in the Specification and insure that DSM, at least in a limited sense, can be applied to products already on the market: metal studs, rack posts, etc., at least where test data is already readily available.
Survey current industry use on members with perforations
Evaluation of existing experimental results
- Gather all existing experimental data
- Elastic FEA of existing experimental results
- Identification of local, distortional, and global modes for perforated members
- Evaluation of existing Direct Strength expressions
Design examples
Specification ballot on Direct Strength Method for conventional members with perforations
Year 2: Tackling theoretical challenges and extending experiments$62,617
Objective: Formalize identification of buckling modes for members with holes. Increase understanding of post-buckling mechanisms for members with holes. Extend DSM as a rational analysis method for any cold-formed member with holes; including unique members with flanged or stiffened holes.
Identification/isolation of buckling modes in a general FEA
- Extension of Adany and Schafer work on modal identification from finite strip to FEA
Extension of existing experimental results
- Nonlinear FEA models of experiments to verify and validate implemented FEA model
- Additional parametric studies for ultimate strength of members with perforations
(focus on unusually large holes, boundary between isolated holes and patterned holes, etc.)
- Examination/extension of Direct Strength expressions
- Detailedevaluation of post-buckling response (using nonlinear FEA)
Evaluation of flanged or stiffened holes
- Collection of existing experimental data, as available
- Complete elastic buckling and nonlinear FEA of typical sections
- Examination/extension of Direct Strength expressions
- Design examples
Specification ballot on members with flanged or stiffened holes
Research tool for the direct identification of buckling modes for members with holes
Draft ballot on rational analysis method for any member with holes
Year 3: Experimental validation and developing open-source tools$81,104
Objective: Experimentally validate DSM as a rational analysis method for any cold-formed member with holes; including unique members with flanged or stiffened holes. Develop open-source tools that engineers may use for easy application of DSM to members with holes.
Experiments on beams and columns with holes
Beam testing would use the existing rig at JHU and column testing would employ the 100 kip universal testing machine. If the DSM beam-column proposal is funded this work would have significant synergy with that proposal – and the developed testing rig from that proposal could be used for the testing here. Since only a 1 year testing program is envisioned the FEA work and existing tests would be used to carefully select a small subset of cross-sections and hole patterns of interest. A total of approximately 36 tests is envisioned, 3 tests at each of 3 lengths for four different cross-sections, this allows the influence of a critical hole to be gauged for local, distortional, and global buckling.
Development of an open-source computational tool for members with perforations
- stand-alone tool for elastic buckling of members with perforations and/or
- post-processing tools for general purpose FE programs (ANSYS, ABAQUS, etc.)
- integrate work on modal identification work from year 2
Design examples
Ballot on rational analysis method for any member with holes
Open-source computational tool for members with holes
The order of the work in years 2 and 3 is somewhat more flexible than year 1. The project may be thought of as consisting of 3 phases, each with the proposed budget provided.
As discussed in the elastic buckling section previously, a key theoretical challenge, addressed in year 2 of the work, is the identification/isolation of individual buckling modes from a general purpose FEA; currently no tool exists for this purpose. Consider a simple model of a lipped channel (without holes) as shown in Figure 4. At a given length, the number of total degrees of freedom determines the number of possible modes. For Figure 4
FSM: 15 nodes × 4 DOF/node = 60 modes, and
FEA: 15 nodes in a section × 9 sections along the length × 4 DOF/node = 540 modes.
In general the situation is actually much worse than this calculation indicates, because FEA is performed on a much longer length model with many more elements. Members with holes require the convenience of FEA to incorporate discontinuities (like holes) along the length but methods are needed for post-processing FEA runs to effectively deal with the large number of modes and to identify modes of actual interest.
Figure 4 FEM/FEA vs. FSM /
Figure 5 FSM analysis with modal constraints
Recently Adany and Schafer (2004) have shown how to use the mechanical assumptions of Generalized Beam Theory (Davies et al. 1994, Silvestre and Camotin 2002a,b) in order to restrain general purpose analysis tools like FSM to the analysis of a single mode. As an example, FSM analysis of a lipped channel in pure compression is shown in Figure 5, the “all-mode” curve is the traditional FSM analysis; note that distortional buckling cannot be readily identified in this curve because no minimum exists. However, in the subsequent curves we have restricted the FSM analysis via a series of mechanical assumptions (deformation constraints) and generated curves which are unique to the traditional buckling modes. These curves provide exact definitions for the individual buckling modes and allow isolation of the modes.
Extension of the technique employed by Adany and Schafer to FEM is possible, and would provide a means to isolate the FEA on individual modes. This would allow one to directly identify the influence of a hole, or series of holes on a particular buckling mode. Without such a tool proper identification of the modes, given a myriad of possible choices in the FEA and the subtle differences between the modes, would rely solely on the experience of the analyst. Such a method is possible in some cases, but can introduce significant approximation and a loss of transparency and generality for the approach.
5.Impact on Industry
The metal stud and rack manufacturing industries, which use perforations in most of their products, need this research to bring the potential short- and long-term advantages of the Direct Strength Method to their industries. Unique innovations involved with perforations in members, such as flanged holes, and slotted holes need this research to provide a recognized design approach in the Specification.
Cold-formed steel members require perforations. The development of the Direct Strength Method provides a means to highly optimize cold-formed sections. Extending the application of the Direct Strength Method to members with holes provides a means for industry to move towards highly optimized sections, more readily embrace high strength steels, and increase the flexibility of cold-formed steel applications.
6.Work Product
Progress reports will be provided to AISI every 6 months at the AISI-COS meetings. A report will be provided upon completion of the project. Upon review and comments from the AISI a final report will be provided. Test data and reports will be made available in electronic format to AISI and other researchers.
7.Budget
A summary of the proposed budget is provided in Table 1. Support for a student and 1 month of faculty salary is requested in all 3 years. In year 1, funding for a computer is requested, and in year 3 funding for performing experiments is requested. A detailed breakdown of the budget is provided in the Appendix.