Appendix 1: Design of Cold-Formed Steel Structural Members by … · 2010-09-22 · DRAFT - January 4, 2002 Œ COMMENTARY: Specification for the Direct Strength Method Appendix 1

DRAFT - JANUARY 7, 2002

COMMENTARY

DIRECT STRENGTH METHOD OF

COLD-FORMED STEEL DESIGN

Contents:

Preface ..................................................................................................... 1 1.1 General Provisions ...................................................................... 1

1.1.1 Applicability........................................................................ 1 1.1.1.1 Pre-qualified Columns..................................................... 2 1.1.1.2 Pre-qualified Beams ........................................................ 4

1.1.2 Elastic Buckling .................................................................. 6 1.1.2.1 Elastic Buckling Numerical Solutions ............................ 7 1.1.2.2 Elastic Buckling Hand Solutions................................... 20

1.1.3 Deflection Determination .................................................. 30 1.2 Columns .................................................................................... 30

1.2.1 Flexural, Torsional, or Flexural-Torsional Buckling ........ 33 1.2.2 Local Buckling .................................................................. 33 1.2.3 Distortional Buckling ........................................................ 33

1.3 Beams ........................................................................................ 34 1.3.1 Lateral-Torsional Buckling ............................................... 36 1.3.2 Local Buckling .................................................................. 36 1.3.3 Distortional Buckling ........................................................ 37

References ............................................................................................. 37

DRAFT - January 4, 2002 COMMENTARY: Specification for the Direct Strength Method

Appendix 1 - 1

COMMENTARY Appendix 1: Design of Cold-Formed Steel Structural Members by the Direct Strength Method

Preface The complexity of the main specification for cold-formed steel design is immediately apparent to all who use it. This complexity is justified by the rather advanced and optimized mechanical behavior that occurs in thin-walled cold-formed steel beams and columns. However, much of the complexity is related to performing detailed elastic buckling calculations, by hand, for the large variety of different buckling modes that exist in cold-formed steel members (much of Chapters B and C of the main Specification are elastic buckling calculations!). As members become more optimized the elastic buckling calculations become more complicated. However, elastic buckling of thin-walled members may be readily completed using numerical methods that are verifiable and robust. Therefore, a design method which employs accurate elastic buckling calculations determined by rational analysis has a chance of eliminating much of the complexity of current design procedures: the Direct Strength Method of this appendix is an attempt at such a method. Currently, the procedures, verification, and calibration for the Direct Strength method have only been completed for beams and columns, but work continues on extending this method to all cold-formed members in all loading cases. This commentary to Appendix 1 provides

• rational analysis procedures and guidelines for elastic buckling, • summary tables of all beams and columns examined, and • discussion and justification of the strength expressions employed.

The bulk of the commentary (section 1.1.2) is devoted to providing procedures and guidelines for elastic buckling calculation. The Specification allows rational analysis for elastic buckling calculation, and thus no specific methods are prescribed. This commentary provides complete hand methods for elastic buckling calculation that are applicable to conventional cold-formed steel members. Additionally, the specification provides background, discussion, examples, references and freely available software for elastic buckling calculation via numerical methods. The remaining commentary is organized to parallel the structure of Appendix 1.

1.1 General Provisions

1.1.1 Applicability Section A1.1 of the main Specification states that rational analysis methods are allowed when the composition or configuration of a component is such that calculations cannot be made in accordance with the main Specification. The provisions of Appendix 1 provide a calibrated rational analysis method for strength prediction of beams and columns which cannot be completed by the main Specification. The reliability of the method is insured by using the calibrated safety and resistance factors within set geometric limits only, and conservative safety and resistance factors for all other configurations. What about those members which ostensibly can be calculated by the main Specification? Can Appendix 1 be used to determine the strength of such members? Yes. Appendix 1 is unique when


Appendix 1 - 2

compared to the typical rational analysis methods of section A1.1(b), because it is explicitly a part of the actual Specification. Therefore, the methods of Appendix 1 are available for all cold-formed steel beams and columns. The applicability of Appendix 1 to all beams and columns implies that in some situations competing methods may exist for strength determination of a member: the main Specification, and Appendix 1. In this situation there is no preferred method. Either method may be used to determine the strength [resistance]. The fact that one method may give a greater or lower strength prediction in a given situation does not imply an increased accuracy for either method. The calibrated safety and resistance factors are designed to insure that both methods reach their target reliability, which for members is a β of 2.5.

1.1.1.1 Pre-qualified Columns An extensive amount of testing has been performed on concentrically loaded, pin-ended, cold-formed steel columns (Kwon and Hancock 1992, Lau and Hancock 1987, Loughlan 1979, Miller and Peköz 1994, Mulligan 1983, Polyzois et al. 1993, Thomasson 1978). Data from these researchers was compiled and used for calibration of the Direct Strength Method. The geometric limitations listed in the Specification are based on these experiments. A summary of the geometry used by each researcher follows:

h

b

d

θ

h

b

d θ

h

di

(a) lipped channel (b) lipped zed (c) lipped channel with web stiffener

and additional lip to lip restraints

(d) lipped channel with web stiffener (e) hat (f) rack sections

Figure 1.1-1 Cross-section of columns used for calibration

Table 1.1-1 Geometry of lipped channel columns h/t b/t d/t h/b d/b fy num

reference max min max min max min max min max min max minLoughlan (1979) 322 91 80 30 33 11 5.0 1.6 0.41 0.36 35 34 33Miller and Pekoz (1994) 170 46 38 18 8 5 4.6 2.5 0.32 0.19 53 44 19Mulligan (1983) 207 93 93 64 16 14 2.9 1.0 0.22 0.16 35 32 13Mulligan (1983) Stubs 353 65 100 33 22 7 3.9 0.7 0.23 0.16 60 29 24Thomasson (1978) 472 207 159 69 32 14 3.0 3.0 0.23 0.19 69 42 13totals 472 46 159 18 33 5 5.0 0.7 0.41 0.16 69 29 102

more lipped channels (subset of U. Sydney tests which fails in distortional mode)


Appendix 1 - 3

h/t b/t d/t h/b d/b fy numreference max min max min max min max min max min max minLau and Hancock (1988) 50 34 37 25 6 4 1.4 1.3 0.18 0.16 69 31 8Kwon and Hancock (1992) 111 110 83 82 6 4 1.3 1.3 0.08 0.05 86 86 5totals 111 34 83 25 6 4 1.4 1.3 0.18 0.05 86 31 13

Table 1.1-2 Geometry of lipped zed columns

h/t b/t d/t h/b d/b fy numreference max min max min max min max min max min max minPolyzois et al. (1993) 137 76 56.1 30 36.2 0 2.74 1.5 0.73 0 50 50 85

Table 1.1-3 Geometry of lipped channel with a web stiffener columns

(a) Additional restraints from lip to lip h/b h/t b/t d/t

max min max min max min max min countThomasson (1978) 3.1 3.0 489 205 160 68 33 14 46

di/d h'/tmax min max min count

no intermediate web stiffeners - - - - 14one intermediate web stiffener 0.94 0.39 222 91 16two intermediate web stiffeners 0.94 0.47 145 57 16

(b) Standard section lipped channel with a web stiffener, width (ws), and depth (ds) all steel fy = 86 ksi

h/t b/t d/t h/b ds/d ws/b numreference max min max min max min max min max min max minKwon and Hancock (1992) 110 105 82 82 13 6 1.3 1.3 2.8 1.4 0.1 0.1 6

Table 1.1-4 Geometry of hat columns

h/t b/t d/t h/b d/b fy numreference max min max min max min max min max min max minLau and Hancock (1988) 50 33 43 34 6 4 1.2 1.0 0.14 0.13 69 31 4

(subset of Lau and Hancock data which failed in distortional mode)

Table 1.1-5 Geometry of rack section columns lipped channel with an outstand (width b2)

h/t b/t d/t h/b b2/d fy numreference max min max min max min max min max min max minLau and Hancock (1988) 46 34 20 12 8 5 2.7 2.3 2.0 1.6 69 31 9lipped channel with an outstand (width b2) and a return lip (width d2)

h/t b/t d/t h/b b2/d d2/d numreference max min max min max min max min max min max minLau and Hancock (1988) 51 34 22 12 7 6 2.9 2.1 2.0 1.6 0.3 0.3 8

(subset of Lau and Hancock data which failed in distortional mode) It is the intent of this Specification that as more cross-sections are verified for use in the Direct Strength method that these tables and sections would be augmented with new information. Those companies with proprietary sections may wish to perform their own testing and follow the details of Chapter F of the main Specification to independently justify the use of higher φ and lower Ω factors for a particular cross-section. Alternatively, members which fall outside the scope of this testing may still use the method, but with the reduced φ and increased Ω factors consistent with any rational analysis method.


Appendix 1 - 4

1.1.1.2 Pre-qualified Beams An extensive amount of testing has been performed on laterally braced beams (Cohen 1987, Ellifritt et al. 1997, LaBoube and Yu 1978, Moreyara 1993, Phung and Yu 1978, Rogers 1995, Schardt and Schrade 1982, Schuster 1992, Shan 1994, Willis and Wallace 1990) and on hats and decks (Acharya 1997, Bernard 1993, Desmond 1977, Höglund 1980, König 1978, Papazian et al. 1994). Data from these researchers was compiled and used for calibration of the Direct Strength Method. The geometric limitations listed in the Specification are based on the experiments performed by these researchers. A summary of the geometry used by each researcher follows (beams are bent about the horizontal axis, as oriented in the figures below):

Cohen (1987)

I II

Ellifritt et al. (1997) Laboube and Yu (1978)

B MB

Moreyra (1993) Rogers (1995) Schardt and Schrade (1982)

Schuster (1992) Shan et al. (1994) Willis and Wallace (1990)

Figure 1.1-2 Cross-section and testing details of of lipped channel and zed beams used for calibration

Table 1.1-6 Geometry of lipped channel beams

section id

name Max of h/t

Min of h/t

Max of b/t

Min of b/t

Max of d/t

Min of d/t

Max of h/b

Min of h/b

Max of d/b

Min of d/b

Max of q

Min of q

Max of t

Min of t

Max of fy

Min of fy n

C Cohen (1987) 128 78 55 33 16 9 2.38 2.34 0.43 0.20 90 89 0.107 0.064 61 54 14Ellifritt et. al. (1997) 126 113 48 47 14 11 2.63 2.36 0.29 0.24 90 90 0.072 0.066 70 60 5LaBoube and Yu (1978) 269 77 75 28 15 11 8.33 1.58 0.44 0.17 90 90 0.052 0.046 54 34 32Moreyra (1993) 124 117 36 33 16 12 3.54 3.45 0.46 0.35 90 90 0.073 0.069 64 57 9Phung and Yu (1978) 321 210 32 31 17 16 10.38 6.47 0.55 0.50 90 90 0.042 0.040 51 51 6Rogers (1995) 228 53 61 15 34 3 13.72 1.49 0.70 0.16 90 90 0.076 0.042 61 44 49Schuster (1992) 168 166 34 33 11 10 5.10 4.95 0.33 0.29 90 90 0.048 0.048 48 39 5Shan (1994) 256 43 58 19 20 6 7.34 1.50 0.41 0.24 90 90 0.073 0.031 65 30 29Willis and Wallace (1990) 131 126 40 38 17 14 3.46 3.18 0.46 0.35 89 87 0.063 0.061 60 54 4

C Total 321 43 75 15 34 3 13.72 1.49 0.70 0.16 90 87 0.107 0.031 70 30 153

Table 1.1-7 Geometry of unlipped channel beams

section id name Max of

h/tMin of

h/tMax of b/t

Min of b/t

Max of d/t

Min of d/t

Max of h/b

Min of h/b

Max of d/b

Min of d/b

Max of q

Min of q

Max of t

Min of t

Max of fy

Min of fy n

C0 Rogers (1995) 225 53 59 12 0 0 17.36 1.67 0.00 0.00 90 90 0.076 0.042 61 44 10

Table 1.1-8 Geometry of lipped channel with edge stiffener at an angle (q) beams


h/tMin of

h/tMax of b/t

Min of b/t

Max of d/t

Min of d/t

Max of h/b

Min of h/b

Max of d/b

Min of d/b

Max of q

Min of q

Max of t

Min of t

Max of fy

Min of fy n

Cs Cohen (1987) 128 78 55 33 16 9 2.38 2.34 0.43 0.20 62 44 0.107 0.064 61 54 22


Appendix 1 - 5

Table 1.1-9 Geometry of lipped channel with a web stiffener beams


h/tMin of

h/tMax of b/t

Min of b/t

Max of d/t

Min of d/t

Max of h/b

Min of h/b

Max of d/b

Min of d/b

Max of q

Min of q

Max of t

Min of t

Max of fy

Min of fy n

Cw1 Phung and Yu (1978) 358 200 58 29 17 14 11.70 5.54 0.56 0.27 90 90 0.042 0.037 51 44 42

Table 1.1-10 Geometry of liped zed beams


h/tMin of

h/tMax of b/t

Min of b/t

Max of d/t

Min of d/t

Max of h/b

Min of h/b

Max of d/b

Min of d/b

Max of q

Min of q

Max of t

Min of t

Max of fy

Min of fy n

Z Ellifritt et. al. (1997) 139 114 39 31 16 13 3.71 3.54 0.43 0.38 41 36 0.072 0.060 67 61 5Schardt and Schrade (1982) 180 178 70 50 10 10 3.58 2.56 0.20 0.14 61 45 0.040 0.040 39 37 8Willis and Wallace (1990) 129 125 40 34 13 11 3.77 3.20 0.40 0.28 55 51 0.080 0.062 59 52 6

Z Total 180 114 70 31 16 10 3.77 2.56 0.43 0.14 61 36 0.080 0.040 67 37 19Zr Schardt and Schrade (1982) 183 179 71 45 16 10 4.06 2.55 0.34 0.15 90 90 0.040 0.039 54 32 29Zr Total 183 179 71 45 16 10 4.06 2.55 0.34 0.15 90 90 0.040 0.039 54 32 29

(a) Acharya/Papazian sections (b) Desmond sections (c) König sections

Figure 1.1-3 Cross-section of hat section beams used in calibration

Table 1.1-11 Geometry of hat section beams

web comp. flange stiffener aspect ratio neutral axis subelement depth/width stiffener yield stress thickness95 h/t bc/t ds/t h/bc bc/(2bt) bc/[(n+1)t] ds/(bc/(n+1)) n fy tmembers max min max min max min max min max min max min max min max min max min max min

Acharya et al. 93 38 393 95 25 6 0.87 0.14 2.40 0.44 79 32 0.35 0.11 4 2 50 45 0.058 0.023 web comp. flange stiffener aspect ratio neutral axis subelement depth/width stiffener yield stress thickness

20 h/t bc/t ds/t h/bc bc/(2bt) bc/[(n+1)t] ds/(bc/(n+1)) n fy tmembers max min max min max min max min max min max min max min max min max min max min

Papazian.. 93 37 398 92 25 10 0.45 0.15 2.47 0.75 80 46 0.37 0.20 4 1 50 45 0.057 0.022 web comp. flange stiffener aspect ratio neutral axis subelement depth/width stiffener yield stress thickness

22 h/t bc/t ds/t h/bc bct/(2btt+bptbp) bc/[(n+1)t] ds/(bc/(n+1)) n fy tmembers max min max min max min max min max min max min max min max min max min max minDesmond et al. 97 68 327 104 24 5 0.83 0.29 0.76 0.48 163 52 0.31 0.05 1 1 46 40 0.069 0.056

web comp. flange stiffener aspect ratio neutral axis subelement depth/width stiffener yield stress thickness32 h/t bc/t ds/t h/bc bc/(2bt) bc/[(n+1)t] ds/(bc/(n+1)) n fy tmembers max min max min max min max min max min max min max min max min max min max min

Konig et al. 94 56 467 167 26 0 0.34 0.20 2.72 1.34 467 59 0.23 0.00 2 0 60 53 0.070 0.025

(a) Bernard deck (b) Höglund deck

Figure 1.1-4 Cross-section of trapezoidal decks used in calibration


Appendix 1 - 6

Table 1.1-12 Geometry of trapezoidal deck beams web comp. flange stiffener aspect ratio neutral axis subelement depth/width stiffener yield stress thickness web angle

27 h/t bc/t ds/t (h/sinθh)/bc bc/(2bt) (bc-ws)/(2t) ds/((bc-ws)/2) n fy t θh

members max min max min max min max min max min max min max min max min max min max min max minBernard et al. 94 94 231 77 16 1 1.25 0.42 1.64 0.55 81 61 0.26 0.01 1 0 95 95 0.023 0.023 78 78

web comp. flange web angle aspect ratio neutral axis tens. flange web comp. flange yield stress thickness98 h/t bc/t θh (h/sinθh)/bc bc/(2bt) stiffeners stiffeners stiffeners fy t

members max min max min max min max min max min max min max min max min max min max minHoglund 203 51 191 61 84 52 1.91 0.70 1.69 0.60 2 0 2 0 2 0 61 43 0.045 0.017

It is the intent of this Specification that as more cross-sections are verified for use in the Direct Strength Method that these tables and sections would be augmented with new information. Those companies with proprietary sections may wish to perform their own testing and follow the details of Chapter F of the main Specification to independently justify the use of higher φ and lower Ω factors for a particular cross-section. Alternatively, members which fall outside the scope of this testing may still use the method of Appendix 1, but with the reduced φ and increased Ω factors consistent with any rational analysis method.

1.1.2 Elastic Buckling Thin-walled cold-formed steel members have multiple elastic buckling modes that are important for design. What is elastic buckling? Elastic buckling is a phenomenon that occurs when the energy associated with out-of-plane deformation response to an in-plane load is equal to the energy for in-plane response to the same in-plane load. The elastic buckling load is the load in which the equilibrium of the member is neutral between the two alternative states: buckled and straight. Using finite strip analysis the elastic buckling response of seven different cold-formed steel members under a variety of different in-plane loads, both flexural and compression, are summarized in Figure 1.1-5 through Figure 1.1-11. The figures demonstrate that there are at least three relevant modes for elastic buckling response of cold-formed steel members: local, distortional, and global. All modes are not guaranteed to occur in all members, but many common members: lipped channels and zeds for example, do include all three modes. Design of cold-formed steel members must include consideration of all relevant buckling modes. Traditionally, the main Specification has only specifically addressed local and global buckling modes. Additionally, the main Specifications approach to local buckling is to conceptualize the member as a collection of elements and investigate local buckling of each element separately. The method of this Appendix provides a means to incorporate local, distortional and global buckling (all relevant modes) into the design process. Further, all buckling modes are determined for the member as a whole rather than element by element. This insures compatibility and equilibrium are maintained at element junctures. Consideration of interaction amongst elements in local buckling is a key difference between the elastic bucking calculations of the main Specification and those of this Appendix. Consider, as an example, the lipped channel shown in pure compression in Figure 1.1-5. This member is the same as design example I-8 from the AISI (1996) design manual. The members local elastic buckling load from the finite strip analysis (see Figure 1.1-5) is:


Appendix 1 - 7

Pcrl = 0.13·50.42 = 6.55 kips . The column has a gross area (Ag) of 0.917 in2, implying a member local buckling stress of

fcrl = Pcrl/Ag = 7.15 ksi . The main Specification determines a k value for each element, then fcr, and finally the effective width. The centerline dimensions (ignoring corner radii) are h = 8.94 in., b = 2.94 in., d = 0.47 in., and t = 0.06 in. For this example, lets consider just the fcr of each element as determined from the main Specification:

lip: k = 0.43, fcrl-lip= 0.43[π2E/(12(1-µ2))](t/d)2 = 186.8 ksi flange: k = 4, fcrl-flange= 4.0[π2E/(12(1-µ2))](t/b)2 = 44.4 ksi web: k = 4, fcrl-web= 4.0[π2E/(12(1-µ2))](t/h)2 = 4.8 ksi

Each element predicts a different buckling stress, even though the member is a connected group of elements. These differences in the buckling stress are ignored in the main Specification. The high flange and lip buckling stresses have little relevance given how low the web buckling stress is. Comparisons to the distortional buckling stress (fcrd) using k from B4.2 of the main Specification do no better (see Schafer and Peköz 1999 for beams and Schafer 2002 for columns). The finite strip analysis, which includes the interaction amongst the elements, shows that the flange aids the web significantly in local buckling, increasing the web buckling stress by 50% over the simply supported value of 4.8 ksi, but the buckling stress in the flange and lip are much reduced due to the same interaction. Whether calculated numerically, or by hand, the Direct Strength Method relies on an accurate determination of the member elastic buckling stress. Thus, using different elastic buckling stresses for each element, and ignoring distortional buckling does not suffice in this approach. On one hand this makes the calculations required greater. On the other hand, elastic buckling is the key basic concept for cold-formed steel structures effort spent on this calculation is rewarded with more efficient members and structures. Further, numerical analysis methods are now freely available for completing this calculation.

1.1.2.1 Elastic Buckling Numerical Solutions A variety of numerical methods: finite element, finite differences, boundary element, generalized beam theory, finite strip analysis, and others may provide accurate elastic buckling solutions for cold-formed steel beams and columns. The Appendix states the basic requirements for these numerical methods: accuracy in classic problems should be no more than 1% greater than exact solutions and compatibility and equilibrium must be maintained, or if neglected done so conservatively. The final comment refers primarily to hand techniques such as the element method for local buckling prediction this method ignores compatibility and equilibrium of the elements, but by using a minimum solution from all the elements, does so in a conservative fashion. Traditional finite element analysis using thin plate or shell elements may be used for elastic buckling prediction. Due to the common practice of using polynomial shape functions, the number of elements required for reasonable accuracy can be significant. Finite element analysis books such as Cook (1989) and Zienkiewicz and Taylor (1989, 1991) explain the basic theory; while a number of commercial implementations, including but not limited to: ABAQUS,


Appendix 1 - 8

MSC/NASTRAN, MARC, ANSYS, and STAGS can all provide accurate elastic buckling answers if implemented with care. Note, in finite element implementations a large number of higher modes may have to be investigated in order to reveal the three fundamental modes of interest for a given member. Finite difference solutions for plate stability are implemented in Harik et al. (1991) and others. The approach is feasible, but it is not known if commercial or academic software is currently available for this method. The boundary element method may also be used for elastic stability (Elzein 1991). Examples demonstrate that the method can provide sufficient accuracy, but it is not known if commercial or academic software is available for this method. Generalized beam theory, developed by Schardt (1989), extended by Davies et al. (1994) and implemented by Davies and Jiang (1996, 1998), and others has been shown to be a useful tool for elastic stability analysis of cold-formed steel members. The ability to separate the different buckling modes makes the method especially amenable to design methods. Generalized beam theory codes have been developed in England with Davies group and in Portugal with Camotims group, but details of their distribution to the general population are unknown. Finite strip analysis is a specialized variant of the finite element method. For elastic stability of cold-formed steel structures it is one of the most efficient and popular methods. Cheung and Tham (1998) explains the basic theory while Hancock et al. (2001) and Schafer (1997) provide specific details for stability analysis with this method. Hancock and his researchers (see Hancock et al. 2001 for full references and descriptions) pioneered the use of finite strip analysis for stability of cold-formed steel members and convincingly demonstrated the important potential of finite strip analysis in both cold-formed steel design and behavior. Hancocks research group distributes the commercial program THIN-WALL for finite strip analysis (www.civil.usyd.edu.au/case/THINWALL.htm). Schafers research group has also used finite strip analysis extensively and make available the academic program, CUFSM, for finite strip analysis (www.ce.jhu.edu/bschafer/cufsm). Also, RSG Softwares program, CFS, performs finite strip analysis (www.rsgsoftware.com) in addition to strength calculations by the AISI Specification. The Direct Strength method of this appendix emphasizes the use of finite strip analysis for elastic buckling determination. Finite strip analysis is a general tool that provides accurate elastic buckling solutions with a minimum of effort and time. Finite strip analysis, as implemented in conventional programs, does have limitations, the two most important ones are

• the model assumes the ends of the member are simply supported, and • the cross-section may not vary along its length (uniform properties longitudinally).

These limitations preclude some analysis from readily being used with the finite strip method, but despite these limitations the tool is useful, and a major advance over plate buckling solutions and plate buckling coefficients ks that only partially account for the important stability behavior of cold-formed steel members. The American Iron and Steel Institute has sponsored research that, in part, has lead to the development of the freely available program, CUFSM, which employs the finite strip method for elastic buckling determination of any cold-formed steel cross-section. The program is available at www.ce.jhu.edu/bschafer/cufsm and runs on any PC with Windows 9x, NT, or 2000. A complete

http://www.civil.usyd.edu.au/case/THINWALL.htm

http://www.ce.jhu.edu/bschafer/cufsm

http://www.rsgsoftware.com/



Appendix 1 - 9

set of tutorials and examples are available online at the same address. Information available online includes:

• Introduction and Common Questions What is CUFSM? What are the system requirements? Why would I use CUFSM? Why do 2 versions of CUFSM exist? Why would I use the Matlab version? CUFSM is free?

What is Finite Strip? What is a buckling curve? What is a buckling mode? What is the half-wavelength? What is the load factor? What are Mcr and Pcr? How can I use finite strip in design?

• Installation • Overview of program features • Tutorials

Manipulating a Cee section in bending SSMA Cee in Compression (Pcrl and Pcrd) LGSI Zee in Bending (Mcrl and Mcrd)

• Advanced Matlab scripting parametric studies • Appendix: Finite Strip Theory

• Advanced Functions Boundary conditions

Constraints Springs Multiple materials Orthotropic material

• Advanced Ideas Defining buckling modes Why define buckling modes? Understanding higher modes Utilizing higher modes Handling indistinct modes

Solution accuracy The information is designed to allow someone with no experience with this type of numerical analysis to become a proficient and conscientious user of the tool. The software and its source code are freely available in an attempt to insure that the solution is not a black box for the user. Additionally, hand methods provided in this commentary may be used as a sanity check on any analysis to insure solutions are at least within expected bounds.

1.1.2.1.1 Numerical, Finite strip analysis notes on Pcrl, Mcrl

Prediction of local buckling modes for a variety of different members in compression and bending using the finite strip analysis program CUFSM are demonstrated in Figure 1.1-5 through Figure 1.1-11. In the finite strip method members are loaded with a reference stress distribution: pure compression for finding Pcr pure bending for finding Mcr. The reference stress distributions are shown in the figures. Determination of the buckling mode requires consideration of the half-wavelength and the mode shape. Half-wavelength Local buckling minima occur at half-wavelengths that are less than the largest characteristic dimension of the member under compressive stresses. For the examples of Figure 1.1-5 through Figure 1.1-11, this length has been demarcated with a short vertical dashed line. For instance, the largest outside dimension for the lipped channel of Figure 1.1-5 is 9 in., therefore the cutoff (and the vertical line) for local buckling is at 9 in. Minima in the buckling curve shown in the Figures that fall at half-wavelengths less than this length may generally be considered as local buckling modes. Buckling modes occurring at longer lengths are either distortional or global in nature. The criteria of limiting the half-wavelength for local bucking to less than the largest outside dimension under compressive stresses is based on the following. Local buckling of a simply


Appendix 1 - 10

supported plate in pure compression occurs in square waves, i.e., it has a half-wavelength that is equal to the plate width (the largest outside dimension). If any stress gradient exists on the plate, or any beneficial restraint is provided to the edges of the plate by other elements, the critical half-wavelength will be less than the width of the plate. Therefore, local buckling, with the potential for stable post-buckling response, is assumed to occur only when the critical half-wavelength is less than the largest potential plate (i.e., outside dimension with compressive stresses applied) in a member. Mode shape Local buckling involves significant distortion of the cross-section, but this distortion involves only rotation, not translation at the fold lines of the member. The mode shapes for members with edge stiffened flanges such as those of Figure 1.1-5 and Figure 1.1-6 provide a direct comparison between the difference between local buckling and distortional bucking. Note the behavior at the flange/lip junction for local buckling only rotation occurs, for distortional buckling translation occurs. Complications Local buckling may be indistinct from distortional buckling in some members, for example Figure 1.1-9 shows a simple angle that may be considered as local buckling by the main Specification, but is considered distortional buckling here, because of the half-wavelength of the mode, and the characteristics of the mode shape. By the definitions of this Appendix no local buckling mode exists for this member. Local buckling may be at half-wavelengths much less than the characteristic dimension if intermediate stiffeners are in place, or if the element undergoes large tension and small compressive stress. For instance Figure 1.1-15 demonstrates local buckling in negative bending that exhibits this trait. Additional examples, and tutorials are available online at www.ce.jhu.edu/bschafer/cufsm.

1.1.2.1.2 Numerical, finite strip analysis notes on Pcrd, Mcrd Prediction of distortional buckling modes for a variety of different members in compression and bending using the finite strip analysis program CUFSM are demonstrated in Figure 1.1-5 through Figure 1.1-11. In the finite strip method members are loaded with a reference stress distribution: pure compression for finding Pcr, pure bending for finding Mcr. The reference stress distributions are shown in the figures. Determination of the buckling mode requires consideration of the half-wavelength and the mode shape. Half-wavelength Distortional buckling occurs at a half-wavelength intermediate to local and global buckling modes, as shown in the figures. The half-wavelength is typically several times larger than the largest characteristic dimension of the member. The half-wavelength is highly dependent on both the loading and the geometry. Mode shape Distortional buckling involves both translation and rotation at the fold line of a member. Distortional buckling involves distortion of one portion of the cross-section and predominately



Appendix 1 - 11

rigid response of a second portion. For instance, the edge stiffened flanges of Figure 1.1-5 and Figure 1.1-6 are primarily responding as one rigid piece while the web is distorting. Complications Distortional buckling may be indistinct (without a minimum) even when local buckling and long half-wavelength (global) buckling are clear - Figure 1.1-5 and Figure 1.1-6 show this basic behavior. For some members distortional buckling may not occur, as in Figure 1.1-8. (Actually, a close examination of higher modes shows the distortional mode in this example, but the low values of the global torsional buckling mode dominate the response at both intermediate and long half-wavelengths.) Further examples, and discussion of handling indistinct buckling modes, understanding higher modes, etc. are all available online at www.ce.jhu.edu/bschafer/cufsm.

1.1.2.1.3 Numerical, finite strip analysis notes on Pcre, Mcre Prediction of global buckling modes for a variety of different members in compression and bending using the finite strip analysis program CUFSM are demonstrated in Figure 1.1-5 through Figure 1.1-11. In the finite strip method members are loaded with a reference stress distribution: pure compression for finding Pcr, pure bending for finding Mcr. The reference stress distributions are shown in the figures. Determination of the buckling mode requires consideration of the half-wavelength and the mode shape. Global bucking modes for columns include: flexure, torsion and flexural-torsional buckling. For beams bent about their strong-axis, lateral-torsional buckling is the global buckling mode of interest. Half-wavelength Global (or Euler) buckling modes: flexural, torsional, or flexural-torsional for columns, lateral-torsional for beams, occur as the minimum mode at long half-wavelengths. Mode Shape Global buckling modes involve translation (flexure) and/or rotation (torsion) of the entire cross-section. No distortion exists in any of the elements in the long half-wavelength buckling modes. Complications Flexure and distortional buckling may interact at relatively long half- wavelengths making it difficult to determine long column modes at certain intermediate to long lengths. Finite strip analysis assumes simply supported ends. When long column end conditions are not simply supported, or when they are dissimilar for flexure and torsion, higher modes may need to be considered, or classical long column calculations performed. Higher modes for the lipped channel of Figure 1.1-5 are given in Figure 1.1-12. Based on the boundary conditions, the effective length, KL, for a given mode can be determined. With KL known then Pcre for that mode may be read directly from the finite strip at a half-wavelength of KL by using the curve that corresponds to the appropriate mode. Mixed flexural and torsional boundary conditions may not be directly treated. Further, in some cases a mode of interest may occur at too high a load factor for practical consideration. In these cases, the classical long column calculations, such as given in section 1.1.2.2.3 of this Appendix may be used.



Appendix 1 - 12

Further examples and discussion are available online at www.ce.jhu.edu/bschafer/cufsm.



Appendix 1 - 13

100

101

102

103

0

0.5

1

1.5

half−wavelength (in.)

Mcr

/ M

y

AISI96 Ex. 1−8

My=135.91kip−in.

Local Mcr

/My=0.67 Distortional M

cr/M

y=0.50

Lateral−torsional

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Pcr

/ P

y

AISI96 Ex. 1−8

Py=50.42kips

Local Pcr

/Py=0.13 Distortional P

cr/P

y=0.20

Flexural−torsional

Figure 1.1-5 Finite Strip Analysis of 9CS3x060 of AISI ’96 Example I-8 in Bending and Compression


Appendix 1 - 14

100

101

102

103

0

0.5

1

1.5


Mcr

/ M

y

AISI96 Ex. 1−10

My=112.82kip−in.

Local Mcr

/My=0.86 Distortional M

cr/M

y=0.65

Lateral−torsional

100

101

102

103

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5


Pcr

/ P

y

AISI96 Ex. 1−10

Py=46.55kips

Local Pcr

/Py=0.16

Distortional Pcr

/Py=0.27

Flexural

Figure 1.1-6 Finite Strip Analysis of 8Z2.5x060 of AISI ’96 Example I-10 in Bending and Compression


Appendix 1 - 15

100

101

102

103

0

0.5

1

1.5

2

2.5

3


Mcr

/ M

y

AISI96 Ex. 1−9

My=20.78kip−in.

Distortional Mcr

/My=1.44

Lateral−torsional

Figure 1.1-7 Finite Strip Analysis of 5.5CU1.25x057 of AISI ’96 Example I-9 in Bending

100

101

102

103

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pcr

/ P

y

AISI96 Ex. 1−11

Py=28.17kips

Local Pcr

/Py=0.49

Torsional

Figure 1.1-8 Finite Strip Analysis of 4LS4x060 of AISI ’96 Example I-11 in Compression


Appendix 1 - 16

100

101

102

103

0

0.5

1

1.5


Mcr

/ M

y

AISI96 Ex. 1−12

My=2.12kip−in.

Distortional Mcr

/My=1.03

Lateral−torsional

100

101

102

103

0

0.5

1

1.5

2

2.5

3

3.5


Mcr

/ M

y

AISI96 Ex. 1−12

My=2.12kip−in.

Distortional Mcr

/My=2.17

Lateral−torsional

Figure 1.1-9 Finite Strip Analysis of 2LU2x060 of AISI ’96 Example I-12 in Positive and Negative Bending


Appendix 1 - 17

100

101

102

103

0

1

2

3

4

5

6

7

8


Mcr

/ M

y

AISI96 Ex. 1−13

My=87.00kip−in.

Local Mcr

/My=3.20

Lateral−torsional

100

101

102

103

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


Pcr

/ P

y

AISI96 Ex. 1−13

Py=95.50kips

Local Pcr

/Py=2.41

Flexural−torsional

Figure 1.1-10 Finite Strip Analysis of 3HU4.5x135 of AISI ’96 Example I-13 in Bending and Compression


Appendix 1 - 18

100

101

102

103

0

0.5

1

1.5


Mcr

/ M

y

AISI96 Approx. Ex. 1−14

My=21.35kip−in.

Local Mcr

/My=0.42

Distortional Mcr

/My=0.68

Lateral−torsional

100

101

102

103

0

0.5

1

1.5


Mcr

/ M

y

AISI96 Approx. Ex. 1−14

My=21.35kip−in.

Local Mcr

/My=0.60

Distortional Mcr

/My=0.52

Lateral−torsional

Figure 1.1-11 Finite Strip Analysis of Wall Section of AISI ’96 Example I-14 in Positive and Negative Bending


Appendix 1 - 19

50 100 150 200 250 300 350 400 450 500 550 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Pcr

/ P

y

AISI96 Ex. 1−8

Py=50.42kips

Flexural

Flexural−torsional (1)

Flexural−torsional (2)

Figure 1.1-12 Finite strip analysis of lipped channel of AISI ’96 example I-8 showing long half-wavelength

and higher buckling modes for examination of global buckling behavior


Appendix 1 - 20

1.1.2.2 Elastic Buckling Hand Solutions

1.1.2.2.1 Local Buckling by Hand: Pcrl, Mcrl

Two hand solution methods for local buckling prediction are provided: (1.1.2.2.1.1) the element method and (1.1.2.2.1.2) the semi-empirical interaction method. The element method provides a simple lowerbound solution with wide applicability, while the semi-empircal interaction method provides a more accurate local buckling solution which includes the interaction of any two elements (e.g., the web and flange), but has limited applicability. The element method (1.1.2.2.1.1) employs plate buckling solutions for the elastic local buckling stress in a manner somewhat similar to Chapter B of the main Specification. An appropriate plate buckling coefficient, or k value, must be found for each element making up a member. Each k value implies a buckling stress for a given element. Thus, the member buckling stress remains unknown as each element provides a different prediction. In Chapter B of the main Specification this fact is ignored, and along with it compatibility and equilibrium of the elements. The element method takes a conservative approach to this problem and assumes the minimum buckling stress of all the elements governs. The element method for local buckling prediction may be excessively conservative. The semi-empirical interaction method (1.1.2.2.1.2) provides a more accurate local buckling prediction by including the equilibrium and compatibility of any two elements, e.g. the flange and the web. This interaction may be quite important, for example, Figure 1.1-13 shows the local plate buckling coefficient for a uniformly compressed flange (b), as influenced by the depth of the web (h), and the type of stress gradient on the web (ξ). Ignoring interaction, the flange plate buckling coefficient k = 4, with interaction, Figure 1.1-13 demonstrates the appropriate k.

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1 2 3 4 5 6 7 8 9 10

ratio of web height to flange width (h/b)

plat

e bu

cklin

g co

effic

ient

for t

he fl

ange

k f

ξ = 3ξ = 2ξ = 1.4ξ = 1

local bucklingfinite strip analysis

ξ = 3

ξ = 2, pure bending

ξ = 1.4

ξ = 1

Figure 1.1-13 Influence of interaction of two elements (flange and web) on local buckling solution


Appendix 1 - 21

1.1.2.2.1.1 Hand: Element Method for Local Buckling The element method is the simplest and most conservative approximation of the elastic

local buckling load or moment for a member. To determine local buckling by the element method: assume all elements of the member are simply supported, calculate the buckling stress of each element, and take the minimum of the predicted element buckling stresses as the buckling stress of the member. The element method provides a lower bound approximation to the actual local buckling stress. For columns, Pcrl = Agfcrl (C-1.1.2-1)

Ag = gross area fcrl = local buckling stress (see below)

For beams, Mcrl = Sgfcrl (C-1.1.2-2)

Sg = gross section modulus to the extreme compression fiber fcrl = local buckling stress at the extreme compression fiber (see below)

fcrl may be found as the minimum of the plate buckling solution of all elements:

( )2

2

2

cr wt

112Ekf

µ−π=

(C-1.1.2-3)

E = Youngs Modulus µ = Poissons ratio t = element thickness w = element flat width k = element (plate) buckling coefficient (see below)

Stiffened or edge stiffened element under stress gradient1:

445.0k 23 +ξ+ξ= (C-1.1.2-4) ξ=(f1-f2)/f1 (C-1.1.2-5)

Stiffened or edge stiffened element in pure compression (f1 = f2, ξ = 0): k = 4 (C-1.1.2-6)

Stiffened or edge stiffened element in pure bending (f1 = -f2, ξ = 2): k =24 (C-1.1.2-7)

Unstiffened element2 under stress gradient: 425.025.04.1k 2 +ξ−ξ= for 0 ≤ ξ ≤ 1.1 (C-1.1.2-8)

801315.6513k 23 −ξ+ξ−ξ= for 1.1 < ξ ≤ 2 (C-1.1.2-9) ξ=(f1-f2)/f1 (C-1.1.2-10)

Unstiffened element2 in pure compression (f1 = f2, ξ = 0): k = 0.425 (C-1.1.2-11)

where f1 and f2 are the stresses on opposite ends of the element, f1>f2, compression stresses are positive and tension stresses are negative.


Appendix 1 - 22

(1) The expression given for the plate buckling coefficient under a stress gradient is slightly different than that currently used in the main Specification; the given expression is accurate over a greater range of stress gradient values and was first presented in Schafer and Peköz (1999). (2) The larger compressive stress (f1) is assumed to be at the simply supported end of the unstiffened element. The main Specification provides no k values for this case; the expression given was first presented in Schafer and Peköz (1999). If the half-wavelength for buckling of the member in this mode is greater than the largest outside width of the member, then local buckling of an unstiffened element should be considered as distortional buckling of the member. Thus, the formulas for unstiffened elements given above are most applicable to lips (serving as edge stiffeners) and small outstands. For unlipped channels or other members in which the entire flange is an unstiffened element, buckling of the unstiffened element is distortional buckling of the member. Rational analysis using a numerical method, such as finite strip, provides a more meaningful answer in these cases. Since each element of a member has a different width, w, one must compare fcr, not k, to determine the element with the minimum stress. Further, for elements under a stress gradient, since buckling stress is typically determined from the flat portion of element, the predicted buckling stress may have to be converted to the extreme compression fiber to compare against other elements. Note, stiffened elements and edge stiffened elements have the same plate buckling coefficients (k) because in local buckling an edge stiffened element behaves as a stiffened element. The mode of buckling, in which the edge stiffener does not fully support the element is termed distortional buckling, and should be calculated using methods given for distortional buckling in the following section. Sub-elements: those elements between a corner and an intermediate stiffener, or between intermediate stiffeners, should be treated as stiffened elements - for the purposes of calculating the local buckling stress. If the distance between the corner and the intermediate stiffener is bs, then bs is the plate width, w, for use in C-1.1.2-3 above. If the sub-element, bs, is under a stress gradient, then the stresses f1 and f2 are the stresses at the ends of bs, for use in C-1.1.2-5 above. For an intermediately stiffened element, buckling of the entire element, with the stiffener included, is a form of distortional buckling, and should be determined as such. Numerical analysis methods, such as finite strip analysis, provide the most accurate elastic buckling values in such cases. Commentary Table C-B2-1 provides additional long plate buckling solutions for k. Upperbound approximations to the local buckling stress may be found by using the fixed boundary condition solutions, instead of the simply supported solutions, as given in the table. Use of the fixed boundary condition solutions for the elements is generally inappropriate for use in the Direct Strength method (this Appendix) as it is difficult to estimate the degree to which the actual member elastic buckling stress has been unconservatively predicted. Therefore, the simply supported k values, given above, are suggested for general use in the element method for elastic buckling prediction.


Appendix 1 - 23

Predictions of the half-wavelength for a buckling mode are not required for strength determination. However, for comparisons to numerical analysis, or to provide more insight in a given situation, knowing the half-wavelength can be useful. The approximate half-wavelength for local buckling of a stiffened or edge stiffened element, based on expressions determined in Schafer (1997), are: L = βw (C-1.1.2-12) 04.040.297.158.006.0 234 +ξ+ξ−ξ+ξ−=β for ξ > 1 (C-1.1.2-13) β = 1.0 for ξ ≤ 1 (C-1.1.2-14) ξ = (f1 - f2)/f1 (C-1.1.2-15)

L = half-wavelength for local buckling w = element flat width f1 and f2 are the stresses on opposite ends of the element, f1>f2, compression stresses are positive and tension stresses are negative.

1.1.2.2.1.2 Hand: Interaction Method for Local Buckling In many cases the element method for local buckling determination gives excessively conservative predictions of the member buckling stress. For limited cases hand solutions are available that improve upon the traditional plate buckling solutions. These solutions consider the local buckling stress of any two connected elements. For instance, a typical lipped channel, consisting of a stiffened element (the web) an edge stiffened element (the flange) and an unstiffened element (the lip) may be considered by finding the local buckling stress for the flange/web combination and the flange/lip combination and taking the minimum of these two values (as opposed to finding individual values for the flange, web, and lip and taking the minimum of those three values as is done in the element method). Specifically:

For columns, Pcrl = Agfcrl (C-1.1.2-16) For beams, Mcrl = Sgfcrl (C-1.1.2-17) where fcrl may be found as the minimum of the plate buckling solutions given below: For any two stiffened or edge stiffened elements of uniform thickness, t, connected together, one of width, h, the second element of width, b, and both elements loaded in pure compression, the local buckling stress of the two combined elements is (Schafer 2002):

( )2

2

2

cr bt

112Ekf

µ−π=

(C-1.1.2-18)

( )( ) ( )24.0 hb4hb2k ⋅−= if h/b ≥ 1 (C-1.1.2-19) ( )( ) 4bh2k 2.0 ⋅−= if h/b < 1 (C-1.1.2-20)


Appendix 1 - 24

For any two stiffened or edge stiffened elements of uniform thickness, t, connected together, one of width, h, the second element of width, b, with element h under a stress gradient (ξ) and element b loaded in pure compression, the local buckling stress of the two combined elements is (Schafer 2001):

( )2

2

2

wcr ht

112Ekf

µ−π=

(C-1.1.2-21)

m

2

1

wssm2wss

w khb

kkkkk +

−= for wss

1

kk

hb < (C-1.1.2-22)

2

1w bhkk

= for wss

1

kk

hb ≥ (C-1.1.2-23)

445.0k 23wss +ξ+ξ= (C-1.1.2-24)

47.7k 2m +ξ= (C-1.1.2-25)

( ) 72.4308.0k 21 +−ξ−= (C-1.1.2-26)

( ) 121 fff −=ξ (C-1.1.2-27) f1 and f2 are the stresses on opposite ends of element h, f1>f2, compression stresses are positive and tension stresses are negative.

For an edge stiffened element of width, b, in which the edge stiffener is a simple lip (unstiffened element) of width, d, (but any angle θ) and element b is under pure compression, and element d is under a stress gradient, the local buckling stress of the two combined elements is (Schafer and Peköz 1999):

( )2

2

2

cr bt

112Ekf

µ−π=

(C-1.1.2-28)

( )( ) ( )( ) 4bd95.359.1bd07.1155.8k 2 ++ξ−+−ξ= for ξ ≤ 1 and d/b ≤ 0.6 (C-1.1.2-29) ( ) 121 fff −=ξ (C-1.1.2-30)

f1 and f2 are the stresses on opposite ends of element d, f1>f2, compression stresses are positive and tension stresses are negative. f1 is the stress at the end of element d, connected to b.

The above formulas allow for an accurate calculation of the local buckling stress of lipped channels and zeds. They are applicable to other cases as well and may be compared (combined) with individual element solutions in order to find the minimum buckling stress. The formulas given here are not intended to preclude the creation, or use of additional local buckling formulas that use rational analysis to arrive at the critical elastic local buckling load or moment.

1.1.2.2.2 Distortional Buckling by Hand: Pcrd, Mcrd Available hand methods for distortional buckling include: members with edge stiffened flanges and elements with intermediate stiffener(s). The first case, members with edge stiffened flanges, includes lipped channels and zeds in compression or bending, and applies to any case where the flange of a member connected to a flat web of width, h, may buckle in the distortional mode. The second case, elements with intermediate stiffener(s), include hats and decks with intermediate


Appendix 1 - 25

stiffeners in the flanges or in the webs, and applies to any stiffened element with single or multiple intermediate stiffener(s) under compression or bending. The available hand formulas for members with edge stiffened flanges calculate distortional buckling as a member phenomena; however, the available formulas for elements with intermediate stiffeners only calculate distortional buckling as an element phenomena. Therefore, the hand solutions of 1.1.2.2.2.1 are directly applicable in the Direct Strength Method, while the hand solutions of 1.1.2.2.2.2 and 1.1.2.2.2.3 may require additional care. Distortional buckling formula require a significant amount of hand calculation to provide an accurate solution. Numerical analysis, such as the finite strip method, provides a useful alternative. For elements with intermediate stiffener(s) (sections 1.1.2.2.2.2 and 1.1.2.2.2.3) member distortional buckling will be well approximated by distortional buckling of the element only if the element governs the distortional buckling mode of the member. This implies that any other element connected to the element with an intermediate stiffener does not require a net rotational or translational restraint prior to buckling. This situation may be approximated by determining the local bucking stress of any connected elements and insuring that the local bucking stress of any connected element is greater than the distortional buckling stress of the element with an intermediate stiffener. If this criteria cannot be met, then numerical analysis should be employed to properly account for the interaction of the elements in the member distortional buckling.

1.1.2.2.2.1 Hand: Distortional Buckling of Members with Edge Stiffened Flanges Based on Schafer and Peköz (1999) for beams and Schafer (2002) for columns, distortional buckling of members with edge stiffened flanges may be predicted by: For columns, Pcrd =Agfcrd (C-1.1.2-31)

Ag = gross area of the member fcrd = distortional buckling stress (see below)

For beams, Mcrd = Sgfcrd (C-1.1.2-32)

Sg = gross section modulus to the extreme compression fiber fcrd = distortional buckling stress at the extreme compression fiber (see below)

where:

wgfg

wefecrd

k~

k~

kkf

φφ

φφ

+

+= (C-1.1.2-33)

( )mcr L,LminL = Flange Rotational Stiffness:

( ) ( ) ( ) f

22

xoyf

2xyf

wf2

xoxf

4

ef GJL

hxII

EEChxEIL

k

π+

−−+−

π=φ (C-1.1.2-34)


Appendix 1 - 26

( ) ( ) ( )

++

++

−−

−

π=φ yfxf2o

2x

yf

xyfxoo

2

yf

xyf2xof

2

gf IIyhII

hxy2II

hxAL

k~ (C-1.1.2-35)

Flexural Member: Critical Length and Web Rotational Stiffness

( ) ( ) ( )4

1

720hhx

II

ChxIt1h4L

442

xoyf

2xyf

wf2

xoxf3

24

cr

π+

−−+−ν−π= (C-1.1.2-36)

( )

π+

π+ν−

=φ 240h

L60h19

Lh3

112Etk

342

2

3

we (C-1.1.2-37)

( )( ) ( )( )

+

π+π

πξ−+

+π+

+ξ−π=φ 42

24

4web

22

2

web2

wg

hL420

hL28

1353Lh448

hL62160145360

13440htk

~ (C-1.1.2-38)

Compression Member: Critical Length and Web Rotational Stiffness

( ) ( ) ( )4

1

2xo

yf

2xyf

wf2

xoxf3

24

cr hxII

ChxIt1h6L

−−+−ν−π= (C-1.1.2-39)

( )2

3

we 1h6Etk

ν−=φ (C-1.1.2-40)

60th

Lk~ 32

wg

π=φ (C-1.1.2-41)

E = Youngs Modulus

G = Shear Modulus ν = Poissons Ratio Lm = Distance between restraints which restrict distortional buckling t = plate thickness h = web depth ξweb = (f1- f2)/f1 = stress gradient in the web, where f1 and f2 are the stresses at the

opposite ends of the web, f1>f2, compression positive, tension negative, calculated on the basis of the gross section, (e.g., pure symmetrical bending, f1=-f2, ξweb = 2)

Af, Ixf, Iyf, Ixyf, Cwf, Jf, xo, and hx, are properties of the compression flange (flange and edge stiffener) about x, y axes. Where the x, y axes are located at the centroid of the flange, with the x-axis measured positive to the right from the centroid, and the y-axis positive down from the centroid. For a flange with a simple lip stiffener (i.e, the flange of a typical Cee or Zee) appropriate expressions from Schafer (1997) are given below.

Af = cross-sectional area of the flange Ixf = x-axis moment of inertia of the flange Iyf = y-axis moment of inertia of the flange Ixyf = product of the moment of inertia of the flange Cwf = warping torsion constant of the flange Jf = St. Venant torsion constant of the flange xo = x distance from the flange/web junction to the centroid of the flange


Appendix 1 - 27

hx = x distance from the centroid of the flange to the shear center of the flange

h

b

d

θ

h

b

d θ

( )

( )( )

( )( )

( )

( )

( )( )

( )xo

2

x

2

oy

2

o

wf

33

1312

133

1of

22

xyf

34

yf

42322

xf

33

133

1f

f

hxbdb2db2bh

db2dyh

db2bx

0CtdbttbI

db4dtbI

db12db4btI

db12dbdtbd4bttI

dtbtJ

tdbA

−=++−=

+−==

+=

=++=

+=

++=

++++=

+=

+=

( )

( ) ( )( )( )

( ) ( ) ( )( )( )

( ) ( )( )( )

( )( )

( )( )

( )( )( )

xo

22

x

2

oy

22

o

wf

33

1312

133

1of

2

xyf

24232234

yf

244223322

xf

33

133

1f

f

hxbdb2

cosddb2bh

db2sindyh

db2cosdbx

0CtdbttbI

db4cosdbsintbdI

db12cosdcosbd4cosbd6db4btI

db12cosddbdtcosbd4bd4bttI

dtbtJ

tdbA

−=+

θ++−=

+θ−==

+θ−=

=++=

+θ+θ=

+θ+θ+θ++=

+θ−++θ−+=

+=

+=

Figure 1.1-14 Geometric flange properties for lipped channel and lipped zeds

Similar distortional buckling hand formulas for columns are available in Lau and Hancock (1987) and for beams in Hancock et al. (1996) and Hancock (1997). For flanges with unusual edge stiffeners (compound lips, etc.) the properties of the flange must be determined (no explicit formula are given). However, in some cases explicit formulas may be found in the literature (e.g., Bambach et al. 1998). Distortional buckling of members with unstiffened flanges (unlipped channels, etc) may be approximated by setting d to 0 in the expressions of Figure 1.1-14.

1.1.2.2.2.2 Hand: Distortional buckling of a stiffened element with an intermediate stiffener under a stress gradient

For flexural members with single intermediate stiffeners in the web, explicit formula for distortional buckling of the element are available based on the work of Schafer (1997).


Appendix 1 - 28

x

y

a

b

f1

f2

f1

f2

c

A

A

AA

Figure 1.1-15 Stiffened element with an intermediate stiffener under a stress gradient

Mcrd = Sgfcrd (C-1.1.2-42) Sg = gross section modulus to the largest compressive edge of the element (f1 end) fcrd = distortional buckling stress at the largest compressive edge of the element (f1 end)

( )2

2

2

dcrd bt

112Ekf

µ−π= (C-1.1.2-43)

E = Youngs Modulus µ = Poissons ratio t = element thickness b = element (web) width kd = element (plate) buckling coefficient for distortional buckling

q2

qs4rrk

2

d

−−−= (C-1.1.2-44)

( )224 EDBGq −+β= (C-1.1.2-45) ( )DIBHG2r 2 ++γβ= (C-1.1.2-46) JH2s +γ= (C-1.1.2-47) α = c/b (C-1.1.2-48) γ = (EIs)/(bDpl), where, Dpl = (Et3)/(12(1-µ2)) (C-1.1.2-49) δ = As/(bt) (C-1.1.2-50) ξ = (f1 f2)/f1 (C-1.1.2-51)

4

1

4ns3

2

17.02.1sin

β+

ξγ

παξ=β (C-1.1.2-52)

for ξ > 1 04.040.297.158.006.0 234ns +ξ+ξ−ξ+ξ−=β (C-1.1.2-53)

for ξ ≤ 1 βns = 1.0 (C-1.1.2-54) ( )ξα−δ−= 12B (C-1.1.2-55) ( )21D ξ−−= (C-1.1.2-56) ( ) ( )2916E πξ−= (C-1.1.2-57) ( ) 122211 SE2SSDG −+= (C-1.1.2-58) 221112 SCSCH += (C-1.1.2-59) 21 CCI += (C-1.1.2-60) 21CCJ = (C-1.1.2-61)


Appendix 1 - 29

( )221 1C β+= (C-1.1.2-62)

( )222 41C β+= (C-1.1.2-63)

( )πα= 211 sinS (C-1.1.2-64)

( )πα= 2sinS 222 (C-1.1.2-65)

( ) ( )παπα= 2sinsinS12 (C-1.1.2-66) where:

c = distance from f1 edge of the element to the centerline of the stiffener Is = moment of inertia of the stiffener1 As = area of the stiffener1 f1, f2 = stresses on opposite ends of element h, f1>f2, compression stresses are positive and tension stresses are negative.

(1) The stiffener moment of inertia may conservatively be taken as the moment of inertia about the stiffener centroidal axis, parallel to the axis of the main element, b. Schafer (1997) provides a more general approach for determining stiffener area and moment of inertia for use in the formulas above. However, for highly complicated stiffener geometries numerical analysis provides a more accurate and efficient result.

1.1.2.2.2.3 Hand: Distortional buckling of a stiffened element with intermediate stiffener(s) under pure compression

The distortional buckling stress of an element under pure compression with intermediate stiffener(s) may be approximated as: Columns, Pcrd =Agfcrd (C-1.1.2-67)

Ag = gross area of the member fcrd = distortional buckling stress of the intermediately stiffened element (see below)

Beams, Mcrd = Sgfcrd (C-1.1.2-68)

Sg = gross section modulus to the stress in the intermediately stiffened element fcrd = distortional buckling stress of the intermediately stiffened element (see below)

( )2

2

2

dcrd bt

112Ekf

µ−π= (C-1.1.2-69)

E = Youngs Modulus µ = Poissons ratio t = element thickness b = element width kd = element buckling coefficient for distortional buckling, Spec. Eq. B5.1.2-2

Main specification section B5 provides the elastic buckling solution (both local and distortional) for elements with intermediate stiffeners in pure compression. The commentary to section B5 of


Appendix 1 - 30

the main specification provides the development of kd, further details can be found in Schafer and Peköz (1998).

1.1.2.2.3 Euler buckling by Hand: Pcre, Mcre The long half-wavelength Euler buckling mode for beams and columns is calculated with some care in the main Specification. Therefore, for both beams and columns, extensive hand expressions are already available and may be used for hand calculation of Pcre and Mcre. See the commentary to main Specification sections C4 and C3 for additional details. For columns, Pcre =Agfcre (C-1.1.2-70)

Ag = gross area of the member fcre = minimum of the elastic critical flexural, torsional, or flexural-torsional buckling

stress. fcre is equal to Fe of Section C4 of the main Specification. The hand methods presented in Specification sections C4.1 through C4.4 provide all necessary formula. Note, C4.4 specifically addresses the long-standing practice that Fe (or fcre) may be calculated by rational analysis. Rational analysis hand solutions to long column buckling are available - see the commentary for main Specification section C4.4 as well as Yu (2001) or Hancock et al. (2001). The hand calculations may be quite lengthy, particular if member properties xo and Cw are unknown.

For beams, Mcre = Sgfcre (C-1.1.2-71)

Sg = gross section modulus to the extreme fiber in compression fcre = elastic critical lateral-torsional buckling stress. fcre is equal to Fe of main

Specification section C3.1.2.1. for open cross-section members and C3.1.2.2 for closed cross-section members. Hand solutions are well established for doubly- and singly-symmetric sections, but not so for point symmetric sections (Zeds). Fe of point-symmetric sections is taken as ½ of the value for doubly-symmetric sections. Rational numerical analysis may be desirable in cases where a more exacting solution is required.

1.1.3 Deflection Determination The provisions of this Appendix use a simplified approach to deflection calculations that assume the moment of inertia of the section for deflection calculations is linearly proportional to the strength of the section, determined at the allowable stress of interest. This approximation avoids lengthy effective section calculations for deflection determination.

1.2 Columns Commentary section C4 provides a complete discussion on the behavior of cold-formed columns as it relates to the main Specification. This commentary addresses the specific issues raised by the use of the Direct Strength Method of Appendix 1 for the design of cold-formed columns. The thin-walled nature of cold-formed columns complicates behavior and design. Elastic buckling analysis (e.g., Figure 1.1-5 through Figure 1.1-11) reveals at least three buckling modes: local, distortional, and Euler (flexural, torsional, or flexural-torsional) that must be considered in


Appendix 1 - 31

design. Therefore, in addition to usual considerations for steel columns: material non-linearity (e.g., yielding), imperfections, and residual stresses; the individual role and potential for interaction of buckling modes must also be considered. The Direct Strength Method of this Appendix emerged through the combination of more refined methods for local and distortional buckling prediction, improved understanding of the post-buckling strength and imperfection sensitivity in distortional failures, and the relatively large amount of available experimental data. Fully effective or compact columns are generally well predicted by conventional column curves (AISC (2001), Galambos (1998), etc.). Therefore, the long column strength Pne, follows the same practice as the main Specification and uses the current AISC (2001) curves for strength prediction. The main Specification provides the long column strength in terms of a stress, Fn (Eq. C4-2 and C4-3). In the Direct Strength method this is converted from a stress to a strength by multiplying times the gross area, Ag, resulting in the formulas for Pne given in Appendix 1. The main Specification calculates the column strength by multiplying the long column stress, Fn, times the effective area, Ae, calculated at Fn. This accounts for local buckling reductions in the actual column strength (i.e., local-global interaction). The Direct Strength Method chooses to break this calculation into two parts: the long column strength without any reduction for local buckling (Pne) and the long column strength considering local-global interaction (Pnl). The strength curves for local and distortional buckling of a fully braced column are presented in Figure 1.2-1. The curves are presented as a function of slenderness which in this case refers to slenderness in the local or distortional mode as opposed to traditional long column slenderness. For non-slender (a.k.a., fully effective, or compact) columns Pcr>>Py and the cross-section is stable and the capacity is equal to the squash load, Py of the column. Slightly more slender columns behave inelastically, with Pn<Pcr and Pn<Py. Slender cross-sections behave primarily elastically, but elastic behavior in a thin-walled column implies post-buckling capacity, with Pn>Pcr. Practical cold-formed steel column sizes exist in all three ranges: fully effective, Pn=Py; inelastic Pn<Pcr and Pn<Py; and post-buckling Pn>Pcr and Pn<Py. Figure 1.2-1 also shows that at the if the local and distortional slenderness is the same, the distortional mode is assumed to have a reduced post-buckling capacity compared with the local mode. The development and calibration of the Direct Strength provisions for columns is reported in Schafer (2000, 2002). The reliability of the column provisions was determined using the test data of section 1.1.1.1 and the provisions of Chapter F of the main Specification. Based on a target reliability, β, of 2.5 a resistance factor, φ, of 0.84 was calculated for all the investigated columns. Based on this information the safety and resistance factors of section 1.2 were determined for the pre-qualified members. Since the range of pre-qualified members is relatively large extensions of the Direct Strength Method to geometries outside the pre-qualified set is allowed. However, given the uncertain nature of this extension reduced resistance factors and increased safety factors are applied in that case. The provisions of Appendix 1, applied to the columns of section 1.1.1.1, are summarized in Figure 1.2-2 below. The controlling strength is either by section 1.2.2 which considers local buckling interaction with long column buckling, or by section 1.2.3 which considers the distortional mode alone. The controlling strength (minimum predicted of the two modes) is


Appendix 1 - 32

highlighted for the examined members by the choice of marker. Overall performance of the method can be judged by examination of Figure 1.2-2. Scatter exists throughout the data set, but the trends in strength are clearly shown, as is the fact that for the same slenderness distortional buckling systematically has a reduced capacity. The scatter shown in the data is similar to that of the main Specification, and in fact since the main Specification has no rules for distortional buckling the Direct Strength Method actually provides better agreement than the main Specification when compared with this test database for many members.

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8

Distortional: Eq. 1.2.3-2

Local: Eq. 1.2.2-2

Elastic

y

n

PP

cry PP sslendernes =

post-buckling reserve

inelastic region

Figure 1.2-1 Direct Strength Method column curves for a fully braced column


Appendix 1 - 33

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8

distortional

local


Local: Eq. 1.2.2-2

crnecrdyd PPor PP =λ=λ

ne

test

dy

test

PP

or

PP

Figure 1.2-2 Direct Strength Method for concentrically loaded pin-ended columns

1.2.1 Flexural, Torsional, or Flexural-Torsional Buckling As discussed in detail above, the strength expressions for long wavelength buckling of columns follow directly from section C4 of the main Specification. These provisions are identical to those used for compact section hot-rolled columns in the AISC Specification (2001) and are fully discussed in the commentary to section C4. The axial capacity Pne, calculated in this section represents the upperbound capacity for a given column. Actual column strength is determined by considering reductions that may occur due to local buckling, and performing a separate check on the distortional mode. See section 1.1.2 for information on rational analysis methods for calculation of Pcre.

1.2.2 Local Buckling The expression selected for local buckling of columns is shown in Figure 1.2-1 and Figure 1.2-2 and discussed above. The potential for local-global interaction is presumed, thus the column strength in local buckling is limited to a maximum of the long column strength, Pne. See section 1.1.2 for information on rational analysis methods for calculation of Pcrl.

1.2.3 Distortional Buckling The expression selected for distortional buckling of columns is shown in Figure 1.2-1 and Figure 1.2-2 and discussed above. Based on experimental test data and on the success of the Australian/New Zealand code (see Hancock et al. 2001 for discussion and Hancock et al. 1994 for further details) the distortional buckling calculation is completed on Py instead of Pne. This


Appendix 1 - 34

presumes that distortional buckling failures are independent of long-column behavior i.e., little if any distortional-global interaction exists. See section 1.1.2 for information on rational analysis methods for calculation of Pcrd.

1.3 Beams Commentary section C3 provides a complete discussion on the behavior of cold-formed beams as it relates to the main Specification. This commentary addresses the specific issues raised by the use of the Direct Strength Method of Appendix 1 for the design of cold-formed beams. The thin-walled nature of cold-formed beams complicates behavior and design. Elastic buckling analysis (e.g., Figure 1.1-5 through Figure 1.1-11) reveals at least three buckling modes: local, distortional, and lateral-torsional buckling (for members in strong-axis bending) that must be considered in design. The Direct Strength Method of this Appendix emerged through the combination of more refined methods for local and distortional buckling prediction, improved understanding of the post-buckling strength and imperfection sensitivity in distortional failures, and the relatively large amount of available experimental data. The lateral-torsional buckling strength, Mne, follows the same practice as the main Specification. The main Specification provides the lateral-torsional buckling strength in terms of a stress, Fc (Eq. C3.1.2.1-2, -3, and -4 ). In the Direct Strength method this is converted from a stress to a moment by multiplying times the gross section modulus, Sg, resulting in the formulas for Mne given in Appendix 1. For beams that are not fully braced and locally unstable the main Specification calculates the beam strength by multiplying the predicted stress for failure in lateral-buckling, Fc, times the effective section modulus, Sc, determined at stress Fc. This accounts for local buckling reductions in the lateral-torsional beam strength (i.e., local-global interaction). The Direct Strength Method chooses to break this calculation into two parts: the lateral-torsional beam strength without any reduction for local buckling (Mne) and the strength considering local-global interaction (Mnl). The strength curves for local and distortional buckling of a fully braced beam are presented in Figure 1.3-1. The curves are presented as a function of slenderness. For non-slender (a.k.a., fully effective, compact, or locally stable) columns Mcr>>My and the cross-section is stable and the capacity is equal to the yield moment, My of the beam. Inelastic reserve capacity is currently not considered in the Direct Strength Method. Slightly more slender beams behave inelastically, with Mn<Mcr and Mn<My. Slender beams behave primarily elastically, but elastic behavior in a thin-walled beam implies post-buckling capacity, with Mn>Mcr. Practical cold-formed steel beams exist in all three ranges: fully effective, Mn=My; inelastic Mn<Mcr and Mn<My; and post-buckling Mn>Mcr and Mn<My. Figure 1.3-1 also shows that at the if the local and distortional slenderness is the same, the distortional mode is assumed to have a reduced post-buckling capacity compared with the local mode.


Appendix 1 - 35

The reliability of the beam provisions was determined using the test data of section 1.1.1.2 and the provisions of Chapter F of the main Specification. Based on a target reliability, β, of 2.5 a resistance factor, φ, of 0.90 was calculated for all the investigated beams. Based on this information the safety and resistance factors of section 1.3 were determined for the pre-qualified members. Since the range of pre-qualified members is relatively large extensions of the Direct Strength Method to geometries outside the pre-qualified set is allowed. However, given the uncertain nature of this extension reduced resistance factors and increased safety factors are applied in that case. The provisions of Appendix 1, applied to the beams of section 1.1.1.2, are summarized in Figure 1.3-2 below. The controlling strength is either by section 1.3.2 which considers local buckling interaction with lateral-torsional buckling, or by section 1.3.3 which considers the distortional mode alone. The controlling strength (minimum predicted of the two modes) is highlighted for the examined members by the choice of marker. Overall performance of the method can be judged by examination of Figure 1.3-2. The scatter shown in the data is similar to that of the main Specification, and in fact since the main Specification has no rules for distortional buckling the Direct Strength Method actually provides better agreement than the main Specification when compared with this test database for many members.

0

0.5

1

1.5

0 1 2 3 4 5


Local: Eq. 1.3.2-2

Elastic Buckling

post-buckling reserve

inelastic region

y

n

MM

cry MM sslendernes =

Figure 1.3-1 Direct Strength method beam strength curves for a laterally braced beam


Appendix 1 - 36

0

0.5

1

1.5

0 1 2 3 4 5

Distortional

Local


Local: Eq. 1.3.2-2

λmax = M My cr

MM

test

y

Figure 1.3-2 Direct Strength Method for Laterally Braced Beams

1.3.1 Lateral-Torsional Buckling As discussed in detail above, the strength expressions for lateral-torsional buckling of beams follows directly from section C3 of the main Specification and are fully discussed in the commentary to section C3. The bending capacity Mne, calculated in this section represents the upperbound capacity for a given beam. Actual beam strength is determined by considering reductions that may occur due to local buckling, and performing a separate check on the distortional mode. See section 1.1.2 for information on rational analysis methods for calculation of Mcre.

1.3.2 Local Buckling The expression selected for local buckling of beams is shown in Figure 1.3-1 and Figure 1.3-2 and discussed above. The use of the Direct Strength Method for local buckling and the development of the empirical strength expression is given in Schafer and Peköz 1998b. The potential for local-global interaction is presumed, thus the beam strength in local buckling is limited to a maximum of the lateral-torsional beam strength, Mne. For fully braced beams the maximum Mne value is the yield moment, My. See section 1.1.2 for information on rational analysis methods for calculation of Mcrl.


Appendix 1 - 37

1.3.3 Distortional Buckling The expression selected for distortional buckling of columns is shown in Figure 1.3-1 and Figure 1.3-2 and discussed above. Based on experimental test data and on the success of the Australian/New Zealand code (see Hancock 2001 for discussion) the distortional buckling calculation is completed on My instead of Mne. This presumes that distortional buckling failures are independent of lateral-torsional buckling behavior i.e., little if any distortional-global interaction exists. See section 1.1.2 for information on rational analysis methods for calculation of Mcrd.

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Appendix 1 - 38

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Hancock, G.J., Rogers, C.A., 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, 125-140, St. Louis, MO.

Harik, I.E., Liu, X., Ekambaram, R. (1991). Elastic stability of plates with varying rigidities. Computers and Structures, 38 (2) 161-168

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Appendix 1 - 39

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Appendix 1 - 40

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Appendix 1: Design of Cold-Formed Steel Structural Members by … · 2010-09-22 · DRAFT - January 4, 2002 Œ COMMENTARY: Specification for the Direct Strength Method Appendix 1