Review The Direct Strength Method of Cold-Formed Steel Member Design
Review The Direct Strength Method of Cold-Formed Steel Member Design
Review The Direct Strength Method of Cold-Formed Steel Member Design
www.elsevier.com/locate/jcsr
Abstract
The objective of this paper is to provide a review of the development and current progress in the Direct Strength Method for cold-formed
steel member design. A brief comparison of the Direct Strength Method with the Effective Width Method is provided. The advantage of methods
that integrate computational stability analysis into the design process, such as the Direct Strength Method, is highlighted. The development of
the Direct Strength Method for beams and columns, including the reliability of the method is provided. Current and ongoing research to extend
the Direct Strength Method is reviewed and complete references provided. The Direct Strength Method was formally adopted in North American
cold-formed steel design specifications in 2004 as an alternative to the traditional Effective Width Method. The appendices of this paper provide
the Direct Strength Method equations for the design of columns and beams as developed by the author and adopted in the North American
Specification.
c 2008 Elsevier Ltd. All rights reserved.
Keywords: Direct Strength Method; Effective Width Method; Cold-formed steel; Stability; Finite strip method; Thin-walled
(a) An effective C-section determined as a (b) Semi-analytical finite strip solution of a C-section in bending showing
composition of effective plates, with the effective local, distortional and lateral-torsional buckling as well as the moment to
width of the flange plate shown along with the cause first yield.
actual flange plate under nonuniform longitudinal
stress.
Fig. 1. Fundamental steps in the strength determination of a C-section by (a) Effective Width Method and (b) Direct Strength Method.
to its effective width, and this reduction from the gross cross- rolled) steel design that it may impede use of the material by
section to the effective cross-section, again as illustrated in some engineers in some situations.
Fig. 1(a), is fundamental to the application of the Effective
Width Method. The effective cross-section (i) provides a clear 2.2. Direct Strength Method
model for the locations in the cross-section where material is
ineffective in carrying load, (ii) cleanly leads to the notion If the effective width (or section) is the fundamental concept
of neutral axis shift in the section due to local-buckling and behind the Effective Width Method, then accurate member
(iii) provides an obvious means to incorporate localglobal elastic stability, as shown in Fig. 1(b) is the fundamental
interaction where reduced cross-section properties influence idea behind the Direct Strength Method. The Direct Strength
global buckling (although specifications often simplify this Method is predicated upon the idea that if an engineer
interaction somewhat). determines all of the elastic instabilities for the gross section,
i.e. local (Mcr` ), distortional (Mcrd ), and global buckling
However, the common two-dimensional nonlinear stress
(Mcre ), and also determines the moment (or load) that causes
distribution that is shown to explain the effective width of
the section to yield (M y ), then the strength can be directly
a plate is itself an approximation, representing the average
determined, i.e. Mn = f (Mcr` , Mcrd , Mcre , M y ). The Direct
of the longitudinal membrane stress and ignoring variation
Strength Method has been mentioned in textbooks and review
in stress through the thickness as well as variation in stress
articles [58]. The method is essentially an extension of the
along the length of the plate. Thus, the true effective width
use of column curves for global buckling, but with application
is far more complicated than typically assumed and existing
to local and distortional buckling instabilities and appropriate
effective width equations only correlate to average membrane
consideration of post-buckling reserve and interaction in these
stress conditions in a plate. Further, the Effective Width Method
modes. The development of, and continued research into, the
(i) ignores inter-element (e.g. between the flange and the Direct Strength Method is explored further in this paper.
web) equilibrium and compatibility in determining the elastic
buckling behaviour, (ii) incorporation of competing buckling 2.3. Long-term goals
modes, such as distortional buckling can be awkward, (iii)
cumbersome iterations are required to determine even basic It is important to recognize in any discussion regarding
member strength and (iv) determining the effective section the Effective Width Method, the Direct Strength Method,
becomes increasingly more complicated as attempts to optimize or other semi-empirical design methods that none of these
the section are made, e.g. folded-in stiffeners add to the plates design methods are theoretically correct. Rather, a complicated
which comprise the section and all plates must be investigated nonlinear problem is simplified in some manner so that
as being potentially partially effective. The Effective Width engineers may have a working model to design from without
Method is a useful design model, but it is intimately tied resorting to testing every individual member. These models
to classical plate stability, and, in general, creates a design serve us well when backed up by the application of reliability
methodology that is different enough from conventional (hot- to incorporate uncertainty in their predictive powers.
768 B.W. Schafer / Journal of Constructional Steel Research 64 (2008) 766778
Fig. 4. Comparison of Direct Strength Method for beams to tests and additional FE results for C and Z sections in (a) local and (b) distortional buckling.
(a) Effective Width Method of [1]. (b) Direct Strength Method of [2].
Fig. 5. Test-to-predicted ratio for (a) the Effective Width Method of [1] and (b) the Direct Strength Method of [2] for all lipped channel columns used in the
development of Direct Strength Method predictor equations plotted as a function of web slenderness (h/t).
and the web), performs accurately over the full range of sections are optimized the Direct Strength Method provides a
web slenderness. Proper inclusion of element interaction is simpler design methodology with wider applicability than the
necessary for accurate strength prediction of these columns. Effective Width Method.
Taken to extremes, inclusion of elastic element interaction
can also work against the Direct Strength Method, making
6. Practical developments
the method overly conservative. This fundamental limitation
of the Direct Strength Method was reported in the first paper
to propose the approach [20]. When one part (element) of the Implementation of the Direct Strength Method has required
cross-section becomes extraordinarily slender that element will a number of practical developments beyond the initial research.
drive the member elastic critical buckling stress to approach This section covers these practical developments as related to
zero. The Direct Strength Method will assume the member the Direct Strength Method adopted in [2]. These developments
strength, like the member elastic critical buckling stress, will focus on three main areas: the definition and use of prequalified
also approach zero. In contrast, the Effective Width Method sections, performing serviceability (deflection) calculations
presumes only that the element itself (not the member) will have using the Direct Strength Method, and design aids developed
no strength in such a situation. Deck or hat sections in bending for engineers employing the Direct Strength Method in practice.
with low yield stress and very slender (wide) compression
flanges without intermediate stiffeners tend to fall in this
category and thus have unduly conservative predictions by the 6.1. Prequalified sections
Direct Strength Method, but quite reasonable predictions via
the Effective Width Method. However, ignoring inter-element
interaction, as the Effective Width Method traditionally does, During the formal codification of the Direct Strength
is not universally a good idea as illustrated for the C-section Method in [2] it was determined that the users of the method
columns in Fig. 5. should be aware of the cross-sections employed to verify the
For optimized deck sections with multiple longitudinal approach. Further, it was decided that the geometrical and
intermediate stiffeners in the web and the flange (see e.g. [34]) material bounds of the cross-sections used in the verification
the Direct Strength Method is highly desirable over the of the Direct Strength approach should be able to use the
Effective Width Method here the benefit is primarily derived factors (Table 1), but new sections falling outside
convenience not theoretical. If a computational solution the boundaries of tested sections should use slightly reduced
is employed for determining the elastic buckling stresses (more conservative) factors. Thus, the idea of prequalified
(moments) an optimized deck section is no more complicated section (or limits) was established, and [2] includes a number
than a simple hat for strength determination; but for the of tables that provide the geometrical and material bounds
Effective Width Method the calculation of effective section for prequalified members. Essentially, the prequalified sections
properties and accurately handling the effective width of the in [2] represent a summary of the experimental database used
numerous sub-elements leads to severe complication without in verifying the Direct Strength Method. It is perhaps worthy
increased accuracy, or worse in the case of many specifications to note that this experimental database is larger than that used
(e.g. [1] or [29]) no design approach is even available for such for determining the Effective Width Method approach of [1]
a section using the Effective Width Method. In general, as or [29].
772 B.W. Schafer / Journal of Constructional Steel Research 64 (2008) 766778
no longer provides a reduction in the strength of this beam (in 7.2. Inelastic reserve capacity in beams
the main Specification [29] this occurs when the stress used
to determine the effective section, Fn is low enough that the Inelastic bending capacity exists in cold-formed steel beams,
section is fully effective at that stress.) Further, the detrimental despite their fundamentally thin-walled nature. For example,
impact of distortional buckling on intermediate length beams is for the experimental results reported in Figs. 3 and 4, of over
shown in Fig. 7. 500 flexural tests on cold-formed steel beams approximately
Additional information on the design of purlins using the 100 tests are found where the bending capacity reaches 95% M y
Direct Strength Method beyond that in [27] is also offered or greater including observations as high as 118% M y , where
in [39]. Further, the behavior of purlins as struts was explored M y is the moment at first yield. Current methods to account for
in [40]; however, comparisons to the Direct Strength Method inelastic reserve capacity, see e.g. [29], are highly involved and
did not incorporate the beneficial influence of rotational restricted in their use. A Direct Strength Method that accounts
restraint to the purlins as discussed in [37] and detailed for inelastic reserve has recently been developed [42].
in [25]. Built-up sections are explored in [37] and in recent Using elementary beam mechanics, and assuming elastic
research [41]. The work reported in [41] has been corrected perfectly plastic material, the inelastic compressive strain at
since its publication and the authors should be contacted for failure is back-calculated for the tested members. Simple
corrected comparisons to the Direct Strength Method. relationships between local and distortional cross-section
As engineers employ the Direct Strength Method on novel slenderness to predict average inelastic strain demands, and
cross-sections one important piece of advice from [37] is that a relationship between average strain demand and inelastic
when in doubt about whether to define a given buckling mode as bending strength are established. These relationships are
local or distortional it is always conservative to assume it is both combined to provide direct design expressions that connect
modes. Such an approach is conservative, but ensures reduced cross-section slenderness in local or distortional buckling with
post-buckling strength at intermediate unbraced lengths (i.e. the the inelastic bending strength of cold-formed steel beams.
distortional reduction) as well as inclusion of interaction effects The tested members are also augmented by a detailed finite
(i.e. localglobal interaction). element study of inelastic local and distortional buckling
and the inelastic strains sustained at failure. The elementary
7. Advancing the Direct Strength Method mechanics models agree well with the finite element models
for the average membrane strains, but peak membrane and
flexural strains can be significantly higher. Thus, the local
A significant amount of research work is ongoing in
strain demands on the section can be significantly higher than
relation to the Direct Strength Method. The following sections
the predicted average inelastic strain demands; nonetheless,
summarize recent research on the Direct Strength Method, most
predicted strain demands remain lower than expected ductility
of the work detailed below has not yet been adopted in the
for commonly used sheet steels.
Specification.
7.3. Members with holes
7.1. Shear
Research is actively underway to extend the Direct Strength
No formal provisions for shear currently exist for the Direct Method to members with holes [3033,4347]. (Note, the work
Strength Method. However, it is recommended in [37] that in [32,33] is an updated version of [30,31].) The primary
existing provisions [29] could be suitably modified. As a complication with extending the Direct Strength Method to
rational analysis extension the existing equations from [29] are members with holes is that the hole introduces the potential for
recast into the Direct Strength format and are suggested for use interactive buckling modes triggered by the hole size, spacing,
geometry, etc. The finite strip method is not well suited to
for v 0.815 Vn = Vy (1) handle members with holes therefore elastic buckling analysis,
for 0.815 < v 1.231
p
Vn = 0.815 Vcr Vy (2) the key input in the Direct Strength Method, must at least in the
for v > 1.231 Vn = Vcr (3) research phase, be completed by general purpose finite element
analysis.
where In [4345] data on existing cold-formed steel columns with
holes is gathered and eigenvalue elastic buckling analysis is
v = Vy /Vcr ,
p
(4) completed using shell element based finite element models that
Vy = Aw 0.60Fy , (5) explicitly include the holes and treat the boundary conditions
Vcr = critical elastic shear buckling force. accurately. Model results, such as shown in Fig. 8 where
distortional buckling occurs near the hole, but local buckling
For members with flat webs where Vcr is determined only for away from the hole are common. The existing Direct Strength
the web, these expressions yield the same results as in [29], Method expressions, but with Pcr` , Pcrd , and Pcre defined as the
for more unique cross-section Vcr can be determined by finite minimum elastic buckling mode that displays characteristics of
element analysis or other methods. Further research to validate local, distortional, and global buckling respectively, were found
these expressions for unique sections is needed. to provide a reasonable and conservative strength prediction.
774 B.W. Schafer / Journal of Constructional Steel Research 64 (2008) 766778
Fig. 8. Mixed local and distortional mode that occurs because of a hole in a
C-section column.
Although angles are geometrically one of the simplest cold- reported in [52] and discussed in [53]. Please note, the results
formed steel members they are not prequalified for use in of Fig. 9 differ from those reported in [53], in which it was
the Direct Strength Method implementation in [2]. Recently, assumed that a linear interaction diagram could be used for the
Rasmussen in [48] extended his work on angles to include Direct Strength Method, and no elastic buckling analysis was
a Direct Strength Method approach. The work explicitly performed for the eccentric loading.
considers eccentricity thus requiring a beamcolumn For any applied combination of P and M (which defines the
approach even for nominally concentrically loaded angle angle in the interaction diagram) the combination that causes
columns. Consistent with the Direct Strength Method the first yield, y , and elastic buckling, cr , (typically determined
developed beamcolumn approach uses the stability of the by finite strip analysis) are constructed. Using the same basic
angle under the applied compression + bending stresses which Direct Strength Method equations as before, but now replacing,
accurately reflects the fact that some eccentricities (away from e.g. Pcr and Py with cr and y the nominal capacity, n ,
the legs) benefit the strength and others (towards the legs) do may be determined. An example of the resulting Direct Strength
Method interaction curve is illustrated in Fig. 9. As discussed
not.
above, the methodology has been applied to angles in [48].
Work performed in [49] examines a Direct Strength Method
Comparison to long-column data is provided in [54] with
approach that ignores eccentricity for angle columns, and also
further discussion and an example in [55]. Further experimental
further explores the relationship between local-plate buckling
and analytical research in this area is currently underway.
and global-torsional buckling of equal leg angle columns; these
authors argue that when one considers the potential for multiple 7.6. Using pure mode analysis from GBT or cFSM
half-waves along the length local-plate and global-torsional
should be treated as unique modes. For now, the Direct Strength Application of the Direct Strength Method is greatly
Method detailed in [48] is the most consistent and rational aided by computational elastic buckling analysis. In fact, the
extension of current design methodologies, though the work development of the Direct Strength Method equations relied on
in [49] may eventually provide a simpler approach. the finite strip method, in particular [10]. However, the finite
strip method does not always provide a definitive identification
7.5. Beamcolumns of the modes (i.e. which result is local, distortional, and/or
global buckling), see [56] for example. Further, the finite
The design of beamcolumns represents an opportunity element method (using plate or shell elements to comprise
for the Direct Strength Method to significantly diverge from the section) provides no definitive method for identifying
current practice. Since the stability of the section can be the modes. The Direct Strength Method requires that the
considered directly under the applied loads (P) and moments modes be positively identified so that the equations may be
(M) the interaction between P and M becomes cross-section applied. Generalized Beam Theory (GBT) [57,58], and now
specific; instead of the invariant interaction equations used the constrained Finite Strip Method (cFSM) [10,59,60] provide
in design specifications such as [29]. A basic methodology methods for definitively separating the buckling modes from
for the application of the Direct Strength Method for beam one another. This not only provides the potential for a cleaner
columns was proposed in [50,51] and a complete design and clearer implementation of the Direct Strength Method, but
example using this methodology provided in [37]. The method goes much further to opening up the possibility of automating
is conceptually summarized in Fig. 9 where a cross-section the strength calculation, which enables optimization efforts,
specific interaction diagram is constructed for the sections such as [61].
B.W. Schafer / Journal of Constructional Steel Research 64 (2008) 766778 775
One word of caution about the application of the pure design aids are now available for engineers who want to
mode solutions of GBT (e.g. [62,63]) or cFSM, they are not apply the Direct Strength Method in design. Expansion of the
identically the same as those used in developing the Direct Direct Strength Method to cover, shear, inelastic reserve, and
Strength Method. As shown in [64,65] the minima in the finite members with holes are all underway. In addition, development
strip method curve (e.g., Fig. 1(b)) include interaction with of a Direct Strength Method for beamcolumns continues
the other modes. In the case of local and global buckling this and will provide cross-section specific interaction with far
interaction generally is small, but in the case of distortional greater accuracy than the simple (essentially linear) interaction
buckling the minima (i.e. Pcrd ) identified by the conventional equations in current use. Much work remains for the continued
finite strip method may be as much as 10% or more lower development of the Direct Strength Method, but the efforts of
than that identified by GBT or cFSM when only focused on many research groups around the world makes it clear that the
distortional buckling. While it may be possible to recalibrate the Direct Strength Method is on path to be a completely viable
Direct Strength Method curves to these pure mode solutions alterative design procedure for cold-formed steel member
for now it is recommended that the GBT or cFSM solutions be design.
used only for determining the critical half-wavelength but the
all mode or conventional finite strip method solution be used Acknowledgments
for determining the elastic buckling load (or moment). The American Iron and Steel Institute is gratefully
acknowledged for their support in nearly all of the research
7.7. Other materials: Stainless steel, hot-rolled steel, alu- presented herein. In addition, the author would like to
minum, plastics acknowledge the National Science Foundation under Grant
No. CMS-0448707 for their funding support. Any opinions,
While not the focus of this review, the application of the findings, and conclusions or recommendations expressed in this
Direct Strength Method to other materials where cross-section material are those of the author and do not necessarily reflect
stability plays an important or dominant role in the strength the views of the National Science Foundation. Finally, recent
determination is underway. For example, in stainless steel research by Tom Sputo and Jennifer Tovar that was shared with
see [66]. for hot-rolled steel see [55], for aluminum see [67 the author lead to the inclusion of Fig. 5.
69], and for thermoplastics see [70]. The basic methodology has
even proved useful in investigating the stability of more unique Appendix A. Direct Strength Method for columns
cross-sections such as the human femur [71].
(As excerpted from Appendix 1 of the North American
7.8. Elevated temperatures Specification for the Design of Cold-Formed Steel Structural
Members, 2004 Supplement to the 2001 Edition.)
Researchers [72,73] have begun to investigate the applica-
1.2.1. Column design
bility of the Direct Strength Method for the design of cold-
formed steel members under fire conditions. The work is in its The nominal axial strength, Pn , is the minimum of Pne , Pn ,
beginning stages and is numerical in nature. Using shell ele- and Pnd as given below. For columns meeting the geometrical
ment based finite element models and appropriately modifying and material criteria of Section 1.1.1.1, c and c are as
E and f y to reflect a simulated elevated temperature both re- follows:
search groups show good agreement with the Direct Strength
USA and Mexico Canada
Method expressions (suitably modified for the lower E and f y ).
Significant research in this area remains, but the initial results c (ASD) c (LRFD) c (LSD)
1.80 0.85 0.80
are promising.
For all other columns, and of Section A1.1(b) apply.
8. Conclusions
1.2.1.1. Flexural, torsional, or torsionalflexural buckling
The Direct Strength Method is a new design methodology The nominal axial strength, Pne , for flexural, . . . or torsional-
for cold-formed steel members. The method has been formally flexural buckling is
adopted as an alternative design procedure in Appendix A of the
for c 1.5Pne = 0.658c Py
2
(1.2.1.1)
North American Specifications for the Design of Cold-Formed
Steel Structural Members, as well as in the Australian/New
0.877
Zealand Standard for cold-formed steel design. The Direct for c > 1.5Pne = Py (1.2.1.2)
2c
Strength Method employs gross cross-section properties, but
requires an accurate calculation of member elastic buckling where
behaviour. Numerical methods, such as the finite strip method c = Py /Pcre
p
(1.2.1.3)
or generalized beam theory, are the best choice for the required
Py = A g Fy (1.2.1.4)
stability calculations. The reliability of the Direct Strength
Method equals or betters the traditional Effective Width Method Pcre = Minimum of the critical elastic column buckling load
for a large database of tested beams and columns. Extensive in flexural, torsional, or torsionalflexural buckling . . . .
776 B.W. Schafer / Journal of Constructional Steel Research 64 (2008) 766778
where
1.2.1.2. Local buckling
The nominal axial strength, Pn , for local buckling is M y = S f Fy , where S f is the gross section modulus
referenced to the extreme fibre in first yield (1.2.2.4)
for ` 0.776 Pn = Pne (1.2.1.5)
Mcre = Critical elastic lateral-torsional buckling moment . . . .
for ` > 0.776
" 0.4 # 0.4
(1.2.1.6)
Pcr` Pcr` 1.2.2.2. Local buckling
Pn = 1 0.15 Pne
Pne Pne The nominal flexural strength, Mn , for local buckling is
where for ` 0.776 Mn = Mne (1.2.2.5)
` = Pne /Pcr`
p
(1.2.1.7) for ` > 0.776
Pcr = Critical elastic local column buckling load . . .
Mcr`
0.4 !
Mcr `
0.4
(1.2.2.6)
Pne is defined in Section 1.2.1.1. Mn = 1 0.15 Mne
Mne Mne
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