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Collapse mechanisms of aluminium structures in fire : thedirect strength methodvan der Meulen, O.R.
Published: 01/01/2010
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The Direct Strength Method
Collapse mechanisms ofaluminium structures in fire
ir. O. R. van der Meulen
August 13, 2010
Oege Ronald van der Meulen is a PhD student at the Eindhoven University ofTechnology, performing research on the stability of aluminium structures in fire.
This research was carried out under project number M81.1.108306 in theframework of the research program of the Materials innovation institute M2i(www.m2i.nl).This work is licensed under the Creative Commons Attribution 3.0 Unported Li-cense. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/or send a letter to Creative Commons, 171 Second Street, Suite 300, San Fran-cisco, California, 94105, USA.When citing this work please use: ‘van der Meulen, O. R. (2010, August) TheDirect Strength Method, Collapse mechanisms of aluminium structures in fire.Available at https://venus.tue.nl/ep-cgi/ep detail.opl?rn=20091711&taal=US’or equivalent.
List of Figures
2.1 Finite strip discretization of a prismatic, thin-walled section . . . 62.2 C-section with lips . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Elastic buckling load for the C-section subjected to bending about
the strong axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Buckled geometries for a beam in bending . . . . . . . . . . . . . 142.5 Buckled geometries for a beam in bending (Scaled) . . . . . . . . 152.6 Elastic buckling load for the C-section subjected to axial compres-
sion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.7 Buckled geometries for a column in compression . . . . . . . . . 162.8 The critical load curve for the C-section in bending according to
FSM and cFSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.9 The deflected and undeflected cross-section at the lowest local
minimum of the cFSM, distortional buckling solution. . . . . . . 202.10 The critical load curve for a C-section with the contribution from
the different buckling modes . . . . . . . . . . . . . . . . . . . . . 22
3.1 The Direct Strength Method compared to test data for thin-walledsteel columns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 The Direct Strength Method compared to test data for thin-walledsteel braced beams. . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 The Direct Strength Method compared to test data and nonlinearFEM results for thin-walled steel C- and Z-sections in bending . 28
3.4 Buckling load as a function of the half-wave length . . . . . . . . 313.6 Normalized beam column interaction diagram . . . . . . . . . . . 363.7 Interaction diagram for local buckling in compression and bending 38
List of Tables
2.1 Mode classification criteria in cFSM . . . . . . . . . . . . . . . . 18
3.1 Resistance factors for LRFD design . . . . . . . . . . . . . . . . . 33
Contents
List of Figures iii
List of Tables v
Contents vii
1 Introduction 1
2 Finite Strip Method 52.1 Conventional finite strip method . . . . . . . . . . . . . . . . . . 5
2.1.1 Elastic stiffness tensor . . . . . . . . . . . . . . . . . . . . 72.1.2 Elastic geometric stiffness tensor . . . . . . . . . . . . . . 82.1.3 Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1.4 Solving the system . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Critical load charts . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 Compression . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Constrained finite strip method . . . . . . . . . . . . . . . . . . . 172.3.1 Modal decomposition . . . . . . . . . . . . . . . . . . . . 182.3.2 Mode contribution calculation . . . . . . . . . . . . . . . . 20
3 Direct Strength Method 233.1 Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 Global buckling . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2 Local buckling . . . . . . . . . . . . . . . . . . . . . . . . 243.1.3 Distortional buckling . . . . . . . . . . . . . . . . . . . . . 24
3.2 Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.1 Global buckling . . . . . . . . . . . . . . . . . . . . . . . . 253.2.2 Local buckling . . . . . . . . . . . . . . . . . . . . . . . . 263.2.3 Distortional buckling . . . . . . . . . . . . . . . . . . . . . 263.2.4 Deflections and serviceability . . . . . . . . . . . . . . . . 273.2.5 Shear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.6 Web crippling . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.7 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Interaction of buckling modes . . . . . . . . . . . . . . . . . . . . 283.4 Beam and column charts . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.1 Local buckling as a function of length . . . . . . . . . . . 293.4.2 Distortional buckling as a function of length . . . . . . . . 30
viii CONTENTS
3.5 Combined axial load and bending . . . . . . . . . . . . . . . . . . 313.5.1 Traditional interaction equations . . . . . . . . . . . . . . 333.5.2 Beam-Columns according to the DSM . . . . . . . . . . . 34
3.6 DSM for inelastic material behaviour . . . . . . . . . . . . . . . . 38
4 Discussion 394.1 Finite strip method . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Constrained finite strip method . . . . . . . . . . . . . . . . . . . 394.3 The Direct Strength Method . . . . . . . . . . . . . . . . . . . . 40
Bibliography 43
Chapter 1
Introduction
One of the most used methods of calculating the strength of cross-sections thatare affected by local buckling, is the effective width approach as pioneered by vonKarman et al. (1932) and later modified to account for imperfections by Winter(1947). The approach simplifies reality by treating the total cross-section assimply supported plates and calculating the critical stress σcr of each of theseplates separately through the well known equation
σcr =Kcrπ
2Et2
12 (1− ν2) b2, (1.1)
where Kcr is the critical buckling factor, its magnitude dependent on the type ofplate and load, equal to 4.0 for plates simply supported on all sides, loaded incompression. t and b are the thickness of the considered plate, respectively, andE and ν are the Young’s modulus and Poisson’s ratio. The effective portion ofthe width of the plate is then calculated by
beb
= α
√σcrσy
(1− β
√σcrσy
), (1.2)
where (i)be is the effective width, σy the yield stress and α and β are constants,determined to be equal to 1 and 0.25, respectively by Winter (1947) for platessimply on all four sides, and equal to 1.19 and 0.0993 for plates simply supportedon three sides by Kalyanaraman et al. (1977). The effective portion of the entirecross-section can thus be found, and its strength and stiffness determined. Butthe procedure is iterative due to the change in effective cross-section affecting thestrain distribution, and can quickly become too complicated for hand-calculationfor cross-sections with more than a few plates.
A second drawback is that considering all plates to be simply supported,neglects the stabilizing influence of neighbour plates. A way of incorporating thisinfluence is by using semi empirical formulae for the value of Kcr for multipleplates interacting, or even the cross-section as a whole. An overview of suchformulae is given in (Rondal and Dubina, 2005), with the most relevant ones forthe types of geometry in aluminium design repeated here. Schafer (2001, 2002a)proposed equations for the flanges of cross-sections interacting with lip stiffenersand the web, respectively.
2 Introduction
For flange-lip
Kcr = −11.07
(d
b
)2
+ 3.95
(d
b
)+ 4, For
d
b< 0.6 (1.3)
For flange-web
Kcr =
4(bh
)2 [2−
(bh
)0.4]. For h
b ≥ 1
4[2−
(hb
)0.2]. For h
b < 1(1.4)
More equations for the critical buckling factor Kcr can be obtained from (Batista,1989), the formulae for the lipped and plain channel are omitted here, but theequation for a hollow rectangular section is given as.
For rectangular hollow section
Kcr = 6.56−5.77
(b2b1
)+8.56
(b2b1
)2
−5.36
(b2b1
)3
, For 0.1 ≤ b2b1< 1.0
(1.5)
where B1 and b2 are the height and width of the profile,respectively.Equations of the type as presented here, only exist for a limited range of
geometries and are as such not a practical solution. The effective width methodis applicable to wide range of cross-sections, but is, as was already mentioned,very conservative in some instances. As the complexity of the cross-section isincreased in an effort to further optimize it with edge and intermediate stiffeners,the assumption of isolated, simply supported, plates becomes even less accurate.
The Direct Strength Method was presented by Schafer and Pekoz (1998) toovercome the posed drawbacks of existing methods. It uses an elastic bucklingsolution for the member as a whole, which is used as an input to the strengthcurve for the entire member. Strength curves were given in (Schafer and Pekoz,1998) for beams and in (Schafer, 2001), for columns.
The Direct Strength Method was not referred to by that name at the time,but it has its roots in the research to distortional buckling by (Hancock et al.,1994; Lau and Hancock, 1987) and has been part of the Australian/New Zealandstandard for thin-steel design (AS/NZ 4600, 2005) since 1996. Hancock himselftraces the origin even further, to the work of Trahair. It shall be clear thatthe Direct Strength Method is by no means an entirely new concept. Theinnovative part however, is its extension to include local buckling. The methodcan now be used to simply calculate the behaviour of members, taking intoaccount, local, distortional and overall buckling phenomena, when the elasticbuckling (Eulerian) solution for the entire member is known. Such a solutioncan be obtained by means of the Finite Element Method (FEM), the GeneralisedBeam Theory (GBT), the (constrained) Finite Strip Method FSM/cFSM, orother less used techniques as boundary value methods. For certain standardgeometries, solutions of the form of equation (1.5) may even be available. Inpractice (c)FSM is the easiest to use as it requires less time and inputs ascompared to FEM, and it has a very convenient, open source, pre-post processorand solver called CUFSM (Schafer and Adany, 2006a), available freely from thewebsite of professor Schafer’s thin-walled structures research group at the Johns
3
Hopkins university. A commercial implementation of FSM also exists, calledThinwall (Papangelis and Hancock, 2005) developed by university of Sydney.
At present the Direct Strength Method has been accepted as an alternativemethod to calculate the behaviour of thin walled steel in the North Americanstandard (AISI S100-2007, 2007), since 2004. It is also been accepted for use in theAustralian/ New Zealand standard for cold-formed steel structures (AS/NZ 4600,2005). Certain issues as, shear, web crippling and members with holes cannotbe dealt with using the Direct Strength Method at the level of development aspresent in the standard, and checks have to be made using the existing methodsin the conventional part of the standard. Since the standard was published,more research was, and is being performed. The mentioned limitations and stepstaken in their resolvement are discussed in (Schafer, 2008).
The present report does not include a description of the Generalised BeamTheory (GBT). This is because the subject of this report is the Direct StrengthMethod and this is used predominantly in conjunction with the finite strip method.It is possible to use GBT with the Direct Strength Method , because there isonly a small difference in the critical loads derived using FSM and GBT (Adanyet al., 2009). The theory behind the GBT is rather complex however, andmay not be used to simulate all types of cross-sections as typically found inaluminium extrusion design. (The theory as defined in (Schardt, 1989) excludesclosed cross-sections and branched members.) The interested reader is referredto (Camotim et al., 2004; Davies and Jiang, 1996, 1998; Schardt, 1989, 1994;Silvestre and Camotim, 2002a,b, 2004; Silvestre and Comatim, 2004). Specificarticles on the subject of GBT in conjunction with round-house materials canbe found in (Goncalves and Camotim, 2004; Goncalves et al., 2004). It is notedthat the range of possible cross-sections for the GBT is being increased (Diniset al., 2004; Goncalves et al., 2004).
Chapter 2
Finite Strip Method
The finite strip method was introduced by Cheung (1968a,b), and was subse-quently extended by Cheung and others. The theory as presented in this sectionis based on the semi-analytical or classical finite strip method, a comprehensivesummary of the method and the theory behind it is given in (Cheung, 1976). Themethod was extended by Hancock (1978) for the solution of the elastic bucklingproblem of open, thin-walled members. Based on these sources, the method wasused by Schafer (1997) to develop a program to calculate the elastic bucklingload of thin-walled steel members called CUFSM (Schafer and Adany, 2006a).The mechanics behind this and their derivation are discussed in section 2.1. Thelatest version of the CUFSM program, also includes an extension to the classicalfinite strip method by (Adany and Schafer, 2008) to allow for the decompositionor the classification of the different buckling modes. The theory behind this isgiven in section 2.3. The classical finite strip method is capable only of simulatingsimply supported. It was later extended from its classical form to be able tocope with more complicated support conditions. This adds to the complexity ofthe method significantly as the longitudinal shape functions are defined now asa series of lines, or splines, leading to significantly larger eigenvalue problems.The method is explained well in Bradford and Azhari (1995). This and otherextensions, along with their derivation are presented in (Cheung and Tham,1999). The conventional finite strip method as used in CUFSM is elaborated onin the next section, following the excellent paper by Schafer and Adany (2006a),with some additions derived from Schafer (1997) and the source code of CUFSM.
2.1 Conventional finite strip method
In the finite strip method, like the more well known finite element method, athin-walled member is discretized into a number of elements. Unlike the finiteelement method, the finite strip method employs only one element in the lengthdirection of the beam, and the elements are referred to as strips. Appropriateshape functions are used to capture the longitudinal variation in displacements.Figure 2.1a shows the discretization of a member into strips. Each strip has eightdegrees of freedom (DOF), as is shown in figure 2.1b, and may be subjected toan edge traction as defined in figure 2.1c. It is noted that the coordinate systemused in the sections concerning the FSM deviates from what is customary; the y
6 Finite Strip Method
(a) Member andstrips
v1v2
b
u1
w1
θ1
u2
w2
θ2
v1v2
v1v2
v1v2
b
u1
w1
θ1
u2
w2
θ2
v1v2
(b) Plate DOF
xy
z
a
T1=f 1 t T2=f 2 t
xy
z
a
T1=f T2=f
(c) Plate traction
Figure 2.1 Finite strip discretization of a prismatic, thin-walled section.From: (Schafer and Adany, 2006a), with minor modifications.
coordinate is specified with the longitudinal axis of the beam, while Z coincideswith the out of plane direction and x the transverse direction. This systemcoincides with the existing literature on the FSM. The vector describing thedisplacement fields u is related to the displacements at the node d by the secondorder tensor N , which contains the shape functions.
u = Nd =
uvw
= N 3x8
u1
u2
v1
v2
w1
θ1w2
θ2
. (2.1)
The shape functions for the membrane displacements u and v are linear in thetransverse (x)-direction, but a harmonic function is used for the longitudinaldirection. As the node for the displacement u is situated in the middle of thestrip, where the displacement is the highest, a sine function is used. The nodesfor the longitudinal displacements v are located at the ends of the strip, and acosine function is used. This choice of longitudinal shape functions is consistentwith simply supported beams.
u = sin(mπy
a
) [1− x
bxb
] [u1
u2
],
v = cos(mπy
a
) [1− x
bxb
] [v1
v2
], (2.2)
where m is the number of half-wavelengths in the length of the beam, whichis invariably equal to 1 in CUFSM. The shape functions for the out-of-planedirection w are cubic and are given by
w = sin(mπy
a
)1− 3x2
b2 + 2x2
b3
x− 2x2
b + x3
b23x2
b2 −2x2
b3x3
b2 −x2
b
T
W1
θ1W2
θ2
. (2.3)
2.1 Conventional finite strip method 7
The nonzero elements of the second order shape function tensor N are thenequal to
N[1,1-2] = sin(mπy
a
) [1− x
bxb
],
N[2,3-4] = cos(mπy
a
) [1− x
bxb
], (2.4)
N[3,5-8] = sin(mπy
a
)1− 3x2
b2 + 2x2
b3
x− 2x2
b + x3
b23x2
b2 −2x3
b3x3
b2 −x2
b
T
.
2.1.1 Elastic stiffness tensor
The strain in the strip is separated into a contribution from membrane (εm)and bending (εb) strains. The chosen longitudinal shape functions make thesequantities uncoupled.
εm =
εxεyγxy
m
=
δuδxδvδy
δuδy + δv
δx
= Bmduv.
εb =
εxεyγxy
b
=
−zδ2wδx2
−z δ2vδy2
2z δ2uδxδy
= Bbdwθ. (2.5)
where the second order tensors Bm, and Bb describe the relationship betweenstrain and nodal displacement, for membrane stresses and Kirchoff thin-platebending, respectively. The second order, elastic stiffness tensor Ke is constructedfrom the decoupled elastic membrane and bending stiffness tensors Kem andKeb, respectively.
Ke =
[Kem 0
0 Keb
], (2.6)
where
Kem =
∫∫∫Bm
TEBm da dbdt,
Keb =
∫∫∫Bb
TEBb dadbdt. (2.7)
which leads to the elastic membrane (Kem) and bending (Keb) stiffness tensorsas given in equations (2.15) and (2.16), respectively. These equations can be
8 Finite Strip Method
found on page 10 and require constants as defined by
km =mπ
a,
E1 =Ex
1− νxνy
,
E2 =Ey
1− νxνy
,
Dx =Ext
3
12 (1− νxνy), (2.8)
Dy =Eyt
3
12 (1− νxνy),
Dxy =Gt3
12,
D1 =νyExt
3
12 (1− νxνy)=
νxEyt3
12 (1− νxνy),
where G is the shear modulus, and νx and νy are the Poisson’s ratio in twoorthogonal directions. It is noted that the last equation (2.8) places a restrictionon valid Poisson’s and Young’s ratio pairs;
νyEx = νxEy. (2.9)
This condition is not mentioned in the documentation for CUFSM. This im-plicit restriction seems to be always present in classical beam theory, see forexample (Reddy, 2007).
2.1.2 Elastic geometric stiffness tensor
The geometric stiffness matrix is obtained by considering the additional potentialenergy gained from a displacement of the edge tractions as defined in figure 2.1cin the longitudinal (y) direction. These tractions vary linearly in magnitudebetween T1 and T2 and the corresponding potential energy function is equal to
U =1
2
∫∫ [T1 − (T1 − T2)
x
b
] [(δuδy
)2
+
(δv
δy
)2
+
(δw
δy
)2]
dxdy, (2.10)
where the tensor resulting from differentiating the shape functions with respectto y is defined here as G [
δuδy
δvδy
δwδy
]T= Gd. (2.11)
Substituting equation (2.11) into equation (2.10) yields an expression for thegeometric stiffness tensor Kg.
U =1
2dT
∫∫ [T1 − (T1 − T2)
x
b
]GTG dx dy d =
1
2dTKgd. (2.12)
Kg may be formed from two uncoupled contributions from membrane(Kgm),and bending deformations (Kgb) according to
Kg =
[Kgm 0
0 Kgb
], (2.13)
2.1 Conventional finite strip method 9
where
Kgm =
∫∫ [T1 − (T1 − T2)
x
b
]Gm
TGm dxdy,
Kgb =
∫∫ [T1 − (T1 − T2)
x
b
]Gb
TGb dx dy. (2.14)
The geometric membrane (Kem) and bending (Keb) stiffness tensors are givenin equations (2.17) and (2.18), respectively, on page 10. Schafer (1997) notesthat the definition of Kg as a stiffness tensor follows directly from the energyderivation of an initial stiffness tensor, and refers the reader to Cook et al. (2001);Crisfield (1996); Ugural and Fenster (2003)1.
1The references have been updated to the latest editions with the cited content
10
FiniteStrip
Method
Kem = t
(aE1
2b +abk2mG
6
)Symmetric(
aKmνxE2
4 − ak2mG4
) (abk2mE2
6 + aG2b
)(−aE1
2b +abk2mG
12
) (−akmνxE2
4 − akmG4
) (aE1
2b +abk2mG
6
)(akmνxE2
4 + akmG4
) (abk2mE2
12 − aG2b
) (−akmνxE2
4 + akmG4
) (abk2mE2
6 + aG2b
)
. (2.15)
Keb = t
(13ab70 k4mDy + 12a
5b k2mDxy
+ 6a5b k
2mD1 + 6a
b3Dx
)Symmetric( 3a
5 k2mD1 + a
5k2mDxy
+ 3ab2Dx + 11ab2
420 k2mDx
) (ab3
210k4mDy + 4ab
15 k2mDxy
+ 2ab15 k
2mD1 + 2a
b Dx
)(
9ab140k
4mDy − 12a
5b k2mDxy
− 6a5b k
2mD1 − 6a
b3Dx
) (13ab2
840 k4mDy − a5k
2mDxy
− a10k
2mD1 − 3a
b2Dx
) (13ab70 k4mDy + 12a
5b k2mDxy
+ 6a5b k
2mD1 + 6a
b3Dx
)(− 13ab2
840 k4mDy + a5k
2mDxy
+ a10k
2mD1 + 3a
b2Dx
) (−ab
3
840k4mDy − ab
15k2mDxy
−ab30k2mD1 + a
bDx
) (− 11ab2
420 k4mDy − a5k
2mDxy
− 3a5 k
2mD1 − 3a
b2Dx
) (ab3
210k4mDy + 4ab
15 k2mDxy
+ 2ab15 k
2mD1 + 2a
b Dx
)
.
(2.16)
Kgm =b (mπ)
2
1680a
70 (3T1 + T2) Symmetric
0 70 (3T1 + T2)70 (T1 + T2) 0 70 (T1 + 3T2)
0 70 (T1 + T2) 0 70 (T1 + 3T2)
. (2.17)
Kgb =b (mπ)
2
1680a
8 (30T1 + 9T2) Symmetric2b (15T1 + 7T2) b2 (5T1 + 3T2)
54 (T1 + T2) 2b (6T1 + 7T2) 24 (3T1 + 10T2)−2b (7T1 + 6T2) −3b2 (T1 + T2) −2b (7T1 + 15T2) b2 (3T1 + 5T2)
. (2.18)
2.1 Conventional finite strip method 11
2.1.3 Assembly
The contributions from the local elastic (ke) and geometric stiffness (kg) tensorsfor each of the individual strips, need to be assembled into the global elasticand geometric stiffness tensor, Ke and Kg, respectively. The strips are in alocal coordinate system, and the coordinates need to be transformed beforeassembly can be performed. The relationship between the global displacements{U, V,W,Θ}, and their local counterparts {u, v, w, θ}, is given by[
Ui
Wi
]=
[cos(α(j)
)− sin
(α(j)
)sin(α(j)
)cos(α(j)
) ] [ui
wi
],[
Vi
Θi
]=
[1 00 1
] [vi
θi
], (2.19)
where i and j refer to the node and strip, respectively, and α(j) is the angle ofstrip j. We may write this more formally as
D(j) = Γ(j)d(j), (2.20)
where
Γ(j) =
cos(α(j)
)0 0 0 − sin
(α(j)
)0 0 0
0 1 0 0 0 0 0 00 0 cos
(α(j)
)0 0 0 − sin
(α(j)
)0
0 0 0 1 0 0 0 0sin(α(j)
)0 0 0 cos
(α(j)
)0 0 0
0 0 0 0 0 1 0 00 0 sin
(α(j)
)0 0 0 cos
(α(j)
)0
0 0 0 0 0 0 0 1
.
(2.21)The elastic and geometric stiffness tensor are transformed to the global coordinatesystem by
Ke(j) = Γ(j)ke
(j)Γ(j)T ,
Kg(j) = Γ(j)kg
(j)Γ(j)T . (2.22)
The stiffness tensors for the member as a whole are then obtained by summationof the stiffness tensors of the individual strips
Ke =n
Aj=1
Ke(j),
Kg =n
Aj=1
Kg(j), (2.23)
where A is defined here as the assembly operator.
2.1.4 Solving the system
The elastic buckling problem is a standard eigenvalue problem, which for a singlemode has the form
KeD = λKgD, (2.24)
12 Finite Strip Method
where λ is the eigenvalue or load factor LF as used in section 3.5.2 and D is theeigenvector containing the global displacements of the degrees of freedom.
Equation (2.24) may be solved by any appropriate eigenvalue algorithm,CUFSM uses the built in MATLAB (2010) solver based on (Anderson et al.,1999). In reality, there are multiple buckling modes and corresponding eigenvalue-eigenvector pairs, CUFSM will output up to ten of the first real solutions, inascending order of eigenvalue. The first of which is nearly allways the dominantmode, as it has the lowest load factor LF.
Both the local elastic (Ke) and geometric stiffness (Kg) tensors have a termfor the beam half-wavelength a, the solutions found for equation (2.24) aretherefore also a function of a. For a complete analysis as shown in figure 2.3 forexample, the problem is solved for a number of different lengths.
The global displacements D(j) may be transformed back to local coordinatesfor post-processing. The required transformation follows from equation (2.20)
d(j) = Γ(j)−1
D(j) = Γ(j)TD(j).(Γ(j)−1
= Γ(j)T)
(2.25)
2.2 Critical load charts
To demonstrate the concept of critical load charts, an example as given in (Schafer,2006) is followed. To allow for an easy comparison between the example cal-culations and the calculations as presented here, the United States customarysystem of units is used for the geometry. A cold formed steel C-section witha geometry according to figure 2.2 is considered. Stresses and strength areidentical to the example case but are expressed in metric units. The materialproperties are identical: A Poisson’s ratio ν = 0.3, a Young’s modulus equalto E = 2.03 · 105 Nmm−2 and a shear modulus G = 7.82 · 104 Nmm−2 isspecified. A yield stress fy = 379 Nmm−2 is specified, but does not influencethe calculations itself as the finite strip method is based on the assumption oflinear elasticity. It does affect the magnitude of the displayed normalized load.
2.2.1 Bending
The geometry of figure 2.2 is subjected to bending about its strong (y) axis. Thefinite strip method based program CUFSM (Schafer and Adany, 2006a) is usedto calculate the elastic buckling load for the cross-section for a large number ofdifferent half-wavelengths. (for a simply supported beam, the half-wavelength isequal to the system length). The results are shown in figure 2.3.
Figure 2.3 shows the elastic buckling load as a function of the half-wavelength.It is important to note that deformation patterns are able to repeat themselvesat scalar multiples of their base half-wavelength, and therefore no increase instrength may be assumed for greater lengths. The critical- and yield momentare denoted as Mcr and My, respectively. Three different modes of bucklingmay be identified,(i) Local buckling at a half-wavelength Lcrl = 126 mm witha normalized moment McrM
−1y = 0.67, (ii) Distortional buckling with Lcrd =
629 mm and McrM−1y = 0.85, and (iii), global buckling for the larger half-
wavelengths. The local minima are used to identify the local and distortionalbuckling points, guidance for less readily identifiable cases is offered in (Schafer,2006). The in-plane deformation patterns for the three buckling modes belonging
0.1875”
9”
2.5”
0.773”
t = 0.059”
z
y
Figure 2.2 C-section with lips. The geometry is taken from (Schafer,2006)[section 3.2.1] and (American Iron and Steel Institute, 2002). Alldimensions are in inches. The dots indicate the position of the nodes usedin the FSM simulation.
101 102 1030
0.5
1
1.5
2
Local,Mcr
My= 0.67
Distortional,Mcr
My= 0.85
Global
Half-wavelength (in)
Mcr
My
Figure 2.3 Elastic buckling load for the C-section as specified in figure 2.2subjected to bending about the strong axis. The abscissa is logarithmic andexpressed in inches. After: (Schafer, 2006).
14 Finite Strip Method
(a) Local (b) Distortional (c) Lateral-torsional
Figure 2.4 Buckled geometries for the beam of figure 2.3 subjected tobending about its strong axis. The half-wavelengths correspond to the localminima of figure 2.3. Proportional scaling was applied to show the bendingmembers side by side.
to the marked points are also given in figure 2.3. The same deformation patternsare also shown, in three dimensions, in figure 2.4. Proportional scaling wasapplied to show the different lengths side by side. To make the beams of figure 2.4more easy to compare, the length is scaled independently, setting it equal to thegreatest in-plain dimension. This is shown in figure 2.5.
2.2.2 Compression
When the same geometry is subjected to an axial force alone, the elastic bucklingloads as given in figure 2.6 are produced by CUFSM. The distortional bucklingmode does not have a local minimum, and the boundary between local anddistortional buckling needs to be identified by manually judging the deformedshapes. The three buckling modes are:(i) Local buckling, at a half-wavelengthLcrl = 168 mm with a normalized load FcF
−1y = 0.12, (ii) Distortional buckling
with Lcrd = 724 mm and FcF−1y = 0.27, and (iii), global buckling for the larger
half-wavelengths. The in-plane deformation patterns for the three buckling modesbelonging to the marked points are shown in three dimensions in figure 2.7.
(a) Local (b) Distortional (c) Lateral-torsional
Figure 2.5 Buckled geometries for the beam of figure 2.3 subjected tobending about its strong axis. The half-wavelengths correspond to the localminima of figure 2.3. The length DRAWN was scaled to make the lengthequal to the greatest cross-section dimension.
101 102 1030
0.1
0.2
0.3
0.4
0.5
Local,Mcr
My= 0.12
Distortional,Mcr
My= 0.27
Global
Half-wavelength (in)
Fcr
Fy
Figure 2.6 Elastic buckling load for the C-section as specified in figure 2.2subjected to axial compression. The abscissa is logarithmic and expressedin inches. After: (Schafer, 2006).
(a) Local (b) Distortional (c) Flexural
Figure 2.7 Buckled geometries for the cross-section of figure 2.2 subjectedto axial compression. The length DRAWN was scaled to make the lengthequal to the greatest cross-section dimension.
2.3 Constrained finite strip method 17
2.3 Constrained finite strip method
Critical elastic-load calculations using the finite strip method (FSM) as given insection 2.1, and the finite element method alike (FEM), do not provide a wayfor distinguishing between he different buckling modes, while this separation isrequired for the Direct Strength Method as given in section 3. It is up to theuser to identify points of ‘pure’ modes, which can be tricky if these points do notlie in clearly distinguishable local minima of the critical load graph. Guidance isoffered in (Schafer, 2006), but the task of identifying modes is left a manual one.One reason for this is that clear mechanical definitions for the different bucklingmodes do not exist. While there is consensus over which modes exist and theseare even discussed in modern design codes, thorough mechanical definitions arelacking (Adany and Schafer, 2008).
A method that is able to distinguish between the different modes, is theGeneralized Beam Theory (Schardt, 1989), or (GBT). While this is a greatadvantage, GBT does have its disadvantages, particularly with regard to gener-ality (Adany and Schafer, 2008). A way was sought to include the GBT’s abilityto distinguish between modes into more general FEM/FSM methods. This ledto the development of the constrained Finite Strip Method, or (cFSM). Theseparation of global and distortional modes is discussed in (Adany and Schafer,2006a,b), and examples of their use and inclusion into CUFSM are providedin (Schafer and Adany, 2005, 2006a,b). Full modal decomposition is presentedin Adany and Schafer (2008). And an overview and comparison of both GBTand cFSM was presented in Adany et al. (2006, 2007, 2009).
Unfortunately, including the classification system of the GBT does placerestrictions on the type of cross-sections that can be simulated. With thecurrent system as described here, only open, non-branched cross-sections maybe simulated with cFSM. For thin-walled steel structures this is a less severerestriction as these are made by folding plates of steel and the laws of economyalready discourage them from being welded into hollow members and I-sections.For aluminium members the restriction is more severe, as extrusion members can,and quite frequently are made into closed shapes with many branches (stiffeners).It is noted GBT is in continuous development and significant progress has beenmade in generalizing it for various geometries such as closed and branchedcross-section (within certain conditions). See for example (Camotim et al., 2004).It may be possible that such improvements find their way into cFSM as well,but at present, no evidence of this being undertaken exist, nor is it known if thisis even possible.
To separate the contributions from the different buckling modes, the criteriaapplied in the GBT are used. These are ‘. . . found to usually be in good agreementwith current engineering classifications’ (Schafer and Adany, 2006a). Moreinformation on the modal identification within GBT can be found in (Silvestreand Camotim, 2002a,b). The separation into (i) Global, (ii)Distortional, (iii)Local, and (iv) Other modes, is performed through the application of the followingthree criteria:
Criterion 1, Vlasov’s hypothesis. A condition derived from classical beamtheory, or (Vlasov, 1959) hypothesis. Certain membrane deformations arerestricted, while warping is allowed.
• γxy = 0, there is no in plane shear strain.
18 Finite Strip Method
Table 2.1 Mode classification criteria in cFSM, based on the criteria 1, 2and 3, the four different buckling modes can be distinguished.
Global Distortional Local Other
Criterion 1-Vlasov’s hypothesis Yes Yes Yes NoCriterion 2-Longitudinal warping Yes Yes NoCriterion 3-Undistorted section Yes No
• εx = 0, there is no transverse strain.
• The axial displacement v is linear within a flat plate. (Flat platesinclude all contiguous strips in a plane.)
Criterion 2, Longitudinal warping. A condition specifying warping to oc-cur. This condition allows us to separate local buckling from other bucklingmechanisms, as local buckling does not induce a deformation at the centreand consequently has a longitudinal displacement v equal to zero.
• v 6= 0, non zero warping displacements must occur.
• εx = 0, transverse extension equal to zero.
• Linear axial displacements v along all of the wall mid-lines, the cross-section is in transverse equilibrium.
Criterion 3, Undistorted section. The distortion of the cross-section is thebasis for the distinction between global and distortional modes
• kxx = 0, There is no transverse flexure.
The three criteria are applied to discriminate between the different bucklingmodes according to table 2.1.
2.3.1 Modal decomposition
The conventional finite strip method finds a solution that includes all bucklingmodes. Using the confined finite strip model, it is possible to derive a solutionbased on just one or more of the separate buckling modes. The total DOFspace D is therefore reduced to the subspace belonging to the buckling mode, orbuckling modes of interest
D = RmDm, (2.26)
where DM is the displacement vector in the reduced space, and RM is the secondorder constraint tensor for the given buckling modes. The subscript M can bereplaced by the subscripts C, D, L, O, or for example GL, for global, distortional,local, other, or a combination of local and global buckling, respectively. Theunity of all modes leads to the original FSM problem, or
RC ∪RD ∪RL ∪RO ≡ I, (2.27)
where I is the identity tensor. The eigenvalue problem for the conventional finitestrip method was given in equation (2.24) for one eigenvalue/vector only. For anumber of eigenvalues/vectors m, the equation is rewritten to
KeΨD = ΛKgΨD, (2.28)
2.3 Constrained finite strip method 19
102 103 1040
0.1
0.2
0.3
0.4
0.5
Local
Distortional
Global
Half-wavelength (in)
Mcr
My
Figure 2.8 The critical load curve for the C-section as was shown in fig-ure 2.3, now obtained with both the constrained (cFSM) and the conventional(FSM) finite strip method. The dashed lines give the solution accordingto cFSM. It is clear that cFSM over-predicts the actual elastic stabilityfor distortional buckling in particular. This is caused by the interaction ofmodes leading to a lower elastic buckling solution
where Λ and ΨD are second order tensors, containing the eigenvalues andeigenvectors; Λ = diag [λ1, λ2, . . . , λm], and ΨD = [D1,D2, . . . ,Dm]. Insertingequation (2.26) into equation (2.28) and pre-multiplying by Rm
T yields
RMTKeRMΨD = ΛRM
TKgRMΨD, (2.29)
which can be rewritten to
Ke,MΨD = ΛKg,MΨD, (2.30)
where Ke,M, and Kg,M are the stiffness tensors of the constrained FSM problem
Ke,M = RMTKeRM,
Kg,M = RMTKeRM. (2.31)
Equation (2.30) is the eigenvalue problem of the cFSM problem, which can againbe solved with any suitable eigenvalue algorithm.
RMis a second order tensor with m rows and mM columns, where mM is thedimension of the reduced DOF space. Consequently, the stiffness matrices of theconstrained FSM problem, are mM by mM tensors, which can be significantlysmaller than the original problem. The modal constrained can thus be considereda model reduction technique, with only the DOFs that are applicable to themode(s) under investigation.
Solving the eigenvalue problem (2.30) for RM equal to RG, RD, RL and RO,consecutively, gives the full critical load chart, but now separated into the fourbuckling modes. This can be seen in figure 2.8
20 Finite Strip Method
Figure 2.9 The deflected and undeflected cross-section at the lowest localminimum of the cFSM, distortional buckling solution. The deflected shapecould also be considered local buckling.
It can be observed from figure 2.8 that the cFSM methods gives higher valuesfor the pure buckling modes than the combined solution according to the FSM.This is because the interaction of buckling modes lowers the bifurcation load ofthe member. A second observation is that the cFSM solution for distortionalbuckling has two local minima, the lowest of which overlaps with the localbucking mode. A closer examination of this lower minimum reveals it to be akind of local buckling, with the lips deflecting. Such a deformation pattern isusually associated with local buckling. A deflected cross-section at this point ispresented in figure 2.9. At this point, an error in CUFSM or its inputs can not beruled out, but the observed behaviour is not reported in any of the publicationsconcerning cFSM. The model decomposition technique of cFSM gives criticalload solutions for all the buckling modes separately. This potentially simplifiesthe usage of design approaches based on these separate critical buckling loadvalues, such as the Direct Strength Method of section 3. It also allows theprocedure to be automatized, as the manual mode classification step is removed.The higher critical load values obtained with cFSM would require a recalibrationof the design methods based on the critical loads. Another concern is thatsolutions according to FSM are a more realistic description of the actual loadthan the pure modes of the cFSM simulation.
2.3.2 Mode contribution calculation
When the modal decomposition technique as outlined in section 2.3.1 is usedto solve the elastic buckling problem, a solution is obtained for the eigenvalueproblem of equation (2.30). The resulting eigenvectors dm lie in a subspace ofthe original DOF space of the FSM problem. The subspaces are denoted G, D, Land O for the Global, Distortional, Local and Other mode buckling respectively.These subspaces are spanned by the second order restrain tensors, RG, RD, RL
and RO, respectively. But the columns of Ψm span the subspaces as well andcan be regarded as their base vectors, with the added property that they areorthogonal to each other with respect to the elastic and the geometric stiffness
2.3 Constrained finite strip method 21
tensor. It is thus possible to write the vector dm as a linear combination of thebase vectors contained in Ψm
dm = Ψmcm, (2.32)
where the vector cm defines the relative contribution of the different modes. (Upto now, the word mode was used as synonymous with buckling mode (G,D,L andO), here the word modes represents the different patters that may develop withina given buckling mode. Usually the first mode is relevant only as they are sortedin order of increasing eigenvector (loadvector)). In analogy with equation (2.26)the displacements d in the local coordinate system within the full DOF space,can be written as a function of the local coordinates in the subspaces
d = Rmdm. (2.33)
Inserting equation (2.32) into equation (2.33) yields
d = RmΨmcm = Ro
mcm, (2.34)
where Ro
mis the constraint tensor, transformed by Ψm to the orthogonal base
system.The eigenvalue problem as posed in equation (2.30) can be solved for the
four buckling modes independently using the modal decomposition technique ofsection 2.3.1, and the resulting eigenvectors ΨG,ΨD, ΨL and ΨO, are then theorthogonal base vectors. It is noted in (Adany and Schafer, 2008) that for eachloading case, a different set of orthogonal base vectors exists, but that whenorthogonality is ‘important’, only the simplest case of uniform compression needsto be considered. Equation (2.34) can then be used to calculate the tensors Ro
m
from the base systems Ψm and the restraint tensors Rm. These tensors Ro
mare
assembled into Ro
GDLO=[Ro
GRo
DRo
LRo
O
], and any local deformation can be
written as a combination of this quantity.
d = Ro
GDLOco
GDLO=[Ro
GRo
DRo
LRo
O
] co
G
co
D
co
L
co
O
. (2.35)
The second order tensor co
GDLOexpresses the relative contributions of the different
buckling modes. It is in fact the sought after modal contribution tensor. It isnot uniquely defined by the orthogonal base system alone as a normalization ofthe orthogonal system is still required. Different option for the normalizationexist ans are discussed in (Schafer and Adany, 2006a), but it was found that anormalization of the buckling modes based on the work performed by the modegives the most physically sound results. This evolves scaling such that
1
2Ri
TKeRi = 1, (2.36)
The example calculation of section 2.2.1 is shown here again, but now the criticalload curve can be shown along with the relative contribution of the differentmodes of buckling. This is shown in figure 2.10 and figure 2.11.
102 103 1040
0.2
0.4
0.6
Half-wavelength L (mm)
Ncr
Ny
Global
Distortional
Local
Other
Figure 2.10 The critical load curve for the C-section in bending as wasshown in figure 2.3. The relative contribution of the various buckling modesis shown, normalized to the strain energy.
102 103 1040
20
40
60
80
100
Half-wavelength L (mm)
%
Global
Distortional
Local
Other
Figure 2.11 The relative contribution of the different buckling mode forthe C-section in bending example as was shown in figure 2.3. Normalizingwas performed according to the strain energy.
Chapter 3
Direct Strength Method
the Direct Strength Method was already introduced in section 1.It is a techniqueto calculate the resistance of the entire cross-section based on the critical,Eulerian buckling loads calculated for the cross-section as a whole. This deviatessignificantly with the effective width approach (von Karman et al., 1932), whichits most widely used alternative and the method used in most design standardworldwide. The advantage of calculating the critical buckling strength for thecross-section as a whole, is that it correctly takes into account the restrainingaction of neighbouring plates, while the effective width approach considers eachplate to be isolated from the rest and simply supported. This leads to a moreaccurate and higher critical strength value. A disadvantage of the Direct StrengthMethod is that the critical strength calculation becomes more complicated thanfor the isolated plates of the effective width theory. This is why A method likethe finite strip method of chapter 2, or an alternative is required.
In this chapter, the Direct Strength Method as it is included in the NorthAmerican standard for thin-walled steel (AISI S100-2007, 2007) is discussed.Additional developments and research are also included, and identified as such
3.1 Columns
The nominal axial strength of a column is denoted as Nn and is equal to theminimum of the global, local and distortional nominal buckling strengths Nne,Nnl and Nnd, respectively:
Nn = min (Nne, Nnl, Nnd). (3.1)
The nominal buckling strength according to the three different modes is calculatedaccording to equations as given in the next three subsections. It is noted thatnot all modes have to be present for a given cross-section, and strength checksfor non-existent modes are omitted.
24 Direct Strength Method
3.1.1 Global buckling
The nominal axial strength Nne for global buckling, i.e. flexural, torsional andtorsional-flexural buckling, is given by
Nne ≡
{Ny0.658λ
2c , For λc ≤ 1.5
Ny0.877λ2c, For 1.5 < λc
(3.2)
where
λc =
√Ny
Ncre
,
Ny = Agfy, (3.3)
and Ncre is the minimum of the critical buckling loads for flexural, torsionaland torsional-flexural buckling, calculated according to the traditional equationsaccording to the main part of the standard (AISI S100-2007, 2007)
3.1.2 Local buckling
The nominal axial strength for local buckling is denoted as Nnl and is given by
Nnl ≡
Nne, For λl ≤ 0.776
Nne
[1− 0.15
(Ncrl
Nne
)0.4](Ncrl
Nne
)0.4, For 0.776 < λl
(3.4)
where
λl =
√Nne
Ncrl
, (3.5)
and Ncrl is the critical elastic local -buckling load, and Nne is the global bucklingload as already defined in equation (3.2). The nominal local buckling curve (3.4)is shown in figure 3.1, in conjunction with test results as collected by Schafer(2002a), used to calibrate the local buckling curve.
3.1.3 Distortional buckling
The nominal axial strength for distortional buckling is equal to Nnd, as definedby
Nnd ≡
Ny, For λd ≤ 0.561
Ny
[1− 0.25
(Ncrd
Ny
)0.6](Ncrd
Ny
)0.6, For 0.561 < λd
(3.6)
where
λd =
√Ny
Ncrd
, (3.7)
and Ncrd is the critical elastic distortional -buckling load, and Ny is the yieldstrength of the gross-area of the cross-section Ag( equation (3.3)). The nominaldistortional buckling curve (3.6) is also shown in figure 3.1, together with testresults as collected by Schafer (2002a), used to calibrate the distortional bucklingcurve.
3.2 Beams 25
0 2 4 6 80
0.5
1
1.5
λd=√
Ny
Ncrd/ λl=
√Nne
Ncrl
( N test
Ny
) d
/( N te
st
Nne
) l Equation (3.4)
Equation (3.6)
local
Distortional
Figure 3.1 The Direct Strength Method compared to test data for thin-walled steel columns. Source: (Loughlan, 1979; Miller and Pekoz, 1994;Mulligan, 1983; Polyzois and Charnvarnichborikarn, 1993; Thomasson, 1978).From: (Schafer, 2002a), with minor modifications: Axes and legend havebeen re-drawn.
3.2 Beams
The nominal flexural strength of a beam is denoted as Mn and is equal to be theminimum of the global, local and distortional nominal flexural strengths Mne,Mnl and Mnd, respectively:
Mn = min (Mne,Mnl,Mnd). (3.8)
The nominal buckling strength according to the three different modes is calculatedaccording to equations as given in the next three subsections. It is noted thatnot all modes have to be present for a given cross-section, and strength checksfor non-existent are omitted
3.2.1 Global buckling
The global buckling mode for flexural members is lateral-torsional buckling, andthe nominal strength of a member for this mode is denoted as Mne and is equalto
Mne ≡
Mcre, For Mcre < 0.56My
109 My
(1− 10My
36Mcre
), For 0.56My ≤Mcre ≤ 2.78My
My, For 2.78My < Mcre
(3.9)
where the yield moment My is equal to the product of the elastic section modulusWel and the yield stress fy. (The original notation for the section modulus, Sf
as used in (AISI S100-2007, 2007) is replaced here)
My = Welfy, (3.10)
where Mcre is the critical elastic lateral-torsional buckling moment.
26 Direct Strength Method
0 1 2 3 4 50
0.5
1
1.5
λmax=√
My
Ncr
Mtest
My
Equation (3.11)
Equation (3.13)
local
Distortional
Figure 3.2 The Direct Strength Method compared to test data for thin-walled steel braced beams. Source: (Loughlan, 1979; Miller and Pekoz, 1994;Mulligan, 1983; Polyzois and Charnvarnichborikarn, 1993; Thomasson, 1978).From: (Schafer, 2002a), with minor modifications: Axes and legend havebeen re-drawn.
3.2.2 Local buckling
The nominal flexural strength for local buckling is denoted as Mnl. The formula-tion is identical to the local buckling strength for the nominal axial strength Pnl,but with all axial forces replaced by their flexural counterparts.
Mnl ≡
Mne, For λl ≤ 0.776
Mne
[1− 0.15
(Mcrl
Mne
)0.4](Mcrl
Mne
)0.4, For 0.776 < λl
(3.11)
where
λl =
√Mne
Mcrl
, (3.12)
and Mcrl is the critical elastic local -buckling load, and Mne is the global bucklingload as already defined in equation (3.9). The nominal local buckling curve forbeams (3.11) is shown in figure 3.2, in conjunction with test results as collectedby Schafer (2002a), used to calibrate the local buckling curve.
3.2.3 Distortional buckling
The nominal flexural strength for distortional buckling is equal to Mnd, as definedby
Mnd ≡
My. For λd ≤ 0.673
My
[1− 0.22
(Mcrd
My
)0.5](Mcrd
My
)0.5. For 0.673 < λd
(3.13)
This equation deviates from its axial strength counterpart slightly, but oldersources have the same equation for both cases.
λd =
√My
Mcrd
, (3.14)
3.2 Beams 27
where Mcrd is the critical elastic distorsional -buckling load, and My is theyield strength multiplied with the elastic section modulus Wel. The nominaldistortional buckling curve (3.13) is also shown in figure 3.2, together with testresults as collected by Schafer (2002a), used to calibrate the distortional bucklingcurve.
3.2.4 Deflections and serviceability
To calculate the deflection at any moment M , an effective second moment ofarea Ieff is used. The effective second moment of area is calculated by
Ieff = IgMd
M, ≤ Ig (3.15)
where Md is equal to the value of Mn as obtained from equation (3.8), with theflexural yield strength My replaced in all underlying equations by the (service)moment M , for which the deformations are to be calculated (M ≤ My). Ig isthe gross second moment of area.
3.2.5 Shear
At present, the Direct Strength Method as formalized in the appendix to theNorth American standard (AISI S100-2007, 2007), does not contain any provisionsfor checking the shear strength of members. The main part of the specificationsdoes contain provisions for shear, and these were casted into a Direct Strengthformat, and suggested for use in (Schafer, 2006).
Vn =
Vy, For λv ≤ 0.815
0.815√VcrVy, For 0.815 < λv ≤ 1.231
Vcr, For 1.231 < λv
(3.16)
where
λv =
√Vy
Vcr
, (3.17)
Vy = 0.6Awfy, (3.18)
and Vcr is the elastic critical shear buckling force. For members with flat webs,Vcr is determined only for the web (Schafer, 2008), and this gives the sameresults as (AISI S100-2007, 2007). For other types of cross-sections, Vcr canbe determined using the finite element method. Recent developments on thesubject of shear design in conjunction with the Direct Strength Method can befound in (Bedair, 2010; Pham and Hancock, 2007a,b,c, 2008, 2009a,b,c,d, 2010;Schafer, 2008).
3.2.6 Web crippling
Web crippling is not accounted for in the Direct Strength Method. It is notpossible to derive or modify equations from the main part of the standard (AISIS100-2007, 2007) for use within the Direct Strength Method framework. This isbecause the existing equations are empirically derived from test data, and anexpansion to other than the tested geometry ranges is difficult and not advisablewithout further testing.
28 Direct Strength Method
0 0.5 1 1.5 2
0.6
0.8
1
1.2
√My
Mcrl
Mn
My
Eq.(3.11)
Tests
FEM
(a) Local
0 0.5 1 1.5 20.4
0.6
0.8
1
√My
Mcrd
Mn
My
Eq.(3.13)
Tests
FEM
(b) Distortional
Figure 3.3 The Direct Strength Method compared to test and nonlinearFEM results for thin-walled steel C- and Z-sections in bending. From: (Yuand Schafer, 2007) , with minor modifications: Axes and legend have beenre-drawn.
3.2.7 Verification
The experimental data used to derive the Direct Strength Method equationsfor local and distortional buckling of beams, and as shown in figure 3.2 wascomplicated to separate into either local or distortional buckling. This is due tothe applied boundary conditions only being able to partially restrain distortionalbuckling (Ziemian, 2010). In a new research program, validation of the DirectStrength Method method was sought; experimental research and finite elementsimulations on local buckling (Yu, 2005; Yu and Schafer, 2003, 2006b, 2007) anddistortional buckling (Yu, 2005; Yu and Schafer, 2004, 2006a,b, 2007) whereperformed and have demonstrated the robustness of the Direct Strength Methodmethod. A comparison of the validation data and the predictions of the DirectStrength Method is shown in figure 3.3.
3.3 Interaction of buckling modes
Buckling can occur in a local, distortional or global mode. These modes caninteract, and local buckling, for example, lowers the effective second momentof area of a cross-section and consequently the global buckling strength of themember. This has been recognized for many decades, and empirical interactionequations for local and global buckling are included in most design standards,the Direct Strength Method being no exception. The influence of the interactionbetween the distortional buckling mode and the other two modes is less wellknown. This is caused by the distortional buckling mode itself only having beenrecognized as a significant phenomena relatively late (Davies, 1999; Davies andJiang, 1998), and there is little fundamental knowledge on the interaction ofdistortional buckling with the other modes. It is stated in (Davies, 1999), that:‘In general, there is little interaction between the distortional and global modesand it is sufficient to consider the critical distortional mode in isolation.’
3.4 Beam and column charts 29
When the Direct Strength Method was derived, the interaction of distortionalbuckling with local and global buckling was considered for inclusion, but thetested approaches led to over conservative results (Schafer, 2000, 2002a). Itwas noted by Schafer (2008) that recent work, by Silvestre et al. (2006); Yangand Hancock (2004), suggested local-distortional buckling interaction should betaken into account in specific cases, where both phenomena produce a similarcritical load. It was shown by (Dinis et al., 2007, 2005) that coupling betweenlocal and distortional buckling modes may influence the post-buckling behaviourand ultimate strength of lipped thin steel channel sections considerably.
3.4 Beam and column charts
The elastic buckling load charts as given in figures 2.3 and 2.6 show the elasticcritical load of members as a function of the half-wavelength. As was alreadynoted, it is possible for buckling phenomena to repeat themselves at scalarmultiples of their base half-wavelength. This means the apparent increase of theelastic buckling load past the local minimum for local buckling as observed infigures 2.3 and 2.6 is deceptive, the local minimum is in fact an upper boundfor all greater lengths. A second effect, which is not visible in the figures asthese are critical load charts, is the interaction of the global and local bucklingmodes with regard to member strength. The interaction between global andlocal buckling leads to a decrease in the nominal local buckling moment Mnl
for nominal global buckling moments Mne lower than the yield moment My.The same holds true for the nominal local buckling force Fnl. To illustrate thebuckling moment and force as a function of the beam length, a beam chartcan be made. Such a graph can be used by a designer to easily lookup thestrength of a given cross-section for a specific length, while the chart itself onlyneeds to be made once by the beam’s manufacturer for example. The contentsof this section follow the example calculation as given in Schafer (2006). Aconvention for notation is used however that deviates from Schafer (2006); whenthe magnitude of the strength as a function of the system length is intended,this is denoted by adding (l), if the local minimum or visually determined ‘pure’buckling critical strength is referred to, this suffix is not present. Because theprocedure to generate the charts is identical for both bending as compressionmembers, both are combined in a single section. It is noted that the procedureas given here is intended for the case of pure bending or compression.
3.4.1 Local buckling as a function of length
Local buckling occurs at a low wavelength as compared to the other modesof buckling. For longer (half-)wavelengths, the pattern repeats itself and thecritical buckling load remains the same. A constant critical local bucklingload, independent of member length, is therefor a realistic approximation. Thisapproximation neglects the higher local buckling loads at half-wavelengths lowerthan the local minimum, but as the half-wavelength is typically very small, thisis no issue in practice. The critical local buckling moment and force as a function
30 Direct Strength Method
of length can thus be expressed by:
Mcrl(L) = Mcrl, (3.19)
Fcrl(L) = Fcrl. (3.20)
3.4.2 Distortional buckling as a function of length
As was the case with the local buckling shape, distortional buckling can simplyrepeat itself for longer half wavelengths. The value for the the critical bucklingstress obtained from the local minimum is therefore to be used for all greaterhalf-wavelengths. The increase in critical distortional buckling loads at smallerlengths, is exploitable however, as members with unbraced lengths shorter thanthe half-wavelength of the local minimum, do exist in practice. To calculatethe value for the critical buckling load at these smaller half-wavelengths, it ispossible to use analytical models to calculate the strength, or in limited cases,read the strength directly from the moment/half-wavelength curve. A thirdoption exists for thin-wall steel sections, based on the work by Yu (2005).
Mcrd (L) =
Mcrd
(LLcrd
)ln LLcrd . For L < Lcrd
Mcrd. For Lcrd ≤ L(3.21)
Fcrd (L) =
Fcrd
(LLcrd
)ln LLcrd . For L < Lcrd
Fcrd. For Lcrd ≤ L(3.22)
While equations 3.21 and 3.22 have been validated for a wide range of C- andZ-sections, it is recommended for use with all thin steel profiles by Schafer (2006).This apparent generality suggests a that it might be the case that the methodmay work in other materials as well. However, this needs to be investigated first.
Global buckling as a function of length
The global buckling load is strongly dependent on the length of the member.Because of limitations to the finite strip model with regard to support conditions(simply supported ends which are free to warp), the traditional equations forglobal buckling modes are used in the Direct Strength Method. It is howeverpossible to make use of the FSM solution for ‘. . . quickly exploring the strengthof different members and avoids recourse to the longer Specification [chapter9 AISI S100-2007 (2007)] equations and the need for dealing with section proper-ties’ (Schafer, 2006). Using the FSM solution in conjunction with a curve fit ofglobal buckling driven part of the buckling load curve, allows the critical globalbuckling load Mcre (L) to be expressed as a function of the length explicitly.When the member considered has equal unbraced lengths for both torsion anddeflection. The curve fit has the form:
Mcre =
√C1
(1
L
)2
+ C2
(1
L
)4
, (3.23)
Fcre =
√C1
(1
L
)2
+ C2
(1
L
)4
, (3.24)
3.5 Combined axial load and bending 31
102 103 1040
1
2
0.67
FSM curve
Local
Distortion Lateral-torsional
Half-wavelength L (mm)
Mcr
My
Figure 3.4 Buckling load as a function of the half-wave length, for theC-profile of section 2.2.1. The FSM load curve from figure 2.3 can be recog-nized, augmented by the length dependent critical moment equations. Af-ter: (Schafer, 2006).
where the constants C1 and C2 can be determined when two arbitrary pointson the FSM buckling curve are known that lie in the global buckling regime.For beams, this is lateral-torsional buckling, and the locations are denoted by(Lcre1,Mcre1) and (Lcre2,Mcre2). For columns the global modes are flexural, tor-sional, and torsional-flexural buckling. The two points are denoted by (Lcre1, Fcre1)and (Lcre2, Fcre2).
C1 =M2
cre1L4cre1−M
2cre2L
4cre2
L2cre1−L2
cre2.
C2 =(M2
cre1L2cre1−M
2cre2L
2cre2)L
2cre1L
2cre2
L2cre2−L2
cre1.
For beams. (3.25)
C1 =F 2
cre1L4cre1−F
2cre2L
4cre2
L2cre1−L2
cre2.
C2 =(F 2
cre1L2cre1−F
2cre2L
2cre2)L
2cre1L
2cre2
L2cre2−L2
cre1.
For columns. (3.26)
Making use of equations (3.19), (3.21), (3.23) and (3.25), allows us to redrawthe buckling load curve derived in section 2.2.1 for a C-profile in bending,but with added curves for the critical, length dependent, load curves for local,distortional and lateral-torsional buckling. This is shown in figure 3.4.
While figure 3.4 illustrates that the critical, length dependent, load does notincrease for lengths greater than the local minimum, the interaction betweenlocal and global buckling, and the post-buckling strength is not yet visible. Toshow this, the Direct Strength Method equations of section 3.1 and 3.2 areapplied, for columns and beams, respectively. The results of which are shown infigures 3.5a and 3.5b.
3.5 Combined axial load and bending
The Direct Strength Method as present in the North American standard (AISIS100-2007, 2007)[Appendix 1] (and the Australian / New Zealand standard(AS/NZ 4600, 2005)), has no explicit provisions for the case of beam-columns.To check for the combined actions of bending and compression, the designer is
0 1 2 3 40
0.5
1
Local
Distortion
Global
Length L (m)
Fn
Fy
(a) Column chart
0 1 2 3 40
0.5
1
Local
Distortion
Global
Length L (m)
Mn
My
(b) Beam chart
Figure 3.5 Column and beam chart for the C-profile of section 2.2.1.The behaviour of the column is completely determined by local buckling,which gives a large reduction in strength, even for short columns. The beamchart reveals distortion buckling to be the dominant buckling mode forbeams approximately 1m in unbraced length. Local buckling dominates forother lengths up to 3m, for greater lengths, global buckling dominates, withno reduction by other modes, and the cross-section is thus fully effective.The moment and axial force are shown normalized to their yield value,but are only valid for the yield stress used in the Direct Strength Methodcalculations (fy = 379 Nmm−1), this deviates from the other graphs in thissection, which are valid for any yield stress. After: (Schafer, 2006).
3.5 Combined axial load and bending 33
Table 3.1 Resistance factors for LRFD design. The LRFD value for φtakes into account the uncertainties and variabilities inherent in the nominalresistance. It is dependent, amongst others, on the reliability of the designmethod
Load type Symbol Value
Bending φb 0.9-0.95Tensile φt 0.95Compression φc 0.85
referred to the traditional interaction equations in the main part of the standard.These equations are first elaborated on here in 3.5.1, while a new, Direct StrengthMethod based approach is introduced in section 3.5.2.
3.5.1 Traditional interaction equations
The North American specification for the design of cold-formed steel structuralmembers (AISI S100-2007, 2007) specifies design rules according to three differentdesign philosophies; Allowable Strength Design (ASD) and Load and ResistanceFactor Design (LRFD) for the United States and Mexico, and Limit StatesDesign (LSD) for use in Canada. In this section only the provisions according toLRFD design are given. Where possible, equations are simplified if they whereoriginally posed for multiple design philosophies, to include only the LRFDportion. Axial loads are denoted in the text by Nc and Nt, which differs fromthe original equations in (AISI S100-2007, 2007), where P and T are used forcompressive and tensile forces, respectively. There is also a difference in thecoordinate system used, the x axis is now parallel to length of the member, whilethe y and z axes lie in the plane of the cross-section, x denoting the horizontaldisplacement.
Tensile axial load and bending
Any combination of tensile and bending forces, must satisfy the set of equa-tions (3.27)
Muy
φbMnyt
+Muz
φbMnzt
+Nu
φtNnt
≤ 1.0,
Muy
φbMnyt
+Muz
φbMnzt
− Nu
φtNnt
≤ 1.0, (3.27)
where Muy,Muz and Nu are the required strengths for bending in the horizontaly and vertical z direction, and axial (tensile) force, respectively. Mny,Mnz andNnt are the nominal strengths (resistance) for bending and tensile forces. φb andφt are the bending and tensile resistance factors according to table 3.1.
Compressive axial load and bending
Any combination of compressive and bending forces, must satisfy the set ofequations (3.28), or alternatively for situations where the compressive stress is
34 Direct Strength Method
low, equation (3.30).
Nuc
φcNnc
+CmyMuy
φbMnyαy
+CmzMyz
φbMnzαz
≤ 1.0,
Nuc
φcNnoc
+CmyMuy
φbMny
+CmzMyz
φbMnz
≤ 1.0, (3.28)
where φc is the compressive resistance factor according to table 3.1. Cmy and Cmz
are equivalent moment factors, that can be taken as unity for simply supportedmembers. Nnoc is the product of the effective area Ae and the yield stress fe.Geometrical second order effects due to the compressive force are accounted forby the amplification factor αy and αz
αy = 1− Nuc
NEy
,
αz = 1− Nuc
NEz
, (3.29)
where NEy and NEz are the Eulerian column buckling loads. For cases where thecompressive stress is modest, Nuc/Nnc ≤ 0.15, inequalities (3.28) are replaced by
Nuc
φcNnc
+CmyMuy
φbMny
+CmzMyz
φbMnz
≤ 1.0. (3.30)
3.5.2 Beam-Columns according to the DSM
A method for the design of beam-columns, using the Direct Strength Method,was presented by (Schafer, 2002b, 2003). The method brakes with the traditionalapproach for calculating beam-columns, which relies on the calculation of thenormal and flexural load resistance independently, and combining these throughsome interaction equation. The proposed method calculates the stability of themember for the actual axial-flexural load combination, using finite strip methodsoftware. Such an approach would in principle produce more accurate strengthcalculations. The potential increase in accuracy is illustrated in (Schafer, 2008)by asking the reader the question ‘for all cross-sections does the maximum axialcapacity exist when the load is concentric’. Traditional interaction formulae arebuilt on the assumption that this is indeed the case, but in reality eccentriccompression forces may alleviate the compressive demand at weak portions ofthe cross-section, leading to a higher total strength. Schafer (2008) goes on tostate that ‘interaction diagrams make some sense for determining when a simplecross-section yields, but stability, this is another matter.’
(Schafer, 2003) investigated short lipped channel sections only, consideringlocal an distortional buckling. Duong and Hancock (2004) extended the methodto longer sections, by including global buckling modes and the geometric secondorder bending moment. In a paper by Rasmussen (2006) the method wasapplied to angle sections, showing convincing results. More underlying detailscan be found in (Rasmussen and Hossain, 2004). An expression for the shift ofeffective centroid was added, based on Stowell’s solution (Stowell, 1948). Theequation for the shifting centroid was derived in (Rasmussen, 2005). At presentaccurate strength curves for global, local and distortional buckling have not beenformulated, impeding the use of the method in general practice. It is noted
3.5 Combined axial load and bending 35
in (Schafer, 2006), that the strength curves according to the column equations(of section 2.2.2) are conservative in all cases.
A paper by Rusch and Lindner (2001) criticized the Direct Strength Methodmethod with regard to beam-column calculations for having no way to accountfor a shifting centroid. In traditional effective width calculations, the effectiveportion of the cross-section changes due to local buckling, and the location of thecentroid may therefore shift. As this changes the eccentricity of the axial force,the moment on the cross-section changes, and this should be taken into account.The paper by Rusch and Lindner (2001) predates the other publications listed inthis section, and falsely assumed a linear interaction equation to be valid for theDirect Strength Method (Schafer, 2008). Furthermore, Rasmussen (2006), hassome critique on the criticism as posted by Rusch and Lindner (2001); particularlythat local buckling does not induce overall bending of columns fixed againstrotations at the ends (Rasmussen and Hancock, 1993; Young and Rasmussen,1998b), and that the shift of the effective centroid may be inaccurately predictedwhen based on the effective cross-section (Young and Rasmussen, 1998a, 1999).However, the test results quoted by Rusch and Lindner (2001) in support ofthe criticism on the Direct Strength Method are used as a benchmark in mostpapers dealing with beam-column buckling using the Direct Strength Method .The method to calculate beam columns using the Direct Strength Method aspresented in (Schafer, 2002b, 2003) was used in a design example of (Schafer,2006).
Method
The elastic-stress distribution in a beam-column can be used as an input intoFSM software like CUFSM to calculate the elastic buckling stress. Second ordereffects are not present in such a calculation, but the increase in moment due to anormal force in a deflected beam, can be applied to the inputted moments in thepre-processing phase. The required moment Mu is therefore multiplied with theequivalent moment factor Cm, and divided by the amplification factor α, analogto the interaction formulae (3.28) for combined bending and axial compressionas specified by (AISI S100-2007, 2007).
Mu2 =CmMu
α, (3.31)
where Mu2 is the required moment as inputted to the finite strip simulation. Anelastic buckling solution is then obtained using the finite strip simulation, forthe case of an axial force Nu, and a moment Mu2 applied to the member at thesame time. The output will be load-factors for the critical load according to thedifferent buckling modes, LFcre, LFcrd and LFcrl, for global, distortional and localbuckling,respectively. In simulations subjected exclusively to either bending orcompression, the load-factor was multiplied with the applied force or momentto derive the critical force or moment. In the case of combined bending andaxial deformation, it is more convenient to use a normalized interaction diagram.Such a diagram is shown in figure 3.6. The required load is situated in point A.The distance from point A to the origin is denoted by βu and is equal to
βu =
√(Nu
Ny
)2
+
(Mu2
My
)2
. (3.32)
36 Direct Strength Method
0 10
1
βy − βu
βu
B
A
σ = fy
Mu2
My
Nu
Ny
Figure 3.6 Normalized beam column interaction diagram. Point A corre-sponds with the stress state (Nu,Mu2), which is situated a distance βu fromthe origin. For the DSM, the distance from the origin to a point lying bothon the yield surface (σ = fy) and the line described by (O-A), is required(point B). This distance is denoted βy.
For the Direct Strength Method, it is also required to know the distance fromthe origin to a point lying on the yield surface at the intersection with the linegoing through the origin and point A. (Point B in figure 3.6.)
βy =
√(Nu
Ny
)2+(Mu2
My
)2Nu
Ny+ Mu2
My
. (3.33)
Load factors obtained from the finite strip simulations, are multiplied withthe normalized combined stress βu, to obtain the critical normalized combinedstresses
βcre = βuLFcre,
βcrd = βuLFcrd, (3.34)
βcrl = βuLFcrl.
Now the normalized combined critical stresses of equation (3.34) and the nor-malized yield stress of equation (3.33) are entered in the conventional DirectStrength Method column buckling curves of section 3.1 in place of their non-normalized counterparts. These modified equations are repeated below. As wasalready discussed, using the column equations is conservative, but dedicatedbeam-column DSM equations would take preference. Research to obtain suchdedicated formulae is reported to be underway (Schafer, 2006), but publicationsremain forthcoming. The normalized combined strength βn is the minimum ofthe normalized combined stresses in the three buckling modes.
βn = min (βne, βnl, βnd). (3.35)
The normalized strength βne for global buckling is given by
βne ≡
{βy0.658λ
2c , For λc ≤ 1.5
βy0.877λ2c, For 1.5 < λc
(3.36)
3.5 Combined axial load and bending 37
where
λc =
√βy
βcre
,
and Ncre is the minimum of the critical buckling loads for flexural, torsionaland torsional-flexural buckling, calculated according to the traditional equationsaccording to the man part of the standard (AISI S100-2007, 2007)
The normalized combined strength for local buckling is denoted as βnl and isgiven by
βnl ≡
βne, For λl ≤ 0.776
βne
[1− 0.15
(βcrl
βne
)0.4](βcrl
βne
)0.4, For 0.776 < λl
(3.37)
where
λl =
√βne
βcrl
, (3.38)
and βcrl is the critical combined elastic local buckling load, and βne is the criticalcombined global buckling load. The nominal axial strength for distortionalbuckling is equal to βnd, as defined by
βnd ≡
βy, For λd ≤ 0.561
βy
[1− 0.25
(βcrd
βy
)0.6](βcrd
βy
)0.6, For 0.561 < λd
(3.39)
where
λd =
√βy
βcrd
, (3.40)
andNcrd is the critical elastic distortional -buckling load, and Ny is the yieldstrength of the gross-area of the cross-section Ag( equation (3.3)). The normalizedcombined buckling resistance βn as given in equation (3.35), is checked againstthe normalized combined strength requirement βu of equation (3.32)
βu
φbcβn
≤ 1.0, (3.41)
where the resistance factor φbc is equal to 0.8. The value of 0.8 is the minimumvalue to be used for LRFD design according to (AISI S100-2007, 2007), in theabsence of more detailed statistical information. While the beam-column DirectStrength Method remains experimental, it combines bending and axial loads ina more physically sound way than the traditional interaction formulae. It wasshown to produce accurate results , most notably for the test data of (Zhang,1989) that was used by Rusch and Lindner (2001) to demonstrate potentialunconservative results when using the Direct Strength Method in conjunctionwith traditional, linear interaction formulae. These results are shown in figure 3.7.It shows the positive influence a positive bending moment can have on the axialstrength of the member. which is caused by the load being transferred fromthe more buckling prone parts of the cross-section to the more stable parts.When the Direct Strength Method is used with the traditional linear interactionequations, the resulting strength curve for figure 3.7 would be symmetric aboutthe ordinate.
38 Direct Strength Method
−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
minor axis bending (Mz/M
zy)
axia
l lo
ad (
N/N
y)
BucklingDirect StrengthYieldingTests (Zhang,1989)
Cy y
z
z
λn λ
yλ
crθ
Figure 3.7 Interaction diagram for local buckling in compression andbending about the minor axis (z) for an unlipped C-section. From: (Schafer,2006), with minor modifications.
3.6 DSM for inelastic material behaviour
While the Direct Strength Method was developed for use with thin steel sections,its area of application is expanded to other materials, including those with aninelastic, round-house type stress-strain law. Examples of such research can befound for stainless steel in (Becque et al., 2008; Becque and Rasmussen, 2008;Lecce and Rasmussen, 2006), for aluminium in (Kim and Pekoz, 2004; Kutanova,2009; Kutanova et al., 2010; Mennink, 2002; Zhu, 2006; Zhu and Young, 2006,2008, 2009), and for steel at elevated temperatures in (Chen and Young, 2006;Ranawaka, 2006; Ranawaka and Mahendran, 2009).
Chapter 4
Discussion
4.1 Finite strip method
The finite strip method is convenient alternative to its cousin the finite elementmethod. The conventional finite strip method as present in CUFSM is unableto cope with support conditions other than simply supported ends that arefree to warp. Loads, supports and the cross-section may not vary along thelength of the beam. However, this also means that the number of requiredinputs is significantly less. Furthermore, No a-priori deformation patterns arerequired, and this all contributes to the fact that a finite strip method problemcan be entered and solved within a fraction of the time of a finite elementsimulation. This, combined with the fact that an implementation of the FSMis freely available and opensource, allows designers and researchers to use themethod with great efficiency and ease. Alternative implementations can be madeby anyone who wishes to do so, based on any of the sources as listed in thebeginning of chapter 2 or even, it is hoped, on the contents of section 2.1 alone.The result of the finite strip analysis is a chart that expresses the buckling loadrelative to the yield stress as a function of the beam half-wavelength. The criticalbuckling loads of the member according to the different buckling modes must beobtained manually from this graph to serve as an input to the Direct StrengthMethod . This last step can be difficult, especially in an efficient cross-sectionwith many stiffeners.
The effective width approach neglects the stabilizing effect adjacent plates inthe cross-section have on each other, while the finite strip method is able to takeinto account this effect in an exact manner. The combination of axial and flexuralforces is possible within the finite strip method in a physically sound way, whilethe effective width approach relies on semi-empirical interaction formulae. Aremark is required on this last statement, the semi-empirical interaction formulaedo have a significant experimental background, while the competing method inFSM/DSM design remains largely untested.
4.2 Constrained finite strip method
The constrained finite strip method, or cFSM, is an extension onto the regularfinite strip method that allows the FSM solution to separated into contributions
40 Discussion
from the varies buckling modes, or alternatively, the stability of a member consid-ering a subset of the total number of buckling modes may be calculated. Whilethis is a great advantage, the identification of especially the critical distortionalbuckling load can be problematic; it suffers from some mayor drawbacks
The procedure for the modal identification/decomposition (as presentlyimplicated) only allows for open, unbranched cross-sections. This meansthat hollow (rectangular) members and I-sections may not be simulated.While this may not be a significant restriction for this-walled steel sections,extruded aluminium sections do tend to fall within the restricted class ofcross-sections.
Modal decomposition, the technique to calculate the buckling strengthof a member for a subset of the total number of buckling modes, canbe used to calculate the required inputs for the Direct Strength Methodautomatically. However, because no nodal interaction is considered, theDirect Strength Method must be re-calibrated to work with the higher‘pure’ critical buckling mode loads. While no research has been performedto validate the claim, it is believed by the author that the interaction ofcritical loads according to the different buckling mechanisms generatesa more physically sound and relevant critical load, than the modes inisolation.
4.3 The Direct Strength Method
The Direct Strength Method is a convenient design standard that allows thestrength of a member to be calculated by explicit equations alone, and without thetedious and complex iterations as required for the effective width approach thatpresently dominates the design standards worldwide. The accuracy of the DirectStrength Method was shown to be equal or maybe even superiour to the effectivewidth approach (Schafer, 2008) for thin walled-steel sections, although the graphsshown to support the last claim are less than convincing. It is believed thereduction in calculatory complexity will foster the development of more efficientand bespoke cross-sections. Calculation methods for the distortional bucklingphenomena remain relatively underdeveloped and complex for the classicalcalculation approaches. The Direct Strength Method offers an explicit functionfor distortional buckling. The effective width approach does have the advantagethat it allows the change in effective centroid location to be calculated in astraightforward manner, with the Direct Strength Method , this is not possibleand complex other methods must be found to describe it for each type of profile.
A mayor disadvantage for the Direct Strength Method is that it uses theelastic buckling solution for the member as a whole to calculate the strengthas a whole. While this can and is a mayor advantage as well, and in factits distinguishing feature, it becomes a mayor disadvantage when there is animbalance in the stiffness of the different parts of the cross-section. The criticalload calculated will be limited by the most slender part, and this limited criticalload leads to a low nominal strength. This low strength cannot be selectivelyapplied to just the more slender part of the cross-section; the whole cross-sectionwill share this share the lowered strength. This is not a problem for properlyoptimized cross-sections, but when this is not the case, the calculated strength
4.3 The Direct Strength Method 41
of the cross-section may be unduly reduced and may become equal to zero justby adding a very slender stiffener in an unstrategic location, that would have noeffect in a effective width style calculation. Aluminium cross-sections, especiallythe extruded variants, tend to be highly optimized, negating the impact of thebefore mentioned drawback at first sight. This may change when the section isexposed to fire and has an uneven temperature distribution across its surface,with a change in loads due to thermal expansion and a change in carryingmechanism. it has not been investigated how great this effect will be, but thisproblem has the potential to disqualify the Direct Strength Method in aluminiumfire design.
The method for calculating the strength of aluminium columns in fire condi-tions as derived by Maljaars (2007, 2008); Maljaars et al. (2009a,b, 2010, 2008) isdependent on the elastic buckling solution of the different constitutive plates ofthe cross-section according to the effective width method. It is therefore believedby the author that the method may be extended to beams and even beams-columns by using the derived equations in conjunction with the Direct StrengthMethod approach. No research into this combined approach has been performed,but may prove to be successful, provided the imbalance of the stability of thedifferent parts of the cross-section does not have a prohibitively big impact.
Bibliography
Adany, S. and B. W. Schafer (2006a, May). Buckling mode decomposition ofsingle-branched open cross-section members via finite strip method: Applica-tion and examples. Thin-Walled Structures 44 (5), 585–600.
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