Collapse mechanisms of aluminium structures in fire : the
direct strength method
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Meulen, van der, O. R. (2010). Collapse mechanisms of aluminium structures in fire : the direct strength method. Eindhoven University of Technology.
Document status and date: Published: 01/01/2010
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Collapse mechanisms of
aluminium structures in fire
ir. O. R. van der Meulen
Oege Ronald van der Meulen is a PhD student at the Eindhoven University of Technology, performing research on the stability of aluminium structures in fire.
This research was carried out under project number M81.1.108306 in the framework 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) The Direct 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.
2.1 Finite strip discretization of a prismatic, thin-walled section . . . 6
2.2 C-section with lips . . . 13
2.3 Elastic buckling load for the C-section subjected to bending about
the strong axis . . . 13
2.4 Buckled geometries for a beam in bending . . . 14
2.5 Buckled geometries for a beam in bending (Scaled) . . . 15
2.6 Elastic buckling load for the C-section subjected to axial
compres-sion. . . 15
2.7 Buckled geometries for a column in compression . . . 16
2.8 The critical load curve for the C-section in bending according to
FSM and cFSM . . . 19
2.9 The deflected and undeflected cross-section at the lowest local
minimum of the cFSM, distortional buckling solution. . . 20
2.10 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-walled
steel columns. . . 25
3.2 The Direct Strength Method compared to test data for thin-walled
steel braced beams. . . 26
3.3 The Direct Strength Method compared to test data and nonlinear
FEM results for thin-walled steel C- and Z-sections in bending . 28
3.4 Buckling load as a function of the half-wave length . . . 31
3.6 Normalized beam column interaction diagram . . . 36
2.1 Mode classification criteria in cFSM . . . 18
List of Figures iii
List of Tables v
Contents vii
1 Introduction 1
2 Finite Strip Method 5
2.1 Conventional finite strip method . . . 5
2.1.1 Elastic stiffness tensor . . . 7
2.1.2 Elastic geometric stiffness tensor . . . 8
2.1.3 Assembly . . . 11
2.1.4 Solving the system . . . 11
2.2 Critical load charts . . . 12
2.2.1 Bending . . . 12
2.2.2 Compression . . . 14
2.3 Constrained finite strip method . . . 17
2.3.1 Modal decomposition . . . 18
2.3.2 Mode contribution calculation . . . 20
3 Direct Strength Method 23 3.1 Columns . . . 23 3.1.1 Global buckling . . . 24 3.1.2 Local buckling . . . 24 3.1.3 Distortional buckling . . . 24 3.2 Beams . . . 25 3.2.1 Global buckling . . . 25 3.2.2 Local buckling . . . 26 3.2.3 Distortional buckling . . . 26
3.2.4 Deflections and serviceability . . . 27
3.2.5 Shear . . . 27
3.2.6 Web crippling . . . 27
3.2.7 Verification . . . 28
3.3 Interaction of buckling modes . . . 28
3.4 Beam and column charts . . . 29
3.4.1 Local buckling as a function of length . . . 29
viii CONTENTS
3.5 Combined axial load and bending . . . 31
3.5.1 Traditional interaction equations . . . 33
3.5.2 Beam-Columns according to the DSM . . . 34
3.6 DSM for inelastic material behaviour . . . 38
4 Discussion 39 4.1 Finite strip method . . . 39
4.2 Constrained finite strip method . . . 39
4.3 The Direct Strength Method . . . 40
Introduction
One of the most used methods of calculating the strength of cross-sections that are affected by local buckling, is the effective width approach as pioneered by von
K´arm´an et al. (1932) and later modified to account for imperfections by Winter
(1947). The approach simplifies reality by treating the total cross-section as
simply supported plates and calculating the critical stress σcr of each of these
plates 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 of
plate and load, equal to 4.0 for plates simply supported on all sides, loaded in compression. t and b are the thickness of the considered plate, respectively, and E and ν are the Young’s modulus and Poisson’s ratio. The effective portion of the width of the plate is then calculated by
be b = α rσ cr σy 1 − β rσ cr σy , (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 plates simply on all four sides, and equal to 1.19 and 0.0993 for plates simply supported on three sides by Kalyanaraman et al. (1977). The effective portion of the entire cross-section can thus be found, and its strength and stiffness determined. But the procedure is iterative due to the change in effective cross-section affecting the strain distribution, and can quickly become too complicated for hand-calculation for 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 this
influence is by using semi empirical formulae for the value of Kcrfor multiple
plates interacting, or even the cross-section as a whole. An overview of such
formulae is given in (Rondal and Dubin˘a, 2005), with the most relevant ones for
the types of geometry in aluminium design repeated here. Schafer (2001, 2002a) proposed equations for the flanges of cross-sections interacting with lip stiffeners and 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 hb2h2 − hb0.4i. For hb ≥ 1 4h2 − hb0.2i. For hb < 1 (1.4)
More equations for the critical buckling factor Kcrcan be obtained from (Batista,
1989), the formulae for the lipped and plain channel are omitted here, but the equation for a hollow rectangular section is given as.
For rectangular hollow section
Kcr= 6.56−5.77 b2 b1 +8.56 b2 b1 2 −5.36 b2 b1 3 , For 0.1 ≤b2 b1 < 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 method is 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 is increased 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 Pek¨oz (1998) to
overcome the posed drawbacks of existing methods. It uses an elastic buckling solution for the member as a whole, which is used as an input to the strength
curve for the entire member. Strength curves were given in (Schafer and Pek¨oz,
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 Zealand standard for thin-steel design (AS/NZ 4600, 2005) since 1996. Hancock himself traces the origin even further, to the work of Trahair. It shall be clear that the Direct Strength Method is by no means an entirely new concept. The innovative part however, is its extension to include local buckling. The method can now be used to simply calculate the behaviour of members, taking into account, local, distortional and overall buckling phenomena, when the elastic buckling (Eulerian) solution for the entire member is known. Such a solution can be obtained by means of the Finite Element Method (FEM), the Generalised Beam Theory (GBT), the (constrained) Finite Strip Method FSM/cFSM, or other less used techniques as boundary value methods. For certain standard geometries, solutions of the form of equation (1.5) may even be available. In practice (c)FSM is the easiest to use as it requires less time and inputs as compared to FEM, and it has a very convenient, open source, pre-post processor
and solver called CUFSM (Schafer and ´Ad´any, 2006a), available freely from the
Hopkins university. A commercial implementation of FSM also exists, called Thinwall (Papangelis and Hancock, 2005) developed by university of Sydney.
At present the Direct Strength Method has been accepted as an alternative method to calculate the behaviour of thin walled steel in the North American standard (AISI S100-2007, 2007), since 2004. It is also been accepted for use in the Australian/ New Zealand standard for cold-formed steel structures (AS/NZ 4600, 2005). Certain issues as, shear, web crippling and members with holes cannot be dealt with using the Direct Strength Method at the level of development as present in the standard, and checks have to be made using the existing methods in the conventional part of the standard. Since the standard was published, more research was, and is being performed. The mentioned limitations and steps taken in their resolvement are discussed in (Schafer, 2008).
The present report does not include a description of the Generalised Beam Theory (GBT). This is because the subject of this report is the Direct Strength Method 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 is
only a small difference in the critical loads derived using FSM and GBT ( ´Ad´any
et al., 2009). The theory behind the GBT is rather complex however, and may not be used to simulate all types of cross-sections as typically found in aluminium extrusion design. (The theory as defined in (Schardt, 1989) excludes closed cross-sections and branched members.) The interested reader is referred to (Camotim et al., 2004; Davies and Jiang, 1996, 1998; Schardt, 1989, 1994; Silvestre and Camotim, 2002a,b, 2004; Silvestre and Comatim, 2004). Specific articles on the subject of GBT in conjunction with round-house materials can
be found in (Gon¸calves and Camotim, 2004; Gon¸calves et al., 2004). It is noted
that the range of possible cross-sections for the GBT is being increased (Dinis
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 section is based on the semi-analytical or classical finite strip method, a comprehensive summary of the method and the theory behind it is given in (Cheung, 1976). The method was extended by Hancock (1978) for the solution of the elastic buckling problem of open, thin-walled members. Based on these sources, the method was used by Schafer (1997) to develop a program to calculate the elastic buckling
load of thin-walled steel members called CUFSM (Schafer and ´Ad´any, 2006a).
The mechanics behind this and their derivation are discussed in section 2.1. The latest version of the CUFSM program, also includes an extension to the classical
finite strip method by ( ´Ad´any and Schafer, 2008) to allow for the decomposition
or the classification of the different buckling modes. The theory behind this is given in section 2.3. The classical finite strip method is capable only of simulating simply supported. It was later extended from its classical form to be able to cope with more complicated support conditions. This adds to the complexity of the method significantly as the longitudinal shape functions are defined now as a series of lines, or splines, leading to significantly larger eigenvalue problems. The method is explained well in Bradford and Azhari (1995). This and other extensions, along with their derivation are presented in (Cheung and Tham, 1999). The conventional finite strip method as used in CUFSM is elaborated on
in the next section, following the excellent paper by Schafer and ´Ad´any (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, a thin-walled member is discretized into a number of elements. Unlike the finite element method, the finite strip method employs only one element in the length direction of the beam, and the elements are referred to as strips. Appropriate shape functions are used to capture the longitudinal variation in displacements. Figure 2.1a shows the discretization of a member into strips. Each strip has eight degrees of freedom (DOF), as is shown in figure 2.1b, and may be subjected to an edge traction as defined in figure 2.1c. It is noted that the coordinate system used in the sections concerning the FSM deviates from what is customary; the y
6 Finite Strip Method
(a) Member and strips v1 v2 b u1 w1 θ1 u2 w2 θ2 v1 v2 v1 v2 v1 v2 b u1 w1 θ1 u2 w2 θ2 v1 v2 (b) Plate DOF x y z a T1=f 1 t T2=f 2 t x y z a T1=f T2=f (c) Plate traction
Figure 2.1 Finite strip discretization of a prismatic, thin-walled section.
From: (Schafer and ´Ad´any, 2006a), with minor modifications.
coordinate is specified with the longitudinal axis of the beam, while Z coincides with the out of plane direction and x the transverse direction. This system coincides with the existing literature on the FSM. The vector describing the displacement fields u is related to the displacements at the node d by the second order tensor N , which contains the shape functions.
u = N d = u v w = N3x8 u1 u2 v1 v2 w1 θ1 w2 θ2 . (2.1)
The shape functions for the membrane displacements u and v are linear in the transverse (x)-direction, but a harmonic function is used for the longitudinal direction. As the node for the displacement u is situated in the middle of the strip, where the displacement is the highest, a sine function is used. The nodes for the longitudinal displacements v are located at the ends of the strip, and a cosine function is used. This choice of longitudinal shape functions is consistent with simply supported beams.
u = sinmπy a 1 −x b x b u1 u2 , v = cosmπy a 1 −x b x b v1 v2 , (2.2)
where m is the number of half-wavelengths in the length of the beam, which is invariably equal to 1 in CUFSM. The shape functions for the out-of-plane direction w are cubic and are given by
w = sinmπy a 1 − 3xb22 + 2x2 b3 x −2x2 b + x3 b2 3x2 b2 − 2x2 b3 x3 b2 − x2 b T W1 θ1 W2 θ2 . (2.3)
The nonzero elements of the second order shape function tensor N are then equal to N[1,1-2]= sin mπy a 1 −x b x b , N[2,3-4]= cos mπy a 1 −x b x b , (2.4) N[3,5-8]= sin mπy a 1 − 3xb22 + 2x2 b3 x −2xb2 +xb32 3x2 b2 − 2x3 b3 x3 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 these
quantities uncoupled. εm= εx εy γxy m = δu δx δv δy δu δy + δv δx = Bmduv. εb= εx εy γxy b = −zδ2w δx2 −zδ2v δy2 2z δ2u δxδy = Bbdwθ. (2.5)
where the second order tensors Bm, and Bbdescribe the relationship between
strain and nodal displacement, for membrane stresses and Kirchoff thin-plate
bending, respectively. The second order, elastic stiffness tensor Keis constructed
from the decoupled elastic membrane and bending stiffness tensors Kem and
Keb, respectively. Ke= Kem 0 0 Keb , (2.6) where Kem= Z Z Z Bm TEB mda db dt, Keb= Z Z Z Bb TEB bda db dt. (2.7)
which leads to the elastic membrane (Kem) and bending (Keb) stiffness tensors
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= νyExt3 12 (1 − νxνy) = νxEyt 3 12 (1 − νxνy) ,
where G is the shear modulus, and νx and νy are the Poisson’s ratio in two
orthogonal directions. It is noted that the last equation (2.8) places a restriction on 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 for example (Reddy, 2007).
2.1.2
Elastic geometric stiffness tensor
The geometric stiffness matrix is obtained by considering the additional potential energy gained from a displacement of the edge tractions as defined in figure 2.1c in the longitudinal (y) direction. These tractions vary linearly in magnitude
between T1 and T2 and the corresponding potential energy function is equal to
U =1 2 Z Z h T1− (T1− T2) x b i " δu δy 2 + δv δy 2 + δw δy 2# dx dy, (2.10)
where the tensor resulting from differentiating the shape functions with respect to y is defined here as G hδu δy δv δy δw δy iT = Gd. (2.11)
Substituting equation (2.11) into equation (2.10) yields an expression for the
geometric stiffness tensor Kg.
U =1 2d T Z Z h T1− (T1− T2) x b i GTG dx dy d = 1 2d TK gd. (2.12)
Kg may be formed from two uncoupled contributions from membrane(Kgm),
and bending deformations (Kgb) according to
Kg=
Kgm 0
0 Kgb
where Kgm = Z Z h T1− (T1− T2) x b i Gm T Gmdx dy, Kgb= Z Z h T1− (T1− T2) x b i Gb T Gbdx dy. (2.14)
The geometric membrane (Kem) and bending (Keb) stiffness tensors are given
in equations (2.17) and (2.18), respectively, on page 10. Schafer (1997) notes
that the definition of Kg as a stiffness tensor follows directly from the energy
derivation of an initial stiffness tensor, and refers the reader to Cook et al. (2001);
Crisfield (1996); Ugural and Fenster (2003)1.
10 Finite Strip Method Kem= t aE1 2b + abk2mG 6 Symmetric aKmνxE2 4 − akm2G 4 abk2 mE2 6 + aG 2b −aE1 2b + abk2mG 12 −akmνxE2 4 − akmG 4 aE1 2b + abk2mG 6 akmνxE2 4 + akmG 4 abk2mE2 12 − aG 2b −akmνxE2 4 + akmG 4 abk2mE2 6 + aG 2b . (2.15) Keb= t 13ab 70 k 4 mDy+ 12a 5b k 2 mDxy +6a 5bk 2 mD1+ 6a b3Dx Symmetric 3a 5k 2 mD1+ a 5k 2 mDxy +3ab2Dx+ 11ab2 420 k 2 mDx ab3 210k 4 mDy+ 4ab 15k 2 mDxy +2ab 15k 2 mD1+ 2a b Dx 9ab 140k 4 mDy− 12a 5bk 2 mDxy −6a 5bk 2 mD1− 6a b3Dx 13ab2 840 k 4 mDy− a 5k 2 mDxy −a 10k 2 mD1− 3a b2Dx 13ab 70 k 4 mDy+ 12a 5b k 2 mDxy +6a5bk2 mD1+ 6a b3Dx −13ab2 840 k 4 mDy+ a 5k 2 mDxy +10ak2 mD1+ 3a b2Dx −ab3 840k 4 mDy− ab 15k 2 mDxy −ab 30k 2 mD1+ a bDx −11ab2 420 k 4 mDy− a 5k 2 mDxy −3a 5k 2 mD1− 3a b2Dx ab3 210k 4 mDy+ 4ab 15k 2 mDxy +2ab15k2 mD1+ 2a bDx . (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) Symmetric 2b (15T1+ 7T2) b 2(5T 1+ 3T2) 54 (T1+ T2) 2b (6T1+ 7T2) 24 (3T1+ 10T2) −2b (7T1+ 6T2) −3b2(T1+ T2) −2b (7T1+ 15T2) b2(3T1+ 5T2) . (2.18)
2.1.3
Assembly
The contributions from the local elastic (ke) and geometric stiffness (kg) tensors
for each of the individual strips, need to be assembled into the global elastic
and geometric stiffness tensor, Ke and Kg, respectively. The strips are in a
local coordinate system, and the coordinates need to be transformed before assembly 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 0 0 1 vi θi , (2.19)
where i and j refer to the node and strip, respectively, and α(j) is the angle of
strip 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 0 0 0 cos α(j) 0 0 0 − sin α(j) 0 0 0 0 1 0 0 0 0 sin α(j) 0 0 0 cos α(j) 0 0 0 0 0 0 0 0 1 0 0 0 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 coordinate system by Ke (j)= Γ(j)k e (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 summation of the stiffness tensors of the individual strips
Ke= n
A
j=1Ke (j), Kg= nA
j=1Kg (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 single mode has the form
12 Finite Strip Method
where λ is the eigenvalue or load factor LF as used in section 3.5.2 and D is the eigenvector 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, in ascending order of eigenvalue. The first of which is nearly allways the dominant mode, as it has the lowest load factor LF.
Both the local elastic (Ke) and geometric stiffness (Kg) tensors have a term
for the beam half-wavelength a, the solutions found for equation (2.24) are therefore also a function of a. For a complete analysis as shown in figure 2.3 for example, the problem is solved for a number of different lengths.
The global displacements D(j)may be transformed back to local coordinates
for post-processing. The required transformation follows from equation (2.20)
d(j)= Γ(j)−1D(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 calcal-culations as presented here, the United States customary system of units is used for the geometry. A cold formed steel C-section with a geometry according to figure 2.2 is considered. Stresses and strength are identical to the example case but are expressed in metric units. The material properties are identical: A Poisson’s ratio ν = 0.3, a Young’s modulus equal
to E = 2.03 · 105 N mm−2 and a shear modulus G = 7.82 · 104 N mm−2 is
specified. A yield stress fy= 379 N mm−2 is specified, but does not influence
the calculations itself as the finite strip method is based on the assumption of linear 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. The
finite strip method based program CUFSM (Schafer and ´Ad´any, 2006a) is used
to calculate the elastic buckling load for the cross-section for a large number of different half-wavelengths. (for a simply supported beam, the half-wavelength is equal 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 themselves at scalar multiples of their base half-wavelength, and therefore no increase in strength may be assumed for greater lengths. The critical- and yield moment
are denoted as Mcr and My, respectively. Three different modes of buckling
may be identified,(i) Local buckling at a half-wavelength Lcrl= 126 mm with
a normalized moment McrMy−1 = 0.67, (ii) Distortional buckling with Lcrd =
629 mm and McrMy−1 = 0.85, and (iii), global buckling for the larger
half-wavelengths. The local minima are used to identify the local and distortional buckling 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” t = 0.059”
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). All dimensions are in inches. The dots indicate the position of the nodes used in the FSM simulation. 101 102 103 0 0.5 1 1.5 2 Local, Mcr My = 0.67 Distortional, Mcr My = 0.85 Global Half-wavelength (in) M cr M y
Figure 2.3 Elastic buckling load for the C-section as specified in figure 2.2
subjected to bending about the strong axis. The abscissa is logarithmic and expressed 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 to
bending about its strong axis. The half-wavelengths correspond to the local minima of figure 2.3. Proportional scaling was applied to show the bending members side by side.
to the marked points are also given in figure 2.3. The same deformation patterns are also shown, in three dimensions, in figure 2.4. Proportional scaling was applied to show the different lengths side by side. To make the beams of figure 2.4 more easy to compare, the length is scaled independently, setting it equal to the greatest 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 buckling loads as given in figure 2.6 are produced by CUFSM. The distortional buckling mode does not have a local minimum, and the boundary between local and distortional buckling needs to be identified by manually judging the deformed shapes. The three buckling modes are:(i) Local buckling, at a half-wavelength
Lcrl= 168 mm with a normalized load FcFy−1= 0.12, (ii) Distortional buckling
with Lcrd= 724 mm and FcFy−1= 0.27, and (iii), global buckling for the larger
half-wavelengths. The in-plane deformation patterns for the three buckling modes belonging to the marked points are shown in three dimensions in figure 2.7.
bending about its strong axis. The half-wavelengths correspond to the local minima of figure 2.3. The length DRAWN was scaled to make the length equal to the greatest cross-section dimension.
101 102 103 0 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 2.2
subjected to axial compression. The abscissa is logarithmic and expressed in inches. After: (Schafer, 2006).
(a) Local (b) Distortional (c) Flexural
Figure 2.7 Buckled geometries for the cross-section of figure 2.2 subjected
to axial compression. The length DRAWN was scaled to make the length equal to the greatest cross-section dimension.
2.3
Constrained finite strip method
Critical elastic-load calculations using the finite strip method (FSM) as given in section 2.1, and the finite element method alike (FEM), do not provide a way for distinguishing between he different buckling modes, while this separation is required for the Direct Strength Method as given in section 3. It is up to the user to identify points of ‘pure’ modes, which can be tricky if these points do not lie in clearly distinguishable local minima of the critical load graph. Guidance is offered 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 buckling modes do not exist. While there is consensus over which modes exist and these are even discussed in modern design codes, thorough mechanical definitions are
lacking ( ´Ad´any and Schafer, 2008).
A method that is able to distinguish between the different modes, is the Generalized Beam Theory (Schardt, 1989), or (GBT). While this is a great advantage, GBT does have its disadvantages, particularly with regard to
gener-ality ( ´Ad´any and Schafer, 2008). A way was sought to include the GBT’s ability
to distinguish between modes into more general FEM/FSM methods. This led to the development of the constrained Finite Strip Method, or (cFSM). The
separation of global and distortional modes is discussed in ( ´Ad´any and Schafer,
2006a,b), and examples of their use and inclusion into CUFSM are provided
in (Schafer and ´Ad´any, 2005, 2006a,b). Full modal decomposition is presented
in ´Ad´any and Schafer (2008). And an overview and comparison of both GBT
and cFSM was presented in ´Ad´any et al. (2006, 2007, 2009).
Unfortunately, including the classification system of the GBT does place restrictions on the type of cross-sections that can be simulated. With the current system as described here, only open, non-branched cross-sections may be simulated with cFSM. For thin-walled steel structures this is a less severe restriction as these are made by folding plates of steel and the laws of economy already 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 been made in generalizing it for various geometries such as closed and branched cross-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 this is even possible.
To separate the contributions from the different buckling modes, the criteria applied in the GBT are used. These are ‘. . . found to usually be in good agreement
with current engineering classifications’ (Schafer and ´Ad´any, 2006a). More
information on the modal identification within GBT can be found in (Silvestre and Camotim, 2002a,b). The separation into (i) Global, (ii)Distortional, (iii) Local, and (iv) Other modes, is performed through the application of the following three criteria:
Criterion 1, Vlasov’s hypothesis. A condition derived from classical beam theory, or (Vlasov, 1959) hypothesis. Certain membrane deformations are restricted, while warping is allowed.
18 Finite Strip Method
Table 2.1 Mode classification criteria in cFSM, based on the criteria 1, 2
and 3, the four different buckling modes can be distinguished.
Global Distortional Local Other
Criterion 1-Vlasov’s hypothesis Yes Yes Yes No
Criterion 2-Longitudinal warping Yes Yes No
Criterion 3-Undistorted section Yes No
• εx= 0, there is no transverse strain.
• The axial displacement v is linear within a flat plate. (Flat plates include 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 buckling mechanisms, as local buckling does not induce a deformation at the centre and 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 the basis 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 buckling modes according to table 2.1.
2.3.1
Modal decomposition
The conventional finite strip method finds a solution that includes all buckling modes. Using the confined finite strip model, it is possible to derive a solution based on just one or more of the separate buckling modes. The total DOF space D is therefore reduced to the subspace belonging to the buckling mode, or buckling modes of interest
D = RmDm, (2.26)
where DMis the displacement vector in the reduced space, and RMis the second
order constraint tensor for the given buckling modes. The subscript M can be replaced 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. The unity 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 finite strip method was given in equation (2.24) for one eigenvalue/vector only. For a number of eigenvalues/vectors m, the equation is rewritten to
102 103 104 0 0.1 0.2 0.3 0.4 0.5 Local Distortional Global Half-wavelength (in) M cr M y
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 according to cFSM. It is clear that cFSM over-predicts the actual elastic stability for distortional buckling in particular. This is caused by the interaction of modes leading to a lower elastic buckling solution
where Λ and ΨD are second order tensors, containing the eigenvalues and
eigenvectors; Λ = diag [λ1, λ2, . . . , λm], and ΨD= [D1, D2, . . . , Dm]. Inserting
equation (2.26) into equation (2.28) and pre-multiplying by Rm
T yields RM T KeRMΨD= ΛRM T KgRMΨD, (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= RM TK eRM, Kg,M= RM T KeRM. (2.31)
Equation (2.30) is the eigenvalue problem of the cFSM problem, which can again be solved with any suitable eigenvalue algorithm.
RMis a second order tensor with m rows and mM columns, where mM is the
dimension of the reduced DOF space. Consequently, the stiffness matrices of the
constrained FSM problem, are mM by mM tensors, which can be significantly
smaller than the original problem. The modal constrained can thus be considered a model reduction technique, with only the DOFs that are applicable to the mode(s) under investigation.
Solving the eigenvalue problem (2.30) for RMequal to RG, RD, RL and RO,
consecutively, gives the full critical load chart, but now separated into the four buckling 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 local
minimum of the cFSM, distortional buckling solution. The deflected shape could also be considered local buckling.
It can be observed from figure 2.8 that the cFSM methods gives higher values for the pure buckling modes than the combined solution according to the FSM. This is because the interaction of buckling modes lowers the bifurcation load of the member. A second observation is that the cFSM solution for distortional buckling has two local minima, the lowest of which overlaps with the local bucking mode. A closer examination of this lower minimum reveals it to be a kind of local buckling, with the lips deflecting. Such a deformation pattern is usually associated with local buckling. A deflected cross-section at this point is presented in figure 2.9. At this point, an error in CUFSM or its inputs can not be ruled out, but the observed behaviour is not reported in any of the publications concerning cFSM. The model decomposition technique of cFSM gives critical load solutions for all the buckling modes separately. This potentially simplifies the usage of design approaches based on these separate critical buckling load values, such as the Direct Strength Method of section 3. It also allows the procedure to be automatized, as the manual mode classification step is removed. The higher critical load values obtained with cFSM would require a recalibration of the design methods based on the critical loads. Another concern is that solutions according to FSM are a more realistic description of the actual load than 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 used to solve the elastic buckling problem, a solution is obtained for the eigenvalue
problem of equation (2.30). The resulting eigenvectors dm lie in a subspace of
the original DOF space of the FSM problem. The subspaces are denoted G, D, L and 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 and
can be regarded as their base vectors, with the added property that they are orthogonal to each other with respect to the elastic and the geometric stiffness
tensor. It is thus possible to write the vector dm as a linear combination of the
base vectors contained in Ψm
dm= Ψmcm, (2.32)
where the vector cm defines the relative contribution of the different modes. (Up
to now, the word mode was used as synonymous with buckling mode (G,D,L and O), here the word modes represents the different patters that may develop within a given buckling mode. Usually the first mode is relevant only as they are sorted in 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= R
o
mcm, (2.34)
where Ro
m is 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 of
section 2.3.1, and the resulting eigenvectors ΨG,ΨD, ΨL and ΨO, are then the
orthogonal base vectors. It is noted in ( ´Ad´any and Schafer, 2008) that for each
loading case, a different set of orthogonal base vectors exists, but that when orthogonality is ‘important’, only the simplest case of uniform compression needs
to be considered. Equation (2.34) can then be used to calculate the tensors Ro
m
from the base systems Ψmand the restraint tensors Rm. These tensors Rom are
assembled into Ro GDLO = R o GR o DR o LR o
O, and any local deformation can be
written as a combination of this quantity.
d = Ro GDLOc o GDLO=R o G R o D R o L R o 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 is not uniquely defined by the orthogonal base system alone as a normalization of the orthogonal system is still required. Different option for the normalization
exist ans are discussed in (Schafer and ´Ad´any, 2006a), but it was found that a
normalization of the buckling modes based on the work performed by the mode gives the most physically sound results. This evolves scaling such that
1
2Ri
TK
eRi= 1, (2.36)
The example calculation of section 2.2.1 is shown here again, but now the critical load curve can be shown along with the relative contribution of the different modes of buckling. This is shown in figure 2.10 and figure 2.11.
102 103 104 0 0.2 0.4 0.6 Half-wavelength L (mm) N cr Ny Global Distortional Local Other
Figure 2.10 The critical load curve for the C-section in bending as was
shown in figure 2.3. The relative contribution of the various buckling modes is shown, normalized to the strain energy.
102 103 104 0 20 40 60 80 100 Half-wavelength L (mm) % GlobalDistortional Local Other
Figure 2.11 The relative contribution of the different buckling mode for
the C-section in bending example as was shown in figure 2.3. Normalizing was performed according to the strain energy.
Direct Strength Method
the Direct Strength Method was already introduced in section 1.It is a technique to calculate the resistance of the entire cross-section based on the critical, Eulerian buckling loads calculated for the cross-section as a whole. This deviates
significantly with the effective width approach (von K´arm´an et al., 1932), which
its most widely used alternative and the method used in most design standard worldwide. The advantage of calculating the critical buckling strength for the cross-section as a whole, is that it correctly takes into account the restraining action of neighbouring plates, while the effective width approach considers each plate to be isolated from the rest and simply supported. This leads to a more accurate and higher critical strength value. A disadvantage of the Direct Strength Method is that the critical strength calculation becomes more complicated than for the isolated plates of the effective width theory. This is why A method like the finite strip method of chapter 2, or an alternative is required.
In this chapter, the Direct Strength Method as it is included in the North American 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 the
minimum 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 calculated according to equations as given in the next three subsections. It is noted that not all modes have to be present for a given cross-section, and strength checks for 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 and
torsional-flexural buckling, is given by
Nne≡ ( Ny0.658 λ2 c, For λ c≤ 1.5 Ny 0.877 λ2 c , For 1.5 < λc (3.2) where λc= r Ny Ncre , Ny= Agfy, (3.3)
and Ncre is the minimum of the critical buckling loads for flexural, torsional
and torsional-flexural buckling, calculated according to the traditional equations according 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 − 0.15Ncrl Nne 0.4 Ncrl Nne 0.4 , For 0.776 < λl (3.4) where λl= r Nne Ncrl , (3.5)
and Ncrl is the critical elastic local -buckling load, and Nne is the global buckling
load 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 defined
by Nnd ≡ Ny, For λd≤ 0.561 Ny 1 − 0.25Ncrd Ny 0.6 Ncrd Ny 0.6 , For 0.561 < λd (3.6) where λd= r Ny Ncrd , (3.7)
and Ncrd is the critical elastic distortional -buckling load, and Ny is the yield
strength of the gross-area of the cross-section Ag( equation (3.3)). The nominal
distortional buckling curve (3.6) is also shown in figure 3.1, together with test results as collected by Schafer (2002a), used to calibrate the distortional buckling curve.
0 2 4 6 8 0 0.5 1 1.5 λd= q N y Ncrd / λl= q Nne Ncrl Ntest N y d / Ntest N ne 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 Pek¨oz, 1994;
Mulligan, 1983; Polyzois and Charnvarnichborikarn, 1993; Thomasson, 1978). From: (Schafer, 2002a), with minor modifications: Axes and legend have been re-drawn.
3.2
Beams
The nominal flexural strength of a beam is denoted as Mnand is equal to be the
minimum 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 calculated according to equations as given in the next three subsections. It is noted that not all modes have to be present for a given cross-section, and strength checks for non-existent are omitted
3.2.1
Global buckling
The global buckling mode for flexural members is lateral-torsional buckling, and
the nominal strength of a member for this mode is denoted as Mne and is equal
to Mne≡
Mcre, For Mcre< 0.56My
10 9My 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 modulus
Weland 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)
26 Direct Strength Method 0 1 2 3 4 5 0 0.5 1 1.5 λmax= qM y Ncr M test 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 Pek¨oz, 1994;
Mulligan, 1983; Polyzois and Charnvarnichborikarn, 1993; Thomasson, 1978). From: (Schafer, 2002a), with minor modifications: Axes and legend have been 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 − 0.15Mcrl Mne 0.4 Mcrl Mne 0.4 , For 0.776 < λl (3.11) where λl= r Mne Mcrl , (3.12)
and Mcrlis the critical elastic local -buckling load, and Mne is the global buckling
load as already defined in equation (3.9). The nominal local buckling curve for beams (3.11) is shown in figure 3.2, in conjunction with test results as collected by 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 defined
by Mnd≡ My. For λd≤ 0.673 My 1 − 0.22Mcrd My 0.5 Mcrd My 0.5 . For 0.673 < λd (3.13)
This equation deviates from its axial strength counterpart slightly, but older sources have the same equation for both cases.
λd=
r My
Mcrd
where Mcrd is the critical elastic distorsional -buckling load, and My is the
yield strength multiplied with the elastic section modulus Wel. The nominal
distortional buckling curve (3.13) is also shown in figure 3.2, together with test results as collected by Schafer (2002a), used to calibrate the distortional buckling curve.
3.2.4
Deflections and serviceability
To calculate the deflection at any moment M , an effective second moment of
area Ieff is used. The effective second moment of area is calculated by
Ieff= Ig
Md
M , ≤ Ig (3.15)
where Mdis equal to the value of Mn as obtained from equation (3.8), with the
flexural yield strength My replaced in all underlying equations by the (service)
moment M , for which the deformations are to be calculated (M ≤ My). Ig is
the gross second moment of area.
3.2.5
Shear
At present, the Direct Strength Method as formalized in the appendix to the North American standard (AISI S100-2007, 2007), does not contain any provisions for checking the shear strength of members. The main part of the specifications does contain provisions for shear, and these were casted into a Direct Strength format, and suggested for use in (Schafer, 2006).
Vn= Vy, For λv≤ 0.815 0.815pVcrVy, For 0.815 < λv≤ 1.231 Vcr, For 1.231 < λv (3.16) where λv= r Vy Vcr , (3.17) Vy= 0.6Awfy, (3.18)
and Vcris the elastic critical shear buckling force. For members with flat webs,
Vcr is determined only for the web (Schafer, 2008), and this gives the same
results as (AISI S100-2007, 2007). For other types of cross-sections, Vcr can
be determined using the finite element method. Recent developments on the subject of shear design in conjunction with the Direct Strength Method can be found 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 not possible to derive or modify equations from the main part of the standard (AISI S100-2007, 2007) for use within the Direct Strength Method framework. This is because the existing equations are empirically derived from test data, and an expansion to other than the tested geometry ranges is difficult and not advisable without further testing.
28 Direct Strength Method 0 0.5 1 1.5 2 0.6 0.8 1 1.2 qM y Mcrl M n M y Eq.(3.11) Tests FEM (a) Local 0 0.5 1 1.5 2 0.4 0.6 0.8 1 q M y Mcrd M n M y Eq.(3.13) Tests FEM (b) Distortional
Figure 3.3 The Direct Strength Method compared to test and nonlinear
FEM results for thin-walled steel C- and Z-sections in bending. From: (Yu and Schafer, 2007) , with minor modifications: Axes and legend have been re-drawn.
3.2.7
Verification
The experimental data used to derive the Direct Strength Method equations for local and distortional buckling of beams, and as shown in figure 3.2 was complicated to separate into either local or distortional buckling. This is due to the applied boundary conditions only being able to partially restrain distortional buckling (Ziemian, 2010). In a new research program, validation of the Direct Strength Method method was sought; experimental research and finite element simulations on local buckling (Yu, 2005; Yu and Schafer, 2003, 2006b, 2007) and distortional buckling (Yu, 2005; Yu and Schafer, 2004, 2006a,b, 2007) where performed and have demonstrated the robustness of the Direct Strength Method method. A comparison of the validation data and the predictions of the Direct Strength 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 can interact, and local buckling, for example, lowers the effective second moment of area of a cross-section and consequently the global buckling strength of the member. This has been recognized for many decades, and empirical interaction equations for local and global buckling are included in most design standards, the Direct Strength Method being no exception. The influence of the interaction between the distortional buckling mode and the other two modes is less well known. This is caused by the distortional buckling mode itself only having been recognized as a significant phenomena relatively late (Davies, 1999; Davies and Jiang, 1998), and there is little fundamental knowledge on the interaction of distortional buckling with the other modes. It is stated in (Davies, 1999), that: ‘In general, there is little interaction between the distortional and global modes
When the Direct Strength Method was derived, the interaction of distortional buckling with local and global buckling was considered for inclusion, but the tested approaches led to over conservative results (Schafer, 2000, 2002a). It was noted by Schafer (2008) that recent work, by Silvestre et al. (2006); Yang and Hancock (2004), suggested local-distortional buckling interaction should be taken into account in specific cases, where both phenomena produce a similar critical load. It was shown by (Dinis et al., 2007, 2005) that coupling between local and distortional buckling modes may influence the post-buckling behaviour and 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 elastic critical load of members as a function of the half-wavelength. As was already noted, it is possible for buckling phenomena to repeat themselves at scalar multiples of their base half-wavelength. This means the apparent increase of the elastic buckling load past the local minimum for local buckling as observed in figures 2.3 and 2.6 is deceptive, the local minimum is in fact an upper bound for all greater lengths. A second effect, which is not visible in the figures as these are critical load charts, is the interaction of the global and local buckling modes with regard to member strength. The interaction between global and
local 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 the
buckling moment and force as a function of the beam length, a beam chart can be made. Such a graph can be used by a designer to easily lookup the strength of a given cross-section for a specific length, while the chart itself only needs to be made once by the beam’s manufacturer for example. The contents of this section follow the example calculation as given in Schafer (2006). A convention for notation is used however that deviates from Schafer (2006); when the 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 the procedure to generate the charts is identical for both bending as compression members, both are combined in a single section. It is noted that the procedure as 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 modes of buckling. For longer (half-)wavelengths, the pattern repeats itself and the critical buckling load remains the same. A constant critical local buckling load, independent of member length, is therefor a realistic approximation. This approximation neglects the higher local buckling loads at half-wavelengths lower than the local minimum, but as the half-wavelength is typically very small, this is 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 simply repeat itself for longer half wavelengths. The value for the the critical buckling stress obtained from the local minimum is therefore to be used for all greater half-wavelengths. The increase in critical distortional buckling loads at smaller lengths, is exploitable however, as members with unbraced lengths shorter than the half-wavelength of the local minimum, do exist in practice. To calculate the value for the critical buckling load at these smaller half-wavelengths, it is possible to use analytical models to calculate the strength, or in limited cases, read the strength directly from the moment/half-wavelength curve. A third option exists for thin-wall steel sections, based on the work by Yu (2005).
Mcrd(L) = Mcrd L Lcrd lnLcrdL . For L < Lcrd Mcrd. For Lcrd≤ L (3.21) Fcrd(L) = Fcrd L Lcrd lnLcrdL . 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- and Z-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 method may 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 for global buckling modes are used in the Direct Strength Method. It is however possible to make use of the FSM solution for ‘. . . quickly exploring the strength of different members and avoids recourse to the longer Specification [chapter 9 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 of global buckling driven part of the buckling load curve, allows the critical global
buckling load Mcre(L) to be expressed as a function of the length explicitly.
When the member considered has equal unbraced lengths for both torsion and deflection. The curve fit has the form:
Mcre= s C1 1 L 2 + C2 1 L 4 , (3.23) Fcre= s C1 1 L 2 + C2 1 L 4 , (3.24)
102 103 104 0 1 2 0.67 FSM curve Local Distortion Lateral-torsional Half-wavelength L (mm) M cr M y
Figure 3.4 Buckling load as a function of the half-wave length, for the
C-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 points
on 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−Mcre22 L4cre2
L2 cre1−L2cre2 . C2= (Mcre12 L2cre1−M2 cre2L 2 cre2)L 2 cre1L 2 cre2 L2 cre2−L2cre1 . For beams. (3.25) C1= F2
cre1L4cre1−Fcre22 L4cre2
L2 cre1−L2cre2 . C2= (Fcre12 L2cre1−F2 cre2L 2 cre2)L 2 cre1L 2 cre2 L2 cre2−L2cre1 . For columns. (3.26)
Making use of equations (3.19), (3.21), (3.23) and (3.25), allows us to redraw the 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 not increase for lengths greater than the local minimum, the interaction between local and global buckling, and the post-buckling strength is not yet visible. To show this, the Direct Strength Method equations of section 3.1 and 3.2 are applied, for columns and beams, respectively. The results of which are shown in figures 3.5a and 3.5b.
3.5
Combined axial load and bending
The Direct Strength Method as present in the North American standard (AISI S100-2007, 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 4 0 0.5 1 Local Distortion Global Length L (m) Fn Fy
(a) Column chart
0 1 2 3 4 0 0.5 1 Local Distortion Global Length L (m) M n M y (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 beam chart reveals distortion buckling to be the dominant buckling mode for beams approximately 1m in unbraced length. Local buckling dominates for other lengths up to 3m, for greater lengths, global buckling dominates, with no 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 Method
calculations (fy= 379 N mm−1), this deviates from the other graphs in this
Table 3.1 Resistance factors for LRFD design. The LRFD value for φ takes into account the uncertainties and variabilities inherent in the nominal resistance. It is dependent, amongst others, on the reliability of the design method
Load type Symbol Value
Bending φb 0.9-0.95
Tensile φt 0.95
Compression 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 Strength Method 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 structural members (AISI S100-2007, 2007) specifies design rules according to three different design philosophies; Allowable Strength Design (ASD) and Load and Resistance Factor Design (LRFD) for the United States and Mexico, and Limit States Design (LSD) for use in Canada. In this section only the provisions according to LRFD design are given. Where possible, equations are simplified if they where originally posed for multiple design philosophies, to include only the LRFD
portion. Axial loads are denoted in the text by Ncand Nt, which differs from
the original equations in (AISI S100-2007, 2007), where P and T are used for compressive and tensile forces, respectively. There is also a difference in the coordinate system used, the x axis is now parallel to length of the member, while the y and z axes lie in the plane of the cross-section, x denoting the horizontal displacement.
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 Nuare the required strengths for bending in the horizontal
y and vertical z direction, and axial (tensile) force, respectively. Mny,Mnz and
Nnt are the nominal strengths (resistance) for bending and tensile forces. φband
φt 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 of equations (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 φcis the compressive resistance factor according to table 3.1. Cmyand Cmz
are equivalent moment factors, that can be taken as unity for simply supported
members. 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 for
by 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 the
compressive 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 traditional approach for calculating beam-columns, which relies on the calculation of the normal and flexural load resistance independently, and combining these through some interaction equation. The proposed method calculates the stability of the member for the actual axial-flexural load combination, using finite strip method software. Such an approach would in principle produce more accurate strength calculations. The potential increase in accuracy is illustrated in (Schafer, 2008) by asking the reader the question ‘for all cross-sections does the maximum axial capacity exist when the load is concentric’. Traditional interaction formulae are built on the assumption that this is indeed the case, but in reality eccentric compression forces may alleviate the compressive demand at weak portions of the cross-section, leading to a higher total strength. Schafer (2008) goes on to state that ‘interaction diagrams make some sense for determining when a simple cross-section yields, but stability, this is another matter.’
(Schafer, 2003) investigated short lipped channel sections only, considering local an distortional buckling. Duong and Hancock (2004) extended the method to longer sections, by including global buckling modes and the geometric second order bending moment. In a paper by Rasmussen (2006) the method was applied to angle sections, showing convincing results. More underlying details can be found in (Rasmussen and Hossain, 2004). An expression for the shift of effective centroid was added, based on Stowell’s solution (Stowell, 1948). The equation for the shifting centroid was derived in (Rasmussen, 2005). At present accurate strength curves for global, local and distortional buckling have not been formulated, impeding the use of the method in general practice. It is noted
