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Computational modelling of geometric imperfections and buckling strength of cold-formed steel
Journal of Constructional Steel Research 78 (2012) 1–7
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Journal of Constructional Steel Research
Computational modelling of geometric imperfections and buckling strength of cold-formed steel Z. Sadovský a,⁎, J. Kriváček a, V. Ivančo b, A. Ďuricová c a b c
ÚSTARCH SAV, Inst. of Construction and Architecture, Slovak Academy of Sciences, 845 03 Bratislava, Slovakia Technical University Košice, Faculty of Mechanical Engineering, Department of Applied Mechanics and Mechatronics, 042 00 Košice, Slovakia Building Testing and Research Institute, 040 00 Košice, Slovakia
a r t i c l e
i n f o
Article history: Received 17 January 2012 Accepted 7 June 2012 Available online 10 July 2012 Keywords: Cold-formed steel Eigenmodes Geometric imperfections Imperfection measures Local-distortional interaction Strength
a b s t r a c t Computational modelling of the buckling strength of cold-formed steel members as influenced by initial geometric imperfections is studied. The geometric imperfections are represented by the member eigenmode shapes. Along with the classical measure — the amplitude of imperfections, an energy measure defined by the square root of the elastic strain energy hypothetically required to distort the originally perfect structural element into the considered imperfect shape is used. Based on the measures, two approaches for the choice of the most unfavourable imperfections are suggested. Normalising imperfections by the amplitude, the energy measure is calculated as indicative parameter of imperfection significance. Vice versa, when adopting normalisation by the energy measure, the amplitude is used as a supporting parameter. The suggestions are illustrated on calculating the strength of an axially compressed steel lipped channel column with eigenmodes exhibiting local-distortional interactions. For eigenvalue and geometrically and materially non-linear strength calculations, the FEM codes MSC.NASTRAN and COSMOS/M are employed. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The strength calculations of cold-formed steel are carried out at several levels of complexity depending on the purpose of its use. For the standardised design of ordinary members the effective width method and the recently developed direct strength method is applied. In conjunction with the latter one, more sophisticated analyses, e.g. the finite strip method or the generalised beam theory, can be applied. The most complete, however at the same time also the most computer time consuming and deep involvement requiring, is the use of analysis by the finite element methods (FEM). Particularly, this relates to the geometrically and materially non-linear FEM analysis of the strength of cold-formed steel with imperfections (GMNIA). As a consequence, the application of FEM GMNIA is mainly aimed at strength calculations of important structural members or parts. A first attempt to codify the use of non-linear FEM for design purposes is given in the rules of EN 1993-1-5, cf. [1]. Currently, the ECCS TC7 working group TWG 7.5 “Practical improvements of Design Guidelines” is preparing a “TC7-recommendation” on the use of finite element methods for thin-walled members in order to further facilitate its use. Inclusion of basic principles, modelling and applications, especially the modelling of geometric imperfections for thin-walled structures is foreseen.
⁎ Corresponding author. Tel.: + 421 2 59309208; fax: + 421 2 54773548. E-mail addresses: [email protected] (Z Sadovský), [email protected] (J Kriváček), [email protected] (V Ivančo), [email protected] (A Ďuricová). 0143-974X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2012.06.005
The development in cold-formed steel may be traced in recent review papers by Hancock [2], Dubina, Ungureanu and Rondal [3], Camotim, Basaglia and Silvestre [4], Schafer [5]. Valuable detailed study on solution sensitivities of computational modelling of elastic buckling and nonlinear collapse analysis for cold-formed steel members is presented by Schafer, Li and Moen [6]. Among the input parameters of the computational model of instability driven strength of cold-formed steel the shape and size of geometric imperfections play the crucial role, e.g. [3]. The imperfections are considered as initial shape deviations of mid-surface from the assumed perfect configuration. Generally, because of the lack of sufficient number of measurements allowing a statistical treatment of imperfection characteristics or in some cases lack of their adequacy, e.g. single component measurements may not be representative for the member imperfection shape, theoretical imperfections are employed. Commonly, eigenmodes of the elastic buckling problem calculated by FEM are used being natural choices associated with the strength problem. Having the same attribute, collapse shapes of the initially perfect member are sometimes applied. Also periodic modes — sine shapes are employed. However, for a complex profile an artificial merging of the imperfections of individual parts may arise. Other shapes are mostly combinations of those mentioned above. The present paper aims at contributing to the guidance on the choice of the most unfavourable geometric imperfections, represented by the eigenmode shapes, for FEM GMNIA. Two imperfection measures are employed. The commonly used amplitude is accompanied with an energy measure derived from the hypothetic elastic strain energy of the
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imperfect surface. Based on the measures, two approaches denoted as A and B are suggested. In the approach A, imperfections are normalised by the amplitude, while the energy measure is shown as a useful parameter indicating the severity of an imperfection. Analogously in the approach B, along with the normalisation by the energy measure, the amplitude is used as a supporting parameter. The ideas presented are illustrated on instance of a lipped channel column in axial compression with initial geometric imperfections exhibiting local-distortional interactions. The imperfections relate to cross-sectional tolerances specified for cold-formed steel in the European Standards EN 1090–2: 2008 + A1 and EN 10162: 2003, particularly to the tolerances for concavity or convexity of the flat sides and to the permitted angular tolerances of the flanges. For the elastic buckling and energy measure calculations as well as for the geometrically and materially nonlinear strength analysis of an imperfect channel, FEM codes MSC.NASTRAN and COSMOS/M are used. The paper is an extended version of the contribution presented by the authors at the EUROSTEEL 2011 Conference in Budapest. The studied example including details on computational modelling by applied FEM codes, adopted measures of geometric imperfections and eigenmode analysis with approximate identification of eigenmodes is described in Section 2. Numerical results related to the approach A, are dealt with in Section 3, while those corresponding to the approach B are presented in Section 4. Short Discussion and Conclusions sections conclude the presentations.
2. Basic setting of the problem 2.1. Considered example For numerical study, the setting of the test specimen ST15A90 of Young and Hancock [7] is adopted. The lipped channel column shown in Fig. 1 has the dimensions a=98.9, b=49.5, c =10.7 and t=1.5 mm. The paper [7] gives the yield stress of the column material as the tensile 0.2% proof stress of 515 MPa (E=210 GPa) resulting from measurements on longitudinal coupons taken from the centre of the web. Since the specimen was produced by press breaking with small bent radius, the roundness of corners and the corner properties (not given) are in computations not considered. Consequently, residual stresses are neglected, cf. [3,6]. The boundary conditions of the specimen of length 1503.7 mm being considered as fixed were realised by precision milling of the ends to ensure full contact with rigid platens transmitting the load to the column [7]. The column failed in distortional mode at the load of 97.3 kN. In computational modelling, precaution about the realisation of fully fixed ends when approaching the collapse load with development of plasticity led to consideration of effective length of L = 800 mm, rather than using the half length of a clamped column.
Fig. 1. Notation of column dimensions.
2.2. Computation tools The MSC.NASTRAN model of the lipped channel column consists of 14,080 quadrilateral four-node linear QUAD4 elements of 85,974 degrees of freedom (DOF) in a uniform mesh of 40, 20 and 4 elements across the widths of the web, flange and lip, respectively; the column length is divided into 160 elements. For eigenvalue calculations, the axial load is imposed by the unit force at one node with uniform translation of the nodes at the column top end, while for GMNIA analysis the axial load is imposed by the uniform translation of the nodes at the top end. All other translations are constrained within the pinned supports model. The geometrically and materially nonlinear strength calculations are carried out applying a displacement-incremental approach. Three values of the increment starting size are used: 0.08 mm is applied for cases with maximal amplitude of geometric imperfections, 0.04 mm for all 354 cases solved and three cases have been recalculated with the 0.02 mm increment size due to the poor solution convergence. The full Newton–Raphson iteration algorithm is applied at each increment with the convergence criteria of 10− 3 and 10 − 7 for displacements and work, respectively. Elastic–plastic material with the work hardening slope of 21 MPa, the yield function criterion of von Mises and isotropic hardening rule are adopted. The increments are summed until the value of the column shortening of 2 mm is reached; the peak of the load-shortening response curve is situated in the interval bounded by this value. The COSMOS/M model uses the SHELL4T elements of the type QUAD4 based on Mindlin's theory. Uniform meshing resulted in altogether 29,992 elements with 181,799 DOF. Along the column length, 320 elements are used. In a cross-section, 42, 20 and 4 elements cover the web, flange and lip, respectively. Pinned supports are modelled by constraining translations similarly as in the former code. The eigenvalue calculations are carried out by the Lanczos method. In the GMNIA analysis, an ideally elastic–plastic material with von Mises yield function criterion and without deformation hardening is adopted. Loading is controlled by scaling the uniform longitudinal translations at the top end proportionally to a pseudo-time. An automatic selection of time steps aimed at minimising the number of iterations needed to reach equilibrium in each increment step is performed. 2.3. Eigenmode analysis and measures of geometric imperfections The geometric imperfections in the eigenmode shapes corresponding to the first 30 elastic buckling loads and their combinations are considered. By FEM eigenvalue calculations the known equations K e φi ¼ λ i K g φi ;
λi ≈P cri ;
i ¼ 1; …; n
ð1Þ
are satisfied. Ke and Kg are the elastic stiffness and geometric matrices, respectively. φi denotes the i-th eigenmode and Pcri the corresponding buckling load. The magnitudes of imperfections derive from two assumed measures. The first one applied is the commonly used amplitude. Based on the collected data, Schafer and Peköz [8] introduced for computational modelling two categories of geometric imperfections: “type 1, maximum local imperfection in a stiffened element and type 2, maximum deviation for a lip stiffened or unstiffened flange”. Two characteristic amplitudes of the channel imply non-uniqueness of the choice of a single imperfection measure. In this paper, the amplitude giving an imperfection magnitude is defined as the maximum of the amplitudes of type 1 and type 2 imperfections, i.e. as the maximum of the web local distortion and of the flange-lip shift parallel to the web. The definition includes the case of equal amplitudes of type 1 and type 2 imperfections, which have been reported for an experimental profile in [9]. As it will be shown below an eigenmode of the channel possesses this kind of distortion.
Z Sadovský et al. / Journal of Constructional Steel Research 78 (2012) 1–7
Another measure derives from the elastic strain energy hypothetically required to distort the originally perfect structural element into the considered imperfect shape [10–12]. Assuming a geometric imperfection w0, its energy measure EM (w0) is defined as the square root of the elastic strain energy functional evaluated for w0. In FEM modelling it is given by the elastic stiffness matrix Ke: ð2Þ
200
180
160
Pcr [kN]
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 T w K w : EM ¼ EM ðw0 Þ ¼ 2 0 e 0
2
1 T 1 T φ K φ ¼ λ i φi K g φ i 2 i e i 2
ð3Þ
can be obtained as a by-product of the elastic eigenvalue analysis in both MSC.NASTRAN and COSMOS/M codes. In MSC.NASTRAN the eigenmodes are normalised by amplitude and the corresponding strain energy is directly available as the code output. COSMOS/M normalises the eigenmodes by the quadratic functional related to the geometric matrix Kg. The strain energy value results from Eq. (3). The hypothetic strain energy of a distorted shape can be directly obtained introducing the imperfection as imposed deformation in the linear elastic FEM model Sadovský et al. [12]. Even incomplete deformation field can be introduced, e.g. suitably chosen set of point deviations from the perfect shape. This can significantly decrease the extent of measurements needed for estimation of the energy measure of a geometric imperfection. The resulting deformation field is physically admissible and can be inserted into FEM strength calculations as initial geometric imperfection. Selected eigenmodes and the corresponding buckling loads obtained by MSC.NASTRAN are shown in Figs. 2 and 3. A working approximate classification of the calculated eigenmodes has been carried out. It is based on the estimated ratio D/L of distortional (D) to local (L) amplitudes. The modes with minute distortional amplitudes or up to 20% of the web amplitude are denoted as local buckling modes. The first three, the fifth and eight eigenmodes (and others) are of the latter kind. The eigenmodes with distortional amplitudes reaching 45% (11th eigenmode),
140
120
The energy measure satisfies the axioms of a norm and by the matrix Ke the corresponding scalar product is generated. Since the eigenmodes of the eigenvalue problem (1) are orthogonal with respect to the elastic stiffness matrix Ke (as well as to the geometric matrix Kg), they form orthogonal basis functions, which can be used in approximations of imperfections by series under the energy norm (2). This feature is not available when normalising imperfections by amplitude. The strain energy of individual eigenmodes: EM ðφi Þ ¼
3
100 Mode 14
Mode 15
Mode 19
Mode 23
Mode 27
Mode 28
>1
>1
Eigenmodes D/L =
>1
0.75
>1
≅1
Fig. 3. Higher buckling loads and the corresponding eigenmodes (MSC.NASTRAN).
75% (15th) and 55% (18th) of the web amplitude are classified as the lower distortional ones. Distortional eigenmodes are found at 14th, 19th, 23rd, 27th and 28th position. The first three exhibit high degree of interaction with local modes possessing amplitudes of 75%, 48% and almost 100% of the distortional amplitude. Global eigenmodes have not been observed. Generally, COSMOS/M code provides slightly lower buckling loads. The shapes and order of eigenmodes correlate quite well with those obtained applying MSC.NASTRAN but for three eigenmode couples: the 5th and 6th, the 14th and 15th and the 27th and 29th eigenmodes, which in COSMOS/M have been found in the reversed order. The buckling loads and eigenmodes used in GMNIA calculations are shown in Fig. 4. Note that the 14th and 15th buckling loads are close using any of the codes, cf. Figs. 3 and 4. From the results of eigenvalue analysis, useful indications on the role of eigenmode imperfections in strength analysis can be drawn. For the eigenmodes normalised by the amplitude, the smaller value of the energy measure means that less energy is required to distort the channel into the eigenmode shape of given amplitude, i.e. that buckling deformability of the channel in that eigenmode shape is greater. On the other hand, high values of the energy measure highlight potential unrealistic amplitudes of severely distorted members. When normalising eigenmodes by the energy measure, higher amplitude of an eigenmode is interpreted as greater buckling deformability of the channel in this shape. Besides, the higher amplitude implies
105
140
100
130
95
120
90
Pcr [kN]
Pcr [kN]
110
85
110 100
80 90
75
80
70 Mode 1
Mode 2
Mode 3
Mode 5
Mode 8
Mode 11
0.2
0.45
Eigenmodes D/L =
< 0.1
< 0.1
< 0.1
0.15
70 Mode 1
Fig. 2. Lower buckling loads and the corresponding eigenmodes (MSC.NASTRAN).
Mode 6
Mode 11
Mode 14
Mode 15
Mode 19
Eigenmodes Fig. 4. Selected buckling loads and the corresponding eigenmodes (COSMOS/M).
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Z Sadovský et al. / Journal of Constructional Steel Research 78 (2012) 1–7
EM [(Nmm)1/2] 350
Pcr - lower distort. Pcr - distortional
200
280
EM - local buckling EM - lower distort. EM - distortional
150
Pcr - local buckling
P cr,P ult [kN]
Pcr [kN]
Pcr - local buckling
250
Pcr - lower distort.
120
max |
0|
= 1.5 [mm]
Pcr - distortional Pult - local buckling Pult - lower distort.
115
Pult - distortional
210
110 100
140
50
70
0
0
105
Buckling mode ~ imperfection mode Fig. 5. Buckling loads and the energy measure of eigenmodes normalised by the amplitude (MSC.NASTRAN).
earlier outset of large deflection effects unfavourably influencing the channel strength. Further, the normalisation by the energy suppresses unrealistic amplitudes of the imperfections. The quadratic functional related to the geometric matrix Kg, i.e. the expression in the right hand side of Eq. (3) except of λi, represents the channel axial shortening corresponding to its deflection in the ith eigenmode. An intuitive deduction suggests that the higher the buckling load, the stiffer (axially) the channel deflected in the corresponding eigenmode shape. However, normalising eigenmodes by the amplitude does not result in elastic shortenings decreasing with the number of the eigenmode. When employing normalisation by the energy measure, the constant left hand side in Eq. (3) implies that the shortenings are inversely proportional to the increasing buckling loads. Thus, for the approach using the energy measure, perfect fit with the intuitive expectation is obtained. These considerations form a background to suggestion of the approaches A and B, in which normalising imperfections by one of the measures considered, another one is employed as a supporting parameter indicating the candidates for the most unfavourable eigenmode imperfections. An additional criterion for the selection of an eigenmode is that the corresponding buckling load has not to be significantly higher than the critical buckling load. 3. Approach A: imperfections normalised by the amplitude The computed critical and higher buckling loads as well as the energy measures EM(φi) of the corresponding eigenmodes φi normalised by the unit amplitude (max |φi| = 1 mm) shown in Fig. 5 are obtained by MSC.NASTRAN. The working classification of eigenmodes to local, lower distortional and distortional is reflected differentiating markers of both buckling loads and EM(φi) values. Fig. 5 shows that the lower values of the energy measure of eigenmodes normalised by the amplitude are obtained for the distortional and lower distortional modes or local buckling modes with perceptible proportion of distortional imperfection. Not distinctly but still distinguishable from the local modes is the energy value for the 18th eigenmode with one central distortional half-wave of L/2 length. The number of half-waves (altogether 21) in the web is responsible for relatively high value of the EM parameter, see Fig. 6. The high values of the strain energy obtained for eigenmodes of large number of local half-waves suggest that their realistic amplitudes should be lower
29
25
27
21
23
19
15
17
11
7
9
5
27
25
23
21
19
29
1
13
i
95
of unit amplitude
3
Buckling mode
17
15
13
9
11
7
5
3
1
100
0
Fig. 7. Buckling and collapse loads of the channel; eigenmode imperfections are normalised by the amplitude (MSC.NASTRAN).
than for modes of smaller number of half-waves, thus questioning the normalisation of imperfections by amplitude. For calculation of the collapse loads Pult, the maximal amplitude considered is of the channel thickness (1.5 mm). This amplitude was recommended by Schafer and Peköz [8] as a conservative estimation of the maximum distortional imperfection — type 2 imperfection for computational modelling of the strength of cold-formed steel members. Note that for the cold rolled sections the corresponding working tolerance (EN 10162: 2003) is of 1.08 mm, while for the cold formed sections the functional manufacturing tolerance of the execution class EXC2 corresponds to distortional amplitude of 1.73 mm (EN 1090–2: 2008 + A1). The tolerances of web flatness are lesser. The collapse loads obtained for the maximal amplitude of imperfections are in Fig. 7 differentiated by the marker shape corresponding to the class of imperfection applied. Comparison with Fig. 5 shows that lower Pult values are generally realised for eigenmodes of lower energy measure and not too high buckling loads, the minimum value of 98.9 kN being reached for the imperfection in the 14th eigenmode, i.e. the first distortional mode. These results suggest that the use of the energy measure as a parameter for indicating the most unfavourable imperfections effectively decreases the number of eigenmodes to be investigated. Referring to the working classification of eigenmodes in Subsection 2.3 and the above discussed Fig. 5, another benefit of the energy measure as a supporting parameter for identification of distortional components in eigenmodes can be highlighted. Particularly, the first distortional mode is important for the Direct Strength Method developed by Schafer and Peköz [9]. The corresponding buckling load is then used in a strength
P ult [kN] Mode 1
120
Mode 5 Mode 11
115
Mode 14 Mode 15
110
Mode 19
105 100 97.3 kN - exp [7]
95 0.0
0.2
0.4
Amplitude of Fig. 6. The 18th eigenmode (MSC.NASTRAN).
0.6 0
0.8
1.0
1.2
1.4
in eigenmode shape [mm]
Fig. 8. Collapse loads related to the amplitude of eigenmode imperfections (MSC.NASTRAN).
Z Sadovský et al. / Journal of Constructional Steel Research 78 (2012) 1–7
P ult [kN]
120
Mode 1
120
5
Mode 6
115
Mode 11
115
Mode 14
110
Mode 19
105
Pult [kN]
Mode 15
110
105 100
100 97.3 kN - exp [7]
95 95 0.0
0.2
0.4
Amplitude of
0.6 0
0.8
1.0
1.2
1.4
90
in eigenmode shape [mm]
Mode 1
formula yielding the distortional strength of a member. Consider mistaken choice of the 11th eigenmode with significant proportion of distortional component (45%) as the first distortional mode instead of the 14th eigenmode referring to the buckling loads drawn in Fig. 7 by symbol line. Supposing that the strength formula displays imperfection sensitivity, its application to buckling load Pcr11 may result in a conservative design. Note that for both imperfections the column failed in distortional mode. The identification problem is still topical as witness the reports on difficulties consisting in “search amongst higher modes” [13] and not exactness “especially when both local and distortional buckling are present in the same eigenmode” [14]. The relationship between collapse loads and imperfection amplitudes is shown in Fig. 8 (MSC.NASTRAN) and Fig. 9 (COSMOS/M). Eigenmode imperfections possessing low energy measure are selected. The collapse loads related to the imperfections having similar shapes in both codes are drawn in Figs. 8 and 9 by the same symbol-line curve, cf. Subsection 2.3. The curves in Figs. 8 and 9 show an overall correlation; generally, the COSMOS/M code provides lower collapse loads. The experimental ultimate load 97.3 kN of the specimen ST15A90 [7] drawn by horizontal dashed line is close below the Pult obtained for the distortional imperfection having the maximal amplitude. The collapse loads and modes obtained by MSC.NASTRAN for the eigenmode imperfections applied in Fig. 8, are for minute imperfection amplitudes (0.01 mm) visualised in Fig. 10 and for imperfection amplitudes of channel thickness in Fig. 11. For minute imperfections, local modes of column failure are obtained. For imperfection amplitudes of channel thickness distortional modes of column failure resulted. The whole group of collapse loads and modes as computed for the first
Mode 14
Mode 15
Mode 19
eigenmode imperfection is shown in Fig. 12, while for the first distortional eigenmode imperfection (the 14th) in Fig. 13. An interesting finding is that for imperfections of minute amplitudes the failure mode corresponding to the lowest Pult value realises in the shape similar to the first eigenmode with dominating local buckles of ten half waves in the web. The failure modes corresponding to higher collapse loads resemble the second eigenmode of eleven local halfwaves, cf. Figs. 2 and 10. 4. Approach B: imperfections normalised by the energy measure Fig. 14 shows the buckling loads and amplitudes of the corresponding eigenmodes normalised by the energy measure. The level of the energy measure is set by the first distortional mode (the 14th eigenmode) of amplitude equalling the channel thickness. For the eigenmode shapes adjusted to the given strain energy level, the higher amplitudes indicate presence of distortional components, see Figs. 2 and 3. The low level amplitudes correspond to the modes of numerous local half-waves; a special case of the 18th eigenmode with simultaneous presence of distortional half-wave is discussed in Section 3, see Fig. 6. The collapse loads of the channel with eigenmode imperfections are shown in Fig. 15 together with the imperfection amplitudes. Distinctly lower collapse loads are obtained for the imperfections with distortional 120 115
115
110
Pult [kN]
Pult [kN]
Mode 11
Fig. 11. Collapse loads and modes obtained for eigenmode imperfections with amplitudes of channel thickness (MSC.NASTRAN).
120
110
Mode 5
Imperfections amplitude of 1.5 mm
Fig. 9. Collapse loads related to the amplitude of eigenmode imperfections (COSMOS/M).
105
105
100
100
95
95
90 0.01
0.075
0.2
0.5
1.0
1.5
Amplitude of Mode 1 imperfection [mm]
90 Mode 1
Mode 5
Mode 11
Mode 14
Mode 15
Mode 19
Imperfections amplitude of 0.01 mm Fig. 10. Collapse loads and modes obtained for eigenmode imperfections of minute amplitudes (MSC.NASTRAN).
Fig. 12. Collapse loads and modes obtained for imperfections in the first eigenmode shape and selected amplitudes (MSC.NASTRAN).
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Z Sadovský et al. / Journal of Constructional Steel Research 78 (2012) 1–7
P ult [kN]
120
Pult - local buckling
130
Pult - lower distort.
EM (
115
0)
= const.
Pult - distortional
125
max | 0| - local buckling max | 0 | - distortional
115
2.0
110
1.5
105
1.0
100
0.5
95
0,0
0
105 100 95
max |
Pult [kN]
120
| [mm]
max | 0 | - lower distort.
110
90 0.01
0.075
0.2
0.5
1.0
1.5
Buckling mode ~ imperfection mode Fig. 13. Collapse loads and modes obtained for imperfections in the 14th eigenmode shape and selected amplitudes (MSC.NASTRAN).
components. Particularly, the lowest Pult is obtained for the first distortional eigenmode, see Fig. 3. The separation of collapse loads related to the imperfections with distortional components from those with prevailing local components is now more obvious than if normalising the imperfections by amplitude, cf. Fig. 7 of Section 3. Also higher separation among collapse loads calculated for imperfections in the shapes of local buckling modes is observed. Normalisation of the imperfections by the energy measure downgrades the amplitudes of severely distorted local buckling modes to significantly smaller amplitudes than are those of distortional modes or those of less distorted local modes. As a result of downgrading the amplitudes, generally significantly higher collapse loads are obtained. The situation can be illustrated also on the above mentioned 18th lower distortional eigenmode, see Fig. 6. Utilising normalisation by the amplitude, the collapse load corresponding to the imperfection “competes” with the lowest Pult, see Fig. 7. However, the energy normalisation assigns to the imperfection mode almost six times smaller amplitude resulting in an unimportant high collapse load, see Fig. 15. Note that employing the energy measure, unrealistic amplitudes of imperfections have been reported in the strength studies on post-buckling of thin plates [11] and stringer-stiffened cylindrical shell [12]. Combination of eigenmodes normalised to a certain level of the energy measure can be readily carried out with combination factors
Pcr - local buckling
P cr [kN]
max |
Pcr - lower distort.
250
EM ( i) = const.
Pcr - distortional
i|
[mm] 2.5
29
25 0
Fig. 15. Collapse loads and amplitudes of the channel eigenmode imperfections normalised by the energy measure (MSC.NASTRAN).
following the Pythagoras rule. For example, coupling of the eigenmodes φ0,11 and φ014 can be given in the form c11φ0,11 + c14φ0,14 with combination factors c11 and c14 ranging within the interval b−1, 1> and satisfying the equation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c11 ¼ 1−ðc14 Þ2 :
ð4Þ
The collapse loads obtained for combinations of the imperfections in 14th eigenmode with 11th and 15th eigenmode are shown in Fig. 16. The level of energy measure applied when calculating data in Figs. 14 and 15 is adopted. Two branches correspond to plus and minus sign of c11 (or c15) following Eq. (4). The least collapse load of 98.3 kN is reached for plus sign and the coupled mode c15φ0,15 + c14φ0,14. Slightly higher minima of 98.4 kN have been obtained in combinations of the first distortional mode (the 14th) with the first and third local modes. Fig. 17 shows the collapse loads as depending on the level of the energy measure of selected imperfections. In comparison with the dependence on amplitude in Fig. 8, the failure loads obtained for the imperfection in the first distortional mode are almost in the whole range of imperfection levels clearly below other failure loads. Local buckling modes of imperfections, particularly the 5th eigenmode and the 1st eigenmode, are decisive only for small and minute magnitudes, respectively. Note that the rate of change of failure loads with the magnitude of
P ult [kN]
Pult(c11ϕ0,11 + c14ϕ0,14)
110
Pult(c15ϕ0,15+ c14ϕ0,14)
max | i | - local
200
27
21
23
19
15
17
13
9
11
5
7
1
3
Amplitude of Mode14 imperfection [mm]
2.0
max | i | - lower dist.
105
max | i | - distortional
150
1.5
100
1.0
50
0.5
95
0
0.0
90
i
29
25
27
21
23
19
15
17
13
9
Buckling mode
11
7
5
1
3
100
normalised by energy measure
Fig. 14. Buckling loads and amplitudes of eigenmodes normalised by the energy measure (MSC.NASTRAN).
EM(ϕ0) = const.
0.0
0.2
0.4
0.6
0.8
1.0
C14 Fig. 16. Collapse loads related to coupled eigenmode imperfections normalised by the energy measure (MSC.NASTRAN).
Z Sadovský et al. / Journal of Constructional Steel Research 78 (2012) 1–7
P ult [kN]
7
6. Conclusions Mode 1
120
Mode 5 Mode 11
115
Mode 14 Mode 15
110
Mode 19
105 100 97.3 kN - exp [7]
95 0
10
20
30
40
Energy measure of ϕ0 in eigenmode shape [(Nmm)1/2 ] Fig. 17. Collapse loads related to the energy measure of eigenmode imperfections (MSC.NASTRAN).
imperfection is significantly higher for small magnitudes than in the rest of interval studied.
Using the first and possibly also the second eigenmode as imperfection patterns for computational modelling of strength of cold-formed steel may not capture the decisive shape, particularly for non-small imperfection sizes. In either of the suggested approaches to the analysis of eigenmodes, useful preliminary information on the choice of the adverse imperfection shapes effectively decreasing the number of candidate modes is obtained. In particular, the presence of distortional component is indicated. As the preferable one, the approach (B) employing normalisation of geometric imperfections by the energy measure and the corresponding amplitude as a supporting parameter is considered. The advantages of the approach are: ➣ Unrealistic imperfections are ruled out; ➣ The eigenmode imperfections are readily combined; ➣ The energy concept may prove useful in establishing a relationship to standards of execution and production tolerances; ➣ Theoretical background is available.
5. Discussion
Acknowledgment
The computed collapse loads presented in Sections 3 and 4 show that both parameters studied in the approaches A and B, i.e. the value of one of the imperfection measures and of the buckling load, are sensible for the choice of the most unfavourable eigenmode shapes. The parameter of imperfection measure reflecting buckling deformability plays more decisive role for middle and high imperfection levels, while for small levels higher weight has to be attributed to the buckling load parameter. For very small imperfection levels, the shape of the first eigenmode overrides the influence of the other modes including those with significant proportion of distortional components. The performance of the approach A appears to be of less significance. Normalising the eigenmodes by the amplitude includes into comparison heavy distorted modes of unrealistically high amplitudes. Moreover, there is a poor correlation between the buckling load parameter and the axial shortenings of the channel with eigenmode deflections of constant amplitude. These facts may be the reasons of less obvious indications of adverse eigenmode shapes. Employing the approach B, the unrealistic amplitudes of imperfections are ruled out by the energy measure normalisation. The axial shortenings of the channel related to eigenmodes are inversely proportional to the corresponding buckling loads. Straightforward combination of eigenmodes allows the study of various different imperfection shapes. Note that an indication of interaction sensitivity of eigenmode shapes may be obtained by Sobol sensitivity indices, however some preliminary statistical information on the occurrence of single modes is required, see Kala [15]. The energy concept of normalisation of geometric imperfections using the amplitude of eigenmode imperfections as a parameter of their influence on channel strength offers an opportunity of setting a relationship to standards of execution and production tolerances. Particularly, the most unfavourable eigenmode shape being in the range of upper amplitude levels suggests itself as the reference shape for adjustment to a tolerance standard. In strength calculations, the other eigenmode shapes have mostly smaller amplitudes and those of possible higher amplitudes are less influential. The study of examples exhibiting higher level of distortional components in the eigenmodes, which is realised by a) halving the length of channel and b) extending the width of flanges and shortening of lips in the former case, is to appear in Procedia Engineering within the Proceedings of the Conference Steel Structures and Bridges 2012.
The authors acknowledge a partial support of their work by grant agency VEGA under grant 2/0101/09 and grant 1/0090/12. The MSC.NASTRAN code has been acquired with the financial assistance of the European Regional Development Fund (ERDF) under the Operational Programme Research and Development/Measure 4.1 Support of networks of excellence in research and development as the pillars of regional development and support to international cooperation in the Bratislava region/Project No. 26240120020 Building the centre of excellence for research and development of structural composite materials — 2nd stage. References [1] Johansson B, Maquoi R, Sedlacek G, Müller C, Beg D. Commentary and worked examples to EN 1993-1-5 “Plated structural elements”. JRC scientific and technical reports. © European Communities; 2007. [2] Hancock GJ. Cold-formed steel structures. J Constr Steel Res 2003;59:473–87. [3] Dubina D, Ungureanu V, Rondal J. Numerical modelling and codification of imperfection for cold-formed steel member analysis. Steel Compos Struct 2005;5 (6): 515–33. [4] Camotim D, Basaglia C, Silvestre N. GBT buckling analysis of thin-walled steel frames: a state-of-the-art report. Thin Wall Struct 2010;48:726–43. [5] Schafer BW. Cold-formed structures around the world. a review on recent advances in applications, analysis and design. Steel Constr 2011;4 (3):141–9. [6] Schafer BW, Li Z, Moen CD. Computational modelling of cold-formed steel. Thin Wall Struct 2010;48:752–62. [7] Young B, Hancock GJ. Compression tests of channels with inclined simple edge stiffeners. J Struct Eng ASCE 2003;129 (10):1403–11. [8] Schafer BW, Peköz T. Computational modeling of cold-formed steel: characterizing geometric imperfections and residual stresses. J Constr Steel Res 1998;47: 193–210. [9] Lecce M, Rasmussen KJR. Finite element modeling and design of cold-formed stainless steel sections. Research rep. no. 845. Sydney: Dept. of Civil Engineering, Univ. of Sydney; 2005. [10] Sadovský Z. A theoretical approach to the problem of the most dangerous initial deflection shape in stability type structural problems. Aplik Mat 1978;23 (4): 248–66. [11] Sadovský Z, Teixeira AP, Guedes Soares C. Degradation of the compressive strength of rectangular plates due to initial deflection. Thin Wall Struct 2005;43:65–82. [12] Sadovský Z, Ďuricová A, Ivančo V, Kriváček J. Imperfection measures of eigen- and periodic modes of axially loaded stringer-stiffened cylindrical shell. Proc. IMechE, 224(G5). Journal of Aerospace Engineering; 2010. p. 601–12. [13] Kwon YB, Kim BS, Hancock GJ. Compression tests of high strength cold-formed steel channels with buckling interaction. J Constr Steel Res 2009;65:278–89. [14] Moen CD, Schafer BW. Experiments on cold-formed steel columns with holes. Thin Wall Struct 2008;46:1164–82. [15] Kala Z. Sensitivity analysis of steel plane frames with initial imperfections. Eng Struct 2011;33 (8):2342–9.
Contents lists available at SciVerse ScienceDirect
Journal of Constructional Steel Research
Computational modelling of geometric imperfections and buckling strength of cold-formed steel Z. Sadovský a,⁎, J. Kriváček a, V. Ivančo b, A. Ďuricová c a b c
ÚSTARCH SAV, Inst. of Construction and Architecture, Slovak Academy of Sciences, 845 03 Bratislava, Slovakia Technical University Košice, Faculty of Mechanical Engineering, Department of Applied Mechanics and Mechatronics, 042 00 Košice, Slovakia Building Testing and Research Institute, 040 00 Košice, Slovakia
a r t i c l e
i n f o
Article history: Received 17 January 2012 Accepted 7 June 2012 Available online 10 July 2012 Keywords: Cold-formed steel Eigenmodes Geometric imperfections Imperfection measures Local-distortional interaction Strength
a b s t r a c t Computational modelling of the buckling strength of cold-formed steel members as influenced by initial geometric imperfections is studied. The geometric imperfections are represented by the member eigenmode shapes. Along with the classical measure — the amplitude of imperfections, an energy measure defined by the square root of the elastic strain energy hypothetically required to distort the originally perfect structural element into the considered imperfect shape is used. Based on the measures, two approaches for the choice of the most unfavourable imperfections are suggested. Normalising imperfections by the amplitude, the energy measure is calculated as indicative parameter of imperfection significance. Vice versa, when adopting normalisation by the energy measure, the amplitude is used as a supporting parameter. The suggestions are illustrated on calculating the strength of an axially compressed steel lipped channel column with eigenmodes exhibiting local-distortional interactions. For eigenvalue and geometrically and materially non-linear strength calculations, the FEM codes MSC.NASTRAN and COSMOS/M are employed. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The strength calculations of cold-formed steel are carried out at several levels of complexity depending on the purpose of its use. For the standardised design of ordinary members the effective width method and the recently developed direct strength method is applied. In conjunction with the latter one, more sophisticated analyses, e.g. the finite strip method or the generalised beam theory, can be applied. The most complete, however at the same time also the most computer time consuming and deep involvement requiring, is the use of analysis by the finite element methods (FEM). Particularly, this relates to the geometrically and materially non-linear FEM analysis of the strength of cold-formed steel with imperfections (GMNIA). As a consequence, the application of FEM GMNIA is mainly aimed at strength calculations of important structural members or parts. A first attempt to codify the use of non-linear FEM for design purposes is given in the rules of EN 1993-1-5, cf. [1]. Currently, the ECCS TC7 working group TWG 7.5 “Practical improvements of Design Guidelines” is preparing a “TC7-recommendation” on the use of finite element methods for thin-walled members in order to further facilitate its use. Inclusion of basic principles, modelling and applications, especially the modelling of geometric imperfections for thin-walled structures is foreseen.
⁎ Corresponding author. Tel.: + 421 2 59309208; fax: + 421 2 54773548. E-mail addresses: [email protected] (Z Sadovský), [email protected] (J Kriváček), [email protected] (V Ivančo), [email protected] (A Ďuricová). 0143-974X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2012.06.005
The development in cold-formed steel may be traced in recent review papers by Hancock [2], Dubina, Ungureanu and Rondal [3], Camotim, Basaglia and Silvestre [4], Schafer [5]. Valuable detailed study on solution sensitivities of computational modelling of elastic buckling and nonlinear collapse analysis for cold-formed steel members is presented by Schafer, Li and Moen [6]. Among the input parameters of the computational model of instability driven strength of cold-formed steel the shape and size of geometric imperfections play the crucial role, e.g. [3]. The imperfections are considered as initial shape deviations of mid-surface from the assumed perfect configuration. Generally, because of the lack of sufficient number of measurements allowing a statistical treatment of imperfection characteristics or in some cases lack of their adequacy, e.g. single component measurements may not be representative for the member imperfection shape, theoretical imperfections are employed. Commonly, eigenmodes of the elastic buckling problem calculated by FEM are used being natural choices associated with the strength problem. Having the same attribute, collapse shapes of the initially perfect member are sometimes applied. Also periodic modes — sine shapes are employed. However, for a complex profile an artificial merging of the imperfections of individual parts may arise. Other shapes are mostly combinations of those mentioned above. The present paper aims at contributing to the guidance on the choice of the most unfavourable geometric imperfections, represented by the eigenmode shapes, for FEM GMNIA. Two imperfection measures are employed. The commonly used amplitude is accompanied with an energy measure derived from the hypothetic elastic strain energy of the
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imperfect surface. Based on the measures, two approaches denoted as A and B are suggested. In the approach A, imperfections are normalised by the amplitude, while the energy measure is shown as a useful parameter indicating the severity of an imperfection. Analogously in the approach B, along with the normalisation by the energy measure, the amplitude is used as a supporting parameter. The ideas presented are illustrated on instance of a lipped channel column in axial compression with initial geometric imperfections exhibiting local-distortional interactions. The imperfections relate to cross-sectional tolerances specified for cold-formed steel in the European Standards EN 1090–2: 2008 + A1 and EN 10162: 2003, particularly to the tolerances for concavity or convexity of the flat sides and to the permitted angular tolerances of the flanges. For the elastic buckling and energy measure calculations as well as for the geometrically and materially nonlinear strength analysis of an imperfect channel, FEM codes MSC.NASTRAN and COSMOS/M are used. The paper is an extended version of the contribution presented by the authors at the EUROSTEEL 2011 Conference in Budapest. The studied example including details on computational modelling by applied FEM codes, adopted measures of geometric imperfections and eigenmode analysis with approximate identification of eigenmodes is described in Section 2. Numerical results related to the approach A, are dealt with in Section 3, while those corresponding to the approach B are presented in Section 4. Short Discussion and Conclusions sections conclude the presentations.
2. Basic setting of the problem 2.1. Considered example For numerical study, the setting of the test specimen ST15A90 of Young and Hancock [7] is adopted. The lipped channel column shown in Fig. 1 has the dimensions a=98.9, b=49.5, c =10.7 and t=1.5 mm. The paper [7] gives the yield stress of the column material as the tensile 0.2% proof stress of 515 MPa (E=210 GPa) resulting from measurements on longitudinal coupons taken from the centre of the web. Since the specimen was produced by press breaking with small bent radius, the roundness of corners and the corner properties (not given) are in computations not considered. Consequently, residual stresses are neglected, cf. [3,6]. The boundary conditions of the specimen of length 1503.7 mm being considered as fixed were realised by precision milling of the ends to ensure full contact with rigid platens transmitting the load to the column [7]. The column failed in distortional mode at the load of 97.3 kN. In computational modelling, precaution about the realisation of fully fixed ends when approaching the collapse load with development of plasticity led to consideration of effective length of L = 800 mm, rather than using the half length of a clamped column.
Fig. 1. Notation of column dimensions.
2.2. Computation tools The MSC.NASTRAN model of the lipped channel column consists of 14,080 quadrilateral four-node linear QUAD4 elements of 85,974 degrees of freedom (DOF) in a uniform mesh of 40, 20 and 4 elements across the widths of the web, flange and lip, respectively; the column length is divided into 160 elements. For eigenvalue calculations, the axial load is imposed by the unit force at one node with uniform translation of the nodes at the column top end, while for GMNIA analysis the axial load is imposed by the uniform translation of the nodes at the top end. All other translations are constrained within the pinned supports model. The geometrically and materially nonlinear strength calculations are carried out applying a displacement-incremental approach. Three values of the increment starting size are used: 0.08 mm is applied for cases with maximal amplitude of geometric imperfections, 0.04 mm for all 354 cases solved and three cases have been recalculated with the 0.02 mm increment size due to the poor solution convergence. The full Newton–Raphson iteration algorithm is applied at each increment with the convergence criteria of 10− 3 and 10 − 7 for displacements and work, respectively. Elastic–plastic material with the work hardening slope of 21 MPa, the yield function criterion of von Mises and isotropic hardening rule are adopted. The increments are summed until the value of the column shortening of 2 mm is reached; the peak of the load-shortening response curve is situated in the interval bounded by this value. The COSMOS/M model uses the SHELL4T elements of the type QUAD4 based on Mindlin's theory. Uniform meshing resulted in altogether 29,992 elements with 181,799 DOF. Along the column length, 320 elements are used. In a cross-section, 42, 20 and 4 elements cover the web, flange and lip, respectively. Pinned supports are modelled by constraining translations similarly as in the former code. The eigenvalue calculations are carried out by the Lanczos method. In the GMNIA analysis, an ideally elastic–plastic material with von Mises yield function criterion and without deformation hardening is adopted. Loading is controlled by scaling the uniform longitudinal translations at the top end proportionally to a pseudo-time. An automatic selection of time steps aimed at minimising the number of iterations needed to reach equilibrium in each increment step is performed. 2.3. Eigenmode analysis and measures of geometric imperfections The geometric imperfections in the eigenmode shapes corresponding to the first 30 elastic buckling loads and their combinations are considered. By FEM eigenvalue calculations the known equations K e φi ¼ λ i K g φi ;
λi ≈P cri ;
i ¼ 1; …; n
ð1Þ
are satisfied. Ke and Kg are the elastic stiffness and geometric matrices, respectively. φi denotes the i-th eigenmode and Pcri the corresponding buckling load. The magnitudes of imperfections derive from two assumed measures. The first one applied is the commonly used amplitude. Based on the collected data, Schafer and Peköz [8] introduced for computational modelling two categories of geometric imperfections: “type 1, maximum local imperfection in a stiffened element and type 2, maximum deviation for a lip stiffened or unstiffened flange”. Two characteristic amplitudes of the channel imply non-uniqueness of the choice of a single imperfection measure. In this paper, the amplitude giving an imperfection magnitude is defined as the maximum of the amplitudes of type 1 and type 2 imperfections, i.e. as the maximum of the web local distortion and of the flange-lip shift parallel to the web. The definition includes the case of equal amplitudes of type 1 and type 2 imperfections, which have been reported for an experimental profile in [9]. As it will be shown below an eigenmode of the channel possesses this kind of distortion.
Z Sadovský et al. / Journal of Constructional Steel Research 78 (2012) 1–7
Another measure derives from the elastic strain energy hypothetically required to distort the originally perfect structural element into the considered imperfect shape [10–12]. Assuming a geometric imperfection w0, its energy measure EM (w0) is defined as the square root of the elastic strain energy functional evaluated for w0. In FEM modelling it is given by the elastic stiffness matrix Ke: ð2Þ
200
180
160
Pcr [kN]
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 T w K w : EM ¼ EM ðw0 Þ ¼ 2 0 e 0
2
1 T 1 T φ K φ ¼ λ i φi K g φ i 2 i e i 2
ð3Þ
can be obtained as a by-product of the elastic eigenvalue analysis in both MSC.NASTRAN and COSMOS/M codes. In MSC.NASTRAN the eigenmodes are normalised by amplitude and the corresponding strain energy is directly available as the code output. COSMOS/M normalises the eigenmodes by the quadratic functional related to the geometric matrix Kg. The strain energy value results from Eq. (3). The hypothetic strain energy of a distorted shape can be directly obtained introducing the imperfection as imposed deformation in the linear elastic FEM model Sadovský et al. [12]. Even incomplete deformation field can be introduced, e.g. suitably chosen set of point deviations from the perfect shape. This can significantly decrease the extent of measurements needed for estimation of the energy measure of a geometric imperfection. The resulting deformation field is physically admissible and can be inserted into FEM strength calculations as initial geometric imperfection. Selected eigenmodes and the corresponding buckling loads obtained by MSC.NASTRAN are shown in Figs. 2 and 3. A working approximate classification of the calculated eigenmodes has been carried out. It is based on the estimated ratio D/L of distortional (D) to local (L) amplitudes. The modes with minute distortional amplitudes or up to 20% of the web amplitude are denoted as local buckling modes. The first three, the fifth and eight eigenmodes (and others) are of the latter kind. The eigenmodes with distortional amplitudes reaching 45% (11th eigenmode),
140
120
The energy measure satisfies the axioms of a norm and by the matrix Ke the corresponding scalar product is generated. Since the eigenmodes of the eigenvalue problem (1) are orthogonal with respect to the elastic stiffness matrix Ke (as well as to the geometric matrix Kg), they form orthogonal basis functions, which can be used in approximations of imperfections by series under the energy norm (2). This feature is not available when normalising imperfections by amplitude. The strain energy of individual eigenmodes: EM ðφi Þ ¼
3
100 Mode 14
Mode 15
Mode 19
Mode 23
Mode 27
Mode 28
>1
>1
Eigenmodes D/L =
>1
0.75
>1
≅1
Fig. 3. Higher buckling loads and the corresponding eigenmodes (MSC.NASTRAN).
75% (15th) and 55% (18th) of the web amplitude are classified as the lower distortional ones. Distortional eigenmodes are found at 14th, 19th, 23rd, 27th and 28th position. The first three exhibit high degree of interaction with local modes possessing amplitudes of 75%, 48% and almost 100% of the distortional amplitude. Global eigenmodes have not been observed. Generally, COSMOS/M code provides slightly lower buckling loads. The shapes and order of eigenmodes correlate quite well with those obtained applying MSC.NASTRAN but for three eigenmode couples: the 5th and 6th, the 14th and 15th and the 27th and 29th eigenmodes, which in COSMOS/M have been found in the reversed order. The buckling loads and eigenmodes used in GMNIA calculations are shown in Fig. 4. Note that the 14th and 15th buckling loads are close using any of the codes, cf. Figs. 3 and 4. From the results of eigenvalue analysis, useful indications on the role of eigenmode imperfections in strength analysis can be drawn. For the eigenmodes normalised by the amplitude, the smaller value of the energy measure means that less energy is required to distort the channel into the eigenmode shape of given amplitude, i.e. that buckling deformability of the channel in that eigenmode shape is greater. On the other hand, high values of the energy measure highlight potential unrealistic amplitudes of severely distorted members. When normalising eigenmodes by the energy measure, higher amplitude of an eigenmode is interpreted as greater buckling deformability of the channel in this shape. Besides, the higher amplitude implies
105
140
100
130
95
120
90
Pcr [kN]
Pcr [kN]
110
85
110 100
80 90
75
80
70 Mode 1
Mode 2
Mode 3
Mode 5
Mode 8
Mode 11
0.2
0.45
Eigenmodes D/L =
< 0.1
< 0.1
< 0.1
0.15
70 Mode 1
Fig. 2. Lower buckling loads and the corresponding eigenmodes (MSC.NASTRAN).
Mode 6
Mode 11
Mode 14
Mode 15
Mode 19
Eigenmodes Fig. 4. Selected buckling loads and the corresponding eigenmodes (COSMOS/M).
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EM [(Nmm)1/2] 350
Pcr - lower distort. Pcr - distortional
200
280
EM - local buckling EM - lower distort. EM - distortional
150
Pcr - local buckling
P cr,P ult [kN]
Pcr [kN]
Pcr - local buckling
250
Pcr - lower distort.
120
max |
0|
= 1.5 [mm]
Pcr - distortional Pult - local buckling Pult - lower distort.
115
Pult - distortional
210
110 100
140
50
70
0
0
105
Buckling mode ~ imperfection mode Fig. 5. Buckling loads and the energy measure of eigenmodes normalised by the amplitude (MSC.NASTRAN).
earlier outset of large deflection effects unfavourably influencing the channel strength. Further, the normalisation by the energy suppresses unrealistic amplitudes of the imperfections. The quadratic functional related to the geometric matrix Kg, i.e. the expression in the right hand side of Eq. (3) except of λi, represents the channel axial shortening corresponding to its deflection in the ith eigenmode. An intuitive deduction suggests that the higher the buckling load, the stiffer (axially) the channel deflected in the corresponding eigenmode shape. However, normalising eigenmodes by the amplitude does not result in elastic shortenings decreasing with the number of the eigenmode. When employing normalisation by the energy measure, the constant left hand side in Eq. (3) implies that the shortenings are inversely proportional to the increasing buckling loads. Thus, for the approach using the energy measure, perfect fit with the intuitive expectation is obtained. These considerations form a background to suggestion of the approaches A and B, in which normalising imperfections by one of the measures considered, another one is employed as a supporting parameter indicating the candidates for the most unfavourable eigenmode imperfections. An additional criterion for the selection of an eigenmode is that the corresponding buckling load has not to be significantly higher than the critical buckling load. 3. Approach A: imperfections normalised by the amplitude The computed critical and higher buckling loads as well as the energy measures EM(φi) of the corresponding eigenmodes φi normalised by the unit amplitude (max |φi| = 1 mm) shown in Fig. 5 are obtained by MSC.NASTRAN. The working classification of eigenmodes to local, lower distortional and distortional is reflected differentiating markers of both buckling loads and EM(φi) values. Fig. 5 shows that the lower values of the energy measure of eigenmodes normalised by the amplitude are obtained for the distortional and lower distortional modes or local buckling modes with perceptible proportion of distortional imperfection. Not distinctly but still distinguishable from the local modes is the energy value for the 18th eigenmode with one central distortional half-wave of L/2 length. The number of half-waves (altogether 21) in the web is responsible for relatively high value of the EM parameter, see Fig. 6. The high values of the strain energy obtained for eigenmodes of large number of local half-waves suggest that their realistic amplitudes should be lower
29
25
27
21
23
19
15
17
11
7
9
5
27
25
23
21
19
29
1
13
i
95
of unit amplitude
3
Buckling mode
17
15
13
9
11
7
5
3
1
100
0
Fig. 7. Buckling and collapse loads of the channel; eigenmode imperfections are normalised by the amplitude (MSC.NASTRAN).
than for modes of smaller number of half-waves, thus questioning the normalisation of imperfections by amplitude. For calculation of the collapse loads Pult, the maximal amplitude considered is of the channel thickness (1.5 mm). This amplitude was recommended by Schafer and Peköz [8] as a conservative estimation of the maximum distortional imperfection — type 2 imperfection for computational modelling of the strength of cold-formed steel members. Note that for the cold rolled sections the corresponding working tolerance (EN 10162: 2003) is of 1.08 mm, while for the cold formed sections the functional manufacturing tolerance of the execution class EXC2 corresponds to distortional amplitude of 1.73 mm (EN 1090–2: 2008 + A1). The tolerances of web flatness are lesser. The collapse loads obtained for the maximal amplitude of imperfections are in Fig. 7 differentiated by the marker shape corresponding to the class of imperfection applied. Comparison with Fig. 5 shows that lower Pult values are generally realised for eigenmodes of lower energy measure and not too high buckling loads, the minimum value of 98.9 kN being reached for the imperfection in the 14th eigenmode, i.e. the first distortional mode. These results suggest that the use of the energy measure as a parameter for indicating the most unfavourable imperfections effectively decreases the number of eigenmodes to be investigated. Referring to the working classification of eigenmodes in Subsection 2.3 and the above discussed Fig. 5, another benefit of the energy measure as a supporting parameter for identification of distortional components in eigenmodes can be highlighted. Particularly, the first distortional mode is important for the Direct Strength Method developed by Schafer and Peköz [9]. The corresponding buckling load is then used in a strength
P ult [kN] Mode 1
120
Mode 5 Mode 11
115
Mode 14 Mode 15
110
Mode 19
105 100 97.3 kN - exp [7]
95 0.0
0.2
0.4
Amplitude of Fig. 6. The 18th eigenmode (MSC.NASTRAN).
0.6 0
0.8
1.0
1.2
1.4
in eigenmode shape [mm]
Fig. 8. Collapse loads related to the amplitude of eigenmode imperfections (MSC.NASTRAN).
Z Sadovský et al. / Journal of Constructional Steel Research 78 (2012) 1–7
P ult [kN]
120
Mode 1
120
5
Mode 6
115
Mode 11
115
Mode 14
110
Mode 19
105
Pult [kN]
Mode 15
110
105 100
100 97.3 kN - exp [7]
95 95 0.0
0.2
0.4
Amplitude of
0.6 0
0.8
1.0
1.2
1.4
90
in eigenmode shape [mm]
Mode 1
formula yielding the distortional strength of a member. Consider mistaken choice of the 11th eigenmode with significant proportion of distortional component (45%) as the first distortional mode instead of the 14th eigenmode referring to the buckling loads drawn in Fig. 7 by symbol line. Supposing that the strength formula displays imperfection sensitivity, its application to buckling load Pcr11 may result in a conservative design. Note that for both imperfections the column failed in distortional mode. The identification problem is still topical as witness the reports on difficulties consisting in “search amongst higher modes” [13] and not exactness “especially when both local and distortional buckling are present in the same eigenmode” [14]. The relationship between collapse loads and imperfection amplitudes is shown in Fig. 8 (MSC.NASTRAN) and Fig. 9 (COSMOS/M). Eigenmode imperfections possessing low energy measure are selected. The collapse loads related to the imperfections having similar shapes in both codes are drawn in Figs. 8 and 9 by the same symbol-line curve, cf. Subsection 2.3. The curves in Figs. 8 and 9 show an overall correlation; generally, the COSMOS/M code provides lower collapse loads. The experimental ultimate load 97.3 kN of the specimen ST15A90 [7] drawn by horizontal dashed line is close below the Pult obtained for the distortional imperfection having the maximal amplitude. The collapse loads and modes obtained by MSC.NASTRAN for the eigenmode imperfections applied in Fig. 8, are for minute imperfection amplitudes (0.01 mm) visualised in Fig. 10 and for imperfection amplitudes of channel thickness in Fig. 11. For minute imperfections, local modes of column failure are obtained. For imperfection amplitudes of channel thickness distortional modes of column failure resulted. The whole group of collapse loads and modes as computed for the first
Mode 14
Mode 15
Mode 19
eigenmode imperfection is shown in Fig. 12, while for the first distortional eigenmode imperfection (the 14th) in Fig. 13. An interesting finding is that for imperfections of minute amplitudes the failure mode corresponding to the lowest Pult value realises in the shape similar to the first eigenmode with dominating local buckles of ten half waves in the web. The failure modes corresponding to higher collapse loads resemble the second eigenmode of eleven local halfwaves, cf. Figs. 2 and 10. 4. Approach B: imperfections normalised by the energy measure Fig. 14 shows the buckling loads and amplitudes of the corresponding eigenmodes normalised by the energy measure. The level of the energy measure is set by the first distortional mode (the 14th eigenmode) of amplitude equalling the channel thickness. For the eigenmode shapes adjusted to the given strain energy level, the higher amplitudes indicate presence of distortional components, see Figs. 2 and 3. The low level amplitudes correspond to the modes of numerous local half-waves; a special case of the 18th eigenmode with simultaneous presence of distortional half-wave is discussed in Section 3, see Fig. 6. The collapse loads of the channel with eigenmode imperfections are shown in Fig. 15 together with the imperfection amplitudes. Distinctly lower collapse loads are obtained for the imperfections with distortional 120 115
115
110
Pult [kN]
Pult [kN]
Mode 11
Fig. 11. Collapse loads and modes obtained for eigenmode imperfections with amplitudes of channel thickness (MSC.NASTRAN).
120
110
Mode 5
Imperfections amplitude of 1.5 mm
Fig. 9. Collapse loads related to the amplitude of eigenmode imperfections (COSMOS/M).
105
105
100
100
95
95
90 0.01
0.075
0.2
0.5
1.0
1.5
Amplitude of Mode 1 imperfection [mm]
90 Mode 1
Mode 5
Mode 11
Mode 14
Mode 15
Mode 19
Imperfections amplitude of 0.01 mm Fig. 10. Collapse loads and modes obtained for eigenmode imperfections of minute amplitudes (MSC.NASTRAN).
Fig. 12. Collapse loads and modes obtained for imperfections in the first eigenmode shape and selected amplitudes (MSC.NASTRAN).
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Z Sadovský et al. / Journal of Constructional Steel Research 78 (2012) 1–7
P ult [kN]
120
Pult - local buckling
130
Pult - lower distort.
EM (
115
0)
= const.
Pult - distortional
125
max | 0| - local buckling max | 0 | - distortional
115
2.0
110
1.5
105
1.0
100
0.5
95
0,0
0
105 100 95
max |
Pult [kN]
120
| [mm]
max | 0 | - lower distort.
110
90 0.01
0.075
0.2
0.5
1.0
1.5
Buckling mode ~ imperfection mode Fig. 13. Collapse loads and modes obtained for imperfections in the 14th eigenmode shape and selected amplitudes (MSC.NASTRAN).
components. Particularly, the lowest Pult is obtained for the first distortional eigenmode, see Fig. 3. The separation of collapse loads related to the imperfections with distortional components from those with prevailing local components is now more obvious than if normalising the imperfections by amplitude, cf. Fig. 7 of Section 3. Also higher separation among collapse loads calculated for imperfections in the shapes of local buckling modes is observed. Normalisation of the imperfections by the energy measure downgrades the amplitudes of severely distorted local buckling modes to significantly smaller amplitudes than are those of distortional modes or those of less distorted local modes. As a result of downgrading the amplitudes, generally significantly higher collapse loads are obtained. The situation can be illustrated also on the above mentioned 18th lower distortional eigenmode, see Fig. 6. Utilising normalisation by the amplitude, the collapse load corresponding to the imperfection “competes” with the lowest Pult, see Fig. 7. However, the energy normalisation assigns to the imperfection mode almost six times smaller amplitude resulting in an unimportant high collapse load, see Fig. 15. Note that employing the energy measure, unrealistic amplitudes of imperfections have been reported in the strength studies on post-buckling of thin plates [11] and stringer-stiffened cylindrical shell [12]. Combination of eigenmodes normalised to a certain level of the energy measure can be readily carried out with combination factors
Pcr - local buckling
P cr [kN]
max |
Pcr - lower distort.
250
EM ( i) = const.
Pcr - distortional
i|
[mm] 2.5
29
25 0
Fig. 15. Collapse loads and amplitudes of the channel eigenmode imperfections normalised by the energy measure (MSC.NASTRAN).
following the Pythagoras rule. For example, coupling of the eigenmodes φ0,11 and φ014 can be given in the form c11φ0,11 + c14φ0,14 with combination factors c11 and c14 ranging within the interval b−1, 1> and satisfying the equation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c11 ¼ 1−ðc14 Þ2 :
ð4Þ
The collapse loads obtained for combinations of the imperfections in 14th eigenmode with 11th and 15th eigenmode are shown in Fig. 16. The level of energy measure applied when calculating data in Figs. 14 and 15 is adopted. Two branches correspond to plus and minus sign of c11 (or c15) following Eq. (4). The least collapse load of 98.3 kN is reached for plus sign and the coupled mode c15φ0,15 + c14φ0,14. Slightly higher minima of 98.4 kN have been obtained in combinations of the first distortional mode (the 14th) with the first and third local modes. Fig. 17 shows the collapse loads as depending on the level of the energy measure of selected imperfections. In comparison with the dependence on amplitude in Fig. 8, the failure loads obtained for the imperfection in the first distortional mode are almost in the whole range of imperfection levels clearly below other failure loads. Local buckling modes of imperfections, particularly the 5th eigenmode and the 1st eigenmode, are decisive only for small and minute magnitudes, respectively. Note that the rate of change of failure loads with the magnitude of
P ult [kN]
Pult(c11ϕ0,11 + c14ϕ0,14)
110
Pult(c15ϕ0,15+ c14ϕ0,14)
max | i | - local
200
27
21
23
19
15
17
13
9
11
5
7
1
3
Amplitude of Mode14 imperfection [mm]
2.0
max | i | - lower dist.
105
max | i | - distortional
150
1.5
100
1.0
50
0.5
95
0
0.0
90
i
29
25
27
21
23
19
15
17
13
9
Buckling mode
11
7
5
1
3
100
normalised by energy measure
Fig. 14. Buckling loads and amplitudes of eigenmodes normalised by the energy measure (MSC.NASTRAN).
EM(ϕ0) = const.
0.0
0.2
0.4
0.6
0.8
1.0
C14 Fig. 16. Collapse loads related to coupled eigenmode imperfections normalised by the energy measure (MSC.NASTRAN).
Z Sadovský et al. / Journal of Constructional Steel Research 78 (2012) 1–7
P ult [kN]
7
6. Conclusions Mode 1
120
Mode 5 Mode 11
115
Mode 14 Mode 15
110
Mode 19
105 100 97.3 kN - exp [7]
95 0
10
20
30
40
Energy measure of ϕ0 in eigenmode shape [(Nmm)1/2 ] Fig. 17. Collapse loads related to the energy measure of eigenmode imperfections (MSC.NASTRAN).
imperfection is significantly higher for small magnitudes than in the rest of interval studied.
Using the first and possibly also the second eigenmode as imperfection patterns for computational modelling of strength of cold-formed steel may not capture the decisive shape, particularly for non-small imperfection sizes. In either of the suggested approaches to the analysis of eigenmodes, useful preliminary information on the choice of the adverse imperfection shapes effectively decreasing the number of candidate modes is obtained. In particular, the presence of distortional component is indicated. As the preferable one, the approach (B) employing normalisation of geometric imperfections by the energy measure and the corresponding amplitude as a supporting parameter is considered. The advantages of the approach are: ➣ Unrealistic imperfections are ruled out; ➣ The eigenmode imperfections are readily combined; ➣ The energy concept may prove useful in establishing a relationship to standards of execution and production tolerances; ➣ Theoretical background is available.
5. Discussion
Acknowledgment
The computed collapse loads presented in Sections 3 and 4 show that both parameters studied in the approaches A and B, i.e. the value of one of the imperfection measures and of the buckling load, are sensible for the choice of the most unfavourable eigenmode shapes. The parameter of imperfection measure reflecting buckling deformability plays more decisive role for middle and high imperfection levels, while for small levels higher weight has to be attributed to the buckling load parameter. For very small imperfection levels, the shape of the first eigenmode overrides the influence of the other modes including those with significant proportion of distortional components. The performance of the approach A appears to be of less significance. Normalising the eigenmodes by the amplitude includes into comparison heavy distorted modes of unrealistically high amplitudes. Moreover, there is a poor correlation between the buckling load parameter and the axial shortenings of the channel with eigenmode deflections of constant amplitude. These facts may be the reasons of less obvious indications of adverse eigenmode shapes. Employing the approach B, the unrealistic amplitudes of imperfections are ruled out by the energy measure normalisation. The axial shortenings of the channel related to eigenmodes are inversely proportional to the corresponding buckling loads. Straightforward combination of eigenmodes allows the study of various different imperfection shapes. Note that an indication of interaction sensitivity of eigenmode shapes may be obtained by Sobol sensitivity indices, however some preliminary statistical information on the occurrence of single modes is required, see Kala [15]. The energy concept of normalisation of geometric imperfections using the amplitude of eigenmode imperfections as a parameter of their influence on channel strength offers an opportunity of setting a relationship to standards of execution and production tolerances. Particularly, the most unfavourable eigenmode shape being in the range of upper amplitude levels suggests itself as the reference shape for adjustment to a tolerance standard. In strength calculations, the other eigenmode shapes have mostly smaller amplitudes and those of possible higher amplitudes are less influential. The study of examples exhibiting higher level of distortional components in the eigenmodes, which is realised by a) halving the length of channel and b) extending the width of flanges and shortening of lips in the former case, is to appear in Procedia Engineering within the Proceedings of the Conference Steel Structures and Bridges 2012.
The authors acknowledge a partial support of their work by grant agency VEGA under grant 2/0101/09 and grant 1/0090/12. The MSC.NASTRAN code has been acquired with the financial assistance of the European Regional Development Fund (ERDF) under the Operational Programme Research and Development/Measure 4.1 Support of networks of excellence in research and development as the pillars of regional development and support to international cooperation in the Bratislava region/Project No. 26240120020 Building the centre of excellence for research and development of structural composite materials — 2nd stage. References [1] Johansson B, Maquoi R, Sedlacek G, Müller C, Beg D. Commentary and worked examples to EN 1993-1-5 “Plated structural elements”. JRC scientific and technical reports. © European Communities; 2007. [2] Hancock GJ. Cold-formed steel structures. J Constr Steel Res 2003;59:473–87. [3] Dubina D, Ungureanu V, Rondal J. Numerical modelling and codification of imperfection for cold-formed steel member analysis. Steel Compos Struct 2005;5 (6): 515–33. [4] Camotim D, Basaglia C, Silvestre N. GBT buckling analysis of thin-walled steel frames: a state-of-the-art report. Thin Wall Struct 2010;48:726–43. [5] Schafer BW. Cold-formed structures around the world. a review on recent advances in applications, analysis and design. Steel Constr 2011;4 (3):141–9. [6] Schafer BW, Li Z, Moen CD. Computational modelling of cold-formed steel. Thin Wall Struct 2010;48:752–62. [7] Young B, Hancock GJ. Compression tests of channels with inclined simple edge stiffeners. J Struct Eng ASCE 2003;129 (10):1403–11. [8] Schafer BW, Peköz T. Computational modeling of cold-formed steel: characterizing geometric imperfections and residual stresses. J Constr Steel Res 1998;47: 193–210. [9] Lecce M, Rasmussen KJR. Finite element modeling and design of cold-formed stainless steel sections. Research rep. no. 845. Sydney: Dept. of Civil Engineering, Univ. of Sydney; 2005. [10] Sadovský Z. A theoretical approach to the problem of the most dangerous initial deflection shape in stability type structural problems. Aplik Mat 1978;23 (4): 248–66. [11] Sadovský Z, Teixeira AP, Guedes Soares C. Degradation of the compressive strength of rectangular plates due to initial deflection. Thin Wall Struct 2005;43:65–82. [12] Sadovský Z, Ďuricová A, Ivančo V, Kriváček J. Imperfection measures of eigen- and periodic modes of axially loaded stringer-stiffened cylindrical shell. Proc. IMechE, 224(G5). Journal of Aerospace Engineering; 2010. p. 601–12. [13] Kwon YB, Kim BS, Hancock GJ. Compression tests of high strength cold-formed steel channels with buckling interaction. J Constr Steel Res 2009;65:278–89. [14] Moen CD, Schafer BW. Experiments on cold-formed steel columns with holes. Thin Wall Struct 2008;46:1164–82. [15] Kala Z. Sensitivity analysis of steel plane frames with initial imperfections. Eng Struct 2011;33 (8):2342–9.