Incorporation of measured geometric imperfections into finite element models for cold rolled aluminium sections

Incorporation of Measured Geometric Imperfections into Finite Element Models for Cold Rolled Aluminium Sections Ngoc Hieu Pham(&), Cao Hung Pham, and Kim J R Rasmussen The University of Sydney, Sydney[.]

Incorporation of Measured Geometric Imperfections into Finite Element Models for Cold-Rolled Aluminium Sections Ngoc Hieu Pham(&), Cao Hung Pham, and Kim J.R Rasmussen The University of Sydney, Sydney, NSW, Australia {ngochieu.pham,caohung.pham, kim.rasmussen}@sydney.edu.au Abstract Geometric imperfections have a significant effect on both buckling and strength capacities of structural members It is essential to accurately measure the geometric imperfections for finite element simulation especially for thin-walled sections This paper presents the procedure to measure and incorporate geometric imperfections into finite element models using ABAQUS software package with the focus of attention for cold-rolled aluminium sections Laser scanners are firstly used to measure geometric imperfections along high-precision tracks while recording the distances to corresponding points on the surface of specimen The measurement lines are located around the cross-section Subsequently, a MATLAB code is developed to incorporate the measured imperfection magnitudes into a perfect mesh of the finite element model The Fourier series approximation is used in the longitudinal direction along measurement lines while the linear interpolation is used for flanges, lips and web in the transverse direction Keywords: Geometric imperfections aluminium sections Á Finite element models Á Cold-rolled Introduction Aluminium alloy structures have been used increasingly in bridges, building walls (Szumigala and Polus 2015) due to its excellent corrosion resistance, high strength to weight ratio and convenience in transportation, extrusion and assembly The conventional way to produce aluminium alloy is extrusion In Australian, the guidelines for the design of aluminium structures AS1664.1 (Australian Standard 1997) are premised on research of extruded sections In recent years, BlueScope Lysaght (2015) has used rollers of steel sections to roll-form successfully aluminium sections with dimensional tolerances to AS/NZS 1734 (Australian Standard 1997) In comparison with extrusion method, roll-forming method is more cost effective, and enhances strength of aluminium alloy The issue of the cold-rolled aluminium section is sensitive to buckling which is significantly affected by geometric imperfections The geometric imperfections are caused by unavoidable disturbances during production, transportation and assembly processes causing imperfect straight of real dimensions of members Thus, the incorporation of geometric imperfections needs to be considered in analyzing cold-rolled aluminium sections © Springer Nature Singapore Pte Ltd 2018 H Tran-Nguyen et al (eds.), Proceedings of the 4th Congrès International de Géotechnique - Ouvrages -Structures, Lecture Notes in Civil Engineering 8, DOI 10.1007/978-981-10-6713-6_15 162 N.H Pham et al Geometric imperfections can be divided into global and sectional imperfections Global imperfections are represented by initial twists and initial deflections whereas the sectional imperfections include displacements of plate elements For channel sections, overall imperfections are presented by three values ðG1 ; G2 ; G3 Þ as shown in Fig where GG ; G2 and G3 are bow, camber and twist of a member respectively The sectional imperfections comprise two values ðd1 ; d2 Þ as also shown in Fig where d1 is the plate out of flatness, and d2 is the plate out of straightness Due to the effect of imperfections, buckling process occurs gradually in lieu of the abruption from pre-buckling, buckling to post-buckling in perfect members, which results in the unclear buckling point For example, a plate is compressed between two rigid frictionless platens and its relationship between the compressive load and the transverse deflection is shown in Fig Although being quite far from curve of the perfect plate in the pre-buckling period, the response of the imperfect plate approaches that of the perfect plate in the post-buckling period Fig Representative values for overall and sectional imperfections Fig Load-deflection relationship of a compression plate Incorporation of Measured Geometric Imperfections into Finite Element Models 163 Several methods were proposed to measure geometric imperfections Dat and Pekoz (1980) and Mulligan (1983) used a dial gauge to provide a global member out-of-straightness at the middle of the web with reference to a straight line between ends of the specimen The specimen lied on a plane surface, and a dial gauge was used to measure the elevation of various points of the specimens as described in Fig 3(a) This device was also used to measure local initial imperfection It includes a ground bar, and fixed and movable support points, where the latter is established at a constant distance from the top of the bar as shown in Fig 3(b) This device measured the out-offlatness of stiffened and edge-stiffened elements, as referenced to their longitudinal edges The local initial imperfection was calculated based on measurements taken with and without the ground bar using an independent 0.001-inch dial gauge For distortional imperfection, a perfectly square ground bar was set up in Fig 3(c) to measure the deviations of the flange element from a right angle established off of the corner of the web Similarly, the imperfection was calculated from 0.001-inch dial gauge measurements taken with and without the square ground bar Dat and Pekoz (1980) also used an alternative method to measure long specimens as shown in Fig The specimen was rested horizontally, and a telescope was placed approximately ten feet away to measure along the specimen axis An optical micro-meter mounted at the end of telescope can be rotated The imperfection was measured by determining the angular deviation of the optical micrometer compared to a fixed reference point The dead weight deflections were also accounted for by subtracting or adding to the initial deflections depending on the direction of the initial deflections In general, measurements just focused on a few sparse points and global out of straightness characters Subsequently, Young (1997) employed transducers attached on an aluminium frame to measure five isolated longitudinal lines along a specimen, and used three transducers to measure the initial deflection and initial twist of the specimen as shown in Fig With the increasing of measurement lines, Young (1997) was able to access both global deviations and cross-section imperfections of a channel section Based on Young (1997)’s idea, Becque (2008) and Niu (2013) used laser scanners instead of transducers to measure geometric imperfections of five and seven lines around a channel cross-section respectively as shown in Fig 6, and provided much finer scale imperfection measurements MCAnallen et al (2014) provided non-contact techniques to take 3D measurements such as photogrammetry In terms of photogrammetry method, sets of photos are taken from multiple viewpoints around the specimen, and then the commercial software is used to identify all the targets as shown in Fig MCAnallen et al (2014) used ringed automatically detected (RAD) targets and dot targets to complete the photogrammetry process of making targets, identifying camera locations, referencing targets, and using an optimization algorithm to process a 3D point cloud In addition, Zhao et al (Zhao and Schafer 2014, 2016; Zhao et al 2015, 2016) constructed a high-throughput and high-accuracy laser measurement platform for complicated structural geometries as shown in Fig The measuring rig consisted of three major components: laser scanner, rotary stage, and linear stage Installed on the rotary stage, the laser could move around the circle to any desired scanning angle The rotary stage was attached on a frame that could drive the laser along the longitudinal direction of the specimen The laser imperfection measurement rig can collected full-field 3D 164 N.H Pham et al point clouds of the specimen Later, the measurement point clouds were converted into operational 3D models for imperfection estimation Aluminium is a soft and lightweight material Cold-rolled aluminium alloys members contain significantly large imperfections Based on the method of Becque (2008) and Niu (2013), this paper presents the laser scanner method in measuring geometric imperfections as described in Sect a) global member out of straightness b) local initial imperfection c) distortional initial imperfection Fig Measuring imperfections using a dial gage (Mulligan 1983) a) Set up b) Detail of optical micrometer Fig Measuring imperfections using a telescope (Dat and Pekoz 1980) Fig Measuring imperfections using transducers (Young 1997) a) Becque’s imperfection rigs (Becque 2008) b) Niu’s imperfection rigs (Niu 2013) Fig Measuring imperfections using laser scanners Incorporation of Measured Geometric Imperfections into Finite Element Models 165 Fig Photogrammetry process (MCAnallen et al 2014) Fig Laser-based imperfection measurement platform (Zhao et al 2014) Measured Geometric Imperfections 2.1 Imperfection Measuring Rigs The imperfection measuring rigs are shown in Fig They include two high precision bars attached on a rigid frame and a trolley was and ran along the bars Laser devices attached to the trolley measure the distances to the surfaces of test specimens while the trolley moves along the bars with a constant speed All measurement data are recorded at points whose distance is mm The locations of nine imperfection measurement lines along the longitudinal direction of specimens are shown in Fig 10 To keep away from the edge and corner region, as well as to allow room for the laser devices, the measurement lines are offset 10 mm from the edges While the global imperfections are measured through lines (3), (4), (6) and (7), the sectional imperfections are obtained through the remaining lines Lines (1), (2), (8) and (9) capture distortional imperfections of two flanges whereas line (5) captures local imperfections of the web Compared to seven measurement laser lines of Niu (2013), nine measurement laser lines in this paper should access imperfections more thoroughly in two lips of the channel sections 166 N.H Pham et al Fig Measuring rigs Fig 10 Measurement line Locations 2.2 Sketching Two Cross-Sections at the Edges Two end cross-sections of the specimen are labeled as A and B Laser scanners run from Section A to Section B The cross-sections at two ends are not perfect It is necessary to account for those non-perfect cross-sections in analysis models These a) Section A b) Section B Fig 11 Non-perfect cross-sections Incorporation of Measured Geometric Imperfections into Finite Element Models 167 Fig 12 Compare the difference between two cross-sections cross-sections are firstly sketched on papers as shown in Fig 11, and are subsequently redrawn in AutoCAD The next step is to compare the difference between two cross-sections of end cross-sections based on the initial twist The initial twist is determined by using imperfections data of lines (4) and (6) Cross-section (B) is compared to cross-section (A) through nine points to determine nine values as shown in Fig 12 Those values are used for the processing method as described in Sect 3 Processing Method After measuring imperfections, raw results of an imperfection line include data of separate points along the length of a specimen In order to introduce this data into numerical models, each discrete data line is converted into Fourier series as given in Eqs (1) and (2) f xị ẳ X nẳ1 Kn ẳ L pnx With 0\ x \ Lị L 1ị pnx ịdx n ẳ 1; 2; 3; ; 1Þ L ð2Þ Kn sin ZL f ðxÞsinð The number of terms in Eq (1) for each specimen length was chosen to encompass short half wave lengths which are more influential to local buckling behavior Typically, number of series terms ranging from 20–35 is used for different specimen length, and depends on the ratio of the length of specimen and a half wave length of local buckling of that section (Becque 2008; Niu 2013) For example, the specimen (AC10030-2000C-1) requires 25 terms in the imperfection line expression A typical imperfection taken from the line (5) of specimen AC10030-2000C-1 as shown in Fig 13, the Fourier expression curve fits very well with the measured imperfection line In order to account for the difference between two cross-sections at two end cross-sections of specimens, values ðai Þ in Fig 12 are transformed into linear equations expressed in Eq (3) to capture the initial twist of the specimen Hence, imperfections lines are expressed by combinations of Fourier series f(x) and linear equations g(x) as described in Eq (4) A MATLAB code (1) is created to determine the coefficients Kn 168 N.H Pham et al Specimen length (mm) Imperfection (mm) -0.3 200 400 600 800 1000 1200 1400 1600 1800 2000 -0.6 Measuring -0.9 Fourier -1.2 -1.5 -1.8 Fig 13 Fourier expression curve and actual imperfection line (5) of the specimen AC10030-2000C-1 of measurement lines based on measured data Fourier series f(x) and linear equations g(x) are introduced into a MATLAB code (2) to incorporate imperfections into finite element models as described in Sect gxị ẳ ặị Fxị ẳ X n¼1 Kn sin xðwith \ x \ Lị L pnx ặị x With \ x \ LÞ L L ð3Þ ð4Þ Incorporation Imperfections into Finite Element Models To incorporate imperfections into Finite Element (FE) models, two end cross-sections (A and B) of the real specimen are sketched on papers and subsequently transferred into AutoCAD The cross-section (A) measured at the beginning of measuring process is used as the original cross-section After being imported into ABAQUS by using the command “File>Import/Sketch”, the original cross-section is extruded to create the perfectly straight specimen as shown in Fig 14(a) Fig 14 The perfect specimen (a) and the imperfect specimen (b) Incorporation of Measured Geometric Imperfections into Finite Element Models 169 Fig 15 The procedure to incorporate geometric imperfections into finite element models With an input file (*.INP) exported from an original model in ABAQUS/Standard – Version 6.14 (2014), the coordinates of all nodes in this input file are reproduced by using a MATLAB code (2) This code can introduce Fourier series curves into nodes along reading lines and interpolate the co-ordinates of intermediate nodes In addition, the difference between two end cross-sections (A and B) is considered to determine the changes of local, distortional imperfection between two end cross-sections in association with initial twist rotation of the specimen Subsequently, the new input file is imported into ABAQUS by using the command “File>Import/Model” Figure 14(b) shows an example of the actual imperfections of the specimen AC10030-2000C-1 after incorporating into the finite element model In comparison with Fig 14(a), (b) shows the changes of local imperfections of the web, distortional imperfections of the flanges as well as initial twist of two end cross-sections The procedure to incorporate geometric imperfections into finite element models is summarized in the flowchart format in Fig 15 170 N.H Pham et al Conclusion This paper provides various methods to measure geometric imperfections of the coldformed sections While methods using a dial gauge or transducers typically measure a few sparse points and global out-of straightness characterization, non-contacts methods described in this paper using laser scanners or photogrammetry provide more accurate data in measuring geometric imperfections With more data collected, geometric imperfections are therefore studied more thoroughly The paper also presents the procedure to introduce geometric imperfections into ABAQUS models Two MATLAB codes are created for implementing this procedure This first code is used to convert the discrete data of imperfection lines into Fourier series The second code aims to introduce Fourier series into reading points, and then interpolate the coordinates of intermediate nodes Acknowledgments Funding provided by the Australian Research Council Linkage Research Grant LP140100563 between BlueScope Lysaght and the University of Sydney has been used to perform this research The authors would like to thank Permalite Aluminum Building Solutions Pty Ltd for the supply of the test specimens and financial support for the project The first author is sponsored by the scholarship provided by Australian Awards Scholarships (AAS) scheme from Australian Government References AS/NZS 1664.1:1997 Australian/New Zealand Standard Part 1: Limit state design aluminium structures Standards Australia, Sydney (1997) AS/NZS 1734: 1997 Australian/New Zealand Standard: Aluminium and aluminium alloys-Flat sheet, coiled sheet and plate Standards Australia, Sydney (1997) ABAQUS/Standard ABAQUS/CAE User’s Manual Dassault Systemes Simulia Corp, Providence, RI, USA (2014) Becque, J.: The interaction of local and overall buckling of cold-formed stainless steel columns 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concrete composite structures Procedia Eng 108, 544–549 (2015) Young, B.: The behaviour and design of cold-formed channel columns Dissertation, University of Sydney (1997) Incorporation of Measured Geometric Imperfections into Finite Element Models 171 Zhao, X., Schafer, B.W.: Laser scanning to develop three-dimensional fields for the precise geometry of cold-formed steel members Recent research and development in cold-formed steel design and construction (2014) Zhao, X., Tootkaboni, M., Schafer, B.W.: Development of a laser-based geometric imperfection measurement platform with application to cold-formed steel construction Exp Mech 55, 1779–1790 (2015) Zhao, X., Schafer, B.W.: Measured geometric imperfections for Cee, Zee, and built-up cold-formed steel members In: Jwei-Wen Yu International Specialty Conference on Cold-Formed Structures, pp 73–87 (2016) Zhao, X., Tootkaboni, M.P., Schafer, B.W.: High fidelity imperfection measurements and characterization for cold-formed steel members In: Proceeding of the 7th International Conference on Coupled Instabilities in Metal Structures (2016) ... imperfect specimen (b) Incorporation of Measured Geometric Imperfections into Finite Element Models 169 Fig 15 The procedure to incorporate geometric imperfections into finite element models With an input... The behaviour and design of cold-formed channel columns Dissertation, University of Sydney (1997) Incorporation of Measured Geometric Imperfections into Finite Element Models 171 Zhao, X., Schafer,... for overall and sectional imperfections Fig Load-deflection relationship of a compression plate Incorporation of Measured Geometric Imperfections into Finite Element Models 163 Several methods