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[.]

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 intofinite element models using ABAQUS software package with the focus of attention for cold-rolled aluminium sections Laser scanners arefirstly 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 Subse-quently, a MATLAB code is developed to incorporate the measured imperfection magnitudes into a perfect mesh of thefinite element model The Fourier series approximation is used in the longitudinal direction along measurement lines while the linear interpolation is used forflanges, lips and web in the transverse direction Keywords: Geometric imperfections
Finite element models
Cold-rolled aluminium sections

1 Introduction

Aluminium alloy structures have been used increasingly in bridges, building walls (Szumigala and Polus2015) due to its excellent corrosion resistance, high strength to weight ratio and convenience in transportation, extrusion and assembly The conven-tional way to produce aluminium alloy is extrusion In Australian, the guidelines for the design of aluminium structures AS1664.1 (Australian Standard1997) 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 Standard1997) In comparison with extrusion method, roll-forming method is more cost effective, and enhances strength of alu-minium alloy The issue of the cold-rolled alualu-minium section is sensitive to buckling which is significantly affected by geometric imperfections The geometric imperfec-tions 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 Congr ès International

de G éotechnique - Ouvrages -Structures, Lecture Notes in Civil Engineering 8,

DOI 10.1007/978-981-10-6713-6_15

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; G3Þ as shown in Fig.1 where GG; G2 and G3 are bow, camber and twist of a member respectively The sectional imperfections comprise two valuesðd1; d2Þ as also shown in Fig.1where d1

is the plate out offlatness, 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.2 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 1 Representative values for overall and sectional imperfections

Fig 2 Load-deflection relationship of a compression plate

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, andfixed 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-of-flatness 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 distor-tional imperfection, a perfectly square ground bar was set up in Fig.3(c) to measure the deviations of theflange element from a right angle established off of the corner of the web Similarly, the imperfection was calculated from 0.001-inch dial gauge measure-ments 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.4 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, measure-ments just focused on a few sparse points and global out of straightness characters Subsequently, Young (1997) employed transducers attached on an aluminium frame to measurefive 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 5 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 imper-fection 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.7 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) con-structed a high-throughput and high-accuracy laser measurement platform for com-plicated structural geometries as shown in Fig.8 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

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.2

a) global member out of straightness b) local initial imperfection c) distortional initial imperfection Fig 3 Measuring imperfections using a dial gage (Mulligan1983)

a) Set up b) Detail of optical micrometer Fig 4 Measuring imperfections using a telescope (Dat and Pekoz1980)

Fig 5 Measuring imperfections using transducers (Young1997)

a) Becque’s imperfection rigs (Becque 2008) b) Niu’s imperfection rigs (Niu 2013)

Fig 6 Measuring imperfections using laser scanners

Fig 7 Photogrammetry process (MCAnallen et al.2014)

Fig 8 Laser-based imperfection measurement platform (Zhao et al.2014)

2 Measured Geometric Imperfections

2.1 Imperfection Measuring Rigs

The imperfection measuring rigs are shown in Fig.9 They include two high precision bars attached on a rigid frame and a trolley was hung 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 1 mm The locations of nine imperfection mea-surement 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 imperfec-tions are obtained through the remaining lines Lines (1), (2), (8) and (9) capture distortional imperfections of twoflanges whereas line (5) captures local imperfections

of the web Compared to seven measurement laser lines of Niu (2013), nine mea-surement laser lines in this paper should access imperfections more thoroughly in two lips of the channel sections

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

Fig 9 Measuring rigs

Fig 10 Measurement line Locations

Fig 11 Non-perfect cross-sections

cross-sections arefirstly 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

3 Processing Method

After measuring imperfections, raw results of an imperfection line include data of sep-arate 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)

fðxÞ ¼X1

n ¼1

Knsinpnx

Kn¼2 L

ZL 0

fðxÞsinðpnx

L Þdx ðn ¼ 1; 2; 3; ; 1Þ ð2Þ

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 Typi-cally, 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, valuesðaiÞ in Fig.12are 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

Fig 12 Compare the difference between two cross-sections

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 intofinite element models as described in Sect.4

gðxÞ ¼ ðÞai

FðxÞ ¼X1

n ¼1

Knsinpnx

L ðÞai

4 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)

-1.8

-1.5

-1.2

-0.9

-0.6

-0.3

0

Specimen length (mm)

Measuring Fourier

Fig 13 Fourier expression curve and actual imperfection line (5) of the specimen AC10030-2000C-1

Fig 14 The perfect specimen (a) and the imperfect specimen (b)

With an inputfile (*.INP) exported from an original model in ABAQUS/Standard – Version 6.14 (2014), the coordinates of all nodes in this inputfile 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 inputfile is imported into ABAQUS by using the command “File>Import/Model” Figure14(b) shows an example of the actual imperfections of the specimen AC10030-2000C-1 after incorpo-rating into thefinite element model In comparison with Fig.14(a), (b) shows the changes

of local imperfections of the web, distortional imperfections of theflanges as well as initial twist of two end cross-sections The procedure to incorporate geometric imper-fections intofinite element models is summarized in the flowchart format in Fig 15 Fig 15 The procedure to incorporate geometric imperfections intofinite element models

5 Conclusion

This paper provides various methods to measure geometric imperfections of the cold-formed 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 Thisfirst 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 andfinancial support for the project The first author

is sponsored by the scholarship provided by Australian Awards Scholarships (AAS) scheme from Australian Government

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