MASTER OF SCIENCE THESIS STOCKHOLM, SWEDEN 2016
Patch loading resistance of welded I-beams -
with respect to misaligned web stiffeners
Dafina Zeka and John-Alexander Boutzas
KTH ROYAL INSTITUTE OF TECHNOLOGY DEPARTMENT OF CIVIL ENGINEERING
3
KTH Architecture and the Built Environment
Patch loading resistance of welded I-beams – with respect to misaligned web stiffeners
Dafina Zeka & John- Alexander Boutzas
June 2016
TRITA-BKN, Master thesis 477 Steel Structures, 2016
ISSN 1103 – 4297
IRSN KTH/BKN/EX – 477 - SE
5 Master of Science thesis
Title Patch loading resistance of welded I-beams –
with misaligned web stiffeners
Author(s) Dafina Zeka and John-Alexander Boutzas
Department Civil engineering
Master Thesis number 477
Supervisor Bert Norlin
Keywords Patch loading, Stiffeners, I-beam, Plated steel
structures, Abaqus, simply supported beam
Abstract
When a concentrated load is introduced perpendicular to the flanges of a steel beam, this con- dition is referred to as Patch loading (Gozzi, 2007). This occurrence is common in many steel structures, for example at supports or during launching of bridges. Because of the usual slen- derness of I-beams and other plated structures, these are sometimes reinforced with stiffeners in order to avoid buckling. Modifications, such as adding stiffeners to a beam, are done to make greater plastic deformations possible before buckling can occur; thereby increasing the resistance against failure.
Transverse stiffeners are added in areas where the beam is exposed to concentrated loads (Lagerqvist, 1994). The descriptions of calculating patch loading in the Eurocode are pre- sented for cases of double stiffeners, with the load applied in between two stiffeners with same distance to each of them, or when there is one single stiffener that is acting in line with the load. In the Eurocode there are also descriptions on how to calculate on the resistance against patch loading when there are no stiffeners added. However, the Eurocode lacks de- scriptions for cases when the stiffeners are misaligned.
The purpose of this paper is the evaluation of the impact from transverse stiffeners to the re- sistance of welded I-beams, when the stiffeners are misaligned and where the length of the beam varies. Because of the complexity of such of problems it is almost impossible to find theoretical solutions (Lagerqvist & Johansson, 1996). Therefore, in this study as well as in almost all studies that aim to predict the ultimate resistances of steel beams subjected to patch loading, the results are gained empirically. The tests herein were done by FE-modeling and the results from the physical experiments done in Lagerkvist’s doctoral thesis were used for validation of the model, as conducting experiments ourselves was not economically possible.
The study was made in two steps. In the first step FE-models were produced under the same circumstances as the results obtained by Lagerqvist (1994). Those analyses were not part of the aim of the study; the intention for making the initial analyses was to strengthen the relia- bility of the results. From there, the final analyses were made with the aim in investigating the influence of stiffeners on the resistance, when these are misaligned. In this step, observations were also made with regards to the impact of the bending moment of the beam on its re- sistance.
The initial analyses, which were made for validation of the modeling, had a satisfying corre- spondence to the physical experiments; hence the final analyses are assumed valid of ac- ceptance. From observations of the results in the final analyses it is noticed that adding stiff- eners is a highly preferred way of increasing the resistance for slender beams. For full utiliza- tion it is however important to have the stiffeners optimally placed, because a small deviation from this position gives an unwanted decrease in resistance.
7
Acknowledgement
This study is a master thesis, done at the Civil engineering department at the Royal Institute of Technology (Kungliga Tekniska Högskolan), in Stockholm. It was produced as the final part of the master’s program in Civil and Architectural engineering in the spring semester of 2016.
Bert Norlin was supervisor and examiner for this thesis.
We would like to gratefully thank our supervisor, Bert Norlin, for his guidance and his good ideas throughout this master thesis; thank you for your patience and great help.
Dafina’s personal thankyou is given to her family and especially to her mother, who has been her biggest support throughout these years of studies at KTH; without your encouragement and positivity these years would have been very hard to get through.
John’s personal thankyou is given to his family for their constant support; thank you for mak- ing these years a pleasure.
Dafina Zeka John-Alexander Boutzas
9 Examensarbete
Titel Bärförmåga vid intryckning av svetsade I-
balkar – med felplacerede avstyvningar
Författare Dafina Zeka och John-Alexander Boutzas
Institution Hus- och Anläggningsteknik
Examensarbete Master nivå 477
Handledare Bert Norlin
Nyckelord Intryckning, Avstyvningar, I-balk, Plåt kon-
struktioner, Abaqus, Fritt upplagd balk
Sammanfattning
När en koncentrerad last införs vinkelrätt mot flänsarna på en stålbalk, benämns detta för in- tryckning. Intryckning av detta slag motsvaran den engelska termen ”Patch loading”. Denna företeelse är ofta förekommande i stålkonstruktioner, till exempel vid stöd eller vid lansering av broar. På grund av den slankhet som I-balkar och andra plåtkonstruktioner har, förekom- mer förstärkning av dessa med avstyvningar för att undvika buckling. När man förstärker bal- kar genom att tillföra avstyvningar så görs detta i avseende att möjliggöra större plastiska de- formationer innan buckling kan inträffa; i och med detta ökas balkens bärförmåga.
Tväravstyvningar används i områden där balken är utsatt för transversella koncentrerade las- ter. I Eurocode presenteras hänvisningar för bärförmågeberäkningar, gällande intryckning med avstyvningar, för begränsade villkor: då balken har dubbla avstyvningar där den vertikala kraften är placerad mittemellan dessa och då en enkel avstyvning är placerad i linje med transversallasten. Eurocode innefattar även hänvisning till beräkningar av bärförmågan för balkar som inte är försedda med avstyvningar. Dock saknar Eurocode beskrivning för de fall av intryckning då avstyvningarna är felplacerade i förhållande till lasten.
Syftet med detta arbete har varit att utvärdera påverkan av avstyvningar på bärförmågan för svetsade I-balkar då avstyvningarna är felplacerade. Längden på de observerade balkarna har valts som en varierande parameter, då momentets inverkan på bärförmågan är av intresse. På grund av komplexiteten i problem av detta slag valdes att ta fram resultaten genom numerisk simulering. Valet var att utföra testerna genom FE-modellering och samtidigt använda resul- taten från Lagerqvists fysiska experiment i dennes doktorsavhandling (1994), i avseende att avgöra tillförlitligheten av FE-modelleringen. Att utföra egna fysiska experiment var inte nå- got alternativ då detta var för omfattande och framförallt inte ekonomiskt möjligt.
Denna studie gjordes i två steg. I det första steget utfördes FE-modeller för samma förhållan- den som Lagerqvist beskriver i sina experiment. Dessa analyser ingick inte i huvudsyftet med arbetet, utan gjordes för att öka tillförlitligheten för de slutgiltiga resultaten. Därefter gjordes de slutgiltiga analyserna i syfte att undersöka inverkan av felplacerade avstyvningar på bär- förmågan. I detta steg utfördes även observationer avseende böjmomentets inverkan på bär- förmågan.
Analyserna i det första steget gav ett tillfredställande resultat, vilket bidrog till att man kunde anta tillförlitliga resultat i steg två. På grund av detta antas de slutgiltiga analyserna som till- förlitliga. Från observationer av de slutgiltiga analyserna så dras slutsatser om att avstyvning- ar är ett mycket tillfredställande sätt att öka bärförmågan för slanka balkar. För maximalt ut- nyttjande är det dock viktigt att dessa är rättplacerade, då en liten avvikelse från den optimala positionen ger upphov till relativt stor minskning av bärförmågan.
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Förord
Detta är ett examensarbete utfört för Institutionen för byggnadsteknik och masterinriktingen Hus- och anläggningsteknik på Kungliga Tekniska Högskolan i Stockholm. Arbetet uträttades som en avslutande del för masterprogrammet och utfördes under vårterminen 2016.
Bert Norlin var handledare och examinator för detta examensarbete.
Vi vill ge ett stort tack till vår handledare Bert Norlin, för hans vägledning och smarta idéer gällande tillvägagångsättet för detta arbete; tack för din hjälp och ditt tålamod.
Dafinas personliga tack går till hennes familj, i synnerhet till hennes mamma som har varit hennes största stöd under åren på KTH; tack för att du, genom din uppmuntran och din optim- ism, underlättat dessa år för mig.
Johns personliga tack går till hans familj, för deras ständiga stöttande; tack för att ni gjort dessa år till ett nöje.
Dafina Zeka John-Alexander Boutzas
13 Contents
1. Introduction ... 3
1.1 Background ... 3
1.2 Main aim ... 4
1.3 Limitations ... 5
1.4 Clarification of the issue ... 5
2. Theory ... 7
2.1 Critical load ... 7
2.2 Yield resistance ... 8
2.3 Resistance function ... 10
3. Previous research ... 11
3.1 Physical patch loading tests ... 13
4. Method ... 15
4.1 Initial numerical analyses ... 20
4.2 Final numerical analyses ... 22
5. Results ... 25
5.1 Initial analyses ... 25
5.2 Final analyses ... 26
5.2.1 Analyses of the results – Patch loading resistance ... 32
5.2.2 Analyses of the results - Moment resistance ... 35
6. Conclusion ... 37
Appendix ... 1
Appendix - A: Patch loading resistance without moment contribution - for beam A61 and A71 ... 1
Appendix - B: Moment resistance – for beam A61 and A71 ... 5
Appendix - C: Force- displacement diagrams ... 9
Works Cited ... 11
1 NOTATIONS
𝑎 Width of the web
𝑏! Width of the flange
𝑡! Thickness of the flange
ℎ! Height of the web
𝑡! Thickness of the web
𝑠! Loading length
𝑙 Length
𝑙!"" Effective length
𝑙! Equivalent loading length
𝐹!" Critical load
𝐹! Ultimate resistance
𝐹! Yielding resistance
𝑓!" Yield strength of the flanges
𝑓!" Yield strength of the web panel
𝐹! Patch loading resistance
𝑘!, 𝑘! Buckling coefficient
𝐸 Module of elasticity
𝜀 Strain
𝑣 Poisson´s ratio
𝜒 Resistance function
𝜆 Slenderness parameter
𝛽 Parameter for the ratio !"
!!!
𝐺𝐾 Torsional stiffness for the flanges
𝐷 Plate stiffness for the web panel
𝛿 Elongation
𝐴! Original cross-sectional area
𝜓 Stress ratio
𝜌 Reduction factor
𝐴!"" Effective section area
𝑀!" Moment resistance
𝑊!" Elastic section modulus
𝑊!" Plastic section modulus
𝐼 Second area moment of inertia
𝐼!"" Effective second area moment of
inertia
𝛾 Partial factor
3
1. Introduction
1.1 Background
One common problem with steel structures is when a steel beam is exposed to concentrated loads. Problems caused by concentrated loads have over the recent sixty years been one of the major and central fields of research regarding steel structures (Lagerqvist, 1994).
Vertical concentrated loads that are applied to one flange, over the loading length, 𝑠!, of a steel girder are referred to as patch loading. The load can in this case be introduced through the upper or the lower flange. Concentrated loads are a common situation in structural appli- cations, for example; at supports, where columns are placed or from crane sheets and during launching of bridges (Chacon, Mirabell, & Real, 2013). Usually patch loading problems are dealt with by applying transverse stiffeners to the web of the beam. This way of increasing the resistance is only suitable when there is a fixed concentrated loading, for example in connec- tions between beams or connections between beam and column. Moreover, for movable con- centrated loads it is difficult to solve the problem by adding transverse stiffeners, example of cases where adding transverse stiffeners is not a solution to the problem is when there are movable loads from crane wheels or launching of bridges. Nevertheless, for movable concen- trated loads, a better solution is adding longitudinal stiffeners or to change the geometry of the cross section by increasing the thickness of the web and thereby making the beam less slen- der. The ultimate resistance against patch loading depends on the following factors: load dis- tribution, flange dimensions, material properties, geometric imperfections and if transverse stiffeners are added or not (Tryland, Hopperstad, & Langseth, 1998).
In EN 1993-1-5, the part of the Eurocode that covers steel structures, patch loading is treated as an instability problem. Information about how to handle these kind of problems are found in the part of Eurocode 3 that covers plated structural elements, and is denoted as Part 1.5.
Patch loading is included in the Eurocode in the same form as other instability problems, with the 𝜒(𝜆) approach. The Eurocode presents three different cases of situations where beams are exposed to concentrated loads, depending on how the load is applied. The three load situa- tions are based on O. Lagerqvist doctoral thesis from 1994 and are the following: when a con- centrated load is applied to one flange, when two concentrated and opposite loads are applied at both flanges in symmetry with each other and when a concentrated load is applied to an unstiffed beam end. For simplification, according to Lagerqvist, these cases are referred to as patch loading, opposite patch loading and end patch loading. The analyses in this study cover the first one, patch loading (Lagerqvist, 1994) (Eurocode, 2003).
Herein, numerical analyses were performed in order to do observations of the bearing re- sistance of beams subjected to patch loading. Additionally, the beams were modelled in the FE- program Abaqus. Focus in the investigations is on two different beams, where the central
difference between these will be in their slenderness. Lagerqvist investigated the behavior of unstiffed beams exposed to different types of patch loading. The results were based on a well- documented experimental investigation which is going to be used for validation for the nu- merical experiments done in this study. The numerical analyses herein are divided into two different steps. The first one includes numerical simulations under exactly the same condi- tions as in Lagerqvist (1994); those were done to validate the FE-model. In this case no stiff- eners are added to the beam. In the second step, when it is clear that the simulations corre- spond to the experimental investigations that Lagerqvist did, stiffeners are added to the beams. In this step, the only difference is that the beam was considerably longer, thus a longer beam was used to observe the influence from a bending moment. (Lagerqvist, 1994).
In the last sixty years, a considerable amount of research as well as a wide variety of studies in steel construction have been focused on patch loading (Lagerqvist & Johansson, 1996). The first models for the ultimate resistance were divided into two different categories based on the type of failure, if the failure is caused by yielding of the material or by instability (Gozzi, 2007). Because of the fact that it is difficult to distinguish between these Lagerqvist suggest- ed, in his doctoral thesis from 1994, that there should instead be a gradual transition in the failure calculations (Lagerqvist, 1994).
1.2 Main aim
The aim of this study is to observe the influence of misalignment between a stiffener and a load. The variation will be from fully effective stiffeners, i.e. the stiffeners are placed in such a way that the resistance is maximized, to when the stiffeners are placed in a way that they do not contribute to the resistance. The analyses were performed by FE-modelling. A specific real life occurrence of this case could be when renovations of a building are made, and the architect for some esthetic reason wants to change the position of an existing column. In addi- tion, this study will also show the importance of not neglecting any possible displacement of the stiffeners in order to get a safe construction.
Before starting with the FE-modeling a literature-study was performed. The purpose of this literature-study was to get more general knowledge regarding steel structures, patch loading and the part of the Eurocode that is specialized in steel structures; but also to get more de- tailed information about the topic. Likewise, the literature study aided in attaining more knowledge about the field of research, and answering questions such as: what type of research has been fulfilled, and what type of research is in progress.
5 1.3 Limitations
Conducting our own physical experiments of beams exposed to patch loading was not an op- tion in this investigation. Such testing is too complicated and above all not economically fea- sible. Instead the choice has been to make numerical simulations and compare these too pre- vious documented physical experiments made by (Lagerqvist, 1994) and also to compare the results with earlier made numerical analyses made by (Tryland, Hopperstad, & Langseth, 1998).
1.4 Clarification of the issue
Adding stiffeners gives an increase to the ultimate resistance for the steel structure. In this study, the influence from misaligned stiffeners with respect to patch loading is investigated.
Questions and topics that are answered and dealt with in this study are:
1. What influence does the misalignment of stiffeners have on the resistance of a welded steel beam, when the beam is exposed to patch loading?
2. A comparison of the influence between smaller and greater misalignments is made.
The misalignment is increasing the distance with 1 centimeter a time, away from the optimal point of action; as shown in Figure 1-1. The optimal point of action is recog- nized as the point that gives the greatest influence from the stiffener to the resistance when a concentrated load is applied.
Figure 1-1 Load acting away from a stiffener.
3. A comparison between beams with smaller and greater slenderness, with the same misalignments, is made. This is done for the slenderness of 40 and 110, which are in (Lagerqvist, 1994) denoted as beams A71 and A61. Those are the beams with the smallest and the greatest slenderness that Lagerqvist (1994) did physical patch loading tests on.
4. Observations of how the length of the beam influences the resistance against patch loading are made, i.e. the influence of bending moments.
7
2. Theory
This chapter deals with the theory behind calculating the resistance to patch loading. There is a presentation of the design models used, and how they were obtained. The design model in- cludes the yield resistance and the critical buckling load leading to the resistance function. All the theory presented in this chapter is based on Lagerqvist (1994), Lagerqvist & Johansson (1996) and Johansson (2005).
“Vertical” concentrated loads acting on a beam give rise to vertical compression stresses in the web panel. The stresses are of larger magnitude close to the flange where the load is acting, and decreases downwards to the opposite flange. The load is also distributed by the flanges, in the longitudinal direction of the beam. These stresses together with a relatively slender cross section lead to buckling. Buckling is characterized as a sudden deformation in the web, which brings failure to the structure. The stresses leading to this buckling failure are smaller than the maximum compressive stresses that the material is capable to withstand before yielding. The failure modes for this type of problems are caused by buckling and yielding. Even though the failure modes are not clearly distinguished from one another usually material yielding is cru- cial for cross sections with thicker web panel, while buckling is the crucial failure mode for slender cross sections. More slender cross sections creates bigger buckles, while a stiffer cross section creates, after plastic deformation, a buckle that is smaller and concentrated to an area close to the exposed flange, also known as crippling (Johansson, 2005).
The following calculations are based on the method used in SS-EN 1993-1-5 and the descrip- tions that are presented by Lagerqvist and Johansson (1996). The method involves finding the critical load of a web panel without imperfections and the load that gives rise to yielding of the material without any buckling. Then, by using these parameters and a resistance function one can calculate the capacity of the beam (Lagerqvist & Johansson, 1996).
2.1 Critical load
When calculating the critical buckling load, first the buckling coefficient has to be calculated.
The buckling coefficient depends on how the load affects the beam, the boundary conditions and the beam geometry. As previously mentioned the appearance can be of three different characters: patch loading, opposite patch loading and end patch loading. The formulas for the buckling coefficients in the Eurocode are based on Lagerqvist doctoral thesis1. For patch load- ing, which is of interest herein, the buckling coefficient, 𝑘!, is obtained from equation (2.1.1).
Were 𝛽 is a parameter that describes the rigidity of the flanges.
𝑘! = 5,82 + 2,1(ℎ!
𝑎 )!+ 0,46 𝛽! (2.1.1)
1 Lagerqvist did a summary of earlier research in his conclusions about the buckling coefficient.
𝛽 = 𝑏!𝑡!!
ℎ!𝑡!! (2.1.2)
The following figure gives approximated formulas to find 𝑘! for patch loading, opposite patch loading and end patch loading. It is these three equations that are presented in the Eu- rocode2.
Figure 2-1 Different cases of patch loading and the corresponding calculations for the buckling coefficients (Lagerqvist & Johansson, 1996)
When the buckling coefficient is known one is able to calculate the critical buckling load, 𝐹!". For a steel plate that is exposed to a concentrated load in the direction of the plane, the critical load can be determined according to the theory of elasticity with equation (2.1.3). Also here a simplification has been made in the Eurocode, from equation (2.1.3) to (2.1.4), assuming 𝜐 = 0,3.
𝐹!" = 𝜋!𝐸 12(1 − 𝑣!)
𝑡!!
ℎ!𝑘! (2.1.3)
𝐹!" = 0,9𝑘!𝐸𝑡!!
ℎ! (2.1.4)
2.2 Yield resistance
There is no sharp definition of the plastic resistance when a concentrated load is applied. The complexity is in making suitable assumptions about the loaded length, 𝑆!. This length is nor- mally assumed to spread through the attached plate at an angle of 45⁰. In the Eurocode as-
9 Figure 2-2 Different loading lengths. (Lagerqvist & Johansson, 1996)
The elastic resistance of the material, 𝐹!", the load that that gives stresses to makes the materi- al yield in the most exposed point, is calculated by equation (2.2.1) (Johansson, 2005). Note that the actual plastic yielding resistance is higher than the one in this equation.
𝐹!" = 𝑓!"𝑡!(𝑠!+ 2𝑡!) (2.2.1)
The maximum resistance is not reached until the web panel buckles, which depends on the slenderness of the web. The plastic resistance of a material with a slender web panel is much higher than the critical buckling load. Ove Lagerqvist presented in 1994 a mechanical model that estimates the plastic resistance of a beam as a further development of Bergfelt’s three hinge model and Roberts´s four hinge model (Roberts, 1981; Bergfelt, 1979). The model is referred to as the Yield model and has a four hinge principle. According to this principle, the flange gets four hinges, two outer and two inner (Lagerqvist, Patch loading: resistance of steel girders subjected to concentrated forces, 1994). Furthermore, the material yields in between these hinges. Also, it is assumed that the inner hinges have the same resistance as the flanges and the outer hinges include a part of the web which is 14% of the height of the web.
Figure 2-3 Four hinge model, used for calculating the plastic resistance.
(Lagerqvist, 1994)
Bringing the forces to equilibrium and computing some further simplifications obtain the fol- lowing equation for the plastic resistance:
𝐹! = 𝑓!"𝑡!𝑙! (2.2.2)
The length 𝑙! is the equivalent loading length and is the length between the two outer hinges in a full yielding model without any web buckling, i.e. 𝐹! is an upper plastic limit for the re- sistance (Norlin, 2015). It is different depending on the loading case analyzed. For patch load-
ing and opposite patch loading the equivalent loading length 𝑙! is calculated in the same way, see equations (2.2.3), (2.2.4) and (2.2.5).
𝑙! = 𝑠! + 2𝑡!(1 + 𝑚!+ 𝑚!) (2.2.3)
𝑚! = 𝑓!"𝑏!
𝑓!"𝑡! (2.2.4)
𝑚! = 0,02 ℎ! 𝑡!
!
𝑖𝑓 𝜆 > 0,5 (2.2.5)
𝑚! = 0 𝑖𝑓 𝜆 ≤ 0,5 (2.2.6)
2.3 Resistance function
When designing the resistance function according to the Eurocode the reduction factor is used in the same way as for other instability problems. The reduction factor is a function of the slenderness parameter, 𝜆, and was obtained empirically by Lagerqvist.
𝐹𝑅=𝐹𝑦𝜒(𝜆) Υ𝑀
(2.3.1)
𝜆 = 𝐹! 𝐹!"
(2.3.2)
In the Eurocode (2.3.3) is simplified to (2.3.4), which is even simpler and slightly more on the safe side.
𝜒 𝜆 = 0,006 +0,47
𝜆 ≤ 1 (2.3.3)
𝜒 𝜆 =0,5 𝜆 ≤ 1
(2.3.4)
11
3. Previous research
The physical tests used for comparison in this study are made by O. Lagerqvist, in his doctor- al thesis from 1994. The reasons for choosing these physical tests is because they are well performed and documented. Lagerqvist made a test series of 48 different tests, where 9 of them were performed as patch loading tests. The tests were done in order to obtain the ulti- mate resistance of welded I-beams and to investigate the behavior at this state. Lagerqvist also did 12 other tests where the tested beams consisted of ordinary steel (Lagerqvist & Johansson, 1996; Lagerqvist, 1994).
All tests were performed with the load applied perpendicular to the flanges and acting in the direction of the height of the beam. The differences between the tests were in their geometry and how the concentrated load was applied to the beam; the concentrated load was applied with different loading lengths and with different loading cases. The loading cases which Lagerqvist based his investigations on were: when a concentrated load was applied to a beam without stiffeners, when two opposite concentrated loads were applied to a beam without stiffeners and when an unstiffed beam-end was exposed to concentrated loading. Ove Lagerqvist named, as previously mentioned, the cases as patch loading, opposite patch load- ing and end patch loading. In this study, only the case of patch loading is taken into considera- tion. (Lagerqvist, 1994)
The loading lengths, s!, used in the physical tests were 40 mm and 80 mm. The beams in the tests had a variation of slenderness between 40 and 110; where the slenderness is the relation between the height of the web and its thickness, h! t!. The throat thickness of the weld that binds together the flanges with the web was 3 mm. The geometrical imperfections, ∆W, were measured at the midpoint of two points, 10 mm from lower flange and 10 mm from upper flange. The rotation of the flanges was represented by ∆H (Lagerqvist, 1994).
Lagerqvist used beams that were double symmetric and made of high strength, general struc- tural steel, Weldox 700. At the ends, the beams had vertical stiffeners in order to make the ends rigid. The rigid ends prevent movements of the flanges in the vertical directions and they also prevent horizontal movements of the web, at the same time as they prevent rotation in the direction of the beam.
In all the physical tests, a load was introduced to the beam through a 50 mm thick steel plate, which was not welded to the flanges. The load was applied by a hydraulic jack with a maxi- mum capacity of 1000 kN. All the geometries of the beams Lagerqvist did tests on are pre- sented in table 3.1, in this table the beams that were tested to end patch loading and opposite patch loading are also included (Lagerqvist, 1994).
Table 3.1 Geometry of beams used for testing
Beam 𝒉𝒘
[mm]
𝒕𝒘 [mm]
𝒉𝒘 𝒕𝒘
𝒃𝒇 [mm]
𝒕𝒇 [mm]
a [mm]
𝒂 𝒉𝒘
A11 239.8 3.8 60 118.5 12.0 1014 4.2
A12 239.8 3.8 60 118.5 12.0 1010 4.2
A13 239.8 3.8 60 118.5 12.0 1008 4.2
A14 239.8 3.8 60 118.5 12.0 1004 4.2
A21 278.1 3.8 70 119.9 12.0 1258 4.5
A22 278.1 3.8 70 119.9 12.0 1260 4.5
A31 319.7 3.9 80 120.1 12.0 1405 4.4
A32 319.7 3.9 80 120.1 12.0 1404 4.4
A41 359.6 3.8 90 120.5 11.9 1315 3.7
A42 359.6 3.8 90 120.5 11.9 1523 4.2
A51 397.7 3.8 100 120.0 12.0 1690 4.2
A52 397.7 3.8 100 120.0 12.0 1900 4.8
A61 439.9 3.8 110 120.0 12.0 1626 3.7
A62 439.9 3.8 110 120.0 12.0 1860 4.2
A71 320.7 7.9 40 120.5 11.9 1405 4.4
A72 320.7 7.9 40 120.5 11.9 1562 4.9
A81 400.5 8.0 50 120.4 12.0 1684 4.2
A82 400.5 8.0 50 120.4 12.0 1471 3.7
Lagerqvist also did tensile tests to predict the mechanical properties of the welded steel beams used in the experiments. The tests were done for steel plates with a thickness of 4 mm, 8 mm and 12 mm, which corresponds to the two types of webs and flanges. Data for the test speci- mens and results obtained are presented in the table 3.2, where the mean values of the yield strength and the ultimate strength for the respective parts are presented in the two rightmost columns. The module of elasticity, E, was 204 GPa. (Lagerqvist, 1994)
Table 3.2 Results from tensile tests.
Specimen Width [mm]
Thickness [mm]
Max.
load [kN]
𝒇𝒚 [MPa]
𝒇𝒖 [MPa]
𝒇𝒖 𝒇𝒚
𝒇𝒚,𝒎𝒆𝒂𝒏 [MPa]
𝒇𝒖,𝒎𝒆𝒂𝒏 [MPa]
AC1 30.09 12.01 318.9 842 882 1.05
844 883 AC2 30.13 12.02 319.9 846 883 1.04
AB1 20.06 7.66 123.1 763 801 1.05
762 802
AB2 20.05 7.65 123.2 761 803 1.06
A1PL1 15.03 3.88 50.00 835 857 1.03
13 material used as web material for beams with the web thickness of ca 4 mm. (Lagerqvist, 1994)
3.1 Physical patch loading tests
For the nine patch loading tests conducted by Lagerqvist, two were with the load applied through a plate with a width of 80 mm and seven with a plate of width of 40 mm. When test- ing, neither the steel plate transmitting the load nor the support plates were fastened to the beam. Table 3.3 gives the important parameters which are measured before the testing was performed. The initial twist of the flanges is referred to as ∆ℎ and the initial out of plane de- formation is referred to as 𝑤!. This gives that 𝑤! 𝑡! is the out of plane deformation of the web in relation to its thickness and is measured under the acting load.
Table 3.3 Initial data for the patch loading tests
Test Beam 𝒔𝒔
[mm]
∆𝒉 [mm]
𝒘𝟎 𝒕𝒘
A13p A13 40 1.7 0.01
A14p A14 80 0.2 0.01
A22p A22 80 0.4 0.33
A32p A32 40 1.0 0.36
A41p A41 40 3.2 0.54
A51p A51 40 0.9 0.39
A61p A61 40 1.4 0.38
A71p A71 40 0.4 0.02
A81p A81 40 0.1 0.08
The data that are underlined in Table 3.3 represent the beams that are analyzed by FE- model- ling in this study. The reason for choosing these beams is because beam A61 is the most slen- der i.e. the most sensitive one, while beam A71 is the most stocky one. Table 3.4 gives the geometrical properties for beam A61 and beam A71.
Table 3.4 Geometrical properties for beam A61 and A71.
Beam 𝒉𝒘 [mm]
𝒕𝒘 [mm]
𝒉𝒘 𝒕𝒘
𝒃𝒇 [mm]
𝒕𝒇 [mm]
a 𝒂
𝒉𝒘
A61 439.9 3.8 110 120.0 12.0 1626 3.7
A71 320.7 7.9 40 120.5 11.9 1405 4.4
15
4. Method
Because physical experiments would have been too comprehensive and complicated, and above all economically not possible the choice was to make the experiments by Finite Ele- ment Modeling, where the program Abaqus/CAE was used. In this chapter the method of the numerical analyses is described for the initial and the final analyses. The input data are pre- sented and how they were obtained. Information about the FE-modeling procedure is de- scribed followed by sections 4.1 and 4.2, which describe the specific conditions for the initial and the final analyses, respectively.
Type of elements
The structures are modeled by shell elements. The reason why shell elements are chosen is because an I-beam contains three different parts which all are three dimensional with one di- mension much shorter than the other two. The advantage of using shell elements instead of solid elements, is that it is time saving due to a reduced number of finite elements, which re- quires less equations to solve. Shell elements can be used in these analyses because the behav- iour in the thickness-dimension is not of importance and can be neglected.
The elements used were of type S4R with linear interpolation functions and reduced integra- tion. Elements of type S4R consists of four nodes and six degrees of freedom per node. The rigid plates: the loading plate and the support plates were modelled by R3D4 elements. When the final analyses were done, the beam length was extended by beam elements of type B31.
Mesh
The beams were partitioned in order to make different sizes of the mesh possible. Smaller mesh sizes were used closer to the load, in the areas where interesting observations can be made. The reason why longer elements where used further from the point where the load is acting is because those areas are not affecting the result that much. Using greater elements in those areas will decrease the required computational time, due to fewer equations to solve for the program. In the figure 4-1 the mesh sizes for each beam are presented.
Beam x1 [mm]
x2 [mm]
y1 [mm]
y2 [mm]
Mesh size [mm]
A B C
A61 500 350 70 200 10 30 60
A71 500 350 80 160 10 30 60
Figure 4-1 Description of the size distribution for the mesh in the two models made.
Type of analyses
In all the analyses, the residual stresses and the initial imperfections were introduced to the model before any final analyses were made. Static and buckle analyses were made for every test.
Buckle analysis
The buckle analysis was made in in order to introduce the buckle modes to the model; which introduces the most disadvantageous initial geometry to the beam. The buckle analyses introduce smooth imperfec- tions with varying amplitude along the length of the web. This is seen in figures 4-2 and 4-3.
Static analysis
The static analysis was made by increasing the deformation at the ref- erence point of the load plate. This deformation was increased in the z- direction only.
17 Figure 4-3 Buckle analyses when stiffeners are added to the beam.
Load application
The load was introduced to the beam through a load plate. There was no physical load mod- elled, instead a boundary condition was applied to the reference point of the load plate in or- der to push the plate downwards. The deformation had to be great enough to bring failure to the structure. In the initial analyses the maximum deformation was set to 50 mm, which was enough for failure to occur in all cases. The reason for choosing an increasing deformation rather than an applied load is that it is easier for Abaqus to obtain a converged solution when passing the top of the load-deflection curve, the top of this curve is the ultimate load (i.e. the resistance).
Stress-strain
The mechanical properties of a material can be obtained by doing tensile testing on a speci- men of the material. When doing the tensile testing, a specimen is subjected to an increasing tensile force. One gets the engineering stress by noting the force, which is uniaxial, and divid- ing it with the original cross-sectional area, 𝐴!. Regarding the engineering tensile strain this is found by noting the elongation, 𝛿, of the specimen and dividing this with the original length, 𝐿!. This is shown by equations (4.0.3) and (4.0.4) (Roylance, 2001). After reaching the maximum stress, the engineering curve leans downwards. The reason for this downward slope is that when reaching this point necking3 of the material starts. When necking starts the cross- sectional area decreases dramatically; in the same time, when calculating the stresses the force applied is dived by a constant area, which is the original cross-sectional area, 𝐴!. The true stresses are the stresses given by dividing the applied force with the actual cross-sectional area.
The properties, regarding stresses and strains, which Abaqus requires in order to get the mod- elling right are the true stresses and strains. This is done by calculating the true stresses and
3 Necking is a type of tensile deformation where large amount of strain in a small region takes place.
strains from the engineering stresses and strains by equations (4.0.1) and (4.0.2). Where 𝜎! and 𝜀!are the engineering values of stress and strain. When inserting the values in Abaqus in order to get the modeling right, this requires also that the elastic strain is excluded; leaving only the plastic part of the strain. The equations used for conversion of the stresses and strains are taken from Eurocode4.
𝜎!"#$ = 𝜎!(1 + 𝜀!) (4.0.1)
𝜀!"#$= ln (1 + 𝜀!) (4.0.2)
𝜎! = 𝐹 𝐴!
(4.0.3)
𝜀! = 𝛿 𝐿!
(4.0.4)
From the total true strain the elastic strain has to be subtracted. Equation (4.0.5) describes how the plastic strain is calculated.
𝜀! = 𝜀!"#$−𝜎!"#$
𝐸 (4.0.5)
19 Residual stresses
When manufacturing a welded I-beam, this is done by welding together the three different parts that the beam consisting of, the flanges and the web. This gives rise to high temperature differences in the steel which in turn gives rise to residual stresses due to uneven cooling (Staflund, 2011). To make the numerical analyses as representative as possible self-stresses were added to the model.
Figure 4-4 The figure in the right shows the distribution of the residual stresses caused by manufacturing. The right figure shows a simplified distribu- tion, which is used in order to have a simple representation of the re- sidual stresses here (Hedmark, 2007; Boverket, 2007).
The simplified distribution was used when adding the residual stresses into the model.
Through equilibrium equations the residual stresses were calculated. The calculated values are given in Figure 4-5, which are the values used when modeling. The same residual stresses are valid for both the upper and the lower flanges.
Figure 4-5 The calculated residual stresses added to the model.
4.1 Initial numerical analyses
Beams with the same conditions as in Lagerkvist’s tests have been redone by numerical anal- yses in Abaqus and are presented in this section. The numerical analyses were done for the tests A61p and A71p in Table 3.3. In the initial analyses the beams are simply supported and have no stiffeners.
Boundary conditions and geometrical imperfections
In the initial analyses, end stiffeners were added to the beams. As mentioned previously in the text, this leads to restrictions in displacements and rotation. The rigid ends could also have been achieved by adding boundary conditions at the ends of the beam, without adding stiffen- ers. Along the upper flange the whole beam is fixed for displacements along the y-direction.
The boundary conditions at the following points:
1. The supports were applied through two rigid plates with one reference point, respectively, where the boundary conditions were added. The refer- ence points are marked with RP in Figure 4-6. In these points the support- plates were fixed in the x-, y- and z-directions, also rotation around the z- axis was prevented. The corresponding points on the flanges are fixed in the y- and z-directions, and for rotation around the z-axis.
2. The reference point on the loading plate has the same restrictions as the corresponding point on the flanges; they are fixed for displacements in the x- and y-directions and for rotation around the z-axis.
3. This point on the lower flange, under the acting load, is fixed along the x- axis.
21 Geometrical imperfections
For the initial analyses, the initial out of plane deformation of the web was set to 1.44 mm for beam A61 and to 0.16 for beam A71. These were the values of the initial imperfections Lagerqvist measured in the experiments (1994). The initial out of plane deformation was measured for a point at the middle of the height of the web plate as the deviation from a straight line between these points (Lagerqvist, 1994).
4.2 Final numerical analyses
In this section, the procedures of the main analyses for this study are presented. The same beams as the ones in the initial analyses are tested but with two main differences; in this step stiffeners are added to the beams at the same time as the lengths are significantly increased.
The extension of the beams was produced by using beam elements of type B31. The end- stiffeners were removed from the initial beams and a reference point was added at the centre of gravity of the cross sectional area for each end. This reference point was tied to every other node on the cross-section by rigid beam elements (MPC BEAM in Abaqus); this is showed in Figure 4-7.
Figure 4-7 The procedure of the beam-extension. The beam has a correctly placed stiffener under the load.
Boundary conditions
The boundary conditions in the final analyses are similar to the boundary conditions in the initial analyses. The exception is that the end-stiffeners and the support plates are replaced by boundary conditions, which are added at a node point on each of the ends of the beam;
marked with point 3 in Figure 4-8. The upper flange is fixed for movements in the y- direction in the same way as for the initial analyses.
Boundary conditions at the marked points - for the extended beam:
23 3. The nodes marked with three in the figure are constrained with boundary
conditions preventing movements in the y- and z-directions and also rotation around the z-direction. For one of the ends, the node is also fixed for move- ments along the x-direction.
Figure 4-8 Boundary conditions used in the final analyses.
It would have been more correct to not fix one of the nodes in point 3 for movements along the x-direction, but since the deviation in resistance results is negligible no modifications have been made.
Geometrical imperfections
For the analyses in this step the geometrical imperfections were calculated according to the guidance in Eurocode 3, part 1:55. Where the geometrical imperfections for a panel with a short span and with a buckling shape has the magnitude of the smallest value between 𝑎 200 and 𝑏 200. Where a is the length of the web and b is the height of it. In the case of a web panel for a beam the length is much greater than the height, b ≪ a; which gives 𝑏 200 as the crucial quotient. Table 4.1 gives the initial out of flatness imperfection of the web panel.
Table 4.1 The initial out of plane deformation
5 Annex C – Fem- calculations, Table C.2 and Figure C.1
Beam 𝒃
[mm]
𝒘𝟎 [mm]
A61 439.9 2.2
A71 320.7 1.6
25
5. Results
5.1 Initial analyses
The results from the patch loading resistances from the initial analyses are presented in Table 5.1 together with the results from earlier work. The second column presents Lagerqvist’s re- sults from physical experiments (1994) and the third column presents the results from FE- simulations, produced by Tryland and colleagues (1998).
Beam Lagerqvist Physical tests
[kN]
Tryland FEM tests
[kN]
Mean value [kN]
Test [kN]
A61 293 299 296 308
A71 931 870 901 837
A mean value is calculated for the earlier produced tests, which is shown in the third column of Table 5.1. For beam A61 the obtained resistance is slightly over the calculated mean value with a negligible deviation, while for beam A71 the resistance is slightly under the mean val- ue. Moreover, the deviation from the physical experiments is of similar type6 as from Try- land’s tests; with the difference that the deviation in these tests is slightly bigger. The failure of beam A61 was caused by buckling of the web, while the failure of A71 was caused by crippling; which is a smaller buckle localized close to the flange where the load is acting and appears after heavy yielding (Tryland, Hopperstad, & Langseth, 1998).
The probable cause for the deviation of the resistance from the physical experiments is the initial imperfections. When doing the buckle analyses, the worst-case scenario of the initial geometry is introduced. For beam A61, which is a slender beam, the choice of initial imper- fections does not have a big influence on the resistance, causing the deviation to be negligible.
Beam A71 is stocky and therefore the imperfections have a greater impact on the resistance.
If, in this case, the shape of the physical beams is more favorable this will lead to a lower re- sistance for the numerical model than in the physical experiments. This is believed to be the underlying cause for the deviation herein.
Other possible indications on the deviation in results include: the difference in the stress-strain curves and differences in the residual stresses. When doing the modification from engineering stresses and strains into true stresses and strains, this modification may have led to differences in the obtained values, in comparison to the stresses and strains in the experiments. Regarding the residual stresses, these are approximated in the model; thus not exactly corresponding to
6 The deviation gives a higher resistance for beam A61 and lower for beam A71.
Table 5.1 A comparison between the results found by FE-modeling and previ- ously made tests.
the actual residual stresses. Furthermore, the residual stresses can be smaller for A71 if the weld dimensions are the same as for A61 even though the web panel is thicker.
The conclusion to be drawn is, if the model could be reproduced with the exactly the same geometry as in the physical experiments, probably the computed resistance would be closer to what was predicted from those. Thus, this indicates that the model is sufficient for this pur- pose.
5.2 Final analyses
The aim of this study was to observe how the resistance against patch loading is affected by misalignment of stiffeners. In these analyses, the length of the beam was chosen as a variable parameter in order to also find the influence from an increasing bending moment. In this sec- tion observations and results concerning these final analyses are presented. The numerical simulations have been produced regarding beam A61 and A71, which are the beams described in section 4.2.
Displacement of the loading point
Before displacing the reference point, i.e. the loading point, observations of the bearing re- sistance against patch loading were made for two cases of each beam; when the beam had no stiffeners and when the beam had a correctly placed stiffener. In the same way as in the previ- ous tests, the load plate had a boundary condition, which was given a downward deformation in the z-direction. The resistances obtained are presented in Table 5.2 and 5.3. Both of the two beam types were tested with four different lengths.
Table 5.2 Patch loading resistances for beam type A61, when no stiffeners are added and when stiffeners are added correctly.
Patch loading resistance, 𝑷𝑹𝒅 Beam type A61
Beam length 3 m 5 m 6,5 m 8 m
No Stiffeners 297 kN 272 kN 253 kN 239 kN
Correctly placed stiffeners
927 kN 600 kN 462 kN 429 kN
Table 5.3 Patch loading resistances for beam type A71, when no stiffeners are added and when stiffeners are added correctly.
27 The misalignment was produced by gradually moving the load plate away from the stiffener.
When doing the tests with misaligned stiffeners the load was moved one centimeter at a time until the contribution from the stiffeners was negligible.
The reason for doing the analyses with different lengths was to make it possible to observe the impact from an increasing moment on the resistance. The results for the resistance against patch loading are presented in Figure 5-2 and 5-3. For beam type A71 no patch loading oc- curred for the lengths of 6.5 and 8 meters. Tables showing the exact values computed follow the diagrams. Different curves are shown in the diagrams depending on a moment resistance ratio; this resistance ratio is the moment resistance when exposed to patch loading, 𝑀!,!!", divided by the moment resistance against bending, 𝑀!,!"!, which is computed from Abaqus.
The moment resistance against patch loading is calculated by equation (5.2.1). Where L is the length of the beam. The obtained values for the moment resistances and the moment re- sistance ratio are presented in Table 5.4 and 5.5.
𝑀!,!!" = 𝑃!"𝐿 4
(5.1.2)
Table 5.4. The moment resistance ratio for every length tested for beam A61.
Beam A61:
Length
𝑴𝑹,𝑷𝑹𝒅 [kNm]
𝑴𝑹,𝑭𝑬𝑴 [kNm]
𝑴𝑹,𝑷𝑹𝒅 𝑴𝑹,𝑭𝑬𝑴
3 m 223 649 0,32
5 m 340 649 0,49
6,5 m 411 649 0,59
8 m 478 649 0,69
Table 5.5. The moment resistance ratio for every length tested for beam A71.
Beam A71:
Length
𝑴𝑹,𝑷𝑹𝒅 [kNm]
𝑴𝑹,𝑭𝑬𝑴 [kNm]
𝑴𝑹,𝑷𝑹𝒅 𝑴𝑹,𝑭𝑬𝑴
3 m 448 575 0,78
5 m 585 575 1,1
From observing Table 5.7 and 5.8 it is seen that the resistance for some eccentricities increas- es slightly although the eccentricity misalignment from the stiffeners increases; this is most surely caused by the initial geometrical imperfections. This, since that the initial imperfec- tions introduced by the buckle analyses are smoothly shaped, with the amplitude varied along the length of the web plate. Buckle analyses were done for every eccentricity tested.
When doing the analyses, the load plate was released for rotation around the y-axis. In real life, the rotation of the load plate is restricted but not completely fixed. Leaving the load plate released for rotation gives the worst case scenario. Tests made in FEM, tell that fixing the
load plate for rotation around the y-axis gives rise to an increased resistance. Figure 5-1 shows the rotation of the load plate.
Figure 5-1 Displaced configuration after failure with the loading plate free for rotation.
29 Table 5.6. Moment ratios with corresponding lengths.
A61 A71
0,32 ≙ 3,0 m 0,78 ≙ 3,0 m
0,49 ≙ 5,0 m 1,1 ≙ 5,0 m
0,59 ≙ 6,5 m 0,69 ≙ 8,0 m
Figure 5-2 Diagram of the decrease of patch loading resistance with in- creased eccentricity from stiffener for beam type A61.
Figure 5-3 Diagram of the decrease of patch loading resistance with in- creased eccentricity from the stiffener for beam type A71.
0 100 200 300 400 500 600 700 800 900 1000
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Resistance [kN]
Eccentricity from stiffeners [cm]
A61 - Patch Loading Resistance
0,32 0,49 0,59 0,69
0 100 200 300 400 500 600 700 800 900 1000 1100
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Resistance [kN]
Eccentricity from stiffeners [cm]
A71 - Patch Loading Resistance
0,78 1,1
Table 5.7 Results from patch loading tests – Beam type A61 Eccentricity from
stiffeners [cm]
𝑷𝑹𝒅,𝟑𝒎 [kN]
𝑷𝑹𝒅,𝟓𝒎 [kN]
𝑷𝑹𝒅,𝟔.𝟓𝒎 [kN]
𝑷𝑹𝒅,𝟖𝒎 [kN]
No stiffeners 297 272 253 239
0 927 600 462 429
1 852 546 452 418
2 651 515 436 383
3 551 448 401 366
4 449 396 367 336
5 421 353 332 308
6 377 329 306 289
7 357 310 289 274
8 339 298 276 264
9 328 290 269 255
10 319 284 263 250
11 313 283 261 248
12 310 281 260 246
13 308 278 258 244
14 307 277 257 243
15 306 276 254 243
31 Table 5.8 Results from patch loading tests – Beam type A71
Eccentricity from stiffeners
[cm]
𝑷𝑹𝒅,𝟑𝒎 [kN]
𝑷𝑹𝒅,𝟓𝒎 [kN]
No stiffeners 597 468
0 1033 606
1 784 5537
2 749 541 8
3 724 502
4 704 498
5 689 496
6 683 485
7 668 485
8 673 486
9 680 483
10 678 483
11 665 483
12 661 485
13 661 482
14 658 485
15 656 483
16 652 482
17 647 482
18 644 484
19 636 482
20 644 483
7 No patch loading occurs.
8 No patch loading occurs.
5.2.1 Analyses of the results – Patch loading resistance
The patch loading resistances for the two beam types, A61 and A71, without any contribution from the length and with no stiffeners added, were computed by the methods described in Chapter 2. The calculated patch loading resistances for the beams are presented in Table 5.9;
with the procedure shown in Appendix A. The rightmost column in Table 5.9 presents the patch loading resistance from the FE-model for each beam without stiffeners.
Table 5.9 Patch loading resistance with small influence from bending and no stiffeners added.
Beam 𝑷𝑹𝒅,𝒉𝒄9
[kN]
𝑷𝑹𝒅,𝑭𝑬𝑴 [kN]
A61 167 308
A71 726 837
In Figures 5-4 and 5-5 the results are presented as diagrams, with a dimensionless parameter on the vertical axis and the eccentricity on the horizontal axis. The vertical parameter repre- sents the ratio between the patch loading resistance for the beam for every eccentricity divid- ed by the patch loading resistance without stiffeners. In Figure 5-4 the patch loading re- sistance without stiffeners is calculated by hand and in Figure 5-5 it is computed from Abaqus. The different curves represent different moment resistance ratios, which depends on the lengths, as described in section 5.2.
Figure 5-4 The patch loading resistance ratio in relation to the eccentricity for beam A61. Where the vertical axis represents the ratio between patch loading resistances for the beam divided by the hand calculated re-
1 2 3 4 5 6
0 5 10 15
PRd/PRd,Hand
calculation
Eccentricity [cm]
A61 - Resistance ratio
0,32 0,49 0,59 0,69
