Plate buckling in design codes

Transcription

1 Plate buckling in design codes The difference between NEN 6771 and NEN-EN Master Thesis

2 Plate buckling in design codes The difference between NEN 6771 and NEN-EN Delft University of Technology The Netherlands Faculty of Civil Engineering and Geosciences Department of Design and Construction Section of Buildings and Civil Engineering Structures Movares Netherlands B.V. Division of Infrastructure Section of Constructions Student number:

3 BOARD OF EXAMINATION: Prof. ir. F.S.K. Bijlaard Delft University of Technology Faculty of Civil Engineering and Geosciences Department of Design and Construction Section of Buildings and Civil Engineering Structures Ir. R. Abspoel Delft University of Technology Faculty of Civil Engineering and Geosciences Department of Design and Construction Section of Buildings and Civil Engineering Structures Dr. ir. M.A.N. Hendriks Delft University of Technology Faculty of Civil Engineering and Geosciences Department of Design and Construction Section of Structural Mechanics Ing. B.H. Hesselink Movares Netherlands B.V. Division of Infrastructure Section of Constructions Ir. L.J.M. Houben Delft University of Technology Faculty of Civil Engineering and Geosciences Secretary 3

4 Preface This thesis is made as a final project of my Civil Engineering master education at Delft University of Technology, where I attended the Structural Engineering master. Most of the work for this thesis has been done at Movares Netherlands B.V. between August 2010 and February I am very grateful for the opportunity to do an internship at large company, which was a very educational experience and a good addition to the knowledge gained at the university itself. The subject of this thesis is plate buckling in steel structures and how various design codes treat this subject. It is interesting to see how such a complicated phenomenon can be approached with relatively simple methods, which still can give accurate design capacities. However, the different methods also give different capacities. In this thesis it is clarified where these differences originate from and which methods should be chosen in varying situations. I would like to thank prof. ir. Bijlaard, ing. Hesselink, ir. Abspoel and dr. ir. Hendriks. Not only for their tips and guidance during the research for this thesis, but also for proof-reading the final result. May 2011, Mark van der Burg 4

5 Abstract Slender steel plates loaded in compression will buckle out of plane before their cross-sectional capacity is reached. Precisely determining the failure load in an analytical way is an impossible exercise. Therefore many researchers have put effort into finding a simple design method, which would still predict the plate buckling load with the required accuracy. Many different methods have been developed, some of which have been adopted in the currently valid design code in the Netherlands. Some preliminary calculations showed that the Eurocode (NEN-EN ) gives much more plate buckling capacity then the Dutch code (NEN 6771). This observation was the start of this thesis, of which the main goal is to find and explain the differences between the two design codes. A secondary goal is to formulate an advice as to what method in the Eurocode is the best to use. Most of the simple methods for plate buckling can be classified as either an effective cross-section method or a reduced stress method. There are fundamental differences between these two methods, which are analyzed and clarified in this thesis. It is found that in general the effective crosssection method delivers a higher buckling capacity, but this method is also more labour-intensive. An important element in every method is the reduction factor to the stress or steel area that is used. Again different researchers came up with different reduction factors, so the influence of those reduction factors is also analyzed. With the use of some example cross-sections the difference in plate buckling capacity between NEN 6771 and NEN-EN in practical design situations is explored. Indeed the Eurocode always gives more capacity, which was to be expected when analyzing the methods in detail. Later in the thesis the plate buckling capacities have been verified using a finite element calculation in Ansys. In all studied cases the design capacity according the Eurocode matched with the capacity according the calculations in Ansys. The fundamental principle of the effective cross-section method is the deformation capacity of slender steel plates. This deformation capacity is needed in built-up members, to let slender plates remain at capacity while more stocky plates deform further to their own buckling load. Ansys calculations have been used in this thesis to confirm this fundamental phenomenon. In analyzing the several plate buckling verification methods in the design codes it came to attention that there are no requirements given for plate edges. This is in sharp contrast to stiffeners, which are analyzed rather thoroughly. With the help of more finite element calculations to clarify this subject, it was found that the global buckling verification ensures stability of the edges of plates in built-up members. As a final subject all understanding gained while working on this thesis is combined in giving a guideline as to what method in NEN-EN is best used in what situation. Which method is best depends on the goal of the plate buckling analysis. 5

6 Table of contents: Chapter 1: Introduction Design codes Plate structures Buckling Verification methods Formulation of problem Chapter 2: Verification methods Effective cross-section vs. reduced stress method Assumptions Procedure Difference in normal force capacity Difference in bending moment capacity Conclusion Reduction factors Euler Von Kármán Winter NEN NEN-EN Overview Verification methods in design codes Inventory of methods NEN NEN-EN Step by step guide Bending moment capacity differences Normal force capacity differences Conclusion

7 Chapter 3: Design codes applied to example cross-sections Capacities according design codes I-shaped girder Box girder Capacity differences Influence of web slenderness I-shaped girder Box girder Conclusion Stiffeners according design codes Capacity at given dimensions of plate and stiffener Minimum stiffener dimensions for full Winter capacity Plate and stiffener dimensions needed for given capacity Conclusion Chapter 4: Finite element calculations Objective Procedure Verification of design code capacities Single plate in compression Very slender plate in compression Single plate in bending Conclusion Capacity of built up sections Deformation capacity Chapter 5: Requirements for plate edges Hypothesis Outline Sensitivity analysis Element and load step size Shape of imperfection Elastically supported plate I-shaped member Conclusion

8 Chapter 6: Conclusion and recommendation Conclusion Recommendation Future research Chapter 7: References Annex A.1 Plate buckling capacity of a plate in bending Annex A.2 Plate buckling capacity of a compressed plate Annex B.1 Cross-sectional capacities calculated with codes Annex B.2 Longitudinal stiffeners according codes Annex C.1 Log-file Ansys of single plate Annex C.2 Log-file Ansys of elastically supported plate Annex C.3 Log-file Ansys of I shaped member Annex D.1 Mathcad sheet for NEN-EN

9 Chapter 1: Introduction Within the field of civil engineering a structural engineer deals with the design and analysis of load bearing structures. These structures can be anything from a simple beam or column to a complete bridge or building. Structural engineers should ensure that their designs meet every given design criteria. Most of these criteria are focused on safety and serviceability. The safety aspect means that a structure should not collapse under the expected loads. When these loads are exceeded a structure should warn that it is about to collapse. Serviceability means that under the design loads with safety factor 1.0, displacements and vibrations should be kept under certain limits, to ensure safe and comfortable use of the structure. Figure 1.1: Some general examples of structural engineering. From top left to bottom right: Burj Khalifa, in Dubai. Golden Gate Bridge, in California. Wembley Stadium, in London. Millau Viaduct, in France. Falkirk Wheel, in Scotland. 1.1 Design codes The loads acting on a structure can include wind, snow, people, traffic, earthquake loads, self weight and so on. Not only in the loads but also in the materials used in construction there is a lot of variation. Even though most structures are built from either steel, concrete or a combination of the two, all kinds of different varieties of steel and concrete exist. Design codes have been composed to provide a guideline, not only as a tool to design a structure, but also for the authorities to check whether a design is safe enough before giving a building permit. In these design codes magnitudes of relevant loads on a structure are stated, as well as properties of materials and methods to analyze a structure. 9

10 Design codes are usually made by a committee of experts. Such a committee translates complicated (statistical) data regarding load levels, research in material engineering and different calculation methods into less complicated design graphs and formulas. An important part of this work is to derive safety factors to ensure a design, made with a simplified analysis, still has the desired level of safety against collapse. Once it is proven that the methods in the design standards provide a reliable design, they can be used in practice by engineers. Now these engineers don t have to prove why a certain method, load or strength is applicable, since it is already done for them. In some situations a more refined analysis might give a less conservative, cheaper design. If however something that is not in the building code is used, the engineer would have to give proof that the method is correct and applicable. Most countries have their own institute to manage the development of new building standards. In the Netherlands this is the Nederlands Normalisatie-instituut, which is also known as NEN (short for NEderlandse Norm ). NEN is a neutral party which keeps track of the standards that are needed, initializes development of standards and publishes them. These standard have a widely various field of application, not only building and structural engineering, but anything from consumer goods to paper sizes or sound levels of headphones. All countries close to the Netherlands have similar agencies: the United Kingdom has the BSI Group (British Standards Institution), Germany has the DIN (Deutsches Institut für Normung), Belgium the NBN (Bureau voor Normalisatie or Bureau de Normalisation) and France the AFNOR (Association Française de Normalisation). Since the formation of the European Union there has been a desire for more cooperation between European industries. The difference in national codes was a barrier in this process. As a solution the CEN (Comité Européen de Normalisation or European Committee for Standardization) was formed to create European Standards (ENs) which would eventually replace national standards. For example: in the Netherlands plate buckling in steel structures was calculated according NEN 6771, but since 31 March 2010 it has been withdrawn and EN should be used. 1.2 Plate structures Many small or medium sized steel structures use hot rolled profiles with a cross-section in the shape of an I, H of U. These shapes are effective both in bending and in shear. Fabricating the profiles is done by guiding steel which is still hot after casting through a series of rollers who successively press it in the desired shape. Well known hot rolled profiles are IPE, HE and UNP sections. IPE HE UNP Figure 1.2: Commonly used hot-rolled sections 10

11 A big advantage of these hot-rolled profiles is their low price compared to built-up members. Disadvantages are the fact there is a maximum size in which they can be fabricated and the low resistance against torsion. Both problems can be solved by creating a cross-section built up from flat steel plates. These plates are welded together to form a single member. It is now possible to create larger cross-sections, or make box sections for torsion members. In principle these sections are more expensive due the amount of welding needed. Plate members are widely used in for example bridges or large span roof structures. A first reason is that the required cross-sectional dimensions might be larger than the largest available hot rolled section. Secondly the cross-section properties can be adjusted to the forces in the member. Where large forces are present, for instance the bending moment at the middle of a span, thicker flanges can be used. This creates a more efficient member, compared to a rolled member with uniform dimensions throughout its length. Another option is to keep the flange thickness constant, while varying the web height to follow the bending moment distribution. This can lead to aesthetically pleasing solutions. 1.3 Buckling Buckling is an effect that occurs in structures which have a high stiffness in one direction and a low stiffness in another. When a compressive load in the stiff direction is gradually increased, the structure suddenly collapses without warning. Structures which are susceptible to buckling are for instance slender columns in buildings or structures built up from thin plates. A very simple example of buckling is a plastic coffee cup, which hardly deforms when gently pressing down on it. However, if the pressure is further increased the cup suddenly buckles and shows excessive deformation. Failure is finally caused by crippling of the cup. Even though a coffee cup is a curved structure and therefore actually shows shell buckling, it is a clear analogue for plate buckling. The same experiment can be done for column buckling with for instance a drinking straw. Slowly applying load Still undeformed Buckling Crippling Figure 1.3: Simple example of shell buckling In a load-displacement diagram buckling is clearly shown. At first a structure behaves linear elastically along the primary load path. When the applied load reaches the buckling load there is a bifurcation point, where the structure starts to follow the secondary load path. This secondary path has a lower stiffness compared to the primary path, which explains the sudden increase in deformation. With columns this secondary load path is almost horizontal, while for most plate 11

12 buckling case it still moves up. This means that a buckled plate can still have some capacity left after initial buckling, also called the post buckling capacity. See also figure 1.4 for a schematic example of a load-displacement diagram when buckling occurs. Load Primary load path Load Primary load path Secondary load path Secondary load path Bifurcation point Elastic buckling load Bifurcation point Elastic buckling load Displacement Displacement Figure 1.4: Load-displacement diagram for elastic buckling (left: plate, right: column) There are many ways to determine the theoretical elastic buckling load. For instance: the direct equilibrium method, the virtual work method or via potential energy. These methods assume a perfectly straight structure, without any initial deformation or residual stresses. In practice however no structure is perfect. After fabrication, either by hot rolling or welding individual plates, every beam or column will have an ever so slight bow or twist. As a result of uneven cooling of the steel there will also be some residual stresses in the material. Both imperfections cause buckling to occur at a lower stress than the theoretical elastic buckling stress, so design formulas have been developed to quantify the influence of the imperfections. These formulas are mostly based on tests with imperfect members. 1.4 Verification methods When analyzing a steel structure preferably all stresses should be known exactly as they appear in the real structure. This way a structural engineer would be able to check for every location if the structure can carry the load. The problem with this method is that it is impossible to exactly calculate the stress distribution, due to all kinds of local effects around welds or corners which introduce a local stress peak. Since steel is a very ductile material these local stress peaks will not cause collapse. When locally the yield stress is reached, redistribution occurs until a state of equilibrium is reached. Even though a member under tension may already have local yielding at a small load, still the final capacity will be the cross-sectional area times the yield stress. In practice this ensures that beams and columns can be modeled by line elements. When the structural forces (bending moment, normal and shear force) are calculated the most severely loaded cross-sections are known. These cross-sectional forces can now be verified to the cross-sectional capacities, which can be determined either elastic or plastic. In case of a cross-section built up from slender plates, the cross-sectional capacity is lower than the expected elastic or plastic capacity, due to local plate buckling. An extensive (finite element) calculation can be carried out to find this local plate buckling capacity, as shown later in this thesis, 12

13 but for daily practice this takes too much time. Also in the time when finite element calculations had not been developed yet, plate buckling capacities where needed. Therefore methods have been developed to reduce the cross-sectional capacity to account for local plate buckling. These generally work by determining a reduction factor, which can then be applied over either the steel area or the yield stress. In chapter three these methods are analyzed. 1.5 Formulation of problem The problem researched in this thesis is the difference in the methods for determining plate buckling capacities between NEN 6771 and NEN-EN A preliminary calculation showed that the Eurocode can give a much higher capacity, in some cases up to several dozens of percent. The goal of this research is to find out why this difference exists and if it is correct. In chapter 2 an inventory is made of the different methods available in both design codes and where they originate from. In chapter 3 it is investigated what the results are for some example cross-sections, to get a view of the difference in capacity in commonly used cross-sections. The fourth chapter deals with verifying the calculations according the codes with a finite element calculation made in Ansys. Within the Eurocode, which is required in the Netherlands nowadays, a user can choose different methods to calculate the plate buckling capacity of a cross-section. These give different values for the capacity and have a different workload for the structural engineer. This thesis will give a recommendation what method to choose in what situation, regarding both capacity and workload. The methods compared are chapter 4 and 10 of NEN-EN and chapter 13 of NEN The last main subject is the requirements in the design codes for plate edges. In both NEN and Eurocode many requirements are given regarding stiffeners. A small sized stiffener will either not be allowed or give very large reductions to the adjacent plate parts. A plate edge on the other hand is considered as a perfect hinged support to a plate, even if this edge is formed by a very small flange. This problem is addressed in chapter 5 using finite element calculations in Ansys. 13

14 Chapter 2: Verification methods Many different methods to determine the plate buckling capacity of steel plate cross-sections exist. All these methods can be divided in two main groups: effective cross-section or effective width methods (Dutch: doorsnede-reductiemethodes) and reduced stress methods (Dutch: spanningsreductiemethodes). Within a method there still is a choice in the reduction factor to be used. This chapter starts with an inventory of the different methods, before investigating how these differences influence the calculated capacity of a steel cross-section. 2.1 Effective cross-section vs. reduced stress method Assumptions The effective cross-section method and the reduced stress method have a fundamental difference in the approach for determining an approximation for the real stress distribution. The reduced stress method checks at which stress level a plate part buckles, if a cross-section is built up from multiple plate parts the lowest stress is governing for the entire cross-section. When the stress does not exceed the critical stress, class 3 section properties can be assumed. The usual cross-section verifications can be used, only with the yield stress multiplied by a reduction factor. The effective cross-section method reduces the area of the cross-section in the parts affected by plate buckling. This method was first introduced by von Kármán (1932), see also chapter The biggest difference with the reduced stress method is that the method assumes a buckled plate part remains at capacity, so other plate-parts can also reach the buckled state. Due to this load shedding there is in general a higher plate buckling capacity in a cross-section than when using the reduced stress method. This effect is best visible in cross-sections with large differences in slenderness between the individual plate parts. Figure 2.1 shows the assumed stress-distribution for an I-shaped cross-section loaded by a bending moment for both discussed methods. d = f y;d d = buck;d f y;d e N d < f y;d Effective cross-section method Reduced stress method Figure 2.1: Assumed stress distributions 14

15 2.1.2 Procedure In general the procedures for the effective cross-section and the reduced stress method have similarities, but also some key differences. The short lists of fundamental steps below clarify these differences. Effective cross-section method: - Divide cross-section in plate parts - Calculate reduction factor for each part - Multiply area of each plate part with its own reduction factor - Use effective part of cross-section in verifications Reduced stress method: - Divide cross-section in plate parts - Calculate reduction factor for each part - Lowest factor is governing - Design stress equals governing factor times yield stress - Use design stress in cross-section verifications It is now clear that the fundamental difference between the two methods is the assumption whether the cross-sectional capacity is reached when the first plate part buckles or not. In the effective crosssection method load shedding is allowed, until all plate parts reach the buckled state. An advantage of using this load shedding effect in the cross-section is that in general a higher capacity is calculated. This capacity will be a more realistic value, since load shedding does occur in cross-sections and buckled plate parts maintain their carrying capacity when buckled. Further attention to this phenomenon is paid in chapter 5.4. A disadvantage is the fact that when using an effective crosssection method one has to recalculate the cross-sectional properties several times. When using a frame program different models have to be made for different checks. In both situations this brings extra workload to the engineer. In daily practice a good procedure might be that a design is made using the reduced stress method, taking advantage of the quicker calculation. If at some locations the design has an insufficient local buckling capacity, instead of adding material a more refined calculation could be made using the effective cross-section method Difference in normal force capacity For a single unstiffened plate the effective cross-section and the reduced stress method should in principle give the same capacity, as long as the reduction factor used is equal. For a load case of pure compression this is easily calculated. The reduced stress method will multiply the reduction factor with the yield stress, while the effective width method multiplies it with the width of the plate. These procedures lead to the same plate buckling load: F buckling = ( f y ) b t = f y ( b) t or: F buckling = d b t = f y A eff Difference in bending moment capacity In a pure bending load case the effective width method will give a slightly higher capacity. The assumed stress distribution (see figure 2.1) is closer to the real stress distribution, which gives a higher capacity. Also the shift of the neutral axis is taken into account. Since the plate now has a gap 15

16 Percentage Master Thesis Plate buckling in design codes in it, no simple formulas can be used as was the case with pure compression. First the location of the neutral axis has to be determined using the first moment of area of the two parts. To calculate the second moment of area with regard to the neutral axis, the rule of Steiner has to be used. These more complicated formulas have been put in a spreadsheet, to plot the difference in capacity. Figure 2.2 shows that when a reduction factor of around 0.6 is applicable, the effective width method gives about 10% more capacity than the reduced stress method using the same reduction factor ,5 0,6 0,7 0,8 0,9 1,0 [-] Figure 2.2: Extra capacity when using effective cross-section method Keep in mind that this difference in capacity for a plate in bending is only due to the assumed stress distribution as shown in figure 2.1, and completely independent of the reduction factor. The reduction factor itself has a much larger influence, which is shown in chapter Conclusion In chapter 2.1 a comparison has been made between the effective cross-section and the reduced stress method. The reduction factor has a large influence on the ultimate capacity, but in this comparison an equal reduction factor for both methods is used, to clarify the difference caused by the assumed stress distribution in the buckled state. For a single plate loaded in compression the buckling load will be identical. When bending is regarded the effective cross-section gives a slightly higher capacity. In a practical situation a plate loaded in bending will be a web in a girder. Since the web only has a small contribution in the bending moment of an I or box section the total bending moment capacity of the cross-section will hardly be influenced by this effect. An effect which does influence the capacity of a cross-section quite a lot is that when using the effective cross-section method a buckled plate part remains at capacity when further loaded. The cross-sectional capacity is no longer governed by the weakest link. This plays a larger role as the difference in plate slenderness increases, but nevertheless the reduced stress method will always give a lower or equal buckling load. 16

17 2.2 Reduction factors Both the effective cross-section and the reduced stress method cannot be used without a way to determine the reduction factor. In history many different forms of reduction factors have been developed. In this chapter these are presented and their origin explained. NEN 6771 uses a formula based on the critical Euler buckling stress, while NEN-EN uses a formula first introduced by Winter, which is based on the effective width theory by von Kármán. These formulas all calculate the reduction factor ( ) using the relative slenderness ( rel ) of the plate. The relative slenderness is defined using only geometrical and material properties: plate slenderness specific slenderness relative slenderness Euler The critical Euler plate buckling stress ( ) is the starting point for every method, since it is used to determine the relative plate slenderness. It is the stress level at which buckling occurs in a perfect plate. No imperfections are taken into account, and linear elastic material is assumed. The critical stress can be derived in many ways, for instance via direct equilibrium equations or a virtual work method. In this thesis a potential energy method is used. The definition of the plate, its properties, relevant axis and displacements can be found in the following figure. n xx x y E,, t b z w y v a z w x u u Figure 2.3: Definition of a single plate 17

18 The most general form of the potential energy equation is: In other words, the elastic potential energy is the half of strain times stress, summed up over every direction and integrated over the entire structure. This equals to the area under a linear stress-strain curve. The potential energy lost by the load is equal to the work done by the load, which is force times displacement. For a minimum of potential energy for every δu it should hold that: In this case the elastic potential energy is due to bending of the plate. For a very small plate part with dimensions dx and dy loaded by distributed bending moments along its edges the following holds for bending in x, bending in y and twisting: dx x y dy z m xx For relating moment to curvature these expressions are available from general plate theory: Adding the three components, integrating over the area of the plate and substituting the moments we get the total elastic potential energy of the plate: 18

19 The potential energy lost by the load is given by: The next step is to assume a displaced shape due to the buckling of the plate. One half sine wave in y direction and m half waves in x direction are used. This equates to: Using this displaced shape implies that a must be larger than b, i.e. a large aspect ratio. Because the buckles are perfectly sinusoidal, only thin plates are covered. All necessary information for calculating the buckling load is present. The next step is to obtain the partial differentials and substitute them in the potential energy equations. Substituting and simplifying gives elastic potential energy: 19

20 Substituting and simplifying gives potential energy in the load: This makes the total potential energy in the structure: There is a minimum of potential energy when the first order variation is set to zero: Since the second order variation will lead to the same equation as above, the equilibrium is classified as indifferent. Further elaboration gives the critical plate buckling stress: This expression for the critical elastic buckling stress can be applied for all types of plates. The factor k is called buckling factor, and is used to take into account the influence of the load case, support conditions and aspect ratio. The here derived buckling factor is valid for a simply supported plate, loaded by pure compression. The influence of the aspect ratio can be found in figure

21 k [-] Master Thesis Plate buckling in design codes m=2 m=1 m=3 m=4 m=5 0,0 1,0 2,0 3,0 4,0 5,0 a [-] Figure 2.4: Influence of aspect ratio on buckling factor In many design codes the aspect ratio is not taken into account in the plate buckling calculations. For a simply supported plate in compression it is taken as 4.0, since the error made is relatively small and on the safe side. In the table below the standard buckling factors for more situations are given. Internal element Outstand element Compression Bending (compression at support) 0.85 (tension at support) Table 2.1: Buckling factors The critical buckling stress can directly be used in a reduced stress method. In some codes however it is preferred to give a formula for a reduction factor expressed in the relative plate slenderness. In the case of the Euler critical stress this leads to: This reduction factor 1/ 2 is widely known as the Euler hyperbola Von Kármán Many researchers showed that the actual ultimate carrying capacity was much larger than the critical load. This was most obvious with the more slender plates. Stockier plates have an ultimate load based on yielding, which can be smaller than the critical plate buckling load. Slender plates on the other hand showed an extra post buckling capacity after passing the initial buckling load. 21

22 The elastic critical buckling stress in is derived using the assumption that the stress distribution stays linear. What actually happens in a buckled plate is stress redistribution away from the buckled part. The real stress distribution is a complex combined action of membrane stress induced by the load, bending stress due to the buckled shape and shear stress because of the rotation at the corners of the plate. There have been developed methods for analyzing a plate with this approach, for simple design purposes however these are too advanced to use. In 1932 Theodore von Kármán introduced the concept of effective width. He stated that (at a given thickness) a fictitious plate with the width of b eff would have the critical stress equal to the yield stress. If the actual plate has larger width, the capacity would be the same as that of the fictitious plate. In a plate the real stress distribution is approximated, or replaced, with two strips which describe the load carrying effective width of the plate. a b b eff /2 b eff /2 cr x f y Figure 2.5: Stress distribution after buckling, the von Kármán assumption for effective width. With the above assumed stress distribution the reduction factor can be derived by replacing the critical stress with the yield stress and the actual width with the effective width. Although von Kármáns method gained reputation as a reliable way to determine the ultimate load of a plate, the method was still based on plates without initial imperfections. When compared to test results it was only valid for large b/t ratios, where initial imperfections have a smaller influence. However, von Kármán still is the first researcher to propose a reduction factor function and has had a large influence on all simplified design methods for plate buckling. 22

23 2.2.3 Winter Since no manufactured plate is without initial deformation and residual stresses many researchers worked at adapting von Kármáns formula. The function introduced by George Winter in 1947 is one of the more widely spreads in design codes. Winter conducted many experiments on cold formed steel sections and came up with: The first suggestion was made with a coefficient of 0.25, but was later changed to the 0.22 used now. It is interesting to see how small the difference is between Winters experimentally based function and the original theoretical function by von Kármán. Even though much research has been conducted since 1947, and the fact that this formula is based only on cold formed sections, it is still the most widely used nowadays NEN 6771 In the Dutch NEN code the main method uses formulas based on the Euler critical buckling stress. To take into account the influence of imperfections on the buckling capacity there is a reduction with respect to the Euler hyperbola in the relative slenderness area 0.7 < rel < Since the ultimate load bearing capacity is higher than the critical load, post critical strength is allowed where < rel < 2.5. At a larger relative slenderness the Euler hyperbola is used again. The reduction factor is: = 1.0 if: 0 < rel 0.7 = rel if: 0.7 < rel < = if: rel 2.5 = if: 2.5 < rel Taking in account the post critical strength is not allowed if one of the longitudinal edges is supported by a stiffener. The rules for stiffeners do not take this enlarged load into account. Even if they would the philosophy is that what will be gained in the plate field by allowing post critical strength is lost in dimensioning the stiffeners to this larger load. So for plates supported by longitudinal stiffeners the reduction factors are: = 1.0 if: 0 < rel 0.7 = rel if: 0.7 < rel = if: < rel NEN 6771 chapter 13 was developed in the years 1978 until 1985, as a successor of the VOSB (Voorschrift Ontwerp Stalen Bruggen, or Regulation Design Steel Bridges). At the time of the development the formula by Winter, with the higher ultimate load, was already known. Nevertheless 23

24 for the main method there has been chosen to use formulas based on critical load, because the NEN 6771 was mainly intended for steel bridges. These designs would be governed by fatigue, so a refined way of calculating plate buckling would not give more economical designs, since it is not the governing criterion. To prevent being too conservative the small amount of post critical strength was added. For structures without fatigue problems the method by Winter wás added to the code, however more modestly (art ) between other sub articles and not as the main method in a separate chapter (art. 13) NEN-EN In the Eurocode the formulas by Winter have been used. A small adjustment to the original formula has been made, to make it dependent of, the stress distribution in the plate. This factor is determined by dividing the smallest stress in the plate by the largest, while taking compressive stresses as positive. For a plate loaded in compression equals unity and the formula is identical to the original by Winter. A plate loaded by bending, 1, has a slightly higher capacity. For plate parts with a free longitudinal edge (outstand element) the reduction factor is: Using the ultimate load according Winter in bridge structures could give problems regarding fatigue. In the normal fatigue verifications the peak stresses are higher because the steel plates will be thinner. In proper fatigue verifications it should become clear whether this is a problem or not. These calculations however do not take into account the deformations out-of-plane of a steel plate. At for instance a web-flange connection in a variably loaded I beam there is high fatigue sensitivity in the welds if the web needs the buckled mode to take up the maximum loads. This phenomenon is called web breathing and NEN-EN takes it into account using: If only normal stresses are present it is clear that they are allowed to be ten percent higher than the critical stress. Or expressed in a reduction factor: x,ed,ser 1.1 cr = f y 24

25 2.2.6 Overview Multiple reductions factors have been discussed, but the best way to compare them is graphically. In figure 2.6 the black lines represent the base formulas, which disregard any imperfection. The blue lines represent the different reduction factors in NEN-EN and the red lines are for NEN 6771 both with and without post critical strength. In NEN 6771 there is a correction factor 1/C, which can slightly adjust the reduction factor (higher or lower) in some cases, but his is not shown in the graph. [-] 1,2 1,1 1,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 [-] 3,5 Figure 2.6: The various discussed reduction factors Von Karman (1/λ) Euler (1/λ²) EN ; Ψ = 1 EN ; Ψ = -1 Web Breathing NEN 6771; incl. NEN 6771; excl. In figure 2.7 it can be found that the largest difference in reduction factor between NEN-EN and NEN 6771 occurs around rel = 1.9. For a plate in pure compression Eurocode will give about 30 percent extra capacity, for a plate in bending this is around 38 percent. Eurocode / NEN [-] 1,40 1,35 1,30 1,25 1,20 1,15 1,10 Ψ = 1 Ψ = 0 1,05 Ψ = -1 1,00 0,5 1,0 1,5 2,0 2,5 0,95 rel [-] Figure 2.7: Extra capacity in Eurocode compared to NEN 25

26 2.3 Verification methods in design codes Inventory of methods To calculate the ultimate bearing capacity of a plate field one needs to choose whether to use the effective cross-section or the reduced stress method. Besides that, a choice must be made between all the different reduction factors. Not all design codes have chosen to use the same method as their default method. As shown in chapter in NEN 6771 the default method is the reduced stress method, with the reduction factors based on the critical stress according Euler. Even though the effective cross-section method and reduction factors according Winter are allowed, they are not prominently presented. In NEN-EN again both reduced stress and effective cross-section method are allowed. Since the effective cross-section method is presented in chapter 4, called plate buckling, it is clear this is the preferred method. The reduced stress method is presented in chapter 10, which consists of only two pages. Both these methods however work with the reduction factors according Winter, as discussed in chapter As shown in figure 2.7 this can lead to much extra capacity compared to the stress reduction method in NEN Since deformation out-of-plane can become a problem using these high capacities the web breathing article is introduced for variably loaded structures. At first hand the stress reduction methods in both design codes look completely different. In NEN 6771 a buckling stress is determined, which is then used instead of the yield stress in the usual crosssection verifications. In NEN-EN the complete stress field of the entire structure is verified in a single formula. Two values have to be determined (a ult and a crit ), which is usually done by a computer calculation. The big advantage of this second method is the speed of the calculation, provided a computer program is available to calculate the a values. If however the equivalent stress or von Mises criterion is used, the simple cross-section verifications can still be deducted NEN 6771 In article of NEN 6771 the most important verification for a plate loaded in compression is given. The formulas (note that plooi is Dutch for plate buckling) in this article combined state: main formula Where is equal to the reduction factor as presented in chapter of this thesis and the correction factor C is determined according articles through of NEN

27 2.3.3 NEN-EN Article 4 of NEN-EN contains the effective cross-section method. In article 4.4 the general formula for effective area is given: Where is the reduction factor for plate buckling, as presented in chapter of this thesis and A c is the steel area in compression. However there is slightly different notation in the Eurocode formulas. The relative plate slenderness, rel, is referred to as, not to be confused with the plate slenderness p. This is due to the fact that in NEN-EN the relative slenderness is defined as, which could be for any kind of element. To emphasize NEN-EN works with plates the index p is added. Compare also to the definition of the specific, plate and relative slenderness at the start of chapter 2.2. The relative plate slenderness used in the Eurocode is simplified to: When the reduction factor is determined it depends on the stress distribution in the plate how it must be applied. Table 2.2 and 2.3 show how Eurocode defines the effective width in different situations. Table 2.2: Internal compression elements 27

28 Table 2.3: Outstand compression elements These layouts of effective areas applied to an I-shaped beam will give the effective cross-sections as in figure 2.8. The parts marked in black are ineffective. The white parts can be used for crosssectional verifications or other structural calculations. d = f y;d d = f y;d Loaded in bending d < f y;d Loaded in compression Figure 2.8: Effective area of an I-shaped cross-section 28

29 In article 10 of NEN-EN the reduced stress method is described. The main formula which governs the plate buckling effect is given as: In which is the minimum load amplifier for the design loads to reach the characteristic value of resistance of the most critical point of the plate. Or in other words, it is the yield stress divided by the largest stress present, since resistance is governed by yielding and the structure is supposed to behave linear elastic without plate buckling. The reduction factor is determined with the same set of formulas as in the effective cross-section method, only now the relative slenderness is defined as: Where is the minimum load amplifier for the design load to reach the critical load of the plate under the complete stress field. Calculating for simple plate structures can be done by hand, but for large structure a finite element buckling (eigen value) analysis is ideal. Since plate buckling in real structures is not equal to the critical load, the extra step through the equations by Winter is made, to take into account imperfections and post buckling strength. In a previous chapter it was already stated that this method with all stresses combined is analogue to the method where separate cross-section verifications are used. This can be shown for any situation using the von Mises criterion, but for simplicity we can also assume stress is present in only one direction. This state can be used for pure bending or compression situations, where only normal stresses are present. The verification formula now becomes: For the reduction factor the following derivation can be made: The relative slenderness is determined in the same manner as in the other methods, and the reduction factor is multiplied with the yield stress to give a maximum allowable stress. Except for the formulas relating and (Euler vs. Winter) there is no difference in the reduced stress methods given in NEN 6771 and NEN-EN Step by step guide In table 2.4 a step by step guide is presented, to give a guideline and overview of the methods described in the last two subsections. Each step identifies where in the relevant code it can be found. 29

30 NEN 6771 Reduced stress method 1 Divide the cross-section in separate plate parts. NEN-EN Reduced stress method Divide the cross-section in separate plate parts. NEN-EN Effective cross-section method Divide the cross-section in separate plate parts. 2 Determine Determine Determine for each compressed flange 3 Determine k ;x Determine k Determine k [fig ] [table ] [table ] 4a Determine E [eq ] 4b Determine i;k;x [eq ] 4c Determine plaat;rel Determine p Determine p [eq ] [art ] [art ] 5 Determine plooi;rel Determine Determine [art ] [art ] [art ] 6a Determine whether Determine whether interaction with column interaction with column buckling is present buckling is present [art ] [art ] 6b Determine C ;x [art ] 7 Determine plooi;d with the smallest plooi;rel [eq a] 8 Determine elastic crosssectional properties (A tot and W el ) Determine max with the smallest [art. 10.5] Determine elastic crosssectional properties (A tot and W el ) Determine capacity (using A tot, W el and max ) Determine the effective area of the flanges [table ] Determine elastic crosssectional properties with effective flanges Determine in web 9 Determine capacity (using A tot, W el and plooi;d ) 10 Determine k, p and (web only) [tabel ] and [art ] 11 Determine whether interaction with column buckling is present [art ] 12 Determine the effective area of the web [table ] 13 Determine effective cross-sectional properties (A eff and W el;eff ) 14 Determine capacitity (using A eff, W el;eff and f y;d ) Table 2.4: Step by step guide for codified plate buckling design Remarks The next steps will have to be repeated for each individual plate part. = 2 / 1 Eurocode table is a little more straightforward. Eurocode takes a more direct approach. EC gives choice from 2 formulas, NEN in total 5. Only for small a Choose from 4 situations. Using the effective cross-section method in NEN-EN there are another two options. Either a single effective cross-section is determined for the actual value of, or both for bending and normal force a separate effective cross-section is calculated. See also the flow chart on the next page. 30

31 Start Effective cross-section method Ch. 4 up to 7 Stress reduction method Ch. 10 A eff for normal force verification ( =1) Single effective crosssection for bending moment and normal force Determine stresses in structure (1 st order, linear elastic) W eff for bending moment verification ( =-1) Determine with actual acting forces Formula 10.1 satisfied no Has e N changed? yes yes Increase cross-sections yes A eff = lim A gross is lim < 0.5? no no Verification according NEN-EN using class 3 properties Use effective crosssections in frame program Use gross crosssections in frame program Choose verification method Determine forces. 1 st order (geometrical linear) Determine forces. 2 nd order (GNL incl. imperfections) Verification of crosssections and (lateral torsional) buckling: NEN-EN using A eff, I eff, W eff Verification of crosssections: NEN-EN Art. 4.6 en 7 Table 2.5: Flow chart for plate buckling in NEN-EN

32 As a part of this thesis the effective cross-section method in NEN-EN has been implemented in a Mathcad sheet. As long as the geometry of a cross-section is as assumed in the sheet, the plate buckling capacity can be calculated. In annex D.1 one of the sheets is given as an example. After a detailed analysis of the different methods studied in this chapter some remarks can be made: - The reduced stress method in NEN 6771 and NEN-EN are very similar. Eurocode has a more compact way of formulating the equations, but the only major difference is in the use of the reduction factor. This can be the only explanation for any deviation between plate buckling capacities. - For more elaborate structures the method in NEN-EN will give an advantage in efficiency, provided that software is used which calculates and. - The effective cross-section method will generally give a higher workload for the engineer, but also more cross-sectional capacity. The workload is caused by the fact that first the effective flanges need to be determined, before finding the stress distribution in the web. This makes that the cross-sectional properties have to be determined several times. The higher capacity is mainly due the fact that each plate part is multiplied with its own reduction factor, instead of choosing the lowest as governing for the entire cross-section. - The forces in a statically undetermined structure are depending on the stiffness of the members. If these members are built up from plates, plate buckling may reduce the stiffness. When calculating with the effective cross-section method this effect is taken into account. Displacements will generally be larger and better predict the behavior of the real structure. A disadvantage is that the iterative procedure which must be made can be quite time consuming. In NEN-EN it is however stated this is only relevant when < lim = 0.5, where lim can also be given in the National Annex. 32

33 M buckling [knm] Master Thesis Plate buckling in design codes Bending moment capacity differences Different calculation methods will give different plate buckling capacities. Figure 2.9 gives the bending moment capacity of a plate with an increasing width, b. The thickness is kept constant and the length of the field, a, is supposed to be long enough not to have an influence in the buckling capacity. These properties give a quadratic increasing elastic capacity, since M el,d = ⅙ f y,d t b 2. Since Winters formula is inversely proportional to the relative slenderness, the capacity calculated by the Eurocode increases linear after initial buckling. The critical stress declines quadratic with the relative slenderness, combined with a quadratic increase of moment of area gives a constant plate buckling bending moment. In this example the thickness is chosen to be 10 mm, the yield stress is 235 N/mm 2. The width of the plate field starts at 0 mm or rel = 0.0, while it stops at 3600 mm or rel = 2.6. As a check if the graph is correct a plate with relative slenderness rel = 2.0 is calculated manually. In annex A.1 this calculation can be found. b 2000 a t, f y x Effective cross-section methods EC: Reduced stress method EC: Web breathing NEN: Red. stress method, with post critical strenght NEN: Red. stress method, no post critical strengh Elastic capacity kn 1429 kn kn ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 rel [-] Figure 2.9: Bending moment capacity for several methods It is clear that especially for more slender plates the difference in capacity can be very large. The difference between the two dotted lines is exactly 10%, which can be related to the factor 1,1 found 33

34 M d,ec / M d,nen [-] Master Thesis Plate buckling in design codes in the web breathing article. The penalty for having a variable loaded structure is quite steep, the capacity falls back from the green line to the dotted red line. Part of this thesis is about understanding why NEN-EN gives more plate buckling capacity then NEN Therefore it is interesting to know how big the difference is in the first place. In figure 2.10 the capacities of a plate in bending calculated by Eurocode are divided by the capacity according NEN This relates to figure 2.9 as the difference between the blue line on the one hand and the red and green on the other. 1,8 1,7 1,6 1,5 1,4 1,3 1,2 1,1 EC: Effective cross-section method EC: Reduced stress method 1,0 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 rel [-] Figure 2.10: Capacity of Eurocode relative to NEN 6771 (plate in bending) The only difference between both reduced stress methods is due to the choice in reduction factor, therefore the dotted line in figure 2.10 is identical to the graph for = -1 in figure 2.7, where only the reduction factors were compared. The solid line in figure 2.10 shows that an NEN-EN calculation can give up to 60% extra capacity compared to a NEN 6771 calculation. The difference is caused by using the effective crosssection method instead of the reduced stress method and again the different reduction factor. Multiplying the graphs in figure 2.2 and 2.7 therefore gives the same result as in figure 2.10, which was made by dividing the graphs shown in figure

35 F buckling [kn] Master Thesis Plate buckling in design codes Normal force capacity differences While in the last subsection the bending moment capacities where compared, in this section the same scheme is used to compare the normal force capacity. Again the thickness is kept constant, while plate width b is increased and length a is long enough not to have influence in the plate buckling capacity. Since the normal force is calculated by N el = f y t b, the elastic capacity shows a linear increase. The effective cross-section methods approach a constant value, which is limited by the effective width the chosen thickness can support, as shown first by von Kármán. The larger the slenderness, the less influence initial imperfections have and the closer the capacity approaches the theoretical capacity by von Kármán. The critical buckling load used in the other methods declines as the slenderness becomes larger. The critical stress decreases quadratic, while the area increases only linear. Even though more steel area is present, the buckling load becomes smaller. In this example the thickness is chosen to be 24 mm, the yield stress is 235 N/mm 2. The width of the plate field increases from zero to 4800 mm or rel = 3,6. As a check if the graph is correct a plate with relative slenderness rel = 2.0 is calculated manually. In annex A.2 this calculation can be found. b a t, f y x Effective cross-section methods and EC red.str.meth. EC: Web breathing NEN: Red. stress method, with post critical strenght NEN: Red. stress method, no post critical strenght Elastic capacity kn 5289 kn ,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0 3,2 3,4 3,6 rel [-] Figure 2.11: Bending moment capacity for several methods Since both reduced stress method and effective cross-section method will give the same result for a compressed plate, the red line in figure 2.11 represents a total of three of the discussed methods. These are both methods in NEN-EN and the effective cross-section method in NEN

36 F d,ec3 / F d,nen [-] Master Thesis Plate buckling in design codes 1,35 1,30 1,25 1,20 1,15 1,10 1,05 1,00 0,95 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 rel [-] Figure 2.12: Capacity of Eurocode relative to NEN 6771 (plate in compression) The graph in figure 2.12 is obtain by dividing the red line from figure 2.11 by the blue line, and represents the extra capacity calculated by NEN-EN compared to NEN Since the only difference is in the choice of reduction factor, the graph is again identical to the one in figure 2.7. In the area around 3.0 the buckling load according Eurocode is almost 2,5 times higher than what NEN predicts. This deviation might seem extremely high but can be explained very well using the contents of this chapter. 36

37 2.3.7 Conclusion The differences between NEN 6771 chapter 13 and NEN-EN chapter 4 as found in this chapter can be summed up to: - Eurocode is based on the ultimate load according Winter, NEN uses formulas based on the critical load. - This choice can give up to dozens of percent extra capacity when using the Eurocode. - When drafting NEN 6771 the choice of using initial buckling theory was deliberately made, since the code was meant for bridge design. These are loaded variably and therefore sensitive for fatigue, which eliminates the need for a refined plate buckling analysis. - When designing bridges according to NEN-EN the buckling stress drops very fast after around rel 1.3, so the most economic structures will have members with plates below this slenderness. Fundamental differences between the reduced stress and the effective cross-section method: - Using the reduced stress method the weakest link in the cross-section is governing for the maximum allowable stress. - Using the effective cross-section method a buckled plate part will remain at capacity when deformed, so other parts can also be loaded up to buckling. This distributing of the load to the stronger parts is called load shedding. - Load shedding (and with it the difference between reduced stress and effective cross-section method) has more influence on the total capacity in cross-section with large differences in slenderness between individual plate parts. 37

38 Chapter 3: Design codes applied to example cross-sections This chapter examines what the impact of the different methods found in the previous chapter on the plate buckling capacity of some example cross-sections is. These examples are chosen such that they represent the order of magnitude used in large bridges in the Netherlands. Not just the capacity is investigated, but also the workload using a certain method and the influence of web-flange slenderness ratios. The last part is about the application of stiffeners. 3.1 Capacities according design codes In this subsection two example cross-sections are analyzed using both NEN 6771 and NEN-EN These sections are an I-shaped girder and box girder, the dimensions are found in figure 3.1. The length of the girders is assumed to be long enough not to be of influence to the plate buckling capacity. Global (or column) buckling is not taken into account. t f = 24 t f = 20 h w = 2000 t w = 15 h t = 2048 h w = 1000 t w = 15 h t = 1015 b f = 1000 b t = 1020 b f = 800 Figure 3.1: Dimensions of example cross-sections All dimensions in: [mm] The plate buckling capacities to be compared are for loading with a normal force, bending moment around the weak axis and around the strong axis. Furthermore the difference in ULS (ultimate limit state) and SLS (serviceability limit state) is distinguished. This implies that a total of 36 calculations is made, of which as example three can be found in the annex B.1. A capacity for the ULS is determined using the normal plate buckling calculations, for NEN 6771 this means the reduced stress method including post buckling strength. The calculation for the SLS according the Eurocode is made with the web breathing article. Since the capacity is severely reduced using this article, it might be the governing situation in a bridge structure, even though no load factors and only the frequent loads are used. To make a fair comparison with NEN 6771 the post buckling capacity has been disregarded. Using only the critical stress comes down to not allowing the web to buckle, and thus preventing breathing fatigue in the web-flange connection. 38

39 3.1.1 I-shaped girder In table 3.1 the results of the capacity of the I-shaped girder can be found. The workload is the total for all six calculations with the same design code. NEN 6771 Reduced stress method NEN-EN Reduced stress method NEN-EN Effective crosssection method SLS ULS SLS ULS SLS ULS Normal force [kn] Bending strong axis [knm] Bending weak axis [knm] Workload 4 pages 2.0 hours 4 pages 1.9 hours 5 pages 2.75 hours Table 3.1: Plate buckling capacities of I-shaped girder In the calculations according NEN 6771 no plate part was in the proper range to adopt post buckling strength. Therefore the capacities found for ULS are equal to those for the SLS. The SLS capacities for both Eurocode calculations are made with the web breathing article, and therefore the same. They have been added in both columns to make a fair comparison between the two methods. Remarkable is that some of the SLS capacities are higher than the ULS capacities. This is a result of the way the web breathing article is formulated. In this case it means that the chosen cross-sectional properties are such that web breathing is not governing for that load case. Even though it has been said several times that the effective cross-section method will give more capacity that the reduced stress method, when bending around the weak axis is regarded it is the other way around. Due to the presence of the web, the neutral line will hardly shift when a part of the flange becomes ineffective. Consequently the distance from neutral line to extreme fiber is still large, and yielding reached sooner. Figure 3.2 gives a graphical representation of this effect. Elastic capacity Reduced stress method Effective cross-section method Figure 3.2: Bending around the weak axis 39

40 3.1.2 Box girder Also for the box girder the hand calculations were carried out. The results can be found on the next page in table 3.2. For some example calculations see annex B.1. In contrast to here there are some small differences in the column for NEN 6771, which means there was a plate part which utilized post buckling capacity. Normal force [kn] Bending strong axis [knm] Bending weak axis [knm] Workload NEN 6771 Reduced stress method NEN-EN Reduced stress method NEN-EN Effective crosssection method SLS ULS SLS ULS SLS ULS pages 4 pages 1.75 hours 1.6 hours Table 3.2: Plate buckling capacities of box girder 5 pages 2.25 hours The workloads have been determined as if every calculation would have been the first one. Many times a relative slenderness or other parameter from a previous calculation could have been used, but to keep the comparison unbiased this is not exploited. Again some of the web breathing capacities are much higher than the normal plate buckling capacities. In bending the webs of the box girder are not sensitive to plate buckling. This made the final step in the Eurocode calculations much faster. Otherwise the second moment of area of a cross-section with a gap in the web had to be calculated, which can be time consuming. Even with this advantage the Eurocode calculations took more time to complete Capacity differences The absolute values of the calculated capacities differ pretty much, but are therefore hard to compare accurate. In table 3.3 four comparisons have been made, similar to situations an engineer might come across in practice. In each column can be found what happens to a capacity when the calculation is switched from an old method to a new one. These switches are: 1: Using article 10 of NEN-EN (reduced stress method) instead of NEN : Using article 4 of NEN-EN (effective cross-section method) instead of NEN : Using article 10 of NEN-EN instead of article 4. 4: Using the web breathing article of NEN-EN instead of NEN 6771 without post critical strength. 40

41 1: 2: 3: 4: Method 1: NEN 6771 NEN 6771 EC: reduced NEN (SLS) stress m. Method 2: EC: reduced stress m. EC: eff. crosssection m. EC: eff. crosssection m. EC (SLS) I-shaped girder: Normal force % % % % Bending strong axis % % % % Bending weak axis % % % % Box girder: Normal force % % % % Bending strong axis % % % % Bending weak axis % % % % Table 3.3: Difference in capacity When regarding the type of load the I-shaped beam is intended for (bending around the strong axis), the Eurocode gives around ten percent extra capacity. When using the Eurocode and choosing between the reduced stress and effective cross-section the difference in capacity is only 1.3 percent, which might not be worth the extra workload. The box girder loaded in compression has up to fifty percent extra capacity. Also choosing the effective cross-section method delivers almost fifteen percent extra capacity over the reduced stress method. Even though these comparisons confirm the large differences between NEN 6771 and NEN-EN , they are only valid for the example cross-sections found in figure 3.1 and not for all girders. In the next chapter the influence of the web slenderness is taken into account to get a more general image of the difference between the design codes. 41

42 M bucklling [knm] Master Thesis Plate buckling in design codes 3.2 Influence of web slenderness In chapter about the differences in calculated capacities of the two example profiles big discrepancies showed up. For a more general understanding the slenderness of thickness of the web is varied. Looking at the basic assumptions for the calculation methods this parameter should have a large influence I-shaped girder In this subsection the same example profile as shown in figure 3.1 is taken, only the thickness of the web plate is varied between 0 and 40 millimeter. For the I-shaped girder only the bending around the strong axis is taken, since this is what the cross-section is designed for. The results can be found in figure 3.3, where also the capacities from table 3.1 are found by checking t w = EC: Effective cross-section method EC: Reduced stress method EC: Web-Breathing NEN 6771 Elastic capacity t 40 w [mm] Figure 3.3: Bending moment capacity knm knm knm knm The most fundamental difference between the reduced stress method and the effective crosssection method is whether the weakest link is found to be governing, or that a buckled plate remains at capacity to give other plate parts the opportunity to also reach the buckling load. In the graph this is illustrated clearly, both stress reduction methods have a capacity reduced to zero as the web becomes thinner. The effective cross-section will always have the bending moment capacity of the effective flanges, even if there is no web. Of course this is not a realistic design situation, but is clearly illustrates the effect of the assumption. Where the reduction of the web is equal to that of the flange, the capacities of both Eurocode methods are almost equal, which is to be expected. The small difference is in the bending moment capacity of the web, but this is not much compared to the entire cross-section. 42

43 Method 2 / Method 1 Master Thesis Plate buckling in design codes Another clear conclusion from figure 3.3 is that web breathing only is of influence in slender webs. This effect is reinforced by the fact that only the frequent load level has to be checked against the web breathing capacity. The slope change of the graph for both reduced stress methods around t w = 15 is caused by the fact that above this web thickness the flange is governing. The maximum allowable stress is not increased after this point, so the only increase in bending moment is possible because the area of steel increases. Since the contribution of the web to the bending moment capacity is rather small the slope is not very steep. In the effective cross-section method it is never either the web or the flange which is governing for the capacity. The slope change around t w = 22 is because the reduction factor for the web has turned to unity. After this point again the increase in bending moment capacity is only due to increase in steel area. The difference between the methods can be found in figure 3.4, were two situations have been calculated. The first is using the reduced stress method according NEN-EN instead of NEN This extra capacity is always present since nowadays the Eurocode has replaced the NEN. The second line is for the choice of method within the Eurocode. This is a choice engineers must make when calculating the capacity of a structure. 1,6 1,5 1,4 1: NEN reduced stress method 2: EC reduced stress method 1: EC reduced stress method 2: EC effective cross-section method 1,3 1,2 1,1 1, t 35 w [mm] Figure 3.4: Difference in bending moment capacity In the spreadsheet file with which the graphs were produced other dimension of for instance the flanges can be substituted. The location of slope changes, slopes, capacities and differences all change, but there always is a point where difference between the Eurocode methods is almost zero. This happens when the slendernesses are such that web flange. If a simple reduced stress method calculation is preferred, this could be a good principle to still have a high capacity in the crosssection. 43

44 Method 2 / Method 1 F buclkling [kn] Master Thesis Plate buckling in design codes Box girder The same comparison as in has also been made for the normal force capacity of the box girder. The result is very similar and therefore only the two figures are shown, the same remarks as before still apply EC: Effective cross-section method EC: Reduced stress method EC: Web breathing NEN 6771 Elastic capacity kn kn kn kn t w [mm] Figure 3.5: Normal force capacity of box girder 1,6 1,5 1,4 1: NEN reduced stress method 2: EC reduced stress method 1: EC reduced stress method 2: EC effective cross-section method 1,3 1,2 1,1 1, t w [mm] Figure 3.6: Difference in normal force capacity Conclusion The most significant conclusion made in this chapter is that by wisely choosing your web-flange slenderness ratio a reduced stress method calculation can give a capacity similar to a more complicated effective cross-section method. 44

45 3.3 Stiffeners according design codes If a steel plate does not have enough capacity to carry the load, it is not unavoidable to increase the thickness of the plate. Adding one or more stiffeners will also increase the capacity of the plate. In many structures adding stiffeners is in the end the cheapest way to built. Even though stiffeners can cost quite a lot to fabricate, the structure itself will be lighter. Which option gives the most economical design cannot be known for sure in advance. In NEN 6771 longitudinal stiffeners are regarded as columns loaded in bending and compression. They are loaded with the plate buckling stress of the adjacent plates, and can either meet the requirements or not. If a stiffener does not meet the requirements the size must be increased, or the length decreased by adding extra transverse stiffeners. In NEN-EN another approach is taken. Depending on the dimensions of the stiffener an extra reduction factor c is determined and multiplied with the area of the neighboring plate parts. A stiffener and plate combination which is not allowed by NEN 6771 can still be analyzed with NEN-EN Since in NEN 6771 it is stated that loading until the full post critical strength of a plate gives such a load on the stiffener, the advantage won in the plate is immediately lost in the stiffener. Therefore we would expect that according Eurocode the stiffeners should be larger, to carry the full load. In three different situations this is investigated. First a given set of dimensions is used to determine the buckling capacity. In the second part it is investigated if a stiffener can be designed to fully make use of the plate buckling capacity by Winter. In the third part a certain design load is assumed and plates and stiffeners are found to match the load Capacity at given dimensions of plate and stiffener Longitudinal stiffeners are regarded as columns loaded in compression and bending in NEN The length of the plate now has a large influence on the final capacity of the plate and stiffener. The first situation investigated is the calculated capacity at some given dimensions. These dimensions can be found in figure mm Plate: t = 12 mm Stiffener: t = 12 mm h = 180 mm 1000 mm Steel grade: S mm Figure 3.7: Dimensions of plate and strip stiffener The plate and stiffener have been analyzed according both NEN 6771 and NEN-EN , of which an example can be found in annex B.2. The results are found in table 3.4 on the next page. 45

46 Buckling normal force [N c;u;d;total ];[kn] Area steel plate [A s;plate ];[mm 2 ] Area stiffener Total area [A s;lv ];[mm 2 ] [A s;total ];[mm 2 ] NEN EC Table 3.4: Normal force capacities of plate and stiffener Average stress [ cr ];[N/mm 2 ] In the calculation according NEN 6771 the unity check of the longitudinal stiffener was 0.74, which means that the stiffener is larger strictly necessary. In table 3.5 the results are found for a stiffener of mm. This gives a unity check of Since the plate is the same, the total capacity drops with N = A cr = = 11.6 kn. The capacity according the Eurocode has a larger decline, since not only the steel area decreases, but also the average stress decreases because the plate is less supported than before Buckling normal force [N c;u;d;total ];[kn] Area steel plate [A s;plate ];[mm 2 ] Area stiffener Total area [A s;lv ];[mm 2 ] [A s;total ];[mm 2 ] NEN EC Table 3.5: Normal force capacities of plate and stiffener Average stress [ cr ];[N/mm 2 ] From the last two tables it can be seen that the Eurocode calculates a higher capacity when a longitudinal stiffener is applied. However this is mainly due to the fact that plates itself already have a higher capacity when using Winters method Minimum stiffener dimensions for full Winter capacity The buckling capacity of the plate itself, calculated in in accordance with NEN-EN , is reduced because of the presence of the stiffener. The buckling load of the plate could be higher if the longitudinal stiffener would provide better support. If the stiffener would provide full support, the buckling load by Winter could be reached in the plate. This would be as much as = 4132 kn, even without the extra contribution of the area of the stiffener. The dimensions needed for the longitudinal stiffener can be deducted from article of NEN-EN To get perfect support, we set the reduction factor c to 1.0 and find rel;lv = 0.2. But to get the relative column slenderness this low we would need a stiffener measuring 800 by 60 millimeters. This would make the total steel area of the stiffener twice as large as that of the plate it is supposed to support. The design philosophy of a stiffener (adding a relatively small steel area to get a relatively large increase in buckling capacity) is completely gone. So when using rationally sized stiffeners, the maximum plate buckling load by Winter will never be reached. The conclusion to be drawn is that there is some truth in the philosophy (no post critical strength when using stiffeners) in NEN Indeed it is not possible to design a stiffener which enables a plate to reach the buckling capacity by Winter. In the stiffeners it is more than lost what is gained in the plate itself. Nevertheless Eurocode does give a higher capacity for a given set of dimensions. 46

47 When a longitudinal stiffener of mm is chosen the reduction factor is equal to = Still not an economical design, but it does reflect what is needed to guarantee near perfect support to the plate Buckling normal force [N c;u;d;total ];[kn] Area steel plate [A s;plate ];[mm 2 ] Area stiffener Total area [A s;lv ];[mm 2 ] [A s;total ];[mm 2 ] EC Table 3.6: Buckling capacity with very large stiffener Average stress [ cr ];[N/mm 2 ] The total buckling load is much larger than the previously calculated 4132 kn, which was for only the plate and perfect support. The area of the stiffener is loaded with the yield stress, because its dimensions make it not sensitive to buckling. Since the stiffener is still 40% of the total area this more than doubles the ultimate load. Compared to the stiffener of 170 by 12 millimeters the structure is now 1.6 times as heavy, but nearly 3 times as much load can be carried. This is reflected in the average stress which becomes 3 / 1.6 = 1.9 times as large. Even though this is quite an academic situation, it does reflect the possibilities by the Eurocode method. An interesting question is whether this 400 by 42 mm longitudinal stiffener really adds capacity to the structure. Therefore it is compared to a single unstiffened plate, with the same total area. The stiffener is as it were smeared out over the entire plate width. The thickness of this plate is / 2000 = 20.4 mm, and the width is 2000 mm. In table 3.7 the capacity can be found according both NEN and the Eurocode. t = 20.4 mm b = 2000 mm Buckling normal force [N c;u;d;total ];[kn] Average stress [ cr ];[N/mm 2 ] NEN EC Table 3.7: Buckling capacity of equivalent plate With the same amount of steel still 60% of the load can be carried. Taking into account the fabrication costs of a stiffener the second option will be more economic. Again the Eurocode will generate more capacity than NEN when using small stiffeners, but increasing the size will indeed cost more than is gained Plate and stiffener dimensions needed for given capacity So far a given set of dimensions was analyzed to compare the capacities in the design codes. In this subsection a total load of 5000 kn is assumed and dimensions to carry the load are sought. This order is more similar to what a structural engineer would come across, so it is a good way of comparing the design codes. The length and total width of the plate field are equal to before, but the plate thickness, number of stiffeners and their dimension are not set. To make an accurate comparison no standard sizes for plate thickness of stiffener are used. Figure 3.8 shows the dimensions of the plate. 47

48 1000 mm Plate: t = mm Stiffener: t = mm h = mm 1000 mm Steel grade: S mm Figure 3.8: Dimensions of plate and stiffener(s) Designing this structure using NEN 6771 the procedure is quite straight forward. First a plate thickness and number of longitudinal stiffeners has to be chosen, with this the critical stress can be calculated. Using the critical stress the dimensions of the stiffener(s) can be determined. The total capacity is the total area times the critical stress. If the capacity is not what was needed, the plate thickness can be adjusted and the calculation remade. Using NEN-EN first a plate thickness is chosen. By adjusting the size of the stiffener any capacity can be created. So if the goal is a fixed total capacity, like in this example, an infinite number of structures can be designed. This is in contrast to NEN 6771, where only a single structure is found. Due to the amount of formulas needed in the Eurocode calculation it is hard to figure out what the most economical design is. Design code Number of stiffeners Plate thickness [t];[mm] Dimensions of stiffener(s) [h st t st ];[mm mm] Steel area Relative steel area Ultimate load [N c;u;d ];[kn] Average stress [ b ];[N/mm 2 ] [A s ];[mm 2 ] NEN {= 0} NEN {= 3800} NEN {= 7220} EC {= 0} EC {= 2320} {= 4620} EC {= 5120} {= 7600} Table 3.8: Dimensions of plate stiffener needed for N u = 5000 kn In table 3.8 the results are shown for the plate with stiffeners and a length of 4000 mm. Since the length has a large influence in the buckling load of the stiffeners, the calculations are repeated with a length a = 2000 mm. The dimensions of the unstiffened plates do not changes, since the aspect ratio a has gone from two to one, which is of no influence on the buckling load. In the stiffened plates with an equal thickness having half the length also means having about half the stiffener area. In table 3.9 the results of this second calculation can be found. 48

49 Design code Number of stiffeners Plate thickness [t];[mm] Dimensions of stiffener(s) [h st t st ];[mm mm] Steel area Relative steel area Ultimate load [N c;u;d ];[kn] Average stress [ b ];[N/mm 2 ] [A s ];[mm 2 ] NEN {= 0 } NEN {= 1410} NEN {= 2700} EC {= 0} EC EC {= 1155} {= 2560} {= 6250} {= 14440} {= 2760} {= 5120} {= 7800} Table 3.9: Dimensions when a = 2000 mm By dividing the length of the plate field in half less steel is needed to carry the load of 5000 kn. This is also reflected in the average stress which is higher. If this will also be the most economical design is not certain, since now a transverse stiffener is needed halfway the original plate. In both situations discussed in this section NEN-EN gives a lighter design with higher average stresses. If however the structure has a variable load and web breathing should be taken into account, capacities similar to that of NEN 6771 are found. Where NEN 6771 automatically allows only reasonable structures, following the EC3 formulas exactly a plate thickness of zero millimeters can be chosen. In this case the slender plate is replaced by a stocky column which carries the entire load. However in reality the longitudinal stiffener needs lateral support by the plate, to support the weak direction. Since this is not checked in the Eurocode some common sense is needed while designing stiffened plate structures Conclusion Between NEN 6771 and NEN-EN there is a fundamental difference in the approach when calculating the capacity of stiffened plates. Where in NEN 6771 a stiffener might be too small to give support to the plate, EC3 just calculates the extra reduction and gives a capacity. This is because NEN 6771 uses a method with rigid stiffeners, where perfect support is needed, while NEN-EN uses a flexible stiffener method, which can always be used. The assumption was that a plate calculated according Eurocode needs a larger stiffener, because post critical strength is taken into account. When starting with an equal plate thickness this is true, but the larger stiffener and higher stress also give a higher design load. When designing a structure for a certain fixed load, the post critical stress used in EC3 ensures a lighter structure can be built. So even while the hypothesis is in principle true, in practical situations NEN is more conservative. 49

50 Chapter 4: Finite element calculations In the plate buckling methods discussed so far, only the ultimate capacity is calculated. Visualized in a load-displacement diagram this will give a horizontal line, since deformations are not known. Using a finite element calculation the entire behavior of the plate under a certain type of loading can be analyzed. In figure 4.1 a schematic view of the expected load-displacement diagrams is shown. Whether effects like plasticity, initial imperfections and second order effects are taken into account decides which graph is followed. The real capacity will be found if all the effects are put into the calculation. Linear elastic x Fx,d [kn] Second order elastic, no imperfections Second order elastic, with initial imperfections u x Ultimate load (Winter) Second order plastic Critical load (Euler) u x [mm] Figure 4.1: Load-displacement diagram In figure 4.2 the displacement out-of-plane is given. This illustrates the difference between columns and plate very well. When the critical load is reached and the displacement out-of-plane starts to set in, a plate still can increase the load and therefore it is stable in its buckled shape. A column on the other hand has almost no increase in capacity and the equilibrium will quickly become instable. The plate has a secondary method to stabilize itself, namely membrane stresses. A column can only generate bending stresses when it is deformed out-of-plane, which are not as affective in handling the second order effects. 50

51 Fx,d [kn] No imperfections Fx,d [kn] No imperfections Critical load With imperfections With imperfections u y [mm] u y [mm] Figure 4.2: Out-of-plane load-displacement diagram If a plate without initial imperfections is regarded, the critical load is clearly distinguished in the loaddisplacement diagram. When the critical load is reached the plate suddenly buckles which is seen as a sharp slope change of the graph. There is a clear distinction between the primary load path and the secondary, this point is called the bifurcation point. When however a plate with a certain amount of initial imperfection is analyzed this clear bifurcation point is not visible. The slope gradually changes and approaches the secondary behavior. If only an analysis of an imperfect plate is available it is quite hard to determine the critical load accurately. This is probably the background of the factor 1,1 found in the web breathing article. Since the critical load is not clearly found in an imperfect plate the committee drafting the Eurocode found the displacements at 10% above the real critical load was the start of web breathing problems. Comparing the ultimate plate buckling load by Winter with a finite element calculation is more straightforward. The ultimate load is the very top of the load-displacement diagram, while its location may be depending on the initial imperfections, it will at least always be clearly visible. 4.1 Objective This thesis so far was mostly about inventorying the plate buckling methods in the design codes which apply to the Netherlands. The differences have been calculated and where possible clarified by researching the background of the methods. However some questions have formed that could not be answered immediately. Three of these will be answered using finite element calculations using Ansys. The questions are: For various cross-section types the plate buckling capacity has been calculated according NEN 6771 as well as NEN How well do these capacities match with a more advanced finite element calculation? In the effective cross-section method it is assumed that a very slender plate will stay at its buckling capacity when further deformed, so other plate parts in the cross-section may also reach their buckling load. How correct is this assumption? The edge of a plate is always modeled as a hinged support, which stays perfectly straight during loading so membrane stresses can develop. A longitudinal stiffener which is designed 51

52 to give the same perfect support needs to be extremely large, chapter demonstrated this. Normally the edges of a plate are formed by other plates, for example the edges of a web plate in an I-shaped beam are formed by the flanges. This leads to the question whether a flange can actually give this assumed perfect support to the web. No requirements at all are given in the Eurocode and NEN for the edges of a plate. A flange with tiny dimensions would still be a perfect edge for the web plate even though it obviously cannot provide the out-of-plane support the web needs. The question to be answered by the finite elements calculations is: why are there no requirements given for plate edges or might they be implied in another article? 4.2 Procedure The answers to the above questions will be sought using the finite element software Ansys. This is a general purpose finite element package, with not only structural capabilities but also for instance magnetism, electricity, heat or flow problems can be addressed. For this thesis only the structural part is used. A big advantage of Ansys is the scripted method of giving input. For instance geometry can be put in parametric, so that when another thickness is analyzed this parameter is changed and the script run again. There is no need to manually redefine the elements, re-mesh the structure and reapply the boundaries and loads, since the script does this for you in a matter of seconds. A disadvantage is that it takes some experience to get the script right, but this does not weigh up to the advantage gained in having a correct script to investigate the influence of parameters. Defining and meshing a rectangular plate in Ansys can for instance be done like this: a=4000 (define parameters a and b, length and width of plate) b=1000 na=80 (define parameters na and nb, the number of elements in each direction) nb=20 k,1,0,0,0 (define four keypoints, k is the command for keypoint, followed by the k,2,b,0,0 keypoint number and the coordinates in x, y and z) k,3,b,a,0 k,4,0,a,0 a,1,2,3,4 ( a generate an area between the given keypoints) lsel,s,tan1,x,1 ( lsel selects lines, in this case those in x direction) lsel,a,tan1,x,-1 lesize,all,,,nb (divides all selected lines and associated area in the given amount) lsel,s,tan1,y,1 (the same for the y direction) lsel,a,tan1,y,-1 lesize,all,,,na amesh,all (automatically meshed all areas, using the latest defined element) Some complete Ansys scripts, stored as a *.log file, can be found in annex C. An accurate calculation of plate buckling loads needs two types of nonlinearities in the analysis. The two types are geometrical nonlinear (second order effects or equilibrium in displaced state) and physical nonlinear (plasticity). The plate itself needs to be modeled accurately so it can have bending stresses trough thickness, membrane stresses in the buckles and shear stress where it is twisted. 52

53 Since the finite element models can be quite complex, it is hard to find the errors in modeling when an unexpected result is obtained. Therefore the model is built up in steps, while checking if the results of each step are correct. This has the extra advantage that it is clear that for each characteristic of the plate what the effect on the load-displacement diagram is. In table 4.1 the different steps taken to build the final model can be found. In the results of each step are discussed briefly. Geometric behavior Physical behavior Imperfections Geometry and objective 1 linear linear no Single unstiffened plate [elastic behavior] 2 linear non-linear no Single unstiffened plate [plastic behavior] 3 non-linear linear no Single unstiffened plate [critical buckling load] 4 non-linear linear yes Single unstiffened plate [elastic buckling behavior] 5 non-linear non-linear yes Single unstiffened plate [elastic-plastic ultimate load and deformation capacity of very slender plates] 6 non-linear non-linear yes I-shaped and box girders [check of design code capacities] 7 non-linear non-linear yes I-shaped and box girders [check of plate edge requirements] Table 4.1: Step by step building of finite element model The elements used in the Ansys models are of the type SHELL181. This element type is the most suitable for the analysis of plate buckling, since it is the most general. The element has four nodes, each nodes is capable of all six displacements. These are the translations in the x, y and z direction and rotations around the x, y and z axis. Furthermore the equations governing the element are suitable for large displacements and strains as well as having geometrical and physical nonlinear capabilities. Having such a complete element increases the time needed for an analysis, so for many simple situations reduced elements have been made. In plate buckling however all these aspects play a role, so the most elaborate element is the only choice. Defining an element in a log file: t=10 et,1,shell181 R,1,t The first command line stores the thickness as a variable, the second line defines element type 1 as SHELL181 and the third line couple the thickness to element type 1. Figure 4.3: Left: SHELL181 geometry Right: defining an element The nonlinear material behavior can be modeled in several ways. These include linear or nonlinear hardening of softening during plasticity, or a fully nonlinear representation of the real stress strain diagram. In the calculation for this thesis plasticity is modeled with a constant yielding plateau, with 53

54 f y = 355 N/mm 2. A perfectly horizontal plateau can give problems with the convergence of the analysis, therefore after yielding the Young s modulus is E y = E/10000 = /10000 = 21 N/mm 2. Figure 4.4 shows the stress-strain diagram as given by Ansys, and the code to create it. Defining and plotting a material: mp,ex,1,2.1e5 mp,prxy,1,0.3 tb,biso,1 tbdata,1,fy,21 tbpl,biso,1 Figure 4.4: Left: stress-strain diagram in Ansys Right: defining and plotting a material Finite element calculations can generally be made in two ways, load controlled or displacement controlled. With the plate buckling problem we expect the load in the plate to drop after the ultimate load is reached. Using a fixed amount of displacement which is slowly added while monitoring the support reaction is the easiest way of getting a good record of the post critical behavior. The biggest advantage of a displacement controlled calculation is that the load does not have to decline after the ultimate capacity is reached, which ensures easier modeling and better convergence. The imperfections can be divided in two general parts, the initial deformation of a plate and its residual stresses, both resulting from the production process. In annex C to NEN-EN there is a guideline to add all imperfections as a single equivalent initial deformation. This is useful since deformations are easier to add then residual stresses. The imperfections should be identical to the plate buckling mode, which is a sinusoidal shape. According to the Eurocode the equivalent initial imperfection should be the smallest plate dimension (length or width) divided by 200. In Ansys an eigenvalue analysis is made to determine the shape of the imperfection, the original geometry can be updated using this shape to add an imperfection. Since the value of the imperfection used in annex C to NEN-EN is tuned to give a design load, the capacities calculated by Ansys can be compared to the capacity according Winter, because this method also determines a design load. The calculation has multiple non-linearities in it, therefore the convergence criteria of importance. Both an energy and a force based convergence criteria has been used, to ensure close approximation of the non-linear effects. 54

55 4.3 Verification of design code capacities Single plate in compression The most basic case is the single unstiffened plate field. In this first section a relatively stocky plate is analyzed. The dimensions of the plate are a*b*t = 2000*1000*20, which makes the plate sensitive to buckling, but with the critical load higher than the ultimate load (N Winter < N Euler ). All edges are simply supported, so the displacement of the edges in z-direction is set to zero. The nodes at y = 0 are supported in y-direction, the sum of support reactions of these nodes are taken as the normal force in the plate. The nodes at y = a are given a fixed displacement which is added in steps. In figure 4.5 all characteristics can also be found. The complete log-file of the Ansys calculation is found in annex C.1. y u y Dimensions: a = 2000 mm b = 1000 mm t = 20 mm z a b x Boundary conditions: all edges: dz = 0 y = 0: dy = 0 y = a: dy = 10 x = y = 0: dx = 0 Figure 4.5: Characteristics of unstiffened plate field Since this is the first finite element calculation the steps given in table 4.1 are briefly discussed. Later in this thesis only the final results are shown. In figure 4.10 the results of the five plate models are given to clarify the influence of the added characteristics. FE model 1 This is the linear elastic model, so the calculations in Ansys are easily verified. Even though calculating the elastic behavior is not something a finite element method is needed for, it is a good way of checking whether boundary conditions, loads, geometry and other modeling features are done correctly. Ansys: y = 0.105e10 N/m 2 Analytical: y = E = E ( l / l ) = (10 / 2000) = 1050 N/mm 2 Ansys: Analytical: N y;total = 0.210e08 N N y;total = y A s = = N Since no yielding is taken into account in the analysis the calculated stress can be higher than the yield stress of 355 N/mm 2. 55

56 FE model 2 This model does include plasticity, but without any imperfections or geometrical nonlinearities. Ansys: Analytical: N y;total = e7 N N y;total = f y A s = = N There is a very small difference in normal force, which is due to the Young s modulus not being zero when yielding and some numerical errors. FE model 3 Using an eigenvalue analysis the exact critical buckling load is calculated. A disadvantage is that using this type of analysis no load-displacement diagram is found. The displacement controlled calculation, geometrical nonlinear but without imperfections, does not converge due to the lack of initial imperfection. Using an imperfection approaching zero (and an increasingly small step size to ensure stability) the critical buckling load is still found. In figure 4.6 the difference is shown between a calculation without imperfections and one with a very small one. The chosen imperfection was e 0 = 0.15 mm, which naturally is very small for a plate of 1000x2000x20 mm. The little imperfection is needed to get the second order effects started, its size is mainly determined by element size and load step size. The consequence of the imperfection is that no exact bifurcation point is found, but using an extrapolation of the post critical area the critical buckling load can still be approximated. Fy [kn] (a) u z [mm] (b) Ansys (eigenvalue analysis): Ansys (displ. contr. analysis): Figure 4.6: (a) no imperfections, clear bifurcation point (b) very small imperfection, extrapolation needed N cr = 6074 kn N cr = 5902 kn Analytical: N cr = = = 6074 kn There is a three percent difference in the displacement controlled calculation and the analytical one. This is caused by the influence of the initial imperfection. Using e 0 = 0.08 mm and decreasing the load step size even further the critical load becomes 5970 kn. It is reasonable to assume the critical load approaches 6074 kn as the initial imperfection approaches 0 mm. 56

57 F y,total [kn] Master Thesis Plate buckling in design codes FE model 4 This model is the same as the previous one, only the initial imperfection has a larger value. According to annex C to NEN-EN the initial imperfection should be e 0 = b/200 = 5 mm, which would cover both fabrication tolerances and residual stresses. The results of the calculation do not show a clear critical load, but they gradually transit to the secondary load path. As a comparison not only the prescribed 5 mm of imperfection is shown in figure 4.7, but also some other values. The displacement used in the graph is the maximum out-of-plane displacement of the plate. FE model : 0.15 mm 2: 2.0 mm 3: 5.0 mm 4: 10.0 mm Euler u z [mm] Figure 4.7: Elastic plate buckling behaviour with different initial imperfections This model has all non-linearity s implemented and can be used to determine the maximum plate buckling capacity of a single unstiffened plate. The initial imperfection is taken as the prescribed 5 mm by the Eurocode. The complete calculation written as Ansys input can be found in annex C.1. The next two figures show the sinusoidal mode shape and the stress distribution in a plate with aspect ratio a = 2. The represented stress in figure 4.9 is the von Mises stress in the middle plane of the plate, which clearly shows how the force flows around the buckled part. Figure 4.8: First mode shape with two sinusoidal buckles 57

58 Figure 4.9: Stress distribution, respectively at 0.3 F max, 0.6 F max and 1.0 F max The capacities are: Ansys: Eurocode: F max = 5297 kn F Winter = 5232 kn At first hand the results are very close together, with the Ansys result slightly higher. However the Ansys result is very sensitive to the initial imperfection. The fact that the two calculated capacities above match that good is because the cold formed sections used by Winter to derive the formula had an equivalent imperfection around the b/200 value used in the Eurocode. Figure 4.10 shows the results of the calculation, again with the out-of-plane displacement and several values of initial imperfection. Figure 4.11 shows the load-displacement diagram with the displacement taken as the shortening of the plate, which is the displacement in y direction of the nodes at y = a. In both graphs the elastic behavior has been added in gray to show the influence of plasticity. 58

59 F y,total [kn] F y,total [kn] Master Thesis Plate buckling in design codes u z [mm] Figure 4.10: Elastic-plastic buckling behavior (in gray: elastic) 1: 0.15 mm 2: 2.0 mm 3: 5.0 mm 4: 10.0 mm Euler Winter Remarkable is the fact that right after the maximum is reached the load falls back rather quickly. Whether in the case of a built up cross-section the capacities of slender and stockier plates can just be added certainly needs to be investigated : 0.15 mm 2: 2.0 mm 3: 5.0 mm 4: 10.0 mm Winter u y [mm] Figure 4.11: Elastic-plastic load-displacement diagram 59

60 F y,total [kn] Master Thesis Plate buckling in design codes Figure 4.12 shows the load-displacement graphs for the five discussed methods. The assumptions made in each method are clearly reflected in the graph Model 1 Model 2 Model 3 Model 4 Model 5 Euler Winter u y [mm] Figure 4.12: Load-displacement diagrams of the five models Concluding it can be said that the ultimate load is estimated very well by the method of Winter, but there is a large influence of the initial imperfection. Using the imperfection given in NEN-EN the finite element calculation in Ansys provided 1.2 % more capacity than the method by Winter. 60

61 F y,total [kn] F y,total [kn] Master Thesis Plate buckling in design codes Very slender plate in compression The plate analyzed had a critical load higher than the ultimate load. The exact same Ansys calculations as before have been made again, only this time with more slender plate. Only the model including geometrical and physical imperfections has been used, since these calculate the ultimate capacity. The dimensions and predicted capacities now are: Dimensions: a*b*t = 2000*1000*10 Capacities: F Euler = 759 kn F Winter = 1474 kn The Ansys analysis gave an ultimate capacity of 1559 kn (with e 0 = 5 mm), which is 5.8 % more than according Winter. Figures 4.13 and 4.14 reflect the buckling behavior of this slender plate. The postcritical strength is visible since after the critical value the load still increases, even if it is with less stiffness. Striking is that in the very slender region the ultimate load is no longer depending on the initial imperfection, since the load is carried mostly by membrane stresses u 6 y [mm] Figure 4.13: Load-displacement diagram of slender plate 1: 0.05 mm 2: 2.0 mm 3: 5.0 mm 4: 10.0 mm Winter Euler u z [mm] Figure 4.14: Out-of-plane behavior of slender plate 1: 0.05 mm 2: 2.0 mm 3: 5.0 mm 4: 10.0 mm Winter Euler 61

62 F buckling / F von Karman [-] Master Thesis Plate buckling in design codes The influence of the slenderness on the ultimate capacity can be found in figure 4.15, which shows the relative buckling load as a function of the slenderness. Here it can be seen that the method by Winter has the best estimation of the buckling load of a imperfect plate. Since a finite element calculation needs dimensions in order to calculate the buckling load, the vertical axis is made dimensionless by dividing by the buckling load according von Kármán. This is the theoretical ultimate load of a perfect (no imperfections) plate, given by: F buckling = (1/ rel ) A s f y. 1,1 1,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0,0 Critical load (Euler) Ultimate load (Winter) NEN 6771 Ansys Elastic load 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 rel [-] Figure 4.15: Buckling load of compressed plate From figure 4.15 it can be concluded that for every plate slenderness the finite element calculation gives more capacity than NEN-EN The largest difference is found in the more slender plates. In the range of the slenderness mostly applied in practice, rel <2.0, the difference is smaller Single plate in bending Unstiffened plates in bending have been analyzed the same way as the compressed plates in section and The results support the same general concept, in a relative stocky plate the ultimate capacity is depending on the initial imperfections, where in more slender plates it does not. Also here the imperfection suggested by NEN-EN annex C gives results close to the capacity calculated by the method of Winter. The analysis was harder to complete, since a lot of deformation was focused in a single point. This was because very soon after buckling, crippling started to occur and that caused large strains and displacements. Crippling was found in the upper left area of the plate, or in figure 4.16 around x = 0 and y = b. This crippling effect is also the cause why the load drops very fast after reaching the ultimate load. Even though a displacement controlled analysis was made, due to the crippling convergence was much harder to achieve that in the previous sections. This resulted in finer meshes, smaller step sizes and larger computation times than the previous analyses. Despite that, the analysis using e 0 = 0.5 mm never converged after reaching the ultimate load. 62

63 M total [knm] M total [knm] Master Thesis Plate buckling in design codes y z Dimensions: a = 4000 mm b = 2000 mm t = 15 mm z b a x Boundary conditions: all edges: dz = 0 x = 0: dx = 0 x = a: rotz = 6/1000 x = y = 0: dy = 0 Figure 4.16: Unstiffened plate in bending /1000 2/1000 3/1000 [mm -1 ] Figure 4.17: Load-displacement diagram 1: 0.5 mm 2: 3.0 mm 3: 5.0 mm 4: 10.0 mm Winter Euler u z [mm] Figure 4.18: Out-of-plane behavior in bending 1: 0.5 mm 2: 3.0 mm 3: 5.0 mm 4: 10.0 mm Winter Euler Using the initial imperfection given in NEN-EN the Ansys calculation gave an ultimate capacity of 3005 knm, while the method by Winter predicts a capacity of 2866 knm. The finite element calculation gives 4.8 % extra capacity compared to the method by Winter. 63

64 The analysis of the plate in bending is also performed with a more slender plate (t = 10 mm). The other dimensions are kept the same as in figure As with the plates in pure compression the dependence on the initial deformation of the ultimate load is reduced severely. The capacity according Winter is 1427 knm, with Ansys calculating 5.0 % more, or 1498 knm Conclusion Four different single unstiffened plates have been analyzed using Ansys, two different slendernesses both in compression and bending. In each situation the capacity calculated by Ansys was higher than the capacity by NEN-EN , ranging from 1.2 % until almost 6 %. Also figure 4.15 shows how the Eurocode slightly underestimates the capacity for any given slenderness. Assuming the finite element calculation is more advanced, the simple methods in the Eurocode are on the safe side. On the other hand there is a dependence of the ultimate capacity on the initial imperfection, which can make a difference of a couple percent of the total load. 64

65 F y,total [kn] Master Thesis Plate buckling in design codes 4.4 Capacity of built up sections In this subsection the capacity of the box girder as calculated in chapter 3.1 is verified. The dimensions of the cross-section and previously calculated capacities are given again in figure Two questions have to be answered here, whether the total capacity of the example cross-section was calculated correctly in the design codes, and if a very slender plate can remain at capacity until a more stocky plate reached buckling. t f = 20 h w = 1000 t w = 10 h t = 1015 Critical (Euler): NEN 6771: reduced stress method: NEN-EN : effective cross-section method: reduced stress method: 4560 kn 6229 kn kn 9046 kn b f = 1000 b t = 1020 All dimensions in: [mm] Figure 4.19: Dimensions and capacities of box girder In figure 4.20 the load-displacement diagram as calculated by Ansys is shown, including the elastic path to show the critical load and the second order path to show the influence of plasticity. The ultimate capacity is kn, which is 0,1 % more than the Eurocode capacity of Ansys Critical load (Euler) NEN: reduced stress method EC3: reduced stress method EC3: effective cross-section method u 5 y [mm] Figure 4.20: Dimensions and capacities of box girder 65

66 F buclkling [kn] Master Thesis Plate buckling in design codes The influence of the web slenderness on the capacities calculated with the different methods was also determined. This is a very clear way of comparing the methods and since Ansys works with input given in a script it is quite easy to recreate the graph. In figure 4.21 the Ansys calculations are given, which confirms that capacity of built up members is calculated most accurately by the Eurocode effective cross-section method EC: Effective cross-section method EC: Reduced stress method EC: Web breathing NEN 6771 Elastic capacity Ansys calculation t w [mm] Figure 4.21: Ansys verification in influence web slenderness 4.5 Deformation capacity In chapter 4.3 two plates have been analyzed using Ansys, both were 1000 mm wide, but the thickness was either 10 or 20 mm. Their respective ultimate capacities were 1559 and 5297 kn. These plates have been combined into a box girder in chapter 4.4, were Ansys calculated an ultimate capacity of kn. Since the expected value was 2 ( )=13712, the deformation capacity of the slender plate seemed to be sufficient. To investigate whether for each slenderness ratio this would be true, a large number of plate dimensions have been put in the Ansys log file. In figure 4.22 the results are compared by plotting the normalized load (F y,total /F max,elastic ) against the principle strain ( l/l). Here the fundamental principle behind any effective cross-section method becomes visible. Due to the influence of the initial imperfections, the stiffness of a plate decreases, ensuring every plate will reach the ultimate load around the same axial strain. In a built-up cross-section all plates undergo the same deformation, so each single plate can reach its plate buckling capacity. 66

67 F y,total / (A s *f y ) Master Thesis Plate buckling in design codes 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, t [mm] a [mm] b [mm] Figure 4.22: Loss of stiffness in slender plates l/l Concluding it can be said that the deformation capacity in a slender plate is not the classical plastic deformation, which is normally regarded in a steel structure. Plates in buckling lose their stiffness as soon as buckling sets in, so the ultimate load is always reached around the same amount of deformation. 67

68 Chapter 5: Requirements for plate edges A steel member can be made up from several plate elements, which together form the cross-section. In the calculation of the plate buckling capacity the various subpanels can be regarded separately. The line along which the subpanel was originally connected to another plate part is a plate edge. Longitudinal stiffeners Plate edges Web: three subpanels Steel cross-section Figure 5.1: Stiffeners and edges in a cross-section When calculating the plate buckling capacity of a stiffened cross-section, there are many requirements given in various available design codes concerning longitudinal and transverse stiffeners. Since a buckled plate develops membrane stresses when it is loaded in-plane, a load outof-plane exists on the stiffeners. These stiffeners must give adequate support to the plate in the outof-plane direction while also having to withstand the in-plane compressive loading itself. All plate edges have to remain straight for the membrane stresses to fully develop. In Eurocode 3 this problem is solved by reducing the area of a plate part which is supported by a longitudinal stiffener. The amount of reduction depends on the geometry of the stiffener. Due to this reduction the design load of the structure decreases. This decreased design load ensures the stresses in the actual structure do not exceed the value at which the stiffener loses its ability to support the plate. In NEN 6771 another approach is chosen. A stiffener is loaded by the plate buckling stress of the adjacent subpanel, assuming simply supported conditions when determining the plate buckling stress. The stiffener is then checked regarding it as a column loaded in bending and compression. If it does not satisfy the requirements the size of the stiffeners or the plate thickness itself needs to be increased. On top of these two methods for checking the out-of-plane properties of a stiffener, there are more requirements given. These regard for instance the torsional buckling capacity and maximum permitted initial deformation. For the edges which are not formed by stiffeners on the other hand, there has not been given any requirement. A plate edge is just a simply supported line which stays perfectly straight during loading. When considering a box girder loaded in compression this naturally feels to be a good assumption. The webs and flanges are perpendicular to each other and can therefore give good support in de out-of-plane direction of the other plate element. 68

69 When considering an I-shaped member this assumption might seem less logical. Now the flanges have to support the web out of its plane when it buckles. If these flanges would be of small dimensions it is easy to see they cannot give enough support to the web, even though according the plate buckling verification methods it is still okay. If the web would continue past the flanges, they suddenly become longitudinal stiffeners and would not meet the requirements, or lead to a very high reduction of the area of the web plate. This leads to the question why flanges in compression don t have to comply with the requirements for stiffeners, and if this is correct or not. 5.1 Hypothesis To give the study some guidance an assumption for the outcome is needed: When calculating the plate buckling capacity of a cross-section built up from steel plates by regarding the different subpanels separately, the plate edges (the location where the other subpanels are connected) can indeed be assumed as simply supported. This assumption can be made because the out-of-plane support capacity of the edges is determined by the global column buckling of the complete member. In practice this would mean that a local plate buckling capacity of the cross-section is determined assuming 100% effective edges. In a situation where these edges can t be regarded fully effective, since they are below the critical stiffness, an extra reduction in capacity is given by the global buckling of the member. In the rest of the chapter this hypothesis is tested using finite element modeling in Ansys. 5.2 Outline In this part of the chapter an outline is given for the study of the edges. The study is divided in three different parts which are described here. In the first part some sensitivity analyses are made. Apart from element and step size the precise shape of the imperfection is also studied. This is an important parameter in the calculations and therefore it is good to know how sensitive the calculations are to this shape. The main reason for this is that when modeling in Ansys it is not possible to add a certain imperfection in a single step. The buckling mode shape of the web is not perfectly sinusoidal when flanges are attached. Therefore we need to know how accurate this imperfection must be modeled to get satisfying results, but in general more accuracy means more time to build the model. This is because the individual parts have to be uncoupled while determining the imperfect shape, and then be coupled again for the analysis of the plate buckling capacity. In the second part a single unstiffened plate is modeled. The simple support which normally constrains the long edge is now replaced by a distributed spring. By varying the spring stiffness and monitoring the change in collapse load, the influence of the supporting edge is studied. See also figure 5.2 on the next page. 69

70 Figure 5.2: Boundary conditions and loads In the third part an I-shaped cross-section is studied. This is mostly the same model as in figure 5.2, only the out-of-plane support of the plate is now given by flanges instead of the distributed elastic foundation. The flanges are chosen in class 3 or lower, mainly to keep modeling of the structure relative simple by excluding an extra imperfection. 5.3 Sensitivity analysis Element and load step size The element size and load step size play an important role in the finite element calculation. When using a displacement controlled, geometrical and physical nonlinear calculation the collapse load will increase while decreasing the load step size. As a result of the smaller steps the top of the loaddisplacement curve is not stepped over. With a larger step size the values itself may be as accurate as with a small step size, because of extra iterations in each load step, but the iterations do not show in the results table in Ansys so the collapse load may be missed. Increasing the number of elements will decrease the collapse load, which makes for a more realistic solution. Due to the extra number of elements the computation time on the other hand increases dramatically. In my example they went from a couple of minutes to well over an hour. This completely limits the ability to quickly study different situations, while the decrease in collapse load was only a couple percent. The goal here is to find an element size which gives reasonably accurate results, combined with manageable computation times. There is no golden rule to reach this and it depends mainly on experience and some trial and error. An example is given in figure 5.3, where the same plate structure is calculated three times. The first calculation is just to have a load-displacement graph to verify the effect of the changes. The second calculation adds a five times smaller load step size to the first, which makes the load-displacement curve much smoother, since intermediate iterations are more regular and stored in the final results. The third analysis is made by using half the element size of the second calculation. 70

71 Load Top of load curve Master Thesis Plate buckling in design codes +0.2% -1.6% Displacement Figure 5.3: Influence of element and load step size 1: Starting point 2: 1/5 load step size 3: 1/2 element size Displacement In most calculations the load step size needs to be quite small to ensure convergence. To see whether it is small enough for an accurate result of the collapse load, you to reference it to the previous and the next value in the load-displacement curve. The difference between these two is a measure for the accuracy of the collapse load itself. The element size is chosen such that computation time is limited to around five minutes. In all calculations a constant element size is maintained, so any inaccuracy will constantly be the same and not influence the results Shape of imperfection The main goal in this subsection is to determine how complicated the model has to be to get accurate results. In Ansys adding imperfection cannot be done with the click of a button, but requires programming with more than a single step. In the regular plate model the buckling mode shapes can be calculated using an eigenvalue analysis. The lowest mode shape can then be scaled to match the required amount of imperfection and the original geometry updated according to this buckled shape. Using the lowest mode shape makes sure the lowest collapse load is found. In a cross-section built up from more than one single steel plate this method is not applicable. Only the most slender plate will deform in the eigenvalue analysis and the rest of the plates will not get an initial imperfection. This problem can be get round by fixing all but one plate in the out-of-plane direction, then calculation its imperfect shape and deleting the extra constraints. This procedure needs to be repeated for each plate part. The problem with this method is that the free plate, from which the buckled shape is calculated, has fixed edges. This leads to an imperfect shape which may not give the lowest collapse load. Another problem is the number of sinus waves in the imperfection. When regarding a plate with an aspect ratio of two (plate length is two times width), the first mode shape should have two sinus shaped imperfections. When the edges have been fixed the first mode shape has three sinus waves. The influence of this shape is studied by using the regular plate model, with the rotation around the y axis as optional constraint. Six calculations are made to compare the different situations. 71

72 Figure 5.4: Constraints in plate model Four different shapes of the imperfection are tested, with and without rotational constraint and for both situations the shape with two and three sinusoidal buckles. Figure 5.5: Mode shape 1 and 2 (roty = free) Figure 5.6: Mode shape 1 and 3 (roty = 0) The results can be found in table 5.1, for each situation is given how the roty constraint of the long edge is varied. The corresponding shapes of the initial imperfection can be found in figures 5.5 and 5.6. The dimensions of the plate are a b t = mm. Which makes the plate buckling capacity according Winter is kn. 72

73 roty in making mode shape Mode shape no. (no. of waves) roty in nonlinear analysis Collapse load [kn] Relative collapse load [-] 1 free 1 (2) free free 2 (3) free fixed 3 (2) free fixed 1 (3) free fixed 3 (2) fixed fixed 1 (3) fixed Table 5.1: Collapse loads for different shapes of imperfection Since the plate is very slender it is expected that the collapse loads according Ansys are higher than the plate buckling capacity according Winter (see also chapter 4.3.1). The interesting conclusion from this study is that the actual shape of the imperfection does not have a very large influence on the collapse load. Another remarkable result is that the lowest collapse load is not reached with the lowest elastic buckling mode, although again the difference is very small. When the rotational degree of freedom of the long edge is still fixed during the non linear analysis, a higher collapse load is reached. This is expected because in transverse direction there the plate is clamped, instead of simply supported. In practice this capacity can t be used, because the crosssection will alternate the direction of the buckles so no fixed edges can be found. Figure 5.7: Plate buckled cross-sections 5.4 Elastically supported plate To get a clear picture what the influence of the support condition of the long edge is, a distributed spring is added to the plate model. The boundary conditions can be found in figure 5.8. The dimensions of the plate are a b t = The spring stiffness (k z ) is variable, since this is the parameter we are interested in. The initial imperfections are chosen according Eurocode Annex C, which states that the local imperfection must be b/200 = 5 mm and the global bow imperfection of the edges must be a/200 = 20 mm. The a value is chosen to be four, so no unexpected stiffness introduced at the support of the short edges of the plate has an influence in the plate buckling capacity. 73

74 Figure 5.8: Boundary conditions For the two extreme values for k z (k z = 0 and k z = ) the load bearing capacity can be calculated using the methods in NEN-EN The finite element calculations can later be confirmed using these values. When k z = 0 the plate can be regarded as a column which can be checked according NEN-EN article 5.3.1: l k = 4000 mm A s = mm 2 I = mm 4 With k = the plate buckling capacity can be calculated according NEN-EN article 4.4: 74

75 F d,buckling [kn] F d,buckling [kn] Master Thesis Plate buckling in design codes The log-file of the calculation in Ansys can be found in annex C.2. Results are listed in the following table and figure. k [N/mm 2 ] F max [kn] Table 5.2: Plate capacity with various k z ,0001 0, k z [N/mm 2 ] Figure 5.9: Plate capacity versus spring stiffness diagram From the figure it can clearly be seen that in the beginning an increase in spring stiffness leads to an increase in collapse load. This is to be expected since the plate element is better supported and more membrane stresses can develop. Failure is governed by a global column buckling mode. After the critical value of the spring stiffness is exceeded, no more increase in collapse load occurs. Here the failure mode is plate buckling, which is independent of the spring stiffness. The capacity for k=0 according to the Eurocode is confirmed by the Ansys calculation, although it is too small to be properly plotted in the graph. It is clearly noticeable that a small area of interaction between the two failure modes occurs. If there would not have been any interaction the transition between k=10 and k=100 would have had a sharp angle , k [N/mm 2 ] Figure 6.10: Region with interaction Nmax Ansys Nmax Eurocode 75

76 5.5 I-shaped member In this part of the chapter the elastic support is replaced by steel flanges. This creates an I-shaped member, from which the length will be varied. When the length is small the plate buckling failure mode will be governing, while at a bigger member length the global buckling capacity will be smaller. In principle the same procedure as with the spring supported plate can be followed, and similar results can be expected. The geometry and boundary conditions of the member can be found in figure u y flanges: b f t f = [mm 2 ] web: h w t w = [mm 2 ] z y l z x Boundary conditions: At y=0: ux = 0 uy = 0 uz = 0 At y=l: ux = 0 uz = 0 Figure 5.11: Boundary conditions and geometry Two important choices have been made to keep the modeling in Ansys relatively simple. Since the model is a 3D plate model, it is easier to give the ends of the members a fixed support. The nodes at y=0 are set to u y = u x = u z = 0 while the nodes at y = l are set to u x = u z = 0 and u y as the prescribed displacement. The buckling length of the member is now of course half of the system length. The other choice is to use flanges of class three or lower. This ensures only two instead of three imperfections must be added. The two remaining imperfections are a global imperfection in the shape of the buckling mode of the member, as seen in figure 5.11, and a local plate buckling imperfection of the web as seen in figure 5.5. Since the member is not supported anywhere except at the ends, global buckling will take place in the direction of the weak axis, which gives displacements in the z-direction. If a global imperfection is given in the positive z-direction, the local imperfection of the web at y = ½ l must be in the same positive z-direction. This gives the least favorable situation for the member. If opposite directions are used collapse loads up to 20% higher can be found, but of course these are not realistic for the structure. Imperfections are added according to Eurocode Annex C, which states that global bow imperfection must be l / 200 and local plate imperfection h w / 200. If the hypothesis is correct a member with a small length will have stable plate edges and therefore reach the plate buckling capacity according Winter. A member with a bigger length will have unstable edges and the collapse load will now be governed by global buckling. These two failure modes can be determined using NEN-EN and

77 Local buckling capacity according NEN-EN : Global buckling capacity according NEN-EN : 77

78 The hypothesis would suggest that when the relative column slenderness ( ) is smaller than 0.2 the local plate buckling mode will be governing. If it is larger the plate edges become unstable and global buckling is the leading failure mode. With the chosen cross-section this transition point will occur at a member length of around 4000 mm, or a buckling length of 2000 mm. The Ansys calculation log file can be found in Annex C.3. The resulting collapse loads for different lengths are summarized in table 5.3. Figure 5.12 and 5.13 shows how the shape of the buckles is affected by the global displacement. Given here is the displacement of the web in the out-of-plane direction, at the moment the ultimate load is reached. The buckles are no longer sinusoidal and equal in size. Figure 5.12: Out-of-plane displacement of web for l = 8000 mm Figure 5.13: Out-of-plane displacement of web for l = 6000 mm l [mm] F y;max [kn] l [mm] F y;max [kn] l [mm] F y;max [kn] l [mm] F y;max [kn] Table 5.3: Collapse loads according Ansys calculations These results are plotted together with the solution according to the Eurocode in figure At first glance the two lines seem to match, which confirms the hypothesis. At the location of where global buckling starts to play role however ( rel = 0.2), some interaction seems to take place in the results according the Ansys calculation. 78

79 Collapse load [kn] Collapse load [kn] Master Thesis Plate buckling in design codes l [mm] Eurocode Ansys rel [-] 0 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,2 2,4 Figure 5.14: Collapse load while increasing l l [mm] Eurocode Ansys ,0 0,1 0,2 0,3 0,4 rel [-] 0,5 Figure 5.15: Detail of region with interaction The jumps in the Ansys results can be explained by the number of buckles present in the imperfection of the web plate. The buckling factor is always taken as k = 4.0 in the Eurocode, while in reality it varies with the aspect ratio a and the number of buckles. Even then the collapse load should not have jumps but only changes in direction. The jumps are created because the aspect ratio where a mode shape (shape of imperfection) succeeds another one is not necessarily the aspect ratio where the two succeeding mode shapes have the same collapse load. This effect is also seen in table 5.1. In figure 5.16 the influence of the number of buckles is shown in even more detail. 79

80 Deviation [%] Master Thesis Plate buckling in design codes m=1 m=2 m=3 m=4 m=5 Figure 5.16: Influence of number of buckles (m) on k and Ansys results Note that in this study no member lengths smaller than 1500 mm have been studied. In the region of very small member lengths the collapse load would have become higher than the collapse load according Winter. This can be seen in the left graph of figure 5.16, where for a < 1 the value of k increases. In Eurocode this effect is accounted for by article 4.5.4, about the column type buckling which can occur in plates with small a s. Except for a small region around rel = 0.4 the collapse loads calculated in Ansys are consistently smaller than what the Eurocode predicts. The difference expressed in per cent can be found in figure Figure 5.17: Deviation Eurocode in relation to Ansys l [mm] 80

81 Collapse load [kn] Deviation [%] Master Thesis Plate buckling in design codes 5.6 Conclusion Even though some deviation exists, the general idea in the hypothesis is correct. Several reasons can be found why the Ansys results are consistently below the Eurocode values: - Inaccurate a- and 0 -values. Here a is not the aspect ratio, but the parameter in the buckling curve in the Eurocode. Only a couple of values can be chosen, according the buckling curves a 0, a, b, c, and d. Intermediate values would give a better fitting. The following values are permitted: a 0 a b c d a Table 5.4: Buckling parameters If a = 0.69 and 0 = 0.29 would have been chosen, the difference for larger length members would be significantly smaller. A big disadvantage is that now an extra curve would be needed to account for the interaction, making the verification code more complex. Most likely a choice for a relatively simple formula has been made in the Eurocode, while making sure the deviation stays within certain reliability limits. Eurocode Ansys l [mm] rel [-] Figure 5.16: Results with a = 0.69 and 0 = 0.29 l [mm] Too much imperfection according Annex C. The magnitude of the imperfection which has to be added according Annex C of Eurocode is h w / 200. This imperfection includes geometrical imperfection as well as residual stress. It appears this value is rounded off to a safe margin. Maybe a detailed study of the real imperfections can lead to a more favorable value and to higher collapse loads. - Rotation of flanges. The buckling of the web plate causes the flanges to rotate. Since the ends of the flanges are fixed, a twisting deformation occurs and strains are introduced in the flanges. Due to these strains collapse is reached sooner than in perfectly flat flanges. 81

82 Chapter 6: Conclusion and recommendation 6.1 Conclusion This thesis started with the question why NEN-EN can give much higher plate buckling capacities then NEN By analyzing the methods used in both codes, and the principles behind these methods the differences in capacity can be explained clearly. The NEN 6771 uses the reduced stress method, with a reduction factor based upon the critical Euler plate buckling stress. For slender plates this critical stress is always lower than the stress at the ultimate failure load. The NEN-EN on the other hand uses the effective cross-section method, first introduced by von Kármán, and the reduction factor according Winter. Since this reduction factor is calibrated using many tests on steel specimens, it predicts the ultimate load very well. When calculating with the reduced stress method in NEN-EN the same reduction formula is used. Due to this the calculated capacity is higher than in a NEN 6771 reduced stress method calculation. Nevertheless, NEN 6771 also contained the effective cross-section method with Winters reduction formula, so strictly speaking there is no extra capacity gained in using NEN-EN However in a variably loaded structure web-breathing might be governing over the effective cross-section method and a stress based upon the critical stress should be used. The ultimate capacities calculated by the effective cross-section method were verified using finite element calculations in Ansys. The finite element models made in Ansys had both geometrical and physical nonlinearities, in order to find the ultimate load of thin steel plates. As well for single plates as for built up sections the capacities of finite element and design code calculations match very precise. The deformation capacity needed in built-up sections was found to be a loss of stiffness due to second order effects. This ensures that when steel plates of different slenderness are combined in a single section, each plate will reach its own buckling capacity at around the same axial strain. This is the fundamental principle behind the effective cross-section method. Plate edges do not have any requirements given for either in NEN 6771 or in NEN-EN Using finite element calculations the hypothesis that the stability of the edges is implied in the global stability of the member was found to be correct. If global stability is guaranteed, the edges of a plate will also remain stable. However in between the stable edges the plate can still experience local plate buckling. If the edges cannot support the adjacent plates, the local plate buckling capacity will not be reached and global buckling is governing for the capacity of the member. 6.2 Recommendation Finally a recommendation is given as to what method to choose when calculating a plate buckling capacity using NEN-EN The reduced stress method given in chapter 10 of NEN-EN is a quick procedure to determine if a structure is below its plate buckling capacity, especially when appropriate software can be used. Also for a quick assessment of the plate buckling capacity of a given cross-section, this is very good method. The capacity will however always be lower than the 82

83 capacity calculated with the effective cross-section method of chapter 4 of NEN-EN , but if the plate slendernesses are chosen such that the reduction factors are of similar value, the capacities will also be close together. If on the other hand such an option for the slenderness is not available, the effective cross-section method will give a higher capacity. How much percent of extra capacity depends on the cross-sectional properties, but several dozens is quite possible. Plate buckling in a member will be accompanied with a loss of stiffness. If this loss of stiffness is expected to have an influence in the force distribution in the structure, the effective cross-section method must be used to take it into account. 6.3 Future research This thesis dealt only with the failure load of slender steel plate cross-sections. At some points the fatigue sensitivity of variably loaded structures was mentioned. Keep in mind that web-breathing is caused by an out-of-plane deformation, while fatigue verifications check in-plane stress cycles. A possibility for future research is how the fatigue verifications are dealt with in the design codes when a structure sensitive for plate buckling is regarded. The effective area is in this case smaller than the gross area, which will increase local stress peaks and therefore decrease fatigue life. Since many effects play a role at the same time it would be interesting to have a better understanding of the fatigue life of plate buckling sensitive structures. In the Netherlands several bridges have suffered from fatigue damage, so it is also a very current subject. The imperfection of the steel plates has a large influence on the final carrying capacity. In all methods treated in this thesis the imperfection is implemented in some way. NEN 6771 has a linear cut-off to the critical Euler stress, the reduction factor in Winters method is calibrated to a large number of tests on cold-formed sections and in a finite element calculation the imperfection is added directly in the geometry. A first thing which would be interesting to analyze is how the calibration by Winter to von Kármáns formula relates to the b/200 value used in NEN-EN and if this can be an explanation for any differences in capacity that is found. A second, more applied research direction can be to the plates fabricated in the steel shops. For instance whether the imperfections of the actual produced plates match with the fabrication tolerances given in the codes, or if they can be much higher or lower. 83

84 Chapter 7: References [1] Abspoel, R., Bijlaard, F.S.K., Stability of Steel Plate Structures, Delft University of Technology, Lecture notes CT4121, [2] Vrouwenvelder, A.C.W.M., Structural stability, Delft University of Technology, Lecture notes CT5144, [3] Gardner, L., Nethercot, D.A., Designers Guide to EN Eurocode 3: Design of steel structures, general rules and rulesfor buildings, London, [4] Hendy, C.R., Murphy, C.J., Designers Guide to EN Eurocode 3: Design of steel structures, Part 2: Steel Bridges, London, [5] Beg, D., et al, ECCS Eurocode Design Manual: Design of Plated Structures. Eurocode 3: Design of Steel Structures. Part-1-5: Design of Plated Structures, Ljubljana, [6] Johansson, B., et al, Commentary and Worked Examples to EN Plated Structural Elements, Luxembourg, [7] Timoshenko, S.P., Gere, J.M., Theory of Elastic Stability, Stanford, [8] Clarin, M., Plate Buckling Resistance Patch Loading of Longitudinally Stiffened Webs and Local Buckling, Luleå University of Technology, Doctoral Thesis 2007:31, [9] Bloom, F., Douglas, C., Handbook of Thin Plate Buckling and Postbuckling, New York, [10] Falzon, B.G., Aliabadi, M.H., Buckling and Postbuckling Structures, London, [11] Yu, W.W., Cold-Formed Steel Design Third Edition, New York, [12] Ghersie, A., et al., Metallic Cold-Formed Thin-Walled Members, London, [13] Paik, J.K., Thayamballi, A.K., Ultimate Limit State Design of Steel-Plated Structures, London, [14] Subramanian, N., Design of Steel Structures, New Delhi, [15] Daley, C., Hermanski, G., Ship Frame Research Program Investigation of finite element analysis boundary conditions, Ocean Engineering Research Center, Report TR , [16] Sedlacek, G., et al., Leitfaden zum DIN Fachbericht 103 Stahlbrücken, Aachen,

85 [n1] [n2] NEN 6700; Technische Grondslagen voor Bouwconstructies TGB 1990 Algemene basiseisen, Delft, NEN 6770; Staalconstructies Basiseisen en basisrekenregels voor overwegend statisch belaste constructies TGB 1990, Delft, [n3] NEN 6771; Technische Grondslagen voor Bouwconstructies TGB 1990 Staalconstructies Stabiliteit, Delft, [n4] NEN 6788; Het Ontwerpen van Stalen Bruggen Basiseisen en Eenvoudige Rekenregels (VOSB 1995), Delft, [n5] NEN-EN 1990 (nl); Eurocode Grondslagen van het Constructief Ontwerp, Delft, [n6] NEN-EN (nl); Eurocode 3: Ontwerp en Berekening van Staalconstructies Deel 1-1: Algemene regels en regels voor gebouwen, Delft, [n7] NEN-EN (en); Eurocode 3: Ontwerp en Berekening van Staalconstructies Deel 1-5: Constructieve Plaatvelden, Delft, [n8] NEN-EN (en); Eurocode 3: Ontwerp en Berekening van Staalconstructies Deel 2: Stalen Bruggen, Delft, [n9] DIN-Fachbericht 103; Stahlbrücken; Ausgabe März [n10] NEN-EN (en); Constructieve Delen van Staal en Aluminium Deel 1: Eisen voor Conformiteitsbeoordeling van dragen delen, Delft, [n11] NEN-EN (en); Het vervaardigen van Staal- en Aluminiumconstructies Deel 2: Technische Eisen voor Staalconstructie (corrected), Delft, [w1] [w2] [w3] [w4] Eurocodes in Nederland, by NEN. (sept. 2010) LUSAS engineering analysis software company, case studies. (sept. 2010) Eurocodes Expert, by the Institition of Structural Engineers. (sept. 2010) ANSYS Simulation Driven Product Development (okt. 2010) 85

86 List of annexes A.1 Plate buckling capacity of a plate in bending A.2 Plate buckling capacity of a compressed plate B.1 Cross-sectional capacities calculated with codes B.2 Longitudinal stiffeners according codes C.1 Log-file Ansys of single plate C.2 Log-file Ansys of elastically supported plate C.3 Log-file Ansys of I shaped member D.1 Mathcad sheet for NEN-EN

87 Annex A.1 Plate buckling capacity of a plate in bending In this annex the plate buckling capacity of a single plate in bending is determined, as support of chapter The same plate is analyzed using NEN 6771, the reduced stress method in NEN-EN and the effective cross-section method in NEN-EN

88 Annex A.2 Plate buckling capacity of a compressed plate In this annex the plate buckling capacity of a single plate in compression is determined, as support of chapter The same plate is analyzed using NEN 6771, the reduced stress method in NEN-EN and the effective cross-section method in NEN-EN