MASTER. Load bearing capacity of stacked civil Poly falsework system. Luu, H.L. Award date: Link to publication

Transcription

1 MASTER Load bearing capacity of stacked civil Poly falsework system Luu, H.L. Award date: 2015 Link to publication Disclaimer This document contains a student thesis (bachelor's or master's), as authored by a student at Eindhoven University of Technology. Student theses are made available in the TU/e repository upon obtaining the required degree. The grade received is not published on the document as presented in the repository. The required complexity or quality of research of student theses may vary by program, and the required minimum study period may vary in duration. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain

2 Eindhoven University of Technology Department of the Built Environment Load bearing capacity of stacked civil Poly falsework system Master s Thesis ABP\SD 2015 H.L. LUU

3 Load bearing capacity of stacked civil Poly falsework system Master s Thesis

4

5 Report A thesis submitted to the Department of the Built Environment in partial fulfillment of the requirements for the degree of Master of Science in Architecture, Building and Planning A O IMPORTANT NOTIFICATION This report provides the results of numerical simulations performed with the finite element software ABAQUS The rotational stiffness of joints was experimentally determined by using the units Nmm/degree. By mistake, this unit was subsequently used in some of the numerical simulations, instead of the correct, converted unit of Nmm/radian. The corresponding graphs and conclusions referring to the (negligible) influence of the rotational spring stiffness on the load bearing capacity of the civil Poly system are therefore incorrect. The correct influence of the rotational stiffness on the load bearing capacity will be determined by means of additional simulations, and the results will be reported shortly in an erratum. PUBLICATION DATE June, 2015 GRADUATION COMMITTEE Prof. dr. ir. Suiker, A.S.J. (Akke) Prof. ir. Snijder, H.H. (Bert) Dr. ir. Hofmeyer, H. (Herm) STUDENT Luu, H.L. (Lin) Student number h.l.luu@student.tue.nl Eindhoven University of Technology Department of the Built Environment Structural Design

6

7 Abstract SAFE BV, a company located in Beek en Donk (The Netherlands) designs, leases and sells modern scaffolding and formwork systems. Their falsework system, known as the Poly falsework system, has improved with the passing of time due to developments in the built environment. The bearing capacity and other properties of the civil Poly elements Middle Piece and Jack are known. What still has been unknown is the exact way how a stacked structure composed of these elements will behave. By examining influencing factors, such as initial imperfections, interaction between elements and boundary conditions in detail, a theoretical model which represents reality was developed to determine the load bearing capacity of a stacked Poly falsework system for higher design loadings. The initial imperfections and eccentricities were investigated on the basis of guidelines given by different design codes. Different design codes provide different calculation methods for determining the initial imperfections. There could be concluded that EN suited the Poly falsework system the most and therefore, this design code is to be continued with for structural modelling. Furthermore, two series of experiments were performed in the Pieter van Musschenbroek Laboratory. While the purpose of the first is to determine the axial strength of a joint, the second s is to determine the bending strength and stiffness of another joint. Both experiments are performed following EN The tensile strength of the connection between a Poly Diagonal and a Poly Ledger is experimentally determined to be 21.32kN. The deformation of the pin hole of the connection is mostly around 4mm (looseness of the joint included). The connection between a Poly Ledger and a Poly Prop was subjected to bending. The moment capacity of the joint is 42.52kNcm and the rotational stiffness is 34.24kNcm/. Additionally, an independent samples T-test was conducted to confirm whether hammering in could be substituted by falling of a solid block with a specific mass from a certain height to attach the ledger onto the middle piece. This was done in order to eliminate human influence. Results suggested that hammering in the wedge by a human being can be substituted by a solid block (m = 1434g) falling from 2.344m through a hollow tube above the wedge. Human influence does not have an effect on the magnitude of the force with which the wedge is being attached. Finally, this report presents results from numerical analyses, performed using finite element software ABAQUS/CAE. Results from the above mentioned laboratory experiments are used as input for the structural model. First, linear buckling analysis (LBA) was done to solve the eigenvalue problem of the system. Then, initial imperfections and eccentricities were taken into account and geometrically nonlinear analysis (GNIA) was done using the arc-length method. Finally, to determine the ultimate load of the Poly falsework system, plasticity is taken into consideration and geometrically as well as materially non-linear analysis (GMNIA) was done. Once the failure loads were defined, design graphs were generated. This study ends with some case studies, leading to several conclusions yet and recommendations for future research. iii

8 iv

9 Preface This report is the results of a Master s thesis on the Load bearing capacity of a stacked Poly falsework system for higher design loadings at Eindhoven University of Technology. This Poly falsework system is designed by SAFE BV in Beek en Donk (The Netherlands) to support formwork. As a graduate intern I have been conducting research at the company and the following topics are discussed, amongst others: initial imperfections given by guidelines of design codes, experimental determination of strength and stiffness of a joint, data processing and elaboration following design codes, statistical analysis of experimental data, creating a finite element model, numerical analysis and finally, determining the ultimate load of the structure. I would like to take this opportunity to express my sincere gratitude to all those who supported me and contributed directly or indirectly to this dissertation. First of all, I would like to extend my appreciation to my graduation committee: Prof. dr. ir. A.S.J. Suiker and Prof. ir. H.H. Snijder for their guidance, patience and valuable advice. Their knowledge and feedback have played a key role in the completion of this research. Furthermore, I would like to acknowledge the assistance of ing. G.F.A.J.M. Joordens, structural engineer at SAFE BV, who provided me valuable suggestions for this study. I am especially grateful for the given opportunity to conduct my graduate thesis at the company, where I have experienced applying knowledge to real world situations. Additionally, I would like to thank the staff members of the Pieter van Musschenbroek Laboratory for their support during the experimental phase, especially T.J. van de Loo. His effort has contributed to a successful completion of the laboratory experiments. Many thanks also go to my friends and my fellow students, with whom I have worked but above all, with whom I have enjoyed my whole study career. Special thanks to Marko Jović for sharing his ABAQUS-software skills, which contributed to expediting the procedure in mastering the software, and Jordan Dorlijn for brainstorming sessions. I thank Stephany Ritoe for being my study buddy and partner in crime from day one at TU/e. And last but definitely not least, I wish to express my heartfelt thanks to my parents and my beloved sister for their unconditionally loving support over the years. Lin Luu Eindhoven, May 2015 v

10 vi

11 Table of contents Abstract Preface Table of contents Nomenclature iii v vii xi 1. Research design Introduction and motivation Problem outline Research objectives Research relevance and contributions Research method 3 2. Overview of literature Description of Poly falsework system Middle Piece Jack Poly Ledger Poly Diagonal Connector Load bearing capacity of a single Poly Prop (van Geijtenbeek, 1993) Load bearing capacity of a coupled Poly system (Kuijer, 1996) Load bearing capacity of a stacked Poly system (Lamerichs, 2011) Effect of specific assumptions and boundary conditions Buckling length reducing mechanism Effect of boundary conditions Effect of different base heights Effect of joint positions Effect of diagonal brace installations Effect of scaffold height Effect of shoring extensions Initial imperfections Design Codes DIN EN NEN 6770/NEN EN Comparison design codes Conclusion Laboratory experiments: axial strength Laboratory experiment Individual Poly elements Purpose Materials and equipment Test method: procedure and implementation Results Elaboration and discussion Conclusion 44 vii

12 5. Laboratory experiments: bending stiffness Laboratory experiment Individual Poly elements Purpose Material and equipment Test Series I: Comparison forces with/without human influence Test method: procedure and implementation Results Elaboration and discussion Conclusion Test Series II and III: Strength and Rotational spring stiffness Test method: procedure and implementation Results Elaboration and discussion: determination of the value of the characteristic resistance Elaboration and discussion: determination of the rotational spring stiffness Conclusion Conclusion Structural modelling Configurations Height variation Varying in number of props coupled Varying in entre-to-center distance between the props Cross section properties Initial imperfections applied on Poly falsework system Boundary conditions and assumptions Structural model in ABAQUS/CAE Part -module Property -module Assembly -module Step -module Interaction -module Mesh -module Load -module Keywords editing Finite element analysis Linear buckling analysis (LBA) Height variation Influence of number of props coupled Influence of center-to-center distance between props Geometrically non-linear analysis Height variation Influence of number of props coupled Geometrically and materially non-linear analysis Height variation Influence of number of props coupled Case studies Replaced diagonal Adjusted center-to-center distance Replaced point of load eccentricity Adding diagonals Translational stiffness of the joint (diagonal - ledger) 104 viii

13 8. Ultimate load of stacked Poly falsework system Design value of resistance Conclusion and discussion Brief summary General conclusions Recommendations Recommendations for future research 119 References 121 Literature 121 Design codes 121 Product information 122 Websites 122 Appendices Appendix A. Initial imperfections Appendix B. Output: force-deformation graphs (axial tests) Appendix C. Output: force-deformation graphs (bending tests) Appendix D. Element properties Appendix E. Python script 3_6200 Appendix F. Validation Ansys-model Lamerichs in ABAQUS Appendix G. Validation rotational spring stiffness Appendix H. Input-file 3_6200_GMNIA Appendix I. Photos experiments I-1. during tests (axial tests) I-2. after failure (axial tests) I-3. during tests (bending tests) I-4. after failure (bending tests) ix

14 x

15 Nomenclature Latin capitals A Cross sectional area mm 2 A nom Cross sectional area mm 2 E lo Energy put in during loading - E ul Energy regained during unloading - F c Ultimate load in deformed state N F E Euler buckling load N I Moment of inertia mm 4 M p Reduced plastic moment for F E Nmm M u;d Elastic moment capacity of a member Nmm N ci Normal force in the full plastic condition N N pl Elastic buckling load N R k Characteristic value of the particular resistance determined with N characteristic or nominal values for the material properties and dimensions R d Design value of resistance N W pl Plastic section modulus mm 3 Latin lower case c k Characteristic value of the stiffness kncm/ d i Inner diameter of a tube mm d o Outer diameter of a tube mm e 0 Bow imperfection mm e Distance between axis of two tubular members meeting end to end mm f Bow imperfection mm f y;a Actual value of the yield stress N/mm 2 f y;d Design value of the yield stress N/mm 2 f y;k Characteristic value of the yield stress N/mm 2 h Height of a member mm k s;k Quantile factor - l Height of a member mm l o Overlap length mm m Mass kg n k Number of loaded columns - n s Number of layers in a framework - n v Number of columns - n Number of columns - r Reduction factor - v x Variation coefficient - w o Bow imperfection mm r u Failure values - s y Standard deviation - y 5 5%-quantile - Greek lower case h Reduction factor for height applicable to columns - m Reduction factor for the number of columns in a row - m Global partial factor for the particular resistance - xi

16 Relative slenderness - Angular deviation from theoretical line 0 Angular impections Abbreviations DIN German Standard Instutite EN European Standard FEM Finite Element Method GMNIA Geometrically and materially non-linear analysis GNIA Geometrically non-linear analysis LBA Linear buckling analysis M Moment NEN Dutch Standard Institute xii

17 xiii

18

19 1. Research design SAFE BV designs, leases and sells modern scaffolding and formwork systems. Their system scaffold, known as Poly falsework system, is continuously developing due to expansion in application field. Experiments on all separate Poly support elements and research of combination of elements have been carried out since the 1970s. With the passing of time and further developments in the field, some of the details in these studies have become outdated and are now questionable. This research project is designed to continue on previous research at the company. This chapter reviews the aims and scope of the thesis, starting with the motivation and problem outline; being followed by the research objectives and contributions; and finally, ending with a research model and the research method Introduction and motivation SAFE BV designs, leases and sells modern scaffolding and formwork systems. The innovative systems developed in-house creatively respond to contemporary insights with regard to economic feasibility and safety. SAFE BV has a significant market position in the slab formwork/supports sector for residential, utility and civil construction with the Poly system. [14] In 1970, the first research into the bearing capacity of steel supporting structures took place (Voorn, 1971). Experiments on all separate Poly support elements and research of combination of elements have also been carried out. For instance, in the ultimate load of a single Poly Prop was investigated by Bertjan van Geijtenbeek (van Geijtenbeek, 1993) and the bearing capacity of horizontally coupled Poly falsework systems was examined by former TU/e student Hilbert Jan Kuijer (Kuijer, 1996), as shown in Figure 1.1. In 2011, Stephanie Lamerichs continued investigating the ideas from Kuijer s research, revised his study and added the stacked Poly falsework systems, which was the main subject of her Master s thesis (Lamerichs, 2011), see Figure 1.1. Figure 1.1(a). Single props (b).horizontally coupled props (c).vertically stacked props With the passing of time and further developments in the field, some of the details in these studies have become outdated and are now questionable. This research project is designed to continue on previous research done at the company. Several points at issue still remain undetermined. This study will focus on the load bearing capacity of stacked Poly falsework systems in civil engineering. Numerical analysis will be part of this graduation project, using the computer software ABAQUS/CAE 1

20 (Complete Abaqus Environment): one of the five software products from the ABAQUS product suite. It is a software application used for both the modelling and analysis of mechanical components and assemblies and visualizing the finite element analysis result Problem outline The ultimate load of a single Poly Prop, the horizontally coupled system, and the vertically stacked Poly falsework system are all determined by former students (van Geijtenbeek, 1993) (Kuijer, 1996) (Lamerichs, 2011). The Poly falsework system has developed and is improved over the years to keep up with the developments in the built environment. From work in residential construction to utility construction, Poly support now also has a bigger share within civil construction. With civil construction, significantly larger forces are involved. To meet these requirements, stacked structures and high-rise falsework are needed. Until now, research has mostly been done for a stacked system of two layers (Lamerichs, 2011). However, highly stacked structures are commonly used in practice. The higher a structure, the more important ensuring structural stability becomes. With present-day developments, new parts of the Poly system have appeared in the market. These parts have been tested and thus the bearing capacity and other properties are now known. However, what is still unknown is the exact way a stacked structure composed of the different parts together will behave. A theoretical model that accurately represents reality is needed to bridge the gap between theory and practice. Based on the problem definition, the research question can then be formulated as follows: What is the load bearing capacity of stacked Poly falsework system for higher design loadings? Different factors can influence the magnitude of the ultimate load and thus the bearing capacity. Therefore, the research question can be divided into sub-questions, each representative for each factor. The result of the sub-questions should lead to the answer to the research question. 1. What are the properties of the Poly falsework elements? 2. What is the magnitude of the initial imperfection to take into account? a. What are the limitations within the design codes? b. What are the experiences from practice? c. What assumptions are made until now? 3. What is the magnitude of the eccentricity of applied load to take into account? a. What are the limitations within the design codes? b. What are the experiences from practice? c. What assumptions are made until now? 4. How should the connections be modelled? a. What influence does the structural model have concerning the connections of the model? b. What are the experiences from practice? c. What assumptions have been made until now? 1.3. Research objectives The situation stated in previous paragraph could be improved if graphs would be generated for the Poly falsework system design in civil engineering applications. The goal within this research is to develop calculation models for stacked structures of the Poly falsework system under higher design loadings. 2

21 1.4. Research relevance and contributions This research is relevant to SAFE BV. This company is specialized in designing scaffolding and formwork systems. Developments in the built environment have led to requirements of stacked structures under higher loadings. SAFE BV would benefit from the creation of a calculation model for designing and calculating the systems that represents the reality as accurately as possible and will lead to an economical easier-to-compose, yet safe, design as desired by the company Research method Figure 1.2 shows a visualization of the research model. This model is based on the sub-questions given in Section 1.2. As can be seen, the approach of this study is built up in steps. Figure 1.2. Research model 3

22 During phase 1, literature about Poly falsework systems will be studied to get familiar with the subject. Furthermore, other literature related to the study will be internalized. Requirements and guidelines given by design codes referring to falsework structures are also of interest. Furthermore, research done by former students must be studied and potentially revised, the assumptions in particular. The result of this literature study will be a literature review, another document associated with this graduation project: Load bearing capacity of stacked Poly falsework system for higher design loadings (Literature review). After literature study, the boundary conditions and requirements within the subject of this study will be determined. Different factors can influence the magnitude of the ultimate load and thus the bearing capacity. Therefore, the research question can be divided into sub-questions, each representing each factor (see Section 1.2). The result of the sub-questions should lead to the answer to the research question. The horizontal coupling of a system scaffold will provide a single prop within the system with stability, by ledgers (see Figure 1.3) acting like a spring constraint with rotational spring stiffness. Until now, conservative assumptions have been made and this effect is therefore being neglected. For this research, this effect is considered to be the most interesting one, and will be studied in detail. Figure 1.3. Example of a configuration to be examined First the significance of the substitution of the hinge by a spring will be examined with an unelaborate calculation model using finite element software ABAQUS. Supposing that changing the connection in the model leads to significant increase in load bearing capacity in the simplified model, the spring stiffness of the connection then needs to be experimentally determined. Whether a spring connection can function as a fixed joint, depends on the magnitude of the spring stiffness. The spring stiffness has to be greater than the critical spring stiffness. If the spring can be seen as a rigid connection, the buckling length can be reduced by half, resulting in a higher ultimate load, see Section If the spring stiffness is smaller than the critical spring stiffness, expected is that the connection will still 4

23 have its positive influence on the load bearing capacity of the Poly Prop and thus the stacked Poly falsework system. Firstly, the rotational spring stiffness of the joint between a Poly Ledger and a Poly Prop (see Figure 1.3) will thus be determined experimentally. Then, this obtained value will be used to do numerical analysis with. Results will be compared with those from conservative approach and the significance of the influence will be examined. Furthermore, focus will also be, although less, on the initial imperfections and load eccentricity. The initial imperfections of the structure can be investigated in more detail. Different design codes provide different calculation methods for determining the initial imperfections, consisting of sway imperfections and bow imperfections. After examining the different design codes, the one that most accurately represents reality will be used to determine the initial imperfections for structural modelling. Next step in this research process shall be numerical analysis, performed using the finite element software ABAQUS. The spring stiffness obtained from experiments will be assigned to the spring connection in the models of different configurations. Figure 1.3 shows one of these configurations. Furthermore, the initial imperfections determined according the chosen design code will also be assigned to the structural model. However, due to the application of new elements, a structural model with the original boundary conditions will also be generated. Subsequently, the influence of the parameters can be accurately determined. The assumptions being defined, forming the boundary conditions of the model, the ultimate load of the system can be determined. This characteristic value of resistance will be found by numerical analysis, using the arc-length method. This process will be done for many different configurations and system lengths. The results will then be put in a graph to define the load bearing capacity for different models and different system lengths. Finally, experiments could be implemented to verify results from numerical analysis. The ultimate load of the system with the improved boundary conditions is expected to be greater than the ultimate load of the same system modelled with original conservative boundary conditions, leading to more economical use of material, from which SAFE BV can benefit. 5

24 6

25 2. Overview of literature Research by former TU/e students on the Poly falsework system is mainly used as literature to gain knowledge on this matter. The successive studies are logically extending the former ones. Although this study on the stacked Poly falsework system will also continue on the previous studies, it focusses on the system designed for higher design loadings. The elements used for this purpose are substituted and the traditional combination of inner tubes and outer tubes will not be regarded. Firstly, this chapter will introduce the Poly falsework system with its elements, limited to the elements to be examined. Furthermore, the results from the studies done by previous students will be given. Regulations regarding temporary structures are viewed as well, but given in the next chapter. Finally, boundary conditions and assumptions made by researchers all over the world are studied Description of Poly falsework system SAFE BV designed a patented falsework system: Poly falsework system. This system consists of different elements. Due to expansion in application field, stronger elements are being developed and jacks and middle pieces can be added to the lists of existing elements: Poly Prop (see Figure 2.1): a telescopic steel prop consisting of an inner tube: used to adjust the height of the Poly Prop. Oval holes are punched in the inner tube, with a center-to-center distance of 100mm; an outer tube: with outer tube rings, allowing assembly between Poly Prop and other elements; and a threaded part consisting of a collar and quick release pin, both for the adjustment of the total height of the Poly Prop. Figure 2.1. Traditional Poly Prop 7

26 Poly Ledger (see Figure 2.2): a hollow circular tube used to horizontally connect the props to create a coupled system; Figure 2.2. Poly Ledger Poly Diagonal (see Figure 2.3): a hollow circular tube used to secure the stability of a coupled system; Figure 2.3. Poly Diagonal Middle Piece Civil (see Figure 2.4): variant of the outer tube with baseplates at both ends, for stacking the props; Figure 2.4. Middle Piece 8

27 Jacks (see Figure 2.5): can be assembled in an outer tube or middle piece, applied in case of high design loads instead of inner tubes. Figure 2.5. Jack The Poly support system is a flexible system. In the commercial construction branch there is a considerable variation in construction designs. As a result, varied floor heights, repeating sheathing, voids and cantilevered balconies and floor levels at random locations are common. Naturally, a flexible system that can anticipate on these variations in one system is then needed. The Poly support system consists of a large number of different parts. These parts all have fixed system dimensions. The most traditional combination of elements used for formwork support includes the inner tube, the outer tube and the threaded part; these parts combined are called a Poly Prop, as shown on Figure 2.1. The Poly Prop is designed to carry vertically applied compressive loads. Poly Props can be coupled using a Poly Ledger (Figure 2.2). Nowadays, Poly Diagonals (Figure 2.3) are used to transfer the horizontal loads. In former times the so-called Poly Frame (Figure 2.6) was responsible for the stability of the coupled system. In addition to coupling the props horizontally, the props can also be vertically stacked. To compose a vertically stacked system, the elements are designed in such a way that all that needs to be done is to mirror the existing props along the horizontal axis. Between the mirrored parts, where the props are put on top of each other, a connector is used to concentrate the joint. Figure 2.6. Poly Frame 9

28 Recently, new parts are in the market: the Middle Piece and the Jack, see Figure 2.4 Figure 2.5. These elements are often used in civil construction, because of the greater bearing capacity. These elements substitute the inner tube, threaded part and outer tube and form the prop for application with higher design loadings. The following subsections describe the elements to be used in this study Middle Piece A middle piece is a variant of the outer tube (see Figure 2.1) with an improved cross section, see Figure 2.7; compared to the cross section described in section of the Outer Tube, this section is slightly more rectangular, and therefore has a greater stiffness. The middle piece has baseplates at both ends as shown in Figure 2.4. These elements are used for stacking the props. The element properties are shown in Table 2.1. Figure 2.7. Comparison cross sections Table 2.1. Element properties Middle Piece Civil Section see Figure 2.7. Comparison cross sections Length 2000mm Thickness 2mm Diameter (outer) 100mm Diameter (inner) variable Steel grade S355 Yield strength 420N/mm Jack Inner tubes, used in traditional scaffold systems, are substituted by jacks (Figure 2.5) when higher design loads are present. Jacks can be assembled in an outer tube or middle piece. These jacks are to be used in combination with Middle Piece 2000 Civil and Middle Piece 4000 Civil. This variant has a total height of 1400mm and an outer diameter of 63.5mm. See Table 2.2 for the element properties of the jack. Table 2.2. Element properties Jack Section trapezoidal thread Length 1400mm Thickness 5mm Diameter (outer) 63.5mm Diameter (inner) 58.5mm Steel grade S355 Yield strength 420N/mm Poly Ledger A horizontally coupled system can be created by adding a Poly Ledger to the singles props (see Figure 1.1). The Poly Ledger is fastened at the outer tube ring of the Poly Prop, see Figure 2.2. Table 2.3 shows 10

29 the element properties of the ledger. The assembly of the Poly Ledger onto the Poly Prop is visualized in Figure 2.8. Table 2.3. Element properties Poly Ledger Section hollow circular tube mm, 500mm, 900mm, 1200mm, Length 1575mm 1, 1800mm, 2100mm, 2400mm, 2700mm, 3000mm, mm Thickness 3.2mm Diameter (outer) 48.3mm Diameter (inner) 41.9mm Steel grade S235 Yield strength 235N/mm 2 1 not standard Figure 2.8. Fixing a Poly Ledger: 1) Hook in; 2) Turn wedge; 3) Secure wedge Poly Diagonal Poly Diagonals are used to stabilize a coupled system (Figure 2.3). These elements can easily be attached to the Poly Ledgers; the assembly of a Poly Diagonal onto a Poly Ledger is visualized in Figure 2.9. Element properties are shown in table 2.4. Table 2.4. Element properties Poly Diagonal Section hollow circular tube 900/850 1 mm, 1200/850mm, 1200/1700mm, 1800/850mm, 1800/1700mm, 2100/850mm, Length 2100/1700mm, 2400/850mm, 2400/1700mm, 3000/850mm, 3300/850 1 mm Thickness 2.65mm Diameter (outer) 42.4mm Diameter (inner) 37.1mm Steel grade S235 Yield strength 235N/mm 2 1 not standard Figure 2.9. Fixing a Poly Diagonal: 1) Fix; 2); Secure; 3) Original position. 11

30 Connector Figure 2.10 shows the connector, see Table 2.5 for the element properties. The connector is placed into the outer tubes or middle pieces (Section 2.1.1), where these elements are stacked on top of each other. A steel bar with a height of 6mm is welded to the tube, which prevents the connector from falling into the outer tube or middle piece and centers the vertical load. Together with the two halves of 150mm, the connector has a total height of 300mm. The nominal outer diameter is equal to the inner tube s: 63.5mm. In case high loadings are present, the baseplates of the outer tube/middle piece are bolted, in order to transfer the loadings. Figure Connector [7] Table 2.5. Element properties connector Section hollow circular tube Length 300mm Thickness 2.5mm Diameter (outer) 63.5mm Diameter (inner) 58.5mm Steel grade S235 Yield strength 235N/mm Load bearing capacity of a single Poly Prop (van Geijtenbeek, 1993) Study of the ultimate load of a single Poly Prop, consisting of an inner tube and an outer tube, was carried out by van Geijtenbeek (1993). The formula given for the ultimate load is as follows:. Eq. 2.1 where F c = ultimate load in deformed state [N] M p = reduced plastic moment for F E [Nmm] w 0 = initial deflection [mm] 12

31 F E = Euler buckling load [N] Van Geijtenbeek (1993) used four different models for analytical analysis. Comparing the results from analytical analysis with those from numerical analysis, there could be concluded that numerical model provides the best results. The best analytical model gives values 20% higher than results from numerical analysis (van Geijtenbeek, 1993). After Kuijer s revision of van Geijtenbeek s study (1993), the boundary conditions are altered. The boundary conditions have influence on the calculation of the buckling load. Kuijer investigated another variant of schematization, see Figure Figure Comparison schematization van Geijtenbeek (1993) Kuijer(1996) By following an iterative procedure the exact value of the ultimate load is approximated. Figure 2.12 shows the ultimate load F c for different lengths of the Poly Props. The graph is confined by a constant load: 125kN. It is experimentally determined that for an axial load F = 125kN, the first plastic hinge appears at a hole of the inner tube. Failure occurs at the pin connection for 125kN. Figure Ultimate load second order analysis (van Geijtenbeek, 1993) 13

32 The influence of different boundary conditions on the Euler buckling load is examined. Figure 2.13 shows the result for three different models, being: Prop hinged at both ends; inner tube hinged, outer tube clamped to create fixed boundary conditions; outer tube hinged, inner tube clamped to create fixed boundary conditions. Figure Ultimate load second order analysis with different boundary conditions (van Geijtenbeek, 1993) A prop rigidly supported at one end gives a significantly larger ultimate load, compared to a prop modelled with hinge supports at both ends. In practice, the props are not hinged at both ends, nor are they rigidly supported. To be on the safe side, these supports are modelled as hinges, while in general they are a spring with varying, unknown spring stiffness. Furthermore, laboratory experiments are done to verify results from numerical analysis. Twelve tests in total were performed, divided into three series. More detail about the test procedure is to be found in Rekenmodel voor de draagkracht van SAFE-stempels (van Geijtenbeek, 1993). The average value of the ultimate load of each series is calculated to compare the results from experiments to the theoretical results. The comparison is graphically visualized in Figure Figure Comparison numerical analysis/laboratory experiments (van Geijtenbeek, 1993) 14

33 From this figure, there can be concluded that graph A (prop hinged at both ends) gives results at the safer side, which is a conservative approach. To approximate the ultimate load more accurately, further study about the boundary conditions is required and recommended Load bearing capacity of a coupled Poly system (Kuijer, 1996) Kuijer continued the study by examining the bearing capacity of coupled Poly Props. The stability of the coupled system is ensured by a Poly Frame, placed between two single Poly Props, which are coupled by the former coupling rods. The ultimate load of one single prop is the lower boundary for the capacity in coupled situation. The bearing capacity of horizontally coupled props depends on two factors, being: length of the prop; number of props coupled in proportion to number of frames applied. Note: The frames and coupling rods are no more in use nowadays. They are replaced by Poly Ledgers and Poly Diagonals. Calculation procedure Kuijer approached the coupled system following calculation procedures from EN 1065 and TGB Staalconstructies. For being aware that EN 1065 Stalen schroefstempels merely gives guidelines for single props, the possibility that different calculation procedures are more suitable for the coupled situation is indicated. Based on two different configuration models, determination of the bearing capacity is done following two working procedures: Finite Element Method, where a non-braced framework is modelled; Analytical analysis done by hand calculations and Finite Element Method, where a supported framework is used. This framework is divided into two parts, being a supportive part and a part being supported. The assumptions used for modelling and analysis are as follows: The ability of transferring bending moments in the connections between a prop and the frame and between a prop and the coupling rod are not taken into account; The formwork on top of the prop is prevented from horizontal displacements. The head of the prop is connected to the formwork, thus also prevented from horizontal translations. In the presence of an axial force in the prop, both head and base of the prop are able to transfer bending moments. The magnitude of bending moment to be possibly transferred depends of the magnitude of the axial force. Both base and head can therefore be seen as spring restraints. This effect is also not taken into consideration; The initial deflection of a prop consists of the initial deflection due to tolerance between the inner tube and the outer tube, and the initial deflection due to imperfections of the elements of the Poly Prop as well. The initial deflection is taken into account following NEN 6771; The overlap length is considered as a homogeneous section with its own section properties; The cross section properties are given in Table 2.6. The material properties are given by the supplier. The yield strength f y;d of the inner tube and outer tube is 420N/mm 2. The yield strength of the frame elements as well as the coupling rod is 235N/mm 2. Table 2.6. Section properties Poly elements (Kuijer, 1996) Area A [mm 2 ] Moment of inertia I [mm 4 ] Plastic section modulus W pl [mm 3 ] Inner tube Outer tube

34 Area A [mm 2 ] Moment of inertia I 1 [mm 4 ] Plastic section modulus W pl [mm 3 ] Overlap area Coupling bar Horizontal frame bar Vertical frame bar Frame Diagonal Given for the weakest axis 2 Given for area with oval pin holes taken into account 3 The axial force in the inner tube is transferred to the outer tube by the quick release pin. Therefore, the cross section area for the overlap area is assumed to be the same as the outer tube s 4 The bending moments and deformations are distributed and resisted by both the inner tube and outer tube, explaining the summation of the moment of inertia and plastic section modulus of both tubes. The configurations modelled are characterized by the number of applied frames in proportion to the number of props coupled. The inner tube is adjustable and therefore the total prop length varies between 2360mm and 3920mm. NEN 6770 art gives the classification of the structure in type of configuration. The falsework system is a framework structure, which can be classified into two classifications, being: non-braced framework, and supported framework. Both were investigated by Kuijer. Nine variants of the non-braced framework were modelled in Ansys and calculated, examples of the configurations are given in Figure Investigating failure of this kind of structure consisted of two parts: determining response of the structure and checking Ultimate Limit State. Figure Variants of configurations (Kuijer, 1996) The load for which the first plastic hinge occurs in the structure has to be defined. This load can be found by an iterative process where the load increases stepwise. The first plastic hinge occurs when the unity check equals 1. Finding the ultimate load using the finite element method proceeds in two steps: 16

35 1. The vertical axial load on the props is increased stepwise. After every step the bending moment and the normal force in every node is determined by geometric non-linear analysis, for which Ansys uses the full Newton-Raphson solution procedure. Next the unity check is done, where M p and N p are the element properties of the element on the nodes. For a unity check < 1 for every node in the structure, next load step is to be checked. This continues until a unity check result 1 is found. 2. The ultimate load has to be found following the bisection method. While the last load step found at 1., for which the unity check result is 1, is the upper bound, the second-to-last load step is the lower bound. Again following an iterative process, the ultimate load is to be found. For every step the load is (upper bound + lower bound)/2. Then a unity check is carried out at every node. If the greatest value of all the results from the unity checks is greater than 1, this will then be the new upper bound. Similarly, if the greatest value of all the results from the unity checks is smaller than 1, this will then be the up-to-date lower bound. This process continues until the difference between the upper bound and lower bound is smaller than a value defined beforehand, chosen to be 0.01kN by Kuijer. For the lowest ultimate load (the ultimate load of the prop with the greatest length = 3920mm) this means a deviation of 0.03%. After calculation of the non-braced framework model, the system was also examined as supported framework, which is shown Figure Figure Supported framework schematization (Kuijer, 1996) The supported framework can be divided into two parts: a supported structure and a supportive structure, see Figure The response of the structure and the checking of the Ultimate Limit State is therefore done separately, where two different failures can cause the whole structure to collapse: 1. failure in the prop attached to a horizontal spring (see Figure 2.17(a)); 2. failure of the field with frame. Figure 2.17(a). Prop attached to horizontal spring (b). Spring stiffness (Kuijer, 1996) 17

36 Kuijer (1996) describes the calculation method with a horizontal spring in more detail. The results from both methods modelling the coupled system both as a non-braced framework as well as a supported framework are rather similar, which makes the design graphs reliable. Results and conclusions The result for all variants of configuration is put into a graph on Figure 2.18, where the bearing capacity is plotted against the prop length. Figure Bearing capacity coupled system (Kuijer, 1996) draagkracht per stempel = capacity of each prop stempellengte = length of the prop stempel = prop vrijstaande stempel = single prop veld met frame = field with frame As van Geijtenbeek experimentally determined that the prop fails for F d = 125kN, Kuijer proved the same following guidelines given by design code EN 1065 art From Kuijer s research the following is concluded: The bearing capacity of the configurations examined is given in the graph on Figure 2.18; The results from modelling the coupled system both as a non-braced framework and a supported framework are rather similar, which makes the design graphs reliable; A horizontally coupled system leads to significantly greater bearing capacity, also for a higher ratio between the number of frames and number of props coupled; For a greater length of the props, the prop fails due to instability; If the total height of the structure is smaller, however, the frame fails due to insufficient strength; The following assumptions were made, while it is not known what influence these simplifications may have on the value of bearing capacity: The prop is hinged at both ends; The coupling rod is eccentrically connected to the prop, but the eccentricity is not taken into account; The tolerance between the frame, coupling rod and props are neglected. 18

37 2.4. Load bearing capacity of a stacked Poly system (Lamerichs, 2011) A Poly falsework system can easily be vertically stacked by mirroring the elements along the horizontal axis. By the time Lamerichs did her research to find the ultimate load for a vertically stacked Poly falsework system (consisting of traditional elements), Poly Frames has no longer been use. This Poly Frame is substituted by Poly Ledgers and Poly Diagonals. Calculation procedure Lamerichs first verified Kuijer s results by remodelling his configurations. Linear buckling analysis is done for the system with Poly Frames, modelled in Ansys with BEAM3-elements. While a BEAM3- element is two-dimensional, a BEAM189-element is three-dimensional. The element has six degrees of freedom and is suitable for analyzing slender to moderately thick beam structures. Results show negligible difference between BEAM3 and BEAM189. Therefore, Lamerichs used the same procedure as Kuijer, but modelled with BEAM189 and different element properties (see Table 2.7). Sensitivity analyses were done to view the influences of the assumptions done in modelling the configurations, concerning the following parameters: initial deflection; structural model; load eccentricity. Table 2.7. Section properties Poly elements (Lamerichs, 2011) Area A [mm 2 ] Moment of inertia I [mm 4 ] Plastic section modulus W pl [mm 3 ] Inner tube Outer tube Overlap area Poly Ledger Poly Diagonal Determined by computer software Solid Works An example of the configurations is shown in Figure Figure Example of configuration (Lamerichs, 2011) Many different models of configurations were regarded due to the continuous variables: The height of the single prop or horizontally coupled system can vary between 2370mm and 3915mm. The systems are modelled in height steps of 100mm, except for the first and last step. Therefore, the different heights being modelled are 2370mm, 2450mm, 2550mm, 2650mm, 2750mm, 2850mm, 2950mm, 3050mm, 3150mm, 3250mm, 3350mm, 3450mm, 3550mm, 19

38 3650mm, 3750mm, 3850mm and 3915mm. For the stacked system the height step is 200mm: 100mm per prop. The number of height steps therefore remains the same; Center-to-center distance of the horizontally coupled system can vary with the length of the Poly Ledger applied. Lamerichs included the influence of the length of the Poly Ledger on the ultimate load of the system in her research, and there can be concluded that there is no influence. All Ledgers are modelled with length = 1800mm, which is most commonly applied in practice. Calculations of the many different configurations were all done with following assumptions: The base of the Poly Props are modelled as hinges, this is a conservative assumption because depending on the amount of normal force in the Poly Prop, the base can bear bending moment; The top of the Poly Props are modelled as vertical rolls; The connection between the Poly Diagonals Poly Ledgers and the Poly Ledgers Poly Prop are all modelled as hinges. This is a conservative approach, because the connection Poly Ledger Poly Props can transfer a small bending moment, which would result in a higher ultimate load than determined in Lamerichs thesis; The load applied with an eccentricity of 5mm, which is prescribed in EN 12812; The initial deflection of the system is the shape of the first buckling mode, with a value of 1/500 of the total height of the system; The configurations are calculated with a center-to-center distance of the Poly Props of 1800mm; The connection between the two Poly Props for the stacked systems is modelled as a fixed connection; The eccentricity between the connector and the outer tubes is not taken into account; The ultimate load can never be higher than 125kN, because this is the load at which failure of the quick release pin occurs. Lamerichs (2011) describes the sensitivity analyses in more detail. The conclusions following from these sensitivity analyses are included in further numerical analyses of the structural model. Results and conclusions Finally, Figure 2.20 shows the ultimate load of the vertically stacked system is determined for the systems modelled. Ultimate load [kn] Total height [mm] Figure Ultimate load vertically stacked systems (Lamerichs, 2011) 20

39 For increasing total height of the system, the ultimate load of the different configurations merges towards circa 20kN. Except for the systems with two props and three props, the graphs of the different configurations all have the same shape. For the shortest lengths of these two systems, the yield stress is reached at the top of the Poly Prop. For configurations with four and more props, the yield stress is reached at the connection between Poly Ledger and Poly Prop, see Figure Figure 2.22 shows that for all greater system heights, the yield stress is reached at the transition between inner tube and outer tube. The yield stress is reached at the same combination of bending moment and normal force for each configuration; this explains the mergence of the graphs in Figure Figure Stress figures at ultimate load Figure Stress figures at ultimate load 2.5. Effect of specific assumptions and boundary conditions Experimental and numerical studies of practical system scaffolds have been done by several researchers in this field. Mainly load capacities and failure modes of falsework systems in various setups in construction are investigated. Many different factors in structural modelling can influence the outcome of analyses of structural behavior. The following research papers published in the last few years were studied: Structural Modeling and Analysis of Modular Falsework Systems (Peng et al., 1997) Analytical and experimental bearing capacities of system scaffolds (Peng et al., 2009) Experimental and numerical studies of practical system scaffolds (Peng et al., 2013) The different parameters for system scaffold modelling are shortly described in the next subsections Buckling length reducing mechanism The influence of a buckling length reducing mechanism is examined in 1972 (Bakker, 1972). Figure 2.23 shows a horizontally coupled falsework system, loaded with a point load P. The props are supported with a ledger halfway its length. The coupling of the props by the ledger is assumed to provide a support force Q (see Figure 2.24), depending on the spring stiffness, following: 21

40 where c = spring stiffness [N/mm] (see Figure 2.23) w = displacement [mm] (see Figure 2.23) Eq. 2.2 Figure Schematization horizontally coupled system (Bakker, 1972) Figure Support force Q (Bakker, 1972) Theory of elasticity If the props are assumed to be centrically loaded, the buckling load can be determined by solving the following equation (Bakker, 1972): sin 1 2 sin cos1 0 Eq where For each spring stiffness c, the buckling load P can be determined. There are two extreme situations: (see Figure 2.25) (for derivation in detail see Bakker, 1972) buckling load buckling length 0 (see Figure 2.25) (for derivation in detail see Bakker, 1972) buckling load buckling length As the spring stiffness is in between, the given expressions of the buckling load are the upper and lower boundaries:, for. 22

41 Figure Buckling mode for two extreme situations (Bakker, 1972) Effect of boundary conditions Both fixed end and hinged end were examined. Changing the joint stiffness of the system scaffold altered the critical load. The fixed end and hinged end can be regarded as the upper bound and lower bound of bearing capacities, respectively, for the system scaffold. The top and bottom boundary conditions of the system scaffold were identical because their base-plates were almost the same. No unequivocal conclusion could be drawn about the significance of the influence of the boundary conditions. The outcomes of two different researches are slightly different. The fact that hinged end condition will lead to reduction of load bearing capacity is undoubtedly, but whether this is significant could not be stated Effect of different base heights Figure 2.26 shows a structural model with different base heights. Figure Varied ground heights (Peng et al., 2013) While the outcome of one study (Peng et al., 2013) suggest that the setup of varied ground heights have insignificant influence on load bearing capacity of system scaffold, it can generally be claimed that it is likely that as the quantity of extended vertical props increases, the critical load of the system scaffold system decreases (Peng et al., 2009). 23

42 Effect of joint positions Figure 2.27 is a drawing of different joint positions in the system scaffold. Figure Different joint positions (Peng et al., 2009) While one study shows that the joint positions should be kept away from story-to-story joints, another study claims that the varying joint has insignificant effect on the load capacity Effect of diagonal brace installations Figure 2.28 shows different bracing installations of a two-story system scaffold. Figure Different bracing installations (Peng et al., 2009) Regardless of the bracing installations, diagonal braces substantially enhance the critical load of a system scaffold. Case B-installation shown in Figure 2.28 is recommended, as this reversed parallel story-to-story diagonal braces-set-up leads to the highest critical load Effect of scaffold height The scaffold height is directly related to the number of stories. Both studies (Peng et al., 2009) (Peng et al., 2013) show that the critical load decreases with the increase of number of stories, and thus with the increase of scaffold height Effect of shoring extensions Figure 2.29 shows a falsework system with and without shoring extensions. 24

43 Figure Two-story modular falsework system with and without shoring extensions (Peng et al., 1997) According to this example (Peng et al., 1997), the use of shoring extensions is the most significant factor that reduces the load-carrying capacity of modular falsework. Large horizontal displacements were observed in the direction along the plan of the modular units. The instability of the total system is aggravated by the horizontal force component introduced by the shore links. The use of shoring extensions thus has a negative influence on the load bearing capacity. 25

44 26

45 3. Initial imperfections Initial imperfections are unavoidable in structures. The second order effect becomes larger with greater initial imperfections. The larger the initial imperfections are, the smaller the ultimate load on the structure is. Therefore, it is favorable to determine the initial imperfection of the Poly system as accurate as possible. Firstly, this chapter describes this matter on the basis of guidelines given by different design codes. Then these design codes will be compared with each other, followed by a conclusion, concluding which code to continue with for structural modeling Design Codes The following design codes are described in this section: DIN 4421, EN 12812, NEN 6770 and EN (Eurocode 3). Two of the design codes being described in this section are no longer in use: DIN 4421 and NEN To internalize the developments of the guidelines on the initial imperfections, however, these codes are as well examined DIN 4421 DIN 4421, a German code published in 1982, gives the following guidelines with regard to geometrical imperfections. Unavoidable imperfections in the geometry of the support frame can influence the stress distribution of the scaffold components significantly. They are therefore to be considered in stability analysis. If accurate confirmation is omitted and appropriate imperfection for individual components are not already defined in standard technical specifications, the following unavoidable eccentricities and imperfections may be assumed. These are valid for columns, including the installations necessary to adjust the column length. Bow imperfection For columns and beams the following bow imperfection may be assumed: where f = bow imperfection [mm], see Figure 3.1 l = height of the member [mm] Eq. 3.1 Figure 3.1. Bow imperfections (DIN 4421) 27

46 This value is allowed to be mitigated with increasing number of n props coupled. For members with equal properties, while neglecting deformations due to systematic influences i.e. deformation within the framework, the following formula can be applied: where n = number of columns Eq. 3.2 Sway imperfection The sway imperfection is given by. Eq. 3.3 for columns with height l < 10m (see Figure 3.2) and;. Eq. 3.4 where for columns with height l > 10m l = height of the column [m] Figure 3.2. Sway imperfections (DIN 4421) Eccentricity of load A load eccentricity of 5mm on top of the column or the jack needs to be taken into account, where there is no centring device EN EN gives guidelines for falsework design in general. This European Standard specifies performance requirements and limit state design methods for two design classes of falsework. The influence of imperfections such as the following shall be taken into account: eccentricities of loads; angular imperfections and eccentricities caused by looseness of the joints; divergence from the theoretical axes (bow imperfections, sway imperfections). Eccentricity of load The load eccentricity at load points shall be taken as a minimum of 5mm where there is no centring device. Where there is a centring device, the eccentricity taken may be reduced to a value consistent with the tolerances of the relevant components. Angular imperfections caused by looseness of the joint For single tubes, angular imperfections, 0, from the theoretical position shall be calculated for loose 28

47 items from the nominal dimensions of the components. The angular imperfection, 0, between two components shall be calculated using Eq. 3.5:. / Eq. 3.5 where 0 = the angle, between two components or loose items, see Figure 3.3 [rad] d i = the specified inner diameter of the tube [mm] d 0 = the specified outer diameter of the spigot of jack spindle [mm] l o = the overlap length [mm] Figure 3.3. Spigot angular imperfections (EN 12812) If there is more than one upright in a row, the angle, at a joint to be used for calculation purposes shall be calculated using Eq. 3.6:. Eq. 3.6 where n v = total number of vertical tubes to be erected side by side Eccentricities caused by looseness of the joint For frame components assembled with spigot joints, the eccentricity, e, between successive vertical frames shall be taken into account. A frame component shall be deemed to consist of at least four members permanently fixed together. For a pair of frames assembled one above another, the value of the eccentricity, e, shall be taken as at least that given by the following equation:. Eq. 3.7 where e = the distance between the axes of two tubular members meeting end to end and all are as shown in Figure 3.4 d i = the specified inner diameter of the tube [mm] d 0 = the specified outer diameter of the spigot of jack spindle [mm] 29

48 Figure 3.4. Spigot eccentricities (EN 12812) Bow imperfections for compression members Compression members shall be assumed to have an initial overall bow imperfection. Stabilizing systems for compression members shall be designed to resist the effect of any bow. This is additional to the member imperfection of a single element, which is defined in EN Figure 3.5 illustrates the overall bow imperfections for a compression member. The bow imperfections used in the design shall not be less than those in EN Figure 3.5. Bow imperfection (EN 12812) The value to be assumed for the lateral displacement of deviation from a true line for a compression member subjected to bending can be calculated as follows: Eq. 3.8 where e = bow imperfection [mm] l = nominal length of the compression member [mm] r = reduction factor = where n v = number of structural components arranged and supported side by side and propped in the same way 30

49 Sway imperfections for compression members The sway imperfection 1 (see Figure 3.6) for a structural component taller than 10m shall be calculated using:. Eq. 3.9 where ϕ = angular deviation from the theoretical line [rad] h = overall height of a compressive member or tower [mm] For structures where h is less than 10m, tan shall be taken as Figure 3.6. Sway imperfection (EN 12812) NEN 6770/NEN 6771 Taking all aspects into account, NEN 6771 is given to examine the developments of the codes over the years. NEN 6771 is the precursor of Eurocode 3, and no longer in use. In the determination of the distribution of the forces the bow imperfection needs to be taken into account, in combination with the sway imperfections. These formulas do not take the residual stresses into account. The influence of residual stresses is included in unity check equations. Bow imperfections The bow imperfection, or the equivalent notional load is given in Figure 3.7. Figure 3.7. Bow imperfections (NEN 6771) geometrische rekenimperfectie = geometric imperfection; equivalente fictieve belasting = equivalent notional load 31

50 It can be determined following Eq. 3.10: ;. ; Eq where w 0 = bow imperfection [mm] k, rel = parameters given in Table 25 by NEN 6770 M u;d = elastic moment capacity of the member [Nmm] f y;d = yield strength [N/mm 2 ] A = cross sectional area [mm 2 ] Sway imperfections The sway imperfection, or the equivalent notional load is given in Figure 3.8. This geometrical imperfection is described by Eq. (3.10). Eq where ϕ = sway imperfections ϕ 0 = 1 250for braced framework; for unbraced framework k 1 = where n s = number of layers in the framework k 2 = where n k = number of loaded columns in which an axial force is of at least 50% of the average normal force in the plane being considered. This value must not be greater than the number of columns that spans the greatest number of floors. Figure 3.8. Sway imperfections (NEN 6771) geometrische rekenimperfectie = geometric imperfection; equivalente fictieve belasting = equivalent notional load EN EN states that appropriate allowanced should be incorporated in the structural analysis to cover the effects of imperfections, including residual stresses and geometrical imperfections such as lack of verticality, lack of straightness, lack of flatness, lack of fit and any minor eccentricities present 32

51 in joints of the unloaded structure. The effects of these imperfections are included in the formulas given in this section. Imperfections for global analysis of frames The assumed shape of global imperfections and local imperfections may be derived from the elastic buckling mode of a structure in the plan of buckling considered. For frames sensitive to buckling in a sway mode the effect of imperfections should be allowed for in frame analysis by means of an equivalent imperfection in the form of an initial sway imperfection and individual bow imperfections of members. The imperfections may be determined from: global initial sway imperfections, see Figure 3.9; relative initial local bow imperfections of members for flexural buckling. Figure 3.9. Equivalent sway imperfections (EN ) Global initial sway imperfections Eq where ϕ = global initial sway imperfection ϕ 0 = basic value = h = reduction factor for height h applicable to columns = ; for α 1.0 where h = height of the structure [m] m = reduction factor for the number of columns in a row = where m = number of columns in a row including only those columns which carry a vertical load N Ed not less than 50% of the average value of the column in the vertical plane considered Relative initial local bow imperfections of members for flexural buckling / Eq where e 0 = bow imperfection [mm] L = member length [mm] The values of e 0 /L may be chosen in the National Annex. Recommended values are given in Table 3.1. Table 3.1. Design values of initial local bow imperfection e 0 /L (EN ) Buckling curve according to Tables Elastic analysis Plastic analysis 23 & 25 given in NEN 6770 e 0 /L e 0 /L a 0 1/350 1/300 a 1/300 1/250 b 1/250 1/200 c 1/200 1/150 d 1/150 1/100 33

52 3.2. Comparison design codes The initial imperfection parameters are given in one table for comparison, see Table 3.2. Table 3.2. Initial imperfections by different design codes l 10m Sway imperfection l > 10m DIN 4421 tan φ 0.01 tan φ 0.1 EN tan φ 0.01 tan φ Bow imperfection Load eccentricity 5mm mm ; NEN 6771 tan φ φ α 0.2 ; EN tan φ φ α α / For explanation of parameters, see Eq. 3.2 to Eq included in geometrical imperfection equations Study of the influence of load eccentricity on the ultimate load was done; there can be concluded that for less eccentricity, the ultimate load increases (Lamerichs, 2011). A detail of the centring device is checked to determine whether it is possible to reduce the eccentricity in modelling. A M12 slot bolt has a nominal diameter of 12mm and the opening in the Alu-beam is 22mm. Therefore, the distance between the center line of the bolt and the Alu-beam can be 5mm, for which the eccentricity of the applied load cannot be reduced to smaller than 5mm (Lamerichs, 2011). As the eccentricity of load is 5mm for the Poly falsework system irrespective of which design code is to be used, load eccentricity is being disregarded in the comparison of the codes. The formulas for determining initial imperfection, including bow imperfection and sway imperfection are put into a graph for comparison, see Figure As can be seen, NEN 6771 is eliminated in the comparison. More detail about this procedure is given in Appendix A. 160 Initial imperfections at 1/2l, e [mm] EN EN DIN Height of structure, l [m] Figure Initial imperfections for different height of structure, according to different design codes 34

53 In general, the German code DIN 4421 is least conservative, followed by EN , while EN takes the largest initial imperfections into consideration. For structures with a height l < 10m, the difference is likely insignificant. However, for structures with a height l > 10, the difference increases up to factor greater than 1.5 at 20m height. This difference might be due to the developments with the passing of time: the regulations have become stricter and experiments are done in practice to support the theory (see Appendix A) Conclusion In order to decide which design code to continue with, some conclusions are stated below, for each design code. DIN 4421 The German code, from 1892, appeared to be the least conservative one. Following this design code will definitely lead to the greatest ultimate load. This design code is, however, no longer valid. Experiments and experience over the years have led to adjustments in the guidelines and this code seemed to be too advantageous. EN The European code for temporary structures, from 2008, appeared to be the most conservative one. Following this design code will lead to the smallest ultimate load. The distinction between different geometrical imperfections (bow, sway, load eccentricity, spigot eccentricities, spigot angular imperfections) is clearly described. NEN 6770/NEN 6771 As not only height of the structure and the number of coupled props, but also several other parameters as elastic moment of elasticity is considered in the determination of the bow imperfection, the implementation becomes complicated. Besides, this Dutch design code is no longer in use today. EN The current European design code, EN , is published for permanent structures. The height of the structure only has its influence between 4m and 9m. The number of props is included in the equation for determining sway imperfections, while the other design codes include the number of props in the formula for determination of the bow imperfections. This Eurocode 3 clearly states that the formulas for determining the initial imperfections include many (see Section 3.1.4) geometrical and materialistic imperfections. As falsework systems are temporary structures and based on the in the above described considerations, the initial imperfections will be determined following EN

54 36

55 4. Laboratory experiments: axial strength Experimental determination of the strength of the connection between a Poly Ledger and a Poly Diagonal Elements of the Poly falsework system are tested in the Pieter van Musschenbroek Laboratory: a sample consisting of a Middle Piece, a Poly Ledger and a Poly Diagonal. The purpose of this test series is to specify the strength of the connection between a ledger and a prop. This structure may fail due to failure in this connection before the capacity of all other elements is reached or failure due to instability occurs Laboratory experiment Figure 4.1 shows a system scaffold built of elements of the Poly falsework system. The connection of the diagonal onto the ledger was tested: the diagonal was axially loaded until failure in the joint occured (it was assumed that the tensile strength of the cross section of the diagonal is greater than the strength of the joint). Figure 4.1. Model of Poly falsework system; framed joint to be tested Individual Poly elements Figures 4.2 to 4.4 show the elements of which the sample to be tested consists of, being parts of: Middle Piece 2000 Civil; Poly Ledger 1200; Poly Diagonal 1200/1700. The element properties of the elements are given in Tables 4.1 to

56 Table 4.1. Element properties Middle Piece Section see Figure 2.7 Length [mm] 2000 Thickness [mm] 2 Diameter (outer) [mm] 100 Diameter (inner) [mm] variable Weight [kg] 10.4 Steel grade S355 Yield strength [N/mm 2 ] 420 Figure 4.2. Middle Piece 2000 Civil Table 4.2. Element properties Poly Ledger Section hollow circular tube Length [mm] 1200 Thickness [mm] 3.2 Diameter (outer) [mm] 48.3 Diameter (inner) [mm] 41.9 Weight [kg] 5.5 Steel grade S235 Yield strength [N/mm 2 ] 235 Figure 4.3. Poly Ledger 1200 Table 4.3. Element properties Poly Diagonal Section hollow circular tube Length [mm] 1200/1700 Thickness [mm] 2.65 Diameter (outer) [mm] 42.4 Diameter (inner) [mm] 37.1 Weight [kg] 5.5 Steel grade S235 Yield strength [N/mm 2 ] 235 Figure 4.4. Poly Diagonal 1200/ Purpose The purpose of the experiment is to determine the strength of the connection between the Poly Diagonal and the Poly Ledger. This connection may fail under loading either before the capacity of other elements is reached or due to instability. The diagonal will be loaded in tension until failure occurs. The number of tests should at least be 5 for reliable results; five tests will be implemented Materials and equipment The test specimen consists of three different Poly elements, all shown in Section Connecting these elements leads to the test sample as shown in Figure 4.5(a). Further equipment required to implement the tests are shown in Figure 4.5(b) to (f): 250kN load cell, see Figure 4.5(b); Mitutoyo clock gauge, see Figure 4.5(c); baseplate as standard (for the specimen), see Figure 4.5(d); bed, see Figure 4.5(e); hydraulic clamp (part of load cell), see Figure 4.5(f). 38

57 Figure 4.5. Test materials and equipment Test method: procedure and implementation Figure 2.9 shows the fixing method of a Poly Diagonal onto a Poly Ledger. The sample of this test series consist of (a part of) a Middle Piece, (a part of) a Poly Ledger (only the part of the connection is needed) and (a part of) a Poly Diagonal, see Figure 4.5(a). The Middle piece is cut at 150mm from the outer tube ring on both ends. As only the connecting part of the ledger is needed, the Poly Ledger is cut at 150mm. The Poly Diagonal is cut at approximately 500mm and connected to the Poly Ledger, which, in turn, is connected to the Middle Piece. Figure 4.6 shows the load application on the sample. Figure 4.6. Load application on sample As the diagonal is attached at an angle of approximately 45, the whole test specimen will be rotated such that the diagonal runs parallel with the vertical axis, see Figure 4.5(a). This is for the purpose of test implementation; by welding a base plate (see Figure 4.5(d)) in advance, the sample can easily be attached to the load cell (see Figure 4.8(a)), so the test could be performed without building a frame to attach the sample to. The test set-up will be explained using both Figure 4.7 and Figure 4.8. First, the sample was connected to the baseplate as shown in Figure 4.5(a). Then, after centrically positioning the diagonal, this sample including the baseplate was fastened to the bed using bolts, see Figure 4.8(b). Furthermore, the 39

58 specimen was retained by a standard (see Figure 4.8(c)), which was fastened to the bed. This is for the purpose of imitating an in-practice situation, where the ledger is coupled to a prop on the other end and thus supported. Figure 4.7. Test set-up The load cell shown on Figure 4.5(b) was used to apply loadings to the diagonal. As can be seen from Figure 4.6, the diagonal was loaded in tension (F). The diagonal was clamped with a hydraulic clamp, (which is part of the load cell,) shown on Figure 4.5(f). As the diagonal has a circular pipe cross section, the pipe was filled with a solid cylinder to enable clamping, see Figure 4.8(d). After positioning the sample and the test equipment, all measuring equipment was set to zero. The load application is displacement controlled in millimetre per minute. The loading happened at 1mm/min and continued until failure of either the diagonal or the joint occurred, where after the sample was unloaded. (Expected was that the joint fails before the diagonal does, the tensile strength of the Poly Diagonal is known.) With a Mitutoyo clock gauge the deformation will be measured, see Figure 4.8(e). To ensure the deformation measured is due to deformation of the pin hole under tensile loading, a Mitutoyo clock gauge is added at the connection, see Figure 4.8(f). This clock gauge is expected to measure a negligible small (approximately zero) deformation. This procedure repeated five times for reliable test results. The force in kilo Newton and the deformation in mm are the output of the tests. 40

59 Figure 4.8. Photos of test set-up 4.2. Results The outputs of the tests were the force in kilo Newton and the deformation in millimeters. Table 4.4 shows the test characteristics and its results. The graph in Figure 4.9 shows an example of the output of this test series. See Appendix B for the force-deformation graphs of all five tests. Table 4.4. Test characteristics and output Test n Speed [mm/min] Comment Force [kn] Deformation 1 [mm] 1 1 Well attached Loosely attached Not hammered in Well attached Strongly hammered in The deformation given in this table is the subtraction of the deformation measured by the Mitutoyo clock gauge in front shown in Figure 4.8(e) by the deformation measured by the Mitutoyo clock gauge positioned at the connection. Section 3.3 gives an explanation in more detail. 41

60 25 20 Force [ kn ] Deformation [ mm ] Figure 4.9. Force-deformation graph; test number 2 Photos were taken during the tests and given in Appendix I-1. Figure shows the sample in undeformed state. Figure Sample in undeformed state Right before failure in the connection occurred, cracks could be observed, see Figure The deformation of the specimen is shown in Figure Photos of all test samples, taken after failure, are given in Appendix I-2. 42

61 Figure Cracks Figure Deformation 4.3. Elaboration and discussion The purpose of this test series is to determine the strength of the joint. All data given from the tests are processed similarly. Table 4.5 shows a part of the output of test number 2. As can be seen, two different deformations were measured. While Mitutoyo(1) measured the total deformation during the tests, Mitutoyo(2) gave results of the deformation due to torsion, as the loading was not applied perfectly centrically. The actual deformation of the connection (at the pin hole) is thus defined as the subtraction of the deformation measured by Mitutoyo(1) by the deformation measured by Mitutoyo(2). The resulting deformation is given in Table 4.5 as well. Table 4.5. Part of output, test number 2; computation of actual deformation Time [s] Force [kn] Mitutoyo(1) 1 [mm] Mitutoyo(2) 2 [mm] Resulting deformation 4 [mm]

62 Time [s] Force [kn] Mitutoyo(1) 1 [mm] Mitutoyo(2) 2 [mm] Resulting deformation 4 [mm] Mitutoyo clock gauge measured the total deformation 2 Mitutoyo clock gauge measured the torsion due to eccentric loading 3 Negative values indicate ingoing direction of the gauge, absolute values are of importance for further computation. 4 Actual deformation defined as the subtraction of the deformation measured by Mitutoyo(1) by the deformation measured by Mitutoyo(2):. The maximum force reached is equal to the capacity of the joint. The mean and standard deviation of the capacity are determined using SPSS Statistics; the results are given in Table 4.6. Table 4.6. Strength of the joint Force [kn] Deformation [mm] Test n Output Mean 1 [kncm/ ] Std. deviation 1 [kncm/ ] Determined using SPSS Statistics Note: From a statistic point of view, the number of tests n should be at least 30 to analyze interval variables. This means that for a number of tests n less than 30, the computation of the mean as well as the standard deviation might lead to unreliable conclusion Conclusion This experiment is implemented to determine the strength of the connection between a Poly Diagonal and a Poly Ledger. The results are given in Table 4.6. As can be seen, the mean value of the capacity of the joint is 21.32kN. As the total number of tests is small, the standard deviation is also acceptable. The deformation of the pin hole of the connection is mostly around 4mm; the mean value is mm. Finally, the standard deviation of the test results is mm, which is also found to be acceptable. 44

63 5. Laboratory experiments: bending stiffness Experimental determination of the strength and rotational spring stiffness of the connection between a Poly Prop and a Poly Ledger The horizontal coupling of a system scaffold will provide a single prop within the system with stability, by ledgers attached to the prop, acting as a spring constraint with rotational spring stiffness. Therefore, this test series is to determine the strength as well as the rotational spring stiffness of the connection between a ledger and a prop. The experiments are carried out following EN Until now, conservative assumptions are made and this effect is therefore being neglected. This change in boundary condition will be studied in detail in this chapter Laboratory experiment Figure 2.1 shows a system scaffold built of elements of the Poly falsework system. The connection of the ledger onto a middle piece was tested. Figure 5.1. Model of Poly falsework system; framed joint to be tested Individual Poly elements Figures 4.2 and 4.3 show the elements to be tested, being: Middle Piece 2000 Civil; Poly Ledger 1200; The element properties of the elements are given in Tables 4.1 and

64 Purpose The purpose of the experiments is to determine the moment capacity of the connection between the Poly Ledger and the Poly Prop, as well as the rotational spring stiffness in this connection. The connection between the ledgers and the props (see Figure 5.1) is modelled as hinges. Substituting this hinge with a spring (local stability) might lead to increase of the ultimate load of the whole system (global stability). In practice, the erection of the system scaffold is done by field workers. In this experiment, human influence will be eliminated. This is done by using a solid block instead of a hammer to attach the ledger onto the prop, see Figure 5.2 and Section 5.2. However, to confirm the comparability of the alternative, both methods are tested first to determine its magnitude of the force. While human influence is eliminated in this experiment to maintain constant circumstances, its effect cannot be neglected in practice. For these reasons, the total number of experiments is divided into the following series (see Table 5.1) : I(a): Force (human hit/human influence); n = 10 I(b): Force (free fall/non-human influence); n = 10 II(a): Strength and rotational spring stiffness joint (negative moment direction); n = 5 II(b): Strength and rotational spring stiffness joint (positive moment direction); n = 5 III(a): Strength and rotational spring stiffness joint (without hammering in); n = 2 III(b): Strength and rotational spring stiffness joint (human hit); n = 1 Note: n = number of tests. The design code states that n should be at least 5 for reliable results. The data obtained from the test need to be, however, approximately equal. Figure 5.2. Attaching the ledger onto the prop with a falling solid block Table 5.1. Laboratory experiments Series I 1 II III 1 a b a b a b Purpose Force to attach the Strength and spring Strength and spring ledger onto the prop stiffness of the joint stiffness of the joint Number of tests Series II is the main test series, series I and III are added for comparison and therefore less number of tests (< 5) is implemented. 46

65 Material and equipment The test specimen consists of only two different Poly elements, both shown in Section Connecting both elements leads to the test sample as shown in Figure 5.3(a). Further equipment required to implement the tests are shown in Figure 5.3 : 250kN load cell, see Figure 5.3(b); hydraulic clamp (part of load cell), see Figure 5.3(c); bracket, see Figure 5.3(d); beam as frame, see Figure 5.3(e); bed, see Figure 5.3(f); inclinometer, see Figure 5.3(g); Mitutoyo clock gauge, see Figure 5.3(h); distance meter/gauge (with greater range), see Figure 5.3(i). Figure 5.3. Test materials and equipment 47

66 5.2. Test Series I: Comparison forces with/without human influence Series I is implemented to measure the force applied when attaching the ledger onto the middle piece. Two different conditions are maintained to exclude human influence. While ten tests are implemented with human influence, series I(b) is done without. The goal is to decide if there is any statistically significant difference Test method: procedure and implementation Figure 2.8 shows the fixing method of a Poly Ledger to a Poly Prop. As can be seen, the wedge is hammered in. This is the case in reality, where field workers hit three times to secure the wedge. To eliminate human influence, however, the wedge could also be secured by a falling block, see Figure 5.2. Thus, series I is divided into two subgroups, representing both conditions: Condition (a): securing the wedge with a hammer; Condition (b): securing the wedge with a falling solid block. The force with which a person hammers on the wedge could not directly be measured. However, the force needed to remove the wedge afterwards can be measured. After securing the wedge, the sample is placed in a 250kN load cell. A displacement controlled test applies a compressive load on the wedge until it detaches; the force at which the wedge detaches, is assumed to be the force with which a field worker hammers in. Figure 5.4 shows this test method. Figure 5.4. Series I; test set-up Series I(b) differs from series I(a) in the method of fixing the ledger. A solid block with mass m = 1.434kg falls from 2.344m (see Figure 5.5 and Figure 5.6) height through a hollow tube on the wedge. The blow on the wedge imitates the blow of a hammer by a human being, see Figure 5.2. Further steps in the test procedure are identical. Note: The design code states that connections using wedges or bolts should be assembled and dismantled three times before assembly for any test. However, used Poly elements were used for these tests; therefore, the connections were not assembled and dismantled three times before assembly for test. 48

67 Figure 5.5. Length of hollow tube Figure 5.6. Mass of solid block Results The graph in Figure 5.7 shows an example of the output of test series I. The force required to remove the wedge from the connection is equal to the maximum force visualized in the graph. The output of all tests is given in Table 5.2. Table 5.2. Output test Series I Force [kn] Test n I(a) I(b)

68 N Force [N] Deformation [mm] Figure 5.7. Output test series I(b), test number Elaboration and discussion The purpose of this test series is to discover if there is any statistically significant difference in the mean value of both conditions (a) and (b). A Student s T-test is done using SPSS Statistics, where condition (a) = human hit/human influence; condition (b) = free fall/non-human influence. Both groups are independent from one and another. While human influence is the independent variable (IV), the magnitude of the force (as well as the stiffness of the connection) forms the dependent variable (DV). The result of a statistical analysis is shown in Table 5.3. Table 5.3. Group Statistics (SPSS Statistics) IVhumaninfluence N Mean Std. Deviation Std. Error Mean DVforce = condition (a): human influence = yes = condition (b): human influence = no The Levene s Test for Equality of Variances is a test that determines if the two conditions have approximately the same or different amounts of variability between scores. A value of Sig. greater than 0.05 means that the variability in the two conditions is approximately the same. See Table 5.4 for the independent-samples test result. From Table 5.4 three can be seen that Sig. = > Thus, the variability in the conditions is essentially the same; it is not significant different. The Sig. (2-tailed) value states if the means of both conditions are statistically different. For Sig. (2- tailed) > 0.05, there is no significant difference between the conditions. As can be seen from Table 5.4, the Sig. (2-tailed) value is > 0.05, which means the differences between the condition s mean values are likely due to chance and not likely due to IV manipulation. 50

69 Table 5.4. Independent Samples test (SPSS Statistics) DVforce Equal variances assumed Equal variances not assumed Levene s Test for F Equality of Variances Sig t df Sig. (2-tailed) T-test for Mean Difference Equality of Means Std. Error Difference % Confidence Interval Lower of the Difference Upper Conclusion An independent samples T-test was conducted to compare the magnitude of force of hammering in in human influence and non-human influence conditions. A significant difference is absent in the scores for independent variable (a) human influence (M = ; SD = ) and independent variable (b) non-human influence (M = ; SD = ); t(18) = , p = These results suggest that hammering in the wedge with a hammer by a human being can be substituted by a solid block (m = 1434g) falling from 2.344m through a hollow tube above the wedge. Human influence does not have an effect on the magnitude of the force with which the wedge is being attached Test Series II and III: Strength and Rotational spring stiffness Series II is implemented according to design code EN :2002. Five tests are done in the positive moment direction and five tests are done in the negative moment direction to determine the strength of the joint described in previous chapters and its rotational spring stiffness Test method: procedure and implementation The samples of this test series consist of a Middle Piece cut at 600mm and a Poly Ledger cut at approximately 500mm connected to each other, see Figure 5.3(a) and Figure 5.8. The usual method to attach the ledger onto the outer tube ring is to hammer in as shown in Figure 2.8. However, since in previous chapter there was concluded that hammering in the wedge with a hammer can be substituted by a solid block falling through a hollow tube, this attaching method will be maintained during test series II. Figure 5.8. Load application on sample 51

70 The test set-up will be explained using Figure 5.9 and Figure First, the beam shown in Figure 5.3(e) was used to attach the specimen to the bed on the 250kN load cell, see Figure 5.10(a). The beam was fastened using bolts. The Middle Piece was placed against the beam and once confirmed to be centrically positioned; the specimen was clamped between the beam and two steel plates, again fastened using bolts. Then, an inclinometer, two Mitutoyo clock gauges and a distance meter were set. The inclinometer was placed at the connection, as near as possible to meticulously measure the rotation at the connection, as can be seen in Figure 5.10(b). The rotation measured at the joint, however, could be caused by rotation of the connection as well as bending of the middle piece due to instability. Thus, to eliminate the uncertainty of correctness of the angle measured by the inclinometer, two Mitutoyo clock gauges are added: one above the center line and the other beneath the center line, both at 75mm from the center line as shown in Figure 5.10(d) and (e). The distance meter has a greater range of measurement, and is added to the set-up to check whether the displacement at the point of force application equals the cell displacement. Finally, the ledger was held by a bracket (see Figure 5.3(d)), through which the sample was loaded, see Figure After two dummy tests, however, compressively loading the sample with help of a bracket proved impractical: because clamping the ledger to the bracket could not be implemented free of stress, compressively loading the sample might lead to incorrect test data. This could be solved by only loading the sample upwards; for downwards loading (negative moment direction) the sample was then placed bottom up. The ledger was then loaded through a rope, as can be seen on both the schematization in Figure 5.9 and the photo in Figure 5.10(e). Note: In case the difference between the cell displacement and the output of the distance meter is constant and negligible small, the check is completed and no further elaboration is needed with the data. Figure 5.9. Test set-up 52

71 Figure Photos of test set-up Figure 5.11(a). Loading through a bracket 53

72 Figure 5.11(b). Loading through a bracket After positioning the sample and the test equipment, all measuring equipment was set to zero. The load application is displacement controlled in mm/min. While loading the sample happened at 10mm/min, unloading happened at 30mm/min. The specimen was cyclically loaded, this continues until failure occurs, see Figure 5.10(f). This procedure was repeated five times for each moment direction for reliable test results. The force in kilo Newton and the rotational angle in degrees associated with the applied deformations in mm are the output of the tests. Note: the speed of loading and unloading was not unfounded, but adjusted after two dummy tests Results The outputs of the tests were the force in kilo Newton, deformation in millimeters as well as rotation in degrees. For further elaboration of the results to define the stiffness, the force has to be converted into moments in kilo Newton-meter. The distance from the point of force application to the center line of the column l is measured during each test. Multiplying this distance with the force applied leads to the moment at that point. Table 5.5 shows the test characteristics of test series II(a). The outputs of test series II(b) is manipulated uniformly. The characteristics of test series II(b) is given in the same table. The graphs in Figure 5.12 and Figure 5.13 show an example of the output of test series II(a) and II(b), respectively, after having computed the moments from the forces. See Appendix C for the force-deformation graphs of all tests. Table 5.5. Output Series II(a) and II(b) Test n Length l [m] Speed [mm/min] loading unloading Force [kn] Rotation 1 [ ] Moment [knm] II(a)_ II(a)_ II(a)_ II(a)_ II(a)_ II(b)_ II(b)_ II(b)_

73 Test n Length l [m] Speed [mm/min] loading unloading Force [kn] Rotation 1 [ ] Moment [knm] II(b)_ II(b)_ The angle measured by the inclinometer is the sum of all rotations. The rotation given in this table is the result after subtracting the rotations due to other effects than stiffness of the joint. See Section 4.3 for more detail Moment [knm] Rotation [ ] Figure Moment-rotation graph of test series II(a): negative direction moment, test number Moment [knm] Rotation [ ] Figure Moment-rotation graph of test series II(b): positive direction moment, test number 4 Photos were taken during the experiments and given in Appendix I-3. Right before failure in the connection occurred, cracks could be observed. Photos of all samples, taken after failure, are given in Appendix I-4. Some examples are given in Figure 5.14 and Figure

74 Figure Photos taken after failure: cracks; test II(a) number 2 Figure Photos taken after failure: cracks; test II(b) number Elaboration and discussion: determination of the value of the characteristic resistance The elaboration of test II(a) number 2 is given below as an example. All data from both test series II and III are elaborated uniformly to discover the moment capacity of the joint. Dissipation of energy The quotient q e shall be calculated from Eq. 5.1:.. Eq where E lo = the energy which is put in during loading, in accordance with Eq. 5.2:., see Figure 5.16 Eq. 5.2 E ul = the energy which can be regained during unloading, in accordance with Eq. 5.3:., see Figure 5.17 Eq. 5.3 For preference, EN gives a suitable approximation function using the method f least square fitting to represent the moment-rotation behavior determined by tests while loading and unloading. While both the loading and unloading curve, M lo ( ) and M ul ( ), respectively, are known, the energy can then be computed using Eq. 5.2 and Eq As can be seen from the formula, the energy which is put in during loading, as well as the energy regained during unloading equal the area beneath the loading and unloading graph, respectively. The energy can therefore be determined using the Riemann sum and by doing this. This approach is implemented by determining the area beneath the exact graph computed from the output of the test data, see Figure 5.16 and Figure 5.17, where 0 and 2 from Eq. 5.2 and Eq. 5.3 are the lower and upper bound and the y-axis is confined to M lo = M ul = M u (see Table 5.5 for values of M u ). 56

75 Figure Energy which is put in during loading (equals area beneath the graph) Figure Energy regained during unloading (equals area beneath the graph) The unloading curve (yellow trendline in Figure 5.17) is determined for the last unloading curve before failure and moved parallel. This is done for all tests and the results are given in Table 5.6. Table 5.6. Dissipation of energy Test Series II(a) Series II(b) n E lo E ul q e R

76 Test Series III(a) Series III(b) n E lo E ul q e R2 3 1 q e 11, see Figure See Eq See Eq = q e, since only one test was performed Figure Partial safety figure R2 as a function of e The partial safety factor R2 depending on the ductility The partial safety factor R2 shall be determined as a function of the quotient e in accordance with Eq. 5.4, also shown graphically in Figure where e = arithmetic mean of the quotients q e determined for a series of identical tests Eq. 5.4,... Eq. 5.5 The ultimate value of the resistance, The ultimate value of the resistance, of the test I shall be taken as the first maximum of the forcedisplacement curve respectively the moment-rotation curve or the force respectively the moment for q e = 11 whatever occurs first, see Figure The values for, are given in Table 5.7. Table 5.7. The ultimate value of resistance, Series II(a) Series II(b) Test n , [kncm] Series III(a) Series III(b) Test n 1 2 1, [kncm]

77 Adjustment of, to, depending on deviations of the dimensions of the cross section The failure values, shall be adjusted to, to account for variations in the actual dimensions of cross-sections from the nominal ones. Reduction of the failure values shall be carried out depending on the deviations of the controlling cross section parameters (e.g. area, bending resistance, moment of inertia) from the nominal values. For longitudinally oriented compressed components, the reduction shall be carried out in accordance with the following given in Table 5.8. Table 5.8. Reduction of failure values depending on deviations of the dimensions of cross section (EN ) Deviation of the controlling parameter Action d 0.01 No reduction required 0.01 d 0.10 Linear reduction 0.10 d Tests with new components required For other components, no reduction is required if the relevant dimensions of the cross sections lie within the specified tolerances. During this test series it is assumed that the dimensions of all test samples were within the boundaries of deviations; Poly elements used for testing meet the requirements of product information. The supposedly adjusted values, are therefore equal to the ultimate values, given in Table 5.7. Adjustment of the ultimate values values, to, depending on the material properties The failure values r, shall be adjusted to r, depending on the proportion of actual to guaranteed material properties, following:,,.. Eq where a = y if where,. / 1.002, / where f y,a = actual value of the yield stress =, / / f y,k = characteristic value of the yield stress = 420N/mm 2 = relative slenderness, calculated from Eq. 5.7: where.. Eq where N pl = the normal force in the full plastic condition, calculated from Eq. 5.8:,. / Eq where A nom = area of the cross section = mm 2 N ci = elastic buckling load (to be determined for the relevant buckling situation in accordance with elastic theory) = / if 0.2 < d M = 1,3 for components made of steel 1,5 for components made of aluminium 1,7 for components made of cast material 59

78 Table 5.9. The ultimate value of resistance, after adjustments Series II(a) Series II(b) Test n , [kncm] [kncm] , Series III(a) Series III(b) Test n 1 2 1, [kncm] , [kncm] Statistical determination of the basic characteristic value of the resistance R k,b The adjusted ultimate values, shall be evaluated statistically to determine the basic characteristic value of the resistance R k,b whereby R k,b is defined as the 5%-quantile for a confidence level of 75%. Figure 5.19 gives values for k sk. Normally, a logarithmic normal distribution may be assumed as following. Figure Quantile factors ksk (quantile: 5%; confidence level: 75%) (EN , Table 4) The values, are transformed to logarithmic values y i using Eq. 5.9:,.. Eq. 5.9 The average value of the values y i is calculated from Eq and the standard deviation from Eq where = average value of the values y i n = number of tests Eq where s y = standard deviation Eq The 5%-quantile is calculated from Eq for the 75% level of confidence. 60

79 ,.... Eq where y 5 = 5%-quantile k s,k = quantile factor = 2.46, see Figure 5.19 The basic characteristic value is then obtained using the reversed logarithmic transformation.,.. Eq where R k,b = characteristic value of the resistance Determination of the nominal characteristic value of the resistance R k,nom The nominal characteristic value of the resistance R k,nom shall be calculated from the basic characteristic value R k,b and the partial safety factor R2 with Eq ,,.. Eq The determination of the nominal characteristic value of the resistance is done following Eq. 5.9 to Eq The mean value and standard deviation of the nominal characteristic value of the resistance are determined using SPSS Statistics. The results are given in Table Table Determination of the nominal characteristic value of the resistance R k,nom Series II(a) Series II(b) Test n , [kncm] y i s y y R k,b [kncm] R k,nom [kncm] Mean [kncm] Std. deviation [kncm] Test n Series III(a) Series III(b), [kncm] y i s y y R k,b [kncm] R k,nom [kncm] = y i, since only one test was performed. 2 Since only one test was performed, no deviation could be computed. 3 The 5% quantile is calculated from Eq. Eq Both quantile factor and standard deviation are not given for n = 1. For further computation, k s,k is assumed to be 2.46, as it is when n = 5 (minimum number of tests required by the design code); s y is assumed to be equal to series III(a) s: Note: The characteristic value of resistance computed in this section is merely an indication, for the number of tests required by the code is not fulfilled. 61

80 Elaboration and discussion: determination of the rotational spring stiffness The rotational spring stiffness is defined as the moment divided by the rotational angle. The angle measured by the inclinometer is the sum of all rotations: the rotation of the connection and the rotation due to instability of the middle piece. Hence, to define the rotational angle ascribed to the stiffness of the joint, the rotation at the middle piece has to be subtracted. The rotation at the middle piece can be computed from the deformation measured by both Mitutoyo clock gauges. Figure 5.20 shows this approach. Figure Defining rotations Mitutoyo clock gauge 1 (see Figure 5.20) showed a smaller displacement than Mitutoyo clock gauge 2, when the ledger was loaded in negative moment direction. However, when loaded in positive moment direction, Mitutoyo clock gauge 1 showed a larger displacement. This can be explained by depression in the surface. Otherwise stated, when loaded in negative moment direction, Mitutoyo clock gauge 1 represents the rotation at the middle piece and this value has to be subtracted from the angle measured by the inclinometer. When loaded in positive moment direction, Mitutoyo clock gauge 2 indicates the rotation at the middle piece. Table 5.11 shows a part of the output from test (II)a number 2, as well as the calculation of the moment and the rotational angle. Table Part of output series II(a)test number 2; computation of moment and rotation Time [s] Force [kn] ADC-02 1 [ ] Mitutoyo(1) 2 [mm] Mitutoyo(2) 3 [mm] Moment 4 [knm] Rotation 5 [ ]

81 Time [s] Force [kn] ADC-02 1 [ ] Mitutoyo(1) 2 [mm] Mitutoyo(2) 3 [mm] Moment 4 [knm] Rotation 5 [ ] Rotational angle measured by inclinometer 2 Mitutoyo clock gauge measured the depression in the surface of the middle piece 3 Mitutoyo clock gauge measured the rotation of the joint at the middle piece 4 Moment applied to the sample, computed by multiplying the force by the length (see Table 4.1) 5 Rotation, defined by rotational angle measured by inclinometer minus the inverse tangent of the displacement measured by Mitutoyo clock gauge 1 in mm divided by 75mm: _ The results from the n th cycle of the cyclic loading where the graph shows no more looseness in the connection were taken for the evaluation of the stiffness. The graph on Figure 5.21 shows these cycles for test II(a) number 2. The greatest moment shall be taken as the ultimate load, or the moment corresponding six times the elastic deformation whatever occurs first (Nather, Lindner & Hertle, 2005). This corresponds with q e = 11, see Section Moment [knm] Rotation [ ] Figure Cycles used to determine the rotational stiffness of test II(a) number 2 Figure 5.22 shows the force-deformation graph. The elastic curve is highlighted, as well as the values corresponding six times the elastic deformation. As can be seen, the greatest force occurs first in this case, for which all cycles before this value are used for the determination of the stiffness. 63

82 Figure Force-deformation graph of test II(a) number 2; determination of ultimate load The slope of the cycles shown in Figure 5.21 equals the stiffness: moment divided by rotation. The stiffness is computed for each cycle. The results are given in Table The mean value of the stiffness of each cycle is the rotational stiffness of n th test. This is done for all tests; tests loaded in positive as well as negative moment direction. The resulting stiffness are shown in Table Table Slope per cycle (test II(a) number 2) Cycle Rotational stiffness [kncm] Mean value of stiffness [kncm] Table Rotational spring stiffness II(a) II(b) Test n Rotational spring stiffness [kncm/ ] Mean [kncm/ ] Std. deviation [kncm/ ] III(a) III(b) Test n Rotational spring stiffness [kncm/ ] Depending on the variation coefficient v x of the stiffness c i (see Eq. 5.15), the characteristic value of the stiffness shall be determined in accordance with the following: 64

83 where v x = variation coefficient s x = standard deviation for the n test results Eq = the mean value of the n test results c p,i, c m,i where the letter p labels the positive load direction and the letter m labels the negative load direction Table Characteristic value of the stiffness c k depending on the variation coefficient v x Variation coefficient v x Characteristic value of the stiffness c k v x 0.10 c 0.10 < v x c 0.20 < v x c 0.30 < v x c 0.40 < v x Configuration to be redesigned The mean and standard deviation of each sub series are computed using SPSS Statictics and the values are given in Table The results of Eq for both series are given in Table 5.15, together with the resulting characteristic value of the stiffness c k. Table Characteristic value of the spring stiffness c k (Series II) Series II(a) II(b) Mean [kncm/ ] Std. deviation s x [kncm/ ] Variation coefficient v x [kncm/ ] < < 0.10 c k [kncm/ ] Comparison of the averaged stiffness in positive and negative load If the linearized averaged inclinations in positive and negative load directions differ by not more than 10%, a straight line can be used for the considered moment of the connection. Eq gives this check:... Eq For 0.29% < 10%, one mean value can be used for the stiffness of the joint. Note: The number of tests for the third series is too small. Therefore, computing the variation coefficient will not lead to meaningful conclusion and in case of series III(b), the variation coefficient cannot be computed. Furthermore, from series II there can be seen that both sub series (a) and (b) have very small deviation. The characteristic value of the stiffness did not have to be adjusted. It is assumed that series III will give the same result, for n = 5. However, tests should be performed according to design code EN to exclusively conclude this matter Conclusion Series II and III were implemented to determine the moment capacity and the rotational spring stiffness of the connection between a Middle Piece and a Poly Ledger. Both are computed following design code EN Series II is subdivided into two sub series: five tests were done in positive moment direction and five tests were done in negative moment direction, as the joint is not symmetric. The third test series was implemented to determine the moment capacity and the 65

84 rotational spring stiffness of the connection between a Middle Piece and Poly Ledger, when loosely attached or hammered in. The purpose of the test is to eliminate the existence of influence from the difference in force with which the ledger is attached onto the middle piece. Only 3 tests were performed in total. All results are given in Table Test n II(a) II(b) III(a) R k,nom [kncm] Mean [kncm] Table Results from test series II Std. deviation [kncm] Rotational spring stiffness [kncm/ ] Mean [kncm/ ] Std. deviation [kncm/ ] III(b) As can be seen from Table 5.16, the mean of the characteristic value of resistance of the joint, when ledger loaded in negative moment direction, is 42.52kNcm. In practice, the structure is mainly loaded in this direction. The moment capacity of the joint in the other direction differs, however, not much from the moment capacity of the joint when loaded in the negative moment direction. From the results there can be concluded that the joint seems to be less strong when loaded in positive moment direction. However, the standard deviation of sub series (b) is greater; therefore, the results might be less accurate. Note: From a statistic point of view, the number of tests n should be at least 30 to analyze interval variables. This means that for a number of tests n less than 30, the computation of the mean as well as the standard deviation do not lead to a meaningful conclusion. Note: The slenderness l of the column is 0.2 < l = 0.29 < 10. The column will most likely not fail due to compression or buckling before failure in the connection occurs. The mean values of the rotational spring stiffness of the joint in both directions are approximately equal. Hence, the resulting rotational spring stiffness of the joint is the mean of the resulting values of both sub series: 34.24kNcm/. The moment capacity from the third test series seems to be smaller than the resulting moment capacity of the same joint, differently attached, from the second series (see Table 5.16). However, the rotational spring stiffness suggests a smaller (or even no) difference, although there is one outlier (24.77kNcm/ ). It is nonetheless unreliable to draw a conclusion, but ostensibly there can be said that the attaching method has no influence on the resulting spring stiffness Conclusion The purpose of the experiments is to determine the moment capacity of the connection between a Poly Ledger and a Middle Piece, as well as the rotational spring stiffness. The connection between the 66

85 ledgers and the props is modelled as hinges; substituting this hinge with a spring might lead to increase of the ultimate load of the whole system. In practice, the erection of the system scaffold is done by field workers. To eliminate human influence, a solid block instead of a hammer is used to attach the ledger onto the middle piece. Both methods are tested to determine the magnitude of the force to confirm the comparability of the alternative. The following can be concluded from test series I. An independent samples T-test was conducted and a significant difference is absent in the scores for independent variable (a) human influence (M = ; SD = ) and independent variable (b) non-human influence (M = ; SD = ); t(18) = , p = These results suggest that hammering in the wedge with a hammer by a human being can be substituted by a solid block (m = 1434g) falling from 2.344m through a hollow tube above the wedge. Human influence does not have an effect on the magnitude of the force with which the wedge is being attached. While human influence is eliminated in this experiment to maintain constant circumstances, its effect cannot be neglected in practice. For this reason, test series III is added. The resulting moment capacity and rotational spring stiffness are given in Table As can be seen, the mean of the characteristic value of resistance of the joint, when ledger loaded in negative moment direction, is 42.52kNcm. In practice, the structure is mainly loaded in this direction. The moment capacity of the joint in the other direction differs, however, not much from the moment capacity of the joint when loaded in the negative moment direction. From the results there can be concluded that the joint seems to be less strong when loaded in positive moment direction. However, the standard deviation of sub series II(b) is greater; therefore, the results might be less accurate. The mean values of the rotational spring stiffness of the joint in both directions are approximately equal. Hence, the resulting rotational spring stiffness of the joint is the mean of the resulting values of both sub series: 34.24kNcm/. The moment capacity resulting from test series III seems to be smaller than the resulting moment capacity of the same joint, differently attached, from test series II. However, the rotational spring stiffness suggests a smaller (or even no) difference, although there is one outlier (24.77kNcm/ ). It is nonetheless unreliable to draw a conclusion, but ostensibly it can be said that the attaching method has no influence on the resulting spring stiffness. 67

86 68

87 6. Structural modelling The basic model consists of three times two props stacked on top of each other, mirrored parallel to the x-axis. These props are connected with Poly Ledgers and the presence of Poly Diagonals serve for the stability of the whole system. Variables in the system lead to a large number of different structural models. These variable parameters are the height of the props, numbers of coupled props and the length of the Poly Ledgers. The emergence of a structural model in the computer software ABAQUS/CAE will be described in this chapter. First the different configurations will be given, followed by the cross section properties of the elements used. Furthermore, the determination of the initial imperfection for the Poly falsework system will be described, following the procedure outlined in Chapter 3. Finally, the boundary conditions and assumptions will then lead to completion of the python script given at the end of this chapter Configurations Figure 6.1 shows the structural model to be examined. This model consists of two parts: the supporting part: composed of middle pieces, jacks, ledgers and diagonals. The diagonals stabilize the structure in the plane in which it is placed. The most common center-to-center distance between the props in practice is 1.200m and 1.800m. Due to conservative considerations, for the supporting part, a ledger length of 1.200m will be used; the supported part: props composed of middle pieces and jacks are coupled to the supporting part through Poly Ledgers. This part derives its stability from the supporting part. The centerto-center distance between the props is chosen to be 1.800m for the supported part, because this is the most unfavorable situation. Figure 6.1. Basic structural model Height variation From this basic model, consisting of three props, a number of variants can be derived. It is expected that for short props (l = 4.500m), the structure will fail due to bearing. As the height of the props increase, buckling will be crucial. Thus, the first variable in the structural system is the height of the props. The height of the props varies between 4.500m to 6.200m, as can be seen on Figure 6.2. The 69

88 structure will be modelled in height steps of 200mm, to discover the prop height at which bearing devolves into buckling. The jacks are the height varying elements, of which its height varies between 250mm and 1100mm, see Figure 6.2. Figure 6.2. Varying height in structural model Varying in number of props coupled Secondly, the number of props coupled to the supporting part of the structure can vary. In fact, the basic model is being extended by adding props connected by ledgers. Figure 6.3 shows the coupling of n props to the supporting part of the structure. It is believed that as more props are coupled to the supporting part, the lower the capacity of the whole system. Figure 6.3. Horizontal coupling: varying number of props coupled The number of configurations is undefined: the number of props coupled will increase until the ultimate load drops below the capacity of the traditional system. This is done in the interest of economical considerations. Figure 6.4 shows the ultimate load capacity of a stacked falsework 70

89 structure composed of traditional elements. Failure of the quick release pin in the pin hole occurs at 125kN, and is governing for systems with relatively smaller height (see Figure 6.4). For this reason, horizontally coupling of the props will continue until the buckling load is lower than 125kN. Characteristic value of resistance [kn] 140 Failure Quick Release Pin Total height [mm] 2X2 3X2 4X2 5X2 6X2 7X2 8X2 QRP Figure 6.4. Buckling load of a stacked falsework system composed of traditional elements (Lamerichs, 2011) Varying in entre-to-center distance between the props Finally, Lamerichs (2011) concluded that the center-to-center distance has no influence on the ultimate load of the system. This parameter was kept constant at the mostly used length in practice (l = 1.800m). However, if the connection between the Poly Ledgers and the props is not a hinge, but a spring constraint with a rotational spring stiffness, it is believed that the center-to-center distance between the props do matter. Figure 6.5 shows a two-dimensional front view (views of all four sides are similar) of a wall structure with 300mm center-to-center distance between the props, which means that the length of the ledgers are also 300mm. The supposition is adopted that such a structure tend to behave as a plate, see Figure 6.5, while the structures shown in Figure 6.6 with longer ledgers is believed to be less stiff. Figure 6.5. Center-to-center 300mm coupled system assumed to behave as rigid plate when connection between prop and ledger has a rotational certain spring stiffness 71

90 Figure 6.6. Horizontally coupled system 6.2. Cross section properties The cross section properties of the Poly elements present in the configurations shown in previous section are shown in Table 6.1. The calculation of these values is given in Appendix D. Table 6.1. Cross section properties of the Poly elements A [mm 2 ] I [mm 4 ] Poly Ledger Poly Diagonal Jack Middle Piece 2000 Civil Overlap length Initial imperfections applied on Poly falsework system The imperfection calculation model given by EN art is assumed to be the most suitable one for the Poly support system (see Chapter 3). The calculation of the imperfection parameters will be given in this section. Eccentricity of load The eccentricity of the applied load cannot be reduced to smaller than 5mm (for more detail, see Lamerichs 2011). However, this only holds for the case in which Alu-beams are used. The distance between the center line of the bolt to connect the beam to the prop and the Alu-beam is 5mm maximum. Therefore, the eccentricity of the applied load can be assumed to be 5mm. Note that use of Poly Props in combination with steel beam elements can lead to a larger eccentricity of loads. Angular imperfections caused by looseness of the joint The Poly Prop has two different joints; first, there is the connection between a jack and a middle piece, and second, the connection between two middle pieces. The erection of the Poly system is implemented such that the first Middle Piece from the ground is always levelled to ensure that this element is not out of plumb. Therefore, this element can be assumed to be positioned perfectly in plan, in level and in orientation, see Figure

91 Figure 6.7. Angular imperfections Poly Prop; first middle piece in line Figure 6.7 shows the connection between a jack and a middle piece. The angular imperfection between these two components is calculated following Eq. 3.5:. /... /. where 0 = the angle, between two components or loose items d i = the specified inner diameter of the tube = 65.5mm [11], see Figure 6.8 d 0 = the specified outer diameter of the jack spindle = 63.5mm [10], see Figure 6.8 l o = the overlap length (depending on total height of structure) = 300mm, see Figure 6.8 Figure 6.8. Angular imperfection joint jack middle piece Figure 6.9. Angular imperfection joint middle piece middle piece 73

92 Figure 6.9 shows the connection between two middles pieces. Again following Eq. 3.5 the angular imperfection can be determined as follows:. /... /. where 0 = the angle, between two components or loose items d i = the specified inner diameter of the tube = 65.5mm, see Figure 6.9 d 0 = the specified outer diameter of the connector = 63.5mm, see Figure 6.9 l o = the overlap length = 150mm, see Figure 6.9 If there is more than one upright in a row, the angle, at a joint to be used for calculation purposes shall be calculated using Eq. 3.6; the formula states that the more props are coupled, the lower the angular imperfection becomes. Thus, a singular prop has the greatest angular imperfection. Eccentricities caused by looseness of the joint Eccentricities caused by looseness of the joint are assumed to be negligible for a Poly system scaffold. The connector has a steel bar with a height of 6mm welded to the tube, which centers the vertical load and prevents the connector from falling into the middle pieces. The dimensions are such that the welded bar fits precisely into the inner diameter of the middle pieces. Therefore, the eccentricities caused by looseness of the joint are inapplicable for the Poly Props, see Figure (See Section for more detail about the connector.) Bow imperfections for compression members The bow imperfection for the prop, consisting of a Jack 1400 and Middle Piece 2000 Civil, can be calculated following Eq The bow imperfection in mm varies for each configuration with different prop length and different number of coupled props. The results are given in Table 6.2. Calculation of the bow imperfection all follow the same procedure (given in EN 12812), shown below the table. The value in the colored cell in Table 6.2 is used for elaboration of the calculation procedure. Table 6.2. Bow imperfection in mm l [m] n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n= where e = bow imperfection [mm] (see Figure 3.5 and Eq. 3.8) l = nominal length of the compression member [mm] = 6200mm r = reduction factor = = where n v = number of structural components arranged and supported side by side and propped in the same way = 3 74

93 Sway imperfections for compression members The height of the structure to be examined, varies between 4.500m and 6.200m (see configurations in Section 6.1), indicating that all models will have a height h less than 10m. For structures where h is less than 10m, tan in Eq. 3.8 shall be taken as The angular deviation from the vertical axis (prop direction) is thus:. The sway imperfection shall be taken as an overall imperfection, this is shown for a lattice tower in Figure 6.10(a). However, where the compression members are not continuous members, as is the case for Poly system scaffolding, the sway imperfection for each individual component (angular imperfection caused by looseness of joint) shall also be taken into account, see Figure 6.10(b). Figure 6.10(a). Modular overall tower (b). Modular independent tower Nevertheless, the code states that the overall sway imperfection and the sway for individual components need not to be considered as simultaneous effects. The line of action is greater if overall sway imperfection is applied. Because this is the most unfavorable situation, overall sway imperfection will be applied to the Poly system scaffold model for conservative results. Although the sway imperfections are determined following equations from EN 12812, the actual value is limited in practice due to a maximum allowance of the deformed structure. The length of a Poly Ledger 1200 is 1200mm 1mm. In a worst case scenario one ledger is 1201mm and the other is 1199mm, which means a maximum allowance of 2mm. Furthermore, the connection at 1.(diagonal to ledger) in Figure 6.11 has a maximum looseness of 1.25mm: the diameter of the pin hole is 17mm [12] and the outer diameter of the pin is 15.75mm [13], see Figure Figure Maximum allowances 75

94 Fastening the diagonal can therefore lead to 1.25mm/2 = 0.75mm maximum looseness on each side. This means that the structure can deviate 1mm + 1mm mm mm = 3.25mm horizontally from the straight line over a height of 1700mm, see Figure The total sway imperfection possible in a Poly system will then be:.. As 0 = rad < 0.01rad, the value for global sway imperfection calculated following the design code cannot take place in this case. A sway imperfection of rad will be applied to the structural model Boundary conditions and assumptions Based on the significance of the influence of the effects of the modelling parameters concluded from earlier research, the following seemingly reasonable assumptions will be made in this study: The base of the Poly props is modelled as a hinge (number 2 on Figure 6.12); Top of the Poly Prop modelled as vertical roll (number 3 on Figure 6.12); Connection between the Poly Diagonal and Poly Ledger will be modelled as a hinge (number 4 on Figure 6.12); Connection between the two Poly Props for the stacked systems modelled as a rigid connection (number 5 on Figure 6.12); The eccentricity between the connector and the middle pieces is not taken into account. Figure Boundary conditions in structural model First a series of configurations will be analyzed with a hinged connection between the Poly Ledgers and the props. However, the final configurations of which the ultimate load will be determined will be modelled with a spring constraint between the ledgers and the props (see number 1 on Figure 6.12). 76

95 6.5. Structural model in ABAQUS/CAE Computer software ABAQUS/CAE 6.12 was used for modelling the structure. The configurations shown in Section 6.1 are modelled, using two-dimensional beam elements. These elements have three degrees of freedom per node: translation in both x and y directions and rotation along the z-axis. Three models were created to do three steps of analyses: linear buckling analysis (LBA), geometrically non-linear analysis (GNIA), as well as geometrically and materially non-linear analysis (GMNIA). The functions within ABAQUS/CAE are organized into modules; the modelling procedure will therefore be explained following these modules in the next subsections. A python script of one of the configurations is given in Appendix E. Note: A structural model modelled by Lamerichs (2011) in Ansys, was modelled in ABAQUS/CAE for the purpose of validation, where after a similar modelling procedure is maintained. More detail on the validation can be found in Appendix F Part -module The geometry is defined in the Part -module. Deformable wire shapes were created in twodimensional planar modeling space. Six different parts were created: Middle Piece, Jack, Poly Ledger(1200), Poly Ledger(1800), Poly Diagonal and finally, Overlap. Overlap is the part where the jack is shoved into the middle piece Property -module The material properties as well as the sections with their profiles are defined in the Property -module. While for linear analysis, only elastic properties are assigned, for non-linear analysis, plasticity is also taken into account, where a bi-linear stress-strain diagram is applied. The jack, ledger and diagonal are hollow circular tubes with a given outer diameter and wall thickness. The middle piece, however, does not have a standard profile (see Figure 2.7 for cross section). This part, and thus also the overlap, are assumed to be hollow rectangular tubes with their original cross section properties given in Table 6.1. This is because for undefined section shape (only I and A are known), no output of stress can be given, so plasticity also cannot be taken into account. The properties of the materials and elements are as given in Table 2.1 to Table Assembly -module The structural model is built in the Assembly -module from the parts generated in the Part -module. The individual parts are placed to create the model shown in Figure As can be seen, the Poly Diagonal is eccentrically attached to the Poly Ledger. Figure 6.14 shows the connection between a Poly Diagonal and a Poly Ledger. The axes of the diagonal and the ledger do not intersect at the node where the axes of the column and the ledger cross. In order to take the eccentricity in the x-direction, as well as the y-direction, the following is assumed: eccentricities of 105.5mm in the x-direction and 37mm in the y-direction result in only a moment in the x-direction of 83mm, see Figure

96 Figure Assembly of individual parts Figure Eccentrically attaching diagonal onto ledger Step -module In the Step -module the analyses are defined. Either a bifurcation analysis is done to discover the eigenvalue and eigenmodes, or a non-linear post-buckling analysis is done using Riks-method. The maximum number of increments is from 3300, depending on the complication level of the 78

97 configurations. The initial arc length increment, minimum and maximum increments are respectively 1e-5, 1e-8 and 1e Interaction -module In the module Interaction the interactions between the parts could be defined. This is done following the assumptions and boundary conditions in previous section. A TIE constraint ties two separate surfaces (or nodes in this case) together so that there is no relative motion between them. This type of constraint is used for the connection between two middle pieces or a middle piece and a jack. Furthermore, a MPC type PIN constraint provides a pinned joint between two nodes. This MPC makes the global displacements equal but leaves the rotations independent of each other and is used for the connection between the prop and a ledger (see Figure 6.15(a)) and between a diagonal and a ledger as well. Finally, reference points were created (see Section 6.5.6) to eccentrically apply loads on top of the props. These reference points are connected to the top nodes of the columns using a COUPLING constraint. This kinematic coupling constraint limits the motion of the reference points to the motion of the top nodes of the props, where the top nodes are the control points and the reference points are the slave nodes. (a) (b) Figure Hinged connection between ledger and prop with (a). zero rotational spring stiffness (b). rotational spring stiffness Within this module one more engineering feature is defined. Two different conditions of the connection between a middle piece and a ledger were modelled: analyses were done with a hinge (pin) connection without rotational spring stiffness as well as the stiffness determined in Chapter 5, see Figure In order to assign rotational spring stiffness to the connection, SPRING2-element is used, where the connectivity type obviously is to connect two points. The degrees of freedom at both nodes are 6, and the rotational spring stiffness is Nmm/ (see Chapter 5). Note: In order to verify the constant value assigned to the rotational spring stiffness, average values from the moment-rotation graphs (from the experiment described in Chapter 5) are as well assigned following a different approach. This verification is given in Appendix G Mesh -module Then, finite element meshes could be created. Beam-elements (type B23) are used, and shear is not taken into consideration. Mesh seeds are defined every 100mm along the two-dimensional parts. (ABAQUS then automatically divides the parts into a number of elements of equal size closest to 100mm). Figure 6.16 shows an example of an assembled configuration after meshing on parts, where the total height is 6200mm. 79

98 Figure Mesh seeds Load -module Finally, a concentrated force has to be applied in the Load -module, as well as some boundary conditions stated in Section 6.4. For the linear analyses, a unit load is to be centrically applied on top of each prop, in the negative y-direction (compressive load). The eigenvalue as a result of the bifurcation analysis is the Euler buckling load, which was eccentrically applied on top of the props in the non-linear analyses. Reference points were created 5mm to the right from the top nodes of the props in the x-direction, see Figure These reference points are coupled to the top nodes as described in previous section. The loads were applied on these reference points. Figure Load application 80

99 The boundary conditions are as well defined in this module. As assumed in Section 6.4, the base of the props is hinged, which means both u1 (x)- and u2 (y)-directions are set, leaving ur3 (rotation along the z-axis) unset. The top of the props is assumed to be modelled as a vertical roll, meaning only u1 (x)-direction is set, while both u2 (y) and ur3 are unset. Having completed all above-described proceedings, a job could be created and the input file (.inp) for one of the configurations is attached in the appendices, see Appendix H. All other configurations created in this research are modelled similarly Keywords editing The initial geometrical imperfections of the prop is the sum of bow and sway imperfection at each node of the model. Adding keyword *IMPERFECTION in the input-file the initial imperfections can be applied to the structural model. The bow imperfection at a node is defined as where e = bow imperfection given in Table 6.2 h = total height of the structure in mm k = y-coordinate of the node (height of the node from base of the prop) The sway imperfection equals. * rad is the maximum sway imperfection determined in Section 6.3 where k = y-coordinate of the node (height of the node from base of the prop) The total geometric imperfection at each node is then the sum of both imperfections at the specific node. 81

100 82

101 7. Finite element analysis A numerical study is conducted to solve stability problems. First, linear buckling analysis (LBA) was done to solve the eigenvalue problem of the system. Then, initial imperfections and eccentricities are taken into account and geometrically non-linear analysis (GNIA) was done using Riks-method. Finally, to determine the ultimate load of the Poly falsework system, plasticity is taken into consideration and geometrically as well as materially nonlinear analysis (GMNIA) was done. Both analyses of models with connections with and without rotational spring stiffness are performed to determine its influence on the ultimate load of a structure globally. Results from Abaqus are discussed in this chapter, and this chapter ends with a few case studies, of which its conclusions will neither be further elaborated in detail, nor included in this study Linear buckling analysis (LBA) Perturbation analysis for the eigenvalue problem of the configurations given in Section 6.1 was conducted. Varying in the different factors, the influence of these factors is studied and discussed in the following section. Modelling of the Poly falsework system is explained in Chapter 6 and the modelling procedure in ABAQUS/CAE is implemented as given in Section Height variation The basic configuration (supporting part with one prop coupled) with varying height is analyzed in this section. Figure 7.1 shows the output of the perturbation analysis. The eigenmode is shown with the respective local stress values. As can be seen, the first buckling mode of a structure with varying height can be different. While for structures with 4.500m (minimum height examined) height h 4.600m, the eigenmode is such as shown on Figure 7.1, for structures with 4.800m h (maximum height examined), the first buckling mode is such as shown on Figure 7.2. Figure 7.1. Output LBA: stress, eigenvalue and eigenmode; height = 4500mm 83

102 Figure 7.2. Output LBA: stress, eigenvalue and eigenmode; height = 6200mm The resulting eigenvalue from LBA is the Euler buckling load. The buckling load belonging to the specific configurations are put into a graph against their corresponding height. This graph is visualized on Figure 7.3. This graph shows that the higher the structure, the lower the Euler buckling load. 300 Euler buckling load [kn] Total height of structure [mm] Figure 7.3. Influence of the height of structure on the Euler buckling load Next, the influence of the substitution of the hinged connection (between ledger and prop) by a connection with rotational spring stiffness is investigated. This is done by performing aforementioned calculations where the interaction between a ledger and a prop is the only varying factor. The rotational spring stiffness used as input for modelling is the stiffness determined experimentally in Chapter 5: 34.24kNcm/. The LBA results of these calculations are shown on Figure 7.4 together with the results from the models without rotational spring stiffness. 84

103 300 n = 3 n = 3, RSS Euler buckling load [kn] Total height of structure [mm] n = total number of props Figure 7.4. LBA results of both conditions Note: The two conditions maintained is referred to as condition I: no RSS and II: RSS, where RSS stands for Rotational Spring Stiffness. All linear buckling analyses of the models with rotational spring stiffness provide more favorable results. However, the difference between the Euler buckling loads of both calculations are minimal, such that the positive influence is negligible. The Euler buckling loads from both analyses are put into Table 7.1 together with the percentage increase. Table 7.1. Percentage increase in LBA results of both conditions Height of the structure [mm] LBA results I: no RSS [kn] LBA results II: RSS [kn] Percentage increase [%] Influence of number of props coupled The second variable factor is the number of props coupled to the supporting part, as explained in Section 6.1. The number of props coupled varies for the linear buckling analyses between 0 and 7, which means a total number of props 2 n 9. The results are given in Figure 7.5 for all heights examined. 85

104 Euler buckling load [kn] h = 4500 h = 4600 h = 4800 h = 5000 h = 5200 h = 5400 h = 5600 h = 5800 h = 6000 h = Total number of props n Figure 7.5. Influence of number of props coupled on the Euler buckling load h = total height of the structure [mm] These configurations are then again analyzed in the second condition: with RSS. The increase of the Euler buckling load for all configurations is again so small, that the influence of the rotational spring stiffness is negligible. The results of all linear buckling analyses are shown in the graph on Figure 7.6. Euler buckling load [kn] Total number of props n Figure 7.6. LBA results of both conditions h = total height of the structure [mm] h = 4500 h = 4500, RSS h = 4600 h = 4600, RSS h = 4800 h = 4800, RSS h = 5000 h = 5000, RSS h = 5200 h = 5200, RSS h = 5400 h = 5400, RSS h = 5600 h = 5600, RSS h = 5800 h = 5800, RSS h = 6000 h = 6000, RSS h = 6200 h = 6200, RSS From the graphs on Figure 7.5 and Figure 7.6, there can be seen that the more number of props are coupled, the lower the Euler buckling load, as expected. Furthermore, the graph shows that the influence on the buckling load becomes greater as the total height of the structure decreases. The graph generated from calculations of structures with a total height of 4.500m shows an offset at three props to the trend line suggested by analyses of structures with higher total number of props. 86

105 The failure mode of this structure also deviates from the failure mode of all other configurations analyzed. The difference shown in Figure 7.1 and Figure 7.2 only apply to structures with one prop coupled to the supporting part. Figure 7.7(a) shows the failure modes of structures with a total height of 4.500m and the failure modes of structures with a total height of 6.200m are given in Figure 7.7(b). (a) (b) Figure 7.7. Instability modes of structures with a height of (a) 4.500m and (b) 6.200m Influence of center-to-center distance between props The last factor of which its influence on the buckling load is to be examined is the center-to-center distance between the props. Instead of the supporting part with a center-to-center distance of 1.200m 87

106 and a Poly Diagonal applied for global stability of the system, the falsework system is built from props with a center-to-center distance of 300mm. In between the props Poly Ledgers of 300mm are applied to horizontally connect the props. In these variants, no diagonals are attached, but the rotational spring stiffness which is determined experimentally in the connection (between ledger and prop) ensures the stability of the system. The supported part of the original configurations is also substituted with props positioned at 300mm horizontal distance. The hypothesis is that these structures will behave as a plate. Results from the linear bucking analyses are given in the graph on Figure 7.8. Results from earlier analyses of the original configurations are added for the comparison of buckling loads n = 11 n = 3 n = 1 (single prop) Euler buckling load [kn] Height of the structure [mm] Figure 7.8. Buckling load on each prop The third graph shown in Figure 7.8 gives the Euler buckling load of one single prop for different heights. Figure 7.9 shows the failure mode of a single prop. Figure 7.9. Output LBA: stress, eigenvalue and eigenmode of where n =1; height = 6200mm 88

107 As can be seen, the buckling load on each single prop in the coupled system is only increased to a negligible small degree in comparison with the Euler buckling load on a single prop. This corresponds to the results stated in previous section: the additional stiffness due to the rotational spring stiffness in the connection is small, such that the effect of coupling can be neglected. See Figure 7.10 for the failure mode of the coupled system: this corresponds to the failure mode of the single prop shown in Figure 7.9 as well. Table 7.2 shows the percentage increase of the Buckling load on each prop. Figure Output LBA: stress, eigenvalue and eigenmode of where n =11; height = 6200mm Table 7.2. Percentage increase in LBA results Height of the structure [mm] LBA results n = 1 [kn] LBA results n = 11 [kn] Percentage increase [%] Unfortunately, the additional rotational stiffness merely provides the structure of global stability. In order to compare the compressive load that the props together in a configuration can carry, the sum of the Euler buckling load of all props placed in the different configurations are plotted against the total height, see Figure The graph shows that from h = 5400mm, applying 11 props with a center-to-center distance of 300mm leads to a higher Euler buckling load compared to the configuration where a diagonal is applied. The difference at the maximum height over two layers (6200mm) is 80kN. In practice, however, this variant means more material and more complex work. In other words, the advantageous increase of the buckling load does not compensate for the adverse practical implementation. For this reason, this variant will no longer be examined in further research. 89

108 n = 11 n = 3 n = 1 (single prop) Euler buckling load [kn] Height of the structure [mm] Figure Sum of buckling loads on each prop 7.2. Geometrically non-linear analysis After linear buckling analysis, a geometrically non-linear analysis has been done. The analysis is implemented by adding initial imperfections to the structural model. Determination of the initial imperfections is given in Sections 6.3 and in detail. Furthermore, load eccentricities are as well taken into account. A non-linear post-buckling analysis is done using Riks-method and the result is shown in Figure 7.12 for the basic configuration LBA GNIA 60 Load [kn] Displacement [mm] Figure Load-displacement graph for basic configuration in condition II: RSS; h = 6.200m As can be seen on Figure 7.12, the result from second-order elastic analysis shows that the load converges towards the eigenvalue resulted from LBA: the Euler buckling load. 90

109 Height variation The same analysis is done for this basic configuration with varying heights. The result for the smallest height calculated (h = 4.600m) is given in Figure LBA GNIA 200 Load [kn] Displacement [mm] Figure Load-displacement graph in condition II: RSS for n = 3, h = 4.600m The results from analyses of different height are all similar to the result described in Section 7.2. From Figure 7.14, however, there can be seen that as the total height of the system increases, the difference between the Euler buckling load and the resulting load from GNIA slightly reduces. In other words, the geometrical imperfection has relatively less influence on systems with a greater total height LBA GNIA 200 Load [kn] Height of the structure [mm] Figure LBA and GNIA results for different heights in condition II: RSS at displacement = 60mm Note: The minimum height for two layers stacked is 4.500m. For this height, however, ABAQUS would not converge and therefore no results can be provided for h = 4.500m. Since this 91

110 height could be applied in reality, depending on the trend of the graph, the loads are be estimated by extrapolating Influence of number of props coupled The next (and last) varying factor is the number of props coupled. The maximum number of props coupled is 3 where total n = 5, because for n = 5 and h = 4.600m, the characteristic ultimate load drops below the capacity of the traditional system, and therefore further coupling of props is no more interesting, see Section This will also be explained in more detail in next section. Results are shown in the graph on Figure Load [kn] LBA, n = 3, RSS LBA, n = 4, RSS LBA, n = 5, RSS LBA, n = 3, no RSS LBA, n = 4, no RSS LBA, n = 5, no RSS GNIA, n = 3, RSS GNIA, n = 4, RSS GNIA, n = 5, RSS GNIA, n = 3, no RSS GNIA, n = 4, no RSS GNIA, n = 5, no RSS Height of the structure [mm] Figure LBA and GNIA results in condition II: RSS at displacement = 60mm For a greater number of props coupled, no deviation from the expected behavior is found. The trend of the graphs follows the expectations arisen from results from previous analyses. Finally, the effect of the rotational spring stiffness is indeed negligible for all configurations examined Geometrically and materially non-linear analysis The final analyses are done with material non-linearity: plasticity of the materials is taken into account. Again a non-linear post-buckling analysis is done using the arc-length method to enable further calculation after failure occurred. Note: Plastic properties were only assigned to the elements (parts), meaning interactions between these elements still have elastic behavior. Figure 7.16 shows the resulting load-displacement graph for the basic configuration. As can be seen, failure occurs and the characteristic ultimate load of this configuration reaches almost 80% of the Euler buckling load, at approximately 70mm. However, before failure of the system due to instability occurs, the force in the diagonal has reached the maximum capacity of its joint. The strength of the connection between the diagonal and the ledger has experimentally been determined and defined in Chapter 5. The section force in both diagonals is plotted in Figure 7.16 as well. The ultimate load of the system therefore has to be reduced, see Figure After failure of the joint at the diagonal in the first layer, the diagonal in the second layer takes over. This second diagonal is loaded in compression, see 92

111 Figure Because the compressive strength of the connection was not determined, failure of the global system is assumed to be when failure of the connection at the first diagonal has occurred. The connection is assumed to be stiffer when loaded in compression. Load [kn] LBA GNIA GMNIA SF-1 SF Displacement [mm] Figure Load-displacement graph in condition II: RSS for n = 3, h = 6.200m While the Euler buckling load is kN, the top of the GMNIA-graph gives 77% of this value for this basic configuration: kN. However, after limiting the ultimate load to failure of the joint at the diagonal, the ultimate load then becomes kN. Figure 7.17 shows the failure mode belonging to this basic configuration. Figure Output LBA: stress, eigenvalue and eigenmode of where n = 3; height = 6200mm Height variation The same analysis is done with varying heights. The result for the smallest height calculated (h = 4.600m) is given in Figure 7.13: again results from LBA, GNIA, GMNIA as well as section force in both diagonals are plotted in one diagram. 93

112 Load [kn] LBA GNIA GMNIA SF-1 SF Displacement [mm] Figure Load-displacement graph in condition II: RSS for n = 3, h = 4.600m For a system with one prop coupled (n = 3), the section force in the diagonal reaches the capacity of the joint before the top of the load-displacement graph as a result of the GMNIA has reached. The ultimate load of the configurations varying in height is determined following the same approach described in Section 7.2 and the result is given in Figure LBA, n = 3, RSS GMNIA, n = 3, RSS 200 Load [kn] Height of the structure [mm] Figure LBA and GMNIA results for different heights in condition II: RSS Influence of number of props coupled Finally, more props are coupled to the basic configuration. The maximum number of props coupled is 3 where total n = 5, because for n = 5 and h = 4.600m, the characteristic ultimate load is kN and thus lower than 125kN, which is the capacity of the traditional system. Further coupling of props is not economical and thus no more interesting. The resulting characteristic ultimate load of all variants is shown in Figure

113 Buckling load [kn] LBA, n = 3, RSS LBA, n = 4, RSS LBA, n = 5, RSS LBA, n = 3, no RSS LBA, n = 4, no RSS LBA, n = 5, no RSS GMNIA, n = 3, RSS GMNIA, n = 4, RSS GMNIA, n = 5, RSS GMNIA, n = 3, no RSS GMNIA, n = 4, no RSS GMNIA, n = 5, no RSS Height of the structure [mm] Figure LBA and GMNIA results for different heights in condition II: RSS Figure 7.21 shows the failure mode belonging to the configuration with the smallest height examined and the most props coupled. Figure Output LBA: stress, eigenvalue and eigenmode of where n = 5; height = 4600mm 7.4. Case studies Aside from the main study which has been finalized in the above, a few case studies have been investigated. Within each case study one single characteristic of the model has been adjusted. The hypothesis is that these adjustments will lead to an increase in ultimate load. Depending on the percentage increase, there will be discussed whether the specific modification of the structure is worth the arrangement. The results of all case studies are compared to the result of the basic configuration in condition I: no 95

114 RSS. To eliminate the influence of height variation and number of props coupled in the conclusions to be drawn, the six most extreme situations were investigated; therefore the different configurations shown in Table 7.3 were modelled. Table 7.3. Combination of height and number of props coupled to be further investigated Total height of the structure h [m] Total number of props n Replaced diagonal The first modification of the original structure is replacing a diagonal, see Figure Replacing the diagonal to create a greater base of support would help improve stability conditions for the system. The element properties of Poly Diagonal 1800/1700 are all equal to Poly Diagonal 1200/1700 s, except for its length. The diagonal is attached onto the ledger 83mm from the end node of the ledger, see Section for more detail. Figure Replaced diagonal to create a greater base of support The resulting LBA and GMNIA results are shown in Table 7.4 with its percentage increase. 96

115 Table 7.4. Results replaced diagonal Original configuration Replaced diagonal Percentage increase LBA [kn] GMNIA [kn] LBA [kn] GMNIA [kn] LBA [%] GMNIA [%] 3_ _ _ _ _ _ As can be seen from Table 7.4, replacing the diagonal to create a greater base of support does have its positive influence on the bearing capacity. While for the basic configuration the percentage increase is negligible (see Figure 7.23), for a configuration with a smaller total height and more props coupled, the percentage increase can be as high as 10%. In order to compare the failure modes, the stress figures of both basic configuration and 5_4600 are given, see Figure 7.24 and Figure 7.25, respectively. The failure modes of original configurations were given in Figure 7.17 and Figure Load [kn] Basic: LBA Basic: GMNIA Replaced diagonal: LBA Replaced diagonal: GMNIA Displacement [mm] Figure Load-displacement graph Figure Output LBA: stress, eigenvalue and eigenmode of where n = 3; height = 6200mm 97

116 Figure Output LBA: stress, eigenvalue and eigenmode of where n = 5; height = 4600mm Adjusted center-to-center distance Next modification of the original structure is the center-to-center distance of the props of the supporting part, as shown in Figure This is for the purpose of a greater base of support as well. All other boundary conditions remain the unchanged. Figure Greater center-to-center distance to create a greater base of support 98

117 The resulting LBA and GMNIA results are shown in Table 7.5 with its percentage increase. Table 7.5. Results center-to-center 1800mm Original configuration c.t.c. 1800mm Percentage increase LBA [kn] GMNIA [kn] LBA [kn] GMNIA [kn] LBA [%] GMNIA [%] 3_ _ _ _ _ _ The table shows a great difference in both LBA as well as GMNIA results between the original configuration and the adjusted one. The linear buckling analyses result in higher Euler buckling loads when increasing the center-to-center distance between the props. Taking imperfections and plasticity into account, however, lead to an even greater percentage increase of the bearing capacity. The smaller the total height of a structure, the greater the influence on the bearing capacity. Furthermore, the percentage increase increases as more props are coupled. The load-displacement graph is shown on Figure Load [kn] Basic: LBA Basic: GMNIA Center-to-center 1800mm: LBA Center-to-center 1800mm: GMNIA Displacement [mm] Figure Load-displacement graph The graphs show that the difference between the results from non-linear calculations is indeed greater than from bifurcation analyses. The stress figures for comparison are shown in Figure 7.28 and Figure

118 Figure Output LBA: stress, eigenvalue and eigenmode of where n = 3; height = 6200mm Figure Output LBA: stress, eigenvalue and eigenmode of where n = 5; height = 4600mm Replaced point of load eccentricity Figure shows the modification in configuration in this case study. As the structure is asymmetric, the point of load eccentricity could make a difference. Figure Replaced point of load eccentricity 100

119 The resulting LBA and GMNIA results are shown in Table 7.6 with its percentage increase. Table 7.6. Replaced point of load eccentricity Original configuration Replaced eccentricity Percentage increase LBA [kn] GMNIA [kn] LBA [kn] GMNIA [kn] LBA [%] GMNIA [%] 3_ _ _ _ _ _ Despite the expected increase in load bearing capacity after modifying the structural model, the results show a decrease in the results of the non-linear analyses. The effect is greater for smaller heights of the structures. For the structures with smaller total heights, the influence of the number of props coupled is little. For the higher structures, the influence becomes greater as the number of props coupled increases. Results from this modelling procedure should be governing. Again the loaddisplacement graph is shown in Figure Load [kn] Basic: LBA Basic: GMNIA Replaced point of load eccentricity: LBA Replaced point of load eccentricity: GMNIA Displacement [mm] Figure Load-displacement graph The graph shows a rather different trend as can be seen on Figure The stress figures in its failure mode are shown in Figure 7.32 and Figure 7.33 for comparison. 101

120 Figure Output LBA: stress, eigenvalue and eigenmode of where n = 3; height = 6200mm Figure Output LBA: stress, eigenvalue and eigenmode of where n = 5; height = 4600mm Adding diagonals Field workers would prefer to apply as little diagonals as possible. Diagonals, however, are indispensable and adding diagonals would even improve the stability condition. Adding a scaffolding tube onto the jack through a swivel is possible, but not usual. The effect on the global system is examined in this subsection. The element properties of the scaffolding tube are shown in Table 7.7. Figure 7.34 shows a scaffolding tube attached onto a jack through a swivel. Table 7.7. Element properties scaffolding tube Section hollow circular tube Length - Thickness 3.2mm Diameter (outer) 48.3mm Diameter (inner) 41.9mm Steel grade S235 Yield strength 235N/mm 2 102

121 Figure Scaffolding tube Applying this element in ABAQUS leads to the structural model shown in Figure The end of the scaffolding tube is attached to the jack approximately 200mm from the head or base plate. Figure Adding diagonals The resulting LBA results are shown in Table 7.8 with its percentage increase. 103