The wear behaviour of arch bridge bearings
The first Van Brienenoord Bridge
Author: Naveen Narain Stud.nr: 4035607 Date: 27/11/2012
Delft University of Technology Faculty of Civil Engineering and Geosciences Master of Science thesis report
Student:
Naveen Narain (4035607) Civil Engineering Structural Engineering Steel and Timber Structures
Graduation committee:
Prof.ir. F.S.K. Bijlaard (Delft University of Technology) Dr. M.H. Kolstein (Delft University of Technology) Dr.ir. M.A.N. Hendriks (Delft University of Technology) Ir. L.J.M. Houben (Delft University of Technology) Dr.Ing. J.S. Leendertz (Rijkswaterstaat)
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Preface
This thesis study was performed in order to finalize my Master of Science program in Civil Engineering with the track ‘Timber and Steel Construction’ within ‘Structural Engineering’ at the Delft University of Technology. To do so, I was offered an opportunity to analyse the wear behaviour of bridge bearings in the main office of Rijkswaterstaat, located in Utrecht, the Netherlands. This opportunity was made possible by Dr. Ing. H. Leendertz, who is part of the graduation committee as well as the committee of the European standards regarding structural bearings.
During my graduation thesis I was able to acquire more knowledge about, for example, finite element modelling, bridges and their bearings. I therefore would like to thank Rijkswaterstaat and the graduation committee, which also provided for support and critical judgment.
Naveen Narain
Delft, November 2012
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Summary
Structural bearings are connections between the substructure and superstructure of the bridge and may allow translations and rotations. They also transmit the external forces, acting on the superstructure, to the foundation. In the past years the traffic loads and intensity on the European traffic system has continuously increased. Bridge bearings are therefore more subjected to the wear.
Structural bearings are not only designed and fabricated according to the European standards NEN- EN 1337, but they are also tested according to these standards. The bearings are however not tested with the forces, translations and rotations which occur in practice. To make a critical judgement regarding these standards, the wear behaviour of a large arch bridge, the first Van Brienenoord Bridge, is analysed. The analysis is performed by means of a linear elastic, finite element model. The bridge model is then subjected to a modified fatigue load model based on the European standards for traffic loads NEN-EN 1991-2 and traffic measurements at the Moerdijk Bridge. Translations and rotation were consequently found along with the simultaneously occurring reaction forces. These results are compared to the qualification tests included in parts 2 (Sliding elements) and 5 (pot bearings) of the European standards for structural bearings.
After the comparison between the obtained results from the finite element model and the tests values from the European standards for structural bearings, there is concluded that: the wear behaviour of arch bridge bearings can successfully be obtain by means of the analysis method used for this study; the slide paths in the tests are reached in one or two year, which is a very short time period; the there is no clear relationship between the test values and the service life of structural bearings; large resistant moment may occur due to restraining of the bearings by means of the friction force. The resistant moment, however, could not be analysed since friction could not be implemented better for the linear elastic model.
According to this study, the following are recommended:
The finite element model constructed for this study can be used for the analysis of dynamic amplifications and is therefore also recommended to do so; It is recommended to simulate the traffic crossing the first Van Brienenoord Bridge to gain more insight about the occurring slide paths since the lorry distance and the number of lorry simultaneously crossing bridge affects the final slide paths; A nonlinear analysis should be performed in order to study the effect of resistant moments on the global behaviour of the superstructure; Even though large slide paths are also found in the direction perpendicular to the main sliding direction, NEN-EN 1337-2 does not include tests for that direction. It is therefore recommended that bearings are tested in two directions; Since the finite element model is constructed such that also a fatigue analysis can be performed, it is recommended that this analysis should indeed be performed.
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Table of contents
1. INTRODUCTION...... 1
2. OBJECTIVE ...... 2
3. PRECONDITIONS AND SCOPE OF THIS MASTER THESIS ...... 3
4. RESEARCH STRATEGY ...... 4
5. LITERATURE STUDY ...... 5
5.1. General ...... 5
5.2. Rijkswaterstaat in brief ...... 5
5.3. Structural bearings in historic perspective (1) ...... 6
5.4. Structural bearings ...... 12 5.4.1. Elastomeric bearings ...... 13 5.4.2. Steel bearings ...... 14 5.4.3. Pot bearings ...... 14 5.4.4. Spherical PTFE bearings ...... 15
5.5. The European standards concerning pot and spherical bearings...... 16 5.5.1. The NEN-EN 1337...... 16 5.5.2. The relevant test for sliding elements (NEN-EN 1337-2) ...... 16 5.5.3. The relevant test for pot bearings (NEN-EN 1337-5) ...... 19
5.6. Earlier studies ...... 20 5.6.1. Mechanical behaviour of the structural bearings of the Dintelhaven Bridge ...... 20 5.6.2. Mechanical behaviour of the structural bearings of a steel plate-girder bridge ...... 20
5.7. Summary...... 21
6. THE FIRST VAN BRIENENOORD BRIDGE ...... 22
6.1. General ...... 22
6.2. Selection of the bridge ...... 22
6.3. Location of the bridge ...... 22
6.4. History of the bridge ...... 23
6.5. Global geometry and description of the bridge ...... 25
6.6. Layout of the bridge deck ...... 28
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6.7. Support system ...... 28
6.8. Summary...... 29
7. THE FINITE ELEMENT MODEL ...... 30
7.1. General ...... 30
7.2. Requirements of the finite element model ...... 30
7.3. Modelling of structural transitions ...... 34
7.4. Modelling the supports ...... 36
7.5. The finished model ...... 37
7.6. Self-weight and superimposed dead loads ...... 38
7.7. Validation of the model ...... 39
7.8. Summary...... 40
8. TRAFFIC LOADS ON THE BRIDGE ...... 41
8.1. General ...... 41
8.2. The applied fatigue load model ...... 41
8.3. Requirements for a static analysis using the influence line theory ...... 43
8.4. Method choice for obtaining influence lines ...... 43
8.5. Modelling axle loads ...... 44
8.6. Summary...... 45
9. LINEAR STATIC ANALYSIS ...... 46
9.1. General ...... 46
9.2. Self-weight, superimposed dead loads and the corresponding results ...... 46
9.3. Influence lines for the unit load ...... 47
9.4. Modifying influence lines for lorries ...... 51
9.5. Translations with and without friction ...... 56
9.6. Modifying the results for pot bearings ...... 61
9.7. Modifying the results for spherical bearings ...... 63
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9.8. The effect of multiple lorries on the bridge ...... 65
9.9. Summary...... 68
10. COMPARISON WITH NEN-EN 1337-2 AND NEN-EN 1337-5 ...... 69
11. CONCLUSIONS AND RECOMMENDATIONS ...... 70
11.1. Conclusions ...... 70
11.2. Recommendations ...... 70
BIBLIOGRAPHY ...... 72
ANNEX A: The Williot diagram ANNEX B: The unit lorry influence lines for the normal support system ANNEX C: The unit lorry influence lines for the fixed support system
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List of figures
FIGURE 1: MAIN WATER SYSTEM (LEFT), HIGHWAY NETWORK (MIDDLE) AND WATERWAYS NETWORK (RIGHT) ...... 5 FIGURE 2: RIJKSWATERSTAAT ORGANIZATION CHART ...... 6 FIGURE 3: EARLY TIMBER BRIDGE RESTING ON WOODEN BEARINGS (1) ...... 7 FIGURE 4: SADDLES ON TOP OF PYLONS (1) ...... 8 FIGURE 5: A BRIDGE FIXEDLY JOINED TO THE BASE (LEFT) AND A BRIDGE RESTING ON ROLLERS (RIGHT) (1)...... 9 FIGURE 6: BEARINGS OF BLAUES WUNDER (1) ...... 10 FIGURE 7: ARRANGEMENT OF THE BEARINGS WITHIN THE BRIDGE SYSTEM (1) ...... 10 FIGURE 8: MULTI-BALL BEARING (LEFT) AND ALL-DIRECTIONALLY MOVABLE MULTI-ROLLER BEARING (RIGHT) (1) ...... 11 FIGURE 9: ARMOURED CONCRETE ROLLER BEARING (1) ...... 11 FIGURE 10: MODERN SPHERICAL BEARING (1) ...... 12 FIGURE 11: REINFORCED ELASTOMER BEARING SYSTEM (3) ...... 13 FIGURE 12: ROLLER BEARING (3) ...... 14 FIGURE 13: LINE ROCKER BEARING (3) ...... 14 FIGURE 14: FIXED POT BEARING (5) ...... 15 FIGURE 15: MULTIDIRECTIONAL POT BEARING (5) ...... 15 FIGURE 16: MULTIDIRECTIONAL SPHERICAL BEARING (5) ...... 16 FIGURE 17: TEST SPECIMEN FOR DIMPLED AND RECESSED PTFE SHEETS (LEFT) AND FOR COMPOSITE MATERIALS (RIGHT) (6)...... 17 FIGURE 18: FRICTION TESTING EQUIPMENT (6) ...... 17 FIGURE 19: TEST SET-UP USING ONE POT BEARING AND A HYDROSTATIC SPHERICAL BEARING (LEFT) AND USING TWO POT BEARINGS AND ONE SLIDING SURFACE (RIGHT) (7) ...... 19 FIGURE 20: VAN BRIENENOORD BRIDGES AT NIGHT WITH OPEN BASCULES (PHOTOGRAPHED BY AUTHOR) ...... 22 FIGURE 21: A PART OF THE RHOMBUS-SHAPED MOTORWAY SYSTEM AROUND ROTTERDAM WITH THE FIRST VAN BRIENENOORD BRIDGE (8) ...... 23 FIGURE 22: THE RHOMBUS-SHAPED MOTORWAY SYSTEM AROUND ROTTERDAM WITH THE FIRST VAN BRIENENOORD BRIDGE (VAN BRIENENOORDBRUG) (9) ...... 23 FIGURE 23: THE VAN BRIENENOORD BRIDGE ANNO 1978 (13) ...... 24 FIGURE 24: A BIRD’S-EYE VIEW OF THE VAN BRIENENOORD BRIDGES (RIJKSWATERSTAAT) ...... 25 FIGURE 25: DRAWING OF THE FINAL DESIGN OF THE ARCH BRIDGE (12) ...... 25 FIGURE 26: GLOBAL DIMENSIONS OF THE SUPERSTRUCTURE ...... 26 FIGURE 27: SIDE VIEW OF THE VAN BRIENENOORD BRIDGES (PHOTOGRAPHED BY AUTHOR) ...... 26 FIGURE 28: GLOBAL DIMENSIONS OF THE BRIDGE CROSS SECTION AT MID-SPAN ...... 27 FIGURE 29: WIDTHS OF THE BRIDGE DECK ...... 27 FIGURE 30: DIRECTION OF TRAFFIC ON THE FIRST VAN BRIENENOORD BRIDGE BEFORE THE SECOND BRIDGE WAS OPENED (9) ...... 28 FIGURE 31: DIRECTION OF TRAFFIC ON THE FIRST VAN BRIENENOORD BRIDGE AFTER THE SECOND BRIDGE WAS OPENED (9)...... 28 FIGURE 32: FIXED BEARING ...... 29 FIGURE 33: ROLLER BEARING ...... 29 FIGURE 34: THE APPLIED SUPPORT SYSTEM WITH THE DIRECTIONS OF ALLOWED TRANSLATION INCLUDED ...... 29 FIGURE 35: NODAL FORCES FOR A QUADRILATERAL PLATE ELEMENT ...... 31 FIGURE 36: NODAL FORCES FOR A TRIANGULAR PLATE ELEMENT ...... 31 FIGURE 37: THE BEAM ELEMENT IN MIDAS CIVIL 2011 ...... 32 FIGURE 38: THE ACTUAL (LEFT) AND MODELLED (RIGHT) BULB SECTIONED STIFFENERS ...... 32 FIGURE 39: CROSS SECTIONAL DATA OF A PART OF THE ARCH ...... 33 FIGURE 40: THE TRAFFIC LANE DESIGNATED FOR LORRIES ...... 34 FIGURE 41: THE DETAILED LORRY LANE ...... 34 FIGURE 42: THE CONNECTION BETWEEN ARCH AND BOTTOM CHORD (LEFT) AND ITS MODEL (RIGHT) ...... 35 FIGURE 43: A MORE DETAILED VIEW OF THE CONNECTION BETWEEN ARCH AND BOTTOM CHORD AND THE STIFFENING ABOVE THE SUPPORT ...... 35
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FIGURE 44: CONNECTION BETWEEN CABLES AND BOTTOM CHORD ...... 36 FIGURE 45: THE NUMBERS OF THE FOUR SUPPORTS IN A TOP VIEW OF THE BRIDGE DECK ...... 36 FIGURE 46: 3D-VIEW OF THE MODEL ...... 38 FIGURE 47: TOP VIEW OF THE FINAL MODEL WITH (BOTTOM) AND WITHOUT (TOP) SUPPORTS ...... 38 FIGURE 48: NORTHBOUND SIDE VIEW OF THE MODEL WITHOUT (LEFT) AND WITH (RIGHT) SUPPORTS ...... 38 FIGURE 49: DEFORMATIONS ACCORDING TO THE WILLIOT DIAGRAM ...... 40 FIGURE 50: DEFORMATIONS ACCORDING TO THE FINITE ELEMENT MODEL DUE TO SELF-WEIGHT ...... 40 FIGURE 51: POSITIONS OF THE NODAL LOADS ON THE TRAFFIC LANE ...... 44 FIGURE 52: THE MODELLED AXLE LOADS OVER THE FULL BRIDGE LENGTH OF THE BRIDGE ...... 44 FIGURE 53: A CLOSER VIEW OF THE MODELLED AXLE LOADS ...... 45 FIGURE 54: THE ORIGIN AND MAIN VALUES OF THE X-AXIS ...... 47 FIGURE 55: UILS OF Z-REACTIONS FOR THE NORMAL SUPPORT SYSTEM ...... 47 FIGURE 56: UILS OF TRANSLATIONS IN THE X-DIRECTION FOR THE NORMAL SUPPORT SYSTEM ...... 48 FIGURE 57: UILS OF TRANSLATIONS IN THE Y-DIRECTION FOR THE NORMAL SUPPORT SYSTEM ...... 48 FIGURE 58: UILS OF ROTATIONS ABOUT THE X- AXIS FOR THE NORMAL SUPPORT SYSTEM ...... 49 FIGURE 59: UILS OF ROTATIONS ABOUT THE Y- AXIS FOR THE NORMAL SUPPORT SYSTEM ...... 49 FIGURE 60: UILS OF ROTATIONS ABOUT THE Z-AXIS FOR THE NORMAL SUPPORT SYSTEM ...... 50 FIGURE 61: UILS OF X-REACTIONS FOR THE FIXED SUPPORT SYSTEM ...... 50 FIGURE 62: UILS OF Y-REACTIONS FOR THE FIXED SUPPORT SYSTEM...... 51 FIGURE 63: SIMPLIFIED LORRY MODEL ...... 51 FIGURE 64: IN CASE THE LOAD MODEL IS PARTLY PRESENT ON THE BRIDGE (LEFT) OR COMPLETELY ABSENT (RIGHT) ...... 52 FIGURE 65: ULILS OF Z-REACTIONS FOR THE NORMAL SUPPORT SYSTEM ...... 53 FIGURE 66: ULILS OF X-TRANSLATIONS FOR THE NORMAL SUPPORT SYSTEM ...... 53 FIGURE 67: ULILS OF Y-TRANSLATIONS FOR THE NORMAL SUPPORT SYSTEM...... 54 FIGURE 68: ULILS OF ROTATIONS ABOUT THE X-AXIS FOR THE NORMAL SUPPORT SYSTEM ...... 54 FIGURE 69: ULILS OF ROTATIONS ABOUT THE Y-AXIS FOR THE NORMAL SUPPORT SYSTEM ...... 55 FIGURE 70: ULILS OF ROTATIONS ABOUT THE Z-AXIS FOR THE NORMAL SUPPORT SYSTEM ...... 55 FIGURE 71: ULILS OF X-REACTIONS FOR THE FIXED SUPPORT SYSTEM ...... 56 FIGURE 72: ULILS OF Y-REACTIONS FOR THE FIXED SUPPORT SYSTEM ...... 56 FIGURE 73: HORIZONTAL X-REACTION LILS FOR SUPPORT 2 AND WHETHER THEY EXCEEDING THE FRICTION FORCE ...... 58 FIGURE 74: HORIZONTAL X-REACTION LILS FOR SUPPORT 3 AND WHETHER THEY EXCEEDING THE FRICTION FORCE ...... 58 FIGURE 75: HORIZONTAL X-REACTION LILS FOR SUPPORT 4 AND WHETHER THEY EXCEEDING THE FRICTION FORCE ...... 59 FIGURE 76: HORIZONTAL Y-REACTION LILS FOR SUPPORT 3 AND WHETHER THEY EXCEEDING THE FRICTION FORCE ...... 59 FIGURE 77: HORIZONTAL Y-REACTION LILS FOR SUPPORT 3 AND WHETHER THEY EXCEEDING THE FRICTION FORCE ...... 60 FIGURE 78: ROTATION OF THE PISTON (16) ...... 61 FIGURE 79: ADDITIONAL DISPLACEMENT DUE TO ROTATION OF A FREE POT BEARING ...... 62 FIGURE 80: ROTATIONAL BEHAVIOUR OF GERMAN FREE SPHERICAL BEARINGS (16) ...... 63 FIGURE 81: ADDITIONAL DISPLACEMENT DUE TO ROTATION OF A FREE SPHERICAL BEARING ...... 64 FIGURE 82: DX ULILS FOR ONE, TWO AND THREE ULS ...... 66 FIGURE 83: ACCUMULATED DX ULILS FOR ONE, TWO AND THREE ULS ...... 66 FIGURE 84: DY ULILS FOR ONE, TWO AND THREE ULS ...... 67 FIGURE 85: ACCUMULATED DY ULILS FOR ONE, TWO AND THREE ULS ...... 67
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List of tables
TABLE 1: FRICTION TEST CONDITIONS (LONG TERM TEST) (6) ...... 18 TABLE 2: CHARACTERISTIC COMPRESSIVE STRENGTH FOR SLIDING MATERIALS (6) ...... 19 TABLE 3: DOFS OF THE NORMAL SUPPORT SYSTEM ...... 37 TABLE 4: DOFS OF THE FIXED SUPPORT SYSTEM ...... 37 TABLE 5: FATIGUE LOAD MODEL M [75/25] (16) ...... 42 TABLE 6: OVERVIEW AND SUMMATION OF THE DEADWEIGHT ...... 46 TABLE 7: VERTICAL REACTIONS DUE TO DEADWEIGHT...... 46 TABLE 8: CALCULATION OF THE WEIGHTED AVERAGE LORRY LENGTH ...... 52 TABLE 9: EXPECTED NUMBER OF LORRIES PER TYPE PER YEAR ...... 57 TABLE 10: TRANSLATIONS AND ROTATIONS PER YEAR WITHOUT TAKING FRICTION INTO ACCOUNT...... 57 TABLE 11: THE POSITION (X-COORDINATE) OF THE LORRIES WHERE THEY INITIATE TRANSLATION ...... 60 TABLE 12: TRANSLATIONS PER YEAR WHEN TAKING FRICTION INTO ACCOUNT ...... 61 TABLE 13: WEAR PATHS ON THE INTERNAL SEAL OF THE POT BEARING PER YEAR ...... 61 TABLE 14: TRANSLATIONS DUE TO THE ROTATIONS OF THE BOTTOM CHORD CROSS SECTION ...... 63 TABLE 15: ADDITIONAL SLIDE PATHS ON THE PLANAR SLIDING PLATE DUE TO ROTATION ...... 63 TABLE 16: FINAL TRANSLATIONS AFTER SUMMATION ...... 63 TABLE 17: THE WEAR PATHS ON THE CONCAVE BEARING SURFACE PER YEAR ...... 64 TABLE 18: TRANSLATIONS DUE TO THE ROTATIONS OF THE BOTTOM CHORD CROSS SECTION ...... 65 TABLE 19: SUBTRACTIVE SLIDE PATHS ON THE PLANAR SLIDING PLATE DUE TO ROTATION ...... 65 TABLE 20: FINAL TRANSLATIONS AFTER SUBTRACTION ...... 65 TABLE 21: ACCUMULATED TRANSLATIONS FOR SUPPORT 4 ...... 68 TABLE 22: REDUCTION OF THE WEAR PATHS ...... 68 TABLE 23: UPPER AND LOWER BOUND VALUES OF THE WEAR PATHS PER YEAR ...... 68 TABLE 24: COMPARISON WITH TEST VALUES FROM NEN-EN 1337-2 ...... 69 TABLE 25: COMPARISON WITH VALUES OF DETERMINATION OF RESISTANT MOMENT TEST FROM NEN-EN 1337-5 ...... 69 TABLE 26: COMPARISON WITH VALUES OF THE LONG TERM ROTATION AND LOAD TEST FROM NEN-EN 1337-5 ...... 69
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List of symbols
Coefficient of friction
sT, Maximum coefficient of friction during a given temperature phase while subjected with static loads D Displacement [mm, m] Dx Displacement in the x-direction [mm, m] Dy Displacement in the y-direction [mm, m] Dz Displacement in the z-direction [mm, m] F Force [N, kN] 2 fk Nominal compressive strength [N/mm ] Fx Reaction force in the x-direction [N, kN] Fy Reaction force in the x-direction [N, kN] Fz Reaction force in the x-direction [N, kN] R Rotation [rad] Rx Rotation about the x-axis [rad] Ry Rotation about the y-axis [rad] Rz Rotation about the z-axis [rad]
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List of abbreviations
CPC Crash barriers, Parapets and Concrete kerbs DOF Degree-of-freedom FLM Fatigue Load Model LIL Lorry Influence Line PTFE Polytetrafluoroethylene RWS Rijkswaterstaat S1 Support 1 S2 Support 2 S3 Support 3 S4 Support 4 UIL Unit Influence Line UL Unit Lorry ULIL Unit Lorry Influence Line LIL Lorry Influence Line
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MSc thesis – Structural Engineering Steel and Timber Structures
1. Introduction In the previous years the traffic loads and the number of vehicles on the European traffic network have increased considerably. For bearings of steel bridges this results is an acceleration of damage, which further threatens the service life of the structural bearings. At the end of their service life, the bearings will have to be renovated or even replaced, which may be very costly operations, especially with the need for the associated jacking up and temporarily supporting the structure.
Structural bearings are connections between the substructure and superstructure of the bridge. The substructure consists of the piers, abutments and foundations, while the superstructure consist of the bridge structure itself. A bearing may allow translations and rotations and transmits the external forces, acting on the superstructure, to the foundation. The combination of loads and displacements at the bearings cause a phenomenon called “wear”. Wear is a result of continuous cyclic displacements together with a certain pressure. Finally, due to this phenomenon the bearings may fail to meet their expected working life or performance level and consequently cause expenses. In addition, the constantly increasing traffic loads and the increasing amount of vehicles provoke a higher impact on the structural bearings, which causes an acceleration of this damage mechanism. Consequently, this will result in a shorter service life of these bearings.
Rijkswaterstaat (RWS) is the executive organization of the Dutch Ministry of Infrastructure and Environment and is, amongst other tasks, responsible for the maintenance of the national infrastructure. Consequently, this organisation is also responsible for the maintenance of bridges. RWS previously carried out studies regarding the relationship between loading and movement of structural bearings for two bridges, which are a steel plate-girder bridge and a concrete box-girder bridge.
Assuming that a plate-girder bridge, a concrete box-girder bridge, an arch bridge and a cable-stayed bridge are representative types of bridges, Rijkswaterstaat now aims at a research of an arch bridge as a following study concerning the load-movement interaction. For this thesis the first Van Brienenoord Bridge has been selected as a typical arch bridge to be analysed. The bridge is located in the Rotterdam port area and has a high traffic intensity. Since an average of more than one hundred thousand vehicles of the northbound traffic cross this arch bridge every day, the first Van Brienenoord Bridge is considered a relevant bridge for the analysis.
The aim of analysing the first Van Brienenoord Bridge is obtaining the loads transferred trough the structural bearings together with the simultaneously occurring translations and rotations and deriving slide paths for pot and spherical bearings. Since these structural bearing types are designed and fabricated according to the NEN-EN 1337, also a comparison will be made with the values of the qualification tests included in parts 2 and 5 of these standards.
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MSc thesis – Structural Engineering Steel and Timber Structures
2. Objective The objective of the study presented in this thesis is to analyse the loads transferred trough the structural bearings with the associated translations and rotations. Two different types of bearings will be considered separately, the pot bearing and the spherical bearing. The bridge for which this analysis is performed is a relevant steel arch bridge. By obtaining the loads and the related transverse, longitudinal and rotational displacements the slide paths can be determined and furthermore also to which extend this has a relationship with the service life of the structural bearings.
Structural bearings are designed and tested according to the European standards NEN-EN 1337. The relation between the test values included in part 2 (Sliding elements) and 5 (Pot bearings) of these standards and the results found after the analysis of the first Van Brienenoord Bridge will therefore also be investigated.
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MSc thesis – Structural Engineering Steel and Timber Structures
3. Preconditions and scope of this Master thesis The preconditions are as follows: - For the determination of the reaction forces and translations and rotations at the structural bearings, the modified fatigue load model FLM M [75/25] proposed by Adriaan Otte [2009] is used; - A 3D Midas Civil, linear elastic, Finite Element model is used to obtain the forces, translations and rotations at the structural bearings; - For this research only the superstructure of the fixed part of the first Van Brienenoord Bridge is considered; - Two types of bearings (pot- and spherical bearings) are considered separately for the analysis of reaction forces with the associated translations and rotations; - The influence of the friction resistance generated by the sliding elements is ignored for the global behaviour of the bridge.
It is assumed that a plate-girder bridge, a concrete box girder bridge, an arch bridge and a cable- stayed bridge are representative types of bridges. Rijkswaterstaat now requires to analyse an arch bridge as a following study concerning the load-movement interaction.
The thermal loads and wind loads are not considered because these do not contribute to relative rapid displacements and therefore will cause relative small wear.
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MSc thesis – Structural Engineering Steel and Timber Structures
4. Research strategy The first step in this research project is to explore what data concerning the Van Brienenoord Bridge and wear behaviour of structural bearings is available on the internet, at Rijkswaterstaat and the Delft University of Technology. These are such as publications, calculations, component specifications, technical papers, technical drawings and thesis reports of similar researches. After making a selection of the available data and thoroughly pursuing them, finally, a model can be constructed in the finite element software Midas Civil. The model is intended for a static analysis, the wear behaviour of the bridge bearings and the steel deck.
When the results of the wear behaviour of the bridge bearings are calculated, wear paths for each type of structural bearing will be calculated as well. Also a comparison is made with the tests as described in NEN-EN 1337 (Sliding elements), NEN-EN 1337-5 (Pot bearings) and NEN-EN 1337-7 (Spherical bearings).
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MSc thesis – Structural Engineering Steel and Timber Structures
5. Literature study
5.1. General This chapter briefly describes the organization Rijkswaterstaat in section 5.2 and how it was founded. Section 5.3 gives a historic overview of the development of structural bearings, while section 5.4 describes four most common types of bearings. Section 5.5 deals with the relevant tests for pot and spherical bearings according to the European standards for structural bearings. Section 5.6 dedicates to earlier studies related to this research. Finally, summary is given in section 5.7.
5.2. Rijkswaterstaat in brief Rijkswaterstaat is the executive organization of the Dutch Ministry of Infrastructure and Environment. It manages and develops the main national infrastructure facilities on behalf of the Minister and State Secretary. Rijkswaterstaat is responsible for the design, construction, management and maintenance of these facilities in the Netherlands.
Rijkswaterstaat was established on May 24, 1798. On that day a major plan was adopted to take control of public works and water management in the Batavian Republic. From that moment on, all matters concerning public works and water management were dealt with in a central way. Now Rijkswaterstaat is in charge of three national infrastructure networks. These are the main water system, the main highway network and the main waterways network. The three networks are illustrated in Figure 1.
Figure 1: Main water system (left), highway network (middle) and waterways network (right)
A national organization that is close to the users of the infrastructure was required to be able to carry out these tasks effectively and efficiently. This is achieved with the aid of ten regional departments (including 19 road districts and 16 water districts) and one project organization (Room for the River). Rijkswaterstaat (RWS) also contains five specialised departments with their own concentrated knowledge. An overview of the organization structure within the RWS can be illustrated with a chart as shown in Figure 2. In this chart all the regional departments, specialised departments and the project organization are included.
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MSc thesis – Structural Engineering Steel and Timber Structures
Figure 2: Rijkswaterstaat organization chart
As can be seen from Figure 2, the Department of Infrastructure (DI) is one of the five specialised departments within RWS. A part of this department is occupied by the sub-department SWI (Steel Construction, Engineering and Installation) and is located in Utrecht, The Netherlands. This Master thesis is executed under the responsibility of the SWI. They focus on the technical integrity of structures and its main attention goes to the investigation, evaluation and purchase of bridges, tunnels, locks, dams, etc.
5.3. Structural bearings in historic perspective (1) The first bridge bearings can be found in early timber bridges and were very simple. To prevent rotting of the timber beams of the load-bearing structure, simple wooden laths (small sleepers) were used. Figure 3 shows a sketch of an early timber bridge were the wooden laths are included. Beams that were used as a base had many advantages. They allowed the load to be spread evenly, they enabled the deflection of the load-bearing structure without edge pressure occurring between bottom chord and masonry substructure and their elasticity could absorb some of the vibrations caused by traffic on the bridge. The latter property was still being used in the early cast-iron girder railroad bridges. These bridges were extremely sensitive to sudden impact. Due to the increasing loads onto the wooden bearings from the trains on the bridge, those forces could be spread by using cast-iron plates.
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MSc thesis – Structural Engineering Steel and Timber Structures
Figure 3: Early timber bridge resting on wooden bearings (1)
From the seventeenth century it was known that materials react to temperature changes with changing volumes. This, however, had a negligible effect on the existing bridges because those were constructed with wood and had a short span. Since the early nineteenth century, when iron became the material to build bridges, concern about the influence of changing temperature on the strength of the material increased. Also the degree of movement caused by this changing temperature was seriously considered along with its effect on the overall stability of the system.
John Rennie the Elder (1761—1821) had his Southwark Bridge build between 1813 and 1819. This bridge had three cast-iron arches with the middle one spanning 73 metres and the other two 64 metres. Direct precautions to counteract temperature related deformations were not taken. Instead Rennie attempted to wedge the complete load-bearing structure between the buttresses to prevent temperature movements causing the abutment masonry to break. After the abutment masonry was rebuild, cracks in both the road and the footpath surfaces were repeatedly caused by periodical movements.
In August of 1820 the engineering entrepreneur Ralph Dodd (1756–1822) finished a bridge crossing the River Chelmer near Springfield. This bridge has a span of only 9 metres and composed of wrought iron bowstring girders which were only resting on tubular cast-iron pillars. The top of those iron pillars contained grooves on which the whole bridge could contract and expand from the various changes of temperature. In that time, this support system was a new technology. Seemingly the Chelmer Bridge contains the first instance of sliding bearings. Even though this bridge appeared to be under-designed and finally had to be rebuild, it pointed in new directions for bridge building, especially with regard to measures coping with temperature changes.
In the year 1822 John Dowell Moxon, a ship owner and merchant from Liverpool, received a patent regarding the “Improvement in the construction of bridges and similar building types”. According to the patent, he advised something like clamped ribs inserted into each other for the arches of his bridges. These clamped ribs would allow less loading due to expansions and contractions.
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MSc thesis – Structural Engineering Steel and Timber Structures
In the mid-1820s the Brothers Leather introduced special bearings into arch-bridge design generally referred to as “hinges”. In multi span structures, such as the Dunham Bridge (finished in 1832), the single elements of the superstructure were separated so each part could react independently to temperature changes. The gaps between these elements were covered with steel plates. Five years before James M. Rendel had similarly used elongated holes, in his bridge crossing the River Lary, to divide the arched superstructure from the buttresses to allow for expansion and contraction. These new structural solutions opened new perspectives for building iron arch-bridges round 1830, and assuring that they can handle temperature changes at the same time.
In the early nineteenth century the spans of chain- and cable bridges were continuously increasing. This led to the introduction of shifting saddles. These saddles were placed on pylons usually made of stone. This is illustrated in Figure 4. The saddles were not only a requirement for the changing lengths of the cables due to temperature changes but also for the changing forms of the cables with respect to the non-uniform live loads moving across the bridge. The support devices ensured that the shape of the cables remained as needed without transmitting large horizontal forces to the pylon tops. This structural principle was acting as precursors for the bearings later on typically applied in ordinary girder bridges.
Figure 4: Saddles on top of pylons (1)
It was in the 1840s when bearings for the girder bridges developed. Before that time, the spans of those bridges were small enough for either timber or cast-iron beams resting on wooden laths. Since puddle iron was introduced as a building material and new girder systems were invented, wider spans could be allowed. This resulted in too high loads on the wooden bearings and therefore level differences which caused bigger impacts than when loads travelled the bridge. Furthermore, the main beams of larger bridges were progressively resting directly on cast-iron bearing plates. This was the beginning of the era of iron bearings.
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Plane bearings were the first iron bridge bearings. The upper and lower part of this bearing shared a planar area of direct contact. Even though engineers were aware of the requirement of movable bearings, as for arched bridges and suspended ones, initially many girders were fixedly joined to bearing plate and base. This is shown on the left hand side of Figure 5. On one hand this led to deformations in the girders or cracks in the masonry structure of the abutments and on the other hand attempts to attain technologies of free supports. The latter was by means of sliding bearings and roller bearings. Figure 5 shows an example of an attempt for roller bearings (right). Rollers had the benefit to decrease friction on the structure, which was known from the shifting saddles of suspension bridges. An attempt towards the sliding bearing was, among others, the application of plane bearings together with balls made from cannon metal. This system was applied in the Conway Bridge (opened in 1848) and two years later in the Britannia Bridge, both built in the United Kingdom.
Figure 5: A bridge fixedly joined to the base (left) and a bridge resting on rollers (right) (1)
Due to a special mechanism deflections of the superstructure could be allowed by rotating bearings. The two options for the mechanism were a rocker bearing and a knuckle bearing. Because of the friction, the rocker bearing was considered better. These bearings were also used with a combined sliding part in Crumlin Viaduct (1853-1857) in southern Wales. The first implementation of roller bearings in Germany was probably in the bridge crossing River Guenz (1853) near Guenzburg where knuckle bearings were used in connection with rollers.
At this time it seemed that the bearing technology was adequate to solve problems caused by temperature changes and constraint deflection. However, superstructures increased in length and width. Bridges with double tracks became a rule. This meant that the deflections in transverse direction also had to be considered. An additional pin was applied to enable the transversal deflection in the bridge crossing the River Moselle near Bullay (1876—1878). Hence the spherical curved sliding bearing was introduced to allow rotation of the superstructure in every direction. The first time the additional row of stilts or rollers to accommodate transverse expansion of the superstructure was applied in the second bridge crossing the River Weichsel near Dirschau in Eastern Prussia (1889—1891). This increased the size of the large bearings even more. Within the project of Blaues Wunder (1891—1893) near Dresden engineer Claus Koepke (1831—1911) found an elegant solution. Implementing a single layer of rollers diagonally enabled the superstructure to cope with temperature related movements both in longitudinal and transversal directions. Pictures of bearings of the Blaues Wunder are shown in Figure 6 while the support system with the diagonal rollers is shown in Figure 7. By using uneven radii for upper and lower part Koepke applied point rocker bearings as a special type of rocker bearings.
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Figure 6: Bearings of Blaues Wunder (1)
Figure 7: Arrangement of the bearings within the bridge system (1)
At this point the disadvantage of all the existing supports was the significant increase of construction height yielding additional costs. This problem was solved by replacing rows of stilts or roller with a sphere which allowed shifts in every direction. So the ball bearing was invented. Documented in 1922, these ball bearings had freedom of movement in all directions to support bridges. By developing a multi-ball bearing Robert Schoenhoefer (1878—1954) improved this type of bearings. A multitude of small balls was chosen over a small number with a large diameter as were previously used. Schoenhoefer’s next step was implementing rollers of a small diameter, sometimes arranged in two rows to permit movability in every direction – the multi-roller bearing. The multi-ball bearing and the multi-roller bearing are presented in Figure 8.
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Figure 8: Multi-ball bearing (left) and all-directionally movable multi-roller bearing (right) (1)
In the 1920s Germany undertook a federal effort to standardize bearings. Due to lack of knowledge regarding the distribution of the loads in a statically indeterminate support system, bearings with more than two rollers were ignored. Because the single roller bearing was so simple they were still being used. Introduced in the 1930s were the armoured concrete roller bearings and the single roller bearings made from stainless steel and in the 1960s it was the single roller bearings with welded applications. The armoured concrete bearing is shown in Figure 9.
Figure 9: Armoured concrete roller bearing (1)
Although rubber, otherwise known as elastomer, bearings had already been used in the nineteenth century, it was only after the Second World War when interlayers could be developed to prevent the soft rubber mass from bulking. So, relatively deep blocks of rubber that did not yield much under
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MSc thesis – Structural Engineering Steel and Timber Structures load could be produced. In 1954 Eugene Freyssinet (1879—1962) came to the idea of equipping rubber blocks with transverse bracings. These reinforced “mats” could only be applied for small loads unless over-sized mats would be used.
By adapting the sand pot bearing, which was already known for centuries, subjecting heavy loads to rubber was made possible. The sand in the sand pot bearing was simply replaced. Rubber pot bearings could only allow rotations of the superstructure. To be able to allow translations in longitudinal and transverse direction these pot bearings had to be combined with a sliding or roller part.
With the introduction of rubber in the bearing technology a new generation of bridge bearings started. When the newly synthesized polytetrafluoroethylene (PTFE or Teflon) was introduced the new generation was completed. The new material has outstanding properties for bridge bearing, especially the low friction coefficient and the wear resistance when lubricated. Teflon-based sliding bearings allowed translation in any direction with a very small construction depth. Even rotations could be accommodated by combining a rocker part. This was the basis of developing the pot sliding bearings, deformation sliding bearings, and point rocker sliding bearings.
Thanks to Teflon the spherical bearing could be developed further by allowing rotations of the superstructure by means of sliding inside a ball joint. The spherical bearing is presented in Figure 10. The radii of the areas action together were increased and the load transfer improved. Now that we are about a half century farther, still there are no similar or better alternatives of bearings for large bridges.
Figure 10: Modern spherical bearing (1)
5.4. Structural bearings Bridges are structures that are not static and are able to translate. The bearings of these structures not only connects the structural members of the superstructure to the supporting units of the substructure (bents, abutments and piers), but they also allow the superstructure to displace over the substructure without forcing the substructure units out of position.
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According to the first part of the European standards for structural bearings (NEN-EN 1337-1) bearings are elements that allow rotation between two parts of a structure and transfer the loads as well as preventing displacements (fixed bearings), allowing unidirectional displacements (guided bearings) or allowing multidirectional displacements (free bearings) as required.
The most commonly used types of structural bearings are summed as follows: 1. Elastomeric bearings 2. Steel bearings 3. Pot bearings 4. Spherical PTFE bearings
5.4.1. Elastomeric bearings Natural rubber and neoprene (synthetic rubber) are both used to construct bridge bearings. The two types of rubber have minor differences. Compared to neoprene, natural rubber has less resistance to ozone and a wide range of chemicals, which makes it less suitable for harsh chemical environments. On the other hand, it tends to stiffen less at low temperatures. (2)
Basically, three types of elastomeric bearings are used. These are: 1. Plain elastomeric pads 3. Fibre reinforced pads 2. Steel reinforced elastomeric pads
Elastomeric plain pad bearings are able to support moderate vertical loads, but can allow limited rotations and translations. The bulging due to the Poisson effect is only restrained by the friction at the pad’s top and bottom surface and is therefore larger than those occurring at the other types of elastomeric bearings. For this reason plain elastomeric pads are relatively thinner and hence tolerate only small translations and rotations. (2)
The behaviour of fiberglass reinforced elastomeric pads is more similar to that of steel reinforced elastomeric bearings. The compressive load capacity, however, is smaller because the fiberglass is weaker, has more ductility, and does not bond to the elastomer as well as the steel reinforcement does. Reinforcement ruptures can lead to sudden failure, which also leads to a smaller compressive load capacity. These pads allow moderate rotations and translations. (2)
Figure 11: Reinforced elastomer bearing system (3)
In the steel reinforced elastomeric bearings the layers of steel and elastomer are spaced uniformly as shown in Figure 11. The deformation of the elastomer allows the bearing to translate and rotate.
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Under a compressive pressure the stiff steel layers restrain the lateral expansion considerably. This is because the steel plates provide stiffness against bulging and therefore increases the compressive load capacity while still accommodating large translations and rotations. (2)
5.4.2. Steel bearings Steel bearings distribute both horizontal and vertical forces through metal-to-metal contact. The most common type is the roller bearing. It consists of three main parts: a top shoe, an expansion roller, and an expansion plate (see Figure 12). The top shoe (sole plate) connects the roller to a beam or girder, the expansion roller accommodates longitudinal movement through a rolling motion and the expansion shoe (masonry plate) connects the expansion roller to the recessed anchor plate. The fixed variation of this type of bearing relies only on a pin or knuckle to allow rotations while restricting the translation. (4)
Figure 12: Roller bearing (3)
Another common type of steel bearings is the rocker bearing, which functions in a similar manner to the roller bearing. A line rocker bearing is illustrated in Figure 13.
Figure 13: Line rocker bearing (3)
5.4.3. Pot bearings A pot bearing essentially consists of a shallow cylinder, or pot, an elastomeric pad, a set of sealing rings and a piston as shown in Figure 14. When this type of bearings allows translations a PTFE sliding surface is applied. Figure 15 illustrates a multidirectional pot bearing where a PTFE sliding surface is included.
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Figure 14: Fixed pot bearing (5)
Figure 15: Multidirectional pot bearing (5)
Compression Vertical loads are transferred trough the piston and the elastomeric pad to the pot of the bearing. Because the elastomer in the pot is deformable and almost incompressible, it behaves hydrostatically under high compressive pressure. In practice this pad has shear stiffness, so the hydrostatic behaviour only applies for relatively slow deformations.
Rotation Pot bearings can tolerate large rotational displacements and may occur about any axes. These displacements are accommodated by the elastomeric pad by compression on one side and expansion on the other. When the piston rotates the contact between the elastomer and the pot causes elastomer abrasion and sometimes contributes to elastomer leakage. In pot bearings cyclic rotation may also damage the sealing rings.
Lateral load Contact between the rim of the piston and the wall of the pot mainly transfer lateral loads from the piston to the pot. The pot wall, on its turn, then transfers the loads to the baseplates. This occurs by means of shear stresses, in the pot wall, parallel to the direction of the loads and bending of the pot wall with respect to base of the pot.
5.4.4. Spherical PTFE bearings These bearings are bearings with a spherical PTFE sliding surface and may accommodate large rotations about any axes. The rotations take place around the centre of radius of the spherical surface. These bearings are fixed against translation unless an additional flat sliding surface is added,
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MSc thesis – Structural Engineering Steel and Timber Structures which is shown in Figure 16. Because the neutral axis of the beam rarely coincides with centre of rotation of the beam, an additional translation occurs due to this eccentricity.
Figure 16: Multidirectional spherical bearing (5)
Spherical bearings are usually relatively expensive bearings and a good alternative when the gravity load or the required rotation is large. They are capable to carry loads up to several tonnes and may rotate more than 5 degrees. These bearings are probably more costly than the pot bearings, but they may be able to accommodate larger rotations.
5.5. The European standards concerning pot and spherical bearings
5.5.1. The NEN-EN 1337 The European standards NEN-EN 1337 cover many types of bearings: elastomeric, pot, rocker, roller, spherical and cylindrical and in addition fixed and movable restraints. In these standards, wear related tests with their setups are describes in order to evaluate materials which are used for structural bearings. When the material can withstand the tests the material itself and thereby also the bearing are considered resistant to the forces, translations and rotations occurring during the bearing’s service life.
5.5.2. The relevant test for sliding elements (NEN-EN 1337-2) NEN-EN 1337-2 describes the long term friction test in its Annex D. This test is intended to determine the friction coefficient of the sliding surfaces of a bearing and consists of two different test specimens; the test specimens for dimpled and recessed PTFE and for composite materials (Figure 17). The first is subjected to a test of 10242 metres total slide path, while the other is subjected to a test of 2066 metres total slide path.
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Figure 17: Test specimen for dimpled and recessed PTFE sheets (left) and for composite materials (right) (6)
The testing equipment depicted in Figure 18 is used to apply a constant vertical force Fz to obtain the specified pressure on the specimen. At the same time, the moving plate is translated horizontally with a constant velocity in a back and forward movement with a constant amplitude. The pressure, friction force and temperature are also continuously measured. Finally, the friction coefficient can be calculated by dividing the horizontal friction force by the applied vertical force.
Figure 18: Friction testing equipment (6)
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Table 1: Friction test conditions (long term test) (6)
Both specimens are alternately tested in type A and B (Table 1). Since the material damage (wear) occurs in the direction perpendicular to the sliding direction, the dimples lubricate the sliding surface as the PTFE thickness decreases.
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Table 2: Characteristic compressive strength for sliding materials (6)
The values for the nominal compressive strength fk can be found in Table 2 for different sliding materials.
5.5.3. The relevant test for pot bearings (NEN-EN 1337-5) In part 5 of NEN-EN 1337 tests for pot bearings are described in Annex D (Determination of resistant moment) and Annex E (Long term rotation and load test). The same specimen can be used for both tests. However, two different test setups are available. As shown in Figure 19, one setup consists of a compression testing machine with one pot bearing and a hydrostatic bearing attached, while the other consists of a compression testing machine with two pot bearings attached.
Figure 19: Test set-up using one pot bearing and a hydrostatic spherical bearing (left) and using two pot bearings and one sliding surface (right) (7)
By means of the compression machine a vertical force Fz is applied. In both cases the attachments will be continuously rotated with ± 0.01 radians by means of a double-acting hydraulic cylinder (Fa) acting on a lever arm. This will case a sinusoidal movement with a frequency adjustable between 0.003 Hz and 0.5 Hz.
Determination of restraint moment When pot bearings rotate, the elastomeric pad deforms accordingly to accommodate the rotation. If the bearing rotates faster than the pad can fully deform, a resistant moment occurs, which can be expressed as a function of the rotation due to permanent loading (α1), the rotation due to the variable load (between α2,min and α2,max) and the diameter of the elastomeric pad (d). The resistant 3 moment can be obtained according to the formula: Me,max = 32 x d x (F0 + (F1 x α1) + (F2 x α2,max))
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The procedure to achieve the values for the factors F0, F1 and F2 is not dealt with. Reference is therefore made to the standard NEN-EN 1337-5. The eventually obtained resistant moment is used to further design the pot bearing.
Long term rotation and load test This test approves the performance of the pot bearing when it is subjected to long term oscillating relative movements between the pot wall and the internal seal and the required ultimate load capacity in rotated state. If no cohesive elastomeric material is extruded after the test and the compression deformation under the test load does not increase for at least 24 hours, a performance of the specimen is confirmed.
The test procedure itself is not dealt with. Therefore, reference is made to the European standard NEN-EN 1337-5.
5.6. Earlier studies Related to the analysis of the wear behaviour of bridge bearings, two studies have been carried out at the Rijkswaterstaat. Both studies were MSc thesis projects by students of the Delft University of Technology. These thesis projects were namely “Mechanical behaviour of the structural bearings of the Dintelhaven Bridge” and “Mechanical behaviour of the structural bearings of a steel plate-girder bridge”. In the next two subsections these studies are discussed in more detail.
5.6.1. Mechanical behaviour of the structural bearings of the Dintelhaven Bridge For this study a concrete box girder bridge was chosen to analyse. This was the west bridge of the Dintelhaven Bridges. During this project the mechanical behaviour of pot and spherical bearings were investigated. In order to investigate this behaviour the bridge was first modelled in a finite element software Midas Civil then subjected to traffic and thermal loads. After performing a linear static analysis in Midas Civil, the results were compared with tests for structural bearings, which are described in the European standards NEN-EN 1337-2 (sliding elements) and NEN-EN 1337-5 (pot bearings).
From the comparison between the results yielding from the finite element analysis and the tests described test in the European Standards, there can be concluded that: - The long term friction test type B (NEN-EN 1337-2) and the durability test for internal seals (NEN-EN 1337-5) are not clear concerning the relation between value of the wear paths and the associated duration; - A great number of results, yielding from the finite element analysis, match the values included in the tests of NEN-EN 1337; - Further research is required concerning the velocity of the translations; - The real transfer of the horizontal forces should be further investigated; - The spacing between lorries and the number of lorries present on the bridge deck greatly affect the wear paths of the structural bearings.
5.6.2. Mechanical behaviour of the structural bearings of a steel plate-girder bridge As the previous study, this study also deals with the finite element analysis of a bridge with regard to the mechanical behaviour of the structural bearings. The biggest difference, however, is that the considered bridge is a steel plate girder bridge. This bridge is the motorway A27 bridge near Hagestein in the Netherlands. The procedure for the finite element analysis and the comparison of
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MSc thesis – Structural Engineering Steel and Timber Structures the results with the European Standards NEN-EN 1337 is similar to that for the Dintelhaven Bridge. Also here the software Midas Civil was used to model the bridge and to perform a linear static analysis. Furthermore, a linear dynamic analysis was performed to investigate whether there are possible dynamic amplifications.
According to the results obtained within this thesis project, the following can be concluded: - A static analysis performs very well to analyse the displacements and vertical reactions of the supports; - The dynamic amplification and frictional resistance can be neglected; - When subjecting an average lorry to the bridge, the results correspond to the parameter values which are included in the long term friction test type B of the European Standard NEN-EN 1337- 2; - According to the results, the parameter values of the durability test for internal seals (NEN-EN 1337-5) only correspond to those of the heaviest lorry; - The accumulated displacements of the results are much higher than those from the tests; - The vertical reaction forces at the intermediate supports increase maximally by 10% and the end supports by 55% due to a single lorry; - The horizontal forces, which are mostly influenced by frictional resistance, are considered unreliable because friction is disregarded for the finite element analysis.
5.7. Summary In this chapter Rijkswaterstaat and how it was founded was described in section 5.2. Furthermore, section 5.3 and 5.4 acknowledged about the development of structural bearings and listed the most used bearings respectively. Section 5.5 dealt with the relevant tests for pot and spherical bearings according to the European standards for structural bearings and section 5.6 summarizes the similar studies performed for Rijkswaterstaat.
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6. The first Van Brienenoord Bridge
Figure 20: Van Brienenoord Bridges at night with open bascules (photographed by author) 6.1. General This chapter acknowledges the reasons for selecting a relevant bridge for this thesis in section 6.2. Section 6.3 describes the location of the first Van Brienenoord Bridge, while section 6.4 deals with its history and section 6.5 with its global dimensions and description with mention of the materials used to construct it. The layout of bridge deck before and after the second Van Brienenoord Bridge was used is explained section 6.6. Both the present and applied support systems are described in section 6.7. Finally, section 6.8 gives summarizes this chapter.
6.2. Selection of the bridge For this study Rijkswaterstaat aimed at a research of an arch bridge. Therefore, a relevant bridge had to be selected in order to build a model for the finite element analysis. Since the first Van Brienenoord Bridges has one of the largest spans in the Netherlands, it seemed an adequate choice for a typical arch bridge.
According to a web application of RWS called MTR+ the daily average number of 106,068 vehicles crossed the first Van Brienenoord Bridge in the year 2011, which is a very high traffic intensity for a bridge. Finally, due to the large span and the high traffic intensity the first Van Brienenoord Bridge was selected as a relevant bridge to be analysed.
6.3. Location of the bridge As can be seen in Figure 20, the Van Brienenoord Bridges are arch bridges situated right next to each other. They are located in Rotterdam, the Netherlands and cross the river Nieuwe Maas as illustrated in Figure 22 and Figure 21. These bridges also belong to the rhombus-shaped system of roads and bridges around Rotterdam, which is otherwise known as “Ruit om Rotterdam” and is shown in Figure 22. It is composed with four sides; the A4 (in the west), the A15 (in the south), the A16 (in the west) and the A20 (in the north). The arch bridges are part of the A16 motorway.
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Figure 21: A part of the rhombus-shaped motorway system around Rotterdam with the first Van Brienenoord Bridge (8)
Figure 22: The rhombus-shaped motorway system around Rotterdam with the first Van Brienenoord Bridge (van Brienenoordbrug) (9)
Originally, there was only one Van Brienenoord Bridge. Later, a new bridge west from the former was constructed to increase the traffic capacity. The bridge on the east side is therefore referred to as the first Van Brienenoord Bridge.
6.4. History of the bridge When the economy of Rotterdam flourished around the middle of the nineteenth century it accelerated its development, which lead to the construction of the New Waterway (1872) and the railroad to Amsterdam (1847) and Antwerp (1877). Connecting the banks of the river Nieuwe Maas
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MSc thesis – Structural Engineering Steel and Timber Structures with the Willem Bridge (1877) and the Maas Tunnel (1942) also contributed a great deal to the development of this city. (10)
During the Second World War the centre of Rotterdam was destroyed and therefore had to be reconstructed, which was a good opportunity to find solutions for the increased traffic intensity. About 37000 vehicles crossed this area each day in 1950 by means of the Willem Bridge and the Maas Tunnel. Also due to the economic growth of the 1960s the number of vehicles increased rapidly until the capacity of the available infrastructure was no longer adequate and the streets both north and south of the river banks could no longer handle the traffic volume. Rijkswaterstaat then came up with the plan for the rhombus-shaped system of roads and bridges around Rotterdam to which the Van Brienenoord Bridge belongs. This system had to bypass the Rotterdam, Schiedam and Vlaardingen traffic agglomerations and manage the increase in traffic volume to and from these urban areas. (10; 11; 12)
The groundwork for the first Van Brienenoord Bridge, started in 1959 followed by construction, which began in 1961. Finally, on February 1, 1965 the bridge and the approach roads were officially opened for traffic by Queen Juliana. Figure 23 gives an impression of the bridge and its surroundings around 1978. (12)
Figure 23: The Van Brienenoord Bridge anno 1978 (13)
The Van Brienenoord Bridge owes its name to the underlying island called Van Brienenoordeiland. Its design was made by ir. W.J. van der Eb and consists of diagonal cable connecting the arches with the deck, which made the superstructure very rigid with the special relations with the arch shape.
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For its time, this design was revolutionary slender and transparent and later inspired the design of many other bridges. (14)
After the opening of the first Van Brienenoord Bridge, it quickly gained national recognition due to the increasing traffic that crossed the bridge. This bridge was mentioned in almost all warnings for traffic jams and so was known throughout the whole country (11). Due to constant traffic jams it was decided in 1987 to construct an almost identical, second Van Brienenoord Bridge, which was built 15 centimetres west from the first and doubled the traffic capacity to 180,000 vehicles per natural day (15). On May 1, 1990 the Minister of Transport, Public Works and Water Management J.R.H. Maij- Weggen officially opened the second Van Brienenoord Bridge. The bird’s-eye view in Figure 24 presents the final structure of the new, almost identical Van Brienenoord Bridge right next to the first.
Figure 24: A bird’s-eye view of the Van Brienenoord Bridges (Rijkswaterstaat)
6.5. Global geometry and description of the bridge The first Van Brienenoord Bridge has a theoretical span of 287 metres and an overall length of 304.40 metres. The drawings in Figure 25 and Figure 26 not only show the dimensions of the lengths, but also the height in the centre of the span, which is approximately 40 metres.
Figure 25: Drawing of the final design of the arch bridge (12)
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Figure 26: Global dimensions of the superstructure
Figure 27: Side view of the Van Brienenoord Bridges (photographed by author)
The bridge consists of two main girders, each composed of a box-sectioned bottom cord and arch with a rigid connection between them by means of gusset plates. The cross section of the arch increases towards the ends. As shown in Figure 27 and Figure 28, the arches are not only connected with their bottom cord, but also with each other through wind bracings. The cables, which are presented the clearest in Figure 27, stretch from the arch to the bottom chord and are attached to the cord by means of cast steel saddles.
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Figure 28: Global dimensions of the bridge cross section at mid-span
The bridge deck is designed as a orthotropic steel deck that consists of a longitudinal and cross beam system with secondary cross girders in between. The main stringer is in the centre of the deck width and extends over its full length. The secondary stringers consist of bulb sections which span between the secondary cross girders and are welded to the deck plate at 30 cm intervals. The deck plate has an overall thickness of 10 mm. As illustrated in Figure 28 and Figure 29, the centre-to-centre distance between the two girders is 24.9 metres while the deck has a total width of 33.5 metres. Figure 28 also indicates how the arches are connected with each other and the deck. It is also worth mentioning that the diagonal cables are 110 millimetres diameter suspension bridge cables.
Figure 29: Widths of the bridge deck
Regarding the material used to construct the bridge, the most used steel grade is Fe 52, which is equivalent to the nowadays commonly used structural steel S355. Both main girders, the orthotropic deck as well as the sidewalks predominantly consist of Fe 52. Only the wind bracings between the arches and the parapets are composed of the steel grade Fe 37, which is equivalent to the structural steel quality S235.
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6.6. Layout of the bridge deck Until the second Van Brienenoord Bridge was opened, all the traffic had to cross the first bridge. This means that the older bridge had to carry the northbound as well as the southbound traffic. The northbound being the direction towards The Hague (Den Haag) and the southbound of course being the direction towards Dordrecht.
Figure 30: Direction of traffic on the first Van Brienenoord Bridge before the second bridge was opened (9)
In that time, the bridge deck had six carriage ways and two tracks at each outer side for mopeds, cycles and pedestrians, which are illustrated in Figure 30 with their traffic direction. However, the current layout is quite different. Now that the new bridge is built on the west side and all vehicles are northbound, the west cycle track is no longer used while the east cycle track was widened to accommodate cyclists in both directions. The bridge deck including the wider east cycle track with the current traffic layout is shown in Figure 31.
Figure 31: Direction of traffic on the first Van Brienenoord Bridge after the second bridge was opened (9)
6.7. Support system The first Van Brienenoord Bridge has two supports at each end. On the north end both bearings are hinged (fixed), while both on the south end are roller bearings. These two types of steel bearings applied in the bridge are shown in Figure 32 and Figure 33.
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Figure 32: Fixed bearing Figure 33: Roller bearing
As stated in the preconditions, for this study it will be assumed that all four bearings are either pot or spherical bearings. Nowadays for nearly each new large bridge one of these two types of bearings is applied, hence the assumption.
To achieve both translational as well as rotational equilibrium by means of restraining as least degrees-of-freedom (DOFs) as possible, the support system depicted in Figure 34 is used. The south- easterly support does not allow any translations while the south-westerly allows it in both longitudinal and transverse direction. On the north side, the westerly also allows both longitudinal and transvers translations whereas the easterly only allows longitudinal translation. The one thing all bearings have in common is that they all have three rotational DOFs.
Figure 34: The applied support system with the directions of allowed translation included
6.8. Summary In this chapter first of all the reasons why the first Van Brienenoord Bridge was selected for this study is in mentioned in section 6.2. The location of this Bridge was described in section 6.3. Next, section 6.4 briefly dealt with the history of the bridge including the effect the second bridge had after it was opened to traffic. A short description and the global dimensions of the bridge are given in section 6.5, while section 6.6 showed the current and former layout of the deck. Furthermore, the present support system under the steel structure and the used support system for the analysis are described in section 6.7.
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7. The finite element model
7.1. General For this study a finite element model of the first Van Brienenoord Bridge is constructed in order to analyse the wear behaviour of its bearings. A proper element choice based on the mutual behaviour between structural elements was therefore required. Of course, the selection was made from the element library of the used finite element software. The available elements are hence briefly described in section 7.2. The proper selection is also made in this same section. Furthermore, section 7.3 explains where and why is it necessary to customize transitions between some structural elements in the model. Modelling the supports in Midas Civil is covered in section 7.4, while the finished finite element model of the first Van Brienenoord Bridge is shown in section 7.5. The self- weight and superimposed dead loads applied for the structure is described in section 7.6. For a better understanding of the bridge's global behaviour hand calculations are made, which results in a Williot-diagram and the deformations of the superstructure. The deformations are furthermore used to validate the finite element model and can be found in section 7.7, while the Williot-diagram is presented in Annex A.
7.2. Requirements of the finite element model For this study the finite element software Midas Civil 2011 v2.1 has been used to model the first Van Brienenoord Bridge. But before the model could actually be constructed, an element choice was necessary for each part of the superstructure while keeping in mind that the aim is to build an efficient model to obtaining accurate enough results. In finite element modelling the more detailed the model is, the more computing time and working memory is required during an analysis. When the model is too detailed, the requirements are excessively higher. The element choice should therefore provide the model with the least nodes possible since these requirements are directly related to the number of DOFs.
Of course, the selection had to be made from the element library of Midas Civil, which consists of the following ten elements:
1. Truss Element: A two-node, uniaxial tension-compression, three-dimensional line element that undergoes axial deformation only. 2. Tension-only Element: A two-node, tension-only, three-dimensional line element that undergoes axial tension deformation only. 3. Cable Element: A two-node, tension-only, three-dimensional line element that is capable of transmitting axial tension force only. 4. Compression-only Element: A two-node, compression-only, three-dimensional line element that undergoes axial compression deformation only. 5. Beam Element/Tapered Beam Element: A two-node, Prismatic/Non-prismatic, three-dimensional beam element that takes the stiffness effects of tension/compression, shear, bending and torsional deformations into account. 6. Plane Stress Element:
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A three- or four-node plane element with no stress components existing in the out-of-plane directions. The strains in the out-of-plane directions can be obtained on the basis of the Poisson’s effects. 7. Plate Element: A three- or four-node plane element that is capable of accounting for in-plane tension/compression, in-plane/out-of-plane shear and out-of-plane bending behaviours. 8. Two-dimensional Plane Strain Element: A three- or four-node plane element with each node in the X-Z plane. When elements are defined, a unit thickness is automatically given. The strains in the out-of-plane directions do not exist and the stress components in the out-of-plane directions can be obtained only based on the Poisson’s Effects. 9. Two-dimensional Axisymmetric Element: A three- or four-node plane element with each node in the X-Z plane. When elements are defined, a unit thickness is automatically given, which has a radial symmetry with the Z-axis being the axis of rotation. The circumferential displacements, shear strains and shear stresses do not exist. 10. Solid Element: A 4, 6 or 8 node element in a three-dimensional space. Each node retains three translational displacement DOFs.
In Midas Civil, the plate element is equivalent to what other finite element software provide as a “Shell element”. It has a three-dimensional displacement field and five DOFs (3 displacements and 2 rotations) in each node, as depicted in Figure 35 and Figure 36. Since Midas Civil named this element type a ‘plate element’, it should be kept in mind that in fact it is a shell element.
Figure 35: Nodal forces for a quadrilateral plate element Figure 36: Nodal forces for a triangular plate element
The elements should be selected such that the behaviour of the model at the bearings corresponds to that of the actual bridge. This means that the orthotropic steel deck should preferably be composed of plate elements, since the in- and out-of-plane shear as well as the in- and out-of-plane bending are of great importance to ensure the proper wear behaviour at the bearings while applying only three or four nodes per element.
The parts of the orthotropic steel deck composed of plate elements are the bottom chords, main and secondary cross girders, main stringer, deck plate and a part of the secondary stringer. These secondary stringers, which are bulb sectioned stiffeners, consist party of a beam element (see Figure
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37). Since the bulb end alone is small compared to the web and behaves quite similar to a beam, it is modelled as a beam element. The web of the stiffeners, on the other hand, is modelled as a plate element, which means that applying beam elements on one edge does not require extra nodes. Consequently, less nodes are used to add the stiffeners to the model. Figure 38 shown how the bulb sectioned stiffeners are modelled. In this figure the red dots, yellow lines and rectangles represent the nodes, the plate elements and the beam elements respectively. Due to simplification the beam element and the bulb end have a minor difference in shape, but still have the same cross sectional area.
Figure 37: The beam element in Midas Civil 2011
Plate element
Beam element
Figure 38: The actual (left) and modelled (right) bulb sectioned stiffeners
Local deformations of the arches and the wind bracing do not have a major influence on the global behaviour of the deck, thus, also not on the required results. These structural element are therefore modelled with use of a somewhat simpler finite element type, the beam element. The arches and bracing are hence not only modelled with fewer elements and nodes, but also with less time and effort.
The cross section of the arches increases towards the ends in terms of height and web thickness. This seems also to be the case for some wind bracings and partly the portal frame, but only with increasing height towards the arches. Luckily, the beam element provides the possibility to add non- prismatic elements. Different cross sections could therefore be defined for each part of the arch, the concerning bracings and part of the portal frame with varying cross sectional properties. As an
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MSc thesis – Structural Engineering Steel and Timber Structures example for the definition of a non-prismatic beam element the cross sectional data of a part of the arch is given in Figure 39.
Figure 39: Cross sectional data of a part of the arch
Now that element types for the bridge deck, aches and wind bracings have been selected, the only remaining parts of the superstructure for which an element needs to be selected are the diagonal cables. These cables are very slender structural elements that are continuously loaded with axial tension. Thus, it is obvious that cable elements are used to model the diagonal cables in the bridge.
According to Figure 31, the current layout of the bridge deck consists of totally six traffic lanes from which the third from the left is designated for lorries. This lane is considered the most important for the analysis of traffic induced movement and load interaction. Since the axle loads are applied as concentrated loads, a relatively wide force distribution is required for the proper behaviour of the model. To provide this better distribution all elements within the area of this traffic lane are longitudinally divided into parts of 205 mm, allowing for a longitudinal division with a factor of ten. Figure 40 shows the precise position and dimension of this lane with elements with a smaller longitudinal dimension. The final detailing of the considered lorry lane is depicted in Figure 41.
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Figure 40: The traffic lane designated for lorries
Figure 41: The detailed lorry lane
Finally, three element types are used to construct the first Van Brienenoord Bridge. The plate (shell) element is used to model most of the deck, while beam elements are used to model the arches, cross bracings and the bulb ends of the stiffeners. Additionally, the diagonal cables are modelled using cable elements.
7.3. Modelling of structural transitions When modelling the first Van Brienenoord Bridge, attention was paid to connections between structural elements to obtain a correct global behaviour of the modelled superstructure. In some cases it was necessary to construct a transition. The most complex is the connection between the bottom chord and the arch. As mentioned in section 6.5 “Global geometry and description of the bridge” and shown in the left image of Figure 42, this connection is extremely rigid. The cross section of the arch at this point is about 9 metres high and has internal stiffening by means of thick plates. A connection with only one node would therefore have been insufficient. To measure up to the rigidness and a proper transmission of forces and moments at the connection, a system of beam and plate elements was constructed from one node on one end to several nodes on the other. It turned out to be pyramid shaped with rectangular 35x35 mm2 solid beam elements running along the edges (except around the base), around half the height and along half the width of the wider side. The
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MSc thesis – Structural Engineering Steel and Timber Structures plate elements used are 32 mm thick and are not only applied on the sides of the transition but also at half the height.
Beam element
Plate elements
Figure 42: The connection between arch and bottom chord (left) and its model (right)
Beam element
Beam element
Plate element
Plate elements Plate element Beam element
Figure 43: A more detailed view of the connection between arch and bottom chord and the stiffening above the support
The other case where a special transition was necessary was the connection between the diagonal cables and the orthotropic steel deck. In reality, these cables are attached to the bottom chord by means of steel saddles, which transfer the cable’s axial tension and distribute it in the deck. Consequently, a proper distribution of this axial tension should be ensured. All cables-chord connections coincide with a main cross girder. It is therefore decided that the cables would be joined at the bottom of the web of the aligned cross girders with the addition of plate elements on each side, which are depicted in Figure 44. In this manner the transfer of the tension in the diagonal cables to the bottom chord is aimed to be reasonably similar to the actual situation.
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Cable element
Plate elements
Figure 44: Connection between cables and bottom chord
Apart from transitions, a somewhat similar adaptation was applied to finite element model. Due to self-weight, superimposed dead loads and traffic loads on the bridge the bottom chord has to transfer large vertical forces to the supports. Since each support is modelled as a node, these forces will concentrate exactly at these points. The drawback of this is that the bottom chord above the supports may deflect inwards. Stiffening this part of the bottom chord is therefore applied. To insure the rigidity of the cross section, 9 solid beam elements were attached to each other and a thick plate was added to the bottom flange right above the support. This stiffening can be seen in Figure 43, where the green/black dot represents the support.
7.4. Modelling the supports When modelling the supports, it is only necessary to analyse the global behaviour of the bridge at the supports. Thus, to obtain the desired results it is not relevant to model the support in detail. Instead, they are included by adding boundary conditions to the concerning nodes, which are simply nodal constraints.
Since the used support system is already described in section 6.7, Table 3 is given to summarise the properties of each support according to the numbering shown in Figure 45. The constraint of each DOF is denoted with the number “1” while the free DOFs are indicated with the number “0”.
Figure 45: The numbers of the four supports in a top view of the bridge deck
To find the translations at the supports due to traffic loading, there should first be considered whether the friction force on the sliding surface of bearings is exceeded, i.e. in the unconstrained translational directing. The bearings can otherwise not translate. To determine the horizontal reactions in the unrestrained translational DOFs, the support system had to be modified. A second
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MSc thesis – Structural Engineering Steel and Timber Structures support system was adopted in the finite element model where all four bearings are constrained in each translational direction while they are free to rotate. It can therefore be verified whether the supports will translate and, if so, from which position of the traveling load this is initiated. From here on out the new support system will be referred to as the “fixed support system” whereas the other will be referred to as the “normal support system”.
Support Node Dx Dy Dz Rx Ry Rz 1 161267 1 1 1 0 0 0 2 69394 0 1 1 0 0 0 3 161264 0 0 1 0 0 0 4 69385 0 0 1 0 0 0 Table 3: DOFs of the normal support system
Table 4 gives a clear overview of the constraints of all bearings based on the fixed support system. Note that all horizontal translations are constrained.
Support Node Dx Dy Dz Rx Ry Rz 1 161267 1 1 1 0 0 0 2 69394 1 1 1 0 0 0 3 161264 1 1 1 0 0 0 4 69385 1 1 1 0 0 0 Table 4: DOFs of the fixed support system
7.5. The finished model In this section figures are shown of the different views of the final finite element model of the First Van Brienenoord Bridge. The difference is support system is needless to show since that would only differ in the green and black coloured nodes.
It is worth mentioning that when the model was being constructed in Midas Civil, eventually the model became so large that after performing even the smallest commands, for example a “pan” command, it took the software about a half minute to respond again. Modelling the remaining part of the bridge was therefore very time consuming and was also associated with other major difficulties such as high frustration and mental stress levels. Finally, the total model consists of 124,616 elements.
Initially, this study would also include a fatigue analysis of the bridge deck. The finite element model was therefore constructed such that accurate results could be obtained for that analysis as well. If fatigue would not have been an item of study, the bridge model could be constructed with larger elements, which would considerably reduce the number of DOFs and consequently also the modelling time. This would furthermore result in a smaller model file, which may have enables the moving load analysis. But since more time was required to model the bridge and more data had to be processed in a spreadsheet software, the analysis of the bearing behaviour consumed so much time that the fatigue analysis had to be cancelled from this study.
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Figure 46: 3D-view of the model
Figure 47: Top view of the final model with (bottom) and without (top) supports
Figure 48: Northbound side view of the model without (left) and with (right) supports
7.6. Self-weight and superimposed dead loads In Midas Civil the self-weight of the structure is calculated automatically when a load case is created with a vertical gravity factor. The superimposed dead loads, on the other hand, were applied manually by means of pressure loads, which are estimated values of 100 kg/m1 (distributed of the edge area) for the crash barriers and parapets, 2.5 kN/m2 for the concrete kerbs below the crash barriers and 1.6 kN/m2 for the ZOAB asphalt (porous asphaltic concrete) layer.
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In total there are three crash barriers and three parapets along the full length of the deck, which together result in a load of: 6 x 1 kN/m1 x 304.345 m = 1826 kN
Even though two different widths are found for the concrete kerbs, the same pressure load of 2.5 kN/m2 remains applicable for both widths. The two wider kerbs are located above the bottom chords and the narrower above the main stringer. Since the dimensions of the kerbs are unknown, a height of 100 mm is assumed for all kerbs whereas the width of the bottom chord (1212 mm) and the distance between the two central longitudinal stiffeners (600 mm) are adopted for the wider and narrower kerbs respectively. These contribute to the following load:
(2 x 1.212) x 304.345 m x 2.5 kN/m2 = 2301 kN
Since the asphalt layer is applied onto the complete area of the bridge deck, it adds the following load to the superimposed dead loads: 304.345 m x 33.5 m x 1.6 kN/m2 = 16313 kN
The total superimposed dead loads are consequently 20,440 kN. When simultaneously applying pressure loads on several plate elements in one plane, attention must be paid to the local axis of the elements. If the local positive z-axes differ from each other, the direction of the pressure load will differ as well because the pressure load input is related to the element’s local axes.
The addition of the self-weight and superimposed dead loads is not relevant for the static analysis because they do not contribute to the wear paths at the bearings. But they are requisite to determine to which extent the wear paths occur since they are directly related to the friction force in the bearings.
7.7. Validation of the model The validation of the model is performed by means of a Williot diagram and comparing its deflection with those found in the finite element software. Due to the longitudinal symmetry, only half of the superstructure is considered and, consequently, half of the self-weight is applied to perform the necessary calculations and draw the diagram. On the finite element model, on the other hand, the total self-weight is considered. According to the Williot diagram the superstructure is deformed as shown in Figure 49, where the translations are drawn 50 times larger compared to the dimensions of the structural elements. The Williot diagram itself and how it was achieved are included in Annex A.
The maximum vertical deflection is found in node “I”, Figure 49, which is 244.0 millimetres whereas the vertical deflection found for the same node in the finite element model is 240.6 millimetres. These values have a difference of only about 1%, which is very small. When comparing the overall deformed shapes of the Williot diagram and the finite element model, Figure 49 and Figure 50, there can be noted that also the shapes are fairly similar to each other. Based on these two aspects the finite element model is considered valid regarding the global behaviour of the superstructure.
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Figure 49: Deformations according to the Williot diagram
Figure 50: Deformations according to the finite element model due to self-weight
7.8. Summary Throughout this chapter, all components of the element library of Midas Civil and the selection for each element types were described in section 7.2. To assure the correct global behaviour of the finite element model several transitions were customized, which were dealt with in section 7.3. Section 7.4 explained how the supports were modelled, while the finished finite element model of the first Van Brienenoord Bridge was shown in section 7.5. Subsequently, the self-weight and superimposed dead loads applied for the structure is described in section 7.6 and the validity of the finite element model was checked by means of a Williot diagram in section 7.7.
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8. Traffic loads on the bridge
8.1. General In this chapter, the used fatigue load model for the analysis of the structural bearings is described in section 8.2, while section 8.3 deals with two methods and their requirements to perform a static analysis based on the influence line theory. The method selection along with its corresponding reasons is included in section 8.4. Section 8.5 explains how the necessary axle loads were modelled. Finally, section 8.6 sums up this chapter.
8.2. The applied fatigue load model The European standards EN 1991 (Eurocode 1) gives many actions which have to be taken into consideration when designing a structure. Part 2 of this standard (EN 1991-2) only deals with traffic loads on bridges. It also states that bridges should be verified for, for example, flexural and fatigue strength by means of traffic load models included in this standard. Different load models are prescribed for different verifications, i.e. load models with maximum expected loads are intended for flexural strength verifications whereas fatigue load models are intended to determine the (remaining) fatigue life of a structure. This study analyses the wear and fatigue of the bearings so fatigue load models are use.
In total 5 different Fatigue Load Models are defined and given in En 1991-2, each intended for another purpose in fatigue life assessment. The Fatigue Load Models (FLMs) 1, 2 and 3 are meant to be used to determine the maximum and minimum stresses resulting from the possible load arrangements on the bridge of any of these models. Fatigue Load Models 4 and 5, on the other hand, are intended to be used to determine stress range spectra resulting from the passage of lorries on the bridge. FLM 4 is composed of sets of standard lorries which together produce effects equivalent to those of typical traffic on European roads and FLM 5 is based on recorded traffic data.
Since only the extreme values of the translations and rotations, and not their paths, can be achieved by the FLMs 1 to 3, they are considered not suitable for this study. Because the application of FLMs 4 and 5 can yield the full path of the wear behaviours, these models could have been used for the analysis of the structural bearings. But Rijkswaterstaat requested to apply a FLM proposed by a TU Delft master student [A. Otte, 2009], the FLM M [75/25]. This fatigue load model is basically a reduced FLM 4 which is suitable for the Dutch motorway bridges since it is based on traffic measurements on the most intensively used motorway in the Netherlands, the Moerdijk Bridge. According to this model, the total expected number of lorries crossing the bridge in one year is 3,000,000.
As shown in Table 5, the FLM M [75/25] consists of six standard trailers, the V11 (truck), V12 (truck), T11O2 (tractor with semi-trailer), T11O3 (tractor with semi-trailer), V11A12 (truck with trailer) and T12O3A2 (tractor with semi-trailer and trailer). The standard trailers as well as their axle distances are based on measurements on the Moerdijk Bridge.
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Table 5: Fatigue Load Model M [75/25] (16)
The numbers in the square brackets of “FLM M [75/25]” are associated with a load set, the first and second associated with load set 1 (lower axle loads) and 2 (higher axle loads) respectively. These load sets can be found in the last two columns of. Each number within the square brackets furthermore represents the percentage of 3 million lorries that cross the bridge in a year. As an
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MSc thesis – Structural Engineering Steel and Timber Structures example, using Table 5, there can be determined how many lorries of the type V12A12 may be expected to cross the bridge in one year. Consequently, a number of 150,000 lorries (5% of 3,000,000) are achieved from which 112,500 (75% of 150,000) weigh 660 kN and the other 37,500 (25% of 150,000) weigh 445 kN.
Both wheel prints and tyre configurations are defined in EN 1991-2 and the thesis report of Otte. For the analysis of the structural bearings of the first Van Brienenoord Bridge, however, these have a minor contribution to the global behaviour of the bridge. The tyre configurations are therefore modified to simplify the processing of analysis data. The area of the wheel prints is, on the other hand, entirely neglected.
8.3. Requirements for a static analysis using the influence line theory Midas Civil contains a powerful built-in feature that allows the user to obtain influence data. This feature is called “the Moving Load Analysis”. In order to implement this option, a traffic line or surface lane should first be added to define the actual traffic lanes on the bridge model over which the loads can “travel”, hence the name Moving Load Analysis. Thereafter, vehicular loads can either be specified manually or can be chosen from many predefined traffic loads which are based on different standards, such as EN 1991-2. The predefined traffic loads, however, are adjustable to enable users to modify the given values according to their own National Annex. The ability to define moving load cases is included in the software to allow different moving loads to simultaneously travel over different traffic lanes. Possible results achieved after performing a moving load analysis are shear and moment diagrams and tables, reactions forces and moments, influence lines and the location of a load that results in a maximum or minimum nodal reaction. However, for this study only the influence lines for reaction forces, displacements and rotations are necessary.
Another method to obtain influence lines is by applying unit axle loads along the entire length of the bridge deck with each axle load in a different static load case. The resulting data should then be entered in a spreadsheet program, where influence lines can be generated with the influences values and the locations of the axle loads. Since these influence lines are based on a unit load, they can easily be scaled for each lorry type. The major drawback of this method is that it is very labour intensive, which makes it less efficient.
8.4. Method choice for obtaining influence lines Since the used finite element software provided a convenient and efficient tool to achieve influence lines, a choice was easily made for the method to analyse the structural bearings. The moving load analysis was favoured over the time consuming method by means of a series of separate axle loads.
As stated in section 8.3, in Midas Civil a traffic lane should first be defined prior to the execution of a moving loads analysis. When performing the necessary steps to define the lane, unfortunately, the software kept showing an error message which notified that the model file was too large for this operation. Decreasing the size of the bridge model by using fewer nodes was not an option since it already contained a minimum of nodes to ensure a proper nodal connectivity between elements and accurate enough results. Realizing that the operation to define the lane was too large for this model, many attempts were made to bypass this problem by reducing the surface area of the traffic lane in different ways. These are, such as narrowing or shortening the lane or selecting fewer nodes to apply the lane on. Sadly, the software still failed to apply the traffic lane. Performing a moving loads
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8.5. Modelling axle loads Since the moving load analysis was no longer an option, a series of axle loads had to be applied instead. Therefore two 50 kN nodal loads were added to model the axle loads, which at the same time function as unit loads of 100 kN. At an approximation of 8.2 metre intervals 38 axle loads are placed over the full length of the bridge. In transverse direction the two nodal loads were positioned at a distance of 2 metres from each other and 50 centimetres from the edges of the traffic lane, as shown in Figure 51. The bridge model with all the 38 axle loads is displayed in Figure 52 whereas Figure 53 shows a closer view on one end of the deck.
Figure 51: Positions of the nodal loads on the traffic lane
Figure 52: The modelled axle loads over the full bridge length of the bridge
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Figure 53: A closer view of the modelled axle loads
If more axle loads were to be applied, the increase in accuracy for the influence lines would be unnoticeable small. It would also have resulted in an unnecessarily high amount of processing time and data and inconvenient load coordinates to work with.
8.6. Summary A general introduction of this chapter was given in section 8.1. Section 8.2 described the used fatigue load model for the analysis of the structural bearings whereas section 8.3 mentioned two methods and their requirements to perform a static analysis based on the influence line theory. The method selection along with its corresponding reasons was dealt with in section 8.4 and section 8.5 explained how the necessary axle loads were modelled.
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9. Linear Static Analysis
9.1. General In this chapter, mostly the results of the linear static analysis are presented. The processed data is included as well. To be more specific, the self-weight and superimposed dead loads with their contribution to the support reactions are dealt with in section 9.2, while section 9.3 presents the influence lines due to unit axle loads. Section 9.4 describes how these influence lines are modified to obtain those for a unit lorry, whereas section 9.5 sums the total translations and rotations without and with taking the friction at the bearings into account. Extra modification of the results was required to obtain the wear behaviour of free pot and spherical bearings. The final outcome can respectively be found in sections 9.6 and 9.7. Until then only single lorries crossing the bridge is considered. The analysis of multiple lorries is therefore performed in section 9.8. Section 9.9 finally gives a summary of this chapter.
9.2. Self-weight, superimposed dead loads and the corresponding results The self-weight of the finite element model can only be provided by the software after running an analysis. Accordingly, the total amount of self-weight resulted in 39547 kN. As for the asphalt loading, a weight of 16313 kN is obtained, which is quite similar to the calculated value of 16313 kN. Since the loads due to the crash barriers, parapets and the concrete kerbs were added in the same static load case, a summed weight of 4126 kN was found in the resulting data. For the simplicity, this load case is called CPC (Crash barriers, Parapets and Concrete kerbs). When comparing it with the calculated value of 4126 kN, very little difference is noticed again. This means that all superimposed dead loads were correctly added to the model.
Load case Fz [kN] Self-weight 39,547 Asphalt 16,313 CPC 4,126 Σ 59,986 Table 6: Overview and summation of the deadweight
As can be seen from Table 6, it gives a clear overview of all mentioned load cases and summarizes their loads as well. Other relevant deadweight related results are the vertical reactions at the supports. These can be found in Table 7. The summation of all vertical reactions should match the total deadweight provided from the analysis results to ensure vertical equilibrium, which is also the case.
Fx [kN] Support Self-weight Asphalt CPC Σ 1 9,901 4,077 1,106 15,083 2 9,873 4,080 1,106 15,058 3 9,901 4,077 957 14,934 4 9,873 4,080 958 14,910 Σ 39,547 16,313 4,126 59,986 Table 7: Vertical reactions due to deadweight
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Note that the reactions due to self-weight of supports 1 and 3 differ from those of supports 2 and 4. This is due to the eccentricity in the width of the superstructure. The cantilevering part at each end has a different length and the composition of the orthotropic steel deck near each end differs as well.
9.3. Influence lines for the unit load As described in chapter 8, axle loads are applied along the length of the bridge to obtain unit influence lines (UILs), which are based on unit loads of 100 kN. The values given in the concerning graphs are therefore based on that load. It is also worth mentioning that the x-coordinates of the applied axle loads are chosen such to ease its use in the influence lines. Accordingly, the origin of the x-axis lies at the southern supports, as shown in Figure 54. This means that the x-coordinate of the southern and northern supports are at 0 and 287 metres respectively.
Figure 54: The origin and main values of the x-axis
When the static analysis was performed, all resulting values for each DOF were transferred to the spreadsheet software Microsoft Excel, where all values were thereafter processed to graphs. Figure 55 to Figure 62, show the necessary resulting graphs for the series of separate axle loads. The first six influence lines are those for the normal support system, while the remaining two are for the fixed support system. As mentioned in section 7.4, the horizontal reactions of the latter support system are required to determine whether and from which x-coordinate each bearing translates.
Vertical (z) reaction forces - Normal supports
70
60
50
40
30 Fz [kN] Fz 20
10
0 -10 40 90 140 190 240 290 -10 x [m]
S1 S2 S3 S4
Figure 55: UILs of z-reactions for the normal support system
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From Figure 55 there can be noticed that supports 1 and 3 are present on one end of the span, while supports 2 and 4 are located at the other end. The vertical reactions of supports on one end, however, differ from each other due to the eccentric position of the traffic lane on the bridge deck.
Horizontal (x) displacements - Normal supports
0.4
0.3
0.2
0.1
0 Dx Dx [mm] -10 40 90 140 190 240 290 -0.1
-0.2
-0.3 x [m]
S2 S3 S4
Figure 56: UILs of translations in the x-direction for the normal support system
Figure 56 shows that the influence regarding the translations in x-direction is considerably higher for support 4. It is therefore only necessary to consider support 4 for accumulated displacement calculations. The same applies for the translations in the y-direction.
Horizontal (y) displacements - Normal supports
0.9 0.8 0.7 0.6
0.5
0.4
0.3 Dy [mm] Dy 0.2 0.1 0 -10 40 90 140 190 240 290 -0.1 x [m]
S3 S4
Figure 57: UILs of translations in the y-direction for the normal support system
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Rotations about x-axis - Normal supports
150
100
50
6 6 rad] - 0
-10 40 90 140 190 240 290 Rx Rx [10^ -50
-100 x [m]
S1 S2 S3 S4
Figure 58: UILs of rotations about the x- axis for the normal support system
According to Figure 58, the influence of the rotations about the x-axis is highest for support 4. Again, considering only support 4 will suffice the accumulated rotation calculations. This is however not the case when the taking rotations about the y-axis into account. Considering only support 3 should in that case suffice, since its graph shows the highest influence.
Rotations about y-axis - Normal supports
80
60
40
20
0 6 6 rad]
- -10 40 90 140 190 240 290 -20
Ry [10^ Ry -40
-60
-80 x [m]
S1 S2 S3 S4
Figure 59: UILs of rotations about the y- axis for the normal support system
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Rotations about z-axis - Normal supports
10 8 6 4 2 0 -10 40 90 140 190 240 290
6 6 rad] -2 - -4
Rz [10^ Rz -6 -8 -10 -12 x [m]
S1 S2 S3 S4
Figure 60: UILs of rotations about the z-axis for the normal support system
The influence of rotations about the z-axis is very scattered, which makes it difficult to choose one support for sufficient the cumulative rotation calculations. All supports should therefore be considered.
Horizontal (x) reaction forces - Fixed supports
80
60
40
20
0
Fx Fx [kN] -10 40 90 140 190 240 290 -20
-40
-60
-80 x [m]
S1 S2 S3 S4
Figure 61: UILs of x-reactions for the fixed support system
Figure 61 and Figure 62 show the horizontal reactions, which are required for analysis concerning the friction force on the sliding surface of the bearings.
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Horizontal (y) reaction forces - Fixed supports
60
40
20
0
Fy Fy [kN] -10 40 90 140 190 240 290 -20
-40
-60 x [m]
S1 S2 S3 S4
Figure 62: UILs of y-reactions for the fixed support system
9.4. Modifying influence lines for lorries Since lorries have multiple axles, the influence lines for the separate axle loads do not present proper trends. Instead they should represent the influence of simultaneously acting axle loads. The unit influence lines are therefore modified for a multi-axle load. Because every type of lorry in the applied fatigue load model consists of different numbers of axes and different lengths, a certain degree of simplification is required to obtain more representative influence lines. To do so, it is firstly assumed that all lorry types can be modelled as three axle loads, i.e. six nodal loads. Secondly, both intervals between the axle loads should be equal. To clarify the two assumptions, Figure 63 is shown.
Figure 63: Simplified lorry model
The weighted average lorry length is used as an estimation of the total length of the simplified lorry model. This average length is based on the total axle spacing of each type of lorry and its occurrence, as shown in Table 8. Consequently, a length of 11.3 metres is found, which means that the interval between the axles is 5.65 metres.
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Lorry type Occurrence [%] Total axle spacing [m] Weighted axle spacing [m] V11 10 5.2 0.52 V12 10 5.1 0.51 T11O2 15 11.7 1.76 T11O3 50 12.0 6.00 V12A12 5 14.8 0.74 T12O3A2 10 17.8 1.78 Weighted average 11.3 Table 8: Calculation of the weighted average lorry length
The applied interval, however, is larger than that of the lorry model. Linear interpolation is therefore used to obtain the influence values for the axle loads with a shorter interval. The UILs can now be converted for the Unit Lorry Influence Lines (ULILs). The procedure applied is basically taking the average of three influence values in a row, while taking a lorry length of 11.3 metres into account. Influence lines are accordingly obtained for a total load of 100 kN, hence the name Unit Lorry Influence Line.
Since the load model now contains three axes, the x-coordinate is set such that it coincides with the middle axis. It hereby follows that the load model is partly present on the bridge deck when the middle axis is on the edge. Attention should also be paid to other cases where the load model is party present on the bridge deck or even absent. For the southern end of the deck, these cases are illustrated in Figure 64. The same applies for the other end. The ULILs are consequently extended at each end with two points; one in case the deck is loaded by only axis and one in case the load model is completely absent.
Figure 64: In case the load model is partly present on the bridge (left) or completely absent (right)
The ULILs are presented in Figure 65 to Figure 72. Due to the spread of the load over three axes, a relatively slight change can be noticed in the peaks of the graphs. The lines are noticably smoother as well.
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Vertical (z) reaction forces - Normal supports
70
60
50
40
30 Fx Fx [kN] 20
10
0 -20 30 80 130 180 230 280 -10 x [m]
S1 S2 S3 S4
Figure 65: ULILs of z-reactions for the normal support system
Horizontal (x) displacements - Normal supports
0.5
0.4
0.3
0.2
0.1 Dx Dx [mm] 0 -20 30 80 130 180 230 280 -0.1
-0.2 x [m]
S1 S2 S3
Figure 66: ULILs of x-translations for the normal support system
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Horizontal (y) displacements - Normal supports
0.9
0.8
0.7
0.6
0.5
0.4
Dy [mm] Dy 0.3
0.2
0.1
0 -20 30 80 130 180 230 280 -0.1 x [m]
S1 S2
Figure 67: ULILs of y-translations for the normal support system
Rotations about x-axis - Normal supports
120
100
80
60
40
20 6 6 rad] - 0 -20 30 80 130 180 230 280
-20 Rx Rx [10^ -40
-60
-80
-100 x [m]
S1 S2 S3 S4
Figure 68: ULILs of rotations about the x-axis for the normal support system
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Rotations about y-axis - Normal supports
100
80
60
40
20
0 6 6 rad] - -20 30 80 130 180 230 280
-20 Ry [10^ Ry -40
-60
-80
-100 x [m]
S1 S2 S3 S4
Figure 69: ULILs of rotations about the y-axis for the normal support system
Rotations about z-axis - Normal supports
6
4
2
0 6 6 rad] - -20 30 80 130 180 230 280
Rz [10^ Rz -2
-4
-6 x [m]
S1 S2 S3 S4
Figure 70: ULILs of rotations about the z-axis for the normal support system
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Horizontal (x) reaction forces - Fixed supports
80
60
40
20
0
Fx Fx [kN] -20 30 80 130 180 230 280 -20
-40
-60
-80 x [m]
S1 S2 S3 S4
Figure 71: ULILs of x-reactions for the fixed support system
Horizontal (y) reaction forces - Fixed supports
60
40
20
0
Fy Fy [kN] -20 30 80 130 180 230 280 -20
-40
-60 x [m]
S1 S2 S3 S4
Figure 72: ULILs of y-reactions for the fixed support system
9.5. Translations with and without friction Now that all the ULILs are found, these can be scaled for each lorry type to obtain their influence on the bearings. The translations and rotations at each bearing can correspondingly be summed, provided that the friction force is not taken into account. Doing so, a summation is made of the wear path for each single lorry type after having it multiplied with the expected number of the corresponding types per year. This expected numbers of each lorry type per year are based on the applied fatigue load model and are presented in Table 9. Consequently, the total translations and rotations are calculated and presented in Table 10.
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Lorry type Occurrence [%] Load [kN] Expected number of lorries per year V11 low 10 110 225,000 V11 high 10 195 75,000 V12 low 10 205 225,000 V12 high 10 355 75,000 T11O2 low 15 205 337,500 T11O2 high 15 315 112,500 T11O3 low 50 300 1125,000 T11O3 high 50 530 375,000 V12A12 low 5 445 112,500 V12A12 high 5 660 37,500 T12O3A2 low 10 545 225,000 T12O3A2 high 10 890 75,000 Table 9: Expected number of lorries per type per year
Support Dx [m] Dy [m] Rx [rad] Ry [rad] Rz [rad] 1 0 0 1,324 1,569 223 2 10,123 0 1,611 290 171 3 4,127 12,762 1,588 2,189 235 4 12,944 16,125 1,961 2,146 209 Table 10: Translations and rotations per year without taking friction into account
The values given in Table 10 are based on the mechanical behaviour of the bearings without taking friction into consideration. However, it is assumed that there is no friction present for the rotations to occur, which means that these values represent the actual rotations whereas the actual translations are much less than the values in the table. Nevertheless, to obtain the actual values, the friction should first be calculated.
The friction force is calculated by multiplying the friction coefficient sT, with the normal force Fz due to permanent action. From Table 2 of the European standard NEN-EN 1337-2 “The coefficients of friction in long term tests of PTFE sheets in combination with austenitic steel used for plane sliding surfaces” a static coefficient of friction is determined for a total slide path of 10,242 metres and a temperature of 10°C, which is the design temperature in the Netherlands. The value obtained is 0.02262. When concerning the normal force, the smaller vertical reaction due to deadweight (Table 7) is chosen as a representative value, which is 14910 kN. The smaller the vertical reaction, the smaller the friction force, the larger the wear path, the more wear damage, hence the choice. The calculated value of the friction force is 337.26 kN.
Figure 73 to Figure 77 show the scaled influence lines for each lorry type, Lorry Influence Lines (LILs). The horizontal red line represents the threshold for translation, which is basically the friction force. The LILs that cross the threshold actually translate, whereas the crossing itself indicates initiation of the translation. Once a bearing starts translating, it behaves correspondingly to the remaining trajectory of the LIL and initiates translation in the other horizontal direction as well. This occurs due to (residual) vibrations caused by the traveling lorry.
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Horizontal (x) reactions - S2 - Fixed supports
200
100
0 -20 30 80 130 180 230 280 -100
-200
Fx Fx [kN] -300
-400
-500 x [m]
V11 low V11 high V12 low V12 high T11O2 low T11O2 high T11O3 low T11O3 high V12A12 low V12A12 high T12O3A2 low T12O3A2 high Threshold
Figure 73: Horizontal x-reaction LILs for support 2 and whether they exceeding the friction force
Horizontal (x) reactions - S3 - Fixed supports
700 600 500 400 300
200 100 Fx Fx [kN] 0 -100 -20 30 80 130 180 230 280 -200 x [m]
V11 low V11 high V12 low V12 high T11O2 low T11O2 high T11O3 low T11O3 high V12A12 low V12A12 high T12O3A2 low T12O3A2 high Threshold
Figure 74: Horizontal x-reaction LILs for support 3 and whether they exceeding the friction force
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Horizontal (x) reactions - S4 - Fixed supports
200 100 0 -100 -20 30 80 130 180 230 280 -200
-300 -400 Fx Fx [kN] -500 -600 -700 x [m]
V11 low V11 high V12 low V12 high T11O2 low T11O2 high T11O3 low T11O3 high V12A12 low V12A12 high T12O3A2 low T12O3A2 high Threshold
Figure 75: Horizontal x-reaction LILs for support 4 and whether they exceeding the friction force
Horizontal (y) reactions - S3 - Fixed supports
100
0 -20 30 80 130 180 230 280 -100
-200
-300 Fy Fy [kN] -400
-500 x [m]
V11 low V11 high V12 low V12 high T11O2 low T11O2 high T11O3 low T11O3 high V12A12 low V12A12 high T12O3A2 low T12O3A2 high Threshold
Figure 76: Horizontal y-reaction LILs for support 3 and whether they exceeding the friction force
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Horizontal (y) reactions - S4 - Fixed supports
100
0 -20 30 80 130 180 230 280 -100
-200
-300 Fy Fy [kN] -400
-500 x [m]
V11 low V11 high V12 low V12 high T11O2 low T11O2 high T11O3 low T11O3 high V12A12 low V12A12 high T12O3A2 low T12O3A2 high Threshold
Figure 77: Horizontal y-reaction LILs for support 3 and whether they exceeding the friction force
As can be noticed from Figure 73 to Figure 77, only the four heaviest lorry types cause translations at the bearings, with each of them initiating translation from a different position on the bridge. Table 11 gives a clear overview of the lorry types that cause translation with their corresponding location at translational initiation. In cases where a LIL crosses the threshold for both the x- as well as the y- direction, only the smallest x-coordinate is sufficient since initiation in one direction activates that of the other direction.
x-cood. for initiation of translation [m] Lorry type S2 S3 S4 T11O3 High 28.1 232.3 V12A12 High 18.2 206.6 T12O3A2 Low 25.7 228.2 T12O3A2 High 18.4 -6.7 25.3 Table 11: The position (x-coordinate) of the lorries where they initiate translation
By means of these values, the actual translations in longitudinal as well as in transverse direction caused by one single lorry can be computed. Since the position (x-coordinate) of one and the same lorry at translational initiation differs between supports, the total displacement caused by that same lorry will differ between supports as well. Depending on how many lorry types causes translation at each support, the total is obtained by multiplying the outcomes by the annually expected number of the concerning lorry types and summing these. For support 3, for example, the translations (Dx and Dy) due to a single lorry of type T11O3 are first determined. These values are found by accumulating displacement changes starting at x= 28.1 metres using the Dx and Dy LILs based on “T11O3 High”. The obtained Dx and Dy should then be multiplied by the expected numbers of lorries per year of type T11O3 High (Table 9). The total Dx and Dy are now considered found for support 3 due to T11O3 High alone. The same procedure should then be applied for lorry types V12A12 High, T12O3A2 Low and T12O3A2 High to find the related Dx’s and Dy’s for support 3. By summing all the obtained Dx’s and Dy’s, the total longitudinal and transverse translations expected per year are
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Support 2 Support 3 Support 4 Lorry Dx [m] Dy [m] Dx [m] Dy [m] Dx [m] Dy [m] T11O3 High 0 0 607 258 1,971 3,113 V12A12 High 0 0 76 102 245 392 T12O3A12 Low 0 0 374 201 1,216 1,928 T12O3A12 High 601 0 263 837 807 1,058 Σ 601 0 1,320 1,399 4,239 6,492 Table 12: Translations per year when taking friction into account
9.6. Modifying the results for pot bearings When pot bearings rotate about their horizontal axes due to deformation of the elastomeric pad, the internal seal moves vertically against the pot wall and will follow the rotation as well. The stress concentrations at the interface between seal and pot wall will eventually lead to wear of the seal. The rotations at the supports should therefore be converted to wear paths. Figure 78 shows how the piston rotates in a pot baring. Since the rotation occurs about the centre at the bottom of the piston, the wear path can be calculated by multiplying the rotation by the radius of the internal seal. The same applies for the rotation about the vertical axis. The wear of the internal seal, however, occurs in a different direction. Instead of moving vertically, the seal moves horizontally against the pot wall.
Figure 78: Rotation of the piston (16)
Since pot bearings are not applied for the first Van Brienenoord Bridge, the dimensions of those present below the bridge deck of the second Van Brienenoord Bridge will be used to calculate the wear paths. From these bearings, the largest piston diameter is considered relevant for the calculations regarding the wear path at the internal seal, which is 1150 millimetres in diameter.
Regarding Rx, Ry and Rz, it is decided that only the rotations of supports 3 and 4 found in Table 10 are considered relevant for further analysis as they undergo the highest amount of rotation. After the multiplication by a seal radius of 0.575 metres the wear paths given in Table 13 are obtained.
About the x-axis About the y-axis About the z-axis S3 S4 S3 S4 S3 S4 R [rad] 1,588 1,961 2,189 2,146 235 209 Slide path [m] 913 1,127 1,258 1,234 135 120 Table 13: Wear paths on the internal seal of the pot bearing per year
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The translations, however, should also be converted based on an application of pot bearings. Since the free pot bearings are not restrained, they follow the deformations of the superstructure. The horizontal displacement is therefore related to the rotation of the bottom chord cross section, which causes an additional contribution to the accumulated sliding distance. The rotation of the support causes a translation at the sliding plate, which was found by multiplying the piston height (58 mm) by the rotation. A sketch of the relation between the rotation and the additional translation is shown in Figure 79, where “r” is the distance between the neutral axis of the bottom chord and the siding plate, while “ΔDy,x” (ΔDy and ΔDx) is the additional displacement. The piston height in figure ‘c’ is sketched much larger than its actual size to give a better understanding about the local rotations in the bearing. Note that the displacements are not drawn proportional to each other. The figures are just for illustration purposes and have no physical meaning, such as extrusion of elastomer etc.
Figure 79: Additional displacement due to rotation of a free pot bearing
The rotation about one axis, however, causes translation in the horizontal direction perpendicular to the rotational axis. Rx, for example, causes a translation in the y-direction, which would be (r x Rx)+(piston height x Rx). Here the additional translation is equal to the piston height multiplied by the rotation Rx. In the output of Midas Civil, however, the found translation at each bearing already contains the effect of the rotation of the bottom chord cross section at the concerning bearing as well as the bearing on the other end. Only the additional translation due to the local rotation should therefore be summed up with the found translations. However, the output of Midas Civil already contains the combination of rotations and translations. The further calculations are therefore based on the translations caused only the rotations at of the bottom chord cross section with a distance of 3250 millimetres between the neutral axis and the planar sliding surface.
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About the x-axis About the y-axis About the z-axis S3 S4 S3 S4 S3 S4 R [rad] 1,588 1,961 2,189 2,146 235 209 Slide path [m] 5,162 6,373 7,113 6,976 765 678 Table 14: Translations due to the rotations of the bottom chord cross section
About the x-axis About the y-axis About the z-axis S3 S4 S3 S4 S3 S4 R [rad] 1,588 1,961 2,189 2,146 235 209 Slide path [m] 92 114 127 124 14 12 Table 15: Additional slide paths on the planar sliding plate due to rotation
About the x-axis About the y-axis About the z-axis S3 S4 S3 S4 S3 S4 Slide path [m] 5,255 6,486 7,240 7,100 779 690 Table 16: Final translations after summation
Table 16 gives the final translations of the planar sliding plate after summing the values of Table 14 with those of Table 15.
9.7. Modifying the results for spherical bearings As for free pot bearings, the found translations and rotations in respectively Table 12 and Table 10 should be modified for free spherical bearings. The horizontal displacement for these bearings are also related to the cross sectional rotation and the theory of the additional contribution to the accumulated sliding distance holds as well. On the contrary, ΔDx and ΔDy should be subtracted from the given values in Table 12. The different centres of rotation (c1 and c2 in Figure 80) cause the convex member to follow the top plate while the bottom chord cross section rotates. The planar bearing surface is therefore less subjected to wear, hence the smaller wear path.
Figure 80: Rotational behaviour of German free spherical bearings (16)
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The concave bearing surface, on the other hand, rotates with the same angle as the bottom chord cross section. No modification is therefore needed to obtain the wear paths on this surface. Since the convex member rotates about its own spherical axis, the arising wear paths on the concave bearing surface are found by multiplication of the rotation by the spherical radius, which is assumed to be 1000 millimetres. By doing so, the outcome given in Table 17 is found about all three axes.
About the x-axis About the y-axis About the z-axis S3 S4 S3 S4 S3 S4 R [rad] 1,588 1,961 2,189 2,146 235 209 Slide path [m] 1,588 1,961 2,189 2,146 94 83 Table 17: The wear paths on the concave bearing surface per year
Sketch ‘b’ depicted in Figure 81 shows that the convex member follows the rotational movement of the top plate, which results in a smaller wear path for the planar bearing surface. The radius r1 is the distance from the neutral axis of the bottom chord cross section to the concave bearing surface, while r2 is the radius of the concave bearings surface. r3, on the other hand, is equal to r2 minus the maximum height of the convex member. According sketch ‘b’ in Figure 81, ΔDx and ΔDy are found by multiplying the associated rotation by r3, which is assumed to be 900 millimetres. To be clear, the subtractive translations on the planar sliding plate is found by the multiplication of Rx and Ry by r3 to find respectively ΔDy and ΔDx.
Figure 81: Additional displacement due to rotation of a free spherical bearing
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The output of Midas Civil hereby also already contains the combination of rotations and translations. The further calculations are therefore again based on the translations caused only the rotations at of the bottom chord cross section with a distance of 3250 millimetres between the neutral axis and the planar sliding surface.
About the x-axis About the y-axis About the z-axis S3 S4 S3 S4 S3 S4 R [rad] 1,588 1,961 2,189 2,146 235 209 Slide path [m] 5,162 6,373 7,113 6,976 765 678 Table 18: Translations due to the rotations of the bottom chord cross section
About the x-axis About the y-axis About the z-axis S3 S4 S3 S4 S3 S4 R [rad] 1,588 1,961 2,189 2,146 235 209 Slide path [m] 1,430 1,765 1,970 1,932 212 188 Table 19: Subtractive slide paths on the planar sliding plate due to rotation
About the x-axis About the y-axis About the z-axis S3 S4 S3 S4 S3 S4 Slide path [m] 3,733 4,608 5,143 5,044 553 490 Table 20: Final translations after subtraction
Table 20Table 16 gives the final translations of the planar sliding plate after subtracting the values of Table 18 with those of Table 19.
9.8. The effect of multiple lorries on the bridge Until now the influence of only single lorries crossing the bridge is analysed. This is, of course, quite different in practice. Many lorries are continuously present on the bridge beck at the same time. It is therefore worth analysing the effect of many lorries on the bridge at the same moment. Fortunately, the same method by means of influence lines can be used. A summation of the influences due to multiple lorries can be made while taking the different lorries positions into account. For this analysis, however, the friction at the bearings is not considered.
To obtain the distance between the lorries, it is first assumed that all 3,000,000 lorries cross the bridge in the busiest time interval, which is a 13 hour time interval from 6 o’clock in the morning until 7 o’clock in the evening. It is furthermore assumed that all lorries travel with a velocity of 90 km/h. This means that it takes a lorry 12.17 seconds to cross the full bridge length (304.345 metres). Within these seconds, 3 vehicles (calculated value of 2.138) cross the bridge with a vehicle distance of 142.35 metres (2.138*304.345).
By using the found vehicle distance, ULILs were calculated of the translations for support 4. Graphs were consequently obtained for the translations based on one single UL, two ULs and three ULs. The accumulated graphs were found as well. Figure 82 to Figure 85 show the ULILs and their accumulated translation.
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Horizontal (x) displacements - S4 - Normal supports
0.7
0.6
0.5
0.4
0.3
0.2 Dx Dx [mm] 0.1
0 -20 80 180 280 380 480 580 -0.1
-0.2 x [m]
One UL Two ULs Three ULs
Figure 82: Dx ULILs for one, two and three ULs
Accumulated horizontal (x) displacements - S4 - Normal supports
3.5
3
2.5
2
1.5
Dx Dx [mm] 1
0.5
0 -20 80 180 280 380 480 580 x [m]
One UL Two ULs Three ULs
Figure 83: Accumulated Dx ULILs for one, two and three ULs
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Horizontal (y) displacements - S4 - Normal supports
0.9 0.8 0.7 0.6
0.5
0.4
0.3 Dy [mm] Dy 0.2 0.1 0 -20 80 180 280 380 480 580 -0.1 x [m]
One UL Two ULs Three ULs
Figure 84: Dy ULILs for one, two and three ULs
Accumulated horizontal (y) displacements - S4 - Normal supports
6
5
4
3
2 Dy [mm] Dy
1
0 -20 80 180 280 380 480 580 x [m]
One UL Two ULs Three ULs
Figure 85: Accumulated Dy ULILs for one, two and three ULs
Table 21 presents a clear overview of the accumulated translations for support 4, while Table 22 gives the percentage of the wear path reduction based on ULILs of a single UL, two and three ULs. The relevant reductions found for Dx and Dy are respectively 16.7% and 0.8%. For this analysis, the upper bound values are those found after accumulating the influences without taking friction into account. Those can be found in Table 10. Table 23 shows the upper and the lower bound slide paths per year for support 4.
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Dx [mm] Dy [mm] One UL 1.30 1.62 Two ULs 2.42 3.23 Three ULs 3.25 4.83 Table 21: Accumulated translations for support 4
Dx [%] Dy [%] One UL 0 0 Two Uls 7.1 0.5 Three Uls 16.7 0.8 Table 22: Reduction of the wear paths
Upper bound value Lower bound value Dx [m] 12,944 10,782 Dy [m] 16,125 15,996 Table 23: Upper and lower bound values of the wear paths per year
9.9. Summary All necessary influence lines and processed data obtained from the linear static analysis can be found in this chapter. Furthermore, section 9.2 dealt with self-weight and superimposed dead loads with their contribution to the support reactions. Section 9.3, on the other hand presented the influence lines due to unit axle loads whereas section 9.4 described how these influence lines were modified to obtain those for a unit lorry. Section 9.5 summed the total translations and rotations with and without taking the friction at the bearings into account. Sections 9.6 and 9.7 presented the final outcome of the wear behaviour for respectively free pot and spherical bearings after the LILs were modified for this purpose. The effect of interacting lorries was also covered for this study in section 9.8.
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10. Comparison with NEN-EN 1337-2 and NEN-EN 1337-5 After the analysis of the wear behaviour of the supports of the first Van Brienenoord Bridge, many results were found. These included such as reaction forces, translations, rotations and wear paths. But to be able to judge the necessary results, comparison was made with the test values included in European standards NEN-EN 1337-2 (Sliding elements) and NEN-EN 1337-5 (pot bearings). However, not all parameters included in the tests were analysed for this study to comparison with.
Values of those parameters that were found during this study are compared with in Table 24 to Table 26. In the former table contains of values concerning tests described in NEN-EN 1337-2, while the latter table presents the “Long term rotation and load test” values described in NEN-EN 1337-5.
Type B (phase 2,4,6) - Variable speed (approximately sinusoidal) According to Parameters EN1337-2 Bridge model Contact pressure of PTFE [N/mm2] 29.7 17.7 Sliding distance [mm] 8 6 Average siding speed [mm/s] 2 - Total slide path Dx for planar PTFE sliding plate [m] 10,000 4,363 (one year) Total slide path Due to Ry for concave PTFE sliding plate [m] 2,000 2,189 (one year) Table 24: Comparison with test values from NEN-EN 1337-2
Determination of restraint moment According to Parameters EN1337-5 Bridge model Contact pressure [N/mm2] 35.0 17.7 Sinusoidal movement through the lever arm [rad] ±0.01 - Simulation of live load rotation α2 [Hz] 0.003 < f < 0.006 - Table 25: Comparison with values of determination of resistant moment test from NEN-EN 1337-5
Long term rotation and load test According to Parameters Bridge EN1337-5 model Contact pressure in rotating state [N/mm2] 35 17.7 Sinusoidal movement through lever arm [rad] ±0.0025 - Frequency of the sinusoidal movement [Hz] 0.25 < f < 2.5 - 500 Compliance with one of the standard accumulated slide path values [m] 1,000 1,258 2,000 Contact pressure of the elastomer in rotated position to check whether 60 - elastomer escapes from the pot [N/mm2] Table 26: Comparison with values of the long term rotation and load test from NEN-EN 1337-5
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11. Conclusions and recommendations
11.1. Conclusions There can be concluded that the wear behaviour of the structural bearings of large arch bridges are successfully obtained since the first Van Brienenoord Bridge is considered a relevant steel arch bridge. Reaction forces, translations and rotations are found for two types of bearings, the pot and spherical bearings.
In generally, the values found for the first Van Brienenoord Bridge according to a finite element analysis are in the same order of magnitude of the test values included in the European standards NEN-EN 1337. This means that the test values are found in one year, which is a very short period of time. The expected service life of structural bearings is around 10 to 25 years. The total wear path in the x-direction on the concave sliding plate, however, is slightly larger than the associated test value. But since the found slide path can be reduced due to multiple lorries on the bridge, the difference is considered negligibly small. The wear path Dx on the planar sliding plate is about half of the test value. The test value is therefore reached in about two years. This is also a very short time period.
It is furthermore not clear what the ration is between the test values and the service life of structural bearings. When in practice the values are reached in one or two year, this does not mean that the bearings fail after that period.
From the threshold value of the friction force it is noticed that the bridge bearings are subjected to large horizontal reactions before translation at the bearings is initiated. Consequently, this causes a large resistant moment. Since friction could not be implemented in the finite element model, no statement can be made regarding the effect of the resistant moment on the global behaviour of the bridge. A nonlinear analysis is therefore required.
11.2. Recommendations From this study most results are comparable with the test values of the European standards for structural bearings. A linear static analysis was sufficient to obtain the necessary results. No further analysis of these results is therefore required. The velocity of the occurring translations and the frequency of the rotations, however, are not obtained and are therefore recommended to be researched more in depth.
A dynamic analysis of the finite element model of the first Van Brienenoord Bridge is recommended as well to study the dynamic behaviour of the bridge and possible dynamic amplification factors.
An estimation of the lorry distance was used to analyse the effect of multiple lorries on the bridge deck. In reality, this value differs continuously. The number of lorries that simultaneously cross the bridge is variable as well. The reduction factor will, accordingly, also differ in practice. When more realistic results are desired, a traffic simulation is needed.
The bearings were analysed such that the friction was obtained from a modified support system and was afterwards combined with results found without taking friction into account. The relation between the horizontal reaction and displacements is clearly more complex and therefore should be analysed more in depth by means of a nonlinear analysis since friction is a nonlinear phenomenon.
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The slide paths at the spherical bearings due to rotations about the longitudinal axis are almost as high as the slide paths caused by rotations about the lateral axis (the main sliding direction). The dimple pattern, however, differs in these two directions, but NEN-EN 1337-2 does not include test for the other direction. It is therefore recommended that both directions (the main sliding direction as well as the direction perpendicular to it) are tested. The dimple pattern, on the other hand, could also be modified such that the sliding surface is lubricated equally in both directions.
Since the finite element model is constructed such that also a fatigue analysis can be performed, it is recommended that this analysis should indeed be performed.
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Bibliography 1. Wetzk, Volker. Department of Architecture. University of Cambridge. [Online] http://www.arct.cam.ac.uk/Downloads/ichs/vol-3-3333-3356-wetzk.pdf.
2. Charles W. Roeder, Ph.D., P.E. and John F. Stanton, Ph.D., P.E. Steel Market Development Instituut. [Online] http://www.smdisteel.org/~/media/Files/SMDI/Construction/Bridges%20- %20All%20-%20Paper%20- %20Steel%20Bridge%20Bearing%20Selection%20and%20Design%20Guide%20- %20%20Part%20I.pdf.
3. Kumar, Shiv. Bridge Bearings. s.l. : Indian Railways Instituut of Civil Engineering, March 2006. PUNE411001.
4. Mark Rossow, PhD, PE. Inspection of Bridge Bearings. CED Engineering. [Online] http://www.cedengineering.com/upload/Inspection%20of%20Bridge%20Bearings.pdf. S02-005.
5. Lecture 15B.10: Bridge Equipment. University of Ljubljana, Faculty of Civil and Geodetic Engineering. [Online] http://www.fgg.uni-lj.si/kmk/esdep/master/wg15b/l1000.htm.
6. Normcommissie 353 068 "Opleggingen". NEN-EN 1337-2. Structural bearings - Part 2: Sliding elements. s.l. : Nederlands Normalisatie-instituut, March 2004.
7. —. NEN-EN 1337-5. Structural Bearings - Part 5: Pot bearings. s.l. : Nederlands Normalisatie- instituut, April 2005.
8. Rijkswaterstaat, directie Zuid-Holland. Verbetering van de veiligheid en de doorstroming van het verkeer. Nota Van Brienenoordbrug. April 1977.
9. Rijkswaterstaat, dirctie Bruggen. Van Brienenoordbrug 2 x zo breed. s.l. : Ballast Nedam Services B.V., February 1989.
10. infrastructuur. ROTTERDAM WORLD PORT WORLD CITY. [Online] http://www.rotterdam.nl/tekst:thema_infrastructuur.
11. De Van Brienenoordbrug 1965. regiocanons.nl. [Online] http://www.regiocanons.nl/zuid- holland/zuid-holland/van-brienenoordbrug.
12. Eb, Ir. W.J. van der. The design and construction of the Van Brienenoord Bridge across the river Nieuwe Maas. 1968.
13. 'brienenoord' grootste brug van land. ROTTERDAM WORLD PORT WORLD CITY. [Online] http://www.rotterdam.nl/tekst:_brienenoord__grootste_brug_van_land.
14. De Van Brienenoordbrug is een brug in de A16 over de Nieuwe Maas aan de Oostkant van Rotterdam ID433362. Rijkswaterstaat. [Online] https://beeldbank.rws.nl/MediaObject/Details/433362.
15. van brienenoordbrug. ROTTERDAM WORLD PORT WORLD CITY. [Online] http://www.rotterdam.nl/tekst:van_brienenoordbrug.
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MSc thesis – Structural Engineering Steel and Timber Structures
16. Bos, V. Traffic induced bearing loads and movements of a steel plate-girder bridge. 2011.
17. Van Brienenoordbrug. Architectuur in Rotterdam. [Online] http://www.architectuurinrotterdam.nl/building.php?buildingid=221&lang=nl&PHPSESSID=175284c d94ea7ed43a3e5d0b49131525.
18. Normcommissie 353 068 "Opleggingen". NEN-EN 1337-1. Structural bearings - Part 1: General design. s.l. : Nederlands Normalisatie-instituut, July 2000.
19. —. NEN-EN 1337-7. Structural bearings - Part 7: Spherical and cylindrical PTFE. s.l. : Nederlands Normalisatie-instituut, March 2004.
20. Boer, M. de. Mechanisch gedrag van brugopleggingen van de Dintelhavenbrug. 2011.
21. Otte, A. Proposal for modified Fatigue Load Model related on EN 1991-2. 2009.
22. Hartsuijker, Coenraad. Toegepaste Mechanica Deel 2. Den Haag : Sdu Uitgevers, 2008. ISBN-10: 90 395 0594 2.
Page 73
Annex A The Williot diagram
The Williot diagram was composed using half of the self-weight of the superstructure. This load was subjected to one arch along with the diagonal cables and half of the deck. The self-weight was furthermore equally distributed over the nodes of the deck by means of concentrated loads.
Firstly, the direction of structural element 2 was fixed when the displacements were determined. Afterwards, the translations were determined when the fixed direction was unconstrained. By summing the two outcomes, the final displacements are found. The results of the displacements in each phase are given in the table below.
With fixed direction Reversed Resulting Nodes Dx [mm] Dz [mm] Dx [mm] Dz [mm] Dx [mm] Dz [mm] B -9,0 -4,8 40,6 -76,7 31,6 -81,5 C 7,6 -25,5 0,0 -112,9 7,6 -138,4 D -27,1 10,2 70,0 -154,5 42,9 -144,3 E 13,8 -0,7 0,0 -195,1 13,8 -195,8 F -48,3 48,0 89,9 -235,2 41,6 -187,2 G 20,2 45,4 0,0 -277,2 20,2 -231,8 H -67,6 107,9 99,8 -317,8 32,2 -209,9 I 26,7 115,3 0,0 -359,3 26,7 -244,0 J -79,5 191,1 99,7 -401,1 20,2 -210,0 K 33,2 209,7 0,0 -441,5 33,2 -231,8 L -78,8 296,4 89,6 -483,7 10,8 -187,3 M 39,6 327,8 0,0 -523,6 39,6 -195,8 N -59,9 420,1 69,5 -564,4 9,6 -144,3 O 45,8 467,3 0,0 -605,8 45,8 -138,5 P -18,9 560,6 39,8 -642,1 20,9 -81,5 Q 53,4 718,7 0,0 -718,7 53,4 0,0
Page 74
The Williot-diagram
Page 75
The resulting displacements are shown in the figure above.
Page 76
Annex B The unit lorry influence lines for the normal support system
Page 77
Support 1 ‐ Normal supports x [m] Fz [kN] Dx [mm] Dy [mm] Rx [10^‐6 rad] Ry [10^‐6 rad] Rz [10^‐6 rad] ‐19.93 0.00 0.00 0.00 0.00 ‐14.27 13.85 ‐23.67 ‐7.33 2.67 ‐8.62 27.13 ‐45.95 ‐9.15 3.04 0.00 40.02 ‐63.95 1.36 ‐0.90 8.14 37.86 ‐47.25 22.83 ‐5.09 16.40 36.10 ‐25.82 40.09 ‐3.53 24.60 34.45 ‐10.25 50.55 ‐1.25 32.80 32.87 ‐3.19 54.07 ‐0.23 41.00 31.42 ‐1.48 52.72 0.00 49.20 30.08 ‐2.05 48.86 0.00 57.40 28.82 ‐2.84 43.97 0.23 65.60 27.64 ‐3.30 38.63 0.79 73.80 26.54 ‐3.86 32.95 1.25 82.00 25.54 ‐4.09 27.38 1.82 90.20 24.62 ‐4.09 21.71 2.05 98.40 23.77 ‐3.86 16.59 2.05 106.60 22.97 ‐3.30 12.04 2.05 114.80 22.26 ‐3.07 7.39 2.05 123.00 21.58 ‐3.07 3.30 2.05 131.20 20.94 ‐2.84 ‐0.23 2.05 139.40 20.31 ‐2.28 ‐3.63 2.05 147.60 19.71 ‐2.05 ‐6.37 2.05 155.80 19.09 ‐2.05 ‐8.97 2.05 164.00 18.44 ‐2.05 ‐11.02 2.05 172.20 17.77 ‐1.82 ‐12.27 2.05 180.40 17.06 ‐1.25 ‐13.29 2.05 188.60 16.27 ‐1.02 ‐14.09 2.05 196.80 15.41 ‐1.02 ‐14.32 1.82 205.00 14.50 ‐1.02 ‐14.32 1.25 213.20 13.50 ‐1.02 ‐14.09 1.02 221.40 12.41 ‐0.79 ‐13.52 1.02 229.60 11.24 ‐0.23 ‐12.84 1.02 237.80 9.98 0.00 ‐11.48 0.79 246.00 8.65 0.00 ‐10.00 0.23 254.20 7.20 0.00 ‐8.41 0.00 262.40 5.63 0.00 ‐6.93 0.00 270.60 4.00 0.00 ‐5.11 0.23 278.80 2.28 0.00 ‐2.84 0.79 287.00 0.10 0.00 0.00 1.02 295.73 ‐1.01 0.00 1.31 0.67 301.38 ‐0.80 0.00 1.00 0.33 307.03 0.00 0.00 0.00 0.00
Page 78 Support 2 ‐ Normal supports x [m] Fz [kN] Dx [mm] Dy [mm] Rx [10^‐6 rad] Ry [10^‐6 rad] Rz [10^‐6 rad] ‐19.93 0.00 0.00 0.00 0.00 0.00 ‐14.27 ‐0.80 ‐0.04 0.00 ‐1.00 ‐0.33 ‐8.62 ‐1.03 ‐0.05 0.00 ‐1.31 ‐0.67 0.00 0.02 0.00 0.00 0.00 ‐1.02 8.14 2.18 0.10 0.00 2.84 ‐0.79 16.40 3.95 0.18 0.00 5.11 ‐0.23 24.60 5.60 0.24 0.00 7.16 0.00 32.80 7.17 0.26 0.00 8.97 ‐0.23 41.00 8.62 0.27 0.00 10.23 ‐0.79 49.20 9.97 0.26 ‐0.23 11.48 ‐1.02 57.40 11.23 0.24 ‐0.79 13.06 ‐1.02 65.60 12.41 0.23 ‐1.02 14.09 ‐1.02 73.80 13.50 0.21 ‐1.02 14.32 ‐1.02 82.00 14.50 0.19 ‐1.02 14.55 ‐1.02 90.20 15.42 0.18 ‐1.25 14.88 ‐1.25 98.40 16.28 0.16 ‐1.82 14.32 ‐1.82 106.60 17.07 0.15 ‐2.05 13.29 ‐2.05 114.80 17.79 0.14 ‐2.05 12.27 ‐2.05 123.00 18.46 0.13 ‐2.28 11.02 ‐2.05 131.20 19.11 0.12 ‐2.84 9.20 ‐2.05 139.40 19.73 0.12 ‐3.07 6.93 ‐2.05 147.60 20.33 0.12 ‐3.07 4.09 ‐2.05 155.80 20.95 0.12 ‐3.30 0.79 ‐2.05 164.00 21.60 0.13 ‐3.86 ‐3.07 ‐2.05 172.20 22.27 0.14 ‐4.09 ‐7.16 ‐2.05 180.40 22.99 0.15 ‐4.09 ‐11.48 ‐2.05 188.60 23.78 0.16 ‐4.32 ‐16.36 ‐2.05 196.80 24.63 0.18 ‐4.88 ‐21.48 ‐1.82 205.00 25.55 0.19 ‐5.11 ‐26.82 ‐1.25 213.20 26.54 0.21 ‐4.88 ‐32.50 ‐1.02 221.40 27.63 0.23 ‐4.09 ‐37.84 ‐0.79 229.60 28.81 0.24 ‐3.07 ‐42.95 ‐0.23 237.80 30.06 0.26 ‐2.05 ‐47.84 0.00 246.00 31.40 0.26 ‐1.25 ‐51.70 0.00 254.20 32.84 0.26 ‐2.17 ‐53.05 0.00 262.40 34.42 0.23 ‐8.89 ‐49.52 0.46 270.60 36.05 0.18 ‐25.82 ‐38.84 2.28 278.80 37.76 0.10 ‐51.59 ‐21.25 4.19 287.00 39.94 0.00 ‐77.11 ‐0.56 2.48 295.73 27.12 ‐0.05 ‐58.35 8.94 ‐0.83 301.38 13.86 ‐0.04 ‐30.67 7.00 ‐1.33 307.03 0.00 0.00 0.00 0.00 0.00
Page 79 Support 3 ‐ Normal supports x [m] Fz [kN] Dx [mm] Dy [mm] Rx [10^‐6 rad] Ry [10^‐6 rad] Rz [10^‐6 rad] ‐19.93 0.00 0.00 0.00 0.00 0.00 0.00 ‐14.27 20.48 0.03 0.20 27.00 ‐11.00 ‐3.33 ‐8.62 40.85 0.05 0.42 53.54 ‐14.19 ‐4.14 0.00 62.28 0.03 0.62 77.80 1.71 ‐0.82 8.14 61.49 ‐0.04 0.46 59.48 35.87 2.25 16.40 60.33 ‐0.09 0.23 29.50 60.88 ‐0.10 24.60 59.06 ‐0.12 0.07 10.02 74.40 ‐1.59 32.80 57.71 ‐0.14 0.02 2.63 78.16 ‐0.79 41.00 56.23 ‐0.14 0.00 1.02 74.76 0.56 49.20 54.66 ‐0.13 0.01 1.25 67.73 1.25 57.40 53.00 ‐0.13 0.01 1.82 59.88 1.82 65.60 51.26 ‐0.12 0.02 2.28 51.36 2.28 73.80 49.43 ‐0.11 0.02 2.84 42.95 2.84 82.00 47.51 ‐0.10 0.02 3.07 35.00 3.07 90.20 45.50 ‐0.09 0.02 3.07 27.61 3.07 98.40 43.44 ‐0.09 0.02 3.07 20.68 3.07 106.60 41.31 ‐0.08 0.02 2.84 14.32 3.07 114.80 39.10 ‐0.07 0.02 2.28 8.41 3.07 123.00 36.85 ‐0.07 0.02 2.05 3.07 3.07 131.20 34.58 ‐0.06 0.01 2.05 ‐1.59 3.07 139.40 32.28 ‐0.06 0.01 2.05 ‐5.34 3.07 147.60 29.96 ‐0.05 0.01 2.05 ‐8.74 3.07 155.80 27.66 ‐0.05 0.01 2.05 ‐11.25 3.07 164.00 25.39 ‐0.04 0.01 2.05 ‐13.29 2.84 172.20 23.14 ‐0.04 0.01 2.05 ‐15.11 2.28 180.40 20.93 ‐0.04 0.01 2.05 ‐16.13 2.05 188.60 18.80 ‐0.03 0.01 2.05 ‐16.36 2.05 196.80 16.73 ‐0.03 0.01 2.05 ‐16.36 2.05 205.00 14.72 ‐0.03 0.01 1.82 ‐16.13 1.82 213.20 12.79 ‐0.03 0.01 1.25 ‐15.34 1.25 221.40 10.96 ‐0.02 0.01 1.02 ‐14.32 1.02 229.60 9.22 ‐0.02 0.01 1.02 ‐13.29 1.02 237.80 7.55 ‐0.02 0.00 1.02 ‐12.04 1.02 246.00 5.96 ‐0.01 0.00 1.02 ‐10.23 0.79 254.20 4.49 ‐0.01 0.00 1.02 ‐8.41 0.23 262.40 3.14 ‐0.01 0.00 0.79 ‐6.70 0.23 270.60 1.85 ‐0.02 0.00 0.23 ‐4.32 0.79 278.80 0.64 ‐0.02 0.00 0.00 ‐2.05 1.02 287.00 ‐0.14 ‐0.03 0.00 0.00 0.23 1.02 295.73 ‐0.32 ‐0.02 0.00 0.00 1.31 0.67 301.38 ‐0.21 ‐0.01 0.00 0.00 1.00 0.33 307.03 0.00 0.00 0.00 0.00 0.00 0.00
Page 80 Support 4 ‐ Normal supports x [m] Fz [kN] Dx [mm] Dy [mm] Rx [10^‐6 rad] Ry [10^‐6 rad] Rz [10^‐6 rad] ‐19.93 0.00 0.00 0.00 0.00 0.00 0.00 ‐14.27 ‐0.20 ‐0.03 0.00 0.00 ‐1.00 ‐0.33 ‐8.62 ‐0.29 ‐0.03 0.00 0.00 ‐1.31 ‐0.67 0.00 ‐0.06 0.02 0.00 0.00 ‐0.23 ‐1.02 8.14 0.73 0.12 0.00 0.23 2.05 ‐1.02 16.40 1.89 0.20 0.00 0.79 4.32 ‐0.79 24.60 3.17 0.25 0.00 1.02 6.70 ‐0.46 32.80 4.52 0.27 0.00 1.02 8.41 ‐0.79 41.00 5.99 0.28 0.00 1.02 10.23 ‐1.02 49.20 7.56 0.28 0.01 1.25 12.04 ‐1.02 57.40 9.23 0.27 0.01 1.82 13.29 ‐1.02 65.60 10.97 0.25 0.01 2.05 14.32 ‐1.25 73.80 12.79 0.24 0.01 2.05 15.34 ‐1.82 82.00 14.72 0.22 0.01 2.05 16.13 ‐2.05 90.20 16.72 0.21 0.01 2.05 16.36 ‐2.05 98.40 18.78 0.19 0.01 2.05 16.36 ‐2.05 106.60 20.91 0.18 0.01 2.05 16.13 ‐2.05 114.80 23.12 0.18 0.01 2.05 15.11 ‐2.28 123.00 25.37 0.17 0.01 2.05 13.29 ‐2.84 131.20 27.64 0.17 0.02 2.05 11.25 ‐3.07 139.40 29.94 0.17 0.02 2.05 8.97 ‐3.07 147.60 32.26 0.18 0.02 2.05 5.91 ‐3.07 155.80 34.56 0.18 0.02 2.05 2.05 ‐3.07 164.00 36.84 0.19 0.02 2.28 ‐2.28 ‐3.07 172.20 39.09 0.21 0.02 2.84 ‐7.39 ‐3.07 180.40 41.29 0.23 0.02 3.07 ‐13.29 ‐3.07 188.60 43.43 0.25 0.02 3.07 ‐19.66 ‐3.07 196.80 45.50 0.27 0.03 3.07 ‐26.82 ‐3.07 205.00 47.50 0.29 0.03 3.30 ‐34.77 ‐3.07 213.20 49.43 0.32 0.03 3.86 ‐42.95 ‐3.07 221.40 51.26 0.35 0.03 3.86 ‐51.13 ‐2.84 229.60 53.01 0.37 0.02 3.30 ‐59.31 ‐2.28 237.80 54.67 0.39 0.01 2.84 ‐67.27 ‐1.82 246.00 56.26 0.40 0.01 2.50 ‐73.97 ‐1.02 254.20 57.74 0.40 0.02 3.99 ‐76.91 0.23 262.40 59.08 0.36 0.08 10.94 ‐72.59 1.36 270.60 60.38 0.27 0.27 33.15 ‐59.06 ‐0.13 278.80 61.59 0.14 0.57 70.96 ‐34.85 ‐3.50 287.00 62.36 ‐0.03 0.79 97.21 ‐1.94 ‐2.02 295.73 40.88 ‐0.10 0.54 67.98 13.31 1.70 301.38 20.49 ‐0.07 0.26 34.33 10.33 2.00 307.03 0.00 0.00 0.00 0.00 0.00 0.00
Page 81 Normal supports x [m] Fz [kN] Ʃ S1 S2 S3 S4 ‐19.93 0.00 0.00 0.00 0.00 0.00 ‐14.27 13.85 ‐0.80 20.48 ‐0.20 33.33 ‐8.62 27.13 ‐1.03 40.85 ‐0.29 66.67 0.00 40.02 0.02 62.28 ‐0.06 102.27 8.14 37.86 2.18 61.49 0.73 102.27 16.40 36.10 3.95 60.33 1.89 102.27 24.60 34.45 5.60 59.06 3.17 102.27 32.80 32.87 7.17 57.71 4.52 102.27 41.00 31.42 8.62 56.23 5.99 102.27 49.20 30.08 9.97 54.66 7.56 102.27 57.40 28.82 11.23 53.00 9.23 102.27 65.60 27.64 12.41 51.26 10.97 102.27 73.80 26.54 13.50 49.43 12.79 102.27 82.00 25.54 14.50 47.51 14.72 102.27 90.20 24.62 15.42 45.50 16.72 102.27 98.40 23.77 16.28 43.44 18.78 102.27 106.60 22.97 17.07 41.31 20.91 102.27 114.80 22.26 17.79 39.10 23.12 102.27 123.00 21.58 18.46 36.85 25.37 102.27 131.20 20.94 19.11 34.58 27.64 102.27 139.40 20.31 19.73 32.28 29.94 102.27 147.60 19.71 20.33 29.96 32.26 102.27 155.80 19.09 20.95 27.66 34.56 102.27 164.00 18.44 21.60 25.39 36.84 102.27 172.20 17.77 22.27 23.14 39.09 102.27 180.40 17.06 22.99 20.93 41.29 102.27 188.60 16.27 23.78 18.80 43.43 102.27 196.80 15.41 24.63 16.73 45.50 102.27 205.00 14.50 25.55 14.72 47.50 102.27 213.20 13.50 26.54 12.79 49.43 102.27 221.40 12.41 27.63 10.96 51.26 102.27 229.60 11.24 28.81 9.22 53.01 102.27 237.80 9.98 30.06 7.55 54.67 102.27 246.00 8.65 31.40 5.96 56.26 102.27 254.20 7.20 32.84 4.49 57.74 102.27 262.40 5.63 34.42 3.14 59.08 102.27 270.60 4.00 36.05 1.85 60.38 102.27 278.80 2.28 37.76 0.64 61.59 102.27 287.00 0.10 39.94 ‐0.14 62.36 102.27 295.73 ‐1.01 27.12 ‐0.32 40.88 66.67 301.38 ‐0.80 13.86 ‐0.21 20.49 33.33 307.03 0.00 0.00 0.00 0.00 0.00
Page 82 Vertical (z) reaction forces ‐ Normal supports
70.00
60.00
50.00
40.00
Page
[kN] 30.00 Fx 83 20.00
10.00
0.00 ‐20.00 30.00 80.00 130.00 180.00 230.00 280.00 ‐10.00 x [m]
S1 S2 S3 S4 Normal supports x [m] Dx [mm] Ʃ S2 S3 S4 ‐19.93 0.00 0.00 0.00 0.00 ‐14.27 ‐0.04 0.03 ‐0.03 ‐0.03 ‐8.62 ‐0.05 0.05 ‐0.03 ‐0.03 0.00 0.00 0.03 0.02 0.05 8.14 0.10 ‐0.04 0.12 0.18 16.40 0.18 ‐0.09 0.20 0.29 24.60 0.24 ‐0.12 0.25 0.36 32.80 0.26 ‐0.14 0.27 0.40 41.00 0.27 ‐0.14 0.28 0.41 49.20 0.26 ‐0.13 0.28 0.40 57.40 0.24 ‐0.13 0.27 0.39 65.60 0.23 ‐0.12 0.25 0.36 73.80 0.21 ‐0.11 0.24 0.34 82.00 0.19 ‐0.10 0.22 0.31 90.20 0.18 ‐0.09 0.21 0.29 98.40 0.16 ‐0.09 0.19 0.27 106.60 0.15 ‐0.08 0.18 0.25 114.80 0.14 ‐0.07 0.18 0.24 123.00 0.13 ‐0.07 0.17 0.23 131.20 0.12 ‐0.06 0.17 0.23 139.40 0.12 ‐0.06 0.17 0.23 147.60 0.12 ‐0.05 0.18 0.24 155.80 0.12 ‐0.05 0.18 0.26 164.00 0.13 ‐0.04 0.19 0.28 172.20 0.14 ‐0.04 0.21 0.30 180.40 0.15 ‐0.04 0.23 0.34 188.60 0.16 ‐0.03 0.25 0.37 196.80 0.18 ‐0.03 0.27 0.41 205.00 0.19 ‐0.03 0.29 0.46 213.20 0.21 ‐0.03 0.32 0.50 221.40 0.23 ‐0.02 0.35 0.55 229.60 0.24 ‐0.02 0.37 0.59 237.80 0.26 ‐0.02 0.39 0.63 246.00 0.26 ‐0.01 0.40 0.65 254.20 0.26 ‐0.01 0.40 0.64 262.40 0.23 ‐0.01 0.36 0.58 270.60 0.18 ‐0.02 0.27 0.44 278.80 0.10 ‐0.02 0.14 0.21 287.00 0.00 ‐0.03 ‐0.03 ‐0.07 295.73 ‐0.05 ‐0.02 ‐0.10 ‐0.17 301.38 ‐0.04 ‐0.01 ‐0.07 ‐0.12 307.03 0.00 0.00 0.00 0.00
Page 84 Horizontal (x) displacements ‐ Normal supports
0.50
0.40
0.30
0.20 Page [mm] 85
Dx 0.10
0.00 ‐20.00 30.00 80.00 130.00 180.00 230.00 280.00
‐0.10
‐0.20 x [m]
S1 S2 S3 Normal supports x [m] Dy [mm] Ʃ S3 S4 ‐19.93 0.00 0.00 0.00 ‐14.27 0.20 0.00 0.20 ‐8.62 0.42 0.00 0.42 0.00 0.62 0.00 0.63 8.14 0.46 0.00 0.47 16.40 0.23 0.00 0.23 24.60 0.07 0.00 0.08 32.80 0.02 0.00 0.02 41.00 0.00 0.00 0.01 49.20 0.01 0.01 0.01 57.40 0.01 0.01 0.02 65.60 0.02 0.01 0.02 73.80 0.02 0.01 0.03 82.00 0.02 0.01 0.03 90.20 0.02 0.01 0.03 98.40 0.02 0.01 0.03 106.60 0.02 0.01 0.03 114.80 0.02 0.01 0.03 123.00 0.02 0.01 0.03 131.20 0.01 0.02 0.03 139.40 0.01 0.02 0.03 147.60 0.01 0.02 0.03 155.80 0.01 0.02 0.03 164.00 0.01 0.02 0.03 172.20 0.01 0.02 0.03 180.40 0.01 0.02 0.03 188.60 0.01 0.02 0.03 196.80 0.01 0.03 0.03 205.00 0.01 0.03 0.03 213.20 0.01 0.03 0.03 221.40 0.01 0.03 0.03 229.60 0.01 0.02 0.03 237.80 0.00 0.01 0.02 246.00 0.00 0.01 0.01 254.20 0.00 0.02 0.02 262.40 0.00 0.08 0.09 270.60 0.00 0.27 0.27 278.80 0.00 0.57 0.57 287.00 0.00 0.79 0.79 295.73 0.00 0.54 0.54 301.38 0.00 0.26 0.26 307.03 0.00 0.00 0.00
Page 86 Horizontal (y) displacements ‐ Normal supports
0.90
0.80
0.70
0.60
0.50
Page [mm] 0.40 Dy 87
0.30
0.20
0.10
0.00 ‐20.00 30.00 80.00 130.00 180.00 230.00 280.00 ‐0.10 x [m]
S1 S2 Normal supports x [m] Rx [10^‐6 rad] Ʃ S1 S2 S3 S4 ‐19.93 0.00 0.00 0.00 0.00 0.00 ‐14.27 ‐23.67 0.00 27.00 0.00 3.33 ‐8.62 ‐45.95 0.00 53.54 0.00 7.59 0.00 ‐63.95 0.00 77.80 0.00 13.86 8.14 ‐47.25 0.00 59.48 0.23 12.46 16.40 ‐25.82 0.00 29.50 0.79 4.47 24.60 ‐10.25 0.00 10.02 1.02 0.79 32.80 ‐3.19 0.00 2.63 1.02 0.46 41.00 ‐1.48 0.00 1.02 1.02 0.56 49.20 ‐2.05 ‐0.23 1.25 1.25 0.23 57.40 ‐2.84 ‐0.79 1.82 1.82 0.00 65.60 ‐3.30 ‐1.02 2.28 2.05 0.00 73.80 ‐3.86 ‐1.02 2.84 2.05 0.00 82.00 ‐4.09 ‐1.02 3.07 2.05 0.00 90.20 ‐4.09 ‐1.25 3.07 2.05 ‐0.23 98.40 ‐3.86 ‐1.82 3.07 2.05 ‐0.56 106.60 ‐3.30 ‐2.05 2.84 2.05 ‐0.46 114.80 ‐3.07 ‐2.05 2.28 2.05 ‐0.79 123.00 ‐3.07 ‐2.28 2.05 2.05 ‐1.25 131.20 ‐2.84 ‐2.84 2.05 2.05 ‐1.59 139.40 ‐2.28 ‐3.07 2.05 2.05 ‐1.25 147.60 ‐2.05 ‐3.07 2.05 2.05 ‐1.02 155.80 ‐2.05 ‐3.30 2.05 2.05 ‐1.25 164.00 ‐2.05 ‐3.86 2.05 2.28 ‐1.59 172.20 ‐1.82 ‐4.09 2.05 2.84 ‐1.02 180.40 ‐1.25 ‐4.09 2.05 3.07 ‐0.23 188.60 ‐1.02 ‐4.32 2.05 3.07 ‐0.23 196.80 ‐1.02 ‐4.88 2.05 3.07 ‐0.79 205.00 ‐1.02 ‐5.11 1.82 3.30 ‐1.02 213.20 ‐1.02 ‐4.88 1.25 3.86 ‐0.79 221.40 ‐0.79 ‐4.09 1.02 3.86 0.00 229.60 ‐0.23 ‐3.07 1.02 3.30 1.02 237.80 0.00 ‐2.05 1.02 2.84 1.82 246.00 0.00 ‐1.25 1.02 2.50 2.28 254.20 0.00 ‐2.17 1.02 3.99 2.84 262.40 0.00 ‐8.89 0.79 10.94 2.84 270.60 0.00 ‐25.82 0.23 33.15 7.56 278.80 0.00 ‐51.59 0.00 70.96 19.36 287.00 0.00 ‐77.11 0.00 97.21 20.10 295.73 0.00 ‐58.35 0.00 67.98 9.63 301.38 0.00 ‐30.67 0.00 34.33 3.67 307.03 0.00 0.00 0.00 0.00 0.00
Page 88 Rotations about x‐axis ‐ Normal supports
120.00
100.00
80.00
60.00
40.00
20.00 Page rad]
6 ‐
89 0.00 [10^ ‐20.00 30.00 80.00 130.00 180.00 230.00 280.00 Rx ‐20.00
‐40.00
‐60.00
‐80.00
‐100.00 x [m]
S1 S2 S3 S4 Normal supports x [m] Ry [10^‐6 rad] Ʃ S1 S2 S3 S4 ‐19.93 0.00 0.00 0.00 0.00 0.00 ‐14.27 ‐7.33 ‐1.00 ‐11.00 ‐1.00 ‐20.33 ‐8.62 ‐9.15 ‐1.31 ‐14.19 ‐1.31 ‐25.96 0.00 1.36 0.00 1.71 ‐0.23 2.84 8.14 22.83 2.84 35.87 2.05 63.59 16.40 40.09 5.11 60.88 4.32 110.40 24.60 50.55 7.16 74.40 6.70 138.81 32.80 54.07 8.97 78.16 8.41 149.62 41.00 52.72 10.23 74.76 10.23 147.93 49.20 48.86 11.48 67.73 12.04 140.10 57.40 43.97 13.06 59.88 13.29 130.21 65.60 38.63 14.09 51.36 14.32 118.40 73.80 32.95 14.32 42.95 15.34 105.56 82.00 27.38 14.55 35.00 16.13 93.06 90.20 21.71 14.88 27.61 16.36 80.56 98.40 16.59 14.32 20.68 16.36 67.96 106.60 12.04 13.29 14.32 16.13 55.79 114.80 7.39 12.27 8.41 15.11 43.18 123.00 3.30 11.02 3.07 13.29 30.68 131.20 ‐0.23 9.20 ‐1.59 11.25 18.64 139.40 ‐3.63 6.93 ‐5.34 8.97 6.93 147.60 ‐6.37 4.09 ‐8.74 5.91 ‐5.11 155.80 ‐8.97 0.79 ‐11.25 2.05 ‐17.39 164.00 ‐11.02 ‐3.07 ‐13.29 ‐2.28 ‐29.66 172.20 ‐12.27 ‐7.16 ‐15.11 ‐7.39 ‐41.93 180.40 ‐13.29 ‐11.48 ‐16.13 ‐13.29 ‐54.20 188.60 ‐14.09 ‐16.36 ‐16.36 ‐19.66 ‐66.47 196.80 ‐14.32 ‐21.48 ‐16.36 ‐26.82 ‐78.97 205.00 ‐14.32 ‐26.82 ‐16.13 ‐34.77 ‐92.04 213.20 ‐14.09 ‐32.50 ‐15.34 ‐42.95 ‐104.87 221.40 ‐13.52 ‐37.84 ‐14.32 ‐51.13 ‐116.81 229.60 ‐12.84 ‐42.95 ‐13.29 ‐59.31 ‐128.40 237.80 ‐11.48 ‐47.84 ‐12.04 ‐67.27 ‐138.62 246.00 ‐10.00 ‐51.70 ‐10.23 ‐73.97 ‐145.88 254.20 ‐8.41 ‐53.05 ‐8.41 ‐76.91 ‐146.78 262.40 ‐6.93 ‐49.52 ‐6.70 ‐72.59 ‐135.74 270.60 ‐5.11 ‐38.84 ‐4.32 ‐59.06 ‐107.33 278.80 ‐2.84 ‐21.25 ‐2.05 ‐34.85 ‐60.98 287.00 0.00 ‐0.56 0.23 ‐1.94 ‐2.28 295.73 1.31 8.94 1.31 13.31 24.88 301.38 1.00 7.00 1.00 10.33 19.33 307.03 0.00 0.00 0.00 0.00 0.00
Page 90 Rotations about y‐axis ‐ Normal supports
100.00
80.00
60.00
40.00
20.00
Page rad]
6
‐ 0.00
91 ‐20.00 30.00 80.00 130.00 180.00 230.00 280.00 [10^
Ry ‐20.00
‐40.00
‐60.00
‐80.00
‐100.00 x [m]
S1 S2 S3 S4 Normal supports x [m] Rz [10^‐6 rad] Ʃ S1 S2 S3 S4 ‐19.93 0.00 0.00 0.00 0.00 0.00 ‐14.27 2.67 ‐0.33 ‐3.33 ‐0.33 ‐1.33 ‐8.62 3.04 ‐0.67 ‐4.14 ‐0.67 ‐2.44 0.00 ‐0.90 ‐1.02 ‐0.82 ‐1.02 ‐3.76 8.14 ‐5.09 ‐0.79 2.25 ‐1.02 ‐4.65 16.40 ‐3.53 ‐0.23 ‐0.10 ‐0.79 ‐4.65 24.60 ‐1.25 0.00 ‐1.59 ‐0.46 ‐3.30 32.80 ‐0.23 ‐0.23 ‐0.79 ‐0.79 ‐2.05 41.00 0.00 ‐0.79 0.56 ‐1.02 ‐1.25 49.20 0.00 ‐1.02 1.25 ‐1.02 ‐0.79 57.40 0.23 ‐1.02 1.82 ‐1.02 0.00 65.60 0.79 ‐1.02 2.28 ‐1.25 0.79 73.80 1.25 ‐1.02 2.84 ‐1.82 1.25 82.00 1.82 ‐1.02 3.07 ‐2.05 1.82 90.20 2.05 ‐1.25 3.07 ‐2.05 1.82 98.40 2.05 ‐1.82 3.07 ‐2.05 1.25 106.60 2.05 ‐2.05 3.07 ‐2.05 1.02 114.80 2.05 ‐2.05 3.07 ‐2.28 0.79 123.00 2.05 ‐2.05 3.07 ‐2.84 0.23 131.20 2.05 ‐2.05 3.07 ‐3.07 0.00 139.40 2.05 ‐2.05 3.07 ‐3.07 0.00 147.60 2.05 ‐2.05 3.07 ‐3.07 0.00 155.80 2.05 ‐2.05 3.07 ‐3.07 0.00 164.00 2.05 ‐2.05 2.84 ‐3.07 ‐0.23 172.20 2.05 ‐2.05 2.28 ‐3.07 ‐0.79 180.40 2.05 ‐2.05 2.05 ‐3.07 ‐1.02 188.60 2.05 ‐2.05 2.05 ‐3.07 ‐1.02 196.80 1.82 ‐1.82 2.05 ‐3.07 ‐1.02 205.00 1.25 ‐1.25 1.82 ‐3.07 ‐1.25 213.20 1.02 ‐1.02 1.25 ‐3.07 ‐1.82 221.40 1.02 ‐0.79 1.02 ‐2.84 ‐1.59 229.60 1.02 ‐0.23 1.02 ‐2.28 ‐0.46 237.80 0.79 0.00 1.02 ‐1.82 0.00 246.00 0.23 0.00 0.79 ‐1.02 0.00 254.20 0.00 0.00 0.23 0.23 0.46 262.40 0.00 0.46 0.23 1.36 2.05 270.60 0.23 2.28 0.79 ‐0.13 3.17 278.80 0.79 4.19 1.02 ‐3.50 2.50 287.00 1.02 2.48 1.02 ‐2.02 2.50 295.73 0.67 ‐0.83 0.67 1.70 2.21 301.38 0.33 ‐1.33 0.33 2.00 1.33 307.03 0.00 0.00 0.00 0.00 0.00
Page 92 Rotations about z‐axis ‐ Normal supports
6.00
4.00
2.00
Page rad]
6
‐ 0.00
93 ‐20.00 30.00 80.00 130.00 180.00 230.00 280.00 [10^
Rz
‐2.00
‐4.00
‐6.00 x [m]
S1 S2 S3 S4
Annex C The unit lorry influence lines for the fixed support system
Page 94
Support 1 ‐ Fixed supports x [m] Fx [kN] Fy [kN] Fz [kN] Rx [10^‐6 rad] Ry [10^‐6 rad] Rz [10^‐6 rad] ‐19.93 0.00 0.00 0.00 0.00 0.00 0.00 ‐14.27 ‐3.00 17.22 13.96 1.00 ‐7.00 3.67 ‐8.62 ‐4.06 35.35 27.27 4.76 ‐8.72 4.81 0.00 ‐0.77 52.13 40.01 11.20 1.59 1.15 8.14 7.78 38.05 37.58 8.06 22.83 ‐4.07 16.40 16.38 17.89 35.66 0.48 39.86 ‐3.53 24.60 22.07 4.98 33.92 ‐2.05 49.75 ‐2.05 32.80 24.82 ‐0.07 32.30 ‐2.05 53.05 ‐1.25 41.00 25.80 ‐1.10 30.86 ‐2.05 51.70 ‐1.02 49.20 25.59 ‐0.74 29.56 ‐2.05 47.84 ‐1.02 57.40 24.58 ‐0.13 28.34 ‐2.05 42.95 ‐0.79 65.60 23.20 0.38 27.20 ‐2.05 37.61 ‐0.23 73.80 21.82 0.68 26.15 ‐2.05 31.93 0.23 82.00 20.43 0.80 25.20 ‐2.05 26.59 0.79 90.20 19.14 0.85 24.33 ‐2.05 21.48 1.02 98.40 18.01 0.92 23.52 ‐1.82 16.36 1.02 106.60 17.11 0.97 22.78 ‐1.25 11.48 1.02 114.80 16.55 0.96 22.10 ‐1.02 7.16 1.25 123.00 16.33 0.96 21.47 ‐1.02 3.07 1.82 131.20 16.48 0.96 20.86 ‐1.02 ‐0.79 2.05 139.40 17.02 0.97 20.28 ‐1.02 ‐4.09 2.05 147.60 17.97 0.97 19.72 ‐0.79 ‐6.93 1.82 155.80 19.34 0.98 19.14 ‐0.23 ‐9.20 1.25 164.00 21.12 1.00 18.54 0.00 ‐11.25 1.02 172.20 23.31 1.02 17.91 0.00 ‐13.06 1.02 180.40 25.91 1.03 17.24 0.00 ‐14.32 1.02 188.60 28.85 1.04 16.49 0.00 ‐15.11 1.02 196.80 32.12 1.05 15.69 0.23 ‐15.34 1.02 205.00 35.63 1.08 14.82 0.79 ‐15.34 0.79 213.20 39.29 1.09 13.87 1.02 ‐15.11 0.23 221.40 42.95 1.11 12.84 1.02 ‐14.32 0.00 229.60 46.34 1.13 11.70 1.02 ‐13.29 0.00 237.80 49.25 1.14 10.49 1.02 ‐12.27 0.00 246.00 51.10 1.14 9.19 1.02 ‐11.02 0.00 254.20 50.60 1.10 7.76 1.02 ‐9.43 ‐0.23 262.40 45.98 0.99 6.15 1.02 ‐7.72 ‐0.56 270.60 36.27 0.82 4.43 1.02 ‐5.34 ‐0.46 278.80 21.15 0.58 2.59 0.79 ‐2.84 ‐0.56 287.00 0.18 0.08 0.13 0.00 0.00 ‐0.23 295.73 ‐10.33 ‐0.23 ‐1.14 ‐0.44 1.31 0.00 301.38 ‐8.08 ‐0.20 ‐0.91 ‐0.33 1.00 0.00 307.03 0.00 0.00 0.00 0.00 0.00 0.00
Page 95 Support 2 ‐ Fixed supports x [m] Fx [kN] Fy [kN] Fz [kN] Rx [10^‐6 rad] Ry [10^‐6 rad] Rz [10^‐6 rad] ‐19.93 0.00 0.00 0.00 0.00 0.00 0.00 ‐14.27 8.06 ‐0.20 ‐0.91 ‐0.33 ‐0.67 0.00 ‐8.62 10.36 ‐0.25 ‐1.17 ‐0.44 ‐0.87 0.00 0.00 0.03 0.04 0.03 0.00 0.00 0.00 8.14 ‐21.00 0.54 2.47 1.02 1.82 0.00 16.40 ‐36.39 0.82 4.38 1.82 3.30 0.00 24.60 ‐46.15 0.99 6.13 2.05 5.11 0.00 32.80 ‐50.67 1.10 7.74 2.05 6.93 0.00 41.00 ‐51.11 1.14 9.18 2.05 8.41 0.00 49.20 ‐49.24 1.14 10.49 2.05 10.00 0.00 57.40 ‐46.32 1.13 11.71 2.05 11.25 0.00 65.60 ‐42.93 1.11 12.84 1.82 12.27 0.00 73.80 ‐39.29 1.09 13.89 1.25 13.06 ‐0.23 82.00 ‐35.65 1.07 14.84 1.02 13.29 ‐0.79 90.20 ‐32.16 1.05 15.71 0.79 13.29 ‐1.02 98.40 ‐28.91 1.03 16.52 0.23 13.29 ‐1.02 106.60 ‐25.98 1.02 17.27 0.00 13.06 ‐1.02 114.80 ‐23.41 1.01 17.94 0.00 12.04 ‐1.02 123.00 ‐21.24 0.99 18.57 ‐0.23 10.23 ‐1.02 131.20 ‐19.47 0.97 19.18 ‐0.79 8.18 ‐1.02 139.40 ‐18.11 0.96 19.76 ‐1.02 5.91 ‐1.02 147.60 ‐17.17 0.97 20.32 ‐1.25 3.07 ‐1.02 155.80 ‐16.64 0.96 20.90 ‐1.82 0.00 ‐1.02 164.00 ‐16.49 0.96 21.51 ‐2.05 ‐3.30 ‐1.02 172.20 ‐16.71 0.98 22.14 ‐2.05 ‐7.39 ‐1.02 180.40 ‐17.26 0.99 22.81 ‐2.05 ‐12.04 ‐1.02 188.60 ‐18.15 0.95 23.55 ‐2.28 ‐16.59 ‐1.02 196.80 ‐19.27 0.90 24.36 ‐2.84 ‐21.71 ‐0.79 205.00 ‐20.54 0.87 25.22 ‐3.07 ‐27.38 ‐0.23 213.20 ‐21.88 0.80 26.17 ‐3.07 ‐32.95 0.23 221.40 ‐23.23 0.56 27.21 ‐3.07 ‐38.63 0.79 229.60 ‐24.56 0.12 28.34 ‐3.07 ‐43.97 1.02 237.80 ‐25.49 ‐0.42 29.55 ‐2.84 ‐48.86 1.25 246.00 ‐25.61 ‐0.82 30.85 ‐2.28 ‐52.49 1.82 254.20 ‐24.55 ‐0.15 32.28 ‐1.82 ‐53.28 1.82 262.40 ‐21.75 4.09 33.90 ‐0.79 ‐49.52 1.25 270.60 ‐15.87 16.02 35.61 4.47 ‐38.84 0.79 278.80 ‐6.92 35.66 37.46 15.42 ‐21.25 ‐0.69 287.00 1.55 50.19 39.91 17.21 ‐0.56 ‐5.01 295.73 4.33 34.46 27.25 6.37 8.94 ‐6.16 301.38 3.04 16.89 13.97 1.00 7.00 ‐4.00 307.03 0.00 0.00 0.00 0.00 0.00 0.00
Page 96 Support 3 ‐ Fixed supports x [m] Fx [kN] Fy [kN] Fz [kN] Rx [10^‐6 rad] Ry [10^‐6 rad] Rz [10^‐6 rad] ‐19.93 0.00 0.00 0.00 0.00 0.00 0.00 ‐14.27 ‐11.39 ‐16.78 20.37 2.67 ‐10.67 ‐3.00 ‐8.62 ‐14.71 ‐34.80 40.71 3.04 ‐13.52 ‐3.70 0.00 ‐0.20 ‐52.20 62.29 2.09 2.50 ‐0.82 8.14 29.04 ‐39.20 61.78 2.92 35.87 1.46 16.40 49.45 ‐19.63 60.77 1.38 60.08 ‐1.13 24.60 62.42 ‐7.07 59.58 ‐0.23 73.38 ‐2.61 32.80 68.72 ‐2.17 58.28 ‐0.79 76.91 ‐2.05 41.00 69.56 ‐1.09 56.79 ‐1.02 73.17 ‐1.02 49.20 67.39 ‐1.31 55.18 ‐1.02 66.24 ‐0.23 57.40 64.02 ‐1.76 53.48 ‐1.02 58.29 0.23 65.60 60.12 ‐2.09 51.69 ‐1.02 50.11 0.79 73.80 55.80 ‐2.19 49.81 ‐1.02 41.93 1.02 82.00 51.49 ‐2.12 47.84 ‐0.79 33.98 1.25 90.20 47.38 ‐1.98 45.79 ‐0.23 26.59 1.82 98.40 43.59 ‐1.86 43.68 0.00 19.66 2.05 106.60 40.23 ‐1.74 41.51 0.00 13.29 2.05 114.80 37.23 ‐1.56 39.26 0.00 7.39 2.05 123.00 34.79 ‐1.38 36.97 0.00 2.28 2.05 131.20 32.80 ‐1.22 34.65 0.00 ‐2.05 2.05 139.40 31.34 ‐1.07 32.31 0.00 ‐5.91 2.05 147.60 30.40 ‐0.90 29.95 0.00 ‐9.20 2.05 155.80 29.95 ‐0.74 27.61 0.23 ‐12.04 2.05 164.00 30.00 ‐0.60 25.29 0.79 ‐14.09 1.82 172.20 30.49 ‐0.45 23.00 1.02 ‐15.57 1.25 180.40 31.45 ‐0.28 20.75 1.02 ‐16.93 1.02 188.60 32.77 ‐0.11 18.57 1.02 ‐17.39 1.02 196.80 34.42 0.06 16.45 1.02 ‐17.39 1.02 205.00 36.30 0.23 14.40 1.02 ‐17.39 0.79 213.20 38.33 0.42 12.42 1.02 ‐17.16 0.23 221.40 40.36 0.60 10.54 1.02 ‐16.36 0.00 229.60 42.19 0.76 8.75 1.02 ‐15.11 0.00 237.80 43.59 0.92 7.04 1.02 ‐13.52 ‐0.23 246.00 44.00 1.07 5.42 1.02 ‐12.04 ‐0.79 254.20 42.54 1.16 3.93 1.02 ‐10.23 ‐1.02 262.40 37.95 1.11 2.62 1.02 ‐8.18 ‐1.02 270.60 28.96 0.95 1.41 1.02 ‐5.91 ‐1.02 278.80 15.26 0.65 0.33 0.79 ‐3.07 ‐0.79 287.00 ‐1.39 0.07 ‐0.18 0.00 0.00 0.00 295.73 ‐8.56 ‐0.29 ‐0.18 ‐0.44 1.31 0.44 301.38 ‐6.36 ‐0.24 ‐0.10 ‐0.33 1.00 0.33 307.03 0.00 0.00 0.00 0.00 0.00 0.00
Page 97 Support 4 ‐ Fixed supports x [m] Fx [kN] Fy [kN] Fz [kN] Rx [10^‐6 rad] Ry [10^‐6 rad] Rz [10^‐6 rad] ‐19.93 0.00 0.00 0.00 0.00 0.00 0.00 ‐14.27 6.33 ‐0.24 ‐0.09 ‐0.33 ‐0.67 0.00 ‐8.62 8.40 ‐0.30 ‐0.14 ‐0.44 ‐0.87 0.00 0.00 0.95 0.02 ‐0.07 0.00 0.00 0.23 8.14 ‐15.82 0.60 0.44 0.79 2.05 0.79 16.40 ‐29.44 0.92 1.45 1.25 4.09 1.02 24.60 ‐38.35 1.10 2.64 1.82 5.91 1.02 32.80 ‐42.87 1.14 3.95 2.05 7.39 0.79 41.00 ‐44.25 1.06 5.43 2.05 9.20 0.23 49.20 ‐43.74 0.91 7.04 1.82 11.02 0.00 57.40 ‐42.28 0.76 8.75 1.25 12.50 0.00 65.60 ‐40.40 0.60 10.53 1.02 14.09 0.00 73.80 ‐38.33 0.42 12.41 1.02 15.11 ‐0.23 82.00 ‐36.27 0.25 14.38 1.02 15.57 ‐0.79 90.20 ‐34.36 0.08 16.43 1.02 15.90 ‐1.02 98.40 ‐32.69 ‐0.08 18.54 1.02 15.57 ‐1.02 106.60 ‐31.35 ‐0.25 20.72 0.79 15.11 ‐1.02 114.80 ‐30.38 ‐0.42 22.96 0.23 14.32 ‐1.25 123.00 ‐29.88 ‐0.57 25.26 0.00 13.06 ‐1.82 131.20 ‐29.81 ‐0.71 27.57 0.00 11.02 ‐2.05 139.40 ‐30.25 ‐0.86 29.91 0.00 8.18 ‐2.05 147.60 ‐31.19 ‐1.03 32.27 ‐0.23 4.88 ‐2.05 155.80 ‐32.65 ‐1.20 34.62 ‐0.79 0.79 ‐2.05 164.00 ‐34.64 ‐1.36 36.93 ‐1.02 ‐4.09 ‐2.05 172.20 ‐37.09 ‐1.55 39.22 ‐1.02 ‐9.43 ‐2.05 180.40 ‐40.09 ‐1.74 41.48 ‐1.02 ‐15.34 ‐2.05 188.60 ‐43.46 ‐1.88 43.65 ‐1.02 ‐21.71 ‐1.82 196.80 ‐47.27 ‐2.02 45.77 ‐1.25 ‐28.63 ‐1.25 205.00 ‐51.40 ‐2.18 47.82 ‐1.82 ‐36.25 ‐1.02 213.20 ‐55.74 ‐2.31 49.80 ‐2.05 ‐44.77 ‐1.02 221.40 ‐60.08 ‐2.27 51.68 ‐2.05 ‐53.18 ‐0.79 229.60 ‐63.97 ‐2.01 53.47 ‐2.05 ‐61.36 ‐0.23 237.80 ‐67.34 ‐1.65 55.18 ‐1.82 ‐69.31 0.23 246.00 ‐69.49 ‐1.39 56.81 ‐1.25 ‐76.01 1.02 254.20 ‐68.59 ‐2.11 58.29 ‐1.02 ‐79.18 2.28 262.40 ‐62.18 ‐6.19 59.60 ‐1.48 ‐75.19 3.63 270.60 ‐49.36 ‐17.79 60.81 ‐1.23 ‐61.11 3.17 278.80 ‐29.49 ‐36.89 61.89 0.54 ‐35.87 1.15 287.00 ‐0.34 ‐50.34 62.40 1.86 ‐1.94 2.86 295.73 14.56 ‐33.94 40.75 3.91 13.75 4.60 301.38 11.40 ‐16.45 20.38 3.33 10.67 3.33 307.03 0.00 0.00 0.00 0.00 0.00 0.00
Page 98 Fixed supports x [m] Fx [kN] Ʃ S1 S2 S3 S4 ‐19.93 0.00 0.00 0.00 0.00 0.00 ‐14.27 ‐3.00 8.06 ‐11.39 6.33 0.00 ‐8.62 ‐4.06 10.36 ‐14.71 8.40 0.00 0.00 ‐0.77 0.03 ‐0.20 0.95 0.00 8.14 7.78 ‐21.00 29.04 ‐15.82 0.00 16.40 16.38 ‐36.39 49.45 ‐29.44 0.00 24.60 22.07 ‐46.15 62.42 ‐38.35 0.00 32.80 24.82 ‐50.67 68.72 ‐42.87 0.00 41.00 25.80 ‐51.11 69.56 ‐44.25 0.00 49.20 25.59 ‐49.24 67.39 ‐43.74 0.00 57.40 24.58 ‐46.32 64.02 ‐42.28 0.00 65.60 23.20 ‐42.93 60.12 ‐40.40 0.00 73.80 21.82 ‐39.29 55.80 ‐38.33 0.00 82.00 20.43 ‐35.65 51.49 ‐36.27 0.00 90.20 19.14 ‐32.16 47.38 ‐34.36 0.00 98.40 18.01 ‐28.91 43.59 ‐32.69 0.00 106.60 17.11 ‐25.98 40.23 ‐31.35 0.00 114.80 16.55 ‐23.41 37.23 ‐30.38 0.00 123.00 16.33 ‐21.24 34.79 ‐29.88 0.00 131.20 16.48 ‐19.47 32.80 ‐29.81 0.00 139.40 17.02 ‐18.11 31.34 ‐30.25 0.00 147.60 17.97 ‐17.17 30.40 ‐31.19 0.00 155.80 19.34 ‐16.64 29.95 ‐32.65 0.00 164.00 21.12 ‐16.49 30.00 ‐34.64 0.00 172.20 23.31 ‐16.71 30.49 ‐37.09 0.00 180.40 25.91 ‐17.26 31.45 ‐40.09 0.00 188.60 28.85 ‐18.15 32.77 ‐43.46 0.00 196.80 32.12 ‐19.27 34.42 ‐47.27 0.00 205.00 35.63 ‐20.54 36.30 ‐51.40 0.00 213.20 39.29 ‐21.88 38.33 ‐55.74 0.00 221.40 42.95 ‐23.23 40.36 ‐60.08 0.00 229.60 46.34 ‐24.56 42.19 ‐63.97 0.00 237.80 49.25 ‐25.49 43.59 ‐67.34 0.00 246.00 51.10 ‐25.61 44.00 ‐69.49 0.00 254.20 50.60 ‐24.55 42.54 ‐68.59 0.00 262.40 45.98 ‐21.75 37.95 ‐62.18 0.00 270.60 36.27 ‐15.87 28.96 ‐49.36 0.00 278.80 21.15 ‐6.92 15.26 ‐29.49 0.00 287.00 0.18 1.55 ‐1.39 ‐0.34 0.00 295.73 ‐10.33 4.33 ‐8.56 14.56 0.00 301.38 ‐8.08 3.04 ‐6.36 11.40 0.00 307.03 0.00 0.00 0.00 0.00 0.00
Page 99 Horizontal (x) reaction forces ‐ Fixed supports
80.00
60.00
40.00
20.00
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[kN] 0.00 Fx 100 ‐20.00 30.00 80.00 130.00 180.00 230.00 280.00 ‐20.00
‐40.00
‐60.00
‐80.00 x [m]
S1 S2 S3 S4 Fixed supports x [m] Fy [kN] Ʃ S1 S2 S3 S4 ‐19.93 0.00 0.00 0.00 0.00 0.00 ‐14.27 17.22 ‐0.20 ‐16.78 ‐0.24 0.00 ‐8.62 35.35 ‐0.25 ‐34.80 ‐0.30 0.00 0.00 52.13 0.04 ‐52.20 0.02 0.00 8.14 38.05 0.54 ‐39.20 0.60 0.00 16.40 17.89 0.82 ‐19.63 0.92 0.00 24.60 4.98 0.99 ‐7.07 1.10 0.00 32.80 ‐0.07 1.10 ‐2.17 1.14 0.00 41.00 ‐1.10 1.14 ‐1.09 1.06 0.00 49.20 ‐0.74 1.14 ‐1.31 0.91 0.00 57.40 ‐0.13 1.13 ‐1.76 0.76 0.00 65.60 0.38 1.11 ‐2.09 0.60 0.00 73.80 0.68 1.09 ‐2.19 0.42 0.00 82.00 0.80 1.07 ‐2.12 0.25 0.00 90.20 0.85 1.05 ‐1.98 0.08 0.00 98.40 0.92 1.03 ‐1.86 ‐0.08 0.00 106.60 0.97 1.02 ‐1.74 ‐0.25 0.00 114.80 0.96 1.01 ‐1.56 ‐0.42 0.00 123.00 0.96 0.99 ‐1.38 ‐0.57 0.00 131.20 0.96 0.97 ‐1.22 ‐0.71 0.00 139.40 0.97 0.96 ‐1.07 ‐0.86 0.00 147.60 0.97 0.97 ‐0.90 ‐1.03 0.00 155.80 0.98 0.96 ‐0.74 ‐1.20 0.00 164.00 1.00 0.96 ‐0.60 ‐1.36 0.00 172.20 1.02 0.98 ‐0.45 ‐1.55 0.00 180.40 1.03 0.99 ‐0.28 ‐1.74 0.00 188.60 1.04 0.95 ‐0.11 ‐1.88 0.00 196.80 1.05 0.90 0.06 ‐2.02 0.00 205.00 1.08 0.87 0.23 ‐2.18 0.00 213.20 1.09 0.80 0.42 ‐2.31 0.00 221.40 1.11 0.56 0.60 ‐2.27 0.00 229.60 1.13 0.12 0.76 ‐2.01 0.00 237.80 1.14 ‐0.42 0.92 ‐1.65 0.00 246.00 1.14 ‐0.82 1.07 ‐1.39 0.00 254.20 1.10 ‐0.15 1.16 ‐2.11 0.00 262.40 0.99 4.09 1.11 ‐6.19 0.00 270.60 0.82 16.02 0.95 ‐17.79 0.00 278.80 0.58 35.66 0.65 ‐36.89 0.00 287.00 0.08 50.19 0.07 ‐50.34 0.00 295.73 ‐0.23 34.46 ‐0.29 ‐33.94 0.00 301.38 ‐0.20 16.89 ‐0.24 ‐16.45 0.00 307.03 0.00 0.00 0.00 0.00 0.00
Page 101 Horizontal (y) reaction forces ‐ Fixed supports
60.00
40.00
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0.00 Fy 102 ‐20.00 30.00 80.00 130.00 180.00 230.00 280.00
‐20.00
‐40.00
‐60.00 x [m]
S1 S2 S3 S4 Fixed supports x [m] Fz [kN] Ʃ S1 S2 S3 S4 ‐19.93 0.00 0.00 0.00 0.00 0.00 ‐14.27 13.96 ‐0.91 20.37 ‐0.09 33.33 ‐8.62 27.27 ‐1.17 40.71 ‐0.14 66.67 0.00 40.01 0.03 62.29 ‐0.07 102.27 8.14 37.58 2.47 61.78 0.44 102.27 16.40 35.66 4.38 60.77 1.45 102.27 24.60 33.92 6.13 59.58 2.64 102.27 32.80 32.30 7.74 58.28 3.95 102.27 41.00 30.86 9.18 56.79 5.43 102.27 49.20 29.56 10.49 55.18 7.04 102.27 57.40 28.34 11.71 53.48 8.75 102.27 65.60 27.20 12.84 51.69 10.53 102.27 73.80 26.15 13.89 49.81 12.41 102.27 82.00 25.20 14.84 47.84 14.38 102.27 90.20 24.33 15.71 45.79 16.43 102.27 98.40 23.52 16.52 43.68 18.54 102.27 106.60 22.78 17.27 41.51 20.72 102.27 114.80 22.10 17.94 39.26 22.96 102.27 123.00 21.47 18.57 36.97 25.26 102.27 131.20 20.86 19.18 34.65 27.57 102.27 139.40 20.28 19.76 32.31 29.91 102.27 147.60 19.72 20.32 29.95 32.27 102.27 155.80 19.14 20.90 27.61 34.62 102.27 164.00 18.54 21.51 25.29 36.93 102.27 172.20 17.91 22.14 23.00 39.22 102.27 180.40 17.24 22.81 20.75 41.48 102.27 188.60 16.49 23.55 18.57 43.65 102.27 196.80 15.69 24.36 16.45 45.77 102.27 205.00 14.82 25.22 14.40 47.82 102.27 213.20 13.87 26.17 12.42 49.80 102.27 221.40 12.84 27.21 10.54 51.68 102.27 229.60 11.70 28.34 8.75 53.47 102.27 237.80 10.49 29.55 7.04 55.18 102.27 246.00 9.19 30.85 5.42 56.81 102.27 254.20 7.76 32.28 3.93 58.29 102.27 262.40 6.15 33.90 2.62 59.60 102.27 270.60 4.43 35.61 1.41 60.81 102.27 278.80 2.59 37.46 0.33 61.89 102.27 287.00 0.13 39.91 ‐0.18 62.40 102.27 295.73 ‐1.14 27.25 ‐0.18 40.75 66.67 301.38 ‐0.91 13.97 ‐0.10 20.38 33.33 307.03 0.00 0.00 0.00 0.00 0.00
Page 103 Vertical (z) reaction forces ‐ Fixed supports
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60.00
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20.00
10.00
0.00 ‐20.00 30.00 80.00 130.00 180.00 230.00 280.00 ‐10.00 x [m]
S1 S2 S3 S4