A pre-study of the dynamic behavior of a single diagonal timber arch bridge

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IN

DEGREE PROJECT THE BUILT ENVIRONMENT, SECOND CYCLE, 30 CREDITS

,

STOCKHOLM SWEDEN 2021

A pre-study of the dynamic

behavior of a single diagonal

timber arch bridge

XIAOQI WANG

SHUFAN YE

KTH ROYAL INSTITUTE OF TECHNOLOGY

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KTH Royal Institute of Technology

School of Architecture and the Built Environment

Department of Civil and Architectural Engineering

Division of Building Materials

SE-100 44 Stockholm, Sweden

TRITA-ABE-MBT-20726

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Abstract

The aim of this Master’s thesis was to study the dynamic behaviour of a special type of pedestrian timber bridge with a single diagonal arch - a design proposal made in a previous student project. The bridge is intended to be built as a gateway to the Alfred Nobel’s Björkborn in the municipality of Karlskoga. The original plan for this thesis was to build and test a downscaled model in order to verify theoretical investigations. The laboratory testing was however not possible to be performed, therefore the study was conducted only by means of analytical and numerical tools. Both a downscaled model and a full-scale bridge model were analysed and compared in order to find proper scale parameters. Different studies were performed on the models by means of the finite element method in order to investigate the influence of relevant parameters on dynamic behaviour of the bridge. A scale factor was determined which allows for the translation of results from the downscaled model to the full-scale model. Results showed that the dynamic behaviour of this type of bridge is rather complicated, and the original design needs to be somewhat modified to meet the comfort criterion for pedestrians. An increase of the width of the arch, a proper arrangement of the cables, and adoption of longitudinal steel beams under the deck were found to be efficient methods to improve the dynamic performance of the bridge. Future work should include experiments on a downscaled model to validate these theoretical solutions.

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Sammanfattning

Syftet med detta examensarbete var att studera det dynamiska beteendet hos en speciell typ av gång- och cykelbro med en diagonal båge - ett designförslag som gjorts i ett tidigare studentprojekt. Bron är tänkt byggas över ett vattendrag vid Alfred Nobels Björkborn i Karlskoga kommun. Den ursprungliga planen för detta examensarbete var att bygga och testa en nerskalad modell för att verifiera teoretiska undersökningar. Laboratorietesterna var dock inte möjliga att utföra, därför genomfördes studien endast med hjälp av analytiska och numeriska verktyg. Både en nerskalad modell och en fullskalemodell analyserades och jämfördes för att hitta rätt skalparametrar. Olika studier utfördes på modellerna med hjälp av finita elementmetoden för att undersöka påverkan av relevanta parametrar på brons dynamiska beteendet. En skalfaktor bestämdes som möjliggör överföringen av resultaten från den nerskalade modellen till fullskalamodellen. Resultaten visade att det dynamiska beteendet hos denna typ av bro är ganska komplicerat och att den ursprungliga designen måste modifieras något för att uppfylla komfortkriteriet för fotgängare. En ökning av bågens bredd, en korrekt placering av kablarna och begagnande av längsgående stålbalkar under brobanan visade sig vara effektiva metoder för att förbättra brons dynamiska prestanda. Framtida arbete bör innehålla experiment på en nerskalad modell för att validera dessa teoretiska lösningar.

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Preface

This master thesis was carried out at KTH Department of Civil and Architectural Engineering, Division of Building Materials, in Stockholm.

In this work, a unique design of bridge by students from Studio 7 at School of Architecture, KTH - Nour Fansa, Kristoffer Hamrin, and Mateusz Szpotowicz, was further studied from an engineering aspect. The design was selected to be constructed in Karlskoga to allow visitors reach the Alfred Nobel museem. It was a great honor to involve with such an actual project case and collaborate with architectural and engineering field.

We would like to thank our supervisor Prof. Roberto Crocetti for introducing this project and giving us the opportunity to participate in, and most important, for his support and valuable guidance throughout this thesis. Also we would like to thank Prof. Jean-Marc Battini as a co-supervisor with illuminating opinions. A parallel study focused on the bridge’s statics are conducted by Lovisa Öhgren and Malin Åström.

We would like to express our deepest gratitude to Daniel Colmenares Herrera for his inspiration and help on the scale factor analysis and modification part, and also for his kind encouragement and support during this period.

Professor Magnus Wålinder was the examiner of this thesis. A gratitude goes to him for his advice and remarks on the thesis. Costin Pacoste and other engineers from ELU should also be thanked for participating our final presentation and give their valuable opinions on our work. Finally, we would like to thank each other for the great collaboration and supports. Due to the COVID-19 situation, it has been a tough time. It was great that we have each in accomplishment of this thesis. The friendship between us will always last. Also thanks to our families and friends for their love, patience and support that always come along with us.

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List of Symbols

𝑎𝑚𝑎𝑥 Maximum acceleration of the bridge deck

d Density of the pedestrian crowd

𝐸0,𝑚𝑒𝑎𝑛 Elastic modulus parallel to grain

𝐸90,𝑚𝑒𝑎𝑛 Elastic modulus perpendicular to grain

𝐸0.05 Fifth percentile value of modulus of elasticity

Ed Design value of the effects of actions

𝐸𝑘 Elastic modulus for steel

𝐹𝑙𝑖𝑛𝑒𝑎𝑟 Linear load used for simplified hand calculations

𝐹𝑚𝑜𝑑𝑎𝑙 Modal force

𝑓𝑎𝑟𝑐ℎ Rise of the arch

𝑓𝑐,0,𝑑 Design compressive strength

𝑓𝑐,0,𝑘 Characteristic compressive strength

𝑓𝑚,𝑑 Design bending strength

𝑓𝑚,𝑘 Characteristic bending strength

𝑓𝑡,0,𝑑 Design tension strength

𝑓𝑡,0,𝑘 Characteristic tension strength

𝑓𝑣,𝑑 Design shear strength

𝑓𝑣,𝑘 Characteristic shear strength

𝑓𝑣 The natural frequency to induce the resonance of the structural

𝑓𝑦,𝑘 Yield strength

𝐺𝑘,𝑗 Characteristic value of the permanent action j

𝐺𝑚𝑒𝑎𝑛 Mean value of shear modulus

𝐼𝑦 Second moment of area about axis y

𝐼𝑧 Second moment of area about axis z

𝑘𝑐𝑟𝑖𝑡 Reduction factor due to the lateral buckling

𝑘𝑐,𝑦 Reduction factors for buckling around the corresponding y-axis

𝑘𝑐,𝑧 Reduction factors for buckling around the corresponding z-axis

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𝑘𝑟 Reduction factor for flexural strength

𝐿𝑎𝑟𝑐ℎ Span of the arch

𝑙𝑒𝑓 Effective length

𝑀𝑚𝑜𝑑𝑎𝑙 Modal mass

𝑁Ed Design tension force

𝑛 Numbers of the pedestrians on the footbridge

𝑄𝑘,𝑖 Characteristic value of the variable action j

𝑞𝑓𝑘 Uniformly distributed pedestrian load

𝑟𝑚 Inner radius of the arch

𝑆 Area of the deck plate

Si Scale factor for the quantity i

𝑢𝑙𝑖𝑚𝑖𝑡 Limitation value for the bridge’s vertical deformation

𝑢𝑚𝑎𝑥 Maximum vertical deformation of the bridge

β Buckling length factor

𝛾𝐺,𝑗 Partial factor for the permanent action j

γ_M0 Partial factor for resistance of cross-sections 𝛾𝑄,𝑖 Partial factor for the variable action i

𝜆𝑦 Slenderness ratio around the y-axis

𝜆𝑧 Slenderness ratio around the z-axis

𝜆𝑟𝑒𝑙,𝑦 Relative slenderness for bending around y-axis

𝜆𝑟𝑒𝑙,𝑧 Relative slenderness for bending around z-axis

𝜌 Density

𝜎𝑐,0,𝑑 Design compressive stress

𝜎𝑚,𝑐𝑟𝑖𝑡 Critical bending stress

𝜎𝑚,𝑑 Design bending stress

𝜎𝑡,0,𝑑 Design tension stress

𝜏𝑑 Design shear stress

𝜈 Poisson’s ratio

𝜉 Critical damping ratio

𝜓 Resonance coefficient

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1 Introduction

1.1 Context

In modern society, a bridge is more than just an engineering structure to bear traffic but also a component of the landscape. Hence the aesthetic requirements of bridges shall be also regarded as a design objective to fulfill. Since pedestrian bridges are more relevant to human scale and often built in urban areas, aesthetic requirements become essential for pedestrian bridges (Rodriguez, S, 2005).

The diagonal arch bridge is a transformation from the tied-arch bridge with an aesthetically pleasing appearance. The diagonal arch obliquely crosses the alignment and is connected to the deck by cables. The centrosymmetric structure creates a very satisfactory order from the visual perspective. With a suitable design of materials and scales, this kind of bridge can be integrated with the environment in a harmonic way. On the other hand, timber has become a trend for construction in recent years with its extraordinary advantage in ecology and economics. One common usage for timber is to be constructed as pedestrian bridges. Predictably, the timber diagonal arch bridge as the pedestrian bridge would contribute to outstanding performance in over-all perspectives.

The municipality of Karlskoga commissioned a new bridge to allow pedestrians to reach the Alfred Nobel museum (hereinafter called Nobel Bridge). Nour Fansa, Kristoffer Hamrin, and Mateusz Szpotowicz (from Studio 7 at School of Architecture, KTH) developed a timber pedestrian bridge design with the form of a single diagonal arch, as shown in Figure 1.1, which was ultimately selected. The bridge span is 45m, and the deck’s width is 3.5m, while further study shall be implemented to determine the specific design and dimension.

Figure 1.1 The model and visualization of the Nobel bridge designed by Kristoffer, H. et al.

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In this thesis, therefore, the timber diagonal arch bridge was studied based on this design. Particularly, the dynamic behavior of this bridge and the corresponding modification methods were studied.

1.2 Background

There are some existing cases of the diagonal arch bridges over the world, including the Hulme bridge in Manchester, Juscelino Kubitschek bridge in Brazil, and Tongtai bridge in China, as shown in Figure 1.2- Figure 1.4.

Figure 1.2 Close view of Hulme Arch Bridge (Mikey, 2008)

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Figure 1.4 A general view of Tongtai Bridge (Wang, H., & Qin, S. , 2014)

It should be noticed that all of these existing bridges were designed as road bridges, and the arches were constructed with steel box profiles. In addtion, the design criterion for these bridges was mainly focused on statics and stability.

Hussain, N., & Wilson, I. (1999) clearly described the design and construction of the Hulme Arch Bridge. The static mechanism was highly investigated for the design, while the dynamic characteristics were not mentioned. Wang, Q., et al. (2011) performed a model study on Tongtai Bridge to verify its static capacity. The experimental data from their results were practically coincident with the theoretical values, proving that the FEM simulation for this unique type of bridge was credible. Another numerical study took the construction phase into consideration, and the results showed that the Tongtai Bridge highly fulfilled the static and stability requirements (Wang, H., & Qin, S, 2014).

Regarding the dynamic aspect, there are only a few research within the field of nature loads. Feiran, M., et al. (2009) showed that Tongtai Bridge had an excellent aseismic performance under numerical simulation. Xun, J.C., et al(2016) stated that the aerodynamic stability of Tongtai Bridge was satisfactory, supported by their wind test model results.

So far, however, no studies have been found on the timber diagonal arch bridge. Also, far too little attention has been paid to the pedestrians’ comfort level for this unique bridge form.

1.3 Aim and objectives

This thesis's overall aim is to understand the behavior of the special-shaped bridge – the timber diagonal arch bridge, particularly regarding its dynamic performance and to investigate efficient methods for improvement. Thus, to provide suggestions for the practical design of the Nobel bridge – the first timber single diagonal arch bridge in Sweden.

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conducted in this stage. Some design parameters, including the rise-span ratio and the arch form, were also studied to optimize the static properties. After that, a final proposal for the scaled-down model was raised. With this proposal, the scaled-down model shall be built for experiments to calibrate the numerical models and draw practical conclusions. However, the tests were unfortunately unable to conduct due to the Covid-19 situation. This thesis's strategy had to alter, and numerical analysis became the principal approach throughout this thesis. The prototype model was then built by amplifying the dimension of the scaled-down model under a specific scale factor(while the material and geometry properties were kept the same). The natural vibration and the resonance induced vibration were studied to evaluate the dynamic performance of the prototype model. The resonance induced vibration was implemented under the instruction of The Footbridge: Assessment of vibration behavior of footbridges under

pedestrian loading (Sétra, 2006). The dynamic response was analyzed by both numerical

approaches and theoretical calculations.

With the dynamic characteristics of the scaled-down model and the prototype model, a scale factor can be reducted and compared with the therotical value, thus to confirm the scaled relation and gain confidence in predicting the fullscale bridge's actual behavior. Dimensioal analysis was applied for the theoretical derivation.

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2 Methods

In this chapter, the methods of this thesis will be explained. In this thesis, a scaled-down model of the single diagonal arch bridge was first designed and verified on its static resistance by hand calculations and FEM analysis. A prototype model was then built to analyze the dynamic performance by numerical software regarding both natural vibration and resonance induced vibration. A scale factor analysis was further implemented based on the dynamic properties of the scaled-down model and the prototype model to gain confidence in predicting the bridge's actual behavior. Ultimately, parametric analyses were used to study the dynamic impacts of different bridge parts and verify efficient improvement methods on the prototype model's dynamic performance.

2.1 Design of the scaled-down model

Figure 2.1 illustrates the flowchart for the design of the scaled-down model. In order to determine the dimension of the scaled-down model, a preliminary hand calculation of the primary bridge components was made to give an approximate result. Finite element analysis was further conducted to ensure the static capacity of the bridge was satisfied regarding Eurocode. The quality of the model was checked by comparing the total value with the analytical calculations, with the limitation between the difference under 1%.

In this step, different parameters, including the arch form and the rise-span ratio, were also tested from the statics' perspective to optimize the design.

Figure 2.1 The methodology for the design of the scaled-down model

2.1.1 Geometry and material properties

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connect the cross beams and limit the arch's movement. Figure 2.2 and Figure 2.3 presents the geometry of the scaled-down model.

Figure 2.2 Front view of the scaled-down model, length in mm

Figure 2.3 Plan view of the scaled-down model, length in mm

The arch was designed to be made of Glulam GL30c, and structural timber C24 was used for the bridge deck and cross beams. The cables and rods were made of steel S235. The detailed material properties are presented in Table 2.1 and Table 2.2 below.

Table 2.1 Material properties for timber

Property Glulam GL30c Structural timber C24

Bending strength parallel to grain 𝒇𝒎,𝒌 [MPa] 30 24

Compression strength parallel to grain 𝒇𝒄,𝟎,𝒌 [MPa] 24.5 21

Tension strength parallel to grain 𝒇_{𝒕,𝟎,𝒌} [MPa] 19.5 14.5

Shear strength 𝒇𝒗,𝒌 [MPa] 3.5 4

Fifth percentile value of modulus of elasticity 𝑬𝟎,𝟎𝟓[MPa] 10800 7400 Elastic modulus parallel to grain 𝑬𝟎,𝒎𝒆𝒂𝒏 [MPa] 13000 11000 Elastic modulus perpendicular to grain𝑬𝟗𝟎,𝒎𝒆𝒂𝒏 [MPa] 300 370

Shear modulus 𝑮𝒎𝒆𝒂𝒏 [MPa] 650 690

Poisson's ratio 𝝂 0 0

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Table 2.2 Material properties for steel S235

Property Steel S235

Elastic modulus 𝑬𝒌 [GPa] 150/210

Yield strength 𝒇𝒚 [MPa] 235

Poisson's ratio 𝝂 0.3

Density 𝝆 [kg/m3_] ₇₈₀₀

It should be noticed that for the cable hangers, the elastic modulus was set as 150GPa to give a more realistic simulation of the sagging phenomenon, while 210GPa was used for the elastic modulus of other steel rods.

2.1.2 Preliminary design of the scaled-down model

In the preliminary design, the arch shape was selected as a parabolic line for simplification. The rise of the arch 𝑓_{𝑎𝑟𝑐ℎ}was set according to a rise-span ratio of 0.2 and span of the arch 𝐿𝑎𝑟𝑐ℎ = 8800𝑚𝑚.

𝑓𝑎𝑟𝑐ℎ = 0.2 ∙ 𝐿𝑎𝑟𝑐ℎ = 0.2 ∙ 8800 = 1760𝑚𝑚

The cross-section profiles and dimensions for each structural component were first chosen under engineering experience and manufacture standards, as shown in Table 2.3. These dimensions could be adjusted through further static analysis.

Table 2.3 Preliminary design of structural components in the scaled-down model

Structural

component Profile sketch Dimension[mm] Material

Arch beam Width: 190

Thickness: 225 Glulam GL30c Crossbeams Width: 90 Thickness: 90 Structural timber C24 Deck Width: 780 Thickness: 70 Structural timber C24

Cable Hangers Diameter: 8 Steel 235

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8 Rods connected to cross beams Diameter: 12 Steel 235 (𝐸𝑘= 210𝐺𝑃𝑎) Rods connected between the arch ends Diameter: 16 Steel 235 (𝐸𝑘= 210𝐺𝑃𝑎)

2.1.3 Loads and section forces • Loads

In this study, only permanent loads and traffic loads were taken into consideration. Self-weight of the bridge and the pavement were included for the permanent loads. As for the traffic loads, the uniformly distributed load was calculated with the magnitude of 𝑞_𝑓𝑘 = 5 𝑘𝑁/𝑚2.

• Load cases

The variable loads were applied in the most unfavorable areas on the bridge deck to obtain the most critical and conservative section forces. The critical areas were determined according to the influence lines for different structural components.

As shown in Figure 2.4, 22 nodes on the arch were chosen to draw the influence lines. Each influence line was obtained with the moving concentrated load 1000N on the central line of the deck. Due to the central-symmetry of the structure, 11 cables and 11 rods connected to one side of the cross beams were studied for their influence lines, as shown in Figure 2.5 below.

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Figure 2.5 Selected cables(above) and rods(below) to draw the influence lines

Influence lines for the arch, cables and rods can be found in Appendix C. Influence lines and load cases. Based on each influence line, the most unfavorable live load’s position was decided as the load case.

The load case for the deck and crossbeams was assumed as the uniformly distributed load 𝑞_𝑓𝑘 = 5 𝑘𝑁/𝑚2 on the whole span.

Figure 2.6 Load case for the deck and crossbeams

• Section forces

Under each load case, the section forces were calculated and then combined to obtain the design value for the section forces in the ultimate state. Matlab scripts were developed to draw the envelope curves of the arch, as shown in Appendix G. Matlab scripts for the envelope curve. The combination of section forces followed the equations below from EN 1990(2002):

𝑬𝒅= 𝑬[∑ 𝜸𝒋 𝑮,𝒋𝑮𝒌,𝒋+ ∑ 𝜸𝒊 𝑸,𝒊𝝍𝟎,𝒊𝑸𝒌,𝒊] ( 2.1a )

𝑬𝒅= 𝑬[∑ 𝝃𝒋 𝒋𝜸𝑮,𝒋𝑮𝒌,𝒋+ 𝜸𝑸,𝟏𝑸𝒌,𝟏+ ∑𝒊>𝟏𝜸𝑸,𝒊𝝍𝟎,𝒊𝑸𝒌,𝒊] （ 2.1b）

where G refers to the permanent loads, and Q refers to the variable loads. The safety coefficients for the equations are summarized in Table 2.4. The reduction factor on the permanent loads ξ was used as 0.89 according to the Swedish amendments to Eurocode.

Table 2.4 Partial coefficients for load combination

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10 2.1.4 Resistance verification

Even though it is a scaled-down model with a span of merely 10m, it shall be verified as a practical bridge regarding the ultimate service limit. The resistances in the arch, cables, crossbeams, and deck were checked to fulfill the ultimate limit state requirements. The critical section forces were obtained under the specific load combination, as mentioned in Section 2.1.3. As for the serviceability limit state, the vertical displacement of the bridge was considered. The criterion for each component and serviceability will be described below.

• Arch

As the primary load-bearing component, the arch is subjected to a combination of axial compression and bending. The following equations were used to verify its resistance according to Eurocode 5 (2004): (𝝈𝒄,𝟎,𝒅 𝒇_{𝒄,𝟎,𝒅}) 𝟐 +𝝈𝒎,𝒚,𝒅 𝒇_{𝒎,𝒚,𝒅}+ 𝒌𝒎∙ 𝝈_{𝒎,𝒛,𝒅} 𝒇_{𝒎,𝒛,𝒅} ≤ 𝟏 ( 2.2a ) (𝝈𝒄,𝟎,𝒅 𝒇_{𝒄,𝟎,𝒅}) 𝟐 + 𝒌𝒎∙ 𝝈_{𝒎,𝒚,𝒅} 𝒇_{𝒎,𝒚,𝒅}+ 𝝈_{𝒎,𝒛,𝒅} 𝒇_{𝒎,𝒛,𝒅} ≤ 𝟏 ( 2.2b ) Where:

𝜎𝑐,0,𝑑 is the design compressive stress;

𝑓_𝑐,0,𝑑 is the design compressive strength;

𝑘_𝑚 is the reduction factor depending on the forms of the cross-sections; k_m= 0.7 is used for the rectangular sections;

𝜎_{𝑚,𝑦,𝑑} and 𝜎_{𝑚,𝑧,𝑑} are the design bending stress around the principle axes, the directions of axes are shown in Figure 2.7;

𝑓_{𝑚,𝑦,𝑑} and 𝑓_{𝑧,𝑦,𝑑} are the corresponding design bending strengths;

Figure 2.7 Defined directions of the axes

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11 ( 𝝈𝒎.𝒚,𝒅 𝒌_{𝒄𝒓𝒊𝒕}∙𝒌_𝒓∙𝒇_{𝒎.𝒚,𝒅}) 𝟐 + 𝝈𝒄,𝟎,𝒅 𝒌_𝒄,𝒛∙𝒇_{𝒄,𝟎,𝒅} ≤ 𝟏 ( 2.3a ) 𝝈𝒄,𝟎,𝒅 𝒌𝒄,z∙𝒇𝒄,𝟎,𝒅+ 𝑘𝑚∙ 𝝈_𝐦,y,d 𝒌𝒓∙𝒇𝒎,y,𝒅≤ 1 ( 2.3b )

For in-plane buckling:

𝝈_{𝒎,𝒚,𝒅} 𝒌𝒓∙𝒇𝒎,𝒚,𝒅+

𝝈_{𝒄,𝟎,𝒅}

𝒌𝒄,𝒚∙𝒇𝒄,𝟎,𝒅≤ 𝟏 ( 2.4 )

Where 𝑘_{𝑐𝑟𝑖𝑡} is a reduction factor due to the lateral buckling, 𝑘_{𝑐𝑟𝑖𝑡} = 1 is used due to a small value of the relative slenderness for bending;

𝑘_𝑟 is a reduction factor for flexural strength, calculated by: 𝑘_𝑟 = 0.76 + 0.001𝑟𝑚

𝑡=0.90

Where the inner radius of the arch 𝑟_𝑚 = 6380𝑚𝑚 and the thickness of the lamination 𝑡 = 45𝑚𝑚.

𝑘_𝑐,𝑦 and 𝑘_c,z are the reduction factors for buckling around the corresponding axis, calculated as follows: 𝑘_𝑐,𝑦 = 1 𝑘𝑦+√𝑘𝑦2−𝜆𝑟𝑒𝑙,𝑦2 and 𝑘_𝑐,𝑧 = 1 𝑘𝑧+√𝑘𝑧2−𝜆𝑟𝑒𝑙,𝑧2 𝑘_𝑦 = 0.5(1 + 𝛽_𝑐(𝜆_{𝑟𝑒𝑙,𝑦}− 0.3) + 𝜆_{𝑟𝑒𝑙,𝑦}2 ) 𝑘𝑧 = 0.5(1 + 𝛽𝑐(𝜆𝑟𝑒𝑙,𝑧− 0.3) + 𝜆𝑟𝑒𝑙,𝑧2 )

Where 𝛽𝑐 is the factor for members within the straightness limits, 𝛽𝑐 = 0.1 for

glulam timber.

𝜆𝑟𝑒𝑙,𝑦 and 𝜆𝑟𝑒𝑙,𝑧 refer to the relative slenderness for bending, calculated by the

following equations: 𝜆𝑟𝑒𝑙,𝑦 = 𝜆𝑦 𝜋 √ 𝑓𝑐,0,𝑘 𝐸0.05 𝜆_{𝑟𝑒𝑙,𝑧} =𝜆𝑧 𝜋 √ 𝑓𝑐,0,𝑘 𝐸0.05

Where 𝐸_0.05 is the fifth percentile value of modulus of elasticity parallel to grain; 𝜆_𝑦 and 𝜆_𝑧 are the slenderness ratio around the corresponding axes, taken as

𝜆_𝑦 = 𝑙_{𝑒𝑓,𝑦}√𝐴_𝐼𝑡𝑜𝑡

𝑦 𝜆𝑧 = 𝑙𝑒𝑓,𝑧√

𝐴𝑡𝑜𝑡

𝐼𝑧

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𝑙_𝑒𝑓 refers to the effective length, 𝑙_𝑒𝑓 = 𝛽 ∙ 𝑠, taking a buckling length factor β into account. For the in-plane bending, 𝛽 = √𝜋2∙8𝑓𝑙

40∙𝑠2, the parameters are presented in

the figure below. For the out-of-plane bending, a suggestion value of β = 0.7 is used.

Figure 2.8 Sketch of a half three-hinged arch

• Cables and rods

Only axial tension occurs in steel cables and rods, and their ULS resistances were checked by the following equation.

𝑵𝑬𝒅≤

𝑨∙𝒇𝒚

𝜸_𝑴𝟎 ( 2.5 )

Where γ_M0 is the partial factor for resistance, γ_M0 = 1 is used; 𝑁_Ed is the design tension force;

𝐴 is the cross-section’s area; 𝑓𝑦 is the yield strength.

• Crossbeams

Crossbeams are also verified as timber cross-sections subjected to combined stresses. For the cross-section subjected to bending and axial compression, the equations 2.2a and 2.2b are used. For the cross-section subjected to bending and axial tension, the following equations are checked: 𝝈𝒕,𝟎,𝒅 𝒇𝒕,𝟎,𝒅+ 𝝈_{𝒎,𝒚,𝒅} 𝒇𝒎,𝒚,𝒅+ 𝒌𝒎∙ 𝝈𝒎,𝒛,𝒅 𝒇𝒎,𝒛,𝒅≤ 𝟏 ( 2.6a ) 𝝈_{𝒕,𝟎,𝒅} 𝒇𝒕,𝟎,𝒅+ 𝒌𝒎∙ 𝝈_{𝒎,𝒚,𝒅} 𝒇𝒎,𝒚,𝒅+ 𝝈_{𝒎,𝒛,𝒅} 𝒇𝒎,𝒛,𝒅≤ 𝟏 ( 2.6b ) Where:

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13 𝑓_𝑡,0,𝑑 is the design tension strength;

Additionally, the shear stresses for the ends of cross beams are checked through the following equation:

𝝉𝒅 ≤ 𝒇_𝒗,𝒅 ( 2.7 )

Where 𝜏_𝑑 is the design shear stress;

𝑓_𝑣,𝑑 is the design shear strength for the actual condition. • Deck

Eurocode 5 is also used to verify the deck's strength, assuming that the deck plate is subjected to combined stresses. For the combination of bending stress and axial compression stress, the equations 2.2a and 2.2b are used; for the combination of bending stress and axial tension stress, the equations 2.6a and 2.6b are used.

• Serviceability limit state

No specified limits for the vertical deformation of footbridges are stipulated in the Eurocode. However, there is a limitation as 1/400 of the theoretical span length in the Swedish publication Krav Brobyggande (Trafikverket, 2011). According to this instruction, the limit value for the vertical deformation is:

1/400 ∙ 10 = 0.025𝑚 = 25𝑚𝑚 ( 2.8 )

2.1.5 Hand calculations

In the hand calculations, the main bridge parts, including the arch, deck plate, crossbeams, and the cables were verified separately. Some assumptions and simplifications were made throughout the hand calculations, and the corresponding difference in verification will be described in the following subsections.

• Deck plate

The deck plate between two crossbeams was simplified as a simply supported beam subjected to the uniformly distributed load 𝑞 = 𝑞_𝑓𝑘 ∙ (𝑊𝑖𝑑𝑡ℎ 𝑜𝑓 𝑑𝑒𝑐𝑘) = 3.9𝑘𝑁/𝑚, as shown in the figure below.

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The bending stress was checked by the equation below:

𝝈𝒎,𝒚,𝒅≤ 𝒇𝒎,𝒚,𝒅 ( 2.9 )

• Crossbeams

The crossbeam is also simplified as a simply supported beam subjected to the uniformly distributed load containing the pedestrian load and the deck’s self-weight within the length equals to the distance between two crossbeams, as shown in Figure 2.10. The shear stress and bending stress are checked by Equation 2.7 and 2.9, respectively.

Figure 2.10 Sketch for the studied crossbeam’s area and its simplification as a simply supported beam

• Arch

The arch is assumed to be loaded directly under the uniformly distributed load along the whole span, as shown in Figure 2.11 Simplified loads on the archbelow. Both the pedestrian loads and the self-weight of the deck are included.

Figure 2.11 Simplified loads on the arch

Since the parabolic line is selected for the arch shape, the cross-section of the arch under uniformly distributed loads merely subjects to compression. The following equation is therefore used for verification:

𝝈_{𝒄,𝟎,𝒅}≤ 𝒇_{𝒄,𝟎,𝒅} ( 2.10 )

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15

In the simple hand calculations, a cable was assumed to bear the loads applied on each side of the crossbeam. For each cable, the length of bearing loads is equal to half of the distance between the two cables. Figure 2.12 presents a simplification sketch of the cable system.

Figure 2.12 Simplification of the cable system (q: distributed load; Q: concentrated load)

Therefore the cable’s axial tension Fn can be derived as the equation below, and verified by

Equation 2.5.

𝑭_𝒏 = (𝒒 ∙𝒔

𝟐+ 𝑸)/ 𝒔𝒊𝒏(𝜶𝒏) (𝒏 = 𝟏, 𝟐, 𝟑 … ) ( 2.11 )

For each cable, the distributed load q contains the pedestrian load and the deck’s self-weight; half of the self-weight of the crossbeam is regarded as the vertical concentrated load.

2.1.6 Parametric analysis

To further study the static impacts of the design parameters and optimize the design, the parametric analysis was implemented based on the preliminary design model. The number of hinges, and rise-span ratio, were discussed.

• Number of hinges

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Figure 2.13 Arch forms for two-hinge arch(above) and three-hinge arch(below)

In the numerical simulation, the crown’s hinge for the three-hinge arch form was implemented by releasing the rotational moment 𝑀_𝑦. The local member axes are shown in Figure 2.14 below. The section forces and envelope curve of the arch were compared.

Figure 2.14 Definition of the local member axes on the crown’s hinge

• Rise-span ratio

The handbook Design of Timber Structures suggests the rise-span ratio of timber bridge to be 0.15 for the reasonable balance between the lateral stability and the horizontal force. However, among the existing diagonal arch bridges, the rise-span ratios appear to be much higher than 0.15. The rise-span ratio of Hulme Bridge is 0.5 (Hussain, N, et al., 1999), and the Tongtai bridge is 0.3451 (Wang, H., & Qin, S., 2014).

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Table 2.5 Rise values for different rise-span ratios

Rise-span ratio Rise 𝒇𝒂𝒓𝒄𝒉 [m]

0.15 1.32

0.2(the preliminary design model) 1.76

0.25 2.20

2.2 FE modeling

The numerical results based on the finite element method present an outcome through the matrix displacement method for thousands of elements. It utilizes the substantial computing power of the computer and provides sufficient accuracy. In this section, the procedure of numerical modeling will be explained. The numerical model was built in FEM software Abaqus following these steps:

1) The model's grid was first built according to the overall geometry, as shown in the figure below.

Figure 2.15 The grid sketch of the model in Abaqus

2) The profiles, materials and member types were assigned to the segments. 3) The parts were assembled into a holistic bridge model.

4) For different study purposes, a specific step was created.

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Figure 2.16 The location of the nodal supports Table 2.6 Definition of nodal supports

Support No.

Axis system

Support Rotational restraint

Ux Uz Uy 𝝋𝒙 𝝋𝒛 𝝋𝒚 1,5 Global system √ √ √ 1’,5’ √ √ 2,4 √ √ 2’,4’ √ 3,3’ √ 6 User-defined system √ √ √ √ √ 6’ √ √ √ √

6) Objective loads were activated in the corresponding steps. Static analysis was performed in the step ‘Static, linear perturbation’, and the step ‘Frequency’ was created for the dynamic analysis.

7) The shell element, the beam element, and the tension element were defined as linear element(S4R), linear beam element(B31), and truss element(T3D2), respectively. For the scaled-down model, 0.1m was selected as the global mesh size. The mesh size of the arch was increased to 0.45m in the prototype model.

8) The job was submitted for analysis, and the solutions were obtained for investigation. For the purpose of model checking, another numerical software RFEM was also used to develop the same model for comparison.

2.3 Prototype model used for dynamic evaluation and parametric

analysis

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19

preliminary design for further tests. However, the experiments were unfortunately delayed due to Covid-19. Thus, the thesis's strategy altered, and the bridge with actual size (hereinafter called the prototype model) was analyzed regarding its dynamic properties by numerical approaches. Following dynamic studies and parametric analyses were based on this prototype model.

The prototype model is obtained by a linear magnification of the scaled-down model's geometry by 4.5 times, as shown in Figure 2.17. Accordingly, the length and width of the bridge deck are 45m and 3.51m, respectively. Table 2.7 summarizes the cross-sections and materials used in the prototype model: the materials keep the same as the scaled-down model, and the dimension of the cross-sections linearly enlarges 4.5 times to the preliminary design of the scaled-down model.

Figure 2.17 Geometry of the prototype model (Front view and plan view), length in m Table 2.7 Cross-sections and materials used in the prototype model

Structural

component Cross-section profile Dimension[mm] Material

Arch beam Width: 855

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Crossbeams Width: 405

Thickness: 405 Structural timber C24

Cable Hangers Diameter: 36 Steel 235

(𝐸𝑘= 150𝐺𝑃𝑎) Rods connected to crossbeams Diameter: 54 Steel 235 (𝐸𝑘= 210𝐺𝑃𝑎) Rods connected

between the arch ends

Diameter: 72 Steel 235 (𝐸𝑘= 210𝐺𝑃𝑎)

Deck Rectangle Width: 3510

Thickness: 315 Structural timber C24

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2.4 Dynamic evaluation

The dynamic performance of footbridges should be evaluated to understand its vibration behavior under the resonance and further investigate responding improvement methods. In this section, the dynamic evaluation method will be clarified. The methodology mainly follows the instructions from the Footbridge: Assessment of vibration behavior of footbridges under

pedestrian loading, 2006 (hereinafter referred to as Sétra), as shown in Figure 2.18 below.

The class of the bridge was first defined according to the specific traffic condition and comfort requirements. With the calculated natural frequencies, the bridge's frequency ranges were defined and lead to the selection of the dynamic load cases. These load cases were activated to simulate the pedestrian load contributing to the resonance, and the amplitudes of the structure’s acceleration were extracted. Then the comfort level was assessed regarding the maximum acceleration value and the comfort criteria.

Figure 2.18 Methodology chart for the dynamic analysis of the footbridge

2.4.1 Footbridge class

The footbridge class specifies the traffic conditions of the bridge regarding its location and usage. Since the bridge was intended to be built to reach the Alfred Nobel Museum adjacent to the Karlskoga City, Class Ⅱ was chosen for this footbridge, indicating an urban footbridge linking populated areas subjected to heavy traffic and that may occasionally be loaded throughout its bearing area.

2.4.2 Frequency ranges

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Figure 2.19 Frequency ranges (Hz) of the vertical and longitudinal vibrations (Sétra, 2006)

Figure 2.20 Frequency ranges (Hz) of the transverse horizontal vibrations (Sétra, 2006)

Range 1 refers to the maximum risk of resonance, and Range 2 refers to the medium risk of resonance. Low risk will occur within the frequency in Range 3, while the risk of resonance is negligible when the natural frequency is located in Range 4.

The natural frequencies of the prototype model can be seen in Table 3.11. According to the results, the natural frequencies with vertical vibrations (modes 1 and 3) can be classified into Range 2 and 3. In contrast, all the frequencies under horizontal modes are under Range 4. Therefore, the dynamic evaluation shall be further performed on the first and third vibration modes.

2.4.3 Dynamic load cases

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Figure 2.21 Selection of load cases

The formulas used to determine the dynamic loads in this research are summarized in Table 2.8.

Table 2.8 Dynamic loads for vertical vibrations

Direction case Load per m2

Vertical 1 𝑑 × (280𝑁) × 10.8 × (𝜉/𝑛)1/2_{× 𝜓 × cos 2𝜋𝑓}

𝑣𝑡 ( 2.12 )

Vertical 3 𝑑 × (70𝑁) × 10.8 × (𝜉/𝑛)1/2_{× 𝜓 × cos 2𝜋𝑓}

𝑣𝑡 ( 2.13 )

where 𝑑: Density of the crowd, which is 0.8 person/m2 for both load cases.

𝑓_𝑣: The natural frequency to induce the resonance of the structural. In this report, the first frequency is used in the formula for case 1 and the third frequency is used in the formula for case 3. It should be noticed that considering the pedestrian will vibrate with the bridge, the mass of the pedestrian shall be included in the total mass of the bridge. The natural frequencies with the adding pedestrian are used here. With the assumption of an average weight 70kg per person, the pedestrian mass is calculated as 0.8*70=56kg/m2

𝑡: The time variable for the periodical load

𝜉: The critical damping ratio, which is 1.5% according to the suggestions from Sétra 𝑛: The numbers of pedestrians on the footbridge, calculated by 𝑛 = 𝑑 ∙ 𝑆 = 126

Where 𝑆 refers to the area of the bridge deck, which is calculated as 157.95 m2

(Width*Span) for the prototype model

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Figure 2.22 Factor ψ in the case of walking, for vertical and longitudinal vibrations, first harmonic on the left, and second harmonic on the right (Sétra, 2006)

Substituting the corresponding values above, the amplitude of the dynamic loads are transformed into the following formulas:

Table 2.9 The amplitudes of the dynamic loads for vertical vibration

Direction case Load per N/m2

Vertical 1 𝜓 × 26.396 Vertical 3 𝜓 × 6.599

These dynamic loads are applied to the whole area of the deck plate as the excited force of the bridge, while the direction of the loads follows the bridge's mode shapes. The application of dynamic loads can be seen in Figure 3.8 and Figure 3.9. The excited force was applied to the deck in 30 second to observe the sufficient convergence of the accelerations

The maximum acceleration of the bridge 𝑎𝑚𝑎𝑥under each dynamic load case could be obtained

through numerical solutions and used to assess the dynamic performance of the bridge. 2.4.4 Comfort criterion

According to CEN – EN 1990, a limitation of the bridge deck’s maximum acceleration is given as 0.7 m/s2. Sétra also presents instructions for the comfort criteria, as shown in Table 2.10 below. The maximum comfort is the highest comfort level described as the acceleration undergone by the structure is practically imperceptible to the users. The mean comfort is defined as the acceleration undergone by the structure that could be merely perceptible to the users. Given that this bridge was intended to be built in front of a museum, there would be some sensitive users like schoolchildren and the elderly. A maximum comfort level should be accessed in this bridge.

Table 2.10 Comfort criteria considered in this research (for vertical vibration)

Acceleration range Comfort criteria

𝑎𝑚𝑎𝑥< 0.5 m/s2 Max. comfort

0.5 m/s2_{< 𝑎}

𝑚𝑎𝑥< 1 m/s2 Mean. comfort

1 m/s2_{< 𝑎}

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25 2.4.5 Hand calculation on dynamics

Apart from the numerical solutions, some simplified hand calculations were also performed to calculate the bridge’s accelerations under the first vibration mode and achieve a better holistic understanding of this unique asymmetric structure. Two theoretical methods were used for the hand calculations.

• Method 1

In this method, the bridge deck was regarded as a simply supported beam. The maximum acceleration was calculated following the equation below, given in Appendix 5 of Sétra.

𝒂 = 𝟏

𝟐⋅𝝃⋅ 𝟒⋅𝑭_{𝒍𝒊𝒏𝒆𝒂𝒓}

𝝆𝑺𝝅 ( 2.14 )

Where:

𝐹𝑙𝑖𝑛𝑒𝑎𝑟 is the linear load applied on the simply supported beam, unit in N/m, calculated

by the amplitude of the dynamic load times the width of the deck. The dynamic load was calculated following the equation given in Table 2.8. It should be noticed that the reduction factor 𝜓 in that equation depends on the frequency of the bridge, here the first frequency obtained from Abaqus is used.

𝜌𝑆 is the linear density, unit in kg/m, including the mass of the bridge and the pedestrian.

ξ is the damping ratio, ξ = 0.015 is also used here. • Method 2

The second method was introduced in Chopra, A. K. (1995), based on the bridge's mode shape. For the purpose of simplification, the displacement of the deck’s central line was caught through the numerical results in Abaqus. Then it was normalized to get the mode shape as a function of the length as y(x), as shown in Figure 2.23 below.

Figure 2.23 First mode shape of the deck, along the central line

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According to the mode shape, the modal force 𝐹_{𝑚𝑜𝑑𝑎𝑙}, modal mass 𝑀_{𝑚𝑜𝑑𝑎𝑙} and the acceleration 𝑎 under the first mode are calculated following the equations below:

𝑭𝒎𝒐𝒅𝒆𝒍= ∫ |𝒚(𝒙)|𝒅𝒙 𝒍 𝟎 ⋅ 𝑭𝒍𝒊𝒏𝒆𝒂𝒓 ( 2.15) 𝑴𝒎𝒐𝒅𝒆𝒍 = ∫ (𝒚(𝒙))𝟐𝒅𝒙 𝒍 𝟎 ⋅ 𝝆𝑺 ( 2.16 ) 𝒂 = 𝟏 𝟐⋅𝝃⋅ 𝑭𝒎𝒐𝒅 𝒆𝒍 𝑴𝒎𝒐𝒅 𝒆𝒍 ( 2.17 )

The linear force 𝐹𝑙𝑖𝑛𝑒𝑎𝑟 and linear density 𝜌𝑆 share the same value as in method 1, calculations

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2.5 Scale factor analysis

For the purpose of having a more accurate deduction from the tested results to the actual bridge, scale factors of some variables were studied to act as a link between the full-scale and the scaled-down model. Due to the omit of experiments, the scale factors were studied based on both numerical models.

2.5.1 Theoretical values for scale factors

The scale factor for the quantity i is defined as: 𝑆_𝑖 = 𝑖𝑝

𝑖𝑚, where the subscripts p and m refer to

the prototype and the scaled-down model, respectively.

The theoretical scale factors of different parameters are derived under the Buckingham’s Pi Theorem(Buckingham, 1914), which suggests that any dimensionally homogeneous equation consists of certain physical quantities (Equation 2.18) can be transformed into an equivalent formula consisting of dimensionless variables (Equation 2.19).

For some physical quantity of interest:

𝑭(𝑿𝟏, 𝑿𝟐, ⋯ , 𝑿𝒏) = 𝟎 ( 2.18 )

can be equivalently expressed in the form:

𝑮(𝝅𝟏, 𝝅𝟐, ⋯ , 𝝅𝒎) = 𝟎 ( 2.19 )

Where the 𝜋 terms are dimensionless products of the n physical variables (𝑋1, 𝑋2, ⋯ , 𝑋𝑛), and

m=n-r, where r is the number of fundamental dimensions. In this thesis, three fundamental

dimensions are Length(L), Force(F), and Time(T). The units of the remaining quantities under study could therefore be described by these fundamental dimensions. The quantities considered in this research and their units are summarized in Table 2.11 below.

Table 2.11 Physical variables and their units

Physical variable Unit Force, Q F Pressure, q FL-2 Acceleration, a LT-2 Linear dimension, l L Frequency, ω T-1 Modulus, E FL-2 Mass density, ρ FL-4_T2 Mass, m FL-1_T2

According to the similarity principle, all the dimensionless terms are the same in both the prototype and the scaled-down model: 𝜋_im= 𝜋_ip, where 𝜋_im is 𝜋_i in the scaled-down model and 𝜋ip is 𝜋i in the prototype (i=1,2,…m). Based on this theory and its derivation, the scale

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detailed derivation procedure can be found in Appendix H. Derivation of the theoretical scale factor

As mentioned in 2.3, the prototype model is is obtained by a linear magnification of the geometry of the scaled-down model by 4.5 times, giving the scale factor for linear dimension 𝑆𝑙 = 4.5. Since the same materials are used for the prototype model and the scaled-down model,

the scale factors for modulus 𝑆_𝐸 and density 𝑆_𝜌 are 1. 2.5.2 Dynamic load for the scaled-down model

In this thesis, scale factors for parameters including mass, frequency and acceleration, are studied and compared with the theoretical values. Therefore, these properties are obtained from Abaqus with regards to both the prototype model and the scaled-down model to calculate their actual scale factors.

Particularly, the scale factor for the responding acceleration is derived under the first mode. For the prototype model, the accelerations acquired under the dynamic load described in 2.4.3 were used. According to the derived results in Table 3.18 , the theoretical scale factor for the pressure loading is 1. Thus to obtain the relevant acceleration in the scaled-down model, the dynamic pressure load keeps the same as the dynamic pressure for the prototype, calculated by Equation 2.12.

2.6 Modification and parametric analysis

According to the dynamic evaluation results in Section 3.3, the full-sacle bridge presents a minimum comfort level and cannot fulfill the requirements, drawing a necessary to improve its dynamic performance.

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Figure 2.24 The methodology of the improvement of the dynamic behavior

Accordingly, several bridge parts and modified approaches were studied regarding their impacts on the natural frequencies and the dynamic response to give suggestions to the actual design, with a detailed explanation in the following subsections. Parametric analyses were carried out to identify these impacts.

All the parametric analyses are based on the original prototype model, as described in Section 2.3. For every single parametric analysis, one particular parameter was selected as the only input variable to the system. The input parameter is set to a list of values, while the other parts are kept the same.

The frequencies and maximum accelerations were extracted as the primary criterion for the parametric analyses. The mass of pedestrian 56kg/m2 was applied along the whole deck. Two load cases were applied to each parametric analysis: the first harmonic of the bridge and the second harmonic of the bridge. Since all these modifications do not significantly change the natural frequencies and the mode shapes, the load’s direction follows the original prototype model, as shown in Figure 3.8 and Figure 3.9. The magnitude of the dynamic load is determined by the calculated frequencies, following Equation 2.12 and 2.13

2.6.1 Extra masses of the pavement

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30 2.6.2 Cross-section of the arch

The arch is the primary load-bearing component of the system. The result in 3.2.2 indicates that the arch significantly contributes to the lower vibration modes’ performance, particularly the out-of-plane bending for the first mode. Thus increasing the arch’s lateral stiffness was assumed to increase the frequencies and modify the vibration response. Three cross-sections were studied to analyze this impact, as shown in the table below. The same material (GL30c) of the arch was used for these three cases.

Table 2.12 Arch’s cross-sections used in the parametric analysis and for the original model(first column), unit in mm

Cross-sections Width*Height [mm*mm]

855*1012.5 1012.5*855 1250*855 1012.5*1012.5

2.6.3 Modification on cables • Stiffness

Both the elastic modulus and the cross-section’s area influence cables' stiffness, and these two parameters are studied separately.

In the preliminary design of the prototype model, the elastic modulus of cables was set as 150GPa to simulate a reduction due to the cables’ sagging phenomena. In the parametric analysis, the elastic modulus is increased to 210GPa to observe the variation trend. In reality construction, this increase can be reached by replacing the cable with bars or inserting prestress on cables. All the cables’ cross-sections dimensions were kept the same as the original prototype model.

The diameter with the range of 30mm to 50mm was studied regarding the cross-section of cables, while the elastic modulus kept as 150GPa.

• Arrangement

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A theoretical proportion of the cable area on both sides of the same crossbeam was then derived. The idea is to obtain an equal vertical stiffness of cables. The cable system is regarded as spring, as shown in Figure 2.25. The cables’ vertical stiffness can be calculated following Equation 2.20 below:

Figure 2.25 The cable system working was spring

Figure 2.26 Force mechanism in the cable-stayed beam system

( 2.20 )

Where h – The vertical height of the cable

𝛼– The inclination between cables and the horizontal plane

E– The elastic modulus of cable A– The cross-section area of cable

A relation between the area of cables on both sides can therefore be derived by setting 𝒌_𝒊= 𝒌_𝒋, as shown in the equation below2.21. The calculation results can be seen in Table 2.13.

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Table 2.13 Calculation on the relation of cable dimensions

Cable No. i Corresponding cable No.j 𝜶𝒊 [°] 𝒉𝒊 [mm] 𝑨𝒊/𝑨𝒋 𝑫𝒊/𝑫𝒋 1 11 22 2651.9 0.2 0.44 2 10 54 3912.2 0.05 0.22 3 9 70 4583.7 0.04 0.21 4 8 53 5335 0.11 0.34 5 7 42 5897.9 0.37 0.61 6 6 34 6417.1 1 1 7 5 30 6894.5 2.7 1.64 8 4 26 7384.4 8.82 2.97 9 3 23 7631.4 23.03 4.8 10 2 21 7904.7 21.42 4.63 11 1 19 8082.5 5.08 2.25

Note: Cable No.1 refers to the shortest cable, and No.11 refers to the longest cable, as shown in Figure 2.5 Due to the bridge's unique geometry, the cables on both sides of the crossbeam adjacent to the supports present more deviation on their heights and inclinations. Moreover, from the calculation results, this deviation leads to a more imbalance in the dimension of cables: under the same load P shown in Figure 2.26, on both sides of one crossbeams, the longer cable’s diameter needs to be even four times of the shorter one to reach an equivalence in the vertical stiffness. Regarding the static resistance of cables, however, this theoretical proportion in Table 2.13 is unrealistic to achieve. A more suitable and simplified method is to set the cables into three groups (Cable No. 1-4 for Group 1, Cable No.5-7 for Group 2, and Cable No. 8-11 for group 3) and assigned with different diameters. Two cases were performed as follows to study the effects of an appropriate assignment of cables:

Table 2.14 Assigned diameters for cables in different groups, unit in mm

Group 1 Group 3 Group 3

Case 1(50-30-25) 25 30 50

Case 2(50-35-25) 25 35 50

• Extra cables

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Figure 2.27 The layout of the extra cables (green lines) on the prototype model

2.6.4 Local enhancements

According to the natural vibration results in 3.2.1, the deck presents vibration combined torsion and bending under the first mode. Due to the torsion trend, the most unfavorable part can be found at the edge of the deck around the quarter-point along the longitudinal direction.

Regarding this unique mode shape and the location of the most unfavorable part of the deck, the local enhancement was proceeded to figure out whether a more comfortable situation could be reached. To be more specific, some support rods were added for local enhancement at the most unfavorable part under the deck, as shown in Figure 2.28. The locations of these rods are illustrated in Figure 2.29. Steel S235 was used for the material of these rods, and the cross-section area was set as 0.005m2. In Abaqus, these rods were simulated as truss elements.

Figure 2.28 The sketch of the support steel rods, length in mm

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34 2.6.5 Substituted longitudinal beams

In order to increase the overall stiffness and restrain the deck's torsion trend, the longitudinal rods connected to the crossbeams were replaced by beams. Three box cross-sections and two H cross-sections are studied for the stiffening beams, as shown in Table 2.15. Among all the cases, steel S235 was used as the material of these beams; the material properties are shown in Table 2.2.

Table 2.15 Cross-sections of the rods connected to crossbeams and substituted stiffening beams

Describtion Profile

Original rods (Diameter=54mm)

400*200 T10

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It should be noticed that an adjustment on the deck was made in this sensitivity analysis: since the steel framework can prove a higher stiffness, there are more possible choices of the deck forms. In this case, the strip wood was adapted to construct the upper deck. Between every strip, a small interval was set, as illustrated in Figure 2.30 Plan view of the prototype model, with the strip woods for the deck. This kind of deck provides convenience for replacing several strips in a small part once damages occurred. Accordingly, the deck's width is changed to 0.7 times of the original deck, as 2.45m in the FEM model, to get a more accurate simulation of this deck form, as shown in Figure 2.31.

500*300 T10

HEB 400

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Figure 2.30 Plan view of the prototype model, with the strip woods for the deck

Figure 2.31 Plan view of the prototype model used in FEM modeling, with the deck’s width of 2.45m

2.6.6 Combined improvements

Above describes every modification parts individually. In order to understand the general effects in combination with different modifications, some modification methods were applied at the same time:

• Case 1: Longitudinal beams + lager elastic modulus of cables

Based on the model with modified longitudinal beams mentioned in 2.6.5, the elastic modulus of cables was further increased to 210 GPa. Three box cross-sections for the longitudinal beams were used: 400*200 T10, 400*300 T10, and 500*300 T10. The details of these cross-sections canbe found inTable 2.15 Cross-sections of the rods connected to crossbeams and substituted stiffening beams.

• Case 2: Longitudinal beams + steel crossbeams

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3 Results and discussions

In this section, the static results of the scaled-down model based on the preliminary design will be illustrated. Both simple hand calculations and FEM models present a satisfactory outcome of the resistance verification. Also, some design parameters, including the arch shape, arch type and the rise-span ratio, will be discussed regarding their impacts on the mechanics.

3.1 Resistance verification and design proposals of the

scaled-down model

3.1.1 Hand calculations

The utilization for the main bridge parts through hand calculations are shown in Table 3.1 below. A more detailed calculation procedure can be found in Appendix B.

Table 3.1 Utilization for primary bridge parts by hand calculations

Type Utilization

Arch 0.406

Deck 0.105

Cable 0.802

Crossbeam 0.546

All the cross-section utilizations are less than 1, indicating sufficient strength. These results promote promising for the preliminary design. However, the assumptions and simplifications under hand calculations lead to a reduction of accuracy. The following sections present the results from the numerical solutions.

3.1.2 Model checking

The comparison between the total mass obtained from numerical models and from hand calculation is given in Table 3.2 below. More detailed hand calculations on the total mass can be found in Appendix A. Hand calculation on the total mass of the scaled-down model. The deviation between the numerical result and the analytical result is merely 1%, proving that the numerical models are sufficiently accurate for further analysis.

Table 3.2 Model checking on the total mass

Type Total mass Difference

Abaqus model 514.02 kg 0.628%

RFEM model 521.80 kg 0.876%

Analytical calculation 517.27 kg -

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The influence lines for 22 nodes on the arch and the corresponding load cases can be seen in Appendix C. Influence lines and load casesFigure 3.1,Figure 3.2, and Figure 3.3 present the envelope curves for the internal forces along the arch span. Along the arch span, the position and corresponding load case that contribute to the peak value for section forces and moments were extracted and combined according to Equation 2.1a and 2.1b. The design section forces for the arch are given in Table 3.3 below.

Figure 3.1 Design envelope for in-plane bending moment My along the arch

Figure 3.2 Design envelope for out-of-plane bending moment Mz along the arch

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Figure 3.3 Design envelope for axial force N along the arch

It can be noticed that unlike the normal arch structure where maximum axial force occurs at the supports, the maximum axial force is observed around the arch’s crown for this bridge. This can be inferred as the inclined cables transmit some horizontal forces to the arch.

Moreover, the most unfavorable in-plane bending moment is much larger than that of the out-of-plane bending moment for both cases. It proves the rationality for choosing a cross-section with a slightly larger dimension on the thickness than the width.

Table 3.3 The design section forces of the arch

𝑵𝑬𝒅[𝑵] 𝑴𝒚.𝑬𝒅 [𝑵 ∙ 𝒎] 𝑴𝒛.𝑬𝒅 [𝑵 ∙ 𝒎] 𝑵𝑬𝒅.𝒎𝒂𝒙 -67158.5 25.68 -94.79 𝑵𝑬𝒅.𝒎𝒊𝒏 -3500.4 338.43 -57.47 𝑴𝒚.𝑬𝒅.𝒎𝒂𝒙 -32693.2 8218.75 1552.53 𝑴𝒚.𝑬𝒅.𝒎𝒊𝒏 -21221.1 -628.46 3079.03 𝑴𝒛.𝑬𝒅.𝒎𝒂𝒙 -26087.2 -50.41 4926.60 𝑴𝒛.𝑬𝒅.𝒎𝒊𝒏 -29324.8 -21.33 -4565.84

Note: My refers to the in-plane bending moment, and Mz refers to the out-of-plane bending moment, as

shown in Figure 2.7.

Each design section force was verified according to Eq 2.2-2.4, and the utilizations are summarized in Table 3.4. A more detailed verification procedure can be found in Appendix D. Resistance verification of the scaled-down model by numerical results. The result indicates that the maximum utilization ratio of the cross-section arises when the in-plane bending moment reaches its maximum value. The arch's highest resistance utilization ratio is merely 0.5, far less than 1, the requirement of resistance for the arch is satisfied.

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The verification of stability also gives a satisfactory result: For each design load case, the stability utilization ratio is around 0.6, less than 1. The design load cases were kept the same, as mentioned in Table 3.3.

Table 3.4 The utilization of the arch regarding the resistance and stability

Resistance Out-of-plane buckling In-plane buckling

Eq 2.2a Eq 2.2b Eq 2.3a Eq 2.3b Eq 2.4 𝑵𝑬𝒅.𝒎𝒂𝒙 0.031 0.032 0.595 0.596 0.302 𝑵𝑬𝒅.𝒎𝒊𝒏 0.020 0.016 0.031 0.045 0.035 𝑴𝒚.𝑬𝒅.𝒎𝒂𝒙 0.500 0.401 0.515 0.622 0.621 𝑴𝒚.𝑬𝒅.𝒎𝒊𝒏 0.168 0.215 0.189 0.213 0.131 𝑴𝒛.𝑬𝒅.𝒎𝒂𝒙 0.219 0.309 0.231 0.233 0.120 𝑴𝒛.𝑬𝒅.𝒎𝒊𝒏 0.203 0.287 0.260 0.261 0.133

3.1.4 Resistance verification of cables and rods

The axial force influence lines for cables and rods and the corresponding load cases can be found in Appendix C. Influence lines and load cases The design section stresses and utilization ratio are summarized in Table 3.5 below. The verification procedure in more detail can be found in Appendix D. Resistance verification of the scaled-down model by numerical results. According to the results, all the cables and rods are sufficient for the capacity.

Table 3.5 Internal stresses and utilization of cables and rods

Type Design stress[MPa] Utilization

Cables 149.05 0.634

Rods connected to cross beams 97.8 0.416

Rods connected to both ends of the arch 78.2 0.333

3.1.5 Resistance verification of crossbeams and the deck • Crossbeams

The design loads for crossbeams are shown in Table 3.6 and Table 3.7 below. All the utilization is smaller than 1, indicating that the cross-sections for crossbeams have sufficient strength for design loads.

Table 3.6 Design loads and the utilization of crossbeams

𝑵𝑬𝒅[𝑵] 𝑴𝒚.𝑬𝒅 [𝑵 ∙ 𝒎] 𝑴𝒛.𝑬𝒅 [𝑵 ∙ 𝒎] Utilization

𝑵𝑬𝒅.𝒎𝒊𝒏 -447.1 132.2 59.7 0.155

𝑵𝑬𝒅.𝒎𝒂𝒙 844.4 434.5 84.7 0.459

Table 3.7 Design shear force and utilization of crossbeams

𝑽𝒅.𝒎𝒂𝒙[kN] Utilization