# Formalising the R of Reduce in a Circular Economy Oriented Design Methodology for Pedestrian and Cycling Bridges

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}emissions and economic costs- for a certain type of structural member. Hence, these optimisation tools allow to improve the sustainability of structural components, as they result in lower material consumption. However, they are not suitable for the optimisation of complete structural typologies.

## 2. Conceptual Design According to Morphological Indicators

- W the indicator of volume
- σ the allowable stress of the constituent material of the structure [N/m²]
- V the total volume of the structure [m³]
- F the total force acting upon the structure [N]
- L the span of the structure [m]
- l
_{i}/L the ratio of the length of member i to the total length of the structure - k
_{i}the portion of the load F present in member i

- Ψ the indicator of buckling
- L the span
- q a form factor defining the cross section of the bars
- E the modulus of elasticity of the material
- σ the allowable stress to which at least one section is dimensioned
- F the total resultant force
- µ the proportion of the buckling length of the compression bars to their geometrical length, depending on the connection type

_{i}and member forces f

_{i}need to be expressed as a function of the overall slenderness L/H. Latteur [31,38] illustrates in his work that with a given Ψ, a value L/H exists with a minimal value of W. This means the volume indicator is not only dependent on the slenderness L/H anymore but is also dependent on the buckling indicator Ψ. Latteur [38] shows that for every structure with a total length L, composed of a material which is fully stressed σ, loaded with a resultant force F working on elements i with a length l

_{i}and following the Rankine buckling curve, W becomes:

_{i,}in the members i will be different. For more information about the theory behind the calculation of these portions of loads, one can consult the scientific literature on MI. The portion of loads k

_{i}will be the same for an even and odd truss structure except for the middle panel, where the portion of loads in the diagonals will be 0 for a Warren truss with an odd number of panels.

- ∆ the indicator of displacement
- E the modulus of elasticity of the used material [N/mm²]
- δ the maximum displacement of the structure [mm]
- σ the allowable stress of the material used for the structure [N/mm²]
- L the span of the structure [mm]
- l
_{i}/L the ratio of the length of member i to the total length of the structure - n
_{i}the portion of the unitary force, applied in a node j, present in member i

_{i}, but also the portion of unitary loads n

_{i}need to be addressed. When ∆ is no longer solely dependent on L/H but also on Ψ, the equation for calculating the displacement indicator ∆ for a Warren truss becomes [33,34,38]:

- σ* (≈$\mathsf{\sigma}/1.40)$ is the stress level in the structure in SLS
- ${\mathrm{f}}_{\mathrm{i}}$ are the natural frequencies of the structure that must be adequately far from the excitation frequencies ${\mathrm{f}}_{\mathrm{e},\mathrm{i}}$ of the external load.

## 3. Design of the Software

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

3S | Strength, Stiffness, Stability |

4R’s | Reduce, Reuse, Recycle, Recover |

A | section area |

B | width of the bridge |

c_{cor} | correction factor |

CE | Circular Economy |

CHS | Circular Hollow Sections |

DfA | Design for Adaptability |

DfD | Design for Disassembly/Deconstruction |

E | modulus of elasticity |

f_{e,i} | excitation frequencies of the external load |

f_{i} | natural frequencies of the structure |

f_{1} | the first natural frequency of the structure |

F | total resultant Force |

FD* | co-vibrating load |

FEM | Finite Element Modelling |

g | gravitational acceleration |

GPa | GigaPascal |

H | Height of a truss |

i | the index of an element |

I | second moment of inertia |

j | the index of a node |

k | static stiffness |

k_{i} | portion of load F present in member i |

kN | kiloNewton |

l_{i} | length of member I [m] |

L | span of a truss |

m | meter |

mm | millimeter |

MI | Morphological Indicator(s) |

MPa | MegaPascal |

n | number of panels of a truss |

n_{i} | the portion of the unitary force, applied in node j, present in member i |

Pa | Pascal |

q | formfactor |

q_{fk} | standard uniform characteristic load for pedestrian found in ‘NBN EN 1991-2—part 2’ |

SLS | Serviceability Limit State |

ULS | Ultimate Limit State |

V | Volume |

V^{(T)} | Volume of the members in tension |

V^{(C)} | Volume of the members in compression |

W | volume indicator |

W^{(T)} | morphological indicator of volume of the members in tension |

W^{(C)} | morphological indicator of volume of the members in compression |

z* | ratio of the co-vibrating load in service limit state to the total load in ultimate limit state |

β | parameter for the stress level in the structure in the calculation of the displacement and first natural frequency indicators |

β_{3} | parameter to define the safety interval around the first natural frequency of the structure |

δ | the maximum displacement in the middle of the truss |

∆ | morphological indicator of displacement |

Θ | morphological indicator of the first natural frequency |

µ | proportion of the buckling length of compression bars over their geometrical length, depending on its connection type |

σ | allowable stress |

σ* | stress level in the structure |

$\Psi $ | morphological indicator of buckling |

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**Figure 7.**Warren truss, span L = 40 m, height H = 7 m, number of panels n = 5, optimal standard sections are indicated.

**Figure 8.**Howe truss, span L = 40 m, height H = 6 m, number of panels n = 6, optimal standard sections are indicated.

**Table 1.**Warren trusses (The most optimal result according to the software tool is indicated in bold, the results in Diamonds that deviate are highlighted in yellow).

Input | Output | Diamonds | |||||
---|---|---|---|---|---|---|---|

L [m] | H [m] | n | γ [°] | W | V [m³] | δ [mm] | V [m³] |

20 | 4 | 2 | 38.7 | 2.62 | 0.050 | 21.7 | 0.0793 |

3 | 50.2 | 2.29 | 0.044 | 18.1 | 0.0544 | ||

5 | 63.4 | 2.31 | 0.044 | 23.6 | 0.0547 | ||

5 | 2 | 45.0 | 2.50 | 0.048 | 18.7 | 0.075 | |

3 | 56.3 | 2.23 | 0.043 | 15.8 | 0.057 | ||

5 | 68.2 | 2.45 | 0.047 | 21.8 | 0.061 | ||

6 | 2 | 50.2 | 2.50 | 0.048 | 16.9 | 0.0740 | |

3 | 60.9 | 2.28 | 0.044 | 14.7 | 0.0575 | ||

5 | 71.6 | 2.71 | 0.052 | 21.2 | 0.0699 | ||

40 | 6 | 5 | 56.3 | 2.80 | 0.215 | 51.4 | 0.2473 |

7 | 64.5 | 2.77 | 0.212 | 59.5 | 0.2494 | ||

9 | 69.7 | 2.78 | 0.220 | 67.1 | 0.2566 | ||

7 | 3 | 46.4 | 2.93 | 0.224 | 37.1 | 0.3045 | |

5 | 60.3 | 2.86 | 0.212 | 46.8 | 0.2468 | ||

7 | 67.8 | 2.83 | 0.217 | 55.4 | 0.2560 | ||

8 | 3 | 50.2 | 2.84 | 0.218 | 33.8 | 0.2935 | |

5 | 63.4 | 2.80 | 0.214 | 43.9 | 0.2614 | ||

7 | 70.3 | 2.97 | 0.228 | 52.9 | 0.2639 | ||

60 | 7 | 7 | 58.5 | 3.14 | 0.541 | 99.8 | 0.5942 |

9 | 64.5 | 3.10 | 0.534 | 109.9 | 0.6174 | ||

11 | 68.7 | 3.15 | 0.542 | 119.4 | 0.5873 | ||

8.4 | 6 | 59.2 | 3.17 | 0.546 | 89.7 | 0.5940 | |

7 | 63.0 | 3.07 | 0.529 | 88.8 | 0.5778 | ||

9 | 68.4 | 3.12 | 0.538 | 99.5 | 0.6016 | ||

10 | 3 | 45.0 | 3.40 | 0.587 | 55.7 | 0.9225 | |

5 | 59.0 | 3.12 | 0.537 | 69.3 | 0.6163 | ||

7 | 66.8 | 3.13 | 0.540 | 81.3 | 0.5305 | ||

80 | 8 | 9 | 60.9 | 3.38 | 1.036 | 157.5 | 1.1215 |

11 | 65.6 | 3.36 | 1.030 | 169.5 | 1.0951 | ||

13 | 69.0 | 3.50 | 1.043 | 180.8 | 1.1287 | ||

9.6 | 7 | 59.2 | 3.36 | 1.029 | 126.5 | 1.1878 | |

9 | 65.2 | 3.31 | 1.014 | 139.7 | 1.0717 | ||

11 | 69.3 | 3.36 | 1.030 | 152.4 | 1.1463 | ||

11 | 5 | 54.0 | 3.48 | 1.066 | 101.6 | 1.2878 | |

7 | 62.5 | 3.32 | 1.018 | 116.1 | 1.1453 | ||

9 | 68.0 | 3.35 | 1.026 | 129.8 | 1.1432 |

**Table 2.**Howe trusses (The most optimal result according to the software tool is indicated in bold, the results in Diamonds that deviate are highlighted in yellow).

Input | Output | Diamonds | |||||
---|---|---|---|---|---|---|---|

L [m] | H [m] | n | γ [°] | W | V [m³] | δ [mm] | V [m³] |

20 | 2 | 8 | 38.7 | 3.04 | 0.058 | 45.6 | 0.0640 |

10 | 45.0 | 2.98 | 0.057 | 47.9 | 0.0658 | ||

12 | 50.2 | 2.98 | 0.057 | 50.2 | 0.0647 | ||

3 | 6 | 42.0 | 2.82 | 0.054 | 31.5 | 0.0634 | |

8 | 50.2 | 2.78 | 0.053 | 34.6 | 0.0624 | ||

10 | 56.3 | 2.85 | 0.055 | 37.9 | 0.0653 | ||

4 | 4 | 38.7 | 2.93 | 0.056 | 23.4 | 0.0678 | |

6 | 50.2 | 2.83 | 0.054 | 26.5 | 0.0637 | ||

8 | 58.0 | 2.95 | 0.056 | 30.5 | 0.0723 | ||

40 | 4 | 10 | 45.0 | 3.34 | 0.256 | 89.6 | 0.2889 |

12 | 50.2 | 3.32 | 0.254 | 94.5 | 0.2781 | ||

14 | 54.5 | 3.34 | 0.256 | 99.4 | 0.2984 | ||

5 | 8 | 45.0 | 3.28 | 0.251 | 71.8 | 0.2748 | |

10 | 51.3 | 3.25 | 0.249 | 77.4 | 0.2623 | ||

12 | 56.3 | 3.29 | 0.252 | 83.1 | 0.2669 | ||

6 | 6 | 42.0 | 3.37 | 0.259 | 58.4 | 0.2776 | |

8 | 50.2 | 3.27 | 0.251 | 64.1 | 0.2806 | ||

10 | 56.3 | 3.32 | 0.254 | 70.5 | 0.2860 | ||

60 | 5 | 12 | 45.0 | 3.73 | 0.643 | 155.2 | 0.7085 |

14 | 49.4 | 3.69 | 0.637 | 161.8 | 0.6974 | ||

16 | 53.1 | 3.70 | 0.638 | 168.2 | 0.7141 | ||

6.6 | 10 | 47.7 | 3.58 | 0.617 | 120.6 | 0.6568 | |

12 | 52.9 | 3.56 | 0.614 | 128.4 | 0.6607 | ||

14 | 57.0 | 3.61 | 0.622 | 136.2 | 0.6975 | ||

8 | 8 | 46.8 | 3.63 | 0.626 | 98.9 | 0.6669 | |

10 | 53.1 | 3.60 | 0.620 | 107.3 | 0.6844 | ||

12 | 58.0 | 3.66 | 0.631 | 116.1 | 0.6826 | ||

80 | 7 | 12 | 46.4 | 3.90 | 1.194 | 193.9 | 1.2908 |

14 | 50.8 | 3.85 | 1.181 | 203.0 | 1.3113 | ||

16 | 54.5 | 3.86 | 1.183 | 211.9 | 1.3302 | ||

8.5 | 10 | 46.7 | 3.86 | 1.181 | 159.8 | 1.2701 | |

12 | 51.9 | 3.81 | 1.166 | 169.8 | 1.2658 | ||

14 | 56.1 | 3.83 | 1.173 | 179.9 | 1.3017 | ||

10 | 8 | 45.0 | 3.94 | 1.208 | 133.5 | 1.3066 | |

10 | 51.3 | 3.85 | 1.180 | 144.0 | 1.2865 | ||

12 | 56.3 | 3.87 | 1.187 | 155.1 | 1.3014 |

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**MDPI and ACS Style**

Anastasiades, K.; Lambrechts, T.; Mennes, J.; Audenaert, A.; Blom, J. Formalising the R of Reduce in a Circular Economy Oriented Design Methodology for Pedestrian and Cycling Bridges. *J* **2022**, *5*, 35-51.
https://doi.org/10.3390/j5010003

**AMA Style**

Anastasiades K, Lambrechts T, Mennes J, Audenaert A, Blom J. Formalising the R of Reduce in a Circular Economy Oriented Design Methodology for Pedestrian and Cycling Bridges. *J*. 2022; 5(1):35-51.
https://doi.org/10.3390/j5010003

**Chicago/Turabian Style**

Anastasiades, Kostas, Thijs Lambrechts, Jaan Mennes, Amaryllis Audenaert, and Johan Blom. 2022. "Formalising the R of Reduce in a Circular Economy Oriented Design Methodology for Pedestrian and Cycling Bridges" *J* 5, no. 1: 35-51.
https://doi.org/10.3390/j5010003