# Life-Cycle Cost Assessment Using the Power Spectral Density Function in a Coastal Concrete Bridge

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Goal and Scope Definition

#### 2.1.1. Model Description

^{3}of concrete [51].

#### 2.1.2. Functional Unit

_{cm}equal to 40 MPa, and a modulus of elasticity E

_{c}equal to 29 GPa. The concrete mix for the bridge has a water-to-cement ratio (w/c) of 0.45 and contains 485.6 kg/m

^{3}of cement.

#### 2.2. Service Life Prediction

#### 2.2.1. Prediction Techniques for Chloride Degradation in RC Rebars

^{2}/years). However, the chloride transport time ($t$) can be calculated using the inverse of Equation (2), as given in Equation (3) [56,58].

#### 2.2.2. Damage Detection using the PSD Method

#### 2.3. Calculation Procedure for LCCA

_{f}, time t

_{k}follows a uniform interval distribution throughout [0, ${t}_{v}$]. The estimated annual percentage of the damage of bridge elements under hazard effects can be written as ($N\left({t}_{v}\right)={\lambda}_{f}\times {t}_{v}$) [37,68].

## 3. Description of the Numerical Model

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**The service life of rebars in an RC structure according to the model by Tuutti [54].

**Figure 6.**The summary of analysis and calculation processes performed on the numerical model of the Arosa bridge.

Properties (Units) | Water (L/m^{3}) | Cement (kg/m^{3}) | Gravel (kg/m^{3}) | Sand (kg/m^{3}) | f_{cm} (MPa) | E_{c} (GPa) | w/c (%) |
---|---|---|---|---|---|---|---|

Amount | 218.5 | 485.6 | 926.7 | 827.9 | 40 | 29 | 0.45 |

Marine Exposure Class | ${\mathbf{C}}_{\mathbf{s}}$ (% of Concrete Weight) | ${\mathbf{V}}_{\mathbf{c}\mathbf{o}\mathbf{r}\mathbf{r}}$ (μm/year) | ${\mathbf{D}}_{\mathbf{0}}$ (×10 ^{−12} m^{2}/s) | ${\mathit{C}}_{\mathit{t}\mathit{h}}$ (%) | $\mathit{d}$ (m) | |
---|---|---|---|---|---|---|

Deck | Piers | |||||

Aerial (IIIa) | 0.14 | 20 | 10.0 | 0.6 | 0.035 | 0.045 |

Submerged (IIIb) | 0.72 | 4 | ||||

In tidal zone (IIIc) | 0.50 | 50 |

**Table 3.**Repair activity cost (EUR) according to Spain data code (2022) [65] for the concrete and coastal Arosa bridge.

No. | Maintenance and Repair Activities | Description/Includes | Price per Unit (EUR) | Unit |
---|---|---|---|---|

1 | Demolition of the unreinforced concrete parts | Debris clearing, loading, and transport of demolished material to authorized manager up to a distance of 60 km | 30.20 | m^{3} |

2 | The reinforced concrete parts demolition | Debris clearing, loading, and transport of demolished material to authorized manager up to a distance of 60 km | 33.07 | m^{3} |

3 | Management, separation, and categorization of debris resulting from destruction | Loading and transport of non-hazardous construction (except earth and stones) consisting of bricks, tiles, and ceramic materials or a mixture of these | 7.78 | ton |

4 | Install and subsequently remove formwork for repair concrete activity | Executed with double-folded, wood cleaning, wetting, application of release agent, and use of complementary elements for its stability and adequate execution | 33.91 | m^{2} |

5 | Repair and replace damaged bar steel with corrugated bar steel with improved ductility characteristics | Including cutting and bending, placement of overlaps, breakout, and binding annealed wire and separators | 1.81 | kg |

6 | Prepare and pour concrete | Concrete activities in the elevation of piles, stirrups, headboards, beams, deck boards, slabs, walls, and frames | 121.35 | m^{3} |

**Table 4.**Identified location, time, and amount of damage, and conventional service life of the Arosa bridge.

Section No. | Location | Service Life and Repair Period (for Damage More Than 20%) | ||||
---|---|---|---|---|---|---|

Damage-Prediction Methods | ||||||

Area No. | The Zone between Points | PSD | Conventional | |||

Damage (%) | Repair Time (year) | Damage (%) | Repair Time (years) | |||

S1 (Deck) | A1 | p1, p2, p6, p7 | - | 100 | 26.67 | 6 |

A2 | p2, p3, p7, p8 | 20.04 | 65 | |||

A3 | p3, p4, p8, p9 | 20.04 | 65 | |||

A4 | p4, p5, p9, p10 | - | 100 | |||

A5 | p1, p6, p56, p58 | 20.05 | 32 | |||

A6 | p56, p57, p58, p59 | 20.16 | 27 | |||

A7 | p5, p10, p57, p59 | 20.05 | 32 | |||

S2 (Deck) | A8 | p6, p7, p11, p12 | 20.04 | 65 | 26.67 | 6 |

A9 | p7, p8, p12, p13 | 20.07 | 51 | |||

A10 | p8, p9, p13, p14 | 20.07 | 51 | |||

A11 | p9, p10, p14, p15 | 20.04 | 65 | |||

A12 | p6, p11, p58, p60 | 20.15 | 42 | |||

A13 | p58, p59, p60, p61 | 20.06 | 36 | |||

A14 | p10, p15, p59, p61 | 20.15 | 42 | |||

S3 (Deck) | A15 | p11, p12, p16, p17 | 20.05 | 32 | 26.67 | 6 |

A16 | p12, p13, p17, p18 | 20.22 | 30 | |||

A17 | p13, p14, p18, p19 | 20.22 | 30 | |||

A18 | p14, p15, p19, p20 | 20.05 | 32 | |||

A19 | p11, p16, p60, p62 | 20.16 | 27 | |||

A20 | p60, p61, p62, p63 | 20.20 | 22 | |||

A21 | p15, p20, p61, p63 | 20.16 | 27 | |||

S4 (Deck) | A22 | p16, p17, p21, p22 | 20.22 | 19 | 26.67 | 6 |

A23 | p17, p18, p22, p23 | 20.95 | 19 | |||

A24 | p18, p19, p23, p24 | 20.95 | 19 | |||

A25 | p19, p20, p24, p25 | 20.22 | 19 | |||

A26 | p16, p21, p62, p64 | 20.50 | 18 | |||

A27 | p62, p63, p64, p65 | 21.14 | 18 | |||

A28 | p20, p25, p63, p65 | 20.50 | 18 | |||

S5 (Deck) | A29 | p21, p22, p26, p27 | 21.77 | 18 | 26.67 | 6 |

A30 | p22, p23, p27, p28 | 22.41 | 18 | |||

A31 | p23, p24, p28, p29 | 22.41 | 18 | |||

A32 | p24, p25, p29, p30 | 21.77 | 18 | |||

A33 | p21, p26, p64, p66 | 21.77 | 18 | |||

A34 | p64, p65, p66, p67 | 22.41 | 18 | |||

A35 | p25, p30, p65, p67 | 21.77 | 18 | |||

S6 (Deck) | A36 | p26, p27, p31, p32 | 21.77 | 18 | 26.67 | 6 |

A37 | p27, p28, p32, p33 | 22.41 | 18 | |||

A38 | p28, p29, p33, p34 | 22.41 | 18 | |||

A39 | p29, p30, p34, p35 | 21.77 | 18 | |||

A40 | p26, p31, p66, p68 | 20.50 | 18 | |||

A41 | p66, p67, p68, p69 | 21.14 | 18 | |||

A42 | p30, p35, p67, p69 | 20.50 | 18 | |||

S7 (Deck) | A43 | p31, p32, p36, p37 | 20.16 | 27 | 26.67 | 6 |

A44 | p32, p33, p37, p38 | 20.20 | 22 | |||

A45 | p33, p34, p38, p39 | 20.20 | 22 | |||

A46 | p34, p35, p39, p40 | 20.16 | 27 | |||

A47 | p31, p36, p68, p70 | 20.05 | 32 | |||

A48 | p68, p69, p70, p71 | 20.22 | 30 | |||

A49 | p35, p40, p69, p71 | 20.05 | 32 | |||

S8 (Deck) | A50 | p36, p37, p41, p42 | 20.15 | 42 | 26.67 | 6 |

A51 | p37, p38, p42, p43 | 20.06 | 36 | |||

A52 | p38, p39, p43, p44 | 20.06 | 36 | |||

A53 | p39, p40, p44, p45 | 20.15 | 42 | |||

A54 | p36, p41, p70, p72 | 20.05 | 84 | |||

A55 | p70, p71, p72, p73 | 20.07 | 51 | |||

A56 | p40, p45, p71, p73 | 20.05 | 84 | |||

S9 (Deck) | A57 | p41, p42, p46, p47 | 20.06 | 36 | 26.67 | 6 |

A58 | p42, p43, p47, p48 | 20.22 | 30 | |||

A59 | p43, p44, p48, p49 | 20.22 | 30 | |||

A60 | p44, p45, p49, p50 | 20.06 | 36 | |||

A61 | p41, p46, p72, p74 | - | 100 | |||

A62 | p72, p73, p74, p75 | 20.99 | 24 | |||

A63 | p45, p50, p73, p75 | - | 100 | |||

S10 (Deck) | A64 | p46, p47, p51, p52 | - | 100 | 26.67 | 6 |

A65 | p47, p48, p52, p53 | 20.07 | 51 | |||

A66 | p48, p49, p53, p54 | 20.07 | 51 | |||

A67 | p49, p50, p54, p55 | - | 100 | |||

A68 | p46, p81, p74, p76 | 20.50 | 32 | |||

A69 | p74, p75, p76, p77 | 20.16 | 27 | |||

A70 | p50, p55, p75, p77 | 20.50 | 32 | |||

S11 (Column20) | A71 | p78, p79, p82, p83, p106, p107, p110, p111 | 21.1 | 14 | 20 | 17 |

A72 | p79, p80, p83, p84, p107, p108, p111, p112 | 21.1 | 14 | |||

A73 | p80, p81, p84, p85, p108, p109, p112, p113 | 21.1 | 14 | |||

S12 (Column20) | A74 | p82, p83, p86, p87, p110, p111, p114, p115 | 24.23 | 15 | 50 | 14 |

A75 | p83, p84, p87, p88, p111, p112, p115, p116 | 24.23 | 15 | |||

A76 | p84, p85, p88, p89, p112, p113, p116, p117 | 24.23 | 15 | |||

S13 (Column20) | A77 | p86, p87, p90, p91, p114, p115, p118, p119 | 26.63 | 16 | 20 | 18 |

A78 | p87, p88, p91, p92, p115, p116, p119, p120 | 26.63 | 16 | |||

A79 | p88, p89, p92, p93, p116, p117, p120, p121 | 26.63 | 16 | |||

S14 (Column20) | A80 | p90, p91, p94, p95, p118, p119, p122, p123 | 21.48 | 17 | 20 | 18 |

A81 | p91, p92, p95, p96, p119, p120, p123, p124 | 21.48 | 17 | |||

A82 | p92, p93, p96, p97, p120, p121, p124, p125 | 21.48 | 17 | |||

S15 (Column20) | A83 | p94, p95, p98, p99, p122, p123, p126, p127 | 26.45 | 19 | 20 | 18 |

A84 | p95, p96, p99, p100, p123, p124, p127, p128 | 26.45 | 19 | |||

A85 | p96, p97, p100, p101, p124, p125, p128, p129 | 26.45 | 19 | |||

S16 (Column20) | A86 | p98, p99, p102, p103, p126, p127, p130, p131 | 22.11 | 19 | 20 | 18 |

A87 | p99, p100, p103, p104, p127, p128, p131, p132 | 22.11 | 19 | |||

A88 | p100, p101, p104, p105, p128, p129, p132, p133 | 22.11 | 19 | |||

S17 (Column21) | A89 | p134, p135, p138, p139, p162, p163, p166, p167 | 21.1 | 14 | 20 | 17 |

A90 | p135, p136, p139, p140, p163, p164, p167, p168 | 21.1 | 14 | |||

A91 | p136, p137, p140, p141, p164, p165, p168, p169 | 21.1 | 14 | |||

S18 (Column21) | A92 | p138, p139, p142, p143, p166, p167, p170, p171 | 24.23 | 15 | 50 | 14 |

A93 | p139, p140, p143, p144, p167, p168, p171, p172 | 24.23 | 15 | |||

A94 | p140, p141, p144, p145, p168, p169, p172, p173 | 24.23 | 15 | |||

S19 (Column21) | A95 | p142, p143, p146, p147, p170, p171, p174, p175 | 26.63 | 16 | 20 | 18 |

A96 | p143, p144, p147, p148, p171, p172, p175, p176 | 26.63 | 16 | |||

A97 | p144, p145, p148, p149, p172, p173, p176, p177 | 26.63 | 16 | |||

S20 (Column21) | A98 | p146, p147, p150, p151, p174, p175, p178, p179 | 21.48 | 17 | 20 | 18 |

A99 | p147, p148, p151, p152, p175, p176, p179, p180 | 21.48 | 17 | |||

A100 | p148, p149, p152, p153, p176, p177, p180, p181 | 21.48 | 17 | |||

S21 (Column21) | A101 | p150, p151, p154, p155, p178, p179, p182, p183 | 26.45 | 19 | 20 | 18 |

A102 | p151, p152, p155, p156, p179, p180, p183, p184 | 26.45 | 19 | |||

A103 | p152, p153, p156, p157, p180, p181, p184, p185 | 26.45 | 19 | |||

S22 (Column21) | A104 | p154, p155, p158, p159, p182, p183, p186, p187 | 22.11 | 19 | 20 | 18 |

A105 | p155, p156, p159, p160, p183, p184, p187, p188 | 22.11 | 19 | |||

A106 | p156, p157, p160, p161, p184, p185, p188, p189 | 22.11 | 19 |

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## Share and Cite

**MDPI and ACS Style**

Hadizadeh-Bazaz, M.; Navarro, I.J.; Yepes, V.
Life-Cycle Cost Assessment Using the Power Spectral Density Function in a Coastal Concrete Bridge. *J. Mar. Sci. Eng.* **2023**, *11*, 433.
https://doi.org/10.3390/jmse11020433

**AMA Style**

Hadizadeh-Bazaz M, Navarro IJ, Yepes V.
Life-Cycle Cost Assessment Using the Power Spectral Density Function in a Coastal Concrete Bridge. *Journal of Marine Science and Engineering*. 2023; 11(2):433.
https://doi.org/10.3390/jmse11020433

**Chicago/Turabian Style**

Hadizadeh-Bazaz, Mehrdad, Ignacio J. Navarro, and Víctor Yepes.
2023. "Life-Cycle Cost Assessment Using the Power Spectral Density Function in a Coastal Concrete Bridge" *Journal of Marine Science and Engineering* 11, no. 2: 433.
https://doi.org/10.3390/jmse11020433