# Embodied Energy Optimization of Buttressed Earth-Retaining Walls with Hybrid Simulated Annealing

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{2}emissions [22,24], or embodied energy [29]. EE optimization has been applied to concrete structures [25,30,31], pre-stressed concrete bridges [32], and tall buildings [33], among other structures; although there are some studies on EE optimization, more research is required in earth-retaining walls.

## 2. Proposed Optimization Problem

_{i}and energy e

_{i}, which are the price and the energy linked with each construction unit (r), are multiplied by the unit’s measurements (m

_{i}) resulting from the optimization procedure. These equations must be minimized to satisfy the constraints problem by Equation (3). The embodied energy data take into account a cradle-to-gate analysis. This means that the energies consumed for every process consider the activities of extracting the raw materials, processing, manufacturing, and the transport of the materials to the construction site. Moreover, the cost takes into account the materials (concrete, reinforcing steel, and formwork) and other activities and elements required to evaluate the total cost of the construction. Data of prices and energy consumption in Table 1 have been collected by the BEDEC database of the Institute of Construction Technology of Catalonia [42]. Table 1 includes all prices and embodied energy for every construction unit. It is assumed that to produce reinforcing steel, the ratio of recycled scrap steel is approximately 40% and the manufacturing process is carried out by an electric arc furnace.

#### 2.1. Design Variables and Parameters

^{3}. The foundation soil maximum bearing capacity considered was 0.3 MPa [43].

_{t}), stem thickness (s

_{t}), toe (t

_{l}) and heel (h

_{l}) lengths, buttress thickness (b

_{t}), distance between them (b

_{d}), angle of buttresses (α), fill slope (β), and foundation depth (h). On the other hand, the variables related to the reinforcement position and amount are R

_{1}to R

_{12}(Figure 3).

_{1}to R

_{4}are related to the flexural bending of the stem, R

_{1}to R

_{3}resist the main bending moment, while R

_{4}acts as a bending reinforcement at the bottom of the stem. The reinforcement related to the resistance of the thermal effects and shrinkage are the longitudinal bars represented by R

_{5}. The buttress needs a longitudinal reinforcement that materializes with the R

_{6}reinforcement. R

_{7}and R

_{8}represent the reinforcement area of the bottom of the buttress. R

_{9}and R

_{11}are the bottom and upper footing reinforcement bars and R

_{12}is the shear reinforcement one. R

_{10}is the reinforcement that resists the longitudinal effects on the footing.

#### 2.2. Structural Analysis

_{b}). The top of the stem works as a cantilever, while the lower part is coerced by the embedding in the foundation at the base of the buttresses. Equations (4) and (5) give the value of the bending moments that appear in the middle section between buttresses:

_{1}represents the pressure in the contact zone of the stem with the footing, M

_{1}is the bending moment in this zone, and M

_{2}is the maximum bending moment produced in the stem slab. Equation (6) shows the value of the shear resistance at the connection of the stem to the footing if the value of the distance between buttresses is less than half the height:

#### 2.3. Optimization Algorithm

^{®}and run using Intel

^{®}Core

^{TM}i7-3820 CPU with 3.60 GHz. The computing time needed to run the algorithm and to obtain one solution was five minutes. In this study, we calculated nine walls for each height and objective function considered [43].

## 3. Results of the Parametric Study

^{2}− 299 H + 1834.1, with a correlation coefficient of R² = 0.9998. In the same way, the embodied energy values obtained by embodied energy optimization adjust to E = 176.15 H

^{2}− 1845.9 H + 8916.5, with a correlation coefficient of R² = 0.9992, as shown in Figure 6. The increase in the cost and the emissions related to the increase in the wall height are not linear and increase faster as the height grows.

_{s}= −0.1853 H

^{2}+ 6.5067 H − 4.755, with a R² = 0.9781 for the stem, and R

_{f}= −0.3959 H

^{2}+ 13.978 H − 52.406, with a R² = 0.9403 for the footing. The expression used to obtain the global reinforcement ratio for 1 m of wall is parabolic and adjusted to R = −0.2165 H

^{2}+ 8.502 H − 19.618, with a R² = 0.9818. All adjustments have a good correlation coefficient, assuring a good prediction of the amount of reinforcing steel per cubic meter of concrete. In addition, the volume of concrete is shown in Figure 10: the trend of adjustment is parabolic, as in the case for reinforcing steel, with the expression of concrete volume for the embodied energy optimization being V

_{cs}= 0.0433 H

^{2}− 0.3137 H + 2.2213, with a R² = 0.9984 for the stem, and V

_{cf}= 0.0881 H

^{2}− 1.294 H + 5.8902, with a R² = 0.9969 for the footing. The expression of the global volume of concrete adjusts to V

_{c}= 0.1313 H

^{2}− 1.6074 H + 8.11, with a R² = 0.9989. You can observe that the embodied energy optimization gives a higher value of concrete volume than the cost optimization in contrast with the reinforcing steel amount, where the embodied energy gives a lower amount. Both optimizations have resulted in a concrete compressive strength of 25 MPa (C25/30) and B500 steel grade for all cases studied. This is because the cost and energy consumed increase as the resistance of the concrete increases and, therefore, varying the geometry and modifying the quantities of materials gives better results than varying the characteristic resistance of the concrete.

_{t}) and in the buttress distance (b

_{d}), while the difference between the foundation geometry obtained by the two optimization objectives is negligible, giving both optimization objectives the same values. Figure 11 shows the values obtained at the stem thickness (s

_{t}). It can be observed that, from 6 to 11 m, the stem thickness obtained is 0.25 m for the two optimizations, but from 11 m of wall height upwards, the embodied energy optimization takes on a parabolic trend equal to s

_{t}= 0.0028 H

^{2}− 0.0607 H + 0.5809, with a R² = 0.9533. The cost optimization shows higher values of stem thickness from 9 m upwards compared with the embodied energy expression. The values obtained from the optimization take the lower limit imposed by the constructive facility of these types of structures; from this point, a greater thickness of wall is needed to resist the efforts. Figure 12 displays the results for the buttress distance, showing that the embodied energy optimization takes lower values for the buttress distances. A shorter distance allows a reduction in the flexural moments and, as a consequence, the need for reinforcing steel at the expense of increasing the concrete amount. This result shows how embodied energy optimization allows us to reduce the amount of reinforcing steel to reach a lower amount of total embodied energy. The expression of the linear trend obtained is b

_{d}= 0.0567 H + 2.4546, with a R² = 0.4435. Furthermore, the results obtained by this optimization have been compared with the Calavera recommendations [41] in Figure 13; as can be seen, the cost optimization is inside the area defined by the geometrical limits defined as the buttresses distance by Calavera (H/3 to H/2), while the energy optimization is only inside from 6 to 10 m.

_{l}) remains constant at 7 m and then it has a linear trend equal to t

_{l}= 0.2114 H − 1.3387, with a good correlation coefficient of R² = 0.9935. Something similar occurs to the heel length (h

_{l}), which remains constant at 9 m of wall height and from this point onwards, adjusts to a linear expression equal to h

_{l}= 0.67 H − 4.2098, with a R² = 0.9798. The results of the analysis show that the main difference between the cost and the embodied energy optimizations is produced in the geometry of the stem and buttresses, which generate a difference in the amount and distribution of materials.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Table 1.**Prices and embodied energy values of the construction units [42].

Unit | Energy (kW·h) | Cost (€) |
---|---|---|

Earth movement | ||

m^{3} of backfill | 78.32 | 14.12 |

m^{3} of backfill over the toe | 76.52 | 12.52 |

m^{3} of earth excavation | 44.65 | 10.56 |

Foundation | ||

kg of steel B400 | 10.39 | 1.11 |

kg of steel B500 | 10.39 | 1.13 |

m^{3} of concrete C25/30 | 413.28 | 90.74 |

m^{3} of concrete C30/37 | 439.51 | 99.44 |

m^{3} of concrete C35/45 | 457.05 | 119.48 |

m^{3} of concrete C40/50 | 477.98 | 122.5 |

m^{3} of concrete C45/55 | 484.49 | 125.45 |

m^{3} of concrete C50/60 | 492 | 127.93 |

m^{2} of cleaning concrete | 24.8 | 9.28 |

m^{2} of formwork | 6.56 | 23.83 |

Stem | ||

kg of steel B400 | 10.4 | 1.29 |

kg of steel B500 | 10.4 | 1.31 |

m^{3} of concrete C25/30 | 427.78 | 101.99 |

m^{3} of concrete C30/37 | 454.01 | 110.69 |

m^{3} of concrete C35/45 | 480.32 | 130.73 |

m^{3} of concrete C40/50 | 511.72 | 133.84 |

m^{3} of concrete C45/55 | 524.48 | 136.91 |

m^{3} of concrete C50/60 | 535.74 | 140.98 |

m^{2} of formwork | 86.57 | 53.26 |

Variable | Unit | Description | Step Size | Lower Bound | Upper Bound |
---|---|---|---|---|---|

f_{t} | cm | Footing thickness | 1 | H/14 | H/6 |

s_{t} | cm | Stem thickness | 1 | 25 | 224 |

tl | cm | Toe length | 1 | 20 | 819 |

h_{l} | cm | Heel length | 1 | 20 | 2019 |

b_{t} | cm | Buttress thickness | 2.5 | 25 | 172.5 |

b_{d} | cm | Distance between buttresses | 5 | 320 | 800 |

f_{ck} | MPa | Concrete compressive strength | 5 | 25 | 50 |

f_{yk} | MPa | Steel yield strength | 100 | 400 | 500 |

R_{1} to R_{10} (n) | Reinforcement number of bars | 1 | 2 | 17 | |

R_{11} to R_{12} (n) | Reinforcement number of bars | 1 | 4 | 10 | |

R_{1} to R_{12} (Ø) | mm | Reinforcement diameter | 6, 8, 10, 12, 16, 20, 25, 32 |

Parameter | Value | |
---|---|---|

Maximum bearing capacity | σ_{adm} | 0.3 MPa |

Fill slope | β | 0° |

Foundation depth | h | 2 m |

Uniform load on top of the fill | q | 10 kN/m^{2} |

Wall-fill friction angle | δ | 0° |

Base friction coefficient | μ | tg 30° |

Safety coefficient against sliding | γ_{fs} | 1.5 |

Safety coefficient against overturning | γ_{to} | 1.8 |

Load safety coefficient | γ_{G} | 1.35 |

Concrete safety coefficient | γ_{c} | 1.5 |

Steel safety coefficient | γ_{s} | 1.15 |

External ambient exposure | IIa |

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**MDPI and ACS Style**

Martínez-Muñoz, D.; Martí, J.V.; García, J.; Yepes, V.
Embodied Energy Optimization of Buttressed Earth-Retaining Walls with Hybrid Simulated Annealing. *Appl. Sci.* **2021**, *11*, 1800.
https://doi.org/10.3390/app11041800

**AMA Style**

Martínez-Muñoz D, Martí JV, García J, Yepes V.
Embodied Energy Optimization of Buttressed Earth-Retaining Walls with Hybrid Simulated Annealing. *Applied Sciences*. 2021; 11(4):1800.
https://doi.org/10.3390/app11041800

**Chicago/Turabian Style**

Martínez-Muñoz, David, José V. Martí, José García, and Víctor Yepes.
2021. "Embodied Energy Optimization of Buttressed Earth-Retaining Walls with Hybrid Simulated Annealing" *Applied Sciences* 11, no. 4: 1800.
https://doi.org/10.3390/app11041800