# Genetic Algorithm for Embodied Energy Optimisation of Steel-Concrete Composite Beams

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Optimising Steel-Concrete Composite Structures

#### 2.1. The Genetic Algorithm

- From input parameters, populations of candidate solutions are randomly generated;
- The performance of a candidate solution within the population are determined against defined fitness functions;
- Repetition; selection of pairs of parent solutions, random crossover to produce candidate solutions, and mutation of offspring solutions;
- Form a new population with these offspring solutions;
- Repeat this process until an optimal solution has been returned.

#### 2.2. Aims of this Study

- Minimisation of the universal beam (UB) section—Objective function 1
- Minimisation of depth of the concrete slab (d
_{slab})—Objective function 2 - Minimisation of overall weight of the composite beam—Objective function 3
- Maximisation of the span length of the composite beam—Objective function 4
- Minimisation of the deflection of the composite beam—Objective function 5

## 3. Methodology for Structural Design and Life Cycle Energy Assessment

#### 3.1. Structural Form

#### 3.2. Actions upon the Structure

^{2}. For the construction stage, permanent action g

_{k}is calculated as the sum of both the steel cross-section and the profiled steel decking. Variable action q

_{k}is the sum of the construction loading and the wet self-weight of the concrete slab. For the composite stage, permanent action is calculated as the sum of the steel cross-section, profiled steel sheeting, dry self-weight of the concrete slab, and an assumed loading for finishes. Variable action is taken as 2.5kN/m

^{2}for a general use office area above ground level [26]. The greatest values for both g

_{k}and q

_{k}are taken as governing and taken forward to calculating a combination of actions (F

_{d}) in accordance to Equations (6) and (10) from Eurocode 0 [27]. Partial factors of safety for the permanent action γ

_{g}is taken as 1.35, for variable action γ

_{q}is taken as 1.5 from the UK National Annexe to Eurocode – Basis of Structural Design BS EN 1990:2002+A1:2005 [28].

#### 3.3. Ultimate Limit State Verification

_{Ed}and shear force V

_{Ed}acting upon the structure, where:

_{pl,Rd}, where:

_{Rd}and degree of shear connection R

_{q}are calculated, where P

_{Rd}is:

_{q}is;

_{Rd}can be determined.

_{pl,Rd}considers the steel section only, and therefore is calculated in accordance with Eurocode 3: Design of Steel Structures BS EN 1993-1-1 [29], where:

#### 3.4. Serviceability Limit State Verification

- At the construction stage, the beam alone is assumed to have insufficient resistance to lateral-torsional buckling and will be fully propped, thus for this scenario, there is no deflection of the beam.
- The beam is assumed to be an internal beam; therefore, relative humidity is assumed as 50%.
- It is assumed that the cement used for the slab is normal hardening, thus class = N.

_{cs}where;

_{cd}is determined by:

_{1}:

_{2.1}:

_{2.2}:

_{2.3}:

#### 3.5. Quantification of Embodied Energy

_{i}of the structure will be quantified as per Equation (49) [32].

_{i}is the quantity of material (i), M

_{i}is the cradle to gate energy content of the material (i) per unity quantity, and E

_{c}is the energy used on-site for construction. As the form of the beam under assessment is not variable (i.e., a single simply supported composite beam), the energy consumption for construction is assumed to be constant, and therefore is omitted from the assessment. Similarly, energy consumption for the transport of materials to the site is assumed to be constant, and therefore is also omitted from assessment [32].

_{i}is to be calculated by the specific component geometries of the candidate designs. For simplification, quantified components are to be limited to the steel universal beam, steel shear connectors, profiled steel sheeting, reinforcing steel, and slab concrete. Supporting columns and connections are assumed to be constant for all candidate designs, and therefore can be omitted from the assessment.

_{J}of the structure, owing to the simplicity of the structure under analysis, it was reasonable to adopt energy per weight as the unit of quantification. It is anticipated that as this work progresses to more complex floor plate structures, it may be more appropriate to utilise more functional units for quantification.

## 4. MATLAB Scripts for Optimisation

#### 4.1. General MATLAB Script for Structural Design and Life Cycle Energy Assessment

_{d}in accordance to Equations (6.10) of Eurocode 0. F

_{d}is calculated applied to the overall floor area supported by the beam, as floor area is required as an input for later functions. A dedicated MATLAB function is utilised for this purpose. Design moment M

_{Ed}, and design shear V

_{Ed}, according to Equations (1) and (2) from Section 3.2 are also calculated in the part of the script.

#### 4.2. Implementing MATLAB Global Optimisation Toolbox GA

[x,fval] = ga(FitFcn,nvars,[],[],[],[],lb,ub,[],options); |

## 5. Optimisation of a Steel-Concrete Composite Beam

#### 5.1. Minimisation of the Universal Beam Section—Objective Function 1

_{Ed}, as well as an outputted, embodied energy content. This was done with the following components:

- A 305 × 102 × 25 universal beam with a span length of 6.0m, and bay spacing of 3.0m;
- A 130 mm deep C25/30 concrete slab cast upon;

_{Ed}output of 119.4 kNm, the moment capacity for full shear connection M

_{pl,Rd}output was calculated as 257.8 kNm. In accordance with Equation (8), the check value was 0.46, less than half the check value of 1.0, implying reduction of the UB is achievable.

function ha = ha_function(x, Npla, dslab, NcSLAB, hc) ha = ((2*(x*10^3))/Npla)-(2*dslab)+((Npla*hc)/NcSLAB); end % %Genetic Algorithm Script for Objective Function 1 - Minimise Universal %Beam Section. % clc, clear, clear all % %Define Parameters hc = 70; dslab = 130; Npla = 987.25; NcSLAB = 1487.5 %Define GA Components FitFcn = @(x)ha_function(x,Npla,dslab,NcSLAB,hc); nvars = 1; lb = 120; ub = 257.79; options = optimoptions(‘ga’,’Generations’,50,... ‘MaxStallGenerations’,Inf,’PlotFcn’,@gaplotbestf); [x,fval] =ga(FitFcn,nvars,[],[],[],[],lb,ub,[],options); x fval |

_{pl,Rd}was set as the variable (x), where other parameters were retained as constants. The GA program calls upon ha_function as the required fitness function. The lower bound for M

_{pl,Rd}was set to M

_{Ed}rounded to the nearest whole number, to constrain the GA to prevent it from determining a depth of beam that would fail ULS checks. The upper bound for M

_{pl,Rd}was set to the computed M

_{pl,Rd}of the initial candidate design. This was to provide a practical upper bound that would prevent a solution having a depth greater than the initial candidate design. With a single variable, nvars was set to 1. Finally, options were set to give a run of 50 generations with MATLAB default population sizes of 50. A stopping criterion of infinite generations (MaxStallGenerations) was also included to ensure convergence during test runs of the script. This option was included for completeness, however, was overridden by setting generations to 50. Finally, the best and mean outputs (fval) per generation were plotted against their respective generation (Figure 4) to visualise the convergence of the GA to a solution.

#### 5.2. Minimisation of Depth of the Concrete Slab—Objective Function 2

_{slab}the subject.

function dslab = dslab_function(x,Npla,NcSLAB,ha,hc) dslab=((x*10^3)/Npla)-(ha/2)+((Npla*hc)/(2*NcSLAB)); end %Genetic Algorithm Script for Objective Function 2 - Minimise depth of %concrete slab. % clc, clear, clear all % %Define Parameters hc = 70; ha = 308.7; Npla = 987.25; NcSLAB = 1487.5 %Define GA Components FitFcn = @(x)dslab_function(x,Npla,NcSLAB,ha,hc); nvars = 1; lb = 120; ub = 257.79; options = optimoptions(‘ga’,’Generations’,10,... ‘MaxStallGenerations’,Inf,’PlotFcn’,@gaplotbestf); [x,fval] =ga(FitFcn,nvars,[],[],[],[],lb,ub,[],options); x fval |

_{pl,Rd}was set as the variable (x), and the remaining parameters retained as constants. The GA program called upon dslab_function as the required fitness function. Lower and upper bounds for M

_{pl,Rd}were the same as for objective function 1 as the benchmark span and beam conditions from the initial candidate design were still valid. With a single variable, nvars was again set to 1. For this objective function, the MATLAB population size of 50 was retained. Initially the number of generations was kept at 50, however, as convergence occurred within 5 generations, this reduced to 10 to enable the convergence to be better graphically visualised (Figure 5).

_{slab}of −9.57. Numerically this follows Equation (51) accurately, however reaping a negative value is an unfeasible design. To determine a feasible solution, the shallowest slab depth in accordance with manufacturer information [34] of 110 mm was run along with the initial candidate design span length and UB section through the structural design script. This structure passed both ULS and SLS criteria and returned a total initial embodied energy of 22534.6 MJ, a 4.1% reduction of embodied energy compared to the initial candidate design. A breakdown of material contribution is included in Table 1.

#### 5.3. Minimisation of Overall Weight of the Composite Beam—Objective Function 3

#### 5.4. Maximisation of the Span Length of the Composite Beam—Objective Function 4

_{pl,Rd}of objective function 3 was calculated at 155.7 kNm, with a F

_{d}of 2076kN imposed on the entire floor area. Rearranging Equation (1) gave a theoretical span length of 7.832 m for a 203 × 102 × 23 UB with a 110 mm concrete slab over a bay spacing of 3.0 m. However, running these inputs through the structural design script, the design failed both the ULS and the SLS criteria. Manually cycling through sections to ensure these criteria were met returned a design with a 254 × 102 × 28 UB. This returned a total initial embodied energy content of 29,410.9 MJ, a 25.2% increase for a 30.4% increase in span, and a proportionally 5.4% increase in total initial embodied energy, assuming a 30.4% increase of 21,408.5 MJ = 27,916.1 MJ.

#### 5.5. Minimisation of the Deflection of the Composite Beam—Objective Function 5

_{total}was returned as 17.4 mm. Running the structural design and LCEA script with the next largest UB section found in the blue book [36] a 203 × 133 × 25, returned a deflection of 16.5 mm, however it also returned a total initial embodied energy content of 21,849.2 MJ, a 2.1% increase when compared to objective function 3. A breakdown of material contribution is included in Table 1.

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

GA | Genetic Algorithm |

UB | Universal Beam |

ULS | Ultimate Limit State |

SLS | Serviceability Limit State |

M_{Ed} | Design Bending Moment |

V_{Ed} | Design Shear Force |

g_{k} | Permanent Action |

q_{k} | Variable Action |

γ_{g} | Partial Factor of Safety for Permanent Actions |

γ_{q} | Partial Factor of Safety for Permanent Actions |

γ_{M0} | Partial Factor for Resistance–Structural Steel |

γ_{c} | Partial Factor for Resistance–Concrete |

γ_{s} | Partial Factor for Resistance–Reinforcing Steel |

γ_{v} | Partial Factor for Resistance–Shear Connectors |

F_{d} | Combined Actions |

h_{a} | Depth of Universal Beam |

b_{a} | Width of Universal Beam |

d | Depth Between Fillets |

t_{w} | Web Thickness |

t_{f} | Flange Thickness |

r | Radius of Root Fillet |

A_{a} | Area of Universal Beam |

W_{pl,y} | Universal Beam Plastic Modulus (y-y axis) |

I_{yy} | Universal Beam Second Moment of Area (y-y axis) |

I_{a} | Universal Beam Second Moment of Area (dominant axis) |

L | Beam Span |

S | Beam Spacing |

d_{slab} | Depth of Slab |

h_{c} | Height of Concrete Above Profile |

h_{p} | Height of Profiled Deck |

b_{1} | Width of Bottom Trough |

b_{2} | Width of Top Trough |

Ø | Nominal Diameter of Shear Connector |

h_{sc} | Height of Shear Connector prior to Welding |

F_{y} | Yield Strength of Structural Steel |

F_{u} | Ultimate Strength of Structural Steel |

F_{yk} | Yield Strength of Reinforcing Steel |

F_{ck} | Cylinder Strength of Concrete |

E_{cm} | Secant Modulus of Elasticity |

b_{eff} | Effective Width of the Compression Flange |

N_{c,slab} | Compression Resistance of the Concrete Slab |

N_{pla} | Tensile Resistance of the Steel Section |

M_{pl,Rd} | Moment Capacity for Full Shear Connection |

P_{Rd} | Design Shear Resistance of a Single Shear Connector |

k_{t} | Deck Shape Influence Factor |

M_{pl,a,Rd} | Plastic Moment Resistance of the Universal Beam |

M_{Rd} | Moment Capacity for Partial Shear Connection |

V_{pl,Rd} | Vertical Shear Resistance of the Composite Beam |

A_{v} | Area of Shear |

A_{sf} | Cross Sectional Area of Reinforcing Steel |

F_{yd} | Yield Strength of Reinforcing Steel |

ε_{cs} | Total Shrinkage Strain |

ε_{cd} | Drying Shrinkage Strain |

ε_{ca} | Autogenous Shrinkage Strain |

f_{cm(t)} | Minimum Concrete Strength for Time (t) |

RH | Relative Humidity |

E_{L} | Effective Modulus of Elasticity of Concrete |

E_{0} | Short Term Effective Modulus of Elasticity of Concrete |

E_{p} | Permanent Effective Modulus of Elasticity of Concrete |

E_{s} | Effective Modulus of Elasticity of Concrete for Shrinkage |

I_{c} | Second Moment of Area of Concrete Flange |

EI_{L} | Effective Flexural Stiffness of Concrete Flange |

EI_{0} | Short Term Effective Flexural Stiffness of Concrete Flange |

EI_{p} | Permanent Effective Flexural Stiffness of Concrete Flange |

EI_{s} | Effective Flexural Stiffness of Concrete Flange for Shrinkage |

Φ_{(t,t0)} | Creep Coefficient |

δ_{i} | ith Deflection Component |

δ_{total} | Total Deflection |

e_{d} | Combined Actions for Serviceability Limit State |

a_{c} | Distance Between Centroidal Axes of Concrete Flange and Universal Beam |

EE_{i} | Initial Embodied Energy Content of Steel-Concrete Composite Beam |

EE_{total} | Total Initial Embodied Energy Content of Steel-Concrete Composite Beam |

EE_{a} | Initial Embodied Energy Content of Universal Beam |

EE_{sc} | Initial Embodied Energy Content of Shear Connectors |

EE_{ps} | Initial Embodied Energy Content of Profiled Deck |

EE_{c} | Initial Embodied Energy Content of Concrete Slab |

EE_{r} | Initial Embodied Energy Content of Reinforcing Steel |

m_{i} | Quantity of Material (i) |

M_{i} | Cradle to Gate Embodied Energy Content for Material (i) |

E_{c} | Embodied Energy Content for Construction Activities |

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Objective Function | UB Section | Slab Depth (mm) | Span (m) | EE_{a} (MJ) | EE_{sc} (MJ) | EE_{ps} (MJ) | EE_{c} (MJ) | EE_{r} (MJ) | EE_{total} (MJ) |
---|---|---|---|---|---|---|---|---|---|

Initial Candidate Design | 305 × 102 × 28 | 130 | 6.0 | 6226.6 | 293.0 | 8143.0 | 4795.2 | 4035.8 | 23,493.6 |

Minimised Universal Beam Section | 203 × 102 × 23 | 130 | 6.0 | 5100.5 | 293.0 | 8143.0 | 4795.2 | 4035.8 | 22,367.5 |

Minimised Depth of Concrete Slab | 305 × 102 × 28 | 110 | 6.0 | 6226.6 | 293.0 | 8143.0 | 3836.2 | 4035.8 | 22,534.6 |

Minimised Weight | 203 × 102 × 23 | 110 | 6.0 | 5100.5 | 293.0 | 8143.0 | 3836.2 | 4035.8 | 21,408.5 |

Maximised Span Length | 254 × 102 × 28 | 110 | 7.823 | 8147.2 | 383.9 | 10617.0 | 5001.7 | 5262.1 | 29,410.9 |

Minimised Deflection | 203 × 133 × 25 | 110 | 6.0 | 5542.1 | 293.0 | 8143.0 | 3836.2 | 4035.8 | 21,849.2 |

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## Share and Cite

**MDPI and ACS Style**

Whitworth, A.H.; Tsavdaridis, K.D.
Genetic Algorithm for Embodied Energy Optimisation of Steel-Concrete Composite Beams. *Sustainability* **2020**, *12*, 3102.
https://doi.org/10.3390/su12083102

**AMA Style**

Whitworth AH, Tsavdaridis KD.
Genetic Algorithm for Embodied Energy Optimisation of Steel-Concrete Composite Beams. *Sustainability*. 2020; 12(8):3102.
https://doi.org/10.3390/su12083102

**Chicago/Turabian Style**

Whitworth, Alex H., and Konstantinos Daniel Tsavdaridis.
2020. "Genetic Algorithm for Embodied Energy Optimisation of Steel-Concrete Composite Beams" *Sustainability* 12, no. 8: 3102.
https://doi.org/10.3390/su12083102