# Non-Iterative Optimal Design Method Based on LM Index for Steel Double-Beam Floor Systems Reinforced with Concrete Panels

^{*}

## Abstract

**:**

_{2}emissions. However, for steel double-beam floor systems, a time-consuming iterative analysis is required to derive an optimal design proposal owing to the rotational constraints in the composite joints between the concrete panel and steel beams. In this study, a non-iterative optimal design method using the LM index is proposed to minimize the embodied CO

_{2}emissions of steel double-beam floor systems. The LM index is a measure that can be used to select the optimal cross-section of the steel beams considering the decreased moment capacity according to the unbraced length. The structural feasibility of the proposed design method was verified by investigating whether safety-related constraints were satisfied by the LM index with respect to the design variables under various gravity loads. The applicability of the proposed optimal design method is verified by comparing the embodied CO

_{2}emissions derived from the proposed and code-based design methods. Applicable design conditions were presented based on the LM index to aid engineers. The proposed design method can provide environmentally-optimized design proposals to ensure structural safety by directly selecting the LM index of steel beams.

## 1. Introduction

_{2}emissions because it can increase the stiffness and strength of the section owing to the composite effect of steel and concrete [12]. Kinderis et al. [13] reported that the composite floor system, the so-called Deltabeam, can reduce the building’s height compared to concrete beams. Du et al. [14] experimentally confirmed that high-strength materials can dramatically improve the flexural performance of the steel–concrete composite floor system. In addition, the flexural performance of composite floor systems can be improved by increasing the degree of composite between steel and concrete [15], but it is difficult to construct in-situ due to complex details. Although the constructability of the steel–concrete composite floor systems can be improved through bolted connections between the beam and column, these systems are disadvantageous in reducing material quantities owing to the increased moment demand at beam members under high gravity loads. To improve rotational constraints on composite connection, Ju et al. [16] proposed a TEC (technical, economical, and convenient) beam having code conforming rigid connections. Amadio et al. [17] showed that steel truss embodied connection effectively increases the flexural performance of the connection. Other studies [18,19] have been conducted to increase the rotational constraints of beam–column connections to save the materials. However, complicated details are still required to secure sufficient rotational constraints at the connections; therefore, there is a disadvantage in that the constructability of these steel–concrete composite floor systems decreases.

_{2}emissions related to material quantity, even under high gravity loads [21]. By applying optimal design methods to steel double-beam floor systems, additional reductions in the embodied CO

_{2}emissions on manufacturing structural members are possible. Szewczyk and Szumigała [22] derived an optimal design proposal that can minimize the cost-related production of materials for a steel–concrete composite beam. Optimal design methods generally require time consuming iterative analyses to minimize objective functions [23,24,25]. In the case of a multi-objective problem [26,27,28,29] considering cost and environmental impacts, a more iterative analysis is required to derive an optimal design proposal that minimizes the objective function while satisfying structural safety. A genetic algorithm (GA) is a good tool to solve the multi-objective problem [30], and the analysis time can be greatly reduced through the improvement of the algorithm [31]. However, when the composition of the steel–concrete beam is changed, it is necessary to calibrate the structural or design parameters constituting the GA. Moreover, because the rotational constraints of composite connections needed to be evaluated using sophisticated numerical models [32,33,34,35] or hybrid models [36,37] to calculate the member forces, the computational cost is significantly increased to derive the optimal design proposal for a steel double-beam floor system reinforced with concrete panels. Data intensive approaches such as back-propagation neural network (BPNN) [38] and particle swarm optimization (PSO) [39] can be used as alternatives. However, because these methods require an extensive database from iterative analysis, they are difficult to utilize in the practical field. Therefore, to design a steel double-beam floor system reinforced with concrete panels, considering the rotational constraints of composite connections in the practical field, a new optimal design method should be developed such that practical engineers can use it without time consuming iterative analyses.

_{2}emissions. The design feasibility of the developed optimal method is verified by comparing it with the material quantity calculated using a code-based design method. Finally, the applicable conditions for designing steel double-beam floor systems are presented based on the LM index with a histogram of the limiting laterally unbraced length for H-beams available in practice.

## 2. Non-Iterative Optimal Design Method Using LM Index

#### 2.1. Description of Steel Double-Beam Floor Systems

^{2}, the material quantity of the steel double-beam system without a concrete panel (DBX) increased owing to the increase in the moment demand of the double-beam (i.e., M

_{1}indicated as red-circled in Figure 1); therefore, the material quantity of the DBX system would be increased compared to the GBG system [21]. A steel double-beam floor system reinforced with a concrete panel (DBO) can reduce the moment demand of the double-beam (i.e., M

_{2}indicated as red-circled in Figure 1) by installing a beam end concrete panel that reduces the effective length of the double-beam and induces negative moments at the beam end.

_{p}is the thickness of the concrete panel (mm), E

_{p}is the elastic modulus of the concrete panel (24,422 MPa; calculated according to ACI 318-19 [41] for compressive strength of 27 MPa), E is the elastic modulus of the double-beam (205,000 MPa), I is the moment of inertia of the double-beam (m

^{4}), L is the length of the double-beam (m), μ is the stiffness ratio, and v is Poisson’s ratio of concrete (0.15).

#### 2.2. Formulation of Objective Function Using LM Index

_{b}M

_{n}) has been defined by multiplying the unbraced length (L

_{b}) and the nominal flexural strength (M

_{n}) of steel beams. As shown in Figure 2a, the LM index has a maximum value in L

_{r}; therefore, the LM index in terms of capacity (LM

_{C}) is defined as Equation (2) using this unique characteristic. Similarly, the LM index in terms of demand (LM

_{D}) was defined by Equation (3) from the results of the structural analysis.

^{i}

_{b}is the unbraced length of the steel beams (m), L

^{i}

_{r}is the limiting laterally unbraced length for the limit state of inelastic lateral-torsional buckling (m), M

^{i}

_{r}is the inelastic bending moment for lateral-torsional buckling (kNm), LM

^{i}

_{C}is the LM index in terms of capacity (kNm

^{2}), and LM

^{i}

_{D}is the LM index in terms of demand (kNm

^{2}). The superscript i denotes i-th steel member.

_{C}. Because the unit weight and the LM

_{C}was proportional, the values of the LM

_{C}can be considered as a weight. Therefore, the objective function for minimizing the material quantity can be established in Equation (4) in terms of the LM index as follows:

_{LM}is the objective function of the material quantity expressed in the LM index (kNm

^{2}), ϕ

_{b}is the flexural strength reduction factor (0.9 suggested from AISC 360-16 [40]), and the superscript i denotes the i-th steel beam.

#### 2.3. Constraint Conditions Conforming Design Codes

^{i}

_{L}is the maximum deflection of the i-th steel member under the service load, L

^{i}

_{s}is the length of the i-th steel beam, M

^{i}

_{u}is the moment demand of the i-th steel beam (kNm), ϕ

_{b}is the flexural strength reduction factor (0.9), and M

^{i}

_{n}is the nominal flexural strength of the i-th steel beam determined by considering the strength reduction according to the unbraced length.

^{i}

_{b}). Then, from the definition of the LM indices in terms of demand and capacity summarized in Equations (2) and (3), the constraint condition for the strength can be expressed for the LM index as shown in Equation (7). Additionally, Equation (8) is employed as a strength-related constraint condition to prevent the violation of the design code.

^{i}

_{p}is the plastic bending moment (kNm).

#### 2.4. Non-Iterative Optimal Design Process

_{C}exceeding the LM

_{D}without repeated structural analysis. Subsequently, the thickness of the concrete panel, which can induce the rotational constraint corresponding to the rigid connection, is calculated using Equation (1).

## 3. Validation of Proposed Optimal Design Method

#### 3.1. Typical Design Conditions of Underground Structures Used for Parking Lots

^{2}in Korea. The dead load was set to be 5.2 kN/m

^{2}including the thickness of the concrete slab of 150 mm and finishing materials from previous work [21]. SM490 with a yield strength of 325 MPa was used for the steel to calculate the capacity of the steel beams and the compressive strength of the concrete was set to 27 MPa. By considering various gravity loads, it is possible to examine whether the developed optimal design method can be used in practical applications and to suggest feasible design conditions under realistic constraints.

#### 3.2. Structural Modeling for Steel Double-Beam Floor Systems

_{C}, which minimizes the difference from the LM

_{D}. A comparison of the optimal design proposal between the code-based and proposed optimal design methods is presented in Section 3.3.

#### 3.3. Design Feasibility of Proposed Optimal Design Method

_{D}for the main beam or double-beam are summarized in Table 2. The model name contains the applied floor system and live load condition; for example, DBOL2 indicates a model with live load condition L2 (i.e., 4 kN/m

^{2}) applied to the DBO system. While the moment demands for the main beam in the GBG system and the double-beam in the DBX system have similar values, the moment demand for the double-beam in the DBO system decreases by 63% compared with that in the DBX system owing to the beam-end constraints of the concrete panel. In other words, the installation of the concrete panel at the ends of the double-beam can lead to a reduction in the material quantity of steel double-beam floor systems.

^{2}derived by the code-based and proposed methods were summarized in Table 3 based on the results of the frame analysis. All structural members designed using both methods had the same cross-section of the H-beam. The optimal design proposal using the proposed method can be derived without repeated analysis by easily selecting the cross-section of the H-beam with an LM

_{C}that minimizes the difference from an LM

_{D}. Therefore, the developed optimal design method has a high potential to be utilized in the practical field while avoiding time consuming repetitive analysis. To further examine the feasibility of the proposed design method, the material quantities designed by the two methods were compared for each floor system under five live loads.

^{2}. This result indicates that the proposed optimal design method for the DBO system provides an optimal design proposal equivalent to the code-based method under high gravity loads exceeding the live loads of 6.0 kN/m

^{2}. Therefore, the proposed optimal design method using the LM index is applicable for three different floor systems by providing a quantity-optimized design output similar to the code-based design method without time consuming iterative analysis.

#### 3.4. Environmental Efficiency of Rotational Constraints

_{2}emissions owing to the addition of the concrete panel has little impact on the total embodied CO

_{2}emissions of structural materials. However, the rotational constraint induced by the concrete panel can significantly improve the environmental performance of the DBO system compared with the GBG system by reducing the embodied CO

_{2}emissions related to steel quantities. To evaluate the environmental efficiency of the rotational constraint induced by the concrete panel, the global warming potential (GWP) was calculated using Equation (9) [47] from the steel quantity derived using the proposed optimal design method. For the DBX and GBG systems, the embodied CO

_{2}emissions required for the calculation of the GWP were determined using Equations (10) and (11) [47] from the steel quantity summarized in Figure 7. For the DBO system, the embodied CO

_{2}emissions were calculated by considering the steel quantity in addition to the concrete quantity for the concrete panel.

_{2}-eq/kg from IPCC 2013 [48]), CO

_{2,Embodied}represents the embodied CO

_{2}emission of whole building materials, CO

_{2,Embodied,j}represents the embodied CO

_{2}emission of the j-th building material, ρ

_{j}is the carbon equivalent emission value for the j-th building material (kgCO

_{2}-eq/kg), and m

_{j}is the mass of the j-th building material (kg). Note that ρ values for the concrete material and the steel material were used as 0.141 kgCO

_{2}-eq/kg and 1.42 kgCO

_{2}-eq/kg, respectively, based on ICE-DATA v2.0 [49,50].

^{2}), the DBX system had a slightly smaller GWP than the GBG system, but the DBO system shows a smaller GWP value compared to the DBX system. In particular, for high live loads (exceeding 6.0 kN/m

^{2}), while the DBX system has a higher GWP value than the GBG system, the DBO system has a smaller GWP value than the GBG system. Therefore, the installation of concrete panels improves the environmental performance of steel double-beam floor systems more effectively under high gravity loads. Furthermore, the proposed optimal design method can be practically utilized in quantity optimization processes that require environmental performance evaluation. Since life cycle assessment (LCA) is becoming important in the construction sector related to buildings and bridges [51], the proposed method could be used in the LCA by evaluating costs comprised of material production and embodied CO

_{2}emissions in future studies.

#### 3.5. Applicable Design Conditions Using LM Index

_{r}and L

_{b}are the same, the designed cross-section of the H-beams based on the LM index satisfies both the safety-related constraint conditions and the objective function of minimizing the steel quantity. Because the ranges of L

_{r}used in the calculation of LM

_{C}and L

_{b}used in the calculation of LM

_{D}were similar in the designed cross-section of the H-beams, the proposed design method could yield reliable optimal design proposals regardless of the applied floor systems.

_{r}ranged from 5.2 m to 7.9 m. By considering the typical dimensions of the parking lots, the range of the L

_{b}was determined from 6.4 m to 8.2 m. Figure 9 shows the histogram of the number of the H-beams according to the L

_{r}to suggest the applicable design length using the LM index. The L

_{r}of the H-beams generally has a smaller value as the cross-sectional area or the moment of inertia is smaller. In a case where L

_{r}is less than 6.4 m (i.e., the lower bound of L

_{b}), there is a high possibility that the safety related constraint conditions will be violated because of a lower nominal flexural strength than the moment demand. However, when L

_{r}exceeds 8.2 m (i.e., the upper bound of L

_{b}), it is difficult to minimize the objective function because the nominal flexural strength of the H-beam significantly exceeds the moment demand. Therefore, in this study, since the range of the L

_{b}was determined from 6.4 m to 8.2 m by considering the typical dimensions of the parking lots, the applicable design length of the H-beams using the LM index can be suggested as 6.4–8.2 m indicated in shaded range of Figure 9.

## 4. Conclusions

- By introducing a new index named the LM index on the design parameters and formulating the objective function, the optimal cross-section of the steel beams can be selected by minimizing the material quantity related to the embodied CO
_{2}emissions. - As a result, considering five categories of live loads ranging from 2.5 to 12.0 kN/m
^{2}, the proposed optimal design method was superb at providing a quantity optimized design option under the high gravity loads with a live load of 6.0 kN/m^{2}or more. - The structural rotational constraint induced by the concrete panel can improve the environmental performance of the steel double-beam floor systems by reducing the GWP compared to the steel beam–girder floor system.
- The applicable design length of steel beams for practical engineers to use the proposed optimal method is suggested from 6.4 m to 8.2 m based on the LM index and plan of the typical underground structure used for parking lots.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Structural plan of steel double-beam floor systems and comparison of bending moment of double-beam according to the absence of concrete panel (unit in mm).

**Figure 2.**Description of the LM index in hot-rolled H-beam. (

**a**) Relationship with the unbraced length. (

**b**) Unit weight according to the capacity of the LM index.

**Figure 3.**Comparison of optimization processes between code-based design and proposed optimal design methods.

**Figure 4.**A representative floor plan of underground structures considering the column spacing for parking lots.

**Figure 5.**Illustration of standard frame models of the typical underground structures applied to three different floor systems (Note that the DBX system has the same configurations as the DBO system except for the concrete panel).

**Figure 6.**Modeling of connections using plate elements with beam-end release conditions in the steel double-beam floor system reinforced with the concrete panel (DBO system).

**Figure 8.**Comparison of global warming potentials (GWPs) between the general beam-girder system (GBG) and the double-beam floor systems with concrete panel (DBO) and without concrete panel (DBX).

**Figure 9.**Applicable design length of steel beams using the LM index with histogram according to the limiting laterally unbraced length of the H-beams available in practice.

**Table 1.**Live loads according to the usages of the underground space [44].

Abbreviation | Usages | Live Load (kN/m^{2}) |
---|---|---|

L1 | Offices | 2.50 |

L2 | Passenger vehicles only | 4.00 |

L3 | Storage warehouses | 6.00 |

L4 | Knowledge industry center | 8.00 |

L5 | Heavy vehicles | 12.00 |

**Table 2.**Results of moment demands and LM index in terms of demand for the main beam or double-beam from frame analysis.

Live Load (kN/m^{2}) | Model | Mu (kNm) | LMD (kNm^{2}) |
---|---|---|---|

2.5 | GBGL1 | 489.5 | 2055.9 |

DBXL1 | 482.5 | 2026.5 | |

DBOL1 | 181.2 | 579.8 | |

4 | GBGL2 | 600.5 | 2522.1 |

DBXL2 | 594.3 | 2496.1 | |

DBOL2 | 222.8 | 713.0 | |

6 | GBGL3 | 747.8 | 3140.8 |

DBXL3 | 740.7 | 3110.9 | |

DBOL3 | 278.6 | 891.5 | |

8 | GBGL4 | 896.5 | 3765.3 |

DBXL4 | 891.3 | 3743.5 | |

DBOL4 | 336.2 | 1075.8 | |

12 | GBGL5 | 1196.4 | 5024.9 |

DBXL5 | 1188.7 | 4992.5 | |

DBOL5 | 447.0 | 1430.4 |

**Table 3.**Optimal design proposal derived from code-based and proposed methods for the steel double-beam floor systems.

Model | Structural Member | Design Proposal | |
---|---|---|---|

Code-Based Method | Proposed Method | ||

DBOL4 | Double-beam | H-400 × 200 × 8 × 13 | H-400 × 200 × 8 × 13 |

Girder | H-482 × 300 × 11 × 15 | H-482 × 300 × 11 × 15 | |

Sub-beam | H-394 × 398 × 11 × 18 | H-394 × 398 × 11 × 18 | |

Concrete Panel | T_{p} = 0.250 m | T_{p} = 0.250 m | |

DBOL5 | Double-beam | H-386 × 299 × 9 × 14 | H-386 × 299 × 9 × 14 |

Girder | H-582 × 300 × 12 × 17 | H-582 × 300 × 12 × 17 | |

Sub-beam | H-594 × 302 × 14 × 23 | H-594 × 302 × 14 × 23 | |

Concrete Panel | T_{p} = 0.320 m | T_{p} = 0.320 m |

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**MDPI and ACS Style**

Choi, I.; Kim, D.; Kim, J.
Non-Iterative Optimal Design Method Based on LM Index for Steel Double-Beam Floor Systems Reinforced with Concrete Panels. *Materials* **2022**, *15*, 4538.
https://doi.org/10.3390/ma15134538

**AMA Style**

Choi I, Kim D, Kim J.
Non-Iterative Optimal Design Method Based on LM Index for Steel Double-Beam Floor Systems Reinforced with Concrete Panels. *Materials*. 2022; 15(13):4538.
https://doi.org/10.3390/ma15134538

**Chicago/Turabian Style**

Choi, Insub, Dongwon Kim, and JunHee Kim.
2022. "Non-Iterative Optimal Design Method Based on LM Index for Steel Double-Beam Floor Systems Reinforced with Concrete Panels" *Materials* 15, no. 13: 4538.
https://doi.org/10.3390/ma15134538