# Development of Optimal Design Method for Steel Double-Beam Floor System Considering Rotational Constraints

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Structural Considerations of Steel Double-Beam Floor System

#### 2.1. System Configurations

^{2}[19]. In the DBO system, a positive moment in the center of the double-beam was decreased, as a negative moment was induced at the ends of the double-beam due to the rotational constraints from the concrete panel. Moreover, the positive moment of the double-beam was additionally decreased due to the shortened effective length as much as the width of the concrete panel. Therefore, the DBO system is a more effective structural system in reducing the material quantity than the DBX system under high gravity loads. The effect of rotational constraints on the reduction of the moment demand in the double-beam is discussed in Section 2.2 to consider that in the structural design of the steel double-beam floor system.

#### 2.2. Reduction of Moment Demand in the Double-Beams

^{2}) is calculated by Equation (1). The moment demands in the double-beam of DBX and DBO systems are determined using Equations (2) and (3), respectively.

_{1}is the moment demand (kNm) of the main beam in the GBG system, P

_{1}is the concentrated load (kN) acting as the center of the main beam, L

_{1}is the length of the main beam and double-beam (8.4 m), A

_{1}is the tributary area of the main beam, which is (10.2 m × (8.4 m/2))/2 × 2EA = 42.84 m

^{2}, w is the arbitrary floor load (kN/m

^{2}), M

_{2}is the moment demand (kNm) of the double-beam in the DBX system, P

_{2}is the concentrated load (kN) acting as the center of double-beam, W

_{1}is the uniformly distributed load (kN/m) from the slab, A

_{2}is the tributary area of the double-beam, which is (8.2 m × (8.4 m/2))/2 × 1EA = 17.22 m

^{2}, D

_{c}is the half-width of the concrete panel, which is 1.0 m, M

_{3}is the moment demand (kNm) of the double-beam in the DBO system, and L

_{2}is the effective length of double-beam in DBO system (6.4 m).

#### 2.3. Rotational Constraints Induced by Concrete Panel

_{R}) to the flexural stiffness of the double-beam (K

_{F}), and the end-fixity factor (r), defined as the fixed end moment to the negative moment generated at the connection. From the experimental study for the steel double-beam floor system [7], the connection comprised of the steel double-beam and the concrete panel corresponding to the connection of the DBO system represents a rotational constraint equivalent to a code conforming rigid connection suggested by ANSI/AISC 360 (American National Standards Institute/American Institute of Steel Construction) [27], as shown in Figure 4. Thus, the connection of the DBO system can be modeled as a rigid connection in the structural analysis based on the experimental result.

#### 2.4. Design Parameters of Steel Double-Beam Floor System with Concrete Panel

_{R}) was generated by the flexural stiffness of the concrete panel (K

_{P}), which is defined by Equation (5) according to the Kirchhoff–Love plate theory.

_{F}is the flexural stiffness of the double-beam, K

_{R}is the rotational stiffness of the connection, E is the elastic modulus of the double-beam (205,000 MPa), I is the moment of inertia of the double-beam (m

^{4}), and L is the length of the double-beam (m).

_{R}and K

_{P}are equal, by substituting K

_{P}in Equation (5) for K

_{R}in Equation (4), the thickness of the concrete panel is calculated as Equation (6). The reinforcement of the concrete panel was designed to meet the minimum ratio of 0.0018, suggested by ACI 318-19 (American Concrete Institute) [28], for preventing cracks due to shrinkage and temperature. Since the reinforcement of the concrete panel has little effect on the flexural stiffness of the concrete panel, the flexural stiffness of the concrete panel was determined considering only the concrete material. The design parameter of the concrete panel considered in this study was the thickness of the concrete panel (T

_{P}), and the design parameters of the double-beam affecting the thickness were the moment of inertia (I) and the length (L). Generally, the length of the beam elements is determined by the distance between columns according to the usage of buildings, so the remaining design parameters in the connection are the thickness of the concrete panel and the moment of inertia of the double-beam. These two design parameters have a dependency as summarized in Equation (6). The variation in the cross-section of the double-beam for quantity optimization leads to a change in the thickness of the concrete panel, and the changed thickness affects the rotational constraint of the connection. In other words, the dependency in the two design parameters of the connections causes the time-consuming iterative analysis in the quantity optimization process as the rotational constraint of the connection modified the moment demand of the double-beam related to the safety. To optimize the material quantity for the steel double-beam floor system in practice, a new optimal design method that can reflect the effect of rotational constraint without the iterative analysis is needed, and details are dealt with in Section 3.

_{P}is the flexural stiffness of the concrete panel (kNm/rad), E

_{P}is the elastic modulus of the concrete panel (MPa), T

_{P}is the thickness of the concrete panel (mm), and v is the Poisson’s ratio of concrete (0.15).

## 3. Optimal Design Method Using LM Index

#### 3.1. Objective Function

_{s}is the density of the steel (78.5 kN/m

^{3}), A

^{i}

_{s}is the cross-sectional area of the i-th steel beam, L

^{i}

_{s}is the length of the i-th steel beam, and M is the total number of the steel beam.

#### 3.2. Constraints Conditions

_{S}/480 in accordance with ASCE 7-16 (American Society of Civil Engineers) [30] under the service load (i.e., 1.0 D.L + 1.0 L.L). The moment demand is calculated under the factored load about the gravity loads (i.e., 1.2 D.L + 1.6 L.L), and the nominal flexural strength of the steel beam is determined by considering the unbraced length (L

_{b}) according to ANSI/AISC 360-16 [27].

^{i}

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

^{i}

_{u}is the design moment (i.e., 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.

#### 3.3. Formulation of Optimization Problem Using LM Index

^{i}

_{b}) to the denominator and numerator on the left side of Equation (9), Equation (10) is yielded. Herein, L

_{b}M

_{n}is named as the LM index. Figure 6 shows the LM index for hot-rolled H-beam according to the unbraced length. Since the LM index has a unique characteristic of having a maximum value in L

_{r}, it can be defined as an index representing the properties of the H-beam. Using this unique characteristic, the LM index in terms of capacity (LM

_{C}) is defined as Equation (11). The LM index in terms of demand (LM

_{D}) is defined as Equation (12).

^{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}). Since LM

_{C}is defined as the product of L

_{r}and M

_{r}as shown in Equation (11), L

_{r}and M

_{r}are calculated from the cross-sectional information of KS D 3502 to derive LM

_{C}for 95 H-beams. As shown in Figure 7a, the cross-sectional area of the H-beam shows a linear proportional relationship to the weight, so it is suitable as an objective function for optimizing the weight. Since the relationship between the LM

_{C}and the weight presents a proportional relation as shown in Figure 7b, the values of LM

_{C}can be considered as a weight. If the difference between LM

_{C}and LM

_{D}is minimized, it is possible to minimize weight as a result. Therefore, the objective function for minimizing material quantity using the LM index as a design variable can be defined as Equation (13).

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

^{2}).

_{C}larger than LM

_{D}.

^{i}

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

#### 3.4. Optimal Design Process Using LM Index

## 4. Application to Steel Buildings in Downtown Area

#### 4.1. Frame Analysis According to Levels of Gravity Load

^{2}including the slab thickness and the floor finishing. The live load was divided into five categories considering the usage of the underground space in the downtown area: 2.5, 4.0, 6.0, 8.0, and 12.0 kN/m

^{2}(see previous work [19] for detailed information about the live loads). The dimensions of the structural plan for the steel double-beam floor systems were summarized in Figure 2, and MIDAS-Gen [32], a commercial structure analysis program, was used for frame analysis. In addition, to examine the reduction of material quantity and the improvement of the environmental performance of the DBO system, the quantity-optimizations for the GBG and DBX systems were performed using the proposed optimal design method under the same design conditions.

_{u}). Depending on the load and boundary conditions of the structural members, the locations of the maximum moment were varied. From the results of the frame analysis for each structural system, the M

_{u}, used as demand in the code-based design, and the LM

_{D}, used as demand in the proposed optimal design, are summarized in Table 1.

_{C}that minimizes the difference from the LM

_{D}summarized in Table 1. For example, the LM

_{D}of the double-beam in a DBO system with a live load of 8 kN/m

^{2}is 1075.8 kNm

^{2}. In this condition, a cross-section of H-400x200x8x13 having the LM

_{C}of 1233.7 kNm

^{2}is simply selected without iteration because it has the minimum LM

_{C}that satisfies the objective function. In addition, the ϕ

_{b}M

_{n}of the selected double-beam (356.3 kNm) exceeds the M

_{u}(336.2 kNm), so it can be seen that the cross-section selected by the LM index also meets the safety-related constraint in this case.

#### 4.2. Verification of Proposed Optimal Design Method

^{2}of live load, is summarized in Table 2. The cross-sections of all members designed by the two methods were equal, and the thickness of the concrete panel by them also equal due to the dependency of design parameters between the double-beam and concrete panel. Thus, the proposed optimal design method using the LM index can optimize the material quantity while avoiding the iterative process. To further investigate the feasibility of the proposed optimal design method in the quantity-optimization, the cross-sections designed by the two methods were compared for the five levels of live load.

^{2}, the design proposal derived by the proposed method showed higher material quantity than that of the code-based method. At relatively low live loads, since the M

_{n}of the H-beam optimized by the LM index tended to be slightly larger than M

_{u}, the design proposal by the proposed optimal design method was analyzed to evaluate the material quantity somewhat higher than the code-based method. This is because the L

_{r}of the optimized section at low live loads was shorter than that of the L

_{b}, so that the section, satisfying the safety-related constraint conditions of the code-based method, did not satisfy the safety-related constraint conditions of the proposed method. For live loads of 6, 8, and 12.0 kN/m

^{2}, the material quantities derived by the two design methods were equal to 80.0, 90.0, and 105.5 ton, respectively. Therefore, the proposed design method yielded a quantity-optimized design proposal similar to the code-based method while avoiding a time-consuming iterative process.

_{r}(5.2~7.9 m) in the selected section and the range of L

_{b}(6.4~8.2 m) in the structural plan are similar. If L

_{r}and L

_{b}are the same, a selected cross-section of H-beam through the LM index can derive a design proposal with the smallest material quantity while satisfying safety. Thus, the proposed optimal design method can be used as an alternative for material quantity optimization if the structural plan used in this study represents a typical steel building located downtown.

#### 4.3. Effect of Rotational Constraints in Environmental Performance

_{2}emissions is very low, the rotation constraints from the concrete panel can greatly improve the overall environmental performance of the double-beam floor system by reducing the material quantity. To evaluate the environmental performance according to the presence or absence of concrete panels, the global warming potential (GWP) was calculated through Equation (16) from the quantity of steel and concrete derived by the proposed optimal design method. In the DBO system, the concrete quantity of the concrete panel was considered to calculate the GWP.

_{2, Embodied}represents the embodied CO

_{2}emission of whole building materials (see detailed calculation process in [19]) and CF is the characterization factor for the GWP (1.00 kgCO

_{2}-eq/kg from IPCC 2013 [33]).

^{2}) and high live load (exceed 6.0 kN/m

^{2}) for each structural system. While the DBX system had a slightly smaller GWP value than the GBG system due to the decrease in the material quantity of sub-beams at low live load, the GWP value of the DBX system increased more than the GBG due to the increase in the material quantity of double beams at high live load. The DBO system exhibited the smallest GWP value compared to the other two structural systems at all live loads. The reason that the GWP value of the DBO system was the smallest is that the rotational constraint induced by the concrete panel greatly influenced the reduction of GWP value in the double-beam. The structural characteristic such as the rotational constraint could improve environmental performance, and the GWP value of the DBO system was 13.1% smaller than the GWP value of the GBG system at high live loads. In other words, the steel double-beam floor system improves environmental performance by adding concrete panels to buildings that require a high live load. In addition, the proposed optimal design method can be effectively used in the optimization process of the material quantity requiring for evaluation in environmental performance.

## 5. Conclusions

- The LM index in terms of capacity (LM
_{C}) was defined as the product of unbraced length (L_{b}) and nominal flexural strength (M_{n}) of the steel beam, and LM_{C}had an intrinsic characteristic of having a maximum value in the limiting unbraced length for inelastic lateral–torsional buckling (L_{r}). Since the value of LM_{C}is proportional to the material quantity, it was used as a design variable for quantity optimization instead of cross-sectional area; - The steel beam sections selected by the LM index satisfied the safety-related constraint conditions. As the LM index reflected the flexural strength considering the lateral torsional-buckling according to the L
_{b}, a cross-section with an LM index that minimizes the objective function automatically met the safety-related constraint conditions if the L_{r}and L_{b}were similar. For the steel double-beam floor system with the concrete panel (DBO system), the design results using the LM index showed that the range of the L_{r}(5.9~7.9 m) for the designed H-beam sections was similar to the range of the L_{b}(6.4~8.2 m) determined by the structural plan. Therefore, the developed design method could provide the optimal design proposals with securing structural safety for the DBO system; - Under the low gravity loads with a live load of 4 kN/m
^{2}or less, the design proposal derived by the developed method showed a higher material quantity of 2.0 ton compared to that of the code-based method. Since the M_{n}of the designed sections as the LM index tended to be larger than the Mu, the developed method evaluated the slightly higher material quantity than that of the code-based method under the low gravity load. On the other hand, for the high gravity load with a live load of 6.0 kN/m^{2}or more, the material quantities derived by the developed and code-based methods were equal. At the high gravity loads, as the M_{u}increased, the difference in M_{n}of the designed section using the LM index decreased, so the developed method suggested the same optimal design proposal as the code-based method. Therefore, the developed design method provided a more optimal design option under the high gravity loads; - The steel double-beam floor system without concrete panel (DBX system) had a higher global warming potential (GWP) than the general beam-girder system (GBG system) at high gravity loads where the live loads exceeded 6.0 kN/m
^{2}. On the other hand, the addition of the concrete panels significantly contributed to the improvement of the environmental performance in the steel double-beam floor system by reducing the GWP up to 13.1% compared to the GBG system under the high gravity loads.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Structural plan for an underground space of buildings used for parking lots according to the applied structural systems: (

**a**) Steel double-beam floor system with concrete panel (DBO system) and (

**b**) general beam-girder system (GBG system).

**Figure 3.**Idealized moment diagram induced by one-way gravity load: (

**a**) Main beam in GBG system, (

**b**) steel double-beam in double-beam floor system without a concrete panel (DBX) system, and (

**c**) steel double-beam in DBO system.

**Figure 4.**Relationship between rotational stiffness (stiffness ratio, μ) and negative moment (end-fixity factor, r) in double-beam floor system with the connection type suggested by ANSI/AISC 360 (American National Standards Institute/American Institute of Steel Construction) [27].

**Figure 6.**Relationship between unbraced length (L

_{b}) and LM index (L

_{b}M

_{n}) in hot-rolled H-beam.

**Figure 7.**Unit weight of hot-rolled H-beam: (

**a**) Cross-sectional area vs. unit weight and (

**b**) capacity of LM index vs. unit weight.

**Figure 10.**Comparison of designed steel quantity for steel double-beam system with concrete panel (DBO system) between code-based method and proposed method according to the level of live load.

**Figure 11.**Comparison of global warming potentials (GWPs) occupied by the structural members in the different structural systems according to the level of live load.

Live Load (kN/m ^{2}) | Model | M_{u} (kNm) | LM_{D} (kNm^{2}) | ||||
---|---|---|---|---|---|---|---|

Main Beam or Double-Beam | Girder | Sub Beam | Main Beam or Double-Beam | Girder | Sub-Beam | ||

2.5 | GBG | 489.5 | 386.2 | 584.7 | 2055.9 | 3939.2 | 5963.9 |

DBX | 482.5 | 614.3 | 372.0 | 2026.5 | 5037.3 | 3050.4 | |

DBO | 181.2 | 230.2 | 372.0 | 579.8 | 1887.6 | 3050.4 | |

4 | GBG | 600.5 | 474.3 | 719.7 | 2522.1 | 4837.9 | 7340.9 |

DBX | 594.3 | 755.5 | 457.5 | 2496.1 | 6195.1 | 3751.5 | |

DBO | 222.8 | 283.2 | 457.5 | 713.0 | 2322.2 | 3751.5 | |

6 | GBG | 747.8 | 595.1 | 897.3 | 3140.8 | 6070.0 | 9152.5 |

DBX | 740.7 | 950.9 | 571.9 | 3110.9 | 7797.4 | 4689.6 | |

DBO | 278.6 | 347.1 | 571.9 | 891.5 | 2846.2 | 4689.6 | |

8 | GBG | 896.5 | 714.1 | 1077.2 | 3765.3 | 7283.8 | 10987.4 |

DBX | 891.3 | 1135.1 | 688.8 | 3743.5 | 9307.8 | 5648.2 | |

DBO | 336.2 | 433.8 | 688.8 | 1075.8 | 3557.2 | 5648.2 | |

12 | GBG | 1196.4 | 951.7 | 1433.3 | 5024.9 | 9707.3 | 14619.7 |

DBX | 1188.7 | 1516.2 | 917.8 | 4992.5 | 12432.8 | 7526.0 | |

DBO | 447.0 | 539.6 | 917.8 | 1430.4 | 4424.7 | 7526.0 |

**Table 2.**Comparison of the optimal design proposal in steel double-beam floor system between code-based design method and proposed design method.

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

Code-Based Method | Proposed Method | |

Double-beam | H-400x200x8x13 | H-400x200x8x13 |

Girder | H-482x300x11x15 | H-482x300x11x15 |

Sub-beam | H-394x398x11x18 | H-394x398x11x18 |

Concrete panel | T_{P} = 0.250 m | T_{P} = 0.250 m |

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## Share and Cite

**MDPI and ACS Style**

Choi, I.; Kim, D.; Kim, J.
Development of Optimal Design Method for Steel Double-Beam Floor System Considering Rotational Constraints. *Appl. Sci.* **2021**, *11*, 3266.
https://doi.org/10.3390/app11073266

**AMA Style**

Choi I, Kim D, Kim J.
Development of Optimal Design Method for Steel Double-Beam Floor System Considering Rotational Constraints. *Applied Sciences*. 2021; 11(7):3266.
https://doi.org/10.3390/app11073266

**Chicago/Turabian Style**

Choi, Insub, Dongwon Kim, and Junhee Kim.
2021. "Development of Optimal Design Method for Steel Double-Beam Floor System Considering Rotational Constraints" *Applied Sciences* 11, no. 7: 3266.
https://doi.org/10.3390/app11073266