Effect of X-Cable Bracing on the Optimized Weight of Planar Steel Frames Under Wind Load: A Parametric Study

Abstract

In designing tall buildings, the primary concern is ensuring an effective lateral load-resisting system in addition to the gravity load system, since it largely governs the overall design. This study investigates the influence of X-cable bracing on the structural weight of tall steel frame buildings subjected to service and wind loading. Three numerical case studies, 10-story, 20-story, and 30-story planar steel frames, were modeled and analyzed using SAP2000, then optimized using Differential Evolution (DE) and Enhanced Colliding Bodies Optimization (ECBO) algorithms. These designs were evaluated under both service and wind load conditions, considering strength and drift constraints. The results indicate that the inclusion of wind loads in addition to service loads leads to a higher total structural weight than considering service loads alone, while cable bracing effectively reduces the overall mass by up to 6%, 38%, and 20% for the 10-story, 20-story, and 30-story frames, respectively, compared to unbraced structures, by improving the internal force distribution among structural components. Strength demands, reflected by the interaction ratio, governed all design cases, while lateral displacement was always less than the maximum limit according to AISC and ASCE requirements. Overall, the results highlight the potential of cable bracing systems to deliver efficient tall building designs; however, further studies are needed to generalize these findings to a broader range of building configurations.

1. Introduction

As cities expand and space in highly populated cities becomes scarcer, space demands are met with the construction of tall buildings. Tall buildings, with their increased height, are challenged with providing lateral load resistance. Such structures are influenced by appreciable wind forces, leading to lateral displacements, shear forces, and bending moments, for which measures must be taken for the stability and safety of the building. In addition to ensuring structural stability, limiting lateral displacement (drift) is crucial for serviceability to maintain occupant comfort and prevent damage to non-structural elements such as cladding and partitions. Thus, the design of efficient lateral load-resisting systems is important for retaining the functionality and the safety of tall buildings. Beyond performance gains, improving structural efficiency also lowers material demand and embodied carbon, which supports wider sustainability efforts within the construction industry. Over time, repeated exposure to lateral loads can result in cumulative damage if the building lacks adequate resistance. There are many types of lateral support systems, such as shear walls, bracing frames, outriggers, diaphragms, cable bracing, and a combination of them. Each system is selected based on many factors, such as building location, height, and architectural requirements [1].

Cable bracing systems date back to the early 20th century but gained significant popularity in the late 20th century due to advances in materials and computational modeling. The introduction of high-strength steel cables, which could carry large tensile forces, provided an alternative to traditional bracing systems made of solid steel members. Tall structures subjected to wind loads benefit greatly from cable bracing systems because of their high strength-to-weight ratio. By distributing lateral forces throughout the structure, they improve stability, reduce deflection, and guard against structural damage without placing an undue amount of weight on the structure. Thus, these systems provide an effective lateral support solution [2].

Cable bracing systems have been investigated because of their potential to reduce a building’s total weight without sacrificing its capacity to withstand lateral forces. Lotfollahi and Alinia [3] compared Tension-Braced Moment-Resisting Frames (TBMRFs) under lateral loads. The paper determined sequential failure modes and concluded that by introducing tie bracing, ductility improves and preferable collapse mechanisms are produced compared to unbraced frames. Saleem [4] proposed an external cable bracing system for tall buildings, inspired by the harped cable arrangement used in cable-stayed bridges. A 60-story building was analyzed under a wind speed of 146 mph (as recommended for Miami, FL by the Florida Building Code) using SAP2000. The study compared rigid frame, shear wall, and shear wall–cable bracing configurations. The results demonstrated that replacing exterior shear walls with cables effectively controlled lateral displacement and inter-story drift within allowable limits, while also reducing construction time and increasing usable interior space. The study also noted that the cable bracing system is more effective for buildings with smaller height-to-width ratios. A study by Jagadish and Doshi [5] compared several bracing systems. Single-diagonal bracing, X-bracing, double X-bracing, K-bracing, and V-bracing were discussed in high-rise steel buildings. STAAD.Pro was used to analyze a G+15-story building with wind loads of 33 m/s. The findings show that X- and double X-bracing performed the best, effectively reducing displacement and improving structural stiffness when K and V systems showed greater displacement due to structural irregularities. Patil et al. [6] compared a G+19 tall building under wind load conditions. The paper noted that forces caused by wind increase stress in beams and columns, leading to higher reinforcement demand and higher bending moments, while story drifts remain within permitted limitations. Similarly, Rajas and Shelke [7] compared reinforced concrete buildings of different bracing schemes under wind loading conditions. The paper concluded that it is not only the type of bracing involved that contributes to lateral resistance, but also its location at the frame, and noted that X-bracing at effective locations produces the best results. Alaghmandan et al. [8] introduced a computational workbench integrating architectural parametric design (AutoLisp), CFD simulation (ANSYS 14.5), and structural analysis (SAP2000) to optimize tall building forms under along-wind effects. The study examined tapering modifications ranging from −3 to 8 degrees on a 360 m, 90-story building using framed tube and diagrid structural systems. The results showed that increasing the tapering angle reduced base shear, base moment, and windward pressure, with the framed tube system exhibiting a weight difference of approximately 14 million kg between the minimum and maximum tapering cases. Diagrid systems, however, showed minimal sensitivity to tapering modifications. Park et al. [9] applied the database-assisted design (DAD) method to investigate wind effects on a 49-story, 238 m tall building with a square cross-section and mid-side columns, modeled similarly to the Citicorp Building. Using wind tunnel pressure data from the Tokyo Polytechnic University database, the study compared structural responses under face winds and corner winds. The results showed that corner winds produced along-wind and across-wind overturning moments approximately 20% and 50% lower, respectively, than face winds. Peak axial forces in the mid-side legs and demand-to-capacity indexes of chevron braces were also 20–30% lower for corner winds, confirming that face winds governed the design of square-plan tall buildings. Fanaie, et al. [2] studied the application of cable–cylinder bracing to reinforce steel moment-resisting frames under seismic loads. The study explains that the cable–cylinder system reduces axial forces in columns, minimizes residual displacements, and improves energy dissipation in comparison with conventional cable cross-bracing. It also provides better drift distribution across building height, helping prevent soft-story failure. The study shows that cable–cylinder bracing offers a more efficient, ductile, and economical retrofit option for enhancing the seismic performance of mid-rise buildings based on nonlinear time history analysis. Giaccu and Caracoglia [10] carried out research on pre-tensioned cable bracing systems. This research explores how slackening impacts structural dynamics, using the Equivalent Linearization Method (ELM) to analyze the problem. A key contribution is the introduction of a performance coefficient (ranging from 0.5 to 1.0) that measures stiffness reduction. According to the study, applying more pressure prevents members from coming loose, improving the stability and predictability of the structure’s response. Fanaie and Zafari [11] suggested a new cable–cylinder bracing system for seismic strengthening of steel frames. The sensitivity analysis reveals that the response modification factor is influenced by cable prestressing. The system demonstrates higher ductility and energy dissipation compared to traditional cross-cable bracing. Naghavi [12] studied three bracing systems: double-channel cross braces, cross cable braces, and cable braces with a cylindrical steel sheath, which are used to retrofit steel moment frames. ABAQUS is used to analyze the seismic performance of the frame under cyclic loading. It is shown that the cylindrical steel sheath preserves ductility and initial stiffness comparable to a moment frame. Moreover, this method stops buckling and lowers column axial forces, making it a complete seismic retrofit solution that does not lose ductility or require major foundation work. Rooshenas and Barghian [13] examined a novel bracing system to enhance seismic performance in moment-resisting frames. The suggested system combines the flexibility of moment-resisting frames with the lateral resistance of cable braces. The numerical analyses of 1-, 3-, and 6-story frames show a reduction in drifts. A study conducted by Mosaddegh [14] shows that cable bracing system with a central steel plate can be cost-effective solution for seismic resilience. However, experimental validation and further studies on diverse configurations are recommended. The results of reinforced concrete and steel frames modeled using SAP2000 demonstrate that the system enhances yielding strength, stiffness, and energy dissipation.

Prior research concerning cable bracing has predominantly centered on seismic applications, specifically investigating systems like cable bracing subjected to cyclic loading. These studies revealed significant enhancements in structural ductility, energy dissipation capabilities, and drift control. Simultaneously, the structural optimization of steel frames employing metaheuristic algorithms has been widely studied; nevertheless, the majority of these investigations have focused on unbraced frames or frames incorporating conventional rigid bracing systems, usually under gravity or seismic loading scenarios. Consequently, the effect of X-cable bracing on the optimized structural weight of tall steel frames subjected to wind loads while simultaneously satisfying strength and serviceability (drift) requirements has not been systematically investigated. Furthermore, the existing literature lacks direct quantitative comparisons between unbraced and cable-braced configurations evaluated under identical wind loading conditions.

To address this gap, this study explores how cable bracing systems could reduce the total weight of tall steel building structures while maintaining or enhancing lateral stability. This study follows previous studies that show X-type bracing as providing the best solution in terms of lateral drift reduction and stiffness increase in comparison with other bracing systems, such as V and K. This work utilizes X-cable bracing as the primary lateral load-resisting system. The analytical work encompasses a structural response analysis of X-cable-braced frames, alongside employing optimization schemes in an effort to achieve weight minimization. One of the main objectives is to promote sustainable design practices by minimizing material use and achieving more economically sound building practices. By comparing optimized unbraced and braced configurations under identical loading and constraint conditions, the study quantifies the weight-saving potential of X-cable bracing and provides insights into member utilization and constraint activity. The findings contribute to more material-efficient and sustainable tall building designs. To the best of the authors’ knowledge, this has not been done before.

2. Materials and Methods

A tall building is generally defined as a structure that exceeds 12 stories or 35–50 m in height. Analyzing and designing tall buildings requires considering the sway at the top of a building due to wind. This lateral deflection may be insignificant compared to the building height; however, those who occupy the top feel it strongly [15]. Thus, in the design process, not only the strength requirements for structural members but also the lateral displacement at the highest node of the structures must be considered.

The adopted methodology follows four main stages: (i) modeling a tall steel structure using SAP2000 (version 23.2.0) and Python 3.14, (ii) applying service (dead and live) and wind loads, (iii) using Differential Evolution (DE) and Enhanced Colliding Bodies Optimization (ECBO) algorithms to find the best solution, and (iv) comparing unbraced and X-cable-braced configurations. It should be noted that this study focuses solely on structural performance and optimization outcomes. The cost of cable bracing materials and installation was not considered. The following sections will go over the steps for each level in detail. This study focuses solely on structural efficiency and weight minimization; thus, material and installation costs are considered outside its scope.

2.1. Structural Modeling

In order to simplify the analysis and accurately capture the fundamental structural behavior under service and lateral loads, the numerical examples were represented as planar frames. Planar frames isolate the fundamental lateral behavior and significantly reduce computational cost compared to full 3D models. This lets metaheuristic algorithms thoroughly explore the design space. For regular buildings, such as the frames considered in this study, where dynamic amplification is low, linear static analysis is the norm for wind-load design. It also fits with the serviceability (drift) limits that apply to tall buildings. SAP2000, a popular finite element analysis (FEA) program, was used to create the models, while the bracing system was modeled as cable elements, which can only withstand tension forces. Namely, each cable is tension-only truss element that automatically becomes inactive in compression to ensure cables add stiffness to the structure only in tension and no additional nonlinear effects (e.g., slackening) were included. Columns and beams were represented as frame elements with assigned W-sections.

Table 1. Member properties.

Member	Element Type	Material	Section or Diameter
Columns and beams	Frame element	ASTM A992 Grade 50 steel	AISC W-shapes [16]
Bracing	Cable element	ASTM A992 Grade 50 steel	30 mm

presents a summary of the structural elements and materials employed in the models, wherein the use of ASTM A992 Grade 50 steel means the yield strength is 345 MPa. The base of the structure was assumed to be fixed, simulating a rigid foundation. In all design examples, the story height was taken as 4 m, and the bay width as 6 m.

2.2. Service and Wind Loads

Loads are significant for keeping the building’s structure stable and secure. In this study, dead load, which stands for the permanent loads due to building self-weight and other permanent components, is assumed as 30 $\mathrm{k}\mathrm{N}/\mathrm{m}$. Live load, which represents the temporary and movable loads such as occupants, furniture, and equipment, is taken as 12 $\mathrm{k}\mathrm{N}/\mathrm{m}$, representing a commercial building. Wind loads as lateral forces are calculated based on a wind speed of 100 $\mathrm{k}\mathrm{m}/\mathrm{h}$ (27.78 $\mathrm{m}/\mathrm{s}$) at 10 m height, building bay of 6 m, and air density of 1.225 $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$. The following equations are used to determine the wind speed profile and velocity pressure [17,18,19]:

$$V(z) = V_{10} {\left({\displaystyle \frac{z}{10}}\right)}^{\alpha }$$

(1)

$$q\left(z\right)=0.5 \rho {V\left(z\right)}^2$$

(2)

where $V$ is the mean wind velocity ($\mathrm{m}/\mathrm{s}$), $z$ is the height ($\mathrm{m}$), $V_{10}$ is the wind velocity at a height of 10 m, $\alpha$ is a dimensionless number that varies with atmospheric conditions and terrain (taken as 0.25), $q$ is the velocity pressure, and $\rho$ is the air density (1.225 $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$).

Table 2. Wind speed, pressure, and panel force for five stories.

Story	z (m)	V (m/s)	q (N/m2)	F (kN)
1	4	22.09	298.95	298.95 × 6× 4 = 7.17
2	8	26.27	422.78	10.15
3	12	29.08	517.80	12.43
4	16	31.24	597.90	14.35
5	20	33.04	668.48	16.04

presents the wind force on five stories only.

2.3. Design Criteria

Two types of constraints are imposed on the structure: strength and displacement constraints. According to AISC [16], symmetric members subjected to axial force and bending must satisfy the interaction ratio (in SAP2000, it is called PMM ratio):

$$\begin{gathered} {\displaystyle \frac{P_u}{\varphi P_n}}+{\displaystyle \frac{8}{9}}\left({\displaystyle \frac{M_{uy}}{{\varphi }_b{M}_{ny}}}\right)-1\le 0 \mathrm{if} {\displaystyle \frac{P_u}{\varphi P_n}}\ge 0.2 \\ {\displaystyle \frac{P_u}{2\varphi P_n}}+\left({\displaystyle \frac{M_{uy}}{{\varphi }_b{M}_{ny}}}\right)-1\le 0 \mathrm{if} {\displaystyle \frac{P_u}{\varphi P_n}}<0.2 \end{gathered}$$

(3)

where $\varphi$ is the resistance factor, with $\varphi$ = 0.85 for compression and $\varphi$ = 0.90 for tension. The flexural resistance factor is $\varphi$ = 0.90. $P_u$ and $P_n$ are the required and nominal axial strengths (tension or compression) in kN. The necessary flexural strength is $M_{uy}$ (kN-m). $M_{ny}$ is the nominal flexural strength in kN-m. At every location along the axis of every member in the structure, the constraints in Equation (3) must be followed. So, the equation shows that there are an endless number of limitations. In the numerical procedure, the restrictions are checked at different positions along the member’s axis, and they are put in place at the point where they have the highest values. Then, the penalty function is checked using these constraint values.

In tall building design, the maximum allowable lateral sway (drift) is controlled by building codes to ensure both structural safety and occupant comfort. ASCE [17], AISC [20], and the International Building Code (IBC) [21] are recommended a maximum allowable lateral drift under wind load between $H/400$ and $H/600$, where $H$ is the total building’s height. The normalized form of this criterion is as follows:

$${\displaystyle \frac{|∆|}{{∆}_{max}}}\le 1$$

(4)

where $∆$ is the horizontal displacement at the top of the structure due to wind load, and ${∆}_{max}$ is the allowable displacement. In this study, ${∆}_{max}$ is taken as $H/500$.

2.4. Simulation and Optimization

This work evaluates and improves the performance of tall steel buildings with X-cable bracing systems using both simulation and optimization methodologies. SAP2000 was used for simulations. As discussed earlier, the model takes into consideration dead, live, and wind loads. To simulate the actual conditions the structure will face throughout its lifetime, wind loads were incorporated. The building’s response to the applied service and wind loads was analyzed using linear static analysis. This type of analysis is appropriate for wind loads on regular structures where dynamic amplification is minimal and serviceability checks (drift) are the primary concern. Linear static analysis is standard in preliminary design and optimization frameworks. While more advanced analyses (e.g., material nonlinearity, nonlinear dynamic response) could be integrated with metaheuristic algorithms, they would drastically increase computational time. The Open Application Programming Interface (OAPI) is employed to connect Python-coded frame design examples with the SAP2000 structural analysis software. The Python scripts automate tasks like optimization, result gathering, and adjusting design parameters. This enables efficient exploration of different design configurations and speeds up the overall design process.

To minimize the structural weight of the building while ensuring sufficient lateral stability, metaheuristic optimization algorithms were applied. Discrete variables and non-differentiable functions can be handled routinely in these methods, and the search is not restricted to the vicinity of the current design. They are sometimes referred to as the global optimization approach [22,23] because they employ random search throughout the design space rather than gradient-based search in the neighborhood of the present point. The AISC [16] W-shapes found in the manufacturer’s catalog are preferred for beams and columns as design variables. According to Arora [24], these design variables fall under the category of linked discrete variables. In other words, all of the cross-sectional characteristics are known from the tables once the section number is known. The goal of this optimization method is to reduce the structure’s overall mass:

$$\mathrm{M}(\mathit{X})={\sum }_{nm=1}^{NM}{m}_{nm}{\sum }_{nk=1}^{NK}{L}_{nk}$$

(5)

where $\mathrm{M}$ is the total weight of the structure, $\mathit{X}$ is the design vector, NM is the total number of member groups for the structure, $m_{nm}$ is the weight per unit length of the members in the nmth group (available in AISC’s tables), NK is the number of members in the nkth group, and $L_{nk}$ is the length of the nkth member. Metaheuristic optimization algorithms deal with unconstrained objective functions to improve designs. One way of treating constraints in metaheuristic algorithms is to combine constraints with the cost function to define a merit function (also called the penalty function) [25,26,27]. The merit function quantifies how much a design solution fails to satisfy the imposed constraints, with larger values indicating greater deviations from the allowable strength or drift limits. In this study, it is defined as follows:

$$Obj\left(\mathit{X}\right)=\mathrm{M}\left(\mathit{X}\right){\left[1+10\times P\left(\mathit{X}\right)\right]}^{\xi }$$

(6)

$$P(\mathit{X})={\sum }_{\mathsf{\kappa }=1}^{\mathsf{{\rm K}}}max(0,p_{\kappa })$$

(7)

where $Obj\left(\mathit{X}\right)$ is the objective function, $P\left(\mathit{X}\right)$ is a constraint violation function, $\xi >1$ is the penalty function exponent (in this study, $\xi =2$), and $max(0,p_{\kappa })\ge 0$ is the violation value of the $\kappa th$ inequality constraint.

Two optimization algorithms are used to find the best design for the numerical examples: Differential Evolution and Enhanced Colliding Bodies Optimization. Both algorithms are multi-agent systems that start with a random population (designs) and continue improving iteratively. These algorithms were selected because of their proven robustness in structural optimization problems and their ability to effectively handle discrete steel sections, nonlinear constraints, and large design spaces [28,29,30,31]. The following subsections briefly discuss both algorithms. Since the main goal of this research is to discuss the effect of bracing on the structural weight of tall buildings, the algorithms’ pseudocode and equations are omitted.

2.4.1. Differential Evolution (DE)

The Differential Evolution (DE) optimization algorithm is a metaheuristic method for solving complicated optimization problems. DE perturbs candidate solutions with scaled vector differences between randomly selected population members, balancing exploration and exploitation to iteratively modify candidate solutions according to evolution differences [32,33]. Through this process, the algorithm is able to strike a balance between exploring new regions of the search space and taking advantage of promising solutions, ultimately leading to the best solution. It is a useful tool in many domains, including engineering, due to its versatility in solving both continuous and discrete optimization problems [34]. Four parameters need to be adjusted for DE: crossover rate, mutation factor, maximum number of iterations, and population size. Following a few trial runs of the problems, these parameters were set to 0.7, 0.5, 100, and 20, respectively, for better results. The number of function evaluations, called SAP2000 for structural analysis, in each iteration is 40 analyses with 100 iterations plus the initial population; therefore, the total number of structural analyses per optimization run is 4020.

2.4.2. Enhanced Colliding Bodies Optimization (ECBO)

Natural collisions are the basis for the Colliding Bodies Optimization (CBO) algorithm, which was created by Kaveh and Mahdavi [27]. When two objects collide in this process, they gravitate toward a minimal energy level. Conceptually, the CBO is straightforward, independent of internal parameters, and memory-free for storing the best-so-far solutions. A memory step is introduced to speed up ECBO’s convergence compared to normal CBO in Enhanced Colliding Bodies Optimization (ECBO), which enhances CBO to produce quicker and more dependable solutions. Altering a few colliding body components will also aid ECBO in escaping local minima [35]. ECBO has demonstrated its effectiveness in solving a wide range of engineering problems [26,36].

ECBO requires three parameters: the escaping from local optima factor (0.25), the maximum number of iterations (200), and the population size (20). Similar to the DE, these parameters are set based on a few trial runs of the problems using ECBO. Consequently, the total number of structural analyses per optimization run amounts to 4020, which is comparable to that of DE. A large number of analyses were necessary for each run to ensure the convergence of the metaheuristic algorithms to a near-global optimum for this complex, discrete, and nonlinear optimization problem.

3. Design Examples

Three design examples are studied to show the advantages of cable bracing systems in tall buildings subjected to wind loads. These case studies are a 10-story, 3-bay steel frame, a 20-story, 5-bay steel frame, and a 30-story, 3-bay steel frame. SAP2000 is used to analyze both examples under service and wind loads and strength and serviceability specifications. Planar frames were utilized to isolate the essential lateral behavior and facilitate effective parametric optimization, a standard initial phase in tall building research. Three-dimensional effects and torsional responses are recognized as significant yet deferred for subsequent investigation.

The DE and ECBO algorithms are utilized to minimize the overall structural weight while ensuring compliance with strength and serviceability requirements. The use of these two well-established algorithms increases confidence in the results. For each frame, results are compared between X-braced and unbraced configurations, highlighting how the inclusion of cable bracing enhances structural efficiency and reduces lateral displacements.

3.1. First Design: 10-Story, 3-Bay Frame

A steel moment frame with three bays and ten stories makes up the first design example. The beams are subjected to uniformly distributed service loads, consisting of a dead load of $30 \mathrm{k}\mathrm{N}/\mathrm{m}$ and a live load of $12 \mathrm{k}\mathrm{N}/\mathrm{m}$. As seen in Figure 1a, wind load is represented as concentrated lateral forces operating at the top of each story (calculated according to the procedure outlined in Section 2.2). As shown in Figure 1b, X-shaped cable bracing is only placed in the middle bay for the braced configuration. There are eleven design variables in the structural model: (i) the first design variable is represented by a single beam group, in which every beam has the same cross-section, (ii) the remaining ten design variables are represented by ten column groups, one group per story (

Table 3. Data for 10-story 3-bay frame designs.

Design Variables	Service Loads Only				Service and Wind Load Without X-Bracing				Service and Wind Load with X-Bracing
	ECBO		DE		ECBO		DE		ECBO		DE
	Run 1	Run 2	Run 1	Run 2	Run 1	Run 2	Run 1	Run 2	Run 1	Run 2	Run 1	Run 2
Beams	W16X36	W14X34	W10X33	W10X33	W21X48	W18X60	W21X44	W16X36	W14X38	W12X35	W10X33	W14X34
Columns 1	W18X76	W18X76	W10X77	W12X72	W18X76	W10X77	W10X88	W18X86	W12X79	W12X79	W12X87	W14X82
Columns 2	W12X65	W18X76	W24X84	W12X79	W14X74	W18X76	W18X76	W16X77	W12X72	W18X76	W16X77	W12X79
Columns 3	W18X76	W14X74	W10X68	W12X72	W24X76	W24X76	W24X84	W24X76	W14X68	W10X60	W14X74	W10X68
Columns 4	W12X53	W12X65	W27X84	W16X67	W14X68	W24X68	W14X68	W21X73	W10X54	W14X68	W12X58	W12X65
Columns 5	W21X62	W21X62	W12X58	W14X61	W18X65	W12X58	W21X73	W14X61	W12X58	W21X62	W8X67	W12X58
Columns 6	W16X67	W12X50	W24X68	W18X60	W12X50	W10X49	W14X48	W14X61	W12X50	W18X65	W10X49	W14X68
Columns 7	W21X55	W12X79	W12X65	W21X48	W21X48	W21X62	W21X62	W8X58	W18X50	W18X60	W8X48	W16X57
Columns 8	W24X55	W8X35	W8X40	W14X43	W16X40	W21X48	W21X48	W8X40	W14X38	W16X40	W12X50	W10X39
Columns 9	W16X67	W10X68	W8X24	W8X28	W10X33	W24X76	W18X46	W8X35	W14X38	W12X40	W14X30	W14X34
Columns 10	W18X71	W10X77	W10X45	W14X53	W16X67	W27X84	W21X44	W18X60	W12X79	W10X33	W16X77	W14X82
Total mass (kg)	24,215	23,846	23,150	22,245	26,620	31,549	26,954	24,001	22,965	23,305	22,483	23,370
Max. PMM ratio	1.00 (C4)	1.00 (C4)	0.99 (C9)	0.97 (C1)	0.99 (C5)	1.00 (C4)	0.99 (C6)	1.00 (C3)	0.99 (C9)	0.99 (C3)	0.97 (B1)	0.98 (C1)
Max. drift (mm)	20 (D)	20 (D)	20 (D)	21 (D)	65 (W)	78 (W)	79 (W)	79 (W)	50 (W)	52 (W)	59 (W)	50 (W)

C = Column; B = beam; number = story level (e.g., C1 = 1st story column, B1 = 1st story beam); D = dead load case; W = wind load case.

). The AISC [16] manufacturer’s catalog lists the lightest 100 W-sections from which member sections are chosen. This strategy guarantees a realistic design. Three configurations are studied: (i) unbraced frame subjected to service loads only and AISC’s strength requirements, (ii) unbraced frame subjected to service and wind loads and AISC’s strength requirements, and (iii) braced frame subjected to service and wind loads with AISC’s strength and displacement requirements $({∆}_{\mathrm{m}\mathrm{a}\mathrm{x}}=(4000\times 10)/500=80 \mathrm{m}\mathrm{m}$), as shown in Table 3. For each case study, two optimization runs were performed for both algorithms to ensure that the best design was obtained. In Table 3, the bold values indicate the best design among the four runs. The last two rows of the table summarize the maximum PMM ratio (with its location) and the maximum drift at the roof level, together with the load case that produced it.

3.2. Second Design: 20-Story, 5-Bay Frame

A steel moment frame with 5 bays and 20 stories is the second design example. Each story has a height of 4 m, and each bay spans 6 m. Service loads are applied as uniformly distributed loads on beams, with a dead load of $30 \mathrm{k}\mathrm{N}/\mathrm{m}$ and a live load of $12 \mathrm{k}\mathrm{N}/\mathrm{m}$. Wind loads are calculated following the procedure in Section 2.2 and applied as concentrated lateral forces at the top of each story (see Figure 2). Similarly to the first example, three different setups and 11 design variables are discussed (

Table 4. Data for 20-story 5-bay frame designs.

Design Variables	Service Loads Only				Service and Wind Load Without Bracing				Service and Wind Load with X-Bracing
	ECBO		DE		ECBO		DE		ECBO		DE
	Run 1	Run 2	Run 1	Run 2	Run 1	Run 2	Run 1	Run 2	Run 1	Run 2	Run 1	Run 2
Beams	W16X45	W16X45	W12X45	W12X45	W27X114	W30X90	W30X90	W30X90	W21X55	W27X94	W24X55	W24X55
Columns 1 and 2	W14X132	W14X145	W27X146	W27X146	W14X145	W14X145	W27X146	W27X146	W14X145	W14X145	W14X145	W14X145
Columns 3 and 4	W18X143	W14X120	W21X132	W21X132	W14X132	W36X150	W36X150	W36X150	W12X136	W27X146	W36X150	W36X150
Columns 5 and 6	W24X117	W18X119	W14X120	W14X109	W24X146	W18X130	W21X122	W27X146	W18X119	W33X141	W36X135	W30X148
Columns 7 and 8	W14X99	W18X119	W14X120	W14X120	W30X124	W27X129	W18X130	W30X124	W24X104	W36X135	W33X141	W33X118
Columns 9 and 10	W14X99	W12X106	W27X114	W18X97	W30X116	W21X101	W18X130	W30X116	W33X130	W30X116	W18X106	W27X102
Columns 11 and 12	W12X72	W10X77	W27X102	W12X96	W24X94	W18X130	W14X99	W24X104	W27X94	W27X94	W10X77	W18X76
Columns 13 and 14	W10X77	W10X77	W12X65	W14X68	W30X90	W30X132	W21X93	W30X90	W12X79	W21X83	W14X61	W10X77
Columns 15 and 16	W12X65	W8X58	W21X68	W24X76	W24X103	W12X120	W24X84	W14X61	W18X71	W24X68	W30X90	W10X49
Columns 17 and 18	W16X45	W12X45	W14X48	W18X55	W21X55	W24X103	W18X65	W21X55	W16X57	W40X149	W24X55	W21X48
Columns 19 and 20	W14X132	W27X146	W24X55	W16X50	W16X45	W16X67	W18X60	W14X53	W18X46	W16X45	W14X145	W14X120
Total mass (kg)	104,255	105,255	108,755	107,612	181,722	168,222	159,578	157,650	119,827	167,579	124,827	118,255
Max. PMM ratio	0.99 (C1)	1.00 (C5)	0.97 (C3)	0.97 (C5)	0.98 (C1)	0.99 (C1)	0.98 (C3)	0.97 (C3)	1.00 (C1)	0.94 (C1)	1.00 (C13)	1.00 (C1)
Max. drift (mm)	52 (D)	51 (D)	50 (D)	50 (D)	119 (W)	156 (W)	149 (W)	141 (W)	134 (W)	99 (W)	131 (W)	160 (W)

C = Column; number = story level (e.g., C5 = 5th story column); D = dead load case; W = wind load case.

). However, since there are 20 stories, the following setups are used: (i) one beam group (all beams have the same section), which is the first design variable; (ii) ten groups of columns, one for two stories, which show the other ten design variables. The lightest 150 W-shapes in the AISC [16] manufacturer’s catalog are chosen for member sections.

The DE and ECBO algorithms are used to find the lightest structure that still meets AISC [16] strength and wind drift limits ${∆}_{max} =$ $(4000\times 20)/500=160 \mathrm{m}\mathrm{m}$). Table 4 shows the best designs for each case, along with the total weight, maximum PMM ratio, and maximum drift for each design.

3.3. Third Design: 30-Story, 5-Bay Frame

The third and tallest design example is a steel moment frame with 5 bays and 30 stories. Each story has a height of 4 m, and each bay spans 6 m, giving a total building height of 120 m. Service loads are applied as uniformly distributed loads on beams, with a dead load of 30 kN/m and a live load of 12 kN/m. Wind loads are calculated following the procedure in Section 2.2 and applied as concentrated lateral forces at the top of each story (see Figure 3a). For the braced configuration, X-shaped cable bracing is placed in the second and fourth bays, as shown in Figure 3b.

The structural model contains 11 design variables: (i) one beam group, where all beams share the same cross-section, representing the first design variable, and (ii) ten column groups, each covering three consecutive stories, representing the remaining ten design variables (

Table 5. Data for 30-story 5-bay frame designs.

Design Variables	Service Loads Only				Service and Wind Load Without Bracing				Service and Wind Load with X-Bracing
	ECBO		DE		ECBO		DE		ECBO		DE
Beams	W21X50	W21X50	W21X50	W14X38	W27X102	W24X103	W24X104	W14X233	W30X90	W36X231	W27X84	W21X83
Columns 1 and 3	W24X207	W14X159	W24X229	W14X211	W27X235	W30X235	W27X235	W27X194	W33X241	W27X194	W30X235	W21X182
Columns 4 and 6	W12X190	W14X145	W12X190	W18X211	W21X201	W27X194	W40X199	W14X193	W30X191	W36X194	W30X191	W30X173
Columns 7 and 9	W12X170	W14X145	W12X170	W24X176	W27X161	W24X176	W27X161	W18X175	W21X182	W14X176	W21X182	W30X235
Columns 10 and 12	W12X152	W12X210	W24X162	W40X215	W24X192	W36X182	W36X182	W18X158	W24X162	W14X176	W27X161	W27X178
Columns 13 and 15	W27X146	W33X141	W18X130	W18X158	W14X145	W40X215	W18X143	W18X130	W12X136	W12X136	W12X136	W18X234
Columns 16 and 18	W14X109	W18X106	W18X119	W36X150	W40X235	W36X232	W36X232	W18X119	W18X119	W24X104	W18X119	W18X130
Columns 19 and 21	W12X96	W12X87	W24X104	W14X99	W24X131	W36X194	W36X135	W27X129	W10X100	W12X87	W14X99	W33X152
Columns 22 and 24	W27X84	W16X100	W10X100	W27X84	W33X141	W24X103	W36X135	W14X90	W24X104	W10X68	W10X88	W36X194
Columns 25 and 27	W27X94	W10X17	W24X104	W30X124	W18X71	W30X90	W16X67	W8X40	W24X84	W21X93	W16X77	W40X211
Columns 28 and 30	W18X55	W27X217	W14X74	W21X101	W24X55	W14X53	W16X50	W27X114	W8X40	W24X68	W14X48	W12X79
Total mass (kg)	206,045	191,258	212,474	207,974	309,549	322,674	309,977	317,157	271,512	244,725	259,511	301,030
Max. PMM ratio	0.99 (C1)	1.00 (C5)	0.97 (C3)	0.97 (C5)	1.00 (C1)	0.99 (C4)	1.00 (C7)	0.99 (C3)	0.99 (C1)	1.00 (C3)	1.00 (C13)	0.98 (C1)
Max. drift (mm)	50 (D)	53 (D)	61 (D)	56 (D)	181 (W)	161 (W)	104 (W)	202 (W)	150 (W)	215 (W)	221 (W)	145 (W)

C = Column; number = story level (e.g., C5 = 5th story column); D = dead load case; W = wind load case.

). Member sections are selected from the lightest 200 W-shapes listed in the AISC [16] manufacturer’s catalog, arranged from lightest to heaviest.

As with the previous examples, three configurations are investigated: (i) unbraced frame subjected to service loads only with AISC’s strength requirements, (ii) unbraced frame subjected to service and wind loads with AISC’s strength requirements, and (iii) braced frame subjected to service and wind loads with both AISC’s strength and displacement requirements (${\mathsf{\Delta }}_{max } = (4000 \times 30)/500 = 240 \mathrm{m}\mathrm{m}$). The DE and ECBO algorithms are used to find the lightest structure that still satisfies AISC [16] strength and wind drift limits. For each case, two optimization runs were performed with both algorithms to ensure the reliability of the best design obtained. Table 5 presents the best designs for each case, along with the total weight, maximum PMM ratio, and maximum drift for each design.

4. Results and Discussion

The inclusion of wind loading in the design had a significant effect on the optimized structural mass of the three frames. In the 10-story frame, the transition from service loads only to combined service and wind loads increased the total mass from 22,245 kg to over 24,001 kg (Table 3). The same trend appeared in the 20-story frame, whereby mass rose from 104,255 kg under service loads alone to in excess of 157,650 kg under wind loads (Table 4). Similarly, the total mass increased from 191,258 kg to 244,725 kg for the 20-story structure. These increases in mass due to the increase in member forces across beams and columns to resist lateral forces, as optimization schemes had to select larger sections to satisfy both strength and drift limitations. The results confirm that wind loads are a governing constraint in tall steel building design and can significantly increase the needs for materials over gravity-only conditions.

Table 6. Effect of wind load and X-cable bracing on structural weight.

Frame Type	Case	Mass (kg)	% Change Relative to the Unbraced Structure Subject to Service Load Only
10-Story	Unbraced (service)	22,245	-
	Unbraced (service + wind)	24,001	+7.9%
	X-braced (service + wind)	22,483	+1.1%
20-Story	Unbraced (service)	104,255	-
	Unbraced (service + wind)	157,650	+51.2%
	X-braced (service + wind)	118,255	+13.4%
30-Story	Unbraced (service)	191,258	-
	Unbraced (service + wind)	309,549	61.9%
	X-braced (service + wind)	244,725	27.9%

shows the comparison between braced and unbraced cases. For the 10-story frame, with wind included, adding X-bracing reduced the optimized mass to near the service-only level, with over a 6% saving versus the unbraced wind case. For a 20-story frame, the impact was greater. X-cable bracing reduced the total mass by nearly 38% compared to the unbraced model under wind load, reducing the increase by 13.4% relative to the service-only base case. For the 30-story frame, including X-bracing reduces the total mass by 34%. The results confirm that while wind loads increase structural loads immensely in many members, cable bracing balances this effect by distributing lateral forces around, whereby beams and columns need reduced size and strength.

The PMM results confirm that strength was the controlling design criterion in all cases. As shown in Table 6 and

Table 7. Comparison of the interaction ratio.

Frame Type	Case	Maximum PMM	Average PMM
10-Story	Unbraced (service)	0.97	0.761
	Unbraced (service + wind)	1.00	0.673
	X-braced (service + wind)	0.98	0.746
20-Story	Unbraced (service)	0.99	0.626
	Unbraced (service + wind)	0.97	0.531
	X-braced (service + wind)	1.00	0.681
30-Story	Unbraced (service)	1.00	0.816
	Unbraced (service + wind)	1.00	0.721
	X-braced (service + wind)	1.00	0.801

, the maximum PMM ratio in every run was either equal to unity (a fully utilized member) or very close to it. This indicates that the optimization process consistently pushed critical members to their maximum capacity limits. That is, sections were governed primarily by strength rather than drift. Namely, strength was the active constraint that governed member selection.

Figure 4 compares the optimized total structural mass of the 10-story frame for three study cases. The figure clearly shows that including wind loads increases the required structural mass in the unbraced frame, while the introduction of X-cable bracing reduces the total mass to a level close to that of the service-load-only case. This demonstrates the ability of cable bracing to offset the material demand introduced by wind loading in mid-rise frames. Figure 5 and Figure 6 present the same comparison for the 20-story and 30-story frames, respectively. In this case, the figure highlights a much larger increase in structural weight when wind loads are considered without bracing. The incorporation of X-cable bracing results in a significant decrease in overall weight, suggesting that the efficacy of bracing has a more pronounced effect on structures of greater height. The provided figure demonstrates that the impact of cable bracing on enhancing structural efficiency intensifies with increasing building height. Together, Figure 4, Figure 5 and Figure 6 visually emphasize the growing benefit of X-cable bracing with increasing building height, showing that cable bracing plays a more critical role in mitigating wind-induced weight penalties in taller frames.

Table 7 provides further insight by comparing the average interaction ratios across configurations. The average PMM is computed by taking the meaning of the interaction ratios across all beam and column elements. Interestingly, mean PMMs are larger in the braced cases compared to the unbraced cases. This suggests an increased overall member capacity mobilization by adding cable bracing. With the redistribution of lateral forces by the bracing system, a larger percentage of the frame members are actually participating in resisting forces. This explains why the average PMM is elevated in braced structures even though the maximum remains near unity. That is, higher average PMM in braced frames indicates that a larger proportion of members are more fully utilized, suggesting improved force redistribution.

For all configurations, these results show that wind-driven drift stayed within permissible amounts even though the interaction ratio seemed to be the determining variable in selecting member sizes. The addition of bracing not only reduced structural weight and drift but also promoted a more efficient distribution of internal forces across the structure, leading to more balanced utilization of the frame elements.

5. Conclusions Remarks

In this study, X-cable bracing systems were investigated to reduce the overall structural mass of high-rise steel structures with wind loading so that the requirements for strength and serviceability were achieved. Three numerical cases, namely 10-story, 20-story, and 30-story high-rise steel frame building, were optimized with Differential Evolution (DE) and Enhanced Colliding Bodies Optimization (ECBO) algorithms. Based on the results given here, the following were formulated:

• Wind loading is a governing factor in tall steel building design. The inclusion of wind forces significantly increased the optimized structural weight compared to the gravity-only load cases.
• The material demands caused by wind loads are lessened by cable bracing. Compared to the unbraced wind-loaded scenario, X-bracing decreased the overall weight and drift in the 10-story frame, 20-story frame, and 30-story frame by approximately 6%, 38%, and 34%, respectively.
• In all design examples, strength, not drift, controlled the optimized designs. The maximum PMM ratios in all cases were at or near unity, indicating that member capacity governed section selection. Braced frames exhibited higher PMM ratios, suggesting that bracing redistributed lateral forces more evenly across members, leading to more balanced utilization of the structural system.

Future research could extend this work by incorporating dynamic wind effects, three-dimensional modeling, and experimental validation to broaden the applicability of the findings.