Strength Behavior of Internally Reinforced Beams Subjected to Structural Optimization Under Simple Bending Loading

Abstract

In this study, we analyzed novel internally reinforced hollow-box beams to evaluate their strength using the finite element method (FEM) in ANSYS Mechanical APDL 18.1. Twelve different FEM models were subjected to static bending loads, and their performance was assessed based on Huber–Mises equivalent strength values. The results show that most optimized models exhibited improved strength compared to their initial versions, with some configurations achieving up to a 470% increase. These findings highlight the effectiveness of structural optimization in enhancing the strength behavior of hollow-box beams, providing valuable insights for engineering applications.

1. Introduction

The strength behavior of internally reinforced beams subjected to structural optimization under simple bending loading is a topic of interest in mechanical and structural engineering. Optimization techniques aim to enhance the mechanical performance of these beams while minimizing material usage and cost. Several studies have explored numerical, analytical, and experimental approaches to improving beam efficiency. Reinforced concrete (RC) beams with optimized reinforcement configurations have been widely studied to achieve superior mechanical properties [1]. Early research on non-prismatic shear beams demonstrated the impact of varying cross-sections on shear behavior [2]. Similarly, standardized beam factors and moment coefficients have been essential for structural analysis [3]. Elementary theories for tapered beams have also been developed, which are relevant for optimization strategies [4]. Efficient methods for deriving shape functions and stiffness matrices have been proposed, vital for non-prismatic beam modeling [5]. Several researchers have reviewed optimal design strategies for RC beams, considering strength and cost efficiency [6]. Stiffness formulations for non-prismatic beams were also developed to enhance analytical precision [7]. More recently, genetic algorithms (GA) have been introduced for optimizing RC beam reinforcement [8]. Structural efficiency in parabolic–haunched beams has been analyzed [9], while micro-GA has been applied for optimal steel channel beam design [10]. Genetic algorithm applications in concrete structure optimization have been extensively explored [11], including nonlinear analysis methodologies [12]. Advanced computational techniques have been utilized to optimize RC frames under nonlinear deformation [13]. The integration of Building Information Modeling (BIM) with GA has further refined reinforcement strategies in structural design [14], with applications extending to RC wall–slab buildings [15]. Cost-effective design solutions have also been automated using GA-based optimization [16]. Novel reinforcement strategies, such as longitudinal spirals, have improved RC beam performance [17]. Finite element model updating techniques have been proposed to refine beam behavior predictions [18]. Sensitivity analyses of internally reinforced beams under bending and torsion loads provide valuable insights [19,20]. Recent contributions include dynamic behavior analyses of thin-walled sandwich beams [21], inertia product calculations in stiffened plates [22], and hollow-box beam mechanical optimization [23]. Experimental validation of thin-walled beam prototypes confirmed theoretical findings [24,25]. Material selection strategies coupled with structural optimization further enhance beam performance [26]. Numerical studies on hollow-box beams subjected to static loads demonstrate improved strength behavior [27]. Feasibility studies on internally reinforced beams for industrial applications highlight the potential of optimized reinforcement strategies [28]. Lastly, research on material–geometry interaction in beams subjected to combined bending and torsion provides fundamental insights for optimization [29]. The present paper deals with the strength analysis of beams previously optimized by an SQP algorithm [23,24], and applied to internally reinforced thin-walled beams, whose strength behavior was, until now, unknown.

2. Numerical Procedure

In this work, linear static analysis was performed in the FEM software ANSYS Mechanical APDL 18.1. The linear static analysis assumes that stress and strain remain proportional to the applied loads, meaning that material nonlinearity and yield effects are not considered. To ensure that the computed stresses remain within safe limits, the Huber–Mises criterion (von Mises yield criterion) is used as a post-processing check. This criterion helps determine whether the stresses in the optimized beams are approaching the material’s yield strength. By evaluating the Huber–Mises stress distribution, the study ensures that no regions within the beams experience plastic deformation under the given loading conditions. Because the considered material is steel, it was considered that for the applied load intensity, the yield strength was significantly below the yield strength of high-strength steels. In total, 12 numerical models were built. The studied beams are composed of two sandwich panels, located at the top and the bottom, and of a reinforcement pattern on both sides [23,24,27,28]. One of the numerical models, with a single concentrated load and degree of freedom constraints, is shown in Figure 1a. The mesh applied to the FEM model is shown in Figure 1b [23,24,27,28]. Figure 1c shows a zoom of the mesh.

The applied concentrated load has an intensity of 1500 N and was applied on the center of the top face, as in Figure 1a. The models are simply supported at their ends. As the type of analysis performed is linear static, the strength limits of the material were not considered for yield. For this reason, ANSYS assumes that the results always vary linearly with the intensity of the applied loads. Because of this, it is very important to analyze strength results by the Huber–Mises criterion to be sure that the loading conditions of the models/geometries under study do not surpass the yield strength of the material. Figure 2 shows the initial model Beam 1 Pattern 1.

The 12 geometries studied, and the methodology used for obtaining the results presented in Figure 5 and using Equation (2), are presented in Figure 3. A simple hollow-box beam, designated as a hollow–solid section and abbreviated as HSS, is used as a reference model, and is also shown in Figure 3. The HSS has similar section dimensions, but with a thickness of 2 mm and without internal reinforcements. The HSS beam was studied using the same conditions as on the sandwich beams. The reference model (HSS beam), presented in Figure 3, is studied as a hollow-box beam with 2 mm of thickness and the same outer section dimensions as all the novel beams, but non-reinforced. The results obtained in Figure 4 are generated by comparing the Huber–Mises strength of the initial models, i.e., with variable values of LG1 = 75, LG2 = 45, and LG3 = 3 mm, with the corresponding optimized model, whose variable values are shown in

Table 1. Values obtained previously in the design optimization of the same beams, patterns 1 and 2 [24,25].

Bending	A1	A2	A3
Pattern 1
LG1f [mm]	48.6	18.0	18.0
LG2f [mm]	77.3	92.6	83.2
LG3f [mm]	3.74	2.79	2.63
Final objective	0.98	0.83	0.79
Pattern 2
LG1f [mm]	18.0	21.7	18.0
LG2f [mm]	101.5	76.1	119.0
LG3f [mm]	2.79	2.99	2.65
Final objective	0.87	0.89	0.81

and

Table 2. Values obtained previously in the design optimization of the same beams, patterns 3 and 4 [24,25].

Bending	A1	A2	A3
Pattern 3
LG1f [mm]	45.0	21.7	18
LG2f [mm]	75.1	90.2	95.2
LG3f [mm]	3.61	2.76	2.58
Final objective	0.97	0.86	0.8
Pattern 4
LG1f [mm]	18.0	21.7	18.0
LG2f [mm]	80.5	77.0	83.6
LG3f [mm]	2.75	3.55	2.76
Final objective	0.80	0.85	0.77

.

The differences of the studied models, shown in Figure 3, are related to both the geometry of the sandwich panel at the top and bottom and the lateral reinforcements. The beam type is related to the geometry of the sandwich panel: Beams A1 have sandwich geometries that are composed of a straight core. Beams A2 are composed of a corrugated core, and Beams A3 are composed of a honeycomb core. All the patterns of the same beam have the same sandwich panel geometry at the top and the bottom. In the case of the patterns, Pattern 1 has straight internal reinforcements without any internal reinforcements at the corners. Pattern 2 is similar to Pattern 1 but has reinforcements on the corners. Pattern 3 has the same reinforcement at the corners as Pattern 2, but in the middle, it has cross reinforcements instead of the straight reinforcements of Patterns 1 and 2. Pattern 4 has a single cross-shaped reinforcement that occupies all the area on both sides. These models represent the novel beams and have already been studied in other aspects [23,24,27,28]. The numerical results show most of the final results are lower than the initial ones, and the Huber–Mises strength remains well below the yield strength of steel for the applied loading conditions and all studied models. The models use the SHELL63 element type, with a quadrilateral free mesh and a mean element size of 2.5 mm. Table 1 and Table 2 show the geometric variables LG1, LG2, and LG3 for the four patterns of the three beams, as well as the value of the objective function. The final values of the variables, as well as the final value of the objective function, are also shown in Table 1 and Table 2.

All of the FEM models presented in this work were previously optimized with the aim of reducing mass, as well as improving stiffness behavior, as presented in [23,24,27,28]. At the time, only deflections were studied, not stresses. The optimization aimed to minimize deflections and total mass. The objective function is better the lower its value is. The objective function for all of the initial models is 1. These geometries, originally presented by Silva and Meireles [23,24,27,28], had their mechanical behavior improved by the use of optimization. For FEM modeling purposes, the material properties are those of the structural steel: Young modulus E = 2.1 × 10¹¹ [Pa], the Poisson coefficient ν = 0.29 [−], and the density ρ = 7890 [kg/m³].

3. Results and Discussion

The comparisons of the Huber–Mises strength results for both initial and final (optimized) FEM models are shown in Figure 4a,b for four different patterns of strengthening of the analyzed beams.

The results shown in Figure 4 represent the initial models (a) or the final/optimized models (b).

For generating the results shown in Figure 5a, where comparisons between HSS and initial or final models were performed, the improvement factors were calculated using the following formula:

$$\mathrm{Imp}={\displaystyle \frac{{\sigma }_i-{\sigma }_f}{{\sigma }_i}}\times 100$$

(1)

where σ_i represents the Huber–Mises strength for the initial beam and σ_f represents the Huber–Mises strength for the final beam.

The results shown in Figure 5b were calculated as follows:

$$\mathrm{Imp}={\displaystyle \frac{{\sigma }_{HSS}-{\sigma }_x}{{\sigma }_{HSS}}}\times 100$$

(2)

where σ_HSS represents the Huber–Mises strength for the HSS (hollow–solid section) (hollow-box beam) and σ_f represents the final models.

In Figure 5a, the improvement of the mechanical behavior in terms of Huber–Mises strength of final models in comparison with initial ones for bending is displayed. These results were obtained by comparing the results of Figure 4a,b. In figure Figure 5b, a reference model of a hollow-box beam, labeled as hollow–solid section (HSS), is used as reference to assess the results of the initial models (Figure 4a).

In Figure 5a, most models show positive values in relation to their counterparts, which means that the initial models already behave better than conventional hollow-box beams. The exceptions are Beam 1 Pattern 1 and Beam 3 Pattern 2, showing a mild negative improvement (worsening). All the models studied present an improvement, according to this metric. In Figure 5b, the best improvement results present Beam 1 pattern 2, reaching a value close to 470%. All the other beams also show a positive improvement values for every model.

4. Conclusions

The optimization process aimed to reduce mass and deflections rather than directly enhancing strength. However, the structural modifications that resulted from optimization, such as changes in reinforcement patterns and geometric alterations, indirectly affected the strength behavior. The study found that while some models exhibited significant improvements in Huber–Mises strength, others showed only moderate gains. This suggests that while stiffness optimization led to improved mechanical performance, the designs were not necessarily optimized for maximum strength. It is expected that if a term related to the strength is considered in a new objective function and all the optimization processes are rerun with that new objective function, the strength behavior would be better, as the values of the design variables LG1, LG2, and LG3 would be different and more directed to strength. However, this study is still interesting, as looking at its results and previous deflection results, we can determine which models behave best considering all stiffness, strength, and mass criteria.

The following conclusions can be drawn from this work:

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The geometric variables discussed here are highly sensitive regarding calculated values of the Huber–Mises strength.

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The optimization code previously developed in MATLAB 2017b is effective in improving the strength behavior under bending loads, even if the models had their mechanical behavior optimized only in terms of deflections and mass.

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The developed beams were already proven to be highly effective under bending loading, according to their stiffness behavior [23,24,27,28]. However, their effectiveness in terms of strength is moderate. This is expected, as the strength behavior was not an objective in the previous optimization routine. From the point of view of working conditions, a lowering in the Huber–Mises strength values represents better safety in terms of machine condition, and it may lead to all parts of good quality as well as to an increase in the production rate.