Perforated and Composite Beam and Arch Design Optimization during Asymmetric Post-Buckling Deformation

Abstract

The structural elements of buildings have recently required the development of efficient design solutions due to increased dynamic and thermal loads. The main solution for improving the efficiency of such elements involves creating lightweight non-uniform beam and arch structures from alloyed steel, which has better mechanical characteristics. The most promising approach is the use of welded beams and arches with perforated partitions and composite beams, which are often used together, for instance, as structural elements of cylindrical shells. The development of an effective cross-sectional shape for perforated beams and crane girders is considered, taking into account the strength, local stability, resistance to flat bending, and fatigue deformation. It has been shown that the effective form for perforated beams is a box-shaped structure made of perforated shvellers. Calculations for selecting a rational design from the assortment of hot-rolled shveller profiles have demonstrated that a significant reduction in the weight of the structure can be achieved by using the proposed cross-sectional shape. An evaluation of the fatigue strength of composite metal crane girders operating in harsh conditions has shown the effectiveness of using hot-rolled I-beams as their upper flange, as well as the necessity of using hot-rolled I-beams to ensure strength in their lower part. When choosing the rational parameters of an arch design, multiple recalculations of its bending with respect to technological cutouts in the thickness are necessary; hence, simplified calculation schemes are commonly used. Some authors simplify this process by replacing an arch with a cutout with a solid arch reduced in height by the cutout radius. We have shown that this model does not accurately describe the actual distribution of forces and displacements, leading to inadequate results. We have developed a simplified methodology for the preliminary calculation of a circular arch with a cutout, which includes correction coefficients calculated by us. A calculation of the flat stress–strain state of an elastic circular metal arch with a central semicircular cutout under various ratios of design parameters and uniform external pressure was conducted. A dependence of the stress concentration coefficient at the cutout’s apex on the ratio of the cutout radius and arch thickness was obtained. These results can be generalized for reinforced non-uniform shells and for the fuzzy application of external influences.

1. Introduction

In recent decades, a need has arisen to develop more efficient solutions for load-bearing elements within buildings and structures of the mining and metallurgical industries. This need arises from the intensification of technological processes, heightened temperature loads, and increased environmental aggressiveness [1,2,3]. A primary strategy to enhance the efficiency of such elements is to design them from economically alloyed steel, which extends the lifespan of structures and mitigates the risk of accidents under elevated temperatures [1,2]. Given that alloyed steels exhibit superior mechanical properties at high temperatures, the prospect of producing lightweight beam structures from such materials is of great interest and aims to reduce material consumption while upholding beam stability and fatigue strength. Notably, the use of welded beams with a perforated wall and composite beams emerges as the most promising approach.

Traditionally, I-sections represent the most effective cross-sectional shape for beams. Numerous studies have explored various designs of shapes of thin-walled elements and shell reinforcement kits [1,4,5,6,7]. However, it is well understood that in the absence of lateral supports, I-beams bent within the plane of the wall may lack sufficient stability. Increased loads beyond certain thresholds can cause these beams to lose stability in flat bending and transition into the asymmetric supercritical bending form, rendering them unable to withstand the load. Some contemporary studies incorporate uncertainties in loading conditions into their calculations and stability assessments [5,6], suggesting ongoing research in this domain. This study relies on traditional deterministic data.

A loss of stability in the thin-walled elements of welded structures can also occur due to structural deviations arising during the manufacturing and operation processes [7,8,9]. However, improving and lightening the shape defined by the section of hot-rolled I-profiles according to DIN 1025 Euronorm 19-57 proves challenging, as achieving strength and stability limits for such beams occurs for similar loads. Despite the increase in the calculated permissible bending loads, creating lightweight beams with perforated webs and composite beams from sections of such profiles necessitates the establishment of constrained bending conditions to prevent flat deformation buckling. Recent work on this topic includes a review of composite beams with openings in partitions and composite cellular beams [10], work on composite beams with openings in partitions with an alternative arrangement of shear connectors [11], and large openings in steel and composite beams [12]. Despite numerous papers addressing the local stability issues of beams with cutouts [1,2,5,6,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], assessing the stability of perforated beams remains an ongoing issue.

Recent work on perforated beams with a box section [23,24] shows that this issue remains poorly researched. The authors note the high resistance of such beams to buckling and the presence of local areas of buckling near and between the holes. Two-dimensional finite elements are used for the associated calculation, and the accuracy of bifurcation load calculation is from 37% to 9%.

Recent studies have been devoted to the issues of the simplified calculation of composite beams with holes [21,22]. Methods for applying correction factors for the calculation of beams with holes during the design process to bring the calculation into compliance with Eurocode 4 are considered in [21]. The effect of holes on beam deflections was assessed using a simple coefficient, which depends on the size and location of the holes. Typical calculation tables for composite I-beams with large openings are presented. In [22], the buckling of the strut web of steel cellular I-beams with regular cutouts was studied. In the FEM calculation, a two-dimensional shell element with four nodes and four integration points was used, and its size was selected after a mesh refinement study. Simplified models for calculating the loss of stability of such beams were obtained, taking into account experimental studies.

Such calculations are possible by means of nonlinear modeling in systems such as SolidWorks [25], which we have previously successfully used in the calculation of complex structures of mining and metallurgical complexes [2]. Existing calculation methods, when compared to experimental data, sometimes exhibit deviations reaching 70% [13,14,18]. The development of a stable perforated web and polybeam structures remains a crucial focus for more rational structural design.

Buckling in perforated beams can occur in three ways: flat bending buckling; loss of local stability in the beam wall, leading to localized wall bulging; and loss of local stability in the beam chord [1,14,19]. Moreover, beams with perforated walls experience a complex stress-strain state with stress concentrations in the notch zone [17]. These factors necessitate refined numerical simulations of such beams’ behavior without the simplification of the assumptions and design schemes. The assessment of fatigue strength and the design of composite metal crane beams operating in harsh conditions also require clarification [1,2,26,27,28].

Arches, like beams, are widely used as load-bearing structural elements and elements of the load-bearing sets of reinforced shells [1,28]. Under load, these elements experience uneven stress distributions, which necessitates detailed analysis to ensure the overall strength and reliability of the structure. Circular arch-based models are commonly employed in complex structures such as dams, ceilings, aircraft, and ships, simplifying the design of load-bearing components [29,30]. In construction, arches find applications in bridges, pavilions, covered markets, hangars, gyms, and other large-scale structures. Arches offer advantages in terms of material efficiency compared to beam and frame systems, with easier manufacturing and installation processes.

The incorporation of cutouts within arch structures is often required when addressing various technological challenges [29,30,31,32,33]. The calculation of arches, including those weakened by cutouts and holes, has been extensively explored in various studies [1,3,4,5,6,7,8,9,13,14,15]. Similar problems encompass the analysis of axisymmetric shells under pressure and the resolution of plane problems in elasticity theory [33,34,35,36,37,38,39]. The findings from both calculations and experiments reveal the heterogeneous deformation of arches. Authors of such studies often emphasize aspects such as the end fastening of arches in the absence of cutouts [36] and the optimization of cutout shapes when present [30,31,32,36]. Stress concentration emerges as a primary factor leading to structural failure during stable deformation [30,31,32,33].

An examination of the existing literature highlights that the simplified schemes [33,34,35] frequently overlook crucial aspects such as the deformation characteristics of arches with cutouts, the stress concentration in such structures, and changes in the bending behavior in comparison to solid arches. These characteristics are particularly notable in arches with relatively large profile heights in the radial direction, such as arch-walls and lightweight arches constructed from welded sheets [32,33], necessitating additional and more detailed analysis.

For tasks involving fuzzy modeling or optimizing design parameters, multiple iterations of elastic bending calculations for the arch are necessary. Consequently, simplified calculation schemes are commonly employed in such structural problems to expedite computer modeling while preserving accuracy [28,32]. Thus, accurately simplifying the development of an arch with a cutout holds significant practical importance.

They are mainly in the form of steel cylindrical wafer casings and have crane runways inside for industrial activities. Each such structure requires an individual calculation, especially at the stage of the preliminary design and selection of a rational structure. Therefore, the creation of new effective forms of all elements of the load-bearing set (composite beams and arches with cutouts) and technological equipment (composite crane beams), as well as new simplified methods for their calculation, is an important integral task. From a technical point of view, the novelty of the proposed solutions lies in the creation of a new effective shape of box beams based on a combination of lightly alloyed steel with mass-produced rolled profiles and their subsequent discrete optimization. The calculation of such beams and composite crane beams as three-dimensional bodies is also new, which allows for the more accurate consideration of stress concentrations and an assessment of the fatigue strength of the structure. The new simplified method of the preliminary calculation of the bearing capacity of an arch with a circular cutout, proposed in this work, makes it possible to significantly reduce the time needed for design calculation without a loss of accuracy.

This paper is organized as follows: Section 2.1 is devoted to the statement of the problem and the calculation of effective perforated welded beams. Section 2.2 describes some peculiarities of the computer modeling of a bent elastic circular arch with a notch. Section 2.3 describes various calculation examples of effective welded crane girders which take the fatigue strength into account. The obtained results are summarized in Section 3.

2. Materials and Methods

2.1. Calculation of Effective Perforated Welded Beams

The analysis of the load-bearing capacity parameters of the beams involved utilizing a design framework for a three-dimensional elastic body subjected to geometrically nonlinear deformation (Figure 1) [2,25].

This approach enables the simultaneous examination of both the local and total strength of the beam, the deformation stability of the walls and flanges, and the maximum deflection of the beam. Loads that did not compromise the bearing capacity across the specified parameters were deemed acceptable. Multiple calculations were conducted for beams of varying dimensions with the objective of identifying the lightest beam that meets the strength and stability criteria at a given length and load. The investigations were focused on a perforated I-beam created by cutting and subsequently welding beams according to a waste-free symmetric scheme, as outlined in DIN 1025 Euronorm 19-57 [17] (Figure 2).

A preliminary assessment conducted to identify a rational design confirmed the limited effectiveness of weight reduction for perforated beams compared to hot-rolled beams of equivalent load-bearing capacity, primarily due to decreased buckling loads in the perforated beams. When comparing the stability coefficient of hot-rolled beams (the ratio of buckling load to actual load, known as the buckling factor of safety, Buckl_FOS [40]) and the factor of safety (FOS), it is evident that their permissible levels almost match for the same profile number, especially for structural steel 13Mn6. Figure 3 illustrates the dependencies of these coefficients for beams spanning 6 m on the serial number of the profile under a uniformly distributed load with an intensity of 1.5 t/m and hinged end connections, constructed from both structural steel 13Mn6 and economically alloyed steel 10G2FB. That is, the use of alloy steel increases the strength coefficient of the beam, but not the stability coefficient. Since the stability coefficient is the main limiting parameter for such a design, replacing the material with a more expensive one makes no sense.

Figure 4 depicts the relationships between the stability and strength coefficients for perforated I-beams, produced through cutting and subsequent welding of DIN 1025 Euronorm 19-57 beams following a waste-free symmetric scheme in relation to the serial number of the profile corresponding to the original hot-rolled beam. As the calculations demonstrate, such an approach only serves to produce a structure that has an excessive strength coefficient while not improving the stability coefficient sufficiently. This constraint limits the feasibility of employing lightweight beams: a beam with perforated walls large enough to withstand the considered load only provides a reduction in weight of 9.2% when compared to a standard hot-rolled beam. Such marginal weight savings may not always justify the additional technological expenses associated with manufacturing welded perforated beams.

As a substitute for the conventional I-beam design, we suggest employing box-shaped welded beams manufactured from hot-rolled shvellers in accordance with standard 8240-89, employing waste-free techniques (see Figure 5). Essentially, this structure constitutes a welded I-beam with a perforated wall, cut along the wall and butt-welded along the flange edges. Computational analysis of these box-shaped welded beams with a perforated wall (see Figure 6) revealed notable advantages over the previously examined options. In drawings created by the SolidWorks system, the movements of the points in the body are displayed in different colors (blue—no or minimal movement, red—maximum movement, the rest of the colors in rainbow order—intermediate values) and are shown on an enlarged scale specified by the user. The loads are shown by arrows, and the intensity determines the length of the arrow in the selected scale. The ends of the beams are cantilevered; this can be seen from the set of reactive forces at the ends and the absence of their displacements.

The stability coefficient for such beams significantly exceeds unity for all considered cases, which indicates the impossibility bearing capacity loss of such beams due to the loss of stability (Figure 7).

In these circumstances, the application of beams with perforated walls offers considerable benefits over their hot-rolled counterparts, particularly when utilizing economically alloyed steels.

Table 1. Comparison of hot rolled and welded box girders.

Beam Type	Profile No.	Strength Factor	Stability Factor	Weight, kg
Hot rolled	22	1.8	1.12	148
Welded box-shaped	16	1.2	4.2	102

presents a comparison of parameters for beams spanning 6 m with the minimum weight, subjected to a load of 1 t/m, constructed from economically alloyed steel 10G2FB. The weight reduction exceeds 31%, rendering beams of this design highly applicable for integration into the construction of modern buildings and structures.

Another benefit of utilizing beams with a perforated wall is their capability to accommodate utilities within them and facilitate access for maintenance and repair tasks. Consequently, this feature enables a reduction in the height of underfloor spaces.

2.2. Certain Characteristics in Computer Modeling of the Bending Behavior of an Elastic Circular Arch with a Notch

This section aims to assess the feasibility of substituting the design scheme of an arch with a circular cutout with an equivalent arch without a cutout but of reduced thickness. To achieve this, we conducted a study consisting of a planar analysis of an elastic circular metal arch in the form of a semiring, the same arch with a central semicircular cutout [38,39,41], and an arch without a cutout, where the section height equals the difference between the original arch’s section height and the cutout radius, all subjected to uniform external pressure. The selection of uniform external pressure as a load is based on its frequent appearance in typical applications of such arches.

Design parameters were varied within specific limits: the arch radius was set to 10 m, the cross-section height ranged from 1 m to 2 m, the cutout radius varied, and the height of the cross-section of the reduced arch was equaled. The arch schemes at $h=1$ m and $r=0.5$ m, utilized for modeling, are depicted in Figure 8.

We examined the effect of a uniform external pressure with the intensity of 500 Pa applied to the outer boundary of the arch. The arch is composed of alloyed structural steel 20KhGSA, with a Young’s modulus of $2.1\times {10}^5$ MPa, yield strength of 0.62 GPa, and Poisson’s ratio of 0.28. The lower section of the arch is either rigidly constrained or supported by a hinge, while the upper section features a sliding constraint, which simulates the symmetry conditions for a semicircular arch. The modeling was conducted using the SolidWorks software package (https://www.solidworks.com) [25]. The mesh was automatically generated and adaptively rebuilt to match the object’s shape, with a minimum of 36 elements along the arc section of the arch and a maximum cell size of 25 mm. The calculation continued with mesh refinement until the desired accuracy was achieved. Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17 depict Von Mises stresses [42] and displacements.

The figures depict the exaggerated deformation of the arch (with displacements exaggerated by a factor of 200), with the arrangement of objects in the figures corresponding to their placement in Figure 8 (hereinafter referred to as models I, II, and III).

Upon analyzing the calculation results for the case of rigid constraint applied to the lower end of the arches (see Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14), it is evident that stresses and displacements exhibit non-uniform distribution along both the length of the arch and across its cross-sectional height. The behavior of an arch with and without a cutout markedly differs for each.

In the case of an arch without a cutout, maximum stresses occur on the inner side of the section in the embedding zone (see Figure 9a,b).

An arch with a cutout also displays significant stress concentration in this area, but the maximum stress is attained at the top of the cutout (see Figure 9c,d). Despite these stresses being concentrated in a very small cross-sectional area (the stress concentration zone is magnified and indicated by an arrow in Figure 9a,b), calculations that consider the plastic properties of the material reveal an expansion in the zone of plastic deformation with increasing load (see Figure 10). This suggests that the failure occurs precisely within the zone of the maximum stress concentration at the top of the notch.

Comparing the displacement fields, as illustrated in Figure 11, Figure 12 and Figure 13), reveals that the deflection pattern of an arch with a cutout alters with increased cutout radius:

-

For smaller values of the cutout radius ($0.05h\le r\le 0.3h$), a deformation pattern with a peak in the vicinity of the arch apex (or the upper end of the analyzed quarter of the arch) predominates (see Figure 11). This deformation pattern closely resembles the movement of an arch without a cutout (models II, III).

-

As the cutout radius increases within the range of $0.4h\le r\le 0.5h$, the region of maximum displacement migrates away from the upper cut (see Figure 13).

-

The value of $0.3h\le r\le 0.4h$ serves as a transition point from one bending pattern to another (see Figure 12).

This shift in deformation pattern is attributed to a reduction in resistance within the cutout zone and a reconfiguration of the stress distribution towards an increase in stress concentration at the top of the cutout. This transformation signifies the transition of the cutout from being a local disturbance factor to a structural element capable of altering the overall deformational behavior of the entire object.

The concentration coefficient for this calculation will be considered as the ratio of the maximum value of the parameter under study for one of the calculation schemes (with cutout (model I), without cutout (model II), reduced (model III), Figure 8) with the maximum value of this parameter for another scheme. By arch thickness, we mean the difference between the outer and inner radii of the arch in the bending plane. The calculation of the stress concentration factor K in an arch with a cutout (model I) and in the reduced arch without a cutout (model III), compared to the maximum stresses of the original arch without a cutout (model II) in the case of rigid constraint on the lower end, is depicted in Figure 14a. Additionally, Figure 14b presents the coefficients representing the ratio of the amplitudes of maximum displacements k in an arch with a cutout (model I) and in the reduced arch without a cutout (model III), compared to the maximum stresses of the original arch without a cutout (model II), under the condition of rigid constraint on the lower end of the arches.

The graphs vividly illustrate a qualitative shift corresponding to the range of alterations in the deformation characteristics of the arch with a cutout.

The calculation of arches with a lower end, constrained by a hinge, as depicted in Figure 15, Figure 16 and Figure 17, exhibits minimal qualitative differences compared to the scenario with the rigid constraint.

The calculation of the stress concentration factor K in an arch with a cutout (model I) and a reduced arch without a cutout (model III), compared to the maximum stresses of the original arch without a cutout (model II) for the case of hinged constraint on the lower end, is depicted in Figure 17a. Additionally, Figure 17b presents the coefficients representing the ratio of the amplitudes of maximum displacements k in the arch with a cutout (model I) and in the reduced arch without a cutout (model III), compared to the maximum stresses of the original arch without a cutout (model II), under the condition of hinged constraint on the lower end of the arches.

The normal stress concentration coefficients for the pure bending of an endless strip with a one-sided semicircular cutout in its plane are provided in [43]. A comparison of the calculation results with this data indicates that for the arch, the stress concentration coefficient is approximately 20% higher compared to the nominal one obtained for the reduced arch without a cutout. This disparity is expected, as [43] does not account for the curvature and finite volume of the structure. Moreover, Ref. [43] lacks data for cutouts in the range of $0.3h\le r\le 0.5h$, which are commonly utilized in the manufacturing of technological holes in lightweight arches.

Based on the calculation data for the considered type of load, we can propose the following approximate formulas for determining the major stress concentration coefficients K in Von Mises criteria as follows:

• $K=3$ for $0.05h\le r\le 0.2h$,
• $K=3.2+2.5{\displaystyle \frac{r}{h}}$ for $0.25h\le r\le 0.5h$.

When calculating the deformation of an arch, one should also calculate the coefficient of increasing maximum displacements for the given arch according to the formula

• $k=1.06+2.5{\displaystyle \frac{r}{h}}$.

Employing these coefficients enables an approximate initial calculation of an arch with a cutout subjected to external pressure, substantially diminishing the required computations when altering parameters. However, it’s crucial to note that the positions of critical points of maximum stresses and displacements differ between a reduced arch and an arch with a cutout. This discrepancy is particularly significant during the optimization of a design with kinematic constraints, such as utilizing the dynamic programming method [28].

2.3. Calculation of Welded Crane Girders with Enhanced Fatigue Strength

It is well known that the longitudinal weld seam connecting the upper flange with the web is the most vulnerable point of a welded crane girder, which is often subject to rigorous operational conditions [26,27]. Placing the welded seam within the highly stressed sub-rail zone constitutes a primary drawback of welded crane beams, as this region experiences the highest shear stress oscillations. Many authors propose relocating the welds from the sub-rail zone of the beams to a distance where shear stress fluctuations diminish significantly, preventing the initiation and propagation of cracks. Their test results validate the superior endurance of beams with belts composed of rolled tees. In beams with T-belts, the weld seam is shifted downward at a considerable distance from the rail-beam belt contact zone, effectively reducing the amplitudes of local stress fluctuations in the weld and minimizing the risk of fatigue crack formation.

However, such beam designs, aimed at enhancing the fatigue strength of longitudinal seams, pose challenges in both manufacturing and operation. The calculations conducted utilized beam schematics and overlooked the intricate stress-strain state near the load transfer zone, necessitating the utilization of a three-dimensional body design scheme.

To develop a rational design for a crane girder and a refined calculation method for such beams, we employed a fatigue calculation module within the SolidWorks complex and a scheme for geometrically nonlinear deformation of a three-dimensional elastic body. Two types of welded beam were examined: a composite beam comprising a hot-rolled I-profile No. 20 and a corresponding welded profile of identical height (see Figure 18a), and a welded I-beam with the same height, matching the width of a hot-rolled profile No. 30 (see Figure 18b).

We analyzed a beam with a length of 6 m and rigidly secured ends, mirroring the operational scenario of the central segment of a continuous crane girder. The beam experienced a load of 4 tons on each of two 15-m sections positioned 1 m apart in the midpoint of the span (Figure 19).

Firstly, an assessment of the static stress-strain condition of the beam under operational loads was conducted to validate the beam’s load-bearing capacity (refer to Figure 20). In the examined beam, the highest stresses, approaching the plastic limit of the steel, were localized within the load zone. These operating conditions pose significant challenges for the crane girder.

The load variation followed a zero-cycle quasi-static loading scheme, transitioning from the absence of load to its maximum level, without accounting for potential dynamic processes. Only the fatigue of the welded seams was considered in this analysis. This decision was made because the fatigue of the upper flange, which bears the highest load, heavily relies on its contact conditions with the rail, which were not adequately represented in this model.

The fatigue calculation for a solid welded beam (see Figure 21) corroborates the findings presented in [27]. A zone of minimal fatigue resistance spans across the entire thickness of the upper flange of the beam and sections of the welded seam within the load zone. Under the specified conditions, akin to static strength conditions, the welded seam can endure only approximately 32 thousand cycles, significantly lower than the standard operational lifespan. This underscores the necessity of assessing the material and welded seams of crane girders for fatigue strength when designing industrial facilities.

The stress-strain condition of a composite beam is essentially similar to that of a welded I-beam, as illustrated in Figure 22.

It is apparent that the welds between the beams are situated within the region of minimal deformations and do not significantly influence the fatigue strength of the structure. This assertion is supported by the fatigue calculation results (refer to Figure 23): the welds in such a configuration endure over 1 million cycles, nearing the design standard.

Currently, the maximum displacements occur within the weld zone of the bottom flange of the welded beam. Their endurance is approximately 300 thousand cycles, which dictates the overall endurance of the structure. Although this endurance falls short of the standard, it is considerably higher than that of the welded I-beam.

Consequently, a comprehensive calculation based on a three-dimensional model reveals that both the upper and lower chords of the crane girder should be replaced with hot-rolled elements. Moreover, to ensure standard fatigue life, it is imperative that the maximum stresses in the beam remain slightly below the calculated maximum stresses in the static analysis. However, this adjustment is likely to result in an increase in the mass of the crane girder.

3. Results and Discussion

Application of the combined method proposed previously for structural analysis of industrial buildings and structures, along with the integration of economically alloyed steels, presents opportunities for the development of novel designs of critical elements [44,45,46,47,48]. It is shown that an effective type of beam with a perforated wall is a box-shaped structure made of perforated hot-rolled channels manufactured using waste-free technology. At a given load, the discrete optimization of the cross-section of a box beam made from a range of hot-rolled channel profiles was carried out. It is shown that through the use of the proposed cross-sectional shape, significant savings in the weight of the structure by more than 31% can be achieved.

In this study, an elastic circular metal arch with a central semicircular cutout was analyzed under uniform external pressure and bending, with various ratios of design parameters being taken into account. Modeling was conducted using the SolidWorks software package. The analysis of the load-bearing capacity parameters of the beams involved utilizing a design framework for a three-dimensional elastic body subjected to geometrically nonlinear deformation. This makes it possible to simultaneously study the local and overall strength of the beam, the deformation stability of the walls and flanges, and the maximum deflection of the beam. During the calculation process, the finite element mesh was refined until the convergence of the beam state parameters was achieved during successive calculations. The investigation yielded relationships between the stress concentration coefficient at the top of the cutout, the dimensions of the cutout, the thickness of the arch, and its radius. The analysis demonstrated that substituting an arch with a cutout with a solid arch featuring a profile with the height reduced by the cutout radius does not provide an accurate depiction of the force distribution and displacement. Such an approach results in an inadequate estimation of the load-bearing ability. For preliminary assessments, a simplified model incorporating the correction factors derived from the study can be utilized.

Recently, new sophisticated [49,50,51,52,53,54] and computer-oriented [53,54] methods for the design and study of complex systems, including the shape and stress state of thin-walled structures [49,52,53,54], have become widespread. This makes it possible to create load-bearing elements based on the individual selection of structures, taking into account the characteristics of their load. Non-uniform grid design with controlled model precision can be made using advanced computer algorithms. This provides an opportunity to automatically adapt such a grid for areas experiencing stress concentration and simplify associated calculations without precision loss.

Another promising avenue of research is the use of fuzzy calculation methods, which have shown their effectiveness in solving real problems [55,56,57]. These methods make it possible to take into account the uncertainty of the loads in and the actual composition of the steel structure, which makes it possible to prevent emergency situations and justify safe overhaul periods for critical equipment. It is clear that in real calculations of equipment and structures, it is necessary to take into account the actual inaccuracy in the dimensions of the structure, the coordinates and direction of the application of the load, the variation in the mechanical properties of the material, and spontaneous temperature fluctuations. All this can have a significant impact on structures susceptible to buckling. Due to the inability to conduct a large set of tests for structures with such technological deviations, the most adequate methods for estimating the bearing capacity in this case are fuzzy calculation methods, the effectiveness of which we have previously shown for other problems. The simplified methods developed in this study make it possible to effectively carry out multiple calculations of structures with the required accuracy and practically implement fuzzy calculations of arches and beams. Random distortions introduced by welding joints are planned to be studied in the future for bending by using fuzzy mathematics. In this work, the effects of welding seams are taken into account when calculating fatigue strength using the automatically built-in SolidWorks tools. It is the reduction in the strength of the area of the weld that dictates the need to use a hot-rolled I-beam as the top chord of the crane beam. Our calculations have shown that the destruction of the weld seam of the lower chord in the form of a welded I-beam also significantly reduces the service life of such a structure. We recommended the use of a hot-rolled I-beam to ensure the strength of its lower part. This is one of the significant conclusions of our work. In combination with inverse methods for assessing the diagnostics of under-rail foundations [54,55], they open up broad prospects for increases in track reliability.

Further improvement of the calculations involves studying the joint deformation of beams and arches of the load-bearing set of reinforced shells with the shell itself [58,59,60,61,62,63]. Controlling the curvature of the design surface and beams also has the potential to create universal building parts with the beneficial orientation of beams. The corresponding reference surfaces are of a constant mean curvature [4,44].

Future research may focus on implementing these developments into real-world projects to optimize costs and improve reliability in challenging mining and metallurgical applications.

4. Conclusions

This paper examines the most important area of rational structural design, namely, the creation of stable perforated box-shaped composite beams and arches. The combination of a new cross-sectional shape and economically alloyed steels opens up opportunities for the development of efficient designs of critical elements. The novelty of the proposed solutions lies in the creation of a new effective form of box beams based on a combination of lightly alloyed steel with mass-produced rolled sections and their subsequent discrete optimization. What is also new is the calculation of such beams and composite crane beams as three-dimensional bodies, which makes it possible to more accurately take into account stress concentrations and evaluate the fatigue strength of the structure. In this work, the effects of welding seams are taken into account when calculating fatigue strength using the automatically built-in SolidWorks tools. It is the reduction in the strength of the area of the weld that dictates the need to use a hot-rolled I-beam as the top chord of the crane beam. Our calculations have shown that the destruction of the weld seam of the lower chord in the form of a welded I-beam also significantly reduces the service life of such a structure. We recommended the use of a hot-rolled I-beam to ensure the strength of its lower part. The new simplified method for the preliminary calculation of the load-bearing capacity of an arch with a circular cutout proposed in this study can significantly reduce the time needed for design calculations without a loss of accuracy.