Evaluation of Load-Bearing Performance and Cost Efficiency in Steel-Welded and Modular Aluminum Rack Structures

Abstract

The profile modular system offers variability, flexibility, ease of assembly, and corrosion resistance as well as non-time-consuming assembly while meeting the required conditions of the customer. It has a broad spectrum of usability. This article compares the results of a stress analysis solution for two variants of a rack structure, namely the original variant made by welding Jäkl profiles and the newly proposed design variant created with aluminum Bosch profiles. The finite element method (FEM) is used in computational analyses. FEM models are created using shell elements. Particular attention is given to the use of shell elements in the FEM and their suitability for finite element analyses of the selected structures. Finally, the advantages and disadvantages of both approaches are evaluated, including a safety assessment and an economic comparison of the variants.

1. Introduction

Racks, in general, form an integral part of production lines, storage spaces, and storage areas in households [1,2,3]. In the production process and industrial settings, racks must meet the specific required load-bearing capacity and the storage capacity for materials and components [4,5,6]. From a design perspective, they must also comply with workplace safety conditions to ensure they do not pose a threat to the operating personnel [7,8,9]. Therefore, it is essential to perform a strength analysis for these constructions [10,11]. The study [12] presents an analysis of 16 aluminum cross-section profiles using finite element analysis (FEA) in ABAQUS, especially for stocky or slender cross-sections. Under the specified loading conditions, a stability loss analysis is necessary to prevent the structure from collapsing, which could result in damage to stored components and, in the worst case, pose a risk to human health [13,14]. When racking systems are in areas with specific conditions, more attention is paid to the analyses themselves than in the study by [15], where a realistic assessment of the effectiveness of a roller-base isolation system in reducing seismic responses and improving the stability of industrial steel storage racks was analyzed [15]. In situations involving the induced loss of stability, this also creates a hazardous condition for both equipment and personnel [16,17]. To verify this risk, it is crucial to perform an Eigenvalue Buckling analysis of the loaded frame structure [18,19,20]. The research [21] identifies distinct failure mechanisms that are dependent on slenderness—local buckling for short columns, combined distortional–flexural buckling for medium-length columns, and global flexural buckling for slender columns. The use of the rack structure is versatile. The modular rack-based construction in Noonan et al.’s study supports scalability, consistent illumination, and compatibility with standard microplate formats, thereby enhancing the efficiency of photosynthetic microorganism screening [22]. The rack-based construction in Yu et al.’s study was implemented to enable the synchronous grafting of six vegetable seedlings, and the design facilitates the precise alignment and coordination of key components, ensuring consistent grafting quality across multiple seedlings simultaneously [23]. An experimental study of the behavior of stiffened, perforated cold-formed steel (CFS) uprights commonly used in industrial storage racks can be found in the study [24].

Most research examines either welded or modular constructions in isolation [9]. The study [25] presents five structural framing methods for building information model (BIM)-based prefabricated MEP racks, demonstrating through pilot testing that optimizing frame components and support spacing can significantly reduce material and labor costs. The article [26] provides a comparative analysis of conventional steel frame warehouses and rack-supported warehouse structures, focusing on aspects such as construction processes, storage capacity, economic viability, and flexibility of usage. The article [27] by Mela and Heinisuo presents a method for optimizing the weight and cost of welded high-strength steel beams, demonstrating that significant material savings and economic benefits can be achieved through advanced structural optimization techniques.

Our work stands out by directly comparing these two approaches, offering insight into their respective advantages and trade-offs. In addition to comparing the two methods, the novelty of this article lies in the addition of an economic analysis. Comparing these two approaches and comparing them from an economic perspective thus allows for a truly comprehensive comparison of both methods, which is the main novelty of this article. Based on the literature review, it can be concluded that the comparison of two strength analysis methods is uncommon and offers the advantage of consolidated information within a single article [28,29,30]. At the same time, this work contributes to the relatively underrepresented scientific literature on rack dimensioning, despite the fact that racks constitute a crucial part of logistics [31,32]. The selection of specific beams and their subsequent analysis enhances the importance of this article for the professional public, as the knowledge presented in this publication can be used for specific cases from practice [33].

2. Methodology and Results

The following text contains the methodology for performing finite element analyses of both the original welded structure and the newly assembled structure, along with the corresponding results. For both constructions, a model was first created, followed by the definition of boundary conditions and the evaluation of critical points. The economic assessment of both variants is presented at the end of the section.

2.1. Assignment Presentation

The requirement of the practice was to design a new racking system that meets the conditions of flexibility, corrosion resistance, and variability. The welded structure constructed from Jäkl profiles is a commercially implemented solution developed for a specific customer, with permission granted for its use in this study. The authors analyzed two different variants of the frame structure. The newly designed variant of the racking system consists of Bosch Rexroth aluminum profiles. The original functional frame structure, installed on the production line, is made of welded Jäkl-type steel profiles. Both variants of the solution will be compared at the end in terms of production costs, material and labor expenses, and the time required for the design and manufacturing process.

The welded structure placed in an automated welding line (Figure 1) was designed and manufactured based on engineers’ experience, with significant over-dimensioning [34,35]. No attention was given to strength analysis. The construction, made from Jäkl profiles and the fact that it is fully welded, makes it a solid, unchanging, and robust storage rack. Over time, a request for a change in access from the rear side arose, which led to the need to create a variable structure while meeting all strength and safety conditions and while maintaining the functional requirements [35,36,37,38].

In the first step, the existing welded steel structure was analyzed (Figure 1). It is primarily a welded construction, consisting mainly of I-shaped profiles with dimensions of 100 × 100 mm. The crossbars on the vertical supports have a rectangular cross-section (Jäkl profiles) with dimensions of 80 × 80 mm. The individual parts of the welded subassemblies are connected using bolted joints.

The rack (4) is positioned between two automated welding towers (1), where parallel friction welding, also known simply as friction welding, takes place. A detailed view of the entire workstation, including the placement of the rack, is shown in Figure 2.

The loading and unloading of welding fixtures from the rack (4) to the welding tower (1) (from the welding tower (1) to the rack (4)) will be carried out automatically using an ABB IRB 6700 industrial robot (2). The production line is designed for manufacturing various semi-finished product variants. These semi-finished products differ primarily in the relative positioning of the workpieces during the welding process, which is ensured by the use of welding fixtures. Each fixture has an average weight of 100 kg, and the rack has the capacity to store up to twenty fixtures. Two vibratory feeders (3) are located beneath the rack, which are accessible to the human operator. Therefore, the personnel can be located beneath the rack structure during the unloading of the fixture by the robot, and thus in the area where the robot has reach. It is crucial to ensure that the structure is sufficiently rigid, as any failure could pose a risk to the operating personnel.

The required flexibility and variability of the new beam structure will be achieved using aluminum profiles from Bosch Rexroth (Figure 3a). Emphasis is also placed on ensuring the necessary load-bearing capacity while maintaining safety in terms of structural strength and resistance to collapse due to stability loss. Additionally, in the case of induced oscillations, an analysis will be conducted to ensure that the structure does not enter its resonant frequency and subsequently lose stability. The original welded structure made of Jäkl profiles is shown in Figure 3b.

2.2. Dimensioning and Strength Material Analysis

In more complex cases of planar and, especially, three-dimensional stress, the issue of finding a material’s critical condition becomes more complicated, as its failure can occur under various combinations of principal stresses ${\sigma }_1, {\sigma }_2, {\sigma }_3$, and, therefore, it is not determined by just one numerical parameter. For the stress corresponding to the limit state, the strength condition for the allowable stress ${\sigma }_D$ is given by Equation (1):

$${\sigma }_D={\displaystyle \frac{R_e}{k}},$$

(1)

where $R_e$ is the yield limit and $k$ is the safety coefficient relative to the yield limit R_e.

Based on these criteria, we can determine a representative stress ${\sigma }_{RED}$, which can be compared with the results of the tensile test. Experimental measurements have shown that the hypothesis based on deformation energy for shape change, HMH (Hubert–Mises–Hencky), provides more accurate results than other well-known hypotheses [39]. The evaluation of results is performed using this hypothesis, Equation (2).

$${\sigma }_{RED}={\displaystyle \frac{\sqrt{2}}{2}}·\sqrt{{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_1-{\sigma }_3\right)}^2+{({\sigma }_2-{\sigma }_3)}^2}<{\sigma }_D,$$

(2)

where

${\sigma }_{RED}$—reduced stress;

${\sigma }_D$—allowable stress;

${\sigma }_1,{\sigma }_2,{\sigma }_3$—principal stress components.

Comparing the analytical solution (Figure 4) with two finite element models allows us to observe the advantages and disadvantages of the chosen solutions. The first model is created using beam elements, and the second uses shell elements.

Consider a simple example of a beam with a length of L = 1000 mm, fixed at one end and loaded with a force F = 500 N at the other end. The beam profile is a Bosch profile with dimensions 80 × 80 mm. The maximum deflection of the profile, f, is calculated as follows:

$$f={\displaystyle \frac{F·{ L}^3}{3·E·I·{10}^4}}={\displaystyle \frac{500 \mathrm{N}·{ 1000}^3 {\mathrm{m}\mathrm{m}}^3}{3·\mathrm{70,000}·132.1·{10}^4}}=1.802 \mathrm{m}\mathrm{m}.$$

(3)

where

$F$—applied force;

$L$—length of the beam;

$E$—modulus of elasticity in tension (Young’s modulus);

$I$—moment of inertia of the beam’s cross-section.

The material characteristics are taken from

Table 1. Percentage deviations of the maximum values of stress and displacement between the numerical and analytical solutions.

Calculation Variant	Maximum Displacement Value [mm]/Deviation	Maximum Stress Value [MPa]/Deviation
Analytical	1.802	15.151
Numerical (shell model)	1.687/6.38%	15.552/2.65%
Numerical (beam model)	1.851/2.72%	15.138/0.09%

. For a comparison of the results, the value of the maximum bending stress, ${\sigma }_{max}$, is important, which we determine from the relation in the analytical solution, Equation (5):

$${\sigma }_{max}={\displaystyle \frac{F·L}{W_o·{10}^3}}={\displaystyle \frac{500·1000}{33· {10}^3}}=15.151 \mathrm{M}\mathrm{P}\mathrm{a}.$$

(4)

The maximum bending stress in the Bosch profile 80 × 80 mm, loaded as shown in Figure 5, will be 15.151 MPa based on the analytical calculation. This value will be compared with the results obtained from the following two methods used for the finite element analysis performed in Ansys. Figure 5 shows a comparison of the stress analysis of the shell and solid elements. Table 1 shows the results for one selected profile to validate the modeling approach using shell elements.

The element size was 7.5 mm for the shell model and 10 mm for the beam model. The highest stresses occur at the point of attachment of the Bosch profile, where the maximum bending moment is present. The maximum values of stress and displacement from both strength calculations and the theoretical calculation are summarized in Table 1.

The degree of agreement in both numerical solutions is sufficient, with the beam model providing more accurate results, as previously mentioned. Beam elements do not provide true stiffness in real body connections. In the case of beam elements, it is not possible to consider the screw preload. For this reason, the authors of the article decided to use shell elements in the modeling process.

3. Assembled Rack Structure

3.1. Design of an Alternative Structure Made of Aluminum Bosch Profiles

The use of Bosch profiles offers numerous advantages. The most fundamental advantage is the flexibility and variability they provide. This is associated with the possibility of reconfiguring according to the current requirements of the workplace and the potential for the repeated use of aluminum profiles. The entire system of fasteners, quick-release connectors, angles (Figure 5a,b), bolts, and universal joints up to 180° with 15° increments is used during the assembly process. A significant benefit is the quick assembly with a small number of assembly tools. The aluminum profile has high resistance to corrosion, reducing the need for regular surface maintenance [40,41]. Modular systems made from these profiles are increasingly preferred in the automotive industry [42].

The shape of the aluminum profiles is optimized and specifically designed to carry the maximum load with the minimum weight [43]. The T-shaped grooves serve a special function when connecting Bosch profiles with other components of the structure. They contain threads, so there is no need to drill holes and weaken the profile.

3.2. Structural Model Made of Aluminum Profiles

At the beginning of every manufacturing process is the structural design, which provides the customer with insight into and vision of the future real construction. The 3D model of the assembled aluminum rack was created in the PTC Creo software, version 10.0.4.0 (Figure 3a). The design adhered to all external dimensions for placement in the welding line so that it could perform the same function as the original welded structure (Figure 3b). The connection of the top storage with the posts is secured through plate joining and fastened with screw connections (Figure 6c). All images in Figure 6 are details of the structural design created in PTC Creo.

Two types of cross-sections, Bosch profile 80 × 80 mm and Bosch profile 80 × 40 mm, form the basis of the aluminum structure, as shown in

Table 2. The shapes of the cross-sections of the Bosch Rexroth profiles used along with the basic dimensions and corresponding material.

Used Profiles	Profile Cross-Section with Dimensions	Cross-Sectional Characteristics	Material
Bosch profile 80 × 80 L		A = 18.2 cm2 Ix = 132.1 cm4 Iy = 132.1 cm4 Wx = 33.0 cm3 Wy = 33.0 cm3 m = 4.9 kg	ISO: AlMgSi 0.5 F25 U.S. equivalent: AW-6063-T66
Bosch profile 80 × 40 L mm		A = 9.9 cm2 Ix = 63.4 cm4 Iy = 17.3 cm4 Wx = 15.9 cm3 Wy = 8.7 cm3 m = 4.9 kg	ISO: AlMgSi 0.5 F25 U.S. equivalent: AW-6063-T66
Connecting plate 25 × 100 mm		A = 25 cm2 Ix = 63.4 cm4 Iy = 17.3 cm4	EN: S235JR (STN: 11 375)

. The remaining structural elements are the bottom anchor plates, connecting plates, and top mounting plates made of structural steel type S235JR.

Material sheets of the Al-Mg-Si alloy are freely available from Bosch Rexroth, and the material characteristics and mechanical properties of the AlMgSi alloy are outlined in

Table 3. Material characteristics and mechanical properties of the AlMgSi alloy.

Material Type	Young’s Modulus in Tension E [GPa]	Poisson’s Ratio μ [-]	Yield Strength Re [MPa]	Ultimate Strength Rm [MPa]
AlMgSi 0.5 F25	70	0.34	200	250

.

The properties of the fastening material of the screws (used strength class 8.8), as well as the structural steel from which the connecting plates, top mounting plates, and anchoring plates are made, are described in

Table 4. Dimensions of the cross-sections and materials of the individual parts of the structure.

Used Profiles	Cross-Section of the Profile with Dimensions	Material
I profile (HEB100)		EN: S235JR (STN: 11 375)
Jäkl profile (80 × 80 × 4)		EN: S235JR (STN: 11 375)
Top plate (25 × 100)		EN: S235JR (STN: 11 375)

.

The maximum allowable stress of individual structural elements made of aluminum alloy, in accordance with Equation (3), will be

$${\sigma }_{D,AlMgSi 0.5 F25}={\displaystyle \frac{R_e}{k}}={\displaystyle \frac{200 \mathrm{M}\mathrm{P}\mathrm{a}}{2}}=100 \mathrm{M}\mathrm{P}\mathrm{a}.$$

(5)

3.3. Finite Element Model of Aluminum Profiles

There are three options when creating a finite element profile. The aluminum beam profile can be modeled using volume elements. With volume elements, it is necessary for the finite element mesh to consist of at least three to five elements across the thickness of the profile (2 mm). In such cases, the number of elements and the number of equations to be solved increase rapidly. From a time perspective, this approach is significantly demanding in terms of both time and computational resources.

Using beam elements representing the prescribed profile is the least demanding in terms of geometry modifications. No geometry modifications are needed for the profile cross-section. This technique is frequently used in practice and provides a high degree of agreement with experimental measurements. However, in the case of the screw pretension, we are considering and assessing its impact on the individual parts of the structure, and this technique cannot be used.

The most suitable modeling technique for our purposes is using shell elements. However, from the model preparation perspective, this is the most time-consuming approach due to the complexity of the cross-sections used. Therefore, Bosch profiles are modeled using shell elements. This approach is scarcely described by other authors.

The shell element creation technique requires geometry modifications, such as removing chamfers and the rounding of edges, so that center planes can be extracted from the volume model of the profile. These geometry modifications were made in the PTC Creo software, as shown in Figure 7.

The difference in the cross-sectional area of the Bosch profile with dimensions of 80 × 80 mm after modification is 0.5%. For the profile with dimensions of 80 × 40 mm, the difference in cross-sectional areas after modification is 10%, which is less significant since these profiles are used as horizontal braces on the uprights and do not form the main load-bearing part of the structure.

The process of modifying the geometry for the angle brackets to use shell elements is shown in Figure 8. The holes for the screws remain intact.

The element size is 5 mm, which is a sufficiently fine mesh for a structure with these dimensions. For the shell model, SHELL181 elements were used, and for the bolt connections, beam elements of type BEAM188 were used (see Figure 9).

The considered friction coefficient values are 0.15–0.2, depending on the type of material in contact, aluminum–steel or steel–steel. For shell elements, the shell thickness is set, which allows for contact detection at an offset distance.

3.4. Boundary Conditions and Loading of the Aluminum Profile Structure

The structure is loaded with its own weight. Axial force is applied to all screw joints in the structure based on the known value of the torque on the wrench used for tightening the screws. There are three groups of screw joints, and based on their size and type, we have the following axial preload values:

1.

Hexagonal screws M14: connecting the storage to the uprights, M_k = 90 N·m. The resulting axial force is

$$F_{Q_1}={\displaystyle \frac{\mathrm{90,000}}{\frac{12.701}{2}·tg\left(arct\frac{2}{\pi 12.701}+arctg\left(0.12\right)\right)+0.1\frac{22+15}{4}}}\cong \mathrm{44,700} \mathrm{N}.$$

(6)

2.

Screws with a cylindrical head M8: connecting the top plates and the bottom seat plates to the storage frame, M_k = 8 N·m. The manufacturer specifies the maximum force $F_{Q_2}\max$ = 7000 N. The resulting axial force will be

$$F_{Q_2}={\displaystyle \frac{8000}{\frac{7.188}{2}·tg\left(arct\frac{1.25}{\pi 7.188}+arctg\left(0.12\right)\right)+0.1\frac{13+8.4}{4}}}\cong 6850 \mathrm{N}.$$

(7)

3.

Flanged screws M8: used in brackets, connecting the profiles, M_k = 8 N·m. Even though the same tightening torque is used as in the previous case, the screws have a larger seating area due to the different shape of the head, resulting in a lower axial force:

$$F_{Q_3}={\displaystyle \frac{8000}{\frac{7.188}{2}·tg\left(arct\frac{1.25}{\pi 7.188}+arctg\left(0.12\right)\right)+0.1\frac{20+18.7}{4}}}\cong 5900 \mathrm{N}.$$

(8)

The weight of the upper fastening plates is F = 30,000 N in the direction of gravity. The defined boundary conditions are as follows: self-weight, fixation of the bottom of the uprights to the floor, loading of the storage, and preloading in the screws during their tightening in the assembly process (Figure 10).

3.5. Results of the Structural Strength Analysis of the Aluminum Profile

The finite element analysis requires at least two time steps. In the first time step, the structure is loaded with its own weight, and the predefined preload in the screws, caused by their tightening, is applied (Figure 11). In the areas where the M14 screw washer makes contact, the stress values range between 40 and 50 MPa.

In the second time step, a continuous load representing the weight of all components stored on the shelf is applied. Three areas with the highest HMH stress values were identified throughout the structure. A detailed view of the first critical area is shown in Figure 12a.

Aside from the small local areas directly affected by the screw preload, the most stressed part of the first “critical area” is the transition between the Bosch profile column and the lower connecting plate. The HMH stress reaches a maximum value of nearly 31 MPa at the top of the profile, which corresponds to 31% of the design strength value.

A closer view of the HMH stress distribution in the second most stressed area is shown in Figure 12b. In this region, bending stress has a significant effect, with the HMH stress in the Bosch profile reaching approximately 34 MPa (34% of the allowed stress value). Stress concentrators are in the areas where the Bosch profiles meet the reach values around 87 MPa. The HMH stress distribution in the third most exposed area is shown in Figure 13a.

The most stressed location in this area is the transition region where the Bosch profile extends over the edge of the seating plate to which it is bolted. The highest HMH stress at this point reaches approximately 57 MPa, which corresponds to 57% of the material’s required strength. An interesting observation is the distribution of HMH stresses in the upper part of the profile in this region, where stress in the most critical areas ranges from 20 to 32 MPa (Figure 13b).

The most stressed M14 screws are at the connection between the shelf and the columns, with stresses ranging from 190 to 495 MPa, where the highest values reach 78% of the yield strength. The maximum directional deformation along the Y-axis is 2.1953 mm.

3.6. Eigenvalue Buckling (Boss of Stability) of the Aluminum Profile

Due to the applied load, the aluminum structure may lose stability. Therefore, it is necessary to investigate the structure subjected to buckling. The first and most likely mode of stability loss is the buckling of the structure from its equilibrium position along the z-axis, shown in Figure 14. In this direction, the structure is not transversely reinforced.

The load multiplier reaches a value of 47.118, which means that an approximately 47 times greater load would need to be applied for the structure to lose stability and buckle out of equilibrium. The designed structure provides a sufficient safety margin.

4. Welded Rack Structure

4.1. Structural Model of the Welded Construction

For the finite element analysis, the model was created in the CAD software PTC Creo version 10.0.4.0 (Figure 3b). It consists of massive wide-flange I profiles of the H series. The horizontal crossbars on the posts are made from thin-walled (so-called Jäkl) profiles. The stiffness of the frame is also significantly influenced by the thick top plates placed at the upper part of the storage space, on which the welding fixtures will be mounted. The dimensions and cross-sections of the used profiles, as well as the corresponding materials, are listed in Table 4.

The mechanical properties of the materials used are described in

Table 5. Mechanical properties of the materials used according to Standard EN 10025-2/04 [45] and ISO 898-1.

Material Type	Young’s Modulus in Tension E [GPa]	Poisson’s Ratio μ [–]	Yield Strength Re [MPa]	Ultimate Strength Rm [MPa]
S235JR	200	0.3	240	360–510
Fastening material, strength class 8.8	200	0.3	640	800

. As can be seen from the previous Table 4, all structural elements of the welded subassemblies are made from the same type of construction steel, designated as S235JR. However, in Table 5, the mechanical properties of the connecting components are also defined according to ISO 898-1 [44], as the screw connections themselves will also be considered in the analysis.

In evaluating the ability of the shelving structure to reliably carry the given load, the relationship (1) will be used, with the minimum value of the safety factor based on practical requirements being considered as k = 2 for the load-bearing elements of the structure. For bolted joints, the safety factor will be considered as k = 1.3. Then, the maximum allowable stress in the individual structural parts will be

$${\sigma }_{D,S235JR}={\displaystyle \frac{240 \mathrm{M}\mathrm{P}\mathrm{a}}{2}}=120 \mathrm{M}\mathrm{P}\mathrm{a},$$

(9)

$${\sigma }_{D,8.8}={\displaystyle \frac{640 \mathrm{M}\mathrm{P}\mathrm{a}}{1.3}}\cong 492 \mathrm{M}\mathrm{P}\mathrm{a}.$$

(10)

To ensure the required strength in the welds, the allowable stress needs to be reduced by 30%, which is achieved by using the welding influence coefficient c:

$${\sigma }_{D,Zv S235JR}=c·{\sigma }_{D,S235JR}=0.7\times 120 \mathrm{M}\mathrm{P}\mathrm{a}=84 \mathrm{M}\mathrm{P}\mathrm{a}.$$

(11)

4.2. Finite Element Shell Model of the Welded Structure

The created structural assembly requires a complete set of modifications to be suitably prepared for the finite element analysis. All I profiles need to be converted to shell elements since they are thin-walled profiles with a thickness of 4 mm, as shown in Figure 15. In finite element analyses using solid elements, the meshing of the profile wall would require three to four elements along the thickness of the profile. The element size used would also need to be maintained along the length of the profile to ensure correct element shapes without singularities. A defined thickness is then assigned to the shell element.

The use of shell elements for this structure, given its dimensions and the profiles used, still appears to be the most suitable option. Their main advantage in this context is the saving of computational time due to the significantly lower number of elements compared to the solid model. Easier mesh creation, faster postprocessing, and a lower probability of negative Jacobian formation in thin structural elements also favor the use of shell elements [46]

On the other hand, the disadvantage of this solution is the more demanding preparation of the model for extracting mid-plane definitions, as shown in Figure 16.

The bolt connections joining the individual columns with the storage assembly will also be modeled using shell elements. The bolts themselves will be represented by beam elements, BEAM 188, with a minimum of two elements ensured. The advantage of using these over volume elements is, similar to the shell case mentioned earlier, a significant reduction in computational time, especially when there are several dozens of them in the model. This method is often used when analyzing the effects of prestress on the behavior of the entire structure under load. The main disadvantage of this modeling approach is that it is not possible to obtain results for the stress distribution along the bolt’s cross-section.

The finite element quadrilateral mesh is created using the shell element type SHELL181. In areas with stress concentrators, the finite element mesh is refined (Figure 17a,b). The entire model consists of 173,217 elements with an element quality of 0.93.

4.3. Boundary Conditions and Loading of the Welded Structure

Based on the real conditions of the placement and loading of the rack structure in space, it is necessary to construct a suitable model of boundary conditions and loads that best approximate this actual state.

The fixation of the model is defined by prescribing a single boundary condition—fixed support—on all surfaces of the anchor plates placed on the individual columns. The loading model is implemented as follows, in two steps:

• Considering the significant weight of the storage, the structure was loaded with its own weight using the “standard earth gravity” function. In the first step, bolt pretension was applied to all bolted joints in the structure using the “bolt pretension” function, which arises due to tightening the joints with a torque wrench. In this case, the known torque value M_k = 70 N·m applied during the tightening of the bolts is used, and the axial force in the bolt is calculated using the formula: [47]

  $$M_K=F_Q·{\displaystyle \frac{d_2}{2}}·tg\left(\gamma +{\varphi }^{\prime }\right)+F_Q·f·r_S.$$

  (12)

The relationship for the axial force $F_Q$ is as follows:

$$F_Q={\displaystyle \frac{M_K}{\frac{d_2}{2}·tg\left(\gamma +{\varphi }^{\prime }\right)+f·r_S}},$$

(13)

where all the variables for the M12 bolt are summarized in

Table 6. Variables in Equations (10) and (11).

Variable	Description	Variable Value
γ	Thread pitch angle	$\gamma =arctg{\displaystyle \frac{p}{\pi ·d_2}}$
$p$	Thread pitch value for the M12 screw	$p$ = 1.75 mm
Φ′	Friction angle considering the thread profile	$tg{\varphi }^{\prime }=f^{\prime }\cong 0.1-0.12$
$f$	Coefficient of friction between the nut (or screw head) and the material	$f\cong 0.1$
$d_2$	Mean thread diameter for the M12 screw	$d_2=10.863 \mathrm{m}\mathrm{m}$
$r_S$	Friction radius	$r_S={\displaystyle \frac{s+\delta }{4}}$
δ	Hole diameter for the screw for M12	$\delta =13 \mathrm{m}\mathrm{m}$
$s$	Hexagonal dimension of the nut or screw head for the M12 screw	$s=19 \mathrm{m}\mathrm{m}$

.

After substituting all the values into Equation (13), we obtain the resulting axial force for the M12 hexagonal screw with a magnitude of

$$F_Q={\displaystyle \frac{\mathrm{70,000}}{\frac{10.863}{2}·tg\left(arctg\frac{1.75}{\pi 10.863}+arctg\left(0.12\right)\right)+0.1·\frac{19+13}{4}}}\cong \mathrm{40,000} \mathrm{N},$$

(14)

A contact is defined between the nut, the head of the screw, and the seating surfaces.

• In the next step, a continuous load of force with a magnitude of F = 30,000 N, acting in the direction of gravity, was applied to the surfaces of the upper plates. This force includes primarily the weight of the welding fixtures when the rack is fully loaded, along with the weight of other structural elements placed on the rack for securing the fixtures (cover plates, seating plates with encoding for recognizing the presence of a specific fixture type, etc.) (Figure 18).

4.4. Results of the Strength Analysis of the Welded Structure

Disregarding small locations that are directly affected by the pretension from the bolted connections, the most stressed areas in the first region are those where the edge of one structural element connects or touches the surface of another structural element. In the upper part of the I-profile, at the point of contact with the connecting plate, the maximum HMH stress of 40.3 MPa occurs, which, considering the weld location, represents 47.98% of the allowable stress (Figure 19a).

In the second critical area, bending stress has a dominant effect, as this is the furthest point from the rack uprights, meaning this part is not supported from below. The maximum HMH stress here reaches 26.7 MPa, which represents 31.79% of the allowable stress, assuming it is a welded area (Figure 19b).

The largest deformations of the profiles occur in the direction of the applied weight of the welded fixtures, and the maximum values of directional deformation along the Y-axis reach 0.34 mm.

4.5. Eigenvalue Buckling (Loss of Stability) of the Welded Structure

For slender structures, it is necessary to perform a stability check in terms of safety and safe operation. The first natural buckling mode occurs in the transverse direction. The shelving structure is not reinforced in this direction and will deflect sideways along the z-axis, as shown in Figure 20. The load multiplier is 167.27, meaning that in order for the structure to lose its equilibrium, the applied load must be 167.27 times greater than the defined load.

5. Cost Comparison from an Economic Perspective

The main comparison criterion is the total cost of both solutions. This includes, of course, the cost of raw materials as well as the production and labor costs. The financial information in Figure 21 is directly obtained from a company that manufactures commercial rack structures. All the main factors that contribute to the price in both solutions are as follows:

(a)

Assembled variant

• Input materials (Bosch profiles, plates, and fastening materials—bolts, angle brackets, T-nuts);
• Cutting profiles to the required dimensions;
• Welding of the structure;
• Milling parts of the structure—drilling cylindrical recesses into plates for bolts with a cylindrical head;
• Assembly of the structure.

(b)

Welded variant

• Input materials (I profiles, Jäkl profiles, plates, and fastening materials—bolts);
• Cutting of semi-finished products to the required dimensions;
• Welding of the structure;
• Milling parts of the structure—the storage (adjusting the top plates and drilling mounting holes);
• Spraying the structure;
• Assembly of the structure.

Figure 21. Price comparison of the proposed variants.

Figure 21. Price comparison of the proposed variants.

The purchase price of the Bosch profile 80 × 80 is approximately 2.5 times higher compared to the rolled HEB100 profile. However, this significant price difference in the profiles, which form the basis of both constructions, surprisingly did not result in a major cost difference between the two variants. Therefore, it can be concluded that the higher purchase price of the Bosch profiles in the context of the entire rack is compensated by savings in costs associated with welding and surface treatment operations.

However, the price of the assembled aluminum construction is still approximately 4% higher compared to the welded steel construction, as shown in Figure 21. Considering the countless advantages that the use of Bosch profiles offers in the field of the automated industry, this difference is relatively negligible. Constructions made from these profiles can be repeatedly, flexibly, and fairly easily reconfigured, as dismountable types of connections are used. The simple handling during assembly is related to the relatively low weight of the profiles. Tasks associated with the requirement for additional protective coverings or any component to be added to an already existing structure are saved from time-consuming manual threading in the required locations. This function is replaced by the profile groove in aluminum Bosch profiles, along with the use of Bosch stones. These mentioned benefits are difficult to quantify financially.

6. Comprehensive Comparison of Variants and Discussion of the Obtained Results

It is important to note that the verification of the constructions’ ability to safely carry the required load was the starting assumption and a fundamental condition for such a comparison. If this fundamental criterion had not been met, the price comparison of variants would not have been justified. The performed strength analyses of the load-bearing parts of the constructions, however, showed satisfactory results in both cases, and no structural modifications were necessary for either variant. Despite this, it is useful to compare the stress distribution of the selected location for both the welded and assembled constructions in order to gain a better understanding of the differences in theoretical strength reserves between these solutions, as shown in Figure 22.

The data from Figure 22 are used in

Table 7. Comparison of the solutions.

Construction Variant	Maximum Value of Equivalent Stress at the Given Location [MPa]	Yield Strength [MPa]	Safety Factor [-]
Welded	13.5	240	17.8
Assembled	24.5	200	8.2

, which also includes the calculated safety factors for both variants at the given location of the structure. The maximum value of equivalent stress in the welded structure is σ = 13.5 MPa (Figure 22a), and in the assembled structure it is σ = 24.5 MPa (Figure 22b). Taking into account the yield strength of the respective materials, the safety factor is determined using Equation (13).

$$k={\displaystyle \frac{R_e}{\sigma }}.$$

(15)

where $R_e$ is the yield strength [MPa] and σ is the maximum value of equivalent stress at the given location [MPa].

From Table 7, it is evident that the welded variant has twice the safety margin compared to the assembled variant at this location of the structure, which is understandable given the predominant bending stress in this area and the robustness of the steel profiles used. However, both solutions are over-dimensioned with sufficient strength reserves, and this comparison serves merely as a brief final summary of the strength calculations for these construction variants.

7. Conclusions

The purpose of this article was to compare and comprehensively analyze two structural design approaches for an industrial rack: the conventional welded structure made of Jäkl profiles and an alternative newly proposed assembled structure using aluminum Bosch Rexroth profiles. The results are applied to a specific profile type that is commonly utilized in practical engineering applications.

To achieve this objective, finite element analysis (FEA) methodologies were proposed and described, followed by a detailed FEM analysis of both solutions in terms of strength, stability loss, and Eigenvalue Buckling. The main contribution of this article lies in the application and validation of an advanced modeling approach using shell elements for aluminum profiles, which required geometric modifications and the extraction of mid-surfaces. A new and comprehensive approach was applied to address the given issue. The advantage of this approach—comparing two methods and conducting an economic analysis—was the ability to perform the most thorough possible assessment. Based on the conclusions of an extensive literature review, it can be stated that no similar comprehensive approach has been identified in any other research. The chosen shell element modeling approach delivers reliable results while reducing computational time and hardware demands.

Another significant component is the consideration of bolt preloading and its effect on the overall stress state in the structure, which provides a more realistic idea of the behavior of assembled systems in operation. The impact of bolt preloading on the overall stress state in the structure was also taken into account. The analysis results were compared.

The authors also conducted an economic analysis of both compared solutions. The results and their thorough analysis indicate that the modular structure made of aluminum profiles also offers significant variability and easy reconfiguration, which aligns with the requirements of modern industrial automation and the concept of “soft automation.” At the same time, the analysis confirmed the sufficient strength and stability of the selected aluminum profiles, while the welded steel structure offers a higher safety margin.

Based on the economic analysis, it can be stated that despite the higher acquisition cost of aluminum profiles, the overall costs of both solutions are comparable, with the aluminum profile solution being approximately 4% more expensive. The analysis took into account the costs of material procurement, processing, labor, and the technology required for assembly. The main contribution of the conducted analyses is the confirmation of the suitability of using the alternative solution based on aluminum profiles. Advantages such as flexibility, easy reconfiguration, and lower weight are difficult to quantify, but from the perspective of maintenance and adaptability, they represent a clear benefit.

The final selection of a specific variant, however, ultimately depends on the designer’s decision based on the given circumstances, as in certain conditions, a structure with a higher safety factor may be more appropriate. The results of this comparison can serve as a methodological basis for making informed decisions regarding the appropriate structural solution for frame constructions in industrial applications.