Numerical Investigation on the Buckling Load Capacity of Novel Compound Cross-Sections Used in Crane Construction

Abstract

Although a crane is exposed to a wide range of loads, there is a growing need for a lighter, more slender design. As a result, double girder cranes are becoming single girder cranes, aiming to make the steel structure as light as possible. The optimization potential of the classic design as a hollow-box girder is approaching its end. In order to meet today’s requirements, a new design was developed, which combines beams with curved panels into a new cross-section to be used as the crane’s main girder. Compound cross-sections pose a challenge for the mechanical engineer as there are usually no comparative data available and designing using numerical methods is complex. For this reason, a scaled model was abstracted for which a load determination will be carried out in 2024. This article deals with the finite element calculations for the design of the test specimen. A global numerical analysis was used to determine the buckling load, and several imperfection patterns were investigated. The results revealed that the buckling loads are similar to each other. This finding may lead to the conclusion that the most damaging imperfection pattern has yet to be found, which supports the need for an accompanying series of tests.

1. Introduction

Reducing dead loads and geometric dimensions are essential when designing any load-bearing structure. These criteria are especially true for crane systems, as every weight reduction represents a potential for additional lifting capacity. From a steel structure’s point of view, a crane consists of a main girder where the trolley moves with the attached load and the supporting pillars on either side, which transfer the loads to the carriages. An example of a crane built to move wood is shown in Figure 1.

Typically, the main girder of a crane is constructed as a box girder composed of thin plates. Because thin plates are prone to buckling, the main girder is reinforced with longitudinal and transversal stiffeners. Due to the need for these additional reinforcements, the optimization of weight not only focuses on the optimization of plate thicknesses but also on the positions and number of individual stiffeners. Even though such an optimized design requires some effort, knowledge about how a box girder must be designed is available in the industry, given the long and successful history of such box-type main girders. Consequently, new design approaches are sparse, and the room for further improvement on the box girder seems limited. These limitations are why the second author created a novel design for main crane girders, leaving behind the classical box girder approach and developing a completely new cross-section. A comparison of a classic box-type design for a crane with two main girders and the novel built-up section design with only one main girder is shown in Figure 2.

However, novel and unique compound cross-sections may challenge engineers’ performing strength assessments. Global and local buckling are most interesting when the compound cross-sections contain plane or curved thin-walled elements. Therefore, the engineer will seek guidance in the available codes and guidelines, hoping to be equipped with the needed tools to ensure the safety of the resulting product.

The stability behavior of beam-type elements can be assessed using the methods given in EN 1993-1-1 [1]. In contrast, the approach to deal with the buckling of plated elements is described in EN 1993-1-5 [2], and shell structures are regulated in EN 1993-1-6 [3]. The stability behavior of beams [4,5,6,7] as well as structures built from curved plates [8,9,10,11,12,13] and shells [14,15,16,17,18,19,20,21,22] have been the interest of research for quite some time, and the available knowledge is extensive. The top and bottom elements of the new cross-section consist of curved elements, which have been the focus of active research as of late [8,9,10,11,12,13]. Some authors have even developed an analytic design approach based on finite element calculations [23].

Unfortunately, this design approach is not directly applicable to the curved elements at hand due to the presumed boundary conditions of the mentioned method. After surveying the literature regarding cases of complex compound cross-sections that combine beam-, plate-, and/or shell-type elements, it can be concluded that such cross-sections have been less investigated than the single components that they are composed of. Nevertheless, as far as crane construction is concerned, the analysis presented in [24,25] shall be mentioned as an example as built-up cross-sections are also analyzed in these investigations.

Extensive experimental investigations on large-scale girders do not appear appropriate to establish a design basis for the novel system. The size of the components alone makes it difficult to set up a suitable test rig. Furthermore, the number of specimens required to generate an empirical basis for design assessment would exceed any economic scope. If no experimental data and no analytic design approach are available, the engineer must rely on an assessment using numerical methods.

Some authors [22,26] even advocate the intensive use of these numerical investigations more than the experimental approach. Given that the experimental data for the novel main girder design is non-existent and data in the literature are rare, designing using geometrically and materially nonlinear analysis with imperfections (GMNIA) may be an alternative approach. By dealing with such GMNIAs, the applied imperfections are the key to reliable results. However, defining the “correct” choice of imperfections may not be easy for plates and shells and may even more difficult for built-up sections.

These difficulties and the rare existence of documented cases of similar cross-sections in the literature have led to the decision to execute an extensive verification program for the novel main girder design. This verification procedure consists of three steps. Firstly, the load-bearing capacity of a main girder’s scaled version is determined using finite element computations; imperfections are assumed based on the literature and engineering judgment. As a next step, several of these scaled versions of the main girder are manufactured, and the buckling loads are determined by destructive testing. After manufacturing, the existing imperfections on the structure are recorded using 3D-laser scanning and used further in another set of GMNIA calculations to obtain the buckling load and calibration factor according to EN 1993-1-6 [3]. Upon completion of the test program planned for 2024, a comparative assessment of the test results and the concurrent computations will be performed.

The motivation for the present investigations results on the one hand from the complete novelty of the investigated cross-section, as well as from the necessity of pre-dimensioning the test specimen for the planned tests. Due to current requirements in terms of weight and sustainability, it can be assumed that new solutions for special applications will often appear in the future, which are not fully covered by any standard. The structural engineer then only has the option of using numerical methods as demonstrated in this contribution.

2. Compound Cross-Sections for Use in Crane Construction

The new design was developed to save weight and simplify production, thus creating a more cost-effective design. Significant improvements were possible by choosing the cross-section shown in Figure 2. Those improvements are discussed briefly. Cranes’ operations at container terminals occur mainly in locations near the sea. Consequently, one desired improvement resulting from the new design is reducing the area exposed to the wind compared with the box-type design. Research partners from academia and specialized consultants strongly supported improving the aerodynamic behavior in the design process. In the first step, an external designer assisted in calculating the improved aeroelastic parameters through a computational fluid dynamic analysis (CFD). In the second step, the associated design parameter was confirmed by scaled wind tunnel tests at RWTH Aachen University [27]. Figure 3 shows one of the scaled models in the wind tunnel.

As the classical design consists of plate elements, local buckling must be avoided using stiffeners (compare with Figure 2). As a crane’s main girder can be as long as 40 m or more, avoiding longitudinal stiffeners can save up to several kilometers of stiffeners for one crane alone. In addition to reducing the overall weight, manufacturing costs can be decreased, and notch classes, which may be difficult to inspect, are entirely avoided. The interested reader can find more information on one specific crane in reference [28].

3. Stress-Based Design

Preliminary calculations were carried out to design the test setup and to determine the necessary forces to be applied during the test. As a first design approach, it may be tempting to consider the shell panel at the top as a part of a cylindrical shell and therefore to apply the stress-based design approach according to EN 1993-1-6 [3]. Unfortunately, this idealization is misleading for two reasons. Firstly, it must be assumed that the load-bearing behavior of the shell panels embedded in the novel cross-section differs from that of a fully cylindrical shell. In addition, the Eurocode framework [3,29] covers cylindrical panels under the condition that the correct boundary conditions are applied, which seems difficult to accomplish for the stress-based design approach. Nevertheless, a first estimation of the load-bearing capacity for the panel under meridional pressure was estimated using the stress-based approach for the most stressed panel between two bulkheads. Based on the geometry’s properties, the panel was classified as medium and the load-bearing behavior was classified as elastic-plastic according to the slenderness.

4. Global Analysis

4.1. Numerical Modeling

Given the planned test execution, the geometry of the girder’s interior was slightly simplified, but the initial stiffness of the bulkheads was maintained. The geometry used in the computations and in the later experimental determination of the buckling load is shown in Figure 4. Only a fourth of the girder is shown due to the inherent double symmetry of the structure. However, the computations were performed using the entire model.

The investigations presented were carried out with the commercial finite element program Ansys, Version 2021R1 [30]. Meshing was performed using SHELL281, an eight-node shell element suitable for analyzing thin to moderately thick shells [31]. Because small imperfections compared with the model’s size must be considered with a sufficient resolution, the upper limit of the mesh size was approximated using Equation (1) [32,33].

$$L_{Element}\approx 0.49·\sqrt{R·t}$$

(1)

With a radius of R = 749 mm and a plate thickness of t = 3 mm, the required mesh size is approximately 23 mm. As the first eigenmode obtained by the linear buckling analysis (LBA) showed a pattern of several short waves (later shown in Figure 8 and Figure 9), the mesh size was further reduced, which also proved beneficial to the mesh quality in the region of the free cuts of the bulkheads. In addition, the mesh size convergence study on the first buckling mode of the linear buckling analysis (LBA) confirmed that the mesh size is sufficient.

Boundary conditions were set according to those commonly used in studies of bending-dominated problems, e.g., [21]. The nodes at the end of the girder were rigidly coupled to a master node, where the constraints were applied. On one side, as shown in Figure 5, all translational degrees of freedom were constrained. In addition, the lateral rotations about the longitudinal (z) and vertical (y) axes were fixed, and the structure was only allowed to rotate about the x-axis. On the other end, the translation of the master node was allowed to shift in the longitudinal direction. It should be emphasized that a crane’s main girder might be under the influence of a complex loading scenario consisting of forces and moments acting around more than one axis. As it is not realistic to represent such a complex loading scenario in a set of experiments, the investigations are focused on the main action of the girder. This main action is the weight of a container, the spreader, and the trolley, which transfer the load on the girder over the trolley’s wheels. For the numerical investigation, this scenario is simplified by introducing the loads directly into the main girder’s web using force elements.

The material behavior is considered elastic-plastic with a Young’s modulus of E = 210 GPa and Poisson’s Ratio of $\nu$ = 0.3. To represent a steel grade of S355, the yield strength was set to $f_y$ = 420 MPa, compensating for the assumed and often higher yield strength. Material hardening was neglected. For path tracing, the implemented version of Riks arclength algorithm was utilized.

Because the solver might miss a bifurcation point, the smallest pivot element was monitored during the solving process. In addition, the eigenvalues of at least one hundred load steps on the pre-buckling path were checked after the solver run.

The load application was organized into three loading steps. Firstly, the imperfections were introduced into the model. After that, the self-weight of the structure was activated before the loading was applied and ramped up as the third step until the defined failure criterion (bifurcation or snap-through) was reached. Additional failure criteria are defined in EN 1993-1-6 [3], which were also tested but not explicitly reported, as bifurcation or the snap-through was always the first to occur. As the loading was ramped up after the initial phase, all results shown later relate to the third loading step. According to [3,34], the amplification factor $R_d$ should be derived from the complete combination of the applied loading. However, the authors believe that, in this case, it is more plausible to compute the amplification factor as a function of the applied loads from the trolley without scaling the self-weight, which is comparable with the situation in the planned experiment. There may also be a need for more discussion regarding self-weight treatment in such an analysis [35,36].

4.2. LA and LBA

To determine the linear elastic stress field of a reference loading state, a linear analysis (LA) was performed, presuming linear elastic material behavior as well as small deformation theory. The resulting membrane stresses in the meridian and circumferential direction, as well as bending and shear stresses, are shown in Figure 6 and Figure 7.

As noted in [34], so-called buckling-relevant membrane stresses must be defined for the stressed-based design. An additional linear buckling analysis (LBA) might be performed to gain knowledge of the buckling modes. The first, third, fifth, and seventh eigenmodes are shown in Figure 8 and Figure 9. As can be deduced from these figures, the first eigenmode shows short waves in a meridional direction starting some distance away from the bulkheads. Consequently, stress peaks in the region of the bulkheads were neglected in the stress-based design approach.

Figure 8. (a) First eigenmode; (b) third eigenmode obtained by LBA.

Figure 8. (a) First eigenmode; (b) third eigenmode obtained by LBA.

Figure 9. (a) Fifth eigenmode; (b) seventh eigenmode obtained by LBA.

Figure 9. (a) Fifth eigenmode; (b) seventh eigenmode obtained by LBA.

4.3. MNA

EN 1993-1-6 [3] suggests undertaking a set of preliminary computations before performing a GMNIA. One recommendation is to run a set of linear buckling (LBA) and materially nonlinear buckling analyses (MNAs) based on the small deformation theory to use the results of these analysis, namely the elastic critical resistance expressed by $R_{cr}$ and the plastic limit factor $R_{pl}$, to derive the overall relative slenderness ${\lambda }_{ov}$ of the structure using Equation (2).

$${\lambda }_{ov}=\sqrt{\frac{R_{pl}}{R_{cr}}}$$

(2)

With the overall relative slenderness, it was possible to categorize the behavior of the shell structure into pure elastic buckling, elastic-plastic, or plastic buckling. Because $R_{cr}$ was already determined, the computation of $R_{pl}$ required only an additional materially nonlinear analysis based on the small deformation theory (MNA). Using the obtained values, the overall relative slenderness yielded a value of ${\lambda }_{ov}$ = 1.09, which still falls into the category of elastic-plastic buckling but is close to being a purely elastic buckling problem (${\lambda }_{ov}$> 1.2).

5. GMNIA

Additionally, the buckling resistance of the perfect structure under consideration of geometrical and material nonlinearity (GMNA) [3,34] is computed. The result of this analysis is useful in two ways: firstly, to demonstrate that the later assumed imperfection is sufficiently damaging; and secondly, to obtain the deformation state at collapse for further use as an imperfection in a set of the following GMNIA [34].

5.1. Amplitudes for Equivalent Imperfections

The amplitudes ${\delta }_{eq}$ for the equivalent imperfections were chosen according to EN 1993-1-6 [3] and applied normally to the surface of the respective perfect shell. Depending on the fabrication quality class, Equations (3)–(5) allow the determination of the amplitude as the maximum of ${\delta }_{eq,1}$ or ${\delta }_{eq,2}$.

$${\delta }_{eq,1}=l_g·U_{n,1}$$

(3)

$${\delta }_{eq,2}=n_i·t·U_{n,2}$$

(4)

$$l_g=l_{gx}=4·\sqrt{R·t}$$

(5)

In Equations (3)–(5) above, $l_g$ is the gauge length, and $U_{n,1}$ and $U_{n,2}$ notate the dimple imperfection amplitudes, which depend on the fabrication tolerance quality class. For this investigation, quality class B was chosen. The parameters R and t are the curved elements’ radius and thickness, and $n_i$ is a multiplication factor. A recommended value for this parameter is $n_i$ = 25 [3]. Besides values from the EN, measured values on the full-scale structures were known as well. However, these values were of little interest as the investigation deals with a scaled version of the full-scale component. However, in a later step, it was possible to use this information on the measured imperfections as a construction-specific imperfection.

5.2. Imperfection Form

Defining the correct imperfection pattern is a crucial step in the computation of the buckling load via GMNIA, and several categories of imperfections are defined in the literature, e.g., in [17,18,19,37]. Estimating meaningful imperfection patterns is complex, even for simple structures such as plate elements or cylindrical shells. Choosing a useful imperfection pattern for a built-up section that combines the above structural elements is even more challenging. As already mentioned, this work deals with the scaled test specimen, and no prior information on the existing imperfection shapes and amplitudes was available when this contribution was written. Therefore, the design computations were performed with imperfection patterns based on the literature and engineering judgment.

The above-mentioned challenge can be approached in several ways: firstly, one could consider the possible imperfections of the single parts (beams and panels) before joining; or secondly, imperfections on the cross-section could be derived from the manufacturing process, e.g., weld depressions or collapse-affine imperfections of the whole assembly. A third option would be to consider possible imperfections from the single parts before performing welding simulations to obtain a realistic idea of the complete assembly’s imperfections. Given that a GMNIA is already quite an effort, the latter approach would not be practical for assembled parts as the overall complexity of such a procedure should also be considered. Therefore, it seemed reasonable to use so-called substitute imperfections at the components or assembly level. I-beams are less challenging than shells if one considers the possible imperfections of the single parts. Typically, the applied imperfections are the scaled eigenmodes of an LBA, where often the first eigenmode is assumed [38] as the most important one. An alternative approach is to use a bow imperfection, for which EN 1993-1-1 [1] provides guidance on the amplitude for lateral and lateral torsional buckling cases.

Regarding the panels, the situation is more complex. Here, eigenmode-affine patterns, as well as patterns derived from the collapse behavior of the perfect shell (quasi-collapse-affine) or shapes derived from post-buckling behavior, might lead to the lowest buckling load. In addition, imperfections resulting from the manufacturing process might have to be considered in a simplified way. An example of such an imperfection is the depression occurring at welds, which was used in the analysis of welded parts in the literature [39,40,41].

On the level of the components, bow-type imperfections were considered globally around the weak axis (z–z) on the webs of the beams. The assumption here is that such bow imperfections should stimulate the phenomena of lateral buckling (LBzz) and lateral torsional buckling (LTB). Because the first eigenmode of a beam under pressure force is close to a bow-type imperfection and by considering that eigenmode-affine imperfections are neglected for the beams, only the bow-type imperfection was applied.

For the top panel, eigenmode-affine imperfections were applied. As the first 300 eigenmodes obtained by an LBA influenced the panels on the top side only, it might be safe to conclude that these eigenmodes are associated with the panels only and not with the beams or the whole structure and that the panels are indeed the weak part of the assembly with respect to the loading situation investigated. It is noted in passing that the latter statement might only be true as long as geometrical nonlinearity is not considered.

Like a cylindrical shell, the eigenmodes showed clustered behavior, meaning several patterns ought to exist at load magnification factors with only a slight difference. On the level of the whole component, depressions near the welds at the bulkheads were also studied. It should be noted that even though the weld depression according to Teng and Rotter [42] has been used in the past, it does not seem appropriate for the case at hand as the bulkhead prevents the deformation of the welded area normal to the surface. What seems more realistic is that the welds between the bulkhead and panels result in a rotation of the panels on one or both sides of the bulkhead. For this scenario, two cases were investigated. Firstly, it was assumed that the panel’s bending due to the bulkhead’s weld was local ($l_{gx}$ = 4·$\sqrt{R·t})$; secondly, it was assumed that the imperfection spanned from bulkhead to bulkhead (l > $l_{gx}$). In addition, collapse and quasi-collapse-affine imperfections and combinations of imperfections of the I-beams and panels were investigated.

5.2.1. Eigenmode-Affine Imperfection Pattern

Imperfections affine to the eigenmodes obtained by an LBA are recommended for consideration in a GMNIA [1,43], especially for structures with a linear pre-buckling path [43,44] or for thin shells where plasticity is not dominant [45]. In the present paper, the eigenmodes shown in Figure 8 and Figure 9 were superimposed on the mesh to create an initial imperfection. In the literature, e.g., [36], it is also recommended to investigate several higher order eigenmodes. Consequently, three higher order modes (modes 20, 22, and 186) were investigated as they showed local buckling inside and outside the two inner bulkheads. After performing the LBA, the first 300 eigenmodes were examined. All eigenmodes showed an influence only on the panel at the top of the girder and no influence on the I-beams or on the whole component.

5.2.2. Depressions near Welds

Because the individual parts were joined by welding, a resulting imperfection near the weld seemed plausible. Therefore, the second imperfection pattern considered is a localized weld depression on both sides of the bulkhead, modeled by a half-sine wave in both the meridional and radial direction. This combination—referred to as pattern 2.1—is exemplarily illustrated in Figure 10. Because it cannot be known in advance where such a depression might take place, it was assumed that it will occur next to every bulkhead. In addition, the worst orientation of the depression (inward/outward) is also unknown; the amplitude was considered again in both directions and for verification purposes, also with an amplitude reduced by 10%.

As mentioned above, a second variation of this pattern (pattern 2.2) was investigated. Therein, the half-sine wave was assumed to extend from bulkhead to bulkhead, as shown in Figure 11.

5.2.3. Quasi-Collapse-Affine Imperfection

The concept of collapse-affine imperfection utilizes the deformation state of the perfect structure at the collapse, as obtained by a GMNA [15]. The main advantage of such a procedure is that this type of imperfection incorporates the nonlinear effect of the geometry and the material [46]. As the collapse deformation includes global deformations of the whole structure, the term quasi-collapse-affine means that the global part of the deformation has to be neglected in order to use only the buckling-relevant part of the deformation [15]. In the contribution at hand, both collapse- as well as quasi-collapse-affine deformations were investigated. First, the collapse deformation state of a GMNA was scaled and applied as an imperfection to the ideal structure, as this deformation state also includes imperfections for the I-beams, similar to a bow-type imperfection, as well as imperfections on the panels, which presumably weaken the whole structure. Figure 12 shows the deformation state at the collapse of the ideal structure (GMNA).

After applying the scaled deformation state, it could be observed that the resulting buckling resistance was relatively high compared with the GMNA’s result, which means that this imperfection pattern is not damaging. The reason for this is the relationship that exists between the global deformation at collapse (bending of the whole component) and the deformation in the region near the bulkheads, which is smaller. Consequently, if the maximum deformation is scaled to fit to the derived amplitude ${\delta }_{eq}$, the result is more or less the deformation of the whole structure due to global bending, and the effect of the local imperfections at the bulkheads is negligible. No severe reduction in the load bearing behavior results from this combination of imperfection pattern and maximum amplitude.

To derive a quasi-collapse-affine imperfection pattern, the local deformation at the bulkheads (visible in Figure 12) is reproduced by perturbing the mesh.

5.2.4. Combined Pattern

As the I-beams may also have imperfections, combined imperfections applied to the I-beam and panels are described below. The investigated combined pattern uses a global bow-type imperfection for the beam web in the direction of the weak axis, as shown in Figure 13 on the left. The amplitude was derived from Equation (6) from prEN 1993-1-14:2020 [47]. With respect to the panels of pattern 2.1 (half-sine wave on both sides of the bulkheads), this imperfection was additionally applied by moving the respective nodes. The combined imperfection shape (half-sine waves on the bulkheads, half-sine waves on the web of the outward-facing beams) is shown in Figure 14.

$$e_{0,d}=\frac{\alpha ·L}{150}\ge \frac{L}{1000}$$

(6)

In Equation (6), $\alpha$ notates the imperfection factor taken from EN 1993-1-1 [1], and L is the length between restraints.

6. Results and Discussion

The results of all computations are summarized in

Table 1. Load bearing capacity.

Number	Analysis	Imperfection	Buckling Mode	LPF	${\mathit{R}}_{\mathbf{GMNIA}}$/${\mathit{R}}_{\mathbf{GMNA}}$
1	LBA	none	-	1.00	-
2	MNA	none	-	1.18	-
3	GMA	none	-	0.96	-
4	GMNIA	eigenmode	1	0.43	0.58
5	GMNIA	eigenmode	3	0.44	0.60
6	GMNIA	eigenmode	5	0.45	0.58
7	GMNIA	eigenmode	7	0.45	0.61
8	GMNIA	eigenmode	20	0.45	0.60
9	GMNIA	eigenmode	22	0.45	0.60
10	GMNIA	eigenmode	186	0.46	0.61
11	GMNIA	collapse-affine	-	0.58	0.78
12	GMNIA	quasi-collapse-affine	-	0.48	0.65
13	GMNIA	post-buckling	-	0.48	0.64
14	GMNIA	weld depression	pattern 2.1	0.46	0.62
15	GMNIA	weld depression	pattern 2.2	0.73	0.98
16	GMNIA	combined pattern	-	0.47	0.64

. The column labeled LPF shows the maximum obtainable load proportional factor that can be obtained in relation to the result of the LBA, and the last column shows—if applicable—the relation of the obtained resistance factor of the imperfect and geometrical and material nonlinear structure $R_{GMNIA}$ to the obtained resistance factor of the perfect shell without imperfections $R_{GMNA}$. Because it is not known a priori whether the amplitude has to be applied inwards or outwards to be most damaging, all calculations were performed probing both possibilities. In addition, all amplitudes were reduced by 10% as required by EN 1993-1-6 [3]. In order to keep Table 1 readable, only the lowest values of the respective computations are given.

Three observations can be made regarding the eigenform-affine imperfections. First, the resulting ratios $R_{GMNIA}$/$R_{GMNA}$ do not show significant differences ($\Delta$ < 7%) for all investigated modes; second, the most damaging imperfection pattern of all investigated patterns seems to be eigenmode one or five; and third, the direction (not shown in Table 1) of the applied eigenform-affine imperfection has an insignificant influence on the resulting resistance values.

Only the run with the collapse-affine imperfections stands out when comparing all results. When the mesh is perturbed according to the deformation state at the collapse in a GMNA, the damaging effect is relatively small compared with all other imperfections. As discussed earlier, the scaling of the whole imperfection results in a (globally) small deformation of the whole structure, which appears to be less damaging. If a quasi-collapse-affine imperfection pattern is derived from this deformation’s state (Figure 12), the result are very similar to the ones obtained by introducing a post-buckling eigenshape obtained from a GMNA as an imperfection.

In [36], as a sanity check, the authors propose to check the relation $R_{GMNIA}$/$R_{MNA}$ against the dimensionless buckling knockdown factor $\chi$ from the axial compression or bending capacity curves in prEN 1993-1-6 [29]. In the present work, this sanity check is performed against the curves in EN 1993-1-6 [3], as this version was used in the calculations. Therefore, the dimensionless slenderness $\lambda$ was estimated to be ($R_{MNA}/R_{LBA}$)${}^{0.5}$, giving a value of ${\lambda }_{check}$ = 1.087. This value was then inserted into the capacity curves in Annex D [3] for the cylinder in bending, which yielded a knockdown factor of ${\chi }_{check}$ = 0.33 for a portion of the main girder spanning from bulkhead to bulkhead. This reference value was then compared with $R_{GMNIA,lowest}$/$R_{MNA}$ = 0.36, which appears to be sufficient.

7. Conclusions

While evaluating a one-piece structure using GMNIA is already a complex task, built-up sections increase the difficulty, even for specialists [35,36]. While guidance can be found in the literature for components such as, e.g., stiffened cylindrical shells [48], completely novel built-up sections pose a challenge to the designer. In addition to the potential sources of error, both the theoretical and practical ones associated with a fully nonlinear calculation, the designer can only be partially confident in the results obtained for a completely new built-up cross-section, as no values exist for direct comparison. Nevertheless, in such a scenario, the design by analysis through GMNIA may be the only choice.

The majority of the presented results are quite similar, suggesting that most of the assumed imperfection patterns are sufficiently stimulating or, more likely, that the most damaging imperfection pattern has not yet been found. In any case, it cannot be ruled out that a different combination of imperfections may occur on the real component that further reduces the ultimate load. Therefore, in addition to numerical analysis, it was decided that the buckling load for the newly developed main girder shall be determined by destructive testing in early 2024.

With regard to the numerical design of new components, an automation of the calculations seems to be imperative. This is especially true for complex compound cross-sections, where a high number of imperfection patterns have to be analyzed. Such an automation strategy was already presented in [26]. From such methodical calculations, simpler design approaches can be derived. However, it remains to be seen whether this approach will prove successful outside universities and research institutes due to the software, computational time, and power requirements.

In practice, a crane main girder is not only loaded by bending about one axis. The influence of multi-axis loads is therefore the focus of future research efforts. A comprehensive parameter study is currently being carried out. Other topics worthy of investigation are the influence of load speed and post-buckling behavior.