Stability analysis of newly developed polygonal cross-sections for lattice wind towers

Abstract

The pursuit for cheaper energy is leading the current wind tower design to increased heights. Common wind turbine tower designs would generate unjustified costs for transportation and erection leading to inefficient use of materials. In order to reduce these costs, several simplified erection methods have been proposed. One of such is the hybrid lattice-tubular steel tower. For economic feasibility, built-up cold-formed polygonal cross-sections have been proposed for the lattice part. This article presents a numerical investigation of the failure modes of closed polygonal cross-sections. The first part contains a presentation of structural systems which incorporate elements composed of plates and cold-formed members. The evaluation of the polygonal sections is done by means of finite element analysis considering local and global geometrical imperfections and residual stresses generated in the fabrication procedure. A comparative study is performed between several finite element models to propose a corresponding European buckling curve for calculating the flexural buckling resistance. The results show that the design of polygonal sections can be done according to European buckling curves methodology.

Keywords

Cold-formed finite element simulation flexural buckling lattice wind tower polygonal sections

Introduction

In the discussions regarding the erection of on-shore wind turbine towers with heights above 100 m, it is commonly agreed among the scientific community that the main limitations of the current tubular steel towers are the transportation and erection conditions.

The on-shore wind turbine towers are thus limited by the transportation conditions. To keep the on-shore wind turbines a competitive option, innovative solutions for the tower design are required. The most common configuration of support structures are the cylindrical and conical tubular shells. Alternatively, lattice structures have been used for the wind turbine towers, mainly due to the ease of transportation. Choosing only one type of configuration (tubular or lattice) one are limited by the disadvantages that these bring. Higher wind turbine towers would require a wider tubular section at the base that would be difficult to manufacture on site or transport if a cylindrical configuration is chosen, in comparison to a lattice tower where the problems would arise due to a time consuming and tedious erection process. As presented by Hau (2013), the conical tubes present a great advantage for heights up 80 m, as shown in Figure 1. However, from this point on, the mass and the costs increase significantly. The presented data do not specify if the prices include assembly costs.

Figure 1.

Tendencies of costs and mass for a 3-MW wind turbine with a rotor diameter of 100 m (2011 steel prices; Hau, 2013).

An alternative to conical section was used to build the modular Steel Shell tower (Kryger and Ryholl, 2013). This tubular tower claimed to have the capacity of reaching heights of more than 150 m. One of such was built to the height of 140 m. However, it did not have a very high impact on the market. The tower is composed of bent plates as shown in Figure 2.

Figure 2.

Plate used in the production of Steel Shell tower (Kryger and Ryholl, 2013).

The cheapest option in this case would be the use of lattice towers. However, these types of towers have been used in the past and present a series of problems such as installation and maintenance costs. Regardless of their known issues, the designs can be improved and adapted to the specificity of the wind turbine. Recently, an 80-m wind turbine tower was suggested by Husemann and Meiners (2013). The idea behind the design lies in the ease of installation. It is a structure composed of cold-formed polygonal elements (see Figure 3) for the legs that can easily be connected with the stabilizing diagonals and horizontal struts. This option becomes even more difficult for higher wind turbine towers due to the necessary number of connections. The individual plates of the cross-section are relatively slender and S355 (mild-strength steel) is recommended. Building higher towers would require an increase in either the cross-section, leading to higher installation and transportation costs, or the strength of the material.

Figure 3.

Plate used for the built-up polygonal cross-section (Stemwede and Bohmte, 2013).

The optimality of a chosen structural system can vary based on different factors such as the number of connections required, the total material used, the installation costs, and the time needed for installation. The complexity of joints increases when dealing with polygonal geometries. However, connections are still feasible without welded solutions using semi-closed sections as shown in Figure 4. The target for investors is normally the total cost, which can vary in a very wide range based on the location of the construction and the fluctuation of the material costs. It is assumed that for the tower to be appealing for the manufacturers a fast execution time is required. Therefore, a reduction of the number of connections is recommended with members as long as possible without secondary bracings.

Figure 4.

Example of joint for polygonal lattice tower (Jovašević et al., 2016).

The resulting towers require a large amount of bolts, which can cause maintenance difficulties. Several solutions have been suggested for this application such as Bobtail^® bolts which are considered to require minimum maintenance. Studies have been made to determine the tension loss of the bolts due to temperature variation (Matos et al., 2017).

A study of the independent plates has been performed by Tran et al. (2016a). The investigated cold-formed plates are made of S650 high-strength steel (HSS). The boundary conditions used in the experiments were considered to be fully fixed. Based on the experimental results, a comparison by means of finite element method was later made (Tran et al., 2016b). It was concluded that a good agreement between ultimate loads can be achieved by considering the imperfections recommended by the EN 1993-1-1 (CEN, 2005).

As stated in the EN 1993-1-1 (CEN, 2005), the verification of the members subjected to pure compression allows for the full use of the cross-sectional area if it is proven to be in the range of classes 1–3. Therefore, the aim would be to use cross-sections at the limit of class 3 to class 4.

This article investigates the behavior of such cross-section considering the plates working as a complete polygonal cross-section, disregarding the contribution of the adjacent plates used for connecting the plates. The methodology for assessing the cross-section class is presented together with the evaluation of the member failure modes. The results are then compared between different cross-section sizes. Finally, the conclusions are presented and discussed.

Methodology

Polygons with nine facets were considered for the analysis. The first step was to identify the limit between the classes of the polygonal cross-sections. The classification provided in EN 1993-1-1 (CEN, 2005) is based on the slenderness of the individual component plates. The classification is done in terms of plate width-to-thickness (b/t) ratio. The plate width was considered to be the distance between the rounded corners, referred to as b_p in Figure 5. Although the reduction of effective area is required only for class 4 sections, according to EN 1993-1-5 (CEN, 2007) a reduction of area occurs in plates with slenderness ${\bar{λ}}_{p} > 0.673$ . A class 4 section is defined by local buckling of the cross-section before reaching the yielding point of the most compressed fiber. The class 4 limit could be defined by the change of the first buckling mode. The change of buckling modes can occur between the Euler buckling mode and the plate buckling mode or the simultaneous buckling of several plates if the bends do not provide a sufficiently stiff rotational constraint. Such behavior has been analyzed by means of generalized beam theory by Gonçalves and Camotim (2014).

Figure 5.

Notations for the geometrical properties of the section and the bent corner.

The critical load of a double-symmetric compressed member with the buckling length, L_cr, and a second moment of area, I, is known to be defined by equation (1) and the critical load of a compressed plate of a given thickness, t, and width, b, is defined by equation (2)

P_{c r, E} = \frac{π^{2} E I}{L_{c r}^{2}}

(1)

σ_{c r} = k \frac{π^{2} E}{12 (1 - ν^{2})} \cdot \frac{t^{2}}{b^{2}}

(2)

The elastic critical load defined by Euler is not valid for intermediate or short members. For values of slenderness of $\bar{λ} < 1.2$ , the yield strength of the material influences considerably the ultimate resistance. The European code proposes the formulation as per equation (3). The equation implies that the resistance is a function of the local and global slenderness and of the yield strength of the material

N_{b, R d} = χ A_{e f f} f_{y}

(3)

The radius of the circle circumscribing the polygonal cross-section, R, and the effective buckling length of the members have been varied to visualize the extent of the potential area reduction if HSS is used. The ultimate purpose would be to determine a cross-section size to which HSS can be efficiently used. As shown in Figure 6, the advantage of the HSS is clear for low member slenderness. Afterwards, the potential advantage is gradually lost due to the instability effects.

Figure 6.

Variation of radius of the circle circumscribing the polygonal cross-section and the normalized critical loads as a function of effective member length for different steel grades.

The second part of the study consisted of analyzing a series of selected cross-sections presented in Tables 1 and 2. The sections were chosen to have approximately equal areas, thus varying the local and global slenderness, by means of the b/t ratio and the second moment of area, I. The plate thickness range was chosen between values that are possible to fabricate using both normal-strength steel (NSS) and HSS, t = 6–10 mm. The section class and the corresponding effective properties were calculated according to EN 1993-1-1 (CEN, 2005).

Table 1.

Geometrical properties of the analyzed struts S355.

R (mm)	L (m)	b (mm)	b/t	Class	P	A (mm²)	A_eff (mm²)
350	5.00 6.43	204	23.80	1	1.00	18,149	18,149
375	7.86 9.29	221	27.52	2	1.00	18,287	18,287
400	10.71 12.14	240	31.97	3	0.986	18,399	18,173
450	13.57 15.00	271	40.58	4	0.854	18,257	15,822

Table 2.

Geometrical properties of the analyzed struts S700.

R (mm)	L (m)	b (mm)	b/t	Class	ρ	A (mm²)	A_eff (mm²)
320	5.00 6.43	179	19.01	1	1.00	18,375	18,375
330	7.86 9.29	180	19.74	2	1.00	18,150	18,150
350	10.71 12.14	204	23.80	3	0.966	18,149	17,626
375	13.57 15.00	221	27.52	4	0.886	18,287	16,459

Rounded corners were considered in the calculation of the geometrical properties. The hardening of the material in the bent corners was considered through the addition of residual stresses.

An initial buckling analysis of the members was performed. The purpose was to identify the buckling modes and to determine an appropriate imperfection pattern that would lead to a minimum failure energy. The requested number of modes was set to 5, in order to eliminate the closely spaced symmetrical modes.

The variation of area with length can be representative if the cross-section will keep the same class for different global and local slenderness. A surface can be plotted by keeping the area constant for different global slenderness as shown in Figures 7 and 8. Thus, the variation of the reduction factors due to flexural buckling χ and local buckling ρ can be observed. The “c” buckling curve was represented with α = 0.49. The slope of the surface in class 4 varies based on the type of steel used.

Figure 7.

Surface plot of the reduction factor based on the flexural and local slenderness for S355.

Figure 8.

Surface plot of the reduction factor based on the flexural and local slenderness for S700.

It can be observed that the range of class 2 and class 3 cross-sections is highly reduced regardless of the steel used.

Finite element model

A three-dimensional (3D) finite element model (FEM) was chosen for the simulation. The model was created using 4-node rectangular shell elements with reduced integration (S4R) and five thickness integration points, integrated using the Simpson method. The ultimate loads were determined using a load-controlled static modified Riks analysis, commonly used in tackling stability problems. The model incorporated material nonlinearity geometrical imperfections and residual stresses (geometrical material nonlinear imperfection analysis (GMNIA)) as presented in Table 3. The size of the mesh elements was set to 20 mm.

Table 3.

FEM combinations.

Model	Global imperfections	Local imperfections	Residual stresses
M1	L/750	b/50	No
M2	L/750	b/200	No
M3	L/750	b/200	Yes
M4	L/1000	b/200	No
M5	L/1000	b/200	Yes

FEM: finite element model.

The static scheme is a simple pinned–pinned strut subjected to compression. The torsional degree of freedom is restrained at one end. The translational degrees of freedom of the end nodes of the model were constrained to a center point to act as a rigid body. The loads and restraints were applied in the two reference points. The geometry of the section took into account the roundness of the corners considering a bending radius r_i = 6t. The thickness was reduced by 1.5% in the rounded corners to account for the thinning due to the manufacturing process. Abvabi et al. (2015) evaluated the bending process and its effects on the longitudinal stresses and on the imperfection of V-shaped sections. The results obtained based on the numerical analysis performed on HSS show that the amplitude of longitudinal stresses is affected based on the thinning percentage of the plates due to cold forming. The residual stress models for the cold-formed sections are largely debated due to the manufacturing process, the b/t ratio of the bent plates, and the type of steel used. Some researchers consider that cold forming introduces mostly transversal residual stresses and that longitudinal residual stresses are almost inexistent. The model of residual stresses chosen is presented and has been applied for NSS and HSS as recommended in Somodi and Kövesdi (2017a, b). Compressive residual stresses were introduced in the flat part and tensile stresses in the corner. As stated by the authors, the application of residual stresses is limited to a certain manufacturing procedure, b/t ratio, plate thickness, and steel type. Some of the investigated elements are slightly outside the range of recommendation, but the hypothesis was considered valid due to simplicity.

The global out-of-straightness imperfection was defined as a sinusoidal shape with the maximum amplitude at the mid-span of the member, as the methodology applied by Beer and Schulz (1970) to validate the European buckling curves. The local imperfections were defined as recommended by EN 1993-1-5 (CEN, 2007) for plate buckling, by considering a sinusoidal deformation applied on the two directions as shown in Figure 9.

Figure 9.

Schematic of applied local imperfections.

The imperfections were introduced by translating the mesh nodes with horizontal displacements as defined by equations (4) and (5) as follows:

Local imperfection

e_{0 w} (x^{'}, z) = {(- 1)}^{n} e_{0 w} \sin (\frac{π x^{'}}{a}) \sin (\frac{π z}{a})

(4)

Global imperfection

e_{0} (z) = e_{0} \sin (\frac{π z}{L})

(5)

The shape of the local imperfections is slightly different from the shape obtained by the buckling analysis. Since the number of corners is odd, the local buckling shape is not symmetrical resulting in a reduced amplitude of imperfection in one of the plates as shown in Figure 10. Thus, the most unfavorable combination of global and local imperfections would be omitted. The amplitude of the local imperfection was kept constant over the entire section and the global imperfection was added opposite to the plates with consecutive identical imperfections.

Figure 10.

First buckling modes.

The amplitude of the imperfections was varied in order to evaluate their effect on the ultimate resistance. Studies conducted by Dubina and Ungureanu (2014) have shown that the imperfections have a higher influence on the resistance of cross-sections when the interaction between modes occurs. The M1 model considers the maximum imperfections that can exist in the structural member as defined by the delivering specifications presented in CEN (2011).

The steel material model is multilinear based on the true stress–strain relationship of coupon tests available at the Steel Structures department at Luleå University of Technology. Two types of steel were chosen for the analysis: the mild steel Ruukki Multisteel S355 and the HSS SSAB Domex 700. Six points of each curve were introduced to describe the material behavior in the FEM (Figure 11). The points were selected to define the plastic region and a modulus of elasticity of E = 210,000 MPa was assigned.

Figure 11.

Tension coupon tests and equivalent FE material.

Results

The comparison between M1 and M2 shows that the local imperfection has a reduced effect on the ultimate resistance of the compressed members with an average value of 8.60%. The maximum difference was recorded to be 10.85% for the NSS class 3 cross-section and 10.47% for the HSS class 4 cross-section. Based on the results, the class that is mostly affected by the local imperfections seems to differ based on the type of steel used. This can be attributed to the interaction of plastic–elastic failure. Due to the hardening of the HSS, the buckling can occur before the steel reaches the softening plateau; however, the NSS reaches the softening plateau earlier, thus causing higher deformations to occur.

For the purpose of common designs, the variation of the global imperfection from L/1000 to L/750 does not affect much the ultimate resistance of the analyzed columns. The comparison between M2 and M4 shows an average difference of 1.88% of the ultimate resistance, with the maximum value of 3.81%. The dispersion of results obtained from compressive experimental tests performed on different types of sections has proven to be much larger (Chan et al., 2015; Li et al., 2016; Wang et al., 2014; Zhou et al., 2013). In numerical models, the choice of amplitude of imperfections can be safely estimated as the recommended fabrication value of EN 1090-2 (CEN, 2011). Higher amplitude of global imperfections would certainly affect the ultimate resistance. However, for industrial application a global imperfection higher than L/750 should not be accepted, as recommended by EN 1090-2 (CEN, 2011).

The modeled residual stresses have been shown to have a small influence on the resistance. The longitudinal bending residual stresses can compensate for the initial geometric imperfection, thus leading to an increased resistance. The overlapping of the two generally occurs on a random basis. The NSS was more affected by the residual stresses due to the nature of the expression, providing higher stresses relatively to the strength of the steel. Columns with higher global imperfection showed a higher sensitivity to residual stresses with a maximum of 2.03% between M2 and M3 and 1.35% between M3 and M4.

Models M2–M5 agree well with the Eurocode buckling curve “c,” providing a small factor of safety (Figure 12). This is more evident for the class 1 and 2 sections. At low global slenderness values $\bar{λ} < 0.6$ , the results obtained with the M1 combination of imperfection tend to be lower than those obtained with the flexural buckling curve. The applied imperfections are exaggerated and serve as a verification of the imperfection sensitivity of the problem.

Figure 12.

Normalized ultimate strength compared to the “c” buckling curve.

Conclusion

The article presents the motivation of choosing to investigate the polygonal cross-sections and the approach of the selected subject. A short introduction and state of the art on the current steel wind turbine towers’ structural typologies are presented. The methodology of the analysis is explained and the selected study cases are specified. The results from the numerical analysis are presented and discussed. Finally, the conclusions of the study are being presented together with the future work.

The EN 1993-1-1 (CEN, 2005) flexural buckling curve “c” can be used for designing class 1–3 polygonal cross-sections if EN 1090-2 (2011) manufacturing standards are applied.

Cold-formed closed polygonal cross-sections in classes 1 and 2 can be designed using curve “c” for flexural buckling for both NSS and HSS. The results have shown that the same curve can be used for HSS class 3 cross-sections if the global slenderness is above 0.6. For lower slenderness as well as for class 4 cross-sections, the curve “c” does not prove to be a safe estimate. This part would require further experimental studies.

By limiting the plate buckling of the individual polygonal facets to the Euler buckling mode, a safe estimate can be made to keep the cross-section in class 3. However, the ultimate resistance of the member becomes sensitive to imperfections. This effect is expected to amplify if a built-up cross-section is used. In this situation, the combination of modes becomes a higher problem.

The model of residual stresses used does not have significant influence on the ultimate resistance of the members. The longitudinal residual stresses obtained from the bending procedure at a 120° angle should be further investigated.

Footnotes

Appendix 1 Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors would like to acknowledge the support of the European Commission’s Framework Program “Horizon 2020,” through the Marie Skłodowska-Curie Innovative Training Networks (ITN) “AEOLUS4FUTURE—Efficient harvesting of the wind energy” (H2020-MSCA-ITN-2014: Grant Agreement No. 643167) to this research project. The numerical simulations were performed using resources provided by the Swedish National Infrastructure for Computing (SNIC) at High Performance Computing Center North (HPC2N).

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