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Investigation of nonlinear dynamic behavior of lattice structure wind turbines
Renewable Energy 97 (2016) 33e46
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Renewable Energy journal homepage: www.elsevier.com/locate/renene
Investigation of nonlinear dynamic behavior of lattice structure wind turbines Davoud Nezamolmolki, Ahmad Shooshtari* Department of Civil Engineering, Ferdowsi University of Mashhad, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 June 2015 Received in revised form 16 May 2016 Accepted 17 May 2016
Wind turbines are kinds of electricity generators, proliferating nowadays due to their consistency with the environment. The rotation of the wind turbine blades duo to wind, burdens some frequencies on the wind turbine tower. Therefore, in addition to common design due to service loads, the wind turbine tower should be checked in the frequency analysis so that its natural frequencies do not coincide with the frequencies caused by blades rotation, resulting in the resonance of the tower and its failure. This makes the design of these structures to be complex. In this research, the frequency analysis of the lattice tower of a three-blade horizontal-axis wind turbine including different sources of nonlinearities, i.e., geometric, material and joint slip effect is carried out and the natural frequencies are obtained using the developed Nonlinear Analysis Software for Towers -NASTower-. It is observed that the joint slip effect can substantially reduce the natural frequencies of the lattice tower, which are significant in the design of these structures. In addition, the wind turbine displacements due to different design load cases are investigated, indicating that incorporating the joint slip effect into the analysis substantially increases the wind turbine displacements, which could affect on its performance. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Lattice tower of wind turbine Nonlinear effects Join slip effect Frequency analysis Natural frequency Mode shape
1. Introduction In recent years, the tendency to renewable energies has been increasing due to the environmental problem caused by the use of fossil fuels. Wind energy is one of renewable energies, extracted from wind, using wind turbines to produce electrical energy. Therefore, the analysis and design of the wind turbines is one of the structural and mechanical engineering problems. The wind turbines are mostly mounted on the tubular and lattice towers and it is an advantage to have a high tower, since wind speeds increase farther away from the ground. In the analysis and design procedure of the wind turbines it is necessary to avoid the coincidence of the blades rotation frequencies with the natural frequencies of the tower, which brings about the resonance of the tower and its failure. This makes the design of the wind turbine towers to be complex. Many of researchers have devoted themselves to the study of the wind turbines. Gebhardt and Roccia presented an aeroelastic model intended for three-blade large-scale horizontal-axis wind
* Corresponding author. E-mail address: [email protected] (A. Shooshtari). http://dx.doi.org/10.1016/j.renene.2016.05.070 0960-1481/© 2016 Elsevier Ltd. All rights reserved.
turbines [1]. This model results from the coupling of an existing aerodynamic model and a structural model based on a segregated formulation derived in an index-based notation that enables combining very different descriptions such as rigid-body dynamics, assumed-modes techniques and finite element methods. Also, a novel design optimization model for placing frequencies of a wind turbine tower/nacelle/rotor structure in free yawing motion was developed and discussed by Karam and Maalawi [2]. The main aim was to avoid large amplitudes caused by the yawing-induced vibrations in the case of horizontal-axis wind turbines or rotational motion of the blades about the tower axis in case of vertical-axis wind turbines. This could be a major cause of fatigue failure and might severely damage the whole tower/nacelle/rotor structure. Jia presented a practical and efficient approach for calculating wind induced fatigue of tubular structures, the effects of the wind direction, across wind and wind grid size on the high cycle fatigue of the structure were addressed [3]. A grid is defined which covers the whole structure. For each node in the grid a time series is generated. This time series contains the mean wind speed and the statistical properties of the fluctuating part of wind components as defined by the one point wind spectra and the coherence functions. In each time step of the dynamic response calculation, the large deformation effects and the wind induced drag forces due to the
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updated structural deformations are taken into account. Moreover, a new design scheme of light weight structure for wind turbine tower was presented by Li and Lu [4]. This design scheme is based on the integration of the nanostructured materials produced by the Surface Mechanical Attrition Treatment (SMAT) process. The objective of this study was to accomplish the weight reduction by optimizing the wall thickness of the tapered tubular structure. Andersen et al. presented a probabilistic approach about the natural frequencies of wind turbines on monopile foundations in clayey soils [5]. A nonlinear stochastic p-y curve was integrated into a finite difference scheme for calculation of the monopole response in which p is the soil resistance and y is the absolute value of the pile deflection. Furthermore, the investigations on the nonlinear behavior of lattice structures are frequently observed in the literature. AlBermani and Kitipornchai presented a nonlinear analytical technique for simulation of the ultimate structural behavior of selfsupporting transmission towers, incorporating both geometric and material nonlinearity effects. Modeling of material nonlinearity for angle members is based on the assumption of lumped plasticity
coupled with the concept of a yield surface in force space. Several transmission towers were modeled and analyzed in developed AK TOWER, which stands for incorporating the presented technique, to simulate their behavior in the full-scale experiments [6e10]. Moreover, Chan and Cho proposed a practical second-order analysis and design method for trusses composed of angle sections. Realistic modeling of semi-rigid connections associated with one- and twobolt end-connections with flexible gusset plate and member imperfections such as initial curvatures and residual stresses is made and load eccentricity is also simulated [11]. Also, Ungkurapinan studied the behavior of such joints, incorporated 36 joint tests, generated joint slip data and developed mathematical expressions to describe slip and load-deformation behavior. In this study, it was concluded that joint slip effect cannot be eliminated and incorporation of the reported joint slip data or mathematical expressions in the tower analysis software will refine their results [12]. Afterwards, Ungkurpinan joint slippage models were employed by Jiang et al. in the modeling of a transmission tower, which resulted in improvement in the displacements results, coinciding with the fullscale test results [13]. In other research, the nonlinear finite
Fig. 1. Nonlinear effects: P-D and P-d
Fig. 2. The detail of a element with fiber section.
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element analysis program NE-NASTRAN had been used by Rao et al. to model the elasto-plastic behavior of transmission towers. Employing the plane elements, the elasto-plastic deformations of the members in the full-scale test were predicted in these studies [14,15]. Finally, Nezamolmolki and Aftabi Sani investigated the dynamic behavior of a Sloping frame, comparing the closed-form solution and the finite element one [16]. In this paper, the frequency analysis of the lattice tower of a three-blade horizontal-axis wind turbine including different sources of nonlinearities, i.e., geometric, material and joint slip effect is carried out and the natural frequencies are obtained, using the developed Nonlinear Analysis Software for Towers -NASTower-. Moreover, since the lattice structures are erected by bolted connections, the accurate modeling of these connections is important. Single-bolted connections accurately could be modeled as hinged connections but double-bolted ones could not be modeled as hinged ones. Herein, this problem is investigated and since most of the end-connections are double-bolted, these connections are modeled as semi-rigid connections. It is observed that the joint slip effect can substantially reduce the natural frequencies, which are effective in the design of these structures. Also, the wind turbine displacements due to different design load cases are investigated, indicating that incorporating the joint slip effect into analysis substantially increases the wind turbine displacements, which could effect on its performance. It should be mentioned that in order to validate the results obtained by the NASTower, some analyses are conducted by the MStower as a valid and reliable commercial software, which is utilizing by many of engineers for design of lattice towers.
received by software and exact material properties are not considered [18]. In this case, the applied load on angle section is not replaced with its components on principle axes, eventuating deflection just in the plane of applied load while accurately, the applied load components on principle axes bring about deflections at orthogonal planes, which are neglected when element with Elastic section is used. On the other hand, in the case of Fiber section, the section is divided into regular shaped meshes and stressstrain relationship is defined for each mesh, dependent to its material. Consequently, forces at integration points are obtained based on the reports from the meshes, resulting in the forces at two ends of element. Contrasting the Elastic section, in the case of Fiber section, the applied load is replaced by its components on principle axes, eventuating deflections at orthogonal planes as actuality. As mentioned before, the element with Fiber section incorporates the material nonlinearity in the analysis. Fig. 2 shows the details of Fiber section. 2.3. Joint slip effect Joint slip effect is one of the most significant nonlinear effects in the lattice transmission towers which is the relative displacement, at a negligible or very low stiffness, of a bolted joint subjected to a shear load. It occurs since boltholes are drilled oversize to provide an erection tolerance of 1.6 mm (1/16 in). For transmission towers, greater slippage is likely to occur as bolt diameters are small, members joined are thin, bearing type joints with a lower clamping force are used, and coefficient of friction of galvanized faying surfaces is low [19]. In order to incorporate joint slip into modeling, nonlinear springs are
2. Nonlinear effects Three major sources of nonlinearity for a latticed structure are identified: geometric nonlinearity, material nonlinearity, and joint flexibility and slippage [10]. 2.1. Geometric nonlinearity The P-Delta effect (P-D) and the P-delta effect (P-d) are the most important geometric nonlinear effects taken into account in the nonlinear analysis as shown in Fig. 1. You may independently include or exclude these two major effects. The P-Delta effect (P-D) occurs when deflections result in displacement of loads, causing additional bending moments that are not computed in linear analysis. Also, the bending stiffness of a member is reduced by axial compression and increased by axial tension called the P-delta effect (P-d) [17]. 2.2. Material nonlinearity This type of nonlinearity arises when the material exhibits nonlinear stress-strain relationship. Recall that for elastic linear finite element analysis the only stress-strain relationship is defined via modulus of elasticity, E. It should be mentioned that common analytical softwares employ the approximate approach of plastic hinge in order to consider the material nonlinearity. The recent approach to consider material nonlinearity is employing the elements with Fiber section in the analysis [18]. 2.2.1. Fiber section - elastic section In general, nonlinear analytical softwares introduce two types of wire elements: wire element with Elastic section and wire element with Fiber section. In the case of Elastic section, some characteristics such as elasticity modulus, section area, moment of inertia, yield stress, shear modulus and polar moment of inertia are
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Fig. 3. Typical bolted joint connections [13].
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Fig. 4. Ungkurapinan joint slippage models [12,13].
employed at connections as shown in Fig. 3, which their forcedisplacement relationships are defined by Ungkurapinan joint slippage models indicated in Fig. 4 drawn for normal clearance as shown in Fig. 5. The Ungkurapinan joint slippage models are independent from the bolt size and material. In addition, there are some approximations in the modeling of the various splices with different angle sizes and bolt numbers. Fig. 5. Normal clearance [13].
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3. Modeling of semi-rigid connections
4. The wind turbine details
Since the end-connections of the current lattice tower of the wind turbine are mostly double-bolted, in this research these endconnections are modeled as semi-rigid connections based on the formulation presented by Chan and Cho [11]. As shown in Fig. 6, the couple, M, formed by the shear forces exerted on the bolts, F, is given by:
In this research, a 100-kW three-blade horizontal-axis wind turbine, which possesses a lattice tower with a 2 m 2 m square base and a height of 30 m from the ground, is studied. The lattice tower is comprised by angle sections with the steel grade of 235 Mpa. Also, the wind turbine has three 13-m blades with a summit distance of 3 m from the center vertical axis of the lattice tower. The details of the wind turbine and its power curve are shown in Figs. 7 and 8, respectively.
M ¼ F d ¼ k:q
(1)
Also the shear stress, t, across the cross-section, A, of the bolt is: 5. Analysis procedure
F t¼ A
(2)
in which A is the shear area and can be taken as 0.9 of the cross sectional area as recommended in most design codes. Also the shear strain, g, of the bolt shank is:
g¼
2d d:q t ¼ ¼ l l G
(3)
where d is the distance between the centroids of the two bolts; q is the rotation of the bolt group; d is the displacement of the bolt; l is the length of the bolt shank and G is the shear modulus of elasticity. Rearranging terms, the rotational stiffness, k, due to the doublebolted connection will be given by:
k¼
GAd2 l
(4)
Fig. 6. The couple formed by shear forces of two bolts [11].
The lattice tower of the wind turbine is modeled in Nonlinear Analysis Software for Towers -NASTower-, developed in this study. This modeling and analytical software, capable of modeling different kinds of towers such as transmission towers, telecommunication towers and wind turbine towers, can take different source of nonlinearity: geometric, material and joint slip effect, into account. Moreover, created models in common commercial softwares like MStower could be imported in NASTower in order to refine modeling by applying joint slippage models and semi-rigidity at connections. It should be mentioned that NASTower employs OpenSees software as processor, including many capabilities in the modeling of different structures with diverse elements. The advantages of NASTower in Comparison with MStower are the capability of modeling with Fiber section, which considers the material nonlinearity, incorporating the joint slippage into analysis and modeling the semi-rigidity of the connections. In the modeling of current lattice towers 3D beam-column elements are employed. Generally, OpenSees includes different 3D beam-column elements such as Elastic Beam Column Element, NonLinear Beam-Column Elements (Force based elements and Displacement based element), Beam with Hinges Element and Nonlinear Beam Column Element. In this study, the Elastic Beam Column Element is utilized in elastic analysis and the Nonlinear Beam Column Element, which is based on the non-iterative (or iterative) force formulation, and considers the spread of plasticity along the element, is utilized in inelastic analysis. The integration along this element is based on Gauss-Lobatto quadrature rule [18]. Also, the Zero-Length Elements are employed at the two ends of each wire element to incorporate the joint slip effect in analysis. Two nodes at the same location define this element. The nodes are connected by multiple UniaxialMaterial objects, which define the stiffness, to represent the force-deformation relationship for the element. In this process the axial DOF stiffness is defined by Ungkurapinan joint slippage models. In addition, the stiffness of the rotational DOF in the orthogonal plane on the bolt shanks at end-connections is defined based on Eq. (4) to consider the semirigidity. In other cases, the DOFs stiffnesses are considered infinite leading to rigidity. In linear analysis procedure, the linear algorithm is applied to construct a Linear algorithm object which takes one iteration to solve the system of equations. In the nonlinear analysis procedure, the load-control Newton algorithm is applied to construct a Newton-Raphson algorithm object, which uses the Newton-Raphson method to advance to the next time step. In this method, the tangent at the current iteration is used to iterate to convergence. It should be mentioned in the analysis procedure, the loads at different cases are applied incrementally. Moreover, the supports are fixed modeled and different sources of nonlinearities, as explained in section 2, are incorporated in each case of the analysis.
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Fig. 7. The details of the wind turbine.
6. Results and discussion In this section, first of all the frequency analysis is carried out and the natural frequencies of the lattice tower of the wind turbine for different cases of analysis, incorporating nonlinear effects and semi-rigidity of the connections, are presented and compared. Afterwards, the first ten mode shapes of the lattice tower of the wind turbine are illustrated. Then, the displacements of the wind turbine due to some critical design load cases for different cases are presented and compared. As mentioned, herein different cases of the analysis are carried out, which incorporate different parameters such as nonlinearity effects and semi-rigidity of the connections into the analysis. The details of these cases are presented in Table 1. As mentioned before, in order to validate the obtained results by NASTower, some analyses are conducted with MStower as a valid and reliable commercial software. As it is clear, the case number “II” represents the model of MStower. The further comparisons are presented at Ref. [20].
Fig. 9. It is necessary to mention that prior to frequency analysis the lattice tower is pre-analyzed due to the weight load of the wind turbine for different cases as presented in Table 1. Each pre-analysis brings about some displacements and deflections. Therefore, the frequency analysis is conducted on a deformed tower with modified stiffnesses in members and nonlinear springs, which model the joint slip effect. The pre-analyses are clear in Fig. 9 and Table 2. In the design procedure of the wind turbines, it is important to design the structure such that the frequencies caused by the rotation of blades and rotor lie at the range between the natural frequencies of the wind turbine structure to avoid the resonance of the structure
6.1. Frequency analysis results The current lattice tower of the wind turbine is modeled in NASTower and its first ten modes natural frequencies for different cases are obtained as presented in Table 2. Also, the column chart of the values of the natural frequencies at each mode is shown in
Fig. 8. The power curve of 100-kW wind turbine.
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Table 1 Details of different cases of the analysis. Case
Definition
Section of the elements
Modeling of double-bolted end-connections
Geometric nonlinearity
Material nonlinearity
Joint slip effect
I
Elastic linear with rigid connections with NASTower Elastic linear with rigid connections with MStower Inelastic nonlinear with rigid connections with NASTower Inelastic nonlinear with semi-rigid connections with NASTower Inelastic nonlinear with joint slip with NASTower Inelastic nonlinear with semi-rigid connections and joint slip with NASTower
Elastic
Rigid
Not considered
Not considered
Not considered
Elastic
Rigid
Not considered
Not considered
Not considered
Fiber
Rigid
Considered
Considered
Not considered
Fiber
Semi-rigid
Considered
Considered
Not considered
Fiber
Rigid
Considered
Considered
Considered
Fiber
Semi-rigid
Considered
Considered
Considered
II III IV V VI
Table 2 The first ten natural frequencies of the wind turbine lattice tower for different cases of pre-analyses. Pre-analysis case
I
II
III
IV
V
VI
First mode Second mode Third mode Fourth mode Fifth mode Sixth mode Seventh mode Eighth mode Ninth mode Tenth mode
0.70 0.70 5.08 7.44 7.45 13.72 19.24 19.28 21.36 25.85
0.70 0.70 5.06 7.44 7.44 13.72 19.23 19.28 21.33 25.71
0.69 0.69 5.05 7.39 7.39 13.61 19.15 19.20 21.33 25.71
0.69 0.69 5.05 7.39 7.39 13.61 19.15 19.20 21.33 25.37
0.50 0.63 3.56 5.76 6.37 12.04 14.28 15.53 15.56 25.68
0.50 0.63 3.56 5.76 6.37 12.04 14.28 15.53 15.56 25.35
and its failure. In the case (I) pre-analysis, the elements with Elastic sections are used and due to the weight load of the wind turbine linear analysis is conducted and then, the frequency analysis is carried out. Herein, the linear trend of the material and linear analysis makes the stiffness remains constant before and after pre-analysis. In other word, the weight of the tower is not effective on the tower stiffness. Case (II) pre-analysis is quite similar to (I) but the tower is modeled in MStower to validate the results by comparison. As it is clear from Table 2 and Fig. 9, the results of this case are in a very good agreement with the results of the case (I) which validates and confirms the accuracy and precision of the
developed NASTower software. In the case (III) pre-analysis, the elements with Fiber sections are used and geometric nonlinearity due to weight load of the wind turbine is considered. In this case, the axial forces in elements change their stiffnesses, reducing tower total stiffness, which makes the natural frequencies to be smaller, though the redaction in natural frequencies is not substantial. The pre-analysis of the case (IV) is similar to Case (III) but in this case the double-bolted end-connections are modeled as semi-rigid connections and the rigidity of the rotational DOF in the orthogonal plane on the bolt shanks is defined based on Eq. (4). As it is clear, considering the semi-rigidity of the doublebolted connections is not effective on the natural frequencies of the structure. The pre-analysis of the case (V) is also similar to case (III) but the joint slip effect is taken into account of this preanalysis. In this case, after the analysis due to weight load of the wind turbine, the nonlinear springs, used to model joint slip effect, are deformed and their stiffnesses based on Ungkurapinan joint slippage models decrease. Therefore, the natural frequencies are reduced considerably in comparison with the other cases due to the slippage of joints, decreasing tower total stiffness, reducing the natural frequencies. Finally, in the case (VI) pre-analysis both semi-rigidity and joint slip of the connections are incorporated. As it is obvious, the natural frequencies of this case are identical with those of the case (V). These results indicate the significance of the joint slip effect on the natural frequencies of the lattice tower of the wind turbine. The weight of the wind turbine components i.e., tower, nacelle and
Fig. 9. The first ten natural frequencies of the wind turbine lattice tower for different cases of pre-analyses.
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Table 3 The chosen critical load cases and their characteristics. Design load case
No. of design load case in GL
Turbine situation
Wind condition
A B C D E F
1.3 1.4 1.5 6.1 6.2 7.1
Power production Power production Power production Parked (standstill or idling) Parked (standstill or idling) Parked plus fault Condition
Extreme coherent gust with direction change Normal wind profile model Extreme direction change Extreme wind speed model Extreme wind speed model Extreme wind speed model
Table 4 The values of the forces and moments produced by critical load cases. Design load case
Wind direction
Vhub [m/s]
Fx [kN]
Fy [kN]
Fz [kN]
Mx [kN m]
My [kN m]
Mz [kN m]
A B C D E F
X X Y Y Y Y
25 13 20 36 45 36
15.602 40.184 25.557 1.129 4.439 0.994
6.450 0.832 7.449 14.148 20.658 13.222
67.500 67.500 67.500 67.500 67.500 67.500
63.280 58.868 7.025 8.791 54.560 26.323
15.128 14.292 40.246 29.228 37.818 11.143
0.003 0.185 0.068 21.465 34.628 18.810
rotor increases the axial forces of the members and brings about the slippage of the connections. In addition, it is concluded that the geometric and material nonlinear effects and incorporation of the double-bolted end-connections semi-rigidity is not effective on the structure natural frequencies. Moreover, the joint slip effect disturbs the symmetry of the modes. As it is obvious in Fig. 9, the modes one and two, four and five and seven and eight, two by two, in the cases (I), (II), (III) and (IV) are symmetric while this symmetry is disturbed in the cases (V) and (VI) due to the joint slip effect. The first ten mode shapes of the tower are presented in Appendix A. As mentioned before, the natural frequencies are significant in analysis and design of wind turbines, so that, their coincidence with the burdened frequencies by the blades rotation leads to resonance
of the structures and its failure. The design of the structure is affected by these natural frequencies in this way that the designer should adjust the stiffness and weight of the structure to avoid the coincidence of frequencies. In design procedure, there are some allowable frequencies intervals, which are defined between the values of the frequencies of the blades rotation in different cases. Moreover, there are some natural frequencies intervals defied between the naturals frequencies of the structure at different modes. The importance of the joint slip effect in the modeling is the reduction of these natural frequencies in this way that if this effect is not incorporated in the analysis the frequency interval for structure designing is different in comparison with the case when this effect is considered.
Fig. 10. The applied forces and moments produced by critical load cases along with correlated wind.
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Fig. 11. The displacements of the wind turbine tower for different cases of analysis at the load case A.
Fig. 13. The displacements of the wind turbine tower for different cases of analysis at the load case C.
Fig. 12. The displacements of the wind turbine tower for different cases of analysis at the load case B.
Fig. 14. The displacements of the wind turbine tower for different cases of analysis at the load case D.
6.2. Displacements of the wind turbine In order to design the tower of the wind turbine, based on GL2010 standard [21], several design load cases are considered. Afterwards, the wind speed and direction related to these load cases are applied on the blades of the wind turbine and forces and moment produced by the rotation of the blades and rotor are obtained. These forces and moments are applied at the top of tower. Also, the correlated wind is applied on the lattice tower body simultaneously. Table 3 presents the chosen critical load cases and their characteristics and Table 4 presents the values of the forces and moments produced by these cases, which are applied on the tower as shown in Fig. 10. It should be mentioned that the FAST program produced by NREL institute is utilized for calculations of the aerodynamic loads. The boundary conditions at the top of the wind turbine are defined as it translates and rotates as the platform (foundation) moves and the tower bends, but it does not yaw with the nacelle. Also, the direction changes are considered according to GL standard. The tower of the wind turbine is modeled and analyzed for the critical load cases in Table 3 and its displacements for different cases, as presented in Table 1, are obtained and shown in Figs. 11e16. Moreover, the displacements of the tower via load
Fig. 15. The displacements of the wind turbine tower for different cases of analysis at the load case E.
factor are shown in Figs. 17e22. As it is obvious, the results of the analysis cases (I), (II), (III) and (IV), are in a good agreement, indicating that at the current rang of loading the geometric and matricals nonlinearities, incorporated in the cases (III) and (IV), and
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Fig. 19. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case C. Fig. 16. The displacements of the wind turbine tower for different cases of analysis at the load case F.
Fig. 20. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case D. Fig. 17. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case A.
Fig. 21. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case E. Fig. 18. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case B.
7. Conclusion also modeling the double-bolted end-connections as semi-rigid connections, incorporated in the analyses case (IV), are not effective on the displacements of the wind turbine. On the other hand, the joint slip effect, incorporated in the analysis cases (V) and (VI), increases the wind turbine displacements approximately twice further than the other cases, emphasizing the role of the joint slip effect in analysis and design, which could affect on the wind turbine performance.
In this paper, the frequency analysis of the lattice tower of a three-blade horizontal-axis wind turbine, including different sources of nonlinearities, i.e., geometric, material and joint slip effect, was carried out and the natural frequencies were obtained using the developed Nonlinear Analysis Software for Towers -NASTower-. Moreover, the double-bolted end-connections were modeled as semi-rigid connections instead of common rigid
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Fig. 22. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case F.
connections. Also, in order to validate the result obtained by the NASTower, some analyses were conducted by the MStower as a valid and reliable commercial software which is utilizing for design of lattice towers. Overall, the main conclusions obtained by the present study can be listed as follows: in contrast to the geometric and material nonlinearities, which are not effective on behavior of the studied wind turbine, the joint slip as an nonlinear effect can substantially affect on the behavior of these structures. The joint slip reduces the natural frequencies of the bare structure up to 40% which is very important in this way that the structure designer should adjust the stiffness and weight of the structure to avoid the coincidence of frequencies, leading to resonance of the structure and its failure. The joint slippage increases the displacements of the structure around 100%, which should be considered especially in performance evaluation of the turbine that could be disturbed by unpredicted displacements. Since the results of the structure modeled by the double-bolted connections as semi-rigid connections are quiet similar with the results of the structure modeled by the double-bolted connections as rigid ones, this kind of connections precisely could be modeled as rigid connections. The agreements of the results relates to cases (I) and (II) which,respectively, are conducted in NASTower and MStower with similar conditions, regarding this point that MStower is a valid commercial software, validates and confirms the accuracy and precision of the developed NASTower software.
Fig. A1. First mode shape.
Acknowledgment This paper is supported by the Sun Air Research Institute (SARI), Ferdowsi University of Mashhad (FUM), as a part of a wind turbine design and development project. In addition, the authors would like to acknowledge Mr. Mohammad Jahanaray, Gam Arak Industrial Company and Gam and Partners L.L.C for their precious cooperation and assistance during this research. Appendix A The first ten mode shapes of the lattice structure of the wind turbine.
Fig. A2. Second mode shape.
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Fig. A3. Third mode shape.
Fig. A5. Fifth mode shape.
Fig. A4. Fourth mode shape.
Fig. A6. Sixth mode shape.
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Fig. A7. Seventh mode shape.
Fig. A9. Ninth mode shape.
Fig. A8. Eighth mode shape.
Fig. A10. Tenth mode shape.
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References [1] C.G. Gebhardt, B.A. Roccia, Non-linear aeroelasticity: an approach to compute the response of three-blade large-scale horizontal-axis wind turbines, Renew. Energy 66 (2014) 495e514, 6//. [2] K.Y. Maalawi, A model for yawing dynamic optimization of a wind turbine structure, Int. J. Mech. Sci. 49 (2007) 1130e1138, 10//. [3] J. Jia, Wind and structural modelling for an accurate fatigue life assessment of tubular structures, Eng. Struct. 33 (2011) 477e491, 2//. [4] Y. Li, J. Lu, Lightweight structure design for wind energy by integrating nanostructured materials, Mater. Des. 57 (2014) 689e696, 5//. [5] L.V. Andersen, M.J. Vahdatirad, M.T. Sichani, J.D. Sørensen, Natural frequencies of wind turbines on monopile foundations in clayey soilsdA probabilistic approach, Comput. Geotechnics 43 (2012) 1e11, 6//. [6] F. Albermani, S. Kitipornchai, R.W.K. Chan, Failure analysis of transmission towers, Eng. Fail. Anal. 16 (2009) 1922e1928, 9//. [7] F. Albermani, M. Mahendran, S. Kitipornchai, Upgrading of transmission towers using a diaphragm bracing system, Eng. Struct. 26 (2004) 735e744, 5//. [8] F.G.A. Albermani, S. Kitipornchai, Numerical simulation of structural behaviour of transmission towers, Thin-Walled Struct. 41 (2003) 167e177, 2//. [9] F.G.A. Al-Bermani, S. Kitipornchai, Nonlinear finite element analysis of latticed transmission towers, Eng. Struct. 15 (1993) 259e269, 7//. [10] F.G.A. Al-Bermani, S. Kitipornchai, Nonlinear analysis of transmission towers, Eng. Struct. 14 (1992) 139e151. //. [11] S.L. Chan, S.H. Cho, Second-order analysis and design of angle trusses Part I:
elastic analysis and design, Eng. Struct. 30 (2008) 616e625, 3//. [12] N. Ungkurapinan, A Study of Joint Slip in Galvanized Bolted Angle Connections, 2000. M.Sc., Manitoba Canada. [13] W.Q. Jiang, Z.Q. Wang, G. McClure, G.L. Wang, J.D. Geng, Accurate modeling of joint effects in lattice transmission towers, Eng. Struct. 33 (2011) 1817e1827, 5//. [14] N. Prasad Rao, G.M.S. Knight, N. Lakshmanan, N.R. Iyer, Investigation of transmission line tower failures, Eng. Fail. Anal. 17 (2010) 1127e1141, 7//. [15] N. Prasad Rao, G.M. Samuel Knight, S.J. Mohan, N. Lakshmanan, Studies on failure of transmission line towers in testing, Eng. Struct. 35 (2012) 55e70, 2//. [16] D. Nezamolmolki, A. Aftabi Sani, Free vibration analysis of a sloping-frame: closed-form solution versus finite element solution and modification of the characteristic matrices, Int. J. Eng. (IJE), Trans. C Aspects 28 (2015) 378e386, 3//. [17] MStower V6 User’s Manual, Engineering Systems (EEC) Limited, England, 2006. [18] S. Mazzoni, F. McKenna, M.H. Scott, G.L. Fenves, OpenSees Command Language Manual, 2007. [19] N. Ungkurapinan, S.R.D.S. Chandrakeerthy, R.K.N.D. Rajapakse, S.B. Yue, Joint slip in steel electric transmission towers, Eng. Struct. 25 (2003) 779e788, 5//. [20] D. Nezamolmolki, The Effect of Joint Slip on Power Transmission Towers Analysis, M.Sc. Thesis, Structural Engineering, Ferdowsi University of Mashhad, 2013. [21] Guideline for the Certification of Wind Turbines, Germanischer Lloyd, Hamburg, Germany, 2010.
Contents lists available at ScienceDirect
Renewable Energy journal homepage: www.elsevier.com/locate/renene
Investigation of nonlinear dynamic behavior of lattice structure wind turbines Davoud Nezamolmolki, Ahmad Shooshtari* Department of Civil Engineering, Ferdowsi University of Mashhad, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 16 June 2015 Received in revised form 16 May 2016 Accepted 17 May 2016
Wind turbines are kinds of electricity generators, proliferating nowadays due to their consistency with the environment. The rotation of the wind turbine blades duo to wind, burdens some frequencies on the wind turbine tower. Therefore, in addition to common design due to service loads, the wind turbine tower should be checked in the frequency analysis so that its natural frequencies do not coincide with the frequencies caused by blades rotation, resulting in the resonance of the tower and its failure. This makes the design of these structures to be complex. In this research, the frequency analysis of the lattice tower of a three-blade horizontal-axis wind turbine including different sources of nonlinearities, i.e., geometric, material and joint slip effect is carried out and the natural frequencies are obtained using the developed Nonlinear Analysis Software for Towers -NASTower-. It is observed that the joint slip effect can substantially reduce the natural frequencies of the lattice tower, which are significant in the design of these structures. In addition, the wind turbine displacements due to different design load cases are investigated, indicating that incorporating the joint slip effect into the analysis substantially increases the wind turbine displacements, which could affect on its performance. © 2016 Elsevier Ltd. All rights reserved.
Keywords: Lattice tower of wind turbine Nonlinear effects Join slip effect Frequency analysis Natural frequency Mode shape
1. Introduction In recent years, the tendency to renewable energies has been increasing due to the environmental problem caused by the use of fossil fuels. Wind energy is one of renewable energies, extracted from wind, using wind turbines to produce electrical energy. Therefore, the analysis and design of the wind turbines is one of the structural and mechanical engineering problems. The wind turbines are mostly mounted on the tubular and lattice towers and it is an advantage to have a high tower, since wind speeds increase farther away from the ground. In the analysis and design procedure of the wind turbines it is necessary to avoid the coincidence of the blades rotation frequencies with the natural frequencies of the tower, which brings about the resonance of the tower and its failure. This makes the design of the wind turbine towers to be complex. Many of researchers have devoted themselves to the study of the wind turbines. Gebhardt and Roccia presented an aeroelastic model intended for three-blade large-scale horizontal-axis wind
* Corresponding author. E-mail address: [email protected] (A. Shooshtari). http://dx.doi.org/10.1016/j.renene.2016.05.070 0960-1481/© 2016 Elsevier Ltd. All rights reserved.
turbines [1]. This model results from the coupling of an existing aerodynamic model and a structural model based on a segregated formulation derived in an index-based notation that enables combining very different descriptions such as rigid-body dynamics, assumed-modes techniques and finite element methods. Also, a novel design optimization model for placing frequencies of a wind turbine tower/nacelle/rotor structure in free yawing motion was developed and discussed by Karam and Maalawi [2]. The main aim was to avoid large amplitudes caused by the yawing-induced vibrations in the case of horizontal-axis wind turbines or rotational motion of the blades about the tower axis in case of vertical-axis wind turbines. This could be a major cause of fatigue failure and might severely damage the whole tower/nacelle/rotor structure. Jia presented a practical and efficient approach for calculating wind induced fatigue of tubular structures, the effects of the wind direction, across wind and wind grid size on the high cycle fatigue of the structure were addressed [3]. A grid is defined which covers the whole structure. For each node in the grid a time series is generated. This time series contains the mean wind speed and the statistical properties of the fluctuating part of wind components as defined by the one point wind spectra and the coherence functions. In each time step of the dynamic response calculation, the large deformation effects and the wind induced drag forces due to the
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updated structural deformations are taken into account. Moreover, a new design scheme of light weight structure for wind turbine tower was presented by Li and Lu [4]. This design scheme is based on the integration of the nanostructured materials produced by the Surface Mechanical Attrition Treatment (SMAT) process. The objective of this study was to accomplish the weight reduction by optimizing the wall thickness of the tapered tubular structure. Andersen et al. presented a probabilistic approach about the natural frequencies of wind turbines on monopile foundations in clayey soils [5]. A nonlinear stochastic p-y curve was integrated into a finite difference scheme for calculation of the monopole response in which p is the soil resistance and y is the absolute value of the pile deflection. Furthermore, the investigations on the nonlinear behavior of lattice structures are frequently observed in the literature. AlBermani and Kitipornchai presented a nonlinear analytical technique for simulation of the ultimate structural behavior of selfsupporting transmission towers, incorporating both geometric and material nonlinearity effects. Modeling of material nonlinearity for angle members is based on the assumption of lumped plasticity
coupled with the concept of a yield surface in force space. Several transmission towers were modeled and analyzed in developed AK TOWER, which stands for incorporating the presented technique, to simulate their behavior in the full-scale experiments [6e10]. Moreover, Chan and Cho proposed a practical second-order analysis and design method for trusses composed of angle sections. Realistic modeling of semi-rigid connections associated with one- and twobolt end-connections with flexible gusset plate and member imperfections such as initial curvatures and residual stresses is made and load eccentricity is also simulated [11]. Also, Ungkurapinan studied the behavior of such joints, incorporated 36 joint tests, generated joint slip data and developed mathematical expressions to describe slip and load-deformation behavior. In this study, it was concluded that joint slip effect cannot be eliminated and incorporation of the reported joint slip data or mathematical expressions in the tower analysis software will refine their results [12]. Afterwards, Ungkurpinan joint slippage models were employed by Jiang et al. in the modeling of a transmission tower, which resulted in improvement in the displacements results, coinciding with the fullscale test results [13]. In other research, the nonlinear finite
Fig. 1. Nonlinear effects: P-D and P-d
Fig. 2. The detail of a element with fiber section.
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element analysis program NE-NASTRAN had been used by Rao et al. to model the elasto-plastic behavior of transmission towers. Employing the plane elements, the elasto-plastic deformations of the members in the full-scale test were predicted in these studies [14,15]. Finally, Nezamolmolki and Aftabi Sani investigated the dynamic behavior of a Sloping frame, comparing the closed-form solution and the finite element one [16]. In this paper, the frequency analysis of the lattice tower of a three-blade horizontal-axis wind turbine including different sources of nonlinearities, i.e., geometric, material and joint slip effect is carried out and the natural frequencies are obtained, using the developed Nonlinear Analysis Software for Towers -NASTower-. Moreover, since the lattice structures are erected by bolted connections, the accurate modeling of these connections is important. Single-bolted connections accurately could be modeled as hinged connections but double-bolted ones could not be modeled as hinged ones. Herein, this problem is investigated and since most of the end-connections are double-bolted, these connections are modeled as semi-rigid connections. It is observed that the joint slip effect can substantially reduce the natural frequencies, which are effective in the design of these structures. Also, the wind turbine displacements due to different design load cases are investigated, indicating that incorporating the joint slip effect into analysis substantially increases the wind turbine displacements, which could effect on its performance. It should be mentioned that in order to validate the results obtained by the NASTower, some analyses are conducted by the MStower as a valid and reliable commercial software, which is utilizing by many of engineers for design of lattice towers.
received by software and exact material properties are not considered [18]. In this case, the applied load on angle section is not replaced with its components on principle axes, eventuating deflection just in the plane of applied load while accurately, the applied load components on principle axes bring about deflections at orthogonal planes, which are neglected when element with Elastic section is used. On the other hand, in the case of Fiber section, the section is divided into regular shaped meshes and stressstrain relationship is defined for each mesh, dependent to its material. Consequently, forces at integration points are obtained based on the reports from the meshes, resulting in the forces at two ends of element. Contrasting the Elastic section, in the case of Fiber section, the applied load is replaced by its components on principle axes, eventuating deflections at orthogonal planes as actuality. As mentioned before, the element with Fiber section incorporates the material nonlinearity in the analysis. Fig. 2 shows the details of Fiber section. 2.3. Joint slip effect Joint slip effect is one of the most significant nonlinear effects in the lattice transmission towers which is the relative displacement, at a negligible or very low stiffness, of a bolted joint subjected to a shear load. It occurs since boltholes are drilled oversize to provide an erection tolerance of 1.6 mm (1/16 in). For transmission towers, greater slippage is likely to occur as bolt diameters are small, members joined are thin, bearing type joints with a lower clamping force are used, and coefficient of friction of galvanized faying surfaces is low [19]. In order to incorporate joint slip into modeling, nonlinear springs are
2. Nonlinear effects Three major sources of nonlinearity for a latticed structure are identified: geometric nonlinearity, material nonlinearity, and joint flexibility and slippage [10]. 2.1. Geometric nonlinearity The P-Delta effect (P-D) and the P-delta effect (P-d) are the most important geometric nonlinear effects taken into account in the nonlinear analysis as shown in Fig. 1. You may independently include or exclude these two major effects. The P-Delta effect (P-D) occurs when deflections result in displacement of loads, causing additional bending moments that are not computed in linear analysis. Also, the bending stiffness of a member is reduced by axial compression and increased by axial tension called the P-delta effect (P-d) [17]. 2.2. Material nonlinearity This type of nonlinearity arises when the material exhibits nonlinear stress-strain relationship. Recall that for elastic linear finite element analysis the only stress-strain relationship is defined via modulus of elasticity, E. It should be mentioned that common analytical softwares employ the approximate approach of plastic hinge in order to consider the material nonlinearity. The recent approach to consider material nonlinearity is employing the elements with Fiber section in the analysis [18]. 2.2.1. Fiber section - elastic section In general, nonlinear analytical softwares introduce two types of wire elements: wire element with Elastic section and wire element with Fiber section. In the case of Elastic section, some characteristics such as elasticity modulus, section area, moment of inertia, yield stress, shear modulus and polar moment of inertia are
35
Fig. 3. Typical bolted joint connections [13].
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Fig. 4. Ungkurapinan joint slippage models [12,13].
employed at connections as shown in Fig. 3, which their forcedisplacement relationships are defined by Ungkurapinan joint slippage models indicated in Fig. 4 drawn for normal clearance as shown in Fig. 5. The Ungkurapinan joint slippage models are independent from the bolt size and material. In addition, there are some approximations in the modeling of the various splices with different angle sizes and bolt numbers. Fig. 5. Normal clearance [13].
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3. Modeling of semi-rigid connections
4. The wind turbine details
Since the end-connections of the current lattice tower of the wind turbine are mostly double-bolted, in this research these endconnections are modeled as semi-rigid connections based on the formulation presented by Chan and Cho [11]. As shown in Fig. 6, the couple, M, formed by the shear forces exerted on the bolts, F, is given by:
In this research, a 100-kW three-blade horizontal-axis wind turbine, which possesses a lattice tower with a 2 m 2 m square base and a height of 30 m from the ground, is studied. The lattice tower is comprised by angle sections with the steel grade of 235 Mpa. Also, the wind turbine has three 13-m blades with a summit distance of 3 m from the center vertical axis of the lattice tower. The details of the wind turbine and its power curve are shown in Figs. 7 and 8, respectively.
M ¼ F d ¼ k:q
(1)
Also the shear stress, t, across the cross-section, A, of the bolt is: 5. Analysis procedure
F t¼ A
(2)
in which A is the shear area and can be taken as 0.9 of the cross sectional area as recommended in most design codes. Also the shear strain, g, of the bolt shank is:
g¼
2d d:q t ¼ ¼ l l G
(3)
where d is the distance between the centroids of the two bolts; q is the rotation of the bolt group; d is the displacement of the bolt; l is the length of the bolt shank and G is the shear modulus of elasticity. Rearranging terms, the rotational stiffness, k, due to the doublebolted connection will be given by:
k¼
GAd2 l
(4)
Fig. 6. The couple formed by shear forces of two bolts [11].
The lattice tower of the wind turbine is modeled in Nonlinear Analysis Software for Towers -NASTower-, developed in this study. This modeling and analytical software, capable of modeling different kinds of towers such as transmission towers, telecommunication towers and wind turbine towers, can take different source of nonlinearity: geometric, material and joint slip effect, into account. Moreover, created models in common commercial softwares like MStower could be imported in NASTower in order to refine modeling by applying joint slippage models and semi-rigidity at connections. It should be mentioned that NASTower employs OpenSees software as processor, including many capabilities in the modeling of different structures with diverse elements. The advantages of NASTower in Comparison with MStower are the capability of modeling with Fiber section, which considers the material nonlinearity, incorporating the joint slippage into analysis and modeling the semi-rigidity of the connections. In the modeling of current lattice towers 3D beam-column elements are employed. Generally, OpenSees includes different 3D beam-column elements such as Elastic Beam Column Element, NonLinear Beam-Column Elements (Force based elements and Displacement based element), Beam with Hinges Element and Nonlinear Beam Column Element. In this study, the Elastic Beam Column Element is utilized in elastic analysis and the Nonlinear Beam Column Element, which is based on the non-iterative (or iterative) force formulation, and considers the spread of plasticity along the element, is utilized in inelastic analysis. The integration along this element is based on Gauss-Lobatto quadrature rule [18]. Also, the Zero-Length Elements are employed at the two ends of each wire element to incorporate the joint slip effect in analysis. Two nodes at the same location define this element. The nodes are connected by multiple UniaxialMaterial objects, which define the stiffness, to represent the force-deformation relationship for the element. In this process the axial DOF stiffness is defined by Ungkurapinan joint slippage models. In addition, the stiffness of the rotational DOF in the orthogonal plane on the bolt shanks at end-connections is defined based on Eq. (4) to consider the semirigidity. In other cases, the DOFs stiffnesses are considered infinite leading to rigidity. In linear analysis procedure, the linear algorithm is applied to construct a Linear algorithm object which takes one iteration to solve the system of equations. In the nonlinear analysis procedure, the load-control Newton algorithm is applied to construct a Newton-Raphson algorithm object, which uses the Newton-Raphson method to advance to the next time step. In this method, the tangent at the current iteration is used to iterate to convergence. It should be mentioned in the analysis procedure, the loads at different cases are applied incrementally. Moreover, the supports are fixed modeled and different sources of nonlinearities, as explained in section 2, are incorporated in each case of the analysis.
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Fig. 7. The details of the wind turbine.
6. Results and discussion In this section, first of all the frequency analysis is carried out and the natural frequencies of the lattice tower of the wind turbine for different cases of analysis, incorporating nonlinear effects and semi-rigidity of the connections, are presented and compared. Afterwards, the first ten mode shapes of the lattice tower of the wind turbine are illustrated. Then, the displacements of the wind turbine due to some critical design load cases for different cases are presented and compared. As mentioned, herein different cases of the analysis are carried out, which incorporate different parameters such as nonlinearity effects and semi-rigidity of the connections into the analysis. The details of these cases are presented in Table 1. As mentioned before, in order to validate the obtained results by NASTower, some analyses are conducted with MStower as a valid and reliable commercial software. As it is clear, the case number “II” represents the model of MStower. The further comparisons are presented at Ref. [20].
Fig. 9. It is necessary to mention that prior to frequency analysis the lattice tower is pre-analyzed due to the weight load of the wind turbine for different cases as presented in Table 1. Each pre-analysis brings about some displacements and deflections. Therefore, the frequency analysis is conducted on a deformed tower with modified stiffnesses in members and nonlinear springs, which model the joint slip effect. The pre-analyses are clear in Fig. 9 and Table 2. In the design procedure of the wind turbines, it is important to design the structure such that the frequencies caused by the rotation of blades and rotor lie at the range between the natural frequencies of the wind turbine structure to avoid the resonance of the structure
6.1. Frequency analysis results The current lattice tower of the wind turbine is modeled in NASTower and its first ten modes natural frequencies for different cases are obtained as presented in Table 2. Also, the column chart of the values of the natural frequencies at each mode is shown in
Fig. 8. The power curve of 100-kW wind turbine.
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Table 1 Details of different cases of the analysis. Case
Definition
Section of the elements
Modeling of double-bolted end-connections
Geometric nonlinearity
Material nonlinearity
Joint slip effect
I
Elastic linear with rigid connections with NASTower Elastic linear with rigid connections with MStower Inelastic nonlinear with rigid connections with NASTower Inelastic nonlinear with semi-rigid connections with NASTower Inelastic nonlinear with joint slip with NASTower Inelastic nonlinear with semi-rigid connections and joint slip with NASTower
Elastic
Rigid
Not considered
Not considered
Not considered
Elastic
Rigid
Not considered
Not considered
Not considered
Fiber
Rigid
Considered
Considered
Not considered
Fiber
Semi-rigid
Considered
Considered
Not considered
Fiber
Rigid
Considered
Considered
Considered
Fiber
Semi-rigid
Considered
Considered
Considered
II III IV V VI
Table 2 The first ten natural frequencies of the wind turbine lattice tower for different cases of pre-analyses. Pre-analysis case
I
II
III
IV
V
VI
First mode Second mode Third mode Fourth mode Fifth mode Sixth mode Seventh mode Eighth mode Ninth mode Tenth mode
0.70 0.70 5.08 7.44 7.45 13.72 19.24 19.28 21.36 25.85
0.70 0.70 5.06 7.44 7.44 13.72 19.23 19.28 21.33 25.71
0.69 0.69 5.05 7.39 7.39 13.61 19.15 19.20 21.33 25.71
0.69 0.69 5.05 7.39 7.39 13.61 19.15 19.20 21.33 25.37
0.50 0.63 3.56 5.76 6.37 12.04 14.28 15.53 15.56 25.68
0.50 0.63 3.56 5.76 6.37 12.04 14.28 15.53 15.56 25.35
and its failure. In the case (I) pre-analysis, the elements with Elastic sections are used and due to the weight load of the wind turbine linear analysis is conducted and then, the frequency analysis is carried out. Herein, the linear trend of the material and linear analysis makes the stiffness remains constant before and after pre-analysis. In other word, the weight of the tower is not effective on the tower stiffness. Case (II) pre-analysis is quite similar to (I) but the tower is modeled in MStower to validate the results by comparison. As it is clear from Table 2 and Fig. 9, the results of this case are in a very good agreement with the results of the case (I) which validates and confirms the accuracy and precision of the
developed NASTower software. In the case (III) pre-analysis, the elements with Fiber sections are used and geometric nonlinearity due to weight load of the wind turbine is considered. In this case, the axial forces in elements change their stiffnesses, reducing tower total stiffness, which makes the natural frequencies to be smaller, though the redaction in natural frequencies is not substantial. The pre-analysis of the case (IV) is similar to Case (III) but in this case the double-bolted end-connections are modeled as semi-rigid connections and the rigidity of the rotational DOF in the orthogonal plane on the bolt shanks is defined based on Eq. (4). As it is clear, considering the semi-rigidity of the doublebolted connections is not effective on the natural frequencies of the structure. The pre-analysis of the case (V) is also similar to case (III) but the joint slip effect is taken into account of this preanalysis. In this case, after the analysis due to weight load of the wind turbine, the nonlinear springs, used to model joint slip effect, are deformed and their stiffnesses based on Ungkurapinan joint slippage models decrease. Therefore, the natural frequencies are reduced considerably in comparison with the other cases due to the slippage of joints, decreasing tower total stiffness, reducing the natural frequencies. Finally, in the case (VI) pre-analysis both semi-rigidity and joint slip of the connections are incorporated. As it is obvious, the natural frequencies of this case are identical with those of the case (V). These results indicate the significance of the joint slip effect on the natural frequencies of the lattice tower of the wind turbine. The weight of the wind turbine components i.e., tower, nacelle and
Fig. 9. The first ten natural frequencies of the wind turbine lattice tower for different cases of pre-analyses.
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Table 3 The chosen critical load cases and their characteristics. Design load case
No. of design load case in GL
Turbine situation
Wind condition
A B C D E F
1.3 1.4 1.5 6.1 6.2 7.1
Power production Power production Power production Parked (standstill or idling) Parked (standstill or idling) Parked plus fault Condition
Extreme coherent gust with direction change Normal wind profile model Extreme direction change Extreme wind speed model Extreme wind speed model Extreme wind speed model
Table 4 The values of the forces and moments produced by critical load cases. Design load case
Wind direction
Vhub [m/s]
Fx [kN]
Fy [kN]
Fz [kN]
Mx [kN m]
My [kN m]
Mz [kN m]
A B C D E F
X X Y Y Y Y
25 13 20 36 45 36
15.602 40.184 25.557 1.129 4.439 0.994
6.450 0.832 7.449 14.148 20.658 13.222
67.500 67.500 67.500 67.500 67.500 67.500
63.280 58.868 7.025 8.791 54.560 26.323
15.128 14.292 40.246 29.228 37.818 11.143
0.003 0.185 0.068 21.465 34.628 18.810
rotor increases the axial forces of the members and brings about the slippage of the connections. In addition, it is concluded that the geometric and material nonlinear effects and incorporation of the double-bolted end-connections semi-rigidity is not effective on the structure natural frequencies. Moreover, the joint slip effect disturbs the symmetry of the modes. As it is obvious in Fig. 9, the modes one and two, four and five and seven and eight, two by two, in the cases (I), (II), (III) and (IV) are symmetric while this symmetry is disturbed in the cases (V) and (VI) due to the joint slip effect. The first ten mode shapes of the tower are presented in Appendix A. As mentioned before, the natural frequencies are significant in analysis and design of wind turbines, so that, their coincidence with the burdened frequencies by the blades rotation leads to resonance
of the structures and its failure. The design of the structure is affected by these natural frequencies in this way that the designer should adjust the stiffness and weight of the structure to avoid the coincidence of frequencies. In design procedure, there are some allowable frequencies intervals, which are defined between the values of the frequencies of the blades rotation in different cases. Moreover, there are some natural frequencies intervals defied between the naturals frequencies of the structure at different modes. The importance of the joint slip effect in the modeling is the reduction of these natural frequencies in this way that if this effect is not incorporated in the analysis the frequency interval for structure designing is different in comparison with the case when this effect is considered.
Fig. 10. The applied forces and moments produced by critical load cases along with correlated wind.
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Fig. 11. The displacements of the wind turbine tower for different cases of analysis at the load case A.
Fig. 13. The displacements of the wind turbine tower for different cases of analysis at the load case C.
Fig. 12. The displacements of the wind turbine tower for different cases of analysis at the load case B.
Fig. 14. The displacements of the wind turbine tower for different cases of analysis at the load case D.
6.2. Displacements of the wind turbine In order to design the tower of the wind turbine, based on GL2010 standard [21], several design load cases are considered. Afterwards, the wind speed and direction related to these load cases are applied on the blades of the wind turbine and forces and moment produced by the rotation of the blades and rotor are obtained. These forces and moments are applied at the top of tower. Also, the correlated wind is applied on the lattice tower body simultaneously. Table 3 presents the chosen critical load cases and their characteristics and Table 4 presents the values of the forces and moments produced by these cases, which are applied on the tower as shown in Fig. 10. It should be mentioned that the FAST program produced by NREL institute is utilized for calculations of the aerodynamic loads. The boundary conditions at the top of the wind turbine are defined as it translates and rotates as the platform (foundation) moves and the tower bends, but it does not yaw with the nacelle. Also, the direction changes are considered according to GL standard. The tower of the wind turbine is modeled and analyzed for the critical load cases in Table 3 and its displacements for different cases, as presented in Table 1, are obtained and shown in Figs. 11e16. Moreover, the displacements of the tower via load
Fig. 15. The displacements of the wind turbine tower for different cases of analysis at the load case E.
factor are shown in Figs. 17e22. As it is obvious, the results of the analysis cases (I), (II), (III) and (IV), are in a good agreement, indicating that at the current rang of loading the geometric and matricals nonlinearities, incorporated in the cases (III) and (IV), and
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Fig. 19. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case C. Fig. 16. The displacements of the wind turbine tower for different cases of analysis at the load case F.
Fig. 20. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case D. Fig. 17. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case A.
Fig. 21. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case E. Fig. 18. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case B.
7. Conclusion also modeling the double-bolted end-connections as semi-rigid connections, incorporated in the analyses case (IV), are not effective on the displacements of the wind turbine. On the other hand, the joint slip effect, incorporated in the analysis cases (V) and (VI), increases the wind turbine displacements approximately twice further than the other cases, emphasizing the role of the joint slip effect in analysis and design, which could affect on the wind turbine performance.
In this paper, the frequency analysis of the lattice tower of a three-blade horizontal-axis wind turbine, including different sources of nonlinearities, i.e., geometric, material and joint slip effect, was carried out and the natural frequencies were obtained using the developed Nonlinear Analysis Software for Towers -NASTower-. Moreover, the double-bolted end-connections were modeled as semi-rigid connections instead of common rigid
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Fig. 22. The displacements of the wind turbine tower via load factor for different cases of analysis at the load case F.
connections. Also, in order to validate the result obtained by the NASTower, some analyses were conducted by the MStower as a valid and reliable commercial software which is utilizing for design of lattice towers. Overall, the main conclusions obtained by the present study can be listed as follows: in contrast to the geometric and material nonlinearities, which are not effective on behavior of the studied wind turbine, the joint slip as an nonlinear effect can substantially affect on the behavior of these structures. The joint slip reduces the natural frequencies of the bare structure up to 40% which is very important in this way that the structure designer should adjust the stiffness and weight of the structure to avoid the coincidence of frequencies, leading to resonance of the structure and its failure. The joint slippage increases the displacements of the structure around 100%, which should be considered especially in performance evaluation of the turbine that could be disturbed by unpredicted displacements. Since the results of the structure modeled by the double-bolted connections as semi-rigid connections are quiet similar with the results of the structure modeled by the double-bolted connections as rigid ones, this kind of connections precisely could be modeled as rigid connections. The agreements of the results relates to cases (I) and (II) which,respectively, are conducted in NASTower and MStower with similar conditions, regarding this point that MStower is a valid commercial software, validates and confirms the accuracy and precision of the developed NASTower software.
Fig. A1. First mode shape.
Acknowledgment This paper is supported by the Sun Air Research Institute (SARI), Ferdowsi University of Mashhad (FUM), as a part of a wind turbine design and development project. In addition, the authors would like to acknowledge Mr. Mohammad Jahanaray, Gam Arak Industrial Company and Gam and Partners L.L.C for their precious cooperation and assistance during this research. Appendix A The first ten mode shapes of the lattice structure of the wind turbine.
Fig. A2. Second mode shape.
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Fig. A3. Third mode shape.
Fig. A5. Fifth mode shape.
Fig. A4. Fourth mode shape.
Fig. A6. Sixth mode shape.
D. Nezamolmolki, A. Shooshtari / Renewable Energy 97 (2016) 33e46
Fig. A7. Seventh mode shape.
Fig. A9. Ninth mode shape.
Fig. A8. Eighth mode shape.
Fig. A10. Tenth mode shape.
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