A simplified model predicting the weight of the load carrying beam in a wind turbine blade

Based on a simplified beam model, the loads, stresses and deflections experienced by a wind turbine blade of a given length is estimated. Due to the simplicity of the model used, the model is well suited for work investigating scaling effects of wind turbine blades. Presently, the model is used to predict the weight of the load carrying beam when using glass fibre reinforced polymers, carbon fibre reinforced polymers or an aluminium alloy as the construction material. Thereby, it is found that the weight of a glass fibre wind turbine blade is increased from 0.5 to 33 tons when the blade length grows from 20 to 90 m. In addition, it can be seen that for a blade using glass fibre reinforced polymers, the design is controlled by the deflection and thereby the material stiffness in order to avoid the blade to hit the tower. On the other hand if using aluminium, the design will be controlled by the fatigue resistance in order to making the material survive the 100 to 500 million load cycles experience of the wind turbine blade throughout the lifetime. The aluminium blade is also found to be considerably heavier compared with the composite blades.


Introduction
A wind turbine blade is a long slender structure where the dominating loads are given by the aerodynamics and the gravitation. Throughout the years, the turbines have grown larger requiring longer blades. The longest blade at the moment is the 88.4 m long and 33.7 tons heavy blade from LM Wind Power [1]. The growth of the turbine size has resulted in cost of energy which for onshore installation pricewise can compete with conventional energy sources. Off-shore, this is still not the case and properly even larger turbines are required before this will be the case. During this upscaling, the material in a wind turbine blade is pushed to the limits, where mainly the stiffness and the fatigue resistance are the key material parameters. Wind turbine blades have until now been based on glass fibre composites and only by improving the fibre properties, the matrix fibre interface and the fibre architecture this has been made possible. A pure carbon fibre based solution is still too costly, while aluminium only is relevant for small blades.
Advanced and specialised models and material characterization methods are used in the effort to optimize the material used in the blade. Nevertheless, seen on the large scale, a wind turbine blade is a simple beam structure with well-defined load and boundary conditions. The beam theory describing the blade can be made at different levels of complexity depending on its use. An example of a more complex version is the BECAS model (BEam Cross section Analysis Software), see e.g. [2]. Even though the presented model is much simpler, it is still bridging the gap from the aerodynamic and gravitational loads to an estimate of the material stresses, deflection and blade weight. These weights are then compared with existing wind turbine blades. This approach has been used to estimate the weight of a glass fibre reinforced wind turbine blade for the DTU 10 MW reference turbine [3] with a blade length of 86 m, and it is found that replacing the material with aluminum will increase the blade weight drastically. On the other hand, using a pure carbon fiber based blade will increase the price of the blade significantly.

Loads on a wind turbine blade
The wind flow around a turbine rotor can simplified be considered following a 1D momentum theory as illustrated in figure 1. Going from upstream () u to downstream () d the wind speed is slowed down from the upstream wind speed u ∞ to the downstream wind speed d u . Due to the decrease of the wind speed, the corresponding cross sectional area of the wind flow will increase from the upstream area u A over the known turbine rotor area, t A to the downstream cross sectional area d A . Based on this information, we can estimate the aerodynamic loads working on the wind turbine blade, see e.g. [4]. The procedure will be described briefly below. Assuming that the air speed changes continuously across the turbine blades and that the pressure far upstream and far downstream are equal to the pressure of the undisturbed airflow the velocity and the pressure drop at the turbine () t with the swept rotor area t A can be found to ½( )   (4) is the downstream wind speed d u describing how much the turbine is slowing down the wind and can be determined optimizing the power t P taken out of the wind. The power is given by the difference between the kinetic energy contained in the wind on the a combination of the equations (1), (4) and (6) will results in the following relations 3 2 (8), the so-called Lanchester-Betz limit, 1 / 3 a = , describes the maximum obtainable power which corresponds to 59.3% of the reference wind power . The factor 1 / 3 a = corresponds to a wind speed at the turbine rotor of 2 / 3 t u u ∞ = and a downstream wind speed of Most modern wind turbines are pitch regulated. Thereby, it is possible to regulate the position of the wind turbine blades in order to achieve the optimal power output. When the wind turbine power is reaching the generator power t G P P = , the blade position will be chosen in order to maintain this production and not exceeding it. Examples of wind turbine power curves can be found in [4], [5]. The incoming wind speed, u ∞ , where the optimal power, t P , just reaches the generator power, G P , is called the rated wind speed, R u and is a design parameter determining the ratio between the swept rotor area, where L is the blade length, and the generator power G P based on the knowledge about the dominating wind speed at a given location. Thus, the parameter a can be written From (10) and (11), the turbine load (9) can be determined as a function of the remote wind speed u ∞ . Thereby, it can be found that the maximum turbine load is found for In (13) and later, the terms () x denote a variable dependent on the axial blade coordinate x measured from the root of the blade. In addition to the aerodynamic load, a gravity load will be present in the wind turbine blade. Assuming now a constant blade material cross section area, blade A , the load intensity in the edge-wise direction is given by with the material density, blade r , and the gravitational constant 2 9.81 / g m s ≈ .

Stresses in the blade material
In order to calculate the stress values in the blade material, a simplification of the cross section of the blade is used. A typical and the simplified cross section are shown in figure 2. The part marked with green is the load carrying part contributing to the overall stiffness of the blade. Using the simplified cross-section shown in figure 2b together with the load intensities defined earlier in equation (13) and (14), the flap and edge-wise moment in the blade, see e.g. [5] can be found as 2( Following (19) will result in blade geometries which are solely material strength given. In reality, the blade geometry is given by a combination of structural and aerodynamic requirements. In   Therefore, based on the parameters listed in table 1, the mass of a specific wind turbine blade can be estimated using equation (21) and (22). Keeping in mind the assumption of a constant cross-section along the blade, the height and width variation is obtained aiming for a constant stress value throughout the blade. In addition to a blade design given by the material strength, the blade should also be stiff enough in order to avoid that the blade hits the tower. In [3], the distance from the blade tip to the tower is given as 18m, this is therefore the maximum allowable deflection max 18m w = for a 86m L = turbine. The distance will scale with the turbine size. In the following the transverse deflection, w , of the blade is predicted using Bernoulli-Euler equations ( ) 1,.... 2

Comparing materials selections for wind turbine blades
Three materials system are considered in the following, see  The curves called "fatigue design" in figure 4 show the resulting blade weight as a function of the blade length in the case the material overall is loaded to the fatigue strength, 0 σ . In figure 4, this weight variation is compared with the weight of existing blade designs and the 10 MW DTU wind energy reference turbine [3]. In figure 5, the maximum deflections found using the difference scheme (24) are shown. It can be seen that the fatigue given blade design for the glass fibre composite case exceeds the maximum allowed tip deflections given in [3]. Therefore, an additional case is shown in figure 4 and 5 called "deflection design". In that case, the flap area flap A is increase with 50% making it possible to stay below max 18m w = for 86m L = . As nearly all the existing wind turbine blades are made of glass fibre composite, this case should be compared with the existing blades. The estimation is found to be 20-40% below the existing turbine blade, which may be considered realistic as only the weight of the load carrying beam in the blade is taken into consideration. In all the estimates, the material cross section, blade A , is for simplicity assumed constant. Nevertheless, in realty, the blade

Conclusion
Based on a simplified beam model of a wind turbine blade, is has been shown possible to make a realistic estimate of the weight of the load carrying laminates in a wind turbine blade. Due to its simplicity, the model is well suited for course work investigating scaling effects of wind turbine blades. This approach has been used as a central element in the materials part of the cousera.org course: "Introduction to Wind Energy" [5]. Even though it turns out, that the design of a glass fibre blade is stiffness given, locally the material can still be loaded to its fatigue strength limits. Therefore, optimizing materials for wind turbine blades, large research activities are going on improving the fatigue resistance, see e.g. [6], [7].