1*MsC. Ingeniería Mecánica, brian.morales@correo.uis.edu.co .
ORCID:
0000-0001-6961-8026.
Universidad Industrial de Santander, Bucaramanga, Colombia
2* PhD. en Ingeniería Mecánica Aplicada
y Computacional, hgsanche@uis.edu.co .
ORCID:
0000-0003-0081-2212.
Universidad Industrial de Santander, Bucaramanga, Colombia
3*PhD. en Ingeniería de Materiales, german.diaz1@uis.edu.co .
ORCID:
0000-0001-6461-8545.
Universidad Industrial de Santander, Bucaramanga,
Colombia
Cómo citar: B. F. Morales-Hernández, H. G. Sánchez-Acevedo, G. A. Díaz-Ramírez, “Incidence of the orientation of the reinforcing fibers in a carbon/epoxy composite shaft, for torsional loads and rotational speed”, Respuestas, vol. 25, no. 3, 29-40, 2020.
© Peer review is the responsibility of the Universidad Francisco de Paula Santander. This is an article under the license CC BY-NC-ND 4.0
Received: March 8, 2020
Approved: July 24, 2020.
Keywords
Composites, Composite shaft, Reinforcing fibers, Critical speed, Ply
Abstract
The study carried out is based on the analysis of the behavior of the angle-ply of a composite shaft, in terms of its resistance to torsional loads and speeds of rotation. For reference, a composite shaft of carbon fiber and an epoxy polymer matrix is optimally designed. The orientations of the laminas in the laminate were selected to ensure their manufacture by the filaments winding method. For the specific case of the study, it can be concluded that the pairs of laminas with angles of 45° and -45° optimally satisfy the design parameters of the shaft. The orientations at 0 ° and 90 °, are useful when the critical speed and torsional buckling are more restrictive.
Palabras claves
Materiales compuestos, Eje compuesto, Fibras de refuerzo, Velocidad crítica, Apilado.
Resumen
El estudio realizado se basa en el análisis del comportamiento de las orientaciones del laminado de un eje compuesto, en cuanto a su resistencia ante cargas torsionales y velocidades de giro. Como referencia se diseña óptimamente un eje compuesto de fibra de carbono y matriz polimérica epóxica. Las orientaciones de las láminas en el apilado se seleccionaron para garantizar su fabricación por el método de bobinado de filamentos. Para el caso específico de estudio se puede concluir que los pares de láminas con ángulos de 45° y -45° satisfacen de forma óptima los parámetros de diseño del eje. Las orientaciones a 0° y 90°, son útiles cuando la velocidad crítica y el pandeo torsional son más restrictivas.
Introduction
Composite materials today have become one of the main alternatives to conventional materials [1]. Characteristics such as its high adaptability to the most demanding design requirements, and its excellent mechanical and physical properties [2], provide engineers with versatility when developing their designs. One of the main reasons for its massive use today lies in the ability to provide high strength and rigidity with low weight, thus generating improvements in energy efficiency and significantly affecting a reduction in manufacturing and operating costs [3].
Some widely used applications of composite materials can be found in the structure of automobiles, aircraft, and boats, in power transmission rotors, in biomedical prostheses, civil structures, and even in applications with additional atypical characteristics, such as in the carbon fibers covered with zinc oxide (ZnO) nanowires, which act as piezoelectric and chemoreceptive sensors, complementing the usual mechanical and structural properties [4].
The design of functional material with excellent properties and characteristics requires considering a significant number of variables or factors, such as, for example, the particularity of composite materials having orthotropic mechanical properties [5], that is, they vary depending on their orientation. This variety of factors that influence design with composite materials motivates the implementation of optimization algorithms, which facilitate the design process and allow finding the most appropriate solution to specific design considerations.
Optimization algorithms are a very useful tool in engineering, not only to facilitate design processes but also as a method of detecting damage to structures [6], [7]. There are various types of optimization algorithms, such as particle swarm algorithms and genetic algorithms, the latter implemented in this study.
Genetic algorithms are an evolutionary optimization method, which emulates the biological processes of gene transfer between different generations in a population. The algorithm successively applies genetic operators to a significant amount of values, until a previously established condition is met, which is the optimal solution [8].
Materials and methods
To analyze the behavior of the orientation of the reinforcing fibers in a composite shaft, the optimal design of a shaft is carried out using a genetic algorithm.
In solidarity with the genetic algorithm, other algorithms are executed as follows:
The way in which the algorithms used are fed back can be seen in figure 1.
With the results obtained from the optimal design, some modifications are made to the algorithms of calculations and macroscopic analysis, to facilitate the modification of the orientations in the laminate obtained from the optimization and to analyze the incidence of the angles of 0°, 90° y 45°/-45° on the final properties of the composite material.
Optimal design
To analyze the incidence of the orientation of the reinforcing fibers in a composite shaft, the optimal design of a tubular shaft was carried out, according to the following considerations:
The design was made with a composite material of carbon fiber and epoxy resin polymer matrix. The properties of the material [9] are shown in table 1.
Optimization algorithm
The optimal design of the shaft was carried out using a genetic algorithm, implemented from the “Global Optimization Toolbox” complement of the MATLAB work environment. The objective function to be minimized corresponds to the weight (W) of the shaft and is given by:
Where L represents the length of the shaft, ρ the density of the composite material and g the gravity. The objective function is delimited by the following variables:
The outer and inner radio of the shaft are closely related to the number of layers by the expression:
Where h represents the total thickness of the laminate, k the number of layers and t the thickness of each layer. The constraints of the optimization algorithm are given by a factor of safety based on a theory of failure, the natural frequencies of bending and torsion, and the critical torque due to torsional buckling.
Considering a large number of possible stacking orientations and sequences in the shaft laminate, a discretization of these variables is performed as suggested by Gürdal, Haftka, and Nagendra in their research [10].
To further facilitate the efficiency of the algorithm, only orientations at 0 °, 90 °, 45 °, and -45 ° are taken into account, these last two orientations as successive pairs to guarantee their possibility of manufacture by the filament winding method [11]. Orientation selection is made considering that the normal stresses and ultimate shear stresses of the material are higher in these orientations.
Security factor
To ensure that the shaft is sufficiently resistant to the torsional loads applied, a safety factor equal to or greater than 3 is established [12]. A high value in the safety factor guarantees that all the layer adequately resist the applied stresses, this due to the variation of the admissible stresses throughout the laminate.
In the particular case of design, the maximum shear stress is given by:
Where T represents the maximum torque, c the distance of the farthest fiber (outer radius) and J the polar moment of inertia
Natural bending frequencies
A criterion of great importance in the design of rotors corresponds to the critical speed, that is, to the first bending mode, which must be sufficiently far from the maximum speed of rotation of the shaft, to avoid entering a state of resonance. To prevent the occurrence of this phenomenon, the value of the natural bending frequency is restricted above 1.5 [13] times the maximum speed of rotation.
Natural torsional frequencies
As with the natural frequencies of bending, the maximum speed of rotation of the shaft must not coincide and must be far enough away from the natural frequencies of torque of the shaft. To avoid the occurrence of this resonance phenomenon, the value of the natural torsion frequencies is restricted to above 1.5 [13] times the maximum speed of rotation.
Torsional buckling
Thin-walled tubular geometry shafts, that is, with slender geometry, can become buckled under the application of torsional loads. In order to avoid the collapse of the shaft during its operation, a critical torque is established, which is restricted to values equal to or greater than the maximum torque.
The critical torque in orthotropic composite materials [14] is given by:
Where 𝑟m is the mean radius and ℎ the thickness of the shaft walls. 𝐸x and 𝐸y correspond to the longitudinal and transverse elastic moduli of the composite material.
Genetic algorithm setup
The most relevant configuration options for the genetic optimization algorithm [15] are:
Macroscopic analysis
Macroscopic analysis of a single layer
To facilitate the analysis of the composite material, it is assumed that the layer that make up the shaft are orthotropic, homogeneous, elastic, of small thickness and that they do not present perpendicular loads. Under these considerations, the stress-strain relationship according to Hooke's general law [9] is:
Where 𝜎1 and 𝜎2 are the longitudinal and transverse normal stresses, 𝜏12 the shear stress in the principal plane, 𝜖1 and 𝜖2 the normal longitudinal and transverse deformations, and 𝛾12 shear deformation in the principal plane. The values 𝑄𝑖𝑗 represent the reduced stiffness coefficients and are given by:
Where 𝐸1, 𝐸2 are the longitudinal and transverse elastic constants, 𝐺12 the shear modulus in the main plane and, 𝑣12 and 𝑣21 the Poisson's ratios
The matrix of reduced stiffness coefficients established in equation (5), is valid only for the particular orientations of the fibers in each sheet, however, for the analysis of the composite material the global orientations of the laminate (x, y, z) are more useful.
To determine the stress-strain relationship in the global axes of the laminate, the Reuter matrix [16] and a transformation matrix are used, which relates the local orientations of each layer with the global orientations of the laminate, which allows obtaining the next relationship:
𝜎𝑥, 𝜎𝑦 and 𝜏12 are the longitudinal and transverse normal stresses and the shear stress in the principal plane, in the global directions. 𝜖𝑥, 𝜖𝑦 and 𝛾𝑥𝑦 are the longitudinal, transverse normal strains and the principal plane shear strain, in the overall directions of the laminate. The elements of the transformed stiffness matrix 𝑄̅𝑖𝑗 are defined by:
θ is the angle of orientation of the reinforcing fibers with respect to the global orientations of the laminate.
As a reference for the analysis, the coordinate system for the laminate is established in figure 2. The terms ℎ𝑘 represent the distance of the outer face of each sheet, with respect to the median plane of the laminate.
To determine the elastic constants of the laminate, it is necessary to establish the stiffness matrices of extension A, coupling B, and bending D, which relate the forces and moments per unit length applied in the principal plane, with the deformations (𝜖𝑥 𝑜 , 𝜖𝑦 𝑜 , 𝛾𝑥𝑦 𝑜 ) and curvatures in the median plane (𝜅𝑥, 𝜅𝑦, 𝜅𝑥𝑦). The matrices are expressed as a unique matrix [17] given by:
The elements of the matrix of extension A, coupling B and bending D, established in equation (17) are given by:
In the particular case of design, there is only one force per unit length corresponding to 𝑁𝑥𝑦 and is given by:
The elements of the inverse matrix of the extension matrix A, denoted as 𝑎𝑖𝑗, allow the calculation of the engineering constants of the laminate and are given by:
Tsai-Wu Failure Criterion
The Tsai-Wu failure theory is the most widely implemented criterion in orthotropic composite materials, due to the close similarity of the theoretical results with the experimental results obtained in various studies[18], [19].
The Tsai-Wu theory [20] states that a sheet fails if:
The components 𝐻 de Tsai-Wu theory are related to the ultimate longitudinal tensile stresses 𝜎𝑡1𝑢 and transversal 𝜎𝑡1𝑢, and ultimate longitudinal compression stresses 𝜎𝑐1𝑢and transversal 𝜎𝑐2𝑢 of the material, and are given by:
Results and Discussion
Optimization algorithms are prone to be trapped in local minima [21], that is, providing an optimal solution for a subregion of the limits of the objective function, but not for the optimal solution of the entire domain of possible solutions. To avoid this particularity, the optimization algorithm is executed several times, obtaining a single solution, which allows inferring that the solution is optimal for the entire domain of possible solutions. The results obtained in an intermediate step of the optimization process and the final results of the optimization algorithm are shown in table 2 and table 3 respectively.
The behavior of the optimization algorithm for the first two executions can be seen in Figures 3 and 4. The figures compare the objective function evaluated for its minimum value (black points) and its average value (green points) on the y axis, with the generations in which the genetic operators were applied on the x-axis.
In the figures, it can be seen that the algorithm actually went through local minima (stepped jumps in the black points), however, it managed to converge in a unique solution for the two executions, whose value corresponding to the weight (objective function) is 2.4650 [N].
When all the fibers are oriented in the same direction and assuming the same geometric configuration obtained from the optimal shaft design, the following results are obtained:
When setting the laminate in a balanced way, that is, with the orientations of 0 °, 90 °, 45 °, and -45 ° in the same proportions and symmetrically, the following results are obtained:
It is important to highlight that the results in table 5 correspond to 4 different stacking combinations that yielded the same results. The stacking combinations analyzed were:
With the results, it can be inferred that, in symmetric and balanced laminates, the properties of the material do not depend on the stacking sequence, but only on the proportion of the orientations.
By maintaining the same proportion of orientations, but this time with a non-symmetric stacking sequence, such as [0, -45, 90, 45, -45, 0, 45, 90] we obtain the following results:
Conclusions
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