Analytical and experimental investigation of bend-twist coupling on the vibrational response of multi-layered stepped composite beams

Abstract

This paper investigates the bend-twist coupling analysis of multi-layered stepped generally orthotropic composite beams subjected to mixed end-of-beam and mid-span supports. Specifically, an analytical closed-form model was developed based on first-order shear deformation theory (FSDT), which discretizes the domain into elements based on the step change of geometry, laminate configuration, or mid-span boundary supports. Hamilton's principle was used to derive the governing equations within each element and connection and boundary equations. The state-Space approach was then utilized to provide an analytical solution. Moreover, an experimental investigation was conducted to validate the mode shapes and natural frequencies of the beam subjected to several mixed boundary conditions. The results are also validated with the literature and a finite element model developed using ANSYS. Comparisons demonstrate the reliability and accuracy of the analytical model.

Introduction

Composite materials have been widely used in industry due to their low weight, high load-bearing capacity, superior corrosion resistance and tailorable material properties. In addition, laminated composite materials provide extensive design flexibility to achieve several design targets, which are unlikely to be succeeded using isotropic materials. Coupling effects are one of the distinguishing characteristics of laminated composites. Coupling leads to responses in directions other than that of the applied load, e.g. bend-twist coupling results in twisting deformation in response to bending loads applied to the laminate. The benefits of Bend-Twist (BT) coupling have been widely utilized in improving the structural design by many industries. For example, it provides wind turbine blades with self-alleviation of abrupt inflow fluctuations, such as gusty or turbulent conditions, decreasing dynamic and fatigue loads and regulating the generated power [1], [2], [3]. Another example is the application in the marine propulsion industry, where performance and maintenance are highly prioritised, it is demonstrated that utilising the BT coupling leads to regulated and less power consumption, decreased blade vibration and noise level, improved propeller efficiency, and less fatigue load [4], [5], [6]. There is a broad range of other applications for utilizing the advantage of bend-twist coupling in tidal energy and other industries, which are discussed elsewhere [7]. The vibrational response is an important, often critical aspect of many designs, whether to avoid resonance or achieve specific natural frequencies. In the case of laminates with significant bend-twist coupling, it is essential to develop tools for accurately and reliably predicting the vibration behaviour of the structure. In order to interpret the complex elastic behaviour of laminated composite beams, various Equivalent Single Layer (ESL) theories have evolved, including classical [8], [9], [10], [11], [12] and shear-deformable [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26] equivalent single-layer beam theories.

Both the numerical and analytical methods are widely used to simulate the bend-twist coupling of generally orthotropic laminated beams. For the most applicable numerical methods, one could refer to finite element [10], [19], [23], boundary elements [27], differential quadrature [9], line element-less method [28] and semi-analytical methods [14]. [10] developed a beam finite element model which includes the bend-twist coupling effect, which shows a good agreement for an ideal beam.

There are usually less simplifying assumptions when implementing numerical methods; however, since the provided solution approximately satisfies the governing equations and boundary conditions, they provide results with relative accuracy, depending on the number of nodes or element size. Additionally, approximate analytical solutions [13], [29] and infinite series solutions [8] provide approximate results depending on the accuracy of the approximating terms and number of truncated terms, respectively. Conversely, exact analytical methods present a closed-form solution that accurately provides the required precision of the results, enabling parametric exploratory studies and design optimisation. Using the method of Lagrange multipliers, [21] developed an analytical solution to the free vibration of generally layered composite beams. [25] developed an analytical method to study the vibrational response of cross-ply laminated beams. [30] used dynamic stiffness matrix to find the exact solution of the free vibration of a three-layer symmetric sandwich beam.

In the case of beams made with laminated generally orthotropic layers, the elastic behaviour of BT coupling is more complicated than bending-extension coupling in cross-ply laminates, as the displacement field is highly dependent on the lateral position along the width of the beam due to the twisting deformation [31]. This limits the application of the basic bend-extension coupled displacement field assumed by ESL theories like Refined Zigzag theory [18], which are not able to capture the bend-twist coupling for generally orthotropic composite beams. There are various simplifying assumptions used to find the vibrational behaviour of structures exhibiting BT coupling. For instance, ignoring the BT coupling [9] or ignoring the contribution of longitudinal-transversal shear stress and the lateral movement due to the twisting motion [23] results in missing the twist-dominated mode shapes and providing less accurate results for the adjacent mode shapes. Additionally, ignoring the effective bending Poisson ratio [32], [33], which plays an important role specifically for fibre orientation closer to ±90°, results in the model not capturing the increase in natural frequencies due to the high $E_1/E_2$ ratio in composite materials.

There is a lack of analytical methods to simulate the bend-twist vibrational behaviour of the beams. [14]. Moreover, most of the works are limited to beams of uniform cross-section, which are less practical, as they cannot adopt reinforcement patches, integrated sensors and actuators, and mid-span supports. Besides, there is a lack of experimental examination to study the BT coupling of the stepped laminated beams. This paper presents a closed-form analytical solution to study the BT coupling of stepped laminated beams subjected to end-of-the-beam and mid-span mixed boundary conditions. Removing the above-mentioned widely used simplifying assumptions, the proposed analytical solution obtains more specific and accurate outcome for a wide range of material, geometrical, and boundary conditions configurations. Results are compared with the literature, and a finite element (FE) model developed using the commercial finite element software ANSYS. Furthermore, since there is lack of published experimental results examining bend-twist behaviour of a more generic beam, an experimental study was performed to find the natural frequencies and mode shapes of a CF/PEEK laminated composite cantilever beam integrated with a pair of macro fibre composite (MFC) piezoelectric actuators subjected to various boundary conditions to examine the model's validity, which could be used as a benchmark in other studies. Finally, a parametric study was conducted using the developed analytical model to assess the effect of step location and laminate configuration on natural frequencies of the structure.