An efficient curved beam element for thermo-mechanical nonlinear analysis of functionally graded porous beams

Abstract

This research is aimed to develop an efficient and high-performance four-node iso-parametric beam element, which is composed of Functionally Graded Material (FGM). In addition, different patterns of material distribution will be considered through the height of element. On the other hand, beam’s imperfection, presented here with porosity, is taken into account by using the rule of mixture. In order to alleviate the shear locking, Mixed Interpolation of Tensorial Components (MITC) is utilized by using tying points. Strain interpolation at some tying points reduces the order of strain functions. Therefore, three Gauss points can be employed for numerical integration instead of four Gauss points. Furthermore, the geometrically nonlinear effects are incorporated by using Green-Lagrange strains. Since the material properties are considered to be thermal-independent, they remain constant during the analysis. Finally, some benchmark problems are solved to illustrate the correctness of formulation and accuracy of the proposed element. Several parameters, including porosity percentage, FGM patterns and corresponding power indices, are investigated in the other examples. It is observed that the proposed element is more accurate for linear and nonlinear analyses of the thin beam with large deformations and rotations, even by using fewer numbers of elements compared to other developed elements. On the other hand, both axial and transverse displacements decrease when the value of the exponent of sigmoid pattern increases. On the contrary, the exponent of power pattern has a different effect on the axial displacement.

Introduction

One of the most applicable elements, which is widely used in skeletal structures, is beam element. In recent years, investigating the nonlinear behavior of the structures is taken into account by many researchers. Due to high applicability of finite element method for nonlinear analyses, several researchers have been focused on formulating new element and improving them. Many studies were implemented to develop elements with high efficiency so that fewer numbers of elements were employed to reach more accurate responses. Although, many papers have been published on proposing new beam element with the different number of nodes, some new approaches are still suggested improving the flexural and shear behaviors of element. Most of these methods have been presented to avoid shear locking, especially for thin beam structures.

It is obvious that beam element is one of the simplest elements, which can be derived by using Lagrangian and Hermitian shape functions. These types of elements were widely developed in some well-known finite element books [3], [51]. In order to improve the flexural and shear behaviors, many studies were implemented based on the FE approach. In 1982, Prathap and Bhashyam proposed a C⁰ continuous element by using shear-flexible formulation based on the Timoshenko beam element. They used reduced integration scheme to eliminate in-plane locking in curved beams [32]. Moreover, a linear two‐node curved beam element was derived by using C⁰ continuous formulation based on the curvilinear deep shell theory [2]. In addition, another valuable research was implemented to propose an iso-parametric curved beam element which can be employed for modeling thick beams [31]. Some analytical researches were also performed on the static and dynamic analysis of curved beams under either thermal or mechanical loading with circular and parabolic form by using explicit stiffness matrix [42], [44], [48], [49], [53]. The other articles were published about presenting new straight and curved element formulations, in which there were different novelty and methods to improve the behavior and accuracy [17], [18], [23], [33]. In most of them, eliminating locking phenomena was the main concern of the researchers while only linear analysis was the base of their formulations.

One of the newest and most applicable formulations, which was employed to remove the locking phenomena, is Mixed Interpolation of Tensorial Components (MITC). This method was first proposed for plate and shell elements [4], [6]. In addition, many researches were carried out on the development of shell elements by using MITC approach [21], [34]. In 2008, this approach was used to propose a two-node tapered beam element [22]. Furthermore, Yoon et al. considered the warping effects in the finite element formulation of 3D beam based on the continuum mechanics [56]. Subsequently, this effect was incorporated in modeling of discontinuous arbitrary cross-section beams by Yoon and Lee [54]. They also extended their approach for nonlinear analysis of beams, in which large twisting behavior was involved [55]. Recently, Carrera et al. developed a higher-order MITC beam element by using Carrera Unified Formulation (CUF) to analyze compact, thin-walled and composite structures [7]. The procedure of MITC method has been extended to nonlinear analysis of beams by Rezaiee-Pajand and coworkers [45]. On the other hand, the MITC method has been recently employed to formulate plane elements [38], [39], [41], [46]. Based on this review, it is expected that MITC scheme can be still useful for development of the beam elements with more accuracy and higher performance.

Another subject of research, which has been widely surveyed, is composite structures. Beams are known as one of the most applicable elements in civil engineering. Many studies were dedicated to analyze composite ones either linearly or nonlinearly , [16], [24], [27], [36], [40], [47], [52]. Among composite materials, researchers have paid more attention to Functionally Graded Materials (FGMs) due to its high performance and capability under both mechanical and thermal loads. This property is arising from the continuously combination of metal and ceramic phases. Recently, these materials have been widely used in plate and shell elements, in which power function was employed to define the distribution of elastic modulus through the thickness [35]. Based on the authors review of the literature, too many efforts have been made to model the behavior of beam structures composed of FGM during the past decade [13], [30], [43], [50], [57]. Most of them are based on the power function, while the other patterns are available to be assigned to the variation of material properties, such as, elastic modulus, Poisson’s ratio and thermal expansion coefficient. For instance, Fallah and Aghdam presented an analytical solution for nonlinear analysis of FG Euler-Bernoulli beam resting on elastic foundation. They used power function as a variation law of elastic modulus and specific mass[15]. In another article, they also investigated the buckling load and nonlinear natural frequencies of FG beams subjected to thermo-mechanical load [14]. Furthermore, the exact second-order stiffness matrix of FG tapered beam-column was analytically obtained based on the Euler-Bernoulli beam theory [37]. Another analytical solution of bi-directional functionally graded beams based on elasticity approach has been obtained by Lü et al. [25]. Accordingly, most of the researchers presumed perfect FG beam in their analyses while this assumption was not compatible with the porosity of structures.

During the past years, several works have been checked the effect of imperfection on the structures’ behavior, especially beams. This imperfection was related to porosity of the material due to the construction process. Although, the distribution of porosity through each material can be variable, many researchers consider a porosity fraction, which is consistently distributed through the structure[20]. Furthermore, elastic buckling and bending of FG porous beams were also studied using Timoshenko beam theory by Chen et al. They considered two different patterns for the distribution of material through the thickness of the beam [9]. In 2016, Ebrahimi et al. investigated the natural frequencies of simply supported FG porous Euler beam under thermal loads. They incorporated the temperature-dependent material properties in their formulation [10]. Moreover, Ebrahimi and Jafari studied the linear vibration of the FG porous beam subjected to thermal loads by using Navier solution method and Reddy beam theory [11]. They also presented a four-variable refined shear-deformation beam theory to obtain the natural frequencies of FG porous beam, in which thermal-dependency of material was considered [12]. Further, thermomechanical analysis of FG porous nanobeams resting on elastic foundation was implemented by using FEM and First Shear Deformation Theory (FSDT) [1].

Based on the elaborated literature review, the authors found out that developing an efficient locking-free beam element to employ in the nonlinear analysis of composite beams can be still taken into account. Consequently, this research deals to perform thermo-mechanical analysis of beams, composed of porous and non-porous FGMs. The main contribution of the present study is to develop a new four-node iso-parametric beam element, based on the MITC approach, in order to alleviate the shear locking, especially in the thin beam. Moreover, the authors used Total-Lagrangian principles to consider large deflection and rotations. Moreover, the variation of material properties (Young’s modulus, Poisson’s ratio and thermal expansion coefficient) through the height of the beam is defined by two patterns, including power and sigmoid. To fulfill all goals of this research, the authors establish the porous FGM formulations based on the rule of mixture approach, firstly. Then, the finite element formulation of the four-node beam element will be developed by using mixed interpolated strain fields. Afterwards, the Green-Lagrange strain formulation is utilized to incorporate large deformations. Finally, some popular benchmarks and new problems will be solved to illustrate the correctness, accuracy and capability of the proposed beam element in geometrically nonlinear thermo-mechanical analyses of beam structures.