Sunday, 18 December 2022

Development of spectral element method for free vibration of axially-loaded functionally-graded beams using the first-order shear deformation theory

 

Development of spectral element method for free vibration of axially-loaded functionally-graded beams using the first-order shear deformation theory


Abstract

In this study, the spectral element method to assess free vibration of axially-loaded functionally-graded beams according to the first-order shear deformation theory is developed. In this approach, the underlying equations of motion and related boundary conditions were determined by using Hamilton's principle. Analytical solutions are presented for simply-simply, clamped-clamped, clamped-free, and clamped-simply supported FG beams. The general solutions corresponding to governing equations determined for axially loaded FG beams are presented in the spectral-element matrix. A comparison is performed to validate the proposed formulation and solution between the obtained results with the existing solutions provided in previous studies. Finally, the parametric study is developed to investigate the impacts of slenderness ratio, end supports, gradient parameter, and axial load value on the free vibration characteristics in the axially-loaded FG beams. A comparison of the results of the developed theory and the results of the existing solutions shows that the proposed solution developed based on the spectral element method is the appropriate accuracy and efficiency in determining the dimensionless natural frequencies parameter. Based on the findings, all four parameters are effective in the free vibration of axially-loaded FG beams.

Introduction

The mechanical characteristics of functionally-graded materials (FGMs) change gradually along with the given directions (Gorji Azandariani et al., 2022; Rajasekaran et al., 2022; Yang et al., 2021). FGMs have begun to find their way into beams due to technological developments and achievements. Considering the broad potential of FG beams in engineering applications, it is of proper great importance to better understand their mechanical behavior to develop these kinds of structures. Consequently, buckling and vibration problems of FG beams have been investigated using various numerical and analytical approaches according to different beam theories (Dangi et al., 2021; Darban et al., 2021; Gorji Azandariani et al., 2021). Also, stability and free vibration problems of FG nanobeam/nanoplate have been investigated using various numerical and analytical approaches according to different beam theories (Ji et al., 2020; Luo et al., 2021; Shen et al., 2020; Van Vinh and Tounsi, 2021, 2022; Vinh, 2022; Vinh et al., 2022).

Aydogdu and Taskin (2007) explored the free vibration of a simply-supported (S–S) FG beam based on classical, parabolic, and exponential shear deformation theories. Sina et al. (2009) provided a new beam theory to study the free vibration of FG beams. These authors set the normal lateral stress of the beam to zero. In a study done on the FG beams, the bending and free vibration problems were investigated using different higher-order shear deformation theories. These developed theories describe the higher-order variations that occurred in the transverse shear strain through the depth of the beam (Thai and Vo, 2012). In another work performed by some scholars, a new first-order shear deformation beam theory (FSDT) was developed for solving the free vibration problem in axially-loaded rectangular FG beams. This study utilized the in-plate stress equilibrium equations to extract the transverse shear stiffness (Nguyen et al., 2013).

Furthermore, the Lagrange multiplier strategy was also employed for evaluating the free vibration problem in the FG beams according to different higher-order beam theories (Şimşek, 2010a, 2010b, 2010c; Şimşek and Kocatürk, 2009). The vibration and buckling behavior of FG sandwich beams have also been analytically investigated by employing the higher-order beam theory developed by Nguyen et al. (2015). A hyperbolic distribution is assumed for transverse shear stress. In two other studies done by the same research group, an efficient and simple analytical method was developed to analyze the dynamic/static behaviors in the FG beams using the theory of elasticity (Li, 2008; Li et al., 2010; Nguyen et al., 2021). The thermoelastic behavior of FG beams was studied using an FSDT-based beam element proposed by Chakraborty et al. (2003). The vibration of axially-loaded FG Timoshenko non-uniform beams was investigated using the novel technique proposed by Huang et al. (2013). In addition to the approaches mentioned above, a new two-node six-degree-of-freedom beam element was also developed to assess the free vibration problem in FG beams, along with a finite element model, constructed to dynamically analyze layered FG beams according to the third-order zigzag theory (Alshorbagy et al., 2011; Kapuria et al., 2008). In two different studies, the classic Euler-Bernoulli theory and Timoshenko beam theory were applied to develop a dynamic stiffness matrix (DSM) for investigating the free vibration problem in the FG beams (Phi et al., 2021; Su et al., 2013; Su and Banerjee, 2015). There have been many studies on the FG beams beyond this article's scope (Barretta et al., 2016; Barretta and Luciano, 2015; Reddy et al., 2020; Vo et al., 2015; Wang, 2021).

According to the studies conducted on the free vibration in the metallic beams, the axial force significantly affects the mode shapes and natural frequencies (Cheng and Tseng, 1973; Howson and Williams, 1973; Lai et al., 2012; Niknam et al., 2014); thus, it needs to be considered in beam vibration analysis. However, the data reported in the literature on the axially-loaded FG beams is still very limited. Kang and Li (2009) presented the Euler-Bernoulli beam model for free vibration of axially-loaded FG beams under S–S boundary conditions. Moreover, Trinh et al. (2016) developed an analytical solution for vibration and buckling of FG beams with various end supports subjected to mechanical loads. Sui et al. (2015) investigated the transverse free vibration of an axial moving beam made of FGM using Timoshenko's beam theory. They studied and evaluated the natural frequencies, vibration modes, and critical velocities of axial moving beams made of FGM. They also used the Hamilton principle to derive the governing equation and used a sophisticated fashion approach to achieve transverse dynamic behaviors such as vibration modes and natural frequencies. In this study, the critical velocity is determined numerically, and its changes are plotted based on the power law, the initial axial stress, and the length-to-thickness ratio. Yao et al. (2020) studied the transverse free vibration and wave propagation of FG microbeams with an axial motion based on a nonlocal theory and the Timoshenko beam model. Yao et al. (2020) hypothesized the properties of FG microbeam performs vary in thickness. Also, the studied parameters were the effects of gradient index, nonlocal parameter, and axial velocity on natural frequencies. In addition, the wave propagation properties of the Timoshenko microbeam were analyzed by FG and studied the significant effects of wave number and other variables on wave propagation frequencies and wave velocities. Zhu et al. (2021) presented vibrational analyzes on axial moving FG nanoplates exposed to hydrothermal environments and developed the governing equation of motion based on the Hamilton principle using nonlocal strain gradient theory. Their studies show that with increasing nonlocal parameters, gradient index, temperature change, humidity concentration, and axial velocity, vibration frequencies have decreased. Also, with increasing the characteristic parameter scale of materials and the aspect ratio, the frequencies have also increased.

According to the literature review, it is clear that several numerical and analytical studies have been performed to analyze the free vibrations of FG structures, including beams, plates, and shells. Although a large number of studies have been performed on the free vibration analysis of FG beams, on the other hand, the development of the spectral element method for free vibration of axial load grading functional beams using first-order shear deformation theory has not been investigated in studies. The spectral element method (SEM) exactly reflects the dynamic behavior of the structural elements since it is developed using the exact frequency-dependent dynamic shape functions satisfying the underlying equations of motion (Gopalakrishnan et al., 1992; Lee and Jang, 2010). Also, SEM, in contrast to the traditional finite element method, allows one to represent the entire uniform structural member as a single element, whatever its length, with no need for dividing the structural members into several fine elements for enhancing the accuracy of the solution. Consequently, the overall number of degree-of-freedoms applied in the dynamic analysis and the costs of the computations are reduced significantly. To the best of the authors' knowledge, no publication that uses the spectral element method to evaluate the free vibration of axial load-bearing functional calibration beams using first-order shear deformation theory is available. Hence, this issue has not been well studied in the literature and needs further study.

The authors know that the vibration problem in the FG beams has not been investigated using the SEM to date. The current study develops the SEM to evaluate the free vibration problems in the axially loaded FG beams based on the first-order shear deformation theory (FSDT). The underlying equations of motion and the corresponding boundary conditions are first determined by employing Hamilton's principle. The general solutions of the underlying equations of motion are then used to formulate the spectral-element matrix. Next, the proposed SEM's accurate performance is evaluated by comparing the captured vibration frequencies with the vibration frequencies reported in previous studies. Finally, the effect of various parameter variations, including material property gradient parameter, slender ratio, and boundary conditions, on the vibration frequencies of FG beams is explored in detail.

Section snippets

Underlying differential equations

For this section, a rectangular FG beam with a length of L, a width of b, and a height of h are considered (Fig. 1). The upper surface of the beam is made of ceramic, while its lower surface is made of metal. Moreover, to simplify the calculations, Poisson's ratio ν is considered constant. Furthermore, the following equations define the Mass density ρ and Young's modulus E (Su and Banerjee, 2015):E(z)=(Ec−Em)(zh+12)k+Emρ(z)=(ρc−ρm)(zh+12)k+ρmwhere c is the ceramic constituent, m is the metallic 

Numerical method and verification

In this section, the proposed method is verified using the obtained results with the existing solutions provided in previous studies. Two comparative studies were carried out as a means to evaluate the reliability of the presented approach. In Table 1, the dimensionless frequencies of the S–S FG beam are compared with the results of Nguyen et al. (2013), which were obtained using an analytical method according to the FSDT. In Table 1, the effects of the axial force on the natural frequencies

Parametric results

This section deals with a parametric study using the presented method of different parameters involved in the FG beams and boundary conditions. In this section, the effects of end support, slender ratios L/h, axial force N0, and gradient index k on the vibration frequencies of the FG beams will be evaluated. The materials used in the upper and lower surfaces of the FG beams and the cross-sectional dimensions are the same as in Ref. (Su and Banerjee, 2015). Eq. (34) is used to non-dimensionalize 

Conclusions

The current paper develops the spectral-element method SEM to evaluate free vibration problems in axially-loaded FG beams. The underlying partial differential equations and the corresponding boundary conditions were determined by employing Hamilton's principle. The general solutions acquired for the underlying equations of axially-loaded FG beams form the basis for developing the spectral element matrix. The natural frequencies were determined using the W–W algorithm as a solution approach. To

Authors contribution statements

Mojtaba Gorji Azandariani: Conceptualization, Investigation, Methodology, Project administration, Writing – original draft, Writing – review & editing. Mohammad Gholami: Data curation, Formal analysis, Methodology, Writing – original draft, Visualization. Elnaz Zare: Data curation, Formal analysis, Visualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (48)

There are more references available in the full text version of this article.

Thanks to 
a
Centre for Infrastructure Engineering, Western SydneyUniversity, Penrith, Australia
b
Department of Civil Engineering, Yasouj University, Yasouj, Iran

Er. SP.ASWINPALANIAPPAN., M.E.,(Strut/.,)
Structural Engineer

http://civilbaselife.blogspot.com


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Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation

 

Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation


Abstract

A third order zigzag theory based model for layered functionally graded beams in conjunction with the modified rule of mixtures (MROM) for effective modulus of elasticity is validated through experiments for static and free vibration response. Two systems, Al/SiC and Ni/Al2O3, fabricated using powder metallurgy and thermal spraying techniques respectively, are considered for the validation. The theoretical predictions for the layered beams with the ceramic content varying from 0% to 40% are compared with the experimental data for the static deflection under simply-supported and cantilever boundary conditions, and for the natural frequencies under cantilever and clamped-clamped boundary conditions. The predictions using the MROM are found to be in close agreement with the experiments for both systems, whereas the linear rule of mixtures based property estimates lead to highly erroneous results. The effect of number of layers on the accuracy of the theoretical model is discussed. The accuracy of the predicted results gives confidence on the values of stress to strain transfer ratio used in the MROM for the two systems in the layered fabrication context, and also demonstrates the capability of the zigzag theory in accurately modelling the mechanics of such beams.

Introduction

The concept of functionally graded material (FGM) emerged from the need to fabricate a new composite for high temperature structural applications by using a heat resistant ceramic on the high temperature side and a metal on the low temperature side to provide mechanical toughness [1]. The gradient compositional variation of the constituents from one surface to the other provides an elegant solution to the problem of high transverse shear stresses that are induced when two dissimilar materials with large difference in material properties are bonded. Since its inception, the concept has found many potential applications like thermal and corrosion barriers, dental and orthopedic implants, plasma facing bio materials, sensors, lightweight armour material with high ballistic efficiency etc. [2], [3], [4], [5]. The gradation of the volume fractions of the constituents along the thickness direction can have a continuous or a step-wise variation [6].

Several studies have been performed to analyze the static and dynamic behavior of functionally graded beams, plates and shells. The various models available for the analysis of FGM structures can be categorized based on two criteria (i) the type of displacement field approximations across the thickness (kinematic modelling) and (ii) the direct use of assumed variations of the material properties across the thickness or use of a material model for computing the effective properties at a point with given volume fractions of the constituents. The accuracy of prediction of response of FGM structures will depend on both the kinematic modelling and the correctness of estimated effective material properties of the two-phase system. Exact analytical solutions based on three dimensional (3D) elasticity have been presented for simply supported functionally graded infinite panel [7] and rectangular plate [8] under mechanical load, and for flat panel under transient thermal load [9], considering an assumed exponential variation of Young’s modulus and thermal expansion coefficient, and constant Poisson’s ratio over the thickness. Vel and Batra [10] have presented a 3D exact solution for free and forced vibrations of simply supported rectangular functionally graded plates, considering the Mori–Tanaka [11] and the self consistent [12] methods for computing the effective Young’s modulus and Poisson’s ratio with power law variation of the volume fractions of the constituents across the thickness. Such 3D analytical solutions are available only for specific geometry and boundary conditions. For analysis of structures or arbitrary shape and boundary conditions, several 2D models for plates and shells and 1D models for beams have been developed. The classical plate theory (CPT) has been employed for static bending [13] and transient [14] nonlinear response of FGM plates, using a power law variation for the volume fractions and employing linear rule of mixtures [15] for computing all effective material properties. The classical theory neglects shear deformation. Reddy and his coworkers have employed first order shear deformation theory (FSDT) (the simplest one to incorporate shear deformation effect) for thermoelastic finite element analysis of FGM plates [16] and for axisymmetric bending of circular FGM plates [17] using the linear rule of mixtures (ROM) for computing effective material properties with a power law variation of the volume fractions. Reddy [18], and Yang and Shen [19] have employed the refined third order theory (TOT) with an assumed power law variation for Young’s modulus, density and thermal expansion coefficient and a constant Poisson’s ratio for static and dynamic response of FGM plates under thermal loading. The same material model has been used by Chakraborty et al. [20], who presented a FSDT based shear locking free element for FGM beams. To the best of the authors’ knowledge, no experimental validation of the theoretical predictions of static and dynamic response of FGM beams and plates has been reported in open literature.

In a recent study, the authors [21] showed that the experimentally obtained values for the modulus of elasticity for Al/SiC composite system with different volume fractions of the constituents are considerably lower than those predicted using the ROM. It was also shown that the modified rule of mixtures (MROM) proposed by Tomota et al. [22] for two-phase systems, which involves a parameter for the stress to strain transfer between the two phases, can be used to accurately predict the Young’s modulus of this system. This parameter was found to be consistent across different volume fractions of Al and SiC.

This paper presents a finite element model for the dynamic analysis of layered FGM beams using an efficient third order zigzag theory [23] for the layerwise mechanics and the MROM for estimating the effective modulus of elasticity, and its experimental validation for the static deflection and natural frequencies of two different FGM systems under various boundary conditions. The two FGM systems considered are Al/SiC and Ni/Al2O3, both of which have generated considerable interest as heat resistent materials [24], [25]. Powder processes are ideally suited for fabricating gradient materials because of the excellent microstructural control and the versatility in these techniques. Several studies have been reported on the fabrication of ceramic–metal gradient materials by different powder metallurgy methods such as consolidation of powder stack [26], [27], consolidation of green sheet lamination [28], [29], powder slurry spraying [30], and combustion synthesis [31]. Reviews of these methods have been recently presented by Kieback et al. [6] and Watanabe et al. [32]. In the present study, Al/SiC FGM beam samples with three and five layers were fabricated using the powder metallurgy route following consolidation of powder stack with cold isostatic pressing (CIP) followed by sintering. The study on the beams with two different numbers of layers illustrates the effect of the gradation step size on the applicability of the theoretical model. Five-layered Ni/Al2O3 beam samples were prepared using combustion powder thermal spray process [33]. The two systems fabricated with different methods ascertain the consistency of the theoretical model.

Section snippets

Third order zigzag beam theory

The zigzag one dimensional (1D) beam theory presented by Kapuria et al. [23] is used to predict the response of the layered FGM beam. Consistent with the fabrication process, the FGM is modelled in this theory as a laminate of multiple perfectly bonded layers of isotropic material of layerwise constant composition. Let the beam be made of L such layers of discretely varying compositions. In general, the volume fractions Vc and Vm of the ceramic and the metal are assumed to vary along the

Effective material properties

The effective Young’s modulus of the two-constituent composite systems is computed using the modified rule of mixtures (MROM), which was originally proposed by Tomota et al. [22] for cemented carbides, and subsequently used for FGMs by many researchers (e.g. [25], [35]). According to this approach, the two-phase material in each layer in the FGM is treated as an isotropic composite for which the uniaxial stress σ and strain ε are related to the average uniaxial stresses σm, σc and strains εm, εc

Specimen geometry and design

FGM beam samples with isotropic homogeneous layers of metal and ceramic with their volume fractions varying from layer to layer were prepared for the experimental investigation. Al/SiC FGM samples were prepared by powder metallurgy route and Ni/Al2O3 FGM samples were made by thermal spray technique. For the Al/SiC system, 99.8% pure aluminum powders of grade 1180 and 95% pure silicon carbide powder with 400 mesh size were used as the raw materials. To study the effect of number of layers on the 

Results and discussions

The material properties of the basic constituents Al, SiC, Ni and Al2O3 of the FGM systems, considered for computing the effective properties of the layers, are listed in Table 2. The overall densities of the samples computed using the ROM and their measured densities are listed in Table 3. It is observed that the measured densities differ from the theoretical densities by a maximum of 3.2% for Al/SiC samples and 5.2% for the Ni/Al2O3 samples. This difference is due to porosity in the samples.

Conclusions

The static and free vibration response of layered FGM beams of Al/SiC and Ni/Al2O3 prepared with powder metallurgy and combustion powder thermal spray processes, respectively, have been studied experimentally as well as theoretically. For the theoretical prediction of response, a finite element based on an efficient zigzag theory is used in conjunction with a modified rule of mixtures for predicting the effective elastic modulii of the layers, with the experimentally determined value (91.6 GPa) 

Acknowledgement

The authors are grateful to International Advanced Research Center for Powder Metallurgy and New Materials, Hyderabad, India, and Metallizing Equipment Co. Pvt. Ltd., Jodhpur, India, for their generous help in fabrication by providing their facilities for sample preparation.

References (39)


Thanks to 
a
Applied Mechanics Department, I.I.T. Delhi, Hauz Khas, New Delhi, India
b
University of Ottawa/CISTECH Corporation, Ottawa, ON, Canada
Er. SP.ASWINPALANIAPPAN., M.E.,(Strut/.,)
Structural Engineer

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Thursday, 15 December 2022

FIELD TESTS ON BRICKS

 

FIELD TESTS ON BRICKS

FIELD TESTS ON BRICKS


It is necessary to check the quality of brick before using it in any construction activities. There are some field tests that we can conduct in the field in order to check the quality of bricks. These tests are as follows.

Water Absorption
Visual inspection
Efflorescence
Dimension
Hardness
Soundness
Structure

1. WATER ABSORPTION

5 bricks are taken and the bricks are weighed dry and the average dry weight of 5 bricks is calculated. Bricks are then immersed in water for a period of 24 hours. After 24 hours of immersion, bricks are weighed again and an average of 5 bricks is calculated. The difference between the final average weight and the initial average weight indicates the amount of water absorbed by the bricks. It should not in any case exceed 20 per cent of the average weight of dry bricks.

2. VISUAL INSPECTION

In this test, bricks are closely inspected for their shape. The bricks of good quality should be uniform in shape and should have truly rectangular shapes with sharp edges.

3. EFFLORESCENCE

This test should be conducted in a well-ventilated room. The brick is placed vertically in a dish 30 cm x 20 cm approximately in size with 2.5 cm immersed in distilled water. The whole water is allowed to be absorbed by the brick and evaporated through it. After the bricks appear dry, a similar quantity of water is placed in the dish, and the water is allowed to evaporate as before. The brick is to be examined after the second evaporation and reported as follows:

Nil: When there is no perceptible deposit of salt
Slight: When not more than 10% of the area of brick is covered with salt
Moderate: When there is a heavy deposit covering 50% of the area of the brick but unaccompanied by powdering or flaking of the surface.
Heavy: When there is a heavy deposit covering more than 50% of the area of the brick accompanied by powdering or flaking of the surface.
Serious: When there is a heavy deposit of salts accompanied by powdering and/or flaking of the surface and this deposition tends to increase in the repeated wetting of the specimen.
Bricks for general construction should not have more than slight to moderate efflorescence.

4. DIMENSIONAL TOLERANCE

Twenty bricks are selected at random to check the measurement of length, width and height. These dimensions are to be measured in one or two lots of ten each as shown in the figure. Variations in dimensions are allowed only within narrow limits, ±3% for class one and ±8% for other classes.

Dimension Test On Bricks
5. HARDNESS

In this test, a scratch is made on a brick surface with the help of a fingernail. If no impression is left on the surface, the brick is treated as to be sufficiently hard.

6. SOUNDNESS

Two bricks are taken, one in each hand, and they are struck with each other lightly. A brick of good quality should not break and a clear ringing sound should be produced.

7. STRUCTURE

A brick is broken and its structure is examined. It should be homogeneous, compact and free from any defects such as holes, lumps etc.



Er. SP.ASWINPALANIAPPAN., M.E.,(Strut/.,)
Structural Engineer

http://civilbaselife.blogspot.com

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Permeable Pavement System In Pavement Construction

 

Permeable Pavement System In Pavement Construction


Permeable pavement system

Permeable paving is a range of sustainable materials and techniques for permeable pavements with a base and sub-base that allow the movement of stormwater through the surface.


In addition to reducing runoff, this effectively traps suspended solids and filters pollutants from the water. Examples include roads, paths, lawns and lots that are subject to light vehicular traffic, such as car/parking lots, cycle paths, service or emergency access lanes, road and airport shoulders, and residential sidewalks and driveways.


Although some porous paving materials appear nearly indistinguishable from nonporous materials, their environmental effects are qualitatively different. Whether previous concrete, porous asphalt, paving stones or concrete or plastic-based pavers, all these pervious materials allow stormwater to percolate and infiltrate the surface areas, traditionally impervious to the soil below.
The goal is to control stormwater at the source, reduce runoff and improve water quality by filtering pollutants in the substrata layers.


Er. SP.ASWINPALANIAPPAN., M.E.,(Strut/.,)
Structural Engineer

http://civilbaselife.blogspot.com

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