A theoretical study on free vibration behavior of prestressed functionally graded material (FGM) beam is carried out. Power law variation of volume fraction along the thickness direction is considered. Geometric nonlinearity is incorporated through von Kármán nonlinear strain–displacement relationship. The governing equation for the static problem is obtained using minimum potential energy principle. The dynamic problem for the prestressed beam is formulated as an eigenvalue problem using Hamiltons principle. Three classical boundary conditions with immovable ends are considered for the present work, namely clamped–clamped, simply supported–simply supported and clamped–simply supported. Four different FGM beams, namely Stainless Steel–Silicon Nitride, Stainless Steel–Zirconia, Stainless Steel–Alumina and Titanium alloy–Zirconia, are considered for generation of results. Numerical results for nondimensional frequency parameters of undeformed beam are presented. The results are presented in nondimensional pressuredisplacement plane for the static problem and in nondimensional frequencydisplacement plane for the dynamic problem. Comparative frequencydisplacement plots are presented for different FGMs and also for different volume fraction indices.
Functionally graded material ; Large deflection ; Timoshenko beam ; Prestressed beam ; Loaded natural frequency
Functionally graded materials (FGMs) are inhomogeneous composites that have smooth and continuous variation of material properties in space. In most of the existing and potential future applications, FGM is considered mainly as a mixture of ceramic and metal in varying proportion. With the strength and toughness of metals, and the thermal and wear resistance of ceramics, FGM components possess good qualities of both the metals and ceramics. This makes it suitable for the FGM structures or components to be used in high temperature environment. FGM components are found in various applications, such as in aerospace, nuclear, automotive, civil, biomechanical, optical, electronic, mechanical, chemical and shipbuilding industries [1] . FGM components have applications in astronautic structures, such as rocket launchpad, space vehicles [2] , etc., because rocket launchpad is subjected to tremendous thermal and mechanical loading, whereas, space vehicles are subjected to extreme thermal conditions. FGMs having excellent thermal and mechanical properties are suitable for such various astronautic structures. It is to be mentioned that the present work deals with FGM beams, which are often found in various structures in the fields of aerospace, mechanical, automotive, civil engineering, etc.
FGM beams are mainly designed for applications under thermal environment. But its behavior under mechanical loadings at ambient condition is also important in order to ascertain its performance when thermal loadings are absent. Knowledge of free vibration behavior of prestressed FGM beams under mechanical loading is important from design point of view. It is known that the amplitude of forced vibration becomes excessively large when the excitation frequency falls in the vicinity of the natural frequency of vibration of a loaded beam. To avoid such undesirable vibration levels, the natural frequency of vibration of the loaded beam must be known to the designer. Hence the present work is meant to investigate such dynamic behavior of FGM beams. The literature review of some related works by other notable researchers are given in the next few paragraphs.
Ke et al. [3] investigated the nonlinear vibration behavior of FGM beams based on Euler–Bernoulli beam theory and von Kármán geometric nonlinearity. Fallah and Aghdam [4] and [5] presented large amplitude free vibration analysis of FGM Euler–Bernoulli beams resting on nonlinear elastic foundation subjected to both mechanical and thermal loadings. Fu et al. [6] carried out nonlinear free vibration analysis of piezoelectric FGM beams under thermal environment employing Euler–Bernoulli beam theory. Lai et al. [7] obtained the accurate analytical solutions for large amplitude vibration of thin FGM beams using Euler–Bernoulli beam theory. Based on Euler–Bernoulli beam theory, Yaghoobi and Torabi [8] studied the nonlinear vibration behavior of geometrically imperfect FGM beams resting on nonlinear elastic foundation subjected to axial force. Hemmatnezhad et al. [9] studied the largeamplitude oscillations of FGM Timoshenko beams using finite element formulation. Rahimi et al. [10] performed free vibration analysis of FGM Timoshenko beams in the vicinity of a buckled equilibrium configuration.
Kapuria et al. [11] presented a theoretical finite element model for vibration analysis of layered FGM beams with experimental validation. Aydogdu and Taskin [12] studied free vibration behavior of simply supported FGM beams using different beam theories. Free vibration characteristics of simply supported FGM beams were investigated by Şimşek and Kocatürk [13] using Lagranges equations under the assumptions of the Euler–Bernoulli beam theory. Thermomechanical vibration analysis of FGM beams resting on variable elastic foundation was carried out by Pradhan and Murmu [14] . Free vibration analysis of FGM beams based on a different first order shear deformation theory was carried out by Sina et al. [15] . Fundamental frequency analysis of FGM beams was carried out by Şimşek [16] using different higherorder beam theories. Giunta et al. [17] addressed free vibration behavior of functionally graded beams via several axiomatic refined theories. Using finite element method, Alshorbagy et al. [18] presented the free vibration characteristics of FGM beams with material graduation axially or transversally through the thickness based on the power law.
Free vibration characteristics of layered functionally graded beams were studied by Wattanasakulpong et al. [19] using Ritz method. Thai and Vo [20] investigated free vibration behavior of FGM beams based on various higherorder beam theories. Free vibration analysis of FGM beams for different boundary conditions was carried out by Pradhan and Chakraverty [21] using Euler–Bernoulli and Timoshenko beam theories. Free vibration behavior of axially loaded rectangular FGM beams was investigated by Nguyen et al. [22] based on the firstorder shear deformation beam theory. The dynamic stiffness method was used by Su et al. [23] to investigate the free vibration behavior of FGM beams. Wattanasakulpong and Mao [24] investigated the dynamic response of Timoshenko FGM beams supported by various classical and nonclassical boundary conditions.
Esfahani et al. [25] studied free vibration behavior of a thermally pre/post buckled FGM beam resting over a nonlinear hardening elastic foundation. Free vibration behavior of a thermoelectrically postbuckled rectangular FGM piezoelectric beams was studied by Komijani et al. [26] . Thermal buckling analysis of FGM beams with temperaturedependent material properties was carried out by Kiani and Eslami [27] and [28] . Esfahani et al. [29] carried out nonlinear thermal stability analysis of temperaturedependent FGM beams resting on nonlinear hardening elastic foundation. Thermoelectrical stability analysis of piezoelectric FGM beams had been carried out by Kiani et al. [30] , Kargani et al. [31] and Komijani et al. [32] , whereas thermal stability analysis of piezoelectric FGM beams was carried out by Kiani et al. [33] .
The present work is based on Timoshenko beam theory, which considers uniform distribution of transverse shear stress across the beam thickness. It is worthwhile to mention some of the research works using higher shear deformation theories (HSDT) developed in the recent years for analysis of plate and beam structures. Tounsi et al. [34] carried out thermoelastic bending analysis of functionally graded sandwich plates using a refined trigonometric shear deformation theory (RTSDT). The thermomechanical bending behavior of FGM plates resting on Winkler–Pasternak elastic foundations was studied by Bouderba et al. [35] using RTSDT. Buckling and free vibration behaviors of exponentially graded sandwich plates were investigated by Ait Amar et al. [36] using simple refined shear deformation theory. Static and dynamic analyses of FGM and sandwich plates had been carried out by Hebali et al. [37] and Mahi et al. [38] using new hyperbolic shear deformation theory. Using higherorder shear deformation theories, wave propagation analysis in porous FGM plates, and bending and vibration analysis of FGM plates, were carried out by Ait Yahia et al. [39] and Belabed et al. [40] respectively. Recently, Bourada et al. [41] developed a refined trigonometric higherorder beam theory to investigate static and dynamic behaviors of FGM beams. In that work, the authors have included stretching deformation effect along the thickness direction and eliminated the need of shear correction factor. Bousahla et al. [42] presented a new trigonometric higherorder theory for the static analysis of FGM plates employing the physical neutral surface concept. Hamidi et al. [43] presented a sinusoidal plate theory for the thermomechanical bending analysis of functionally graded sandwich plates. Bessaim et al. [44] developed a new higherorder shear and normal deformation theory for investigating the bending and free vibration behavior of sandwich plates with functionally graded isotropic face sheets. Thermoelastic bending analysis of functionally graded sandwich plates was carried out by Bouchafa et al. [45] using a refined hyperbolic shear deformation theory. Houari et al. [46] , using a new higherorder shear and normal deformation theory, simulated the thermoelastic bending of FGM sandwich plates.
From the literature review presented, it is clear that an exhaustive study on free vibration behavior of transversely loaded beam for different FGM materials and different classical boundary conditions is scarce. Most of the published works are involved with either free vibration behavior of undeformed FGM beam or large amplitude free vibration behavior of FGM beam. Hence, in the present work, free vibration frequencies of FGM beam are computed for different prestressed configurations under uniform transverse pressure. Prestressed configurations are obtained through a geometrically nonlinear static analysis. The linear vibration frequency of the prestressed beam, hereafter termed as loaded natural frequency, is then computed through an eigenvalue problem that includes the effect of prestressing using the displacement fields of the static problem. The effect of geometric nonlinearity is included using von Kármán nonlinear strain–displacement relationship. Timoshenko beam theory is used to consider the effects of shear deformation for the static problem and of rotary inertia for the subsequent dynamic problem. Suitable energybased variational principles are used to derive the governing equations for both parts of the problem. Four different functionally graded materials and three different immovable classical boundary conditions are considered to show the prestressed dynamic behavior of beams.
The present work aims at finding loaded natural frequency of prestressed FGM Timoshenko beam. A uniform rectangular beam with length L , height h and width b is considered. A beam with symbolic dimensions is shown in Fig. 1 , where, x , y and z denote the coordinate axes along the length, width and thickness directions respectively. As mentioned earlier, two distinct but interrelated problems are formulated and solved to obtain the desired solution. The purpose of the first one, the static problem, is to obtain the prestressed configuration of the beam under the application of uniform transverse pressure. And the second problem, named as the dynamic problem, is utilized to obtain the loaded natural frequency of the deformed beam. It must be mentioned that the static configurations for different loadings are obtained through a geometrically nonlinear analysis to address the large deflection effect.

Fig. 1. Beam with dimensions and coordinate axes.

The present work considers continuous variation of graded material properties across the beam thickness. A continuous variation of volume fraction of ceramic ( ) and metal ( ) constituents across the thickness is assumed in accordance with the powerlaw given by and respectively [47] . Here, is the volume fraction index. Hence the effective material property of any FGM layer is determined using the Voigt model, which is given by , where and are the material properties of the ceramic and metal constituents respectively. So any effective material property at any layer z is given by . For the present FGM beam model, the top layer ( ) is purely ceramic and the bottom layer ( ) is purely metal. It is to be mentioned that the subscripts m and c refer to the metal and ceramic constituents respectively.
Any material property of the individual constituent is temperaturedependent and such temperature dependence is considered using the relationship , where is the temperature in K and , , , and are the coefficients of temperature. The relevant effective material properties for the present problem are elastic modulus , shear modulus , Poissons ratio and density . The temperature coefficients [47] of the material properties for the ceramic or metal constituents considered are given in Table 1 . For the present problem, the beam is assumed to be at ambient temperature , which is also considered to be the temperature at which thermal stress is zero. Hence all the required material properties are calculated at .
Constituent material  Property 




 

SUS304  (Pa)  201.04 × 10^{9}  0  3.079 × 10^{−4}  −6.534 × 10^{−7}  0  

0.3262  0  −2.002 × 10^{−4}  3.797 × 10^{−7}  0  
(kg m^{−3} )  8166  0  0  0  0  
Ti6Al4V  (Pa)  122.56 × 10^{9}  0  −4.586 × 10^{−4}  0  0  

0.2884  0  1.121 × 10^{−4}  0  0  
(kg m^{−3} )  4429  0  0  0  0  
Si_{3} N_{4}  (Pa)  348.43 × 10^{9}  0  −3.070 × 10^{−4}  2.160 × 10^{−7}  −8.946 × 10^{−11}  

0.2400  0  0  0  0  
(kg m^{−3} )  2370  0  0  0  0  
ZrO_{2}  (Pa)  244.27 × 10^{9}  0  −1.371 × 10^{−3}  1.214 × 10^{−6}  −3.681 × 10^{−10}  

0.2882  0  1.133 × 10^{−4}  0  0  
(kg m^{−3} )  3000  0  0  0  0  
Al_{2} O_{3}  (Pa)  349.55 × 10^{9}  0  −3.853 × 10^{−4}  4.027 × 10^{7}  −1.673 × 10^{−10}  

0.2600  0  0  0  0  
(kg m^{−3} )  3750  0  0  0  0 
The governing equation of the static problem is obtained using the minimum potential energy principle [48] given by,

( 1) 
where is the strain energy developed due to external loadings, is the potential energy of the external loadings, and is the variational operator. The strain energy consists of two parts, i.e., , where and are the strain energies due to axial strain and shear strain respectively. Using the proportionality of stress and strain (i.e., linear elastic material), and using suitable strain–displacement relationships, the strain energies can be expressed in terms of the displacement fields. The three displacement fields considered for the present problem are the following: , the inplane displacement field, , the transverse displacement field, and , the rotational field of beam cross section due to bending. Here , and are defined at the midplane of the beam and are functions of the axial coordinate x .
The expressions of axial strain and shear strain are given by,

( 2) 
and

( 3) 
It is to be mentioned that the first term of Eq. (2) is von Kármán type nonlinear strain–displacement relationship. Hence the strain energies and are given by,

( 4) 
and

( 5) 
The stiffness coefficients used in Eqs. (4) and (5) are given below:
and . In Eq. (5) , is the shear correction factor, which is taken to be 5/6 for rectangular cross section. It is to be mentioned that the shear modulus is given by, .
The potential energy of the applied uniform transverse pressure is given by,

( 6) 
where is defined as the force per unit length of the beam.
Following Ritz method, the displacement fields are approximated as finite linear combinations of admissible functions and unknown coefficients given as,

( 7) 
Here, , and are set of orthogonal admissible functions for the displacement fields , and respectively; and , and are the number of functions used to approximate , and respectively. It is to be noted that is the set of unknown coefficients, which are to be determined from the governing equations. The admissible functions satisfy the boundary conditions of the beam. The lowest order functions for each of the displacement fields are selected suitably and the corresponding higherorder functions are developed numerically following Gram–Schmidt orthogonalization scheme. Three boundary conditions with immovable ends are considered for the present work. And these are clamped–clamped (CC), simply supported–simply supported (SS) and clamped–simply supported (CS). The selected lowest order admissible functions for each of the displacement fields are given in Table 2 for all the three boundary conditions considered.
Displacement field  Boundary  Conditions  Function  


CC 

 
SS  
CS  

CC 

 
SS  
CS  

CC 

 
SS 

 
CS 


Using the expression of various potential energies, given by Eqs. (4) , (5) and (6) into Eq. (1) and using the approximate displacement fields, given by Eq. (7) , the governing algebraic equations are obtained in the form given below:

( 8) 
where and are the stiffness matrix and load vector, respectively, each of dimension . The elements of and are given in the Appendix. It can be seen that the set of governing equations is nonlinear in nature as the stiffness matrix is a function of the unknown coefficients. To solve this set of nonlinear equations, a multidimensional secant method known as Broydens method [49] and [50] is used. The solution of Eq. (8) gives the statically deflected configuration of a prestressed beam. The next stage of the problem is now to determine the loaded natural frequency of the prestressed beam and its mathematical formulation is discussed in the next section.
The governing equation of the dynamic problem is derived using Hamiltons principle [48] given by,

( 9) 
where is the kinetic energy of the vibrating beam and is the time. The present work is a free vibration problem of a prestressed beam, in which the prestressed configuration is already obtained in the previous step of static problem. Hence the potential energy of the external loadings is zero in this case. Taking as the effective density of any FGM layer, the expression of is as follows:

( 10) 
where the inertia coefficients and are given by . The expression of remains the same as given for the static problem.
The approximate dynamic displacement fields, which are assumed to be separable in space and time, are given by,

( 11) 
where and are a new set of unknown parameters for the dynamic problem. The complete set of the space part of the dynamic displacements, i.e., , and , is the same as taken for the static problem. In Eq. (11) , denotes the natural frequency of vibration of the beam.
Using the expressions of strain energy (Eqs. (4) and (5) ), kinetic energy (Eq. (10) ) and dynamic displacements (Eq. (11) ), the governing equation is obtained as follows:

( 12) 
where and are the stiffness matrix and mass matrix respectively. The elements of are the same as given in the Appendix and the elements of are given below:

( 13) 
The offdiagonal elements of are zero. Eq. (12) is an eigenvalue problem, in which the square root of the eigenvalues gives the natural frequency of vibration of various vibration modes and is the corresponding eigen vectors used to obtain the vibration mode shapes. It must be noted that the stiffness matrix given in Eq. (12) is nonlinear in nature. But as the dynamic problem is to be formulated at the prestressed beam configuration, the nonlinear terms of the stiffness matrix are updated with the prestressed beam displacement fields obtained from the static problem [51] . Hence the solution of Eq. (12) gives the linear loaded natural frequency of vibration of the prestressed beam.
The present work is carried out to determine the natural frequency of the first mode of vibration of prestressed FGM beams. It is to be mentioned that the frequency determined is of small amplitude vibration of prestressed beam. So the dynamic behavior is presented graphically in nondimensional plane, where is the nondimensional loaded natural frequency of first vibration mode and is the normalized maximum transverse deflection of prestressed beam. Hence such plots present the dynamic behavior in terms of loaded natural frequency of vibration as a function of maximum beam deflection. On the other hand, the static equilibrium path of the beam is presented graphically in plane, where is the nondimensional uniform transverse pressure. The nondimensional parameters are defined as: , , and , where is the elastic modulus of metal constituent, is the maximum transverse deflection, is the density of metal constituent, is the cross sectional area, and is the area moment of inertia of the beam cross section about the centroidal axis. The results are generated for and .
Four different functionally graded materials are considered for the present work, namely Stainless Steel (SUS304)–Silicon Nitride (Si_{3} N_{4} ), Stainless Steel–Zirconia (ZrO_{2} ), Stainless Steel–Alumina (Al_{2} O_{3} ) and Titanium alloy (Ti6Al4V)–Zirconia, and hereafter these are termed as FGM 1, FGM 2, FGM 3 and FGM 4 respectively. Using the temperature coefficients for the constituents of these FGM compositions given in Table 1 , the various material properties are calculated at . These are presented in Table 3 . The static and dynamic behaviors of these FGM beams are presented for three boundary conditions, i.e., CC, SS and CS.
Material property  SUS304  Ti6Al4V  Si_{3} N_{4}  ZrO_{2}  Al_{2} O_{3}  

(GPa)  207.79  105.70  322.27  168.06  320.24  

0.318  0.298  0.240  0.298  0.260  
(kg m^{−3} )  8166  4429  2370  3000  3750 
The nondimensional frequency parameter of undeformed FGM beam is compared with the results of Ref. [9] for different volume fraction indices and also for different boundary conditions. The comparison is made for Steel–Alumina FGM beam for a lengththickness ratio L/h = 20. The material properties used for comparison purpose are as follows: , , , , , and . The comparison is presented in Table 4 . The comparison shows good agreement of the present results with Ref. [9] . This validates the free vibration dynamic behavior of undeformed FGM beam analyzed by the present method.
Boundary condition  Nondimensional frequency parameter,  

k = 0  k = 0.1  k = 0.2  k = 0.5  k = 1  k = 2  k = 5  
CC  Present  6.4864  6.2664  6.0896  5.7504  5.4617  5.2340  5.0333 
Ref. [9] .  6.4971  6.2737  6.1001  5.7575  5.4713  5.2413  5.0390  
SS  Present  4.3311  4.1980  4.0653  3.8402  3.6652  3.5203  3.3863 
Ref. [9] .  4.3371  4.1889  4.0753  3.8554  3.6742  3.5244  3.3803  
CS  Present  5.4099  5.2240  5.0820  4.8056  4.5385  4.3759  4.2091 
Ref. [9] .  5.4086  5.2228  5.0786  4.7951  4.5590  4.3688  4.1990 
The validation plots of prestressed Stainless Steel–Zirconia (FGM 2) beam are presented in Fig. 2 (a–b) for k = 2.0. Fig. 2 (a) presents the static equilibrium path in plane, whereas Fig. 2 (b) presents the prestressed free vibration behavior in plane. The validation is carried out with finite element package ANSYS (version 10.0). The comparison plots are presented for CC, SS and CS boundary conditions. The comparative plots in Fig. 2 show very good agreement of the present method with ANSYS for both the static and dynamic behaviors. The finite element model in ANSYS is created using BEAM 188 element, and the results are generated with 30 elements. It must be noted that a layered variation of material properties is used to create the finite element model in ANSYS.

Fig. 2. Validation plots of prestressed beam: (a) static behavior and (b) free vibration behavior.

The nondimensional frequency parameter of undeformed CC beam for all the four functionally graded materials considered is presented in Table 5 for different volume fraction indices. The list presented in Table 5 includes results for L/h = 10 and L/h = 25. Similar lists for SS and CS FGM beams are presented in Table 6 and Table 7 respectively. As seen from these tables, the nondimensional frequency parameter decreases with increase in values of k , and this is true for all the four FGMs considered. Only exception to this occurs for FGM 2 with L/h = 25, where increases from k = 20 to k = 50. It is to be mentioned that higher k values indicate more metal content in the beam. It is also seen that the nondimensional frequency parameter increases with L/h ratio but the change is very little. The effect of L/h ratio on nondimensional frequency of vibration remains insignificant for prestressed beam also. Hence the results, as presented in the following section, for prestressed FGM beams are generated for L/h = 25.
k  L/h = 10  L/h = 25  

FGM 1  FGM 2  FGM 3  FGM 4  FGM 1  FGM 2  FGM 3  FGM 4  
0.0  48.595  31.128  38.477  32.142  51.212  32.872  40.537  33.943 
0.1  43.041  29.308  35.828  30.774  45.334  30.915  37.723  32.470 
0.2  39.411  28.051  33.864  29.715  41.448  29.597  35.751  31.378 
0.5  33.475  25.815  30.290  27.617  35.312  27.260  31.870  29.106 
1.0  29.326  24.070  27.502  25.862  30.803  25.413  28.969  27.268 
2.0  26.288  22.657  25.320  24.425  27.742  23.896  26.728  25.802 
5.0  23.853  21.563  23.445  23.102  25.246  22.712  24.771  24.430 
10.0  22.697  21.198  22.499  22.358  24.001  22.348  23.799  23.581 
20.0  21.953  21.052  21.847  21.802  23.204  22.190  23.099  23.055 
50.0  21.390  20.984  21.350  21.350  22.579  22.135  22.552  22.547 
k  L/h = 10  L/h = 25  

FGM 1  FGM 2  FGM 3  FGM 4  FGM 1  FGM 2  FGM 3  FGM 4  
0.0  22.457  14.410  17.789  14.880  22.393  14.596  17.804  15.072 
0.1  19.873  13.573  16.532  14.234  20.200  13.920  16.718  14.514 
0.2  18.208  12.998  15.637  13.759  18.303  13.270  15.796  14.043 
0.5  15.522  11.969  14.046  12.829  15.692  12.186  14.249  12.915 
1.0  13.635  11.159  12.794  12.048  13.772  11.279  12.989  12.230 
2.0  12.243  10.500  11.803  11.399  12.250  10.472  12.031  11.613 
5.0  11.098  9.999  10.909  10.746  11.234  10.140  11.133  10.924 
10.0  10.566  9.819  10.461  10.401  10.561  9.986  10.586  10.497 
20.0  10.168  9.747  10.147  10.119  10.258  9.811  10.310  10.286 
50.0  9.934  9.704  9.886  9.889  10.042  9.873  10.152  9.953 
k  L/h = 10  L/h = 25  

FGM 1  FGM 2  FGM 3  FGM 4  FGM 1  FGM 2  FGM 3  FGM 4  
0.0  34.326  22.013  27.203  22.730  35.330  22.785  28.021  23.527 
0.1  30.411  20.740  25.306  21.783  31.326  21.558  25.969  22.512 
0.2  27.831  19.853  23.940  21.017  29.007  20.403  24.488  21.737 
0.5  23.681  18.276  21.416  19.527  24.481  18.867  22.181  19.990 
1.0  20.728  17.036  19.472  18.310  21.457  17.507  20.152  18.861 
2.0  18.622  16.034  17.934  17.298  19.166  16.534  18.662  18.033 
5.0  16.889  15.252  16.615  19.373  17.299  15.773  17.022  17.098 
10.0  16.094  14.998  15.934  15.845  16.395  15.463  16.504  16.321 
20.0  15.553  14.899  15.482  15.444  19.120  15.330  16.050  16.026 
50.0  15.146  14.842  15.113  15.112  15.649  15.321  15.572  15.662 
The static deflection behavior of CC beam in plane is presented in Fig. 3 (a–d) for FGM 1, FGM 2, FGM 3 and FGM 4 respectively. In each of these figures, the static equilibrium path is presented for a set of volume fraction indices, i.e., for k = 0.0, 0.1, 0.2, 0.5, 1.0, 2.0, 5.0, 10.0, 20.0, 50.0. The plots are useful in finding the nondimensional pressure value corresponding to a static deflection level, at which the loaded natural frequency is found out. As seen from Fig. 3 , the beam exhibits nonlinear hardening type loaddeflection behavior. This is due to stiffening effect induced in the beam as a result of generation of tensile membrane forces due to immovable ends. With increase in k , i.e., the increase in metal content, FGM 1, FGM 3 and FGM 4 beam shows decreased stiffness levels as the elastic modulus of ceramic constituent is greater than its metal counterpart, as can be seen from Table 3 . The trend is completely reverse for FGM 2 beam as the elastic modulus of its ceramic constituent is lesser than that of its metal part.

Fig. 3. Nondimensional pressuredeflection behavior for different volume fraction indices of CC beams: (a) FGM 1, (b) FGM 2, (c) FGM 3 and (d) FGM 4.

The loaded natural frequency versus maximum transverse deflection plots in nondimensional plane is presented in Fig. 4 (a–f) for k = 0.0, 0.2, 0.5, 5.0, 20.0 and 50.0, respectively, for CC FGM beam. In each of the figures, free vibration behavior is shown for FGM 1, FGM 2, FGM 3 and FGM 4 beams. In accordance with the static behavior, the loaded natural frequency is shown to be increasing with increased deflection level as a result of enhanced stiffening effect. With regard to the comparative behavior among different FGMs considered, FGM 1 shows highest frequency of vibration with FGM 3, FGM 4 and FGM 2 coming next in order of exhibiting decreasing vibration frequency at any common deflection levels. This is true irrespective of the values of the volume fraction index. It is also seen that the relative differences in frequencydeflection behavior of various FGMs diminish with increase in k values. At higher k values, the dynamic behavior becomes almost identical for all the FGMs considered.

Fig. 4. Nondimensional frequencydeflection behavior of different CC FGM beams: (a) k = 0.0, (b) k = 0.2, (c) k = 0.5, (d) k = 5.0, (e) k = 20.0 and (f) k = 50.0.

It is also important to study the dynamic behavior of prestressed FGM beam for different volume fraction indices. Fig. 5 (a–d) shows such nondimensional frequencydeflection plots for FGM 1. FGM 2, FGM 3 and FGM 4, respectively, each showed comparative behavior for k = 0.0, 0.1, 1.0, 10.0, 50.0. It can be seen that the loaded natural frequency decreases with increase in k values for any particular deflection level. This is true for all the FGMs considered. The explanation for the variation of loaded natural frequency with k cannot be given by seeing only the static deflection behavior as shown in Fig. 3 . Because in predicting the dynamic behavior, both the relative density values and the relative elastic modulus values of the constituents are to be considered.

Fig. 5. Nondimensional frequencydeflection behavior of CC beams for different volume fraction indices: (a) FGM 1, (b) FGM 2, (c) FGM 3 and (d) FGM 4.

The nondimensional static deflection behavior for different volume fraction indices is presented in Fig. 6 (a–d) for SS beams each for different functionally graded materials. Also the nondimensional frequencydeflection plots of SS beams are shown in Fig. 7 (a–f) for different k values and in Fig. 8 (a–d) for different FGMs. As for CS beams, static deflection behavior is presented in Fig. 9 (a–d), whereas the dynamic behavior in terms of frequencydeflection plots is shown in Fig. 10 (a–f) for different k values and in Fig. 11 (a–d) for different FGMs. For both these boundary conditions, the nature of static and dynamic behavior is similar in nature as described for CC beams. The relative behavior for these three different boundary conditions, although not shown in a single plot, differs obviously due to the stiffness effects contributed from the support conditions. Because it is known that CC beam exhibits the highest transverse stiffness, with CS and SS beam being next in order of decreasing stiffness levels.

Fig. 6. Nondimensional pressuredeflection behavior for different volume fraction indices of SS beam: (a) FGM 1, (b) FGM 2, (c) FGM 3 and (d) FGM 4.


Fig. 7. Nondimensional frequencydeflection behavior of different SS FGM beams: (a) k = 0.0, (b) k = 0.2, (c) k = 0.5, (d) k = 5.0, (e) k = 20.0 and (f) k = 50.0.


Fig. 8. Nondimensional frequencydeflection behavior of SS beams for different volume fraction indices: (a) FGM 1, (b) FGM 2, (c) FGM 3 and (d) FGM 4.


Fig. 9. Nondimensional pressuredeflection behavior for different volume fraction indices of CS beam: (a) FGM 1, (b) FGM 2, (c) FGM 3 and (d) FGM 4.


Fig. 10. Nondimensional frequencydeflection behavior of different CS FGM beams: (a) k = 0.0, (b) k = 0.2, (c) k = 0.5, (d) k = 5.0, (e) k = 20.0 and (f) k = 50.0.


Fig. 11. Nondimensional frequencydeflection behavior of CS beams for different volume fraction indices: (a) FGM 1, (b) FGM 2, (c) FGM 3 and (d) FGM 4.

An energy based mathematical model is presented to study the free vibration behavior of prestressed FGM Timoshenko beams. The entire work is carried out in solving two different but interrelated problems, namely the static problem and the dynamic problem. The static problem is used to determine the prestressed configuration of FGM beam under uniform transverse pressure. And the dynamic problem, formulated as an eigenvalue problem, is used to determine the loaded natural frequency of the prestressed beam. Four different FGMs, namely Stainless Steel–Silicon Nitride, Stainless Steel–Zirconia, Stainless Steel–Alumina and Titanium alloy–Zirconia, are used to generate results for different volume fraction indices. Numerical results for nondimensional frequency parameters of undeformed beam are presented for different functionally graded materials with CC, SS and CS boundary conditions. Effects of material as well as the volume fraction index on nondimensional frequencydeflection behavior of prestressed beams are studied. The results are presented for three boundary conditions, i.e., CC, SS and CS. Static equilibrium paths in nondimensional plane are also presented in order to relate the applied pressure with loaded natural frequency through the static deflection level. The results can serve as benchmarks for further study in this field.







Published on 10/04/17
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