1 Introduction

Functionally graded material (FGM) was first proposed by Japanese scientists to decrease the thermal stresses in propulsion and airframe structural systems of astronautical flight vehicles [33]. Functionally graded materials (FGMs) are considered a special class of composite materials characterized by smooth and continuous distribution of material properties. They have comparatively smaller residual stresses, lower stress concentrations, and higher bonding strength than conventional composite laminates. Due to these advantages, FGMs are being used increasingly in many applications, such as, in aircraft industry, space vehicles, reactor vessels, automobile, electronics, optics, chemistry, biomedical engineering, nuclear engineering and mechanical engineering [23].

In the microscopic scale, FGM is considered a heterogeneous material, which should be homogenized accurately to be easily modeled computationally. Anthoine [2] presented a second-order homogenization of functionally graded materials. Lee and Kim [35] exploited Voigt rule (VR) and Mori–Tanaka method (MTM) to detect the effective material properties of heterogeneous micromechanical FGM. Effective properties of FG panels are investigated for the two types of models, namely, power function (P-FGM) and sigmoid function (S-FGM) functions.

Chi and Chung [17] suggested a sigmoid function, a combination of two type of P-FGM functions, to reduce the stress intensity factors in a crack structure. In 2003, Chakraborty et al. [16] studied static, free vibration and wave propagation of exponential and power-law FGM Timoshenko beam (TBT) by finite element method (FEM). Li et al. [36] proposed a unified approach to the formulation of TBT and Euler–Bernoulli’s (EB). Kapuria et al. [32] fabricated multi-layered FG beam using powder metallurgy and thermal spraying techniques, and then experimentally validated the results on static and free vibration. Benatta et al. [8] presented an analytical solution for static bending of simply supported FGM hybrid beams using higher-order shear deformation (HOSD) beam theory. Ben-Oumrane et al. [10] presented a theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams according to different beam theories. Simsek [52] investigated static analysis of FG simply-supported beam subjected to a uniformly distributed load using Ritz method within the framework of TBT and the HOSD theories. Mahi et al. [42] presented exact solutions to study the free vibration of a unified HOSD beam with temperature-dependent material properties. The materials distributions are varied continuously through the thickness according to an exponential function (E-FGM) or S-FGM. Giunta et al. [25] proposed several axiomatic refined theories for the linear static analysis of beams made of materials whose properties are graded along one or two directions. Huang and Li [30] studied free vibration of non-uniform axially FG beams by transforming the governing differential equation to Fredholm integral equations.

Alshorbagy et al. [1] investigated the free vibration characteristics of P-FGM distributed through beam thickness or beam length using FEM. Birsan et al. [11] employed the direct approach to analyze the deformation of curved FG rods and beams. Shahba and Rajasekaran [51] studied free vibration and stability of axially FG tapered Euler–Bernoulli beams by conventional differential transform method (DTM). Menaa et al. [43] derived analytical solutions for static shear correction factor of FG rectangular beams using the energy equivalence principle. Mohanty et al. [45] presented the evaluation of static and dynamic behavior of P-FGM ordinary and sandwich beam using FEM and TBT. Li and Batra [38] derived analytical relations between the critical buckling load of an FGM Timoshenko beam and that of the corresponding homogeneous Euler–Bernoulli beam subjected to axial compressive load for different boundary conditions. Pradhan and Chakraverty [48] investigated free vibration of Euler and Timoshenko P-FGM beams using Rayleigh–Ritz method. Li et al. [39] derived analytically the bending solutions of FGM of TBT in terms of the homogenous Euler–Bernoulli beams. Daouadji et al. [18] derived elasticity solution of a cantilever FG beam with P-FGM and E-FGM subjected to linearly distributed load. Duc and Cong [20] presents a proposal to investigate the nonlinear dynamic response of imperfect symmetric thin S-FGM subjected to mechanical loads. Wang and Wang [57] presented static analysis of higher order sandwich beams by weak form quadrature element method. Liu and Shu [40] investigated analytically free vibrations of E-FGM Euler–Bernoulli beams with a single delamination. Sarkar and Ganguli [50] derived closed-form solutions for axially-loaded FG Timoshenko beams having uniform cross-section and fixed–fixed boundary condition. Jin and Wang [31] studied free vibration of FG Euler–Bernoulli beam using a weak form of DQM. Bourada et al. [14] presented a new simple shear and normal deformations theory for FG beams. Nguyen et al. [47] studied lateral buckling analysis of thin-walled functionally graded open-section beams. For a nano-scale beam, Eltaher et al. [24] presented a review for bending, buckling, vibrations, and wave propagation of nanoscale FG beams. Hamed et al. [26] investigated free vibration behavior of symmetric and sigmoid functionally graded nanobeams. Ebrahimi and Barati [21] developed a dynamic model of smart shear-deformable heterogeneous piezoelectric nanobeams. Ebrahimi and Salari [22] presented dynamic behavior of piezo-thermo-electrically excited FG nanobeams. The material distribution is assumed to be sigmoid and power-law graded. Azimi et al. [3] studied thermo-mechanical vibration of rotating axially FG nonlocal Timoshenko beam.

The analysis of FG plate structures are considered by many authors, such as, Meziane et al. [44] presented a simple refined theory for buckling and free vibration of exponentially FG plates. Zidi et al. [58] studied hygro-thermo-mechanical bending of FG plate using higher order shear theory. Bousahla et al. [15] and Bellifa [7] considered the neutral axis effect on the bending and vibration of FG plate with higher order shear theory. Hebali et al. [28] and Mahi et al. [42] developed 3D hyperbolic HOSD for the bending and free vibration of P-FGM plates. Yahia et al. [56] studied wave propagation in porous FG plates using HOSD. Tounsi et al. [54], Bouderba et al. [12] and Hamidi et al. [27] presented a sinusoidal plate theory including stretching effect for thermomechanical bending of FG sandwich plates. By dividing the transverse displacement into bending, shear, and thickness stretching parts, Belabed et al. [4] and Bennoun et al. [9] developed refined plate theory to study vibrations of sandwich FGM plates. Tounsi et al. [55] and Houari et al. [29] proposed 3-unknown non-polynomial shear deformation theory for static, buckling and vibration analyses FGM sandwich plates. Beldjelili et al. [5] studied hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations. For a nano-plate, Belkorissat et al. [6] and Bounouara et al. [13] developed a nonlocal zeroth-order shear deformation theory for free vibration of FG nanoscale plates resting on elastic foundation.

According to the author’s knowledge and survey, the static deflection and stress distribution of symmetric power function and sigmoid function are not yet addressed in literature. Thus, the present paper illustrates the bending behavior of rectangular FG beam subjected to transverse loading with different material distributions. The material consists of ceramics (Al2O3) and metal (Al) phases varying through the thickness direction. The Equilibrium equations of FG beam are derived using Euler–Bernoulli beam theory and virtual work principle. The paper is organized as follows: The problem formulation is addressed in Sect.  2 . The finite element (FE) model and stiffness matrices are derived in Sect.  3 . Code validation and numerical results are discussed in Sect.  4 . Summarized concluding remarks are presented in Sect.  5 .

2 Problem formulation

In this section, material distribution through the beam thickness according to Voigt rule is assumed, constitutive equations are illustrated and equilibrium equation of FG Euler-beam is derived.

2.1 Spatial material graduation functions

FGMs are manufactured by mixing different material phases continuously though a specific spatial direction. To estimate the effective properties at micromechanics level, the simplest homogenization techniques are Voigt rule Mori and Tanaka [46] and Mori–Tanaka model, Tomota et al. [53]. Although, Voigt rule does not consider the interaction among adjacent inclusions, Mori–Tanaka method (MTM) considered these interactions of neighboring phases at microscopic level, Lee and Kim [35]. According to Li et al. [37] and Komijani et al. [34], the volume fraction of FG materials can be expressed as

$${V_{\text{c}}} = {\left( {\frac{1}{2} + \frac{z}{h}} \right)^k}\left( {0 \leqslant k < \infty } \right)$$
(1a)
$${V_{\text{c}}} + {V_{\text{m}}} = 1$$
(1b)

where \(V\) is the volume fraction, \(k\) is a nonnegative power exponent, and subscripts c and m represent the ceramic and metal, respectively. In the current analysis, three functions are assumed to describe the spatial distribution of materials through the thickness direction. The first and simplest on is the power law function P-FGM, which is described by [1, 16, 18, 23, 31]

$$E\left( z \right) = {E_{\text{m}}}{V_{\text{m}}} + {E_{\text{c}}}{V_{\text{c}}} = {E_{\text{c}}} + \left( {{E_{\text{m}}} - {E_{\text{c}}}} \right){\left( {\frac{1}{2} + \frac{z}{h}} \right)^k}\quad\left( {\frac{{ - h}}{2} \leqslant z \leqslant \frac{h}{2}} \right)$$
(2)

The modified symmetric power-law function S-P-FGM has the following form [41]

$$E\left( z \right) = {E_{\text{c}}} + \left( {{E_{\text{m}}} - {E_{\text{c}}}} \right){\left( {\frac{{ - 2z}}{h}} \right)^k}\quad\left( {\frac{{ - h}}{2} \leqslant z \leqslant 0} \right)$$
(3a)
$$E\left( z \right) = {E_{\text{c}}} + \left( {{E_{\text{m}}} - {E_{\text{c}}}} \right){\left( {\frac{{2z}}{h}} \right)^k}\quad\left( {0 \leqslant z \leqslant \frac{h}{2}} \right)$$
(3b)

The Sigmoid function S-FGM is assumed to be two power-law functions to ensure smooth distribution of stresses [10, 20, 26, 35, 43]. This function is depicted by

$$E\left( z \right) = {E_{\text{m}}}{V_{\text{m}}} + {E_{\text{c}}}{V_{\text{c}}} = {E_{\text{m}}} + \frac{1}{2}\left( {{E_{\text{c}}} - {E_{\text{m}}}} \right){\left( {1 + \frac{{2z}}{h}} \right)^k}\quad\left( {\frac{{ - h}}{2} \leqslant z \leqslant 0} \right)$$
(4a)
$$E\left( z \right) = {E_{\text{m}}}{V_{\text{m}}} + {E_{\text{c}}}{V_{\text{c}}} = {E_{\text{c}}} - \frac{1}{2}\left( {{E_{\text{c}}} - {E_{\text{m}}}} \right){\left( {1 - \frac{{2z}}{h}} \right)^k}\quad\left( {0 \leqslant z \leqslant \frac{h}{2}} \right)$$
(4b)

The FG beam in the current manuscript is composed of aluminum metal [\({E_{\text{m}}}\) = 70 GPa, and Poisson’s ratio \(\left( {{\upsilon _{\text{m}}}} \right)\)of 0.3] and a ceramic of alumina [\({E_{\text{c}}}\) = 380 GPa, and \({\upsilon _{\text{c}}} =\) 0.3]. Delale and Erdogan [19] proved that the effect of Poisson’s ratio on the deformation is much less than Young’s modulus. So, the Poisson’s ratio is assumed to be constant in this analysis. The distribution of Young modulus through the beam thickness for the three assumed functions is shown in Fig. 1.

Fig. 1
figure 1

The variation of Young Modulus through the beam thickness. a Power-Law Function (P-FGM). b Symmetric Power-Law Function (S-P-FGM). c Sigmoid Function S-FGM

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The power function distribution shown in Fig. 1a shows an increase in the power index from 0 (the metal phase) to k = 10 (ceramics rich phase) as the Young modulus increases. Un-symmetric distribution of the material characteristics through the mid-plane (z = 0) is the main feature of this function. So, the mid-plane is not coincident with neutral plane, Eltaher et al. [23]. But the other two distributions (S-P-FGM and S-FGM) distribution, as illustrated in Fig. 1b, c, are symmetric and anti-symmetric with respect to mid-plane, respectively, and their mid-planes are coincident with the neutral plane.

2.2 Constitutive equations

Based on the Euler–Bernoulli theory, that plane sections perpendicular to the axis of the beam before deformation remain (a) plane, (b) rigid, and (c) rotate such that they remain perpendicular to the (deformed) axis after deformation. The assumptions amount to neglecting the Poisson effect and transverse strains, Reddy [49]. The displacement field can be assumed as:

$$u\left( {x,z} \right) = {u_0}\left( x \right) - z\frac{{{\text{d}}{w_0}\left( x \right)}}{{{\text{d}}x}}$$
(5a)
$$w\left( {x,z} \right) = {w_0}\left( x \right)$$
(5b)

where \(u\) and \(w\) are the total displacements along the coordinate \(\left( x \right)\), and \({u_0}\) and \({w_0}\) denote the axial and transverse displacements of a point on the neutral axis. According to Euler hypothesis, the only nonzero strain is

$${\varepsilon _{xx}}\left( {x,z} \right) = \frac{{\text{d}}}{{{\text{d}}x}}\left[ {{u_0}\left( x \right) - z\frac{{{\text{d}}{w_0}\left( x \right)}}{{{\text{d}}x}}} \right] = \frac{{{\text{d}}{u_0}\left( x \right)}}{{{\text{d}}x}} - z\frac{{{{\text{d}}^2}{w_0}\left( x \right)}}{{{\text{d}}{x^2}}} = \varepsilon _{xx}^0 + z\varepsilon _{xx}^1$$
(6)

And the nonzero stress can be described by

$${\sigma _{xx}}\left( {x,z} \right) = E\left( z \right){\varepsilon _{xx}}\left( {x,z} \right) = E\left( z \right)\left[ {\varepsilon _{xx}^0 + z\varepsilon _{xx}^1} \right]$$
(7)

Axial and bending moment can be written as

$${N_{xx}} = \mathop \smallint \limits_A {\sigma _{xx}}dA = {A_{11}}\varepsilon _{xx}^0 + {B_{11}}\varepsilon _{xx}^1$$
(8a)
$${M_{xx}} = \mathop \smallint \limits_A z{\sigma _{xx}}dA = {B_{11}}\varepsilon _{xx}^0 + {D_{11}}\varepsilon _{xx}^1$$
(8b)

where

$$\left[ {A_{{11}} ,~B_{{11}} ,~D_{{11}} } \right] = \mathop \smallint \limits_{A} E\left( z \right)~\left[ {1,~z,~z^{2} } \right]dA$$
(9)

Based on the principle of virtual work which states that “if a body is in equilibrium, the total virtual work done by internal as well as external forces through their respective virtual displacements is zero”. The analytical form of virtual work is represented by

$$\delta W_T^e = \delta W_I^e - \delta W_E^e = 0$$
(10)

where the internal strain energy is given by

$$\begin{aligned} \delta W_I^e = & \mathop \smallint \limits_V \delta {\varepsilon _{xx}} {\sigma _{xx}}dV = \mathop \smallint \limits_0^L \mathop \smallint \limits_A \left( {\delta \varepsilon _{xx}^0 + z\delta \varepsilon _{xx}^1} \right){\sigma _{xx}}dAdx \\ = \mathop \smallint \limits_0^L \mathop \smallint \limits_A \left[ {\left( {\frac{{d\delta {u_0}}}{{dx}} + \frac{{d{w_0}}}{{dx}}\frac{{d\delta {w_0}}}{{dx}}} \right) - z\frac{{{d^2}\delta {w_0}\left( x \right)}}{{d{x^2}}}} \right]{\sigma _{xx}}dAdx \\ = \mathop \smallint \limits_0^L \left[ {\left( {\frac{{d\delta {u_0}}}{{dx}} + \frac{{d{w_0}}}{{dx}}\frac{{d\delta {w_0}}}{{dx}}} \right){N_{xx}} - \frac{{{d^2}\delta {w_0}\left( x \right)}}{{d{x^2}}}{M_{xx}}} \right]dx \\ \end{aligned}$$
(11)

and work done by external forces is

$$\delta W_E^e = \mathop \smallint \limits_0^L q\delta {w_{0 }}{\text{d}}x + \mathop \smallint \limits_0^L f\delta {u_{0 }}{\text{d}}x + Q\delta \Delta$$
(12)

in which \(q\left( x \right)\) is the distributed transverse load, \(f\left( x \right)\) is the distributed axial force, \(Q\) is the applied point load, and \(\delta \Delta\) is the virtual displacement. By substituting Eqs. (11 and 12) into Eq. (10), and performing integration by parts, yields

$$\begin{gathered} \mathop \smallint \limits_0^L \left\{ {\left( {\frac{{{\text{d}}{N_{xx}}}}{{{\text{d}}x}} - f} \right)\delta {u_0} - \left[ {\frac{{\text{d}}}{{{\text{d}}x}}\left( {\frac{{{\text{d}}{w_0}}}{{{\text{d}}x}}{N_{xx}}} \right) + \frac{{{{\text{d}}^2}{M_{xx}}}}{{{\text{d}}{x^2}}} + q} \right]\delta {w_0}} \right\}{\text{d}}x + \hfill \\ \,\,\,\,\left[ {{N_{xx}}\delta {u_0} + \left( {\frac{{{\text{d}}{w_0}}}{{{\text{d}}x}}{N_{xx}} + \frac{{{\text{d}}{M_{xx}}}}{{{\text{d}}x}}} \right)\delta {w_0} - {M_{xx}}\frac{{{\text{d}}\delta {w_0}}}{{{\text{d}}x}}} \right]_0^L - Q\delta \Delta = 0 \hfill \\ \end{gathered}$$
(13)

Thus, the governing equilibrium equations in terms of forces and moments can be described by

$$\delta {u_0}:\frac{{{\text{d}}{N_{xx}}}}{{{\text{d}}x}} - f = 0$$
(14a)
$$\delta {w_0}:\frac{{\text{d}}}{{{\text{d}}x}}\left( {\frac{{{\text{d}}{w_0}}}{{{\text{d}}x}}{N_{xx}}} \right) + \frac{{{{\text{d}}^2}{M_{xx}}}}{{{\text{d}}{x^2}}} + q = 0$$
(14b)

Equilibrium equations in terms of displacements can be described by

$${A_{11}}\frac{{{d^2}{u_0}}}{{d{x^2}}} + {B_{11}}\frac{{{d^3}{w_0}}}{{d{x^3}}} - f = 0$$
(15a)
$${A_{11}}\left( {\frac{{d{u_0}}}{{dx}}\frac{{{d^2}{w_0}}}{{d{x^2}}} + \frac{{{d^2}{u_0}}}{{d{x^2}}}\frac{{d{w_0}}}{{dx}}} \right) + {B_{11}}\left( {\frac{{{d^3}{u_0}}}{{d{x^3}}} + \frac{{{d^2}{w_0}}}{{d{x^2}}}\frac{{{d^2}{w_0}}}{{d{x^2}}} + \frac{{d{w_0}}}{{dx}}\frac{{{d^3}{w_0}}}{{d{x^3}}}} \right) + {D_{11}}\frac{{{d^4}{w_0}}}{{d{x^4}}} + q = 0$$
(15b)

3 Numerical FE formulation

The displacement field at the mid-plane of the FG beam can be assumed as

Axial displacement,

$${u_0}\left( x \right) = \mathop \sum \limits_{j = 1}^2 {u_j}{\psi _j}\left( x \right)$$
(16a)

Transverse displacement,

$${w_0}\left( x \right) = \mathop \sum \limits_{j = 1}^4 {w_j}{\phi _j}\left( x \right)$$
(16b)

where \({\psi _j}\) is the linear Lagrange interpolation function, and \({\phi _j}\) is the Hermite cubic interpolation functions. \({u_j}\) and \({w_j}\) are the axial and transverse nodal displacement, respectively. According to Eq. (13), the virtual work statements can be expressed in terms of generalized displacement \(\left( {{\text{ }}{u_0},{w_0}} \right)\) as

$$\int\limits_0^L {\int\limits_A {E\left( z \right)\left[ {\left( {\frac{{d\delta {u_0}}}{{dx}} + \frac{{d{w_0}}}{{dx}}\frac{{d\delta {w_0}}}{{dx}}} \right) - z\frac{{{d^2}\delta {w_0}\left( x \right)}}{{d{x^2}}}} \right]\left[ {\frac{{d{u_0}\left( x \right)}}{{dx}} - z\frac{{{d^2}{w_0}\left( x \right)}}{{d{x^2}}}} \right]dAdx} } - \mathop \smallint \limits_0^L q\delta {w_{0 }}dx + \mathop \smallint \limits_0^L f\delta {u_{0 }}dx + Q\delta \Delta = 0$$
(17)

Substituting Eq, (16) into Eq. (17) results in the following equilibrium equation in matrix form:

$$\left[ {{K_{uu}} + {K_{uw}} + {K_{wu}} + {K_{ww}}} \right]\left\{ d \right\} = \left\{ f \right\} + \left\{ q \right\} + \left\{ Q \right\}$$
(18)

where \({K_{uu}}\) is the axial stiffness, \({K_{uw}}\) and \({K_{wu}}\) is the coupling axial-bending stiffness, \({K_{ww}}\) is the bending stiffness, \(\left\{ f \right\}\), \(\left\{ q \right\}\) and \(\left\{ Q \right\}\) are the axial, transverse distributed loads and concentrated load vectors, respectively.

4 Numerical results

The results of the proposed model are compared with previous published works to validate the code and model in the first subsection. After that, the effects of elasticity ratio and power index exponent, on the transverse deflection and axial stresses are investigated for the three proposed function distributions.

4.1 Validation

The geometrical dimensions of the beam in this subsection are proposed by Simsek [52], for a slenderness ratio L/h = 16. The present results are illustrated in Table 1. As concluded from this table, the present results for P-FGM and those of Simsek [52] are close. A small deviation is observed (5% as a maximum), which is there for two reasons. The first is the difference in proposed theories in analysis (Euler in current analysis, Timoshenko in Simsek [52]). The second reason is due to the dimension of the beam (L/h = 16), which is a moderately thick beam, and Euler beam is more efficient for thin beams (L/h > 20).

Table 1 Maximum non-dimensional transverse deflection of the beam for various power exponent
Full size table

Figure 2 illustrates the effect of material distribution parameter (k) on the normal stress distribution through the beam thickness for different function behaviors. The stress results obtained by Simsek [52] is also compared with the proposed power function as shown in Fig. 2a. The obtained results are approximately identical to those obtained by Simesk [52]. The small deviation is again due to the difference in the proposed theories.

Fig. 2
figure 2

Maximum Non-dimensional axial stress of the beam for various power exponent. a Power-Law Function (P-FGM). b Symmetric Power-Law Function (S-P-FGM). c Sigmoid Function S-FGM

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4.2 Effect of power index (k)

It is noted from Table 1 that, as power index increases the deflection decreases for both P-FGM and S-P-FGM. Because the ceramics phase which has a higher modulus increases in constituent of the beam, the maximum deflection for S-P-FGM is higher than P-FGM. That means the distribution of ceramics phase in S-P-FGM is less than the distribution in P-FGM at the same power index value. By comparing S-FGM with P-FGM, it is noted that the deflection increases with increasing the power index for S-FGM. However, an opposite effect is observed for P-FGM. As shown in Fig. 2, the higher stress is observed for P-FGM with relative the two other distribution. A symmetric and uniform stress distribution is noted for S-P-FGM, which gives advantage for symmetric power law distribution compared to other functions.

4.3 Effect of elasticity ratio (E1/E2)

Tables 2, 3 and 4 present the maximum deflection of FG beam as the elasticity ratio ranges from 1 to 20 and the material distribution parameter k changes from 0.2 to 10 for the three function distributions: FGM, S-P-FGM and S-FGM, respectively. First of all, when E1/E2 = 1, the isotropic material is rendered. Fixing material distribution parameter k and varying the elasticity ratio results in a significant decrease in deflection for three function distributions. However for the case in hand, the material distribution parameter has the same effect on P-FGM and S-P-FGM while it has a counter effect on S-FGM at a specific elasticity ratio. For example, at E1/E2 = 5, the static deflection decreases from 0.56098 to 0.23782 (approximately half value) as the power exponent increases from 0.2 to 10 for P-FGM, as seen in Fig. 2. The same observation shows for S-P-FGM, where the deflection decreases from 0.8 to 0.245 (approximately one-third) as the power exponent increases. However, deflection increases from 0.33466 to 0.36513 as the material distribution parameter increases, as shown in Table 4. The previous observation gives the designer a facility to select the proper distribution and the proper function for his/her application.

Table 2 The effect of elasticity ratio and for power exponent on non-dimensional max deflection for P-FGM
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Table 3 The effect of elasticity ratio and for power exponent on non-dimensional max deflection for S-P-FGM
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Table 4 The effect of elasticity ratio and for power exponent on non-dimensional max deflection for S-FGM
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Figure 3 shows the effect of elasticity ratio on the axial stress distribution along the thickness direction for the three proposed functions at a specific power exponent k = 2.0. As shown in Fig. 3a for P-FGM, as the elasticity ratio increases, the stress increases at the lower surface (\(\frac{z}{h} = \frac{{ - 1}}{2}\)) and decreases at the upper surface (\(\frac{z}{h} = \frac{1}{2}\)). However, for S-P-FGM, by increasing the elasticity ratio, the stress increases relatively at \(\frac{z}{h} = \frac{{ \pm 1}}{4}\) and decreases at upper and lower surfaces, as seen in Fig. 3b. The effect of elasticity ratio on stress distribution for S-FGM is the same as P-FGM. It increases at the lower surface and decreases at the upper one. However, the elasticity ratio is more significant in case of S-FGM than in P-FGM, especially at lower surfaces. Uniform and symmetric distributions are the main features of S-P-FGM.

Fig. 3
figure 3

Non-dimensional axial stress distributions for various elasticity ratios at k = 2.0. a Power-Law Function (P-FGM). b Symmetric Power-Law Function (S-P-FGM). c Sigmoid Function S-FGM

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5 Conclusions

Static bending analysis of an FG simply-supported beam modeled according to Euler–Bernoulli theory and subjected to a uniform distributed load is investigated. Three functions are assumed to describe the variation of metal and ceramics phases through the beam thickness, namely, power function, symmetric power function, and sigmoid function. The equilibrium equations are derived in details according to virtual work and the weak form is proposed to derive the element stiffness matrices and force vectors using finite element method. The model and results are validated compared to previous work. Numerical results show significant effects of the elasticity ratio and material distribution exponent on the static deflection and stress distribution. Results illustrate a reduction of axial stress for a symmetric power distribution function than for other functions. The proposed model gives designers and engineers a facility to select the proper distributions and proper functions for their application.