_{1}

Modeling of the behavior for Functionally Graded Beam (FGB) under thermal loading is introduced in the present work. The material properties are assumed to vary according to power function along the thickness of the beam. The effects of several parameters such as thermal expansion parameter, thermal conductivity and modulus of elasticity on the resultant axial stress of the FG beam have been studied. For thermal loading the steady state of heat conduction with power and exponentially variations through the thickness of FGB, is considered. The results obtained show that temperature distribution plays very important parameter controlling thermal resultant distribution of stresses and strains.

Functionally Graded Materials (FGMs) are new advanced heat resistant materials that are used in modern technologies as smart structures. Besides excellent thermal properties, they are erosion and corrosion resistant and have high fracture resistance. The fundamental theory is to mix metal and ceramic such that the material properties continuously vary from one basic material to the other. Actually, the governing equations for the temperature and stress distribution are coordinate dependent. As well, other material properties are functions of position.

There is massive amount of research in both of functionally graded beams and the thermal behaviors of beams. An elastic solution is given for a functionally graded beam by Snaker [

Esfahani et al. [

Cheng et al. [

This study is intended to introduce a mathematical analysis to the functionally graded beam under thermal loading. In this study the material properties are assumed to vary according to a power law while the Poisson’s ratio is considered to be constant along the beam height. The heat distribution along the beam height is assumed to have two behaviors: power distribution (P-state) and exponential distribution (E-state).

The concept of functionally gradient materials is to make a composite material by varying the microstructure from one material to another one with a specific gradient. This enables the material to have the best interaction of both materials. The transition between the two materials can usually be approximated by means of a power- law function (P-FGB), exponential function (E-FGB), or sigmoid function (S-FGB). In this study, the material properties distribution is defined by power function through the thickness direction of the beam (i.e. z-direction) along the height (h) from z = 0 until z = h (

where E, a and k are the modulus of elasticity, the thermal expansion and the thermal conductivity of the beam, respectively. The constants E_{o}, a_{o}, k_{o}, ω, b and l can be obtained by boundary conditions.

In accordance with the first order shear deformation theory, a point with a distance z from lower surface (z = 0) will be moved, after deformation with certain displacement [

where u_{o} is the displacement at the lower surface (i.e. z = 0) and w is the transverse deformation, ∂w/∂x and f_{y} are the rotations and all of them are independent of z-direction [

The stress-strain relations for the FGB which subjected to thermal load, and plane strain condition are,

where

the boundary conditions and the equilibrium equations.

When the beam is in equilibrium, the axial resultant forces in the x-direction must be zero value, which yields; (assuming the thermal loading only along distribution in z-direction);

where, _{Ty});

where I_{y} is the moment of inertia about Y-axis, M_{My} is the mechanical moment about the Y-axis and M_{Ty} is the moment due to the thermal affect which could expressed as:

At this instant, Equation (5) and Equation (6) give two of the three required equations for determination the three unknowns in Equation (4-b). The third one could be provided through the boundary conditions which are, for the simply supported beam:

where

Substituting from Equation (11) in Equation (5), Equation (6), Equation (10); yields, the following form;

It is clear that A_{2} is directly obtained, while the coefficient A_{1} and A_{3} could be given as follows:

Therefore, using Equation (13), the axial stress could be obtained as:

Moreover, Equation (14) could introduce the axial stress distribution taking into consideration the temperature distribution along the beam height, which will be discussed next.

The thermal bending occurring in the x-z plane could be presented as follows; [

where t_{xz} is the transverse shear stress. Rearranging Equation (15) gives:

Using Equation (14) into Equation (16), yields;

For the simply supported beam it is obvious that the shear stress is going to zero value because of absent of second derivatives of u_{o}, f_{y} and the third derivatives of w.

Firstly starting with the power distribution of temperature (P-State). The governing equation of heat conduction; assuming one-dimensional transient heat conduction is given as; [

Thus the heat distribution, T(z) through the thickness h, in the direction z is, according to Equation (1-c);

where constants B_{1} and B_{2} could be obtained from the thermal boundary conditions, which are:

and

Therefore, the thermal stress distribution σ_{xx}(z) along the height of the beam (h) could be obtained.

Whereas for the exponential distribution of temperature (E-State); the governing equation of heat conduction could be obtained; referring to Equation (18) but with the exponential distribution of temperature. The heat distribution, T(z) through the thickness h, in the direction z is,

where D_{1} and D_{2} could be obtained from the thermal boundary conditions; assume again

and

Moreover, the thermal stress distribution σ_{xx}(z) along the height of the beam (h) could be obtained with the exponential distribution of temperature.

Consider an elastic square cross section beam (

where T_{1} and T_{2} present the temperature of stress free state i.e. temperatures at surface z = 0 and surface z = h, respectively.

The effect of the parameters (ω, b and l) on the normalized thermal stress _{xx}(z) divided by σ_{n}) distributions are illustrated Figures 2-6, where

The final expression of the axial stress, Equation (14), as a function of A_{1}, A_{2} and A_{3} can be obtained by using a MATLAB program to perform the integrations of the expressions of Equation (11-b).

The effect of the positive values of parameters (ω, b and l) on the normalized axial stress (

^{4} MPa with (ω) equals to 4 at h = z. Moreover, the effect of the parameter (b) on the normalized stress with constant values of ω and l (ω = l = 3) is shown in ^{4} at b = 4).

Whereas the effect of the thermal conductivity parameter (l) on the normalized stress with constant values of b and ω (b = ω = 3) is shown in

After Consider the elastic square cross section beam (_{1} and T_{2} are indicated the temperature of stress free state i.e. temperatures at surface z = 0 and surface at z = h, respectively. Taking a FGB with material properties given in _{2}O_{3}).

With dividing the thermal responses of beams to two parts, one part for explaining the thermal behavior of beam without any end supports (referred as part A) and another part for representing the thermal behavior of beam with considering the end support influence (referred as part B) when the beam is exposed to P-state or E-state, for the temperature distribution. Thus for these two parts and considering Equation (14), the following can be obtained;

where the σ^{A}, σ^{B} and e^{A}, e^{B} are the stress and strain resultants of part A and part B, respectively. Then the total stresses and strains are as follows;

Furthermore, comparing the beam in

Alternatively,

Moreover,

Material | r (kg/m^{3}) | E (GPa) | Ν | C_{v} (J/kg K) | k (W/m・K) | Melting point (K) | a (/K ´ 10^{−6}) | |
---|---|---|---|---|---|---|---|---|

Al_{2}O_{3 } | 3970 | 393 | 0.3 | 775 | 30.1 | 2323 | 8.8 | |

Ni | 8900 | 199.5 | 0.3 | 444 | 90.7 | 1730 | 13.3 | |

In addition,

A mathematical analysis was introduced to study the functionally graded beam under thermal loading. In this study the material properties were assumed to change according to a power law while the Poisson’s ratio is considered constant along the beam height. The heat distribution along the beam height is assumed for two cases: P-state and E-state. The form of temperature distribution has an enormous effect on the resultant thermal stress distributions. It was found that the exponential distribution of thermal loading has more effect than the power distribution on the thermal stress distribution. The functionally graded beam was compared with a traditional beam which also could be called as a Timoshenko’s beam. The analysis introduced in the present work could be extended to any functionally graded beam with power function material property and different beams dimensions.

Abla El-Megharbel, (2016) A Theoretical Analysis of Functionally Graded Beam under Thermal Loading. World Journal of Engineering and Technology,04,437-449. doi: 10.4236/wjet.2016.43044