# Thermal–Structural Linear Static Analysis of Functionally Graded Beams Using Reddy Beam Theory

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermal–Structural Problem Description

- The FG beam has a rectangular cross-section of width b and thickness t, as shown in Figure 1, and the beam’s length is L.
- The top and bottom surfaces, as seen in Figure 1, are exposed to the temperatures ${T}_{top}$ and ${T}_{bot}$, respectively, where ${T}_{top}>{T}_{bot}$. The temperature only varies through the z coordinate, and it remains constant along the other directions.
- There is no internal heat generation, and convection heat transfer between the beam’s surfaces and the surrounding media is not considered.
- In this case, a uniform distributed load ${q}_{0}$ is applied to the FG beam, as shown in Figure 1. However, the distributed load can also be function of the x coordinate.

#### 2.1. Mechanical Properties

#### 2.2. Temperature Distribution

## 3. Finite Element Model

#### 3.1. Principle of Virtual Work

#### 3.2. Displacement Vector

#### 3.3. Stiffness Matrix and Generalized Force Vector

## 4. Numerical Results

#### 4.1. Dependence Study of Parameter $\eta $

#### 4.2. Static Analysis

#### 4.3. Thermal–Structural Analysis of Isotropic Beams

- For a C-F isotropic beam:$$w\left(x\right)=-{\displaystyle \frac{\alpha \left({T}_{top}-{T}_{bot}\right)}{2t}}{x}^{2}.$$
- For an S-R isotropic beam:$$w\left(x\right)=-{\displaystyle \frac{\alpha \left({T}_{top}-{T}_{bot}\right)}{2t}}\left(xL-{x}^{2}\right).$$

#### 4.4. Thermal–Structural Analysis of FG Beams

- $L/t=3$: $1.98\%$ for $n=0.5$ and 1, and $4.45\%$ for $n=5$ and 10, respectively.
- $L/t=5$: $0.93\%$ for $n=0.5$ and 1, and $2.79\%$ for $n=5$ and 10, respectively.
- $L/t=10$: $0.45\%$ for $n=0.5$ and 1, and $1.51\%$ for $n=5$ and 10, respectively.
- $L/t=20$: $0.31\%$ for $n=0.5$ and 1, and $0.88\%$ for $n=5$ and 10, respectively.

- $L/t=3$: $2.38\%$ for $n=0.5$ and 1, and $5.70\%$ for $n=5$ and 10, respectively.
- $L/t=5$: $1.39\%$ for $n=0.5$ and 1, and $3.54\%$ for $n=5$ and 10, respectively.
- $L/t=10$: $0.65\%$ for $n=0.5$ and 1, and $1.89\%$ for $n=5$ and 10, respectively.
- $L/t=20$: $0.26\%$ for $n=0.5$ and 1, and $1.07\%$ for $n=5$ and 10, respectively.

#### 4.5. Thermal–Structural Analysis of FG Beams, for $n=1$

- $L/t=3$: $1.34\le {\epsilon}_{r}\le 2.26\%$.
- $L/t=5$: $0.64\le {\epsilon}_{r}\le 0.88\%$.
- $L/t=10$: $0.18\le {\epsilon}_{r}\le 0.44\%$.
- $L/t=20$: $0.02\le {\epsilon}_{r}\le 0.29\%$.

- $L/t=3$: $0.06\le {\epsilon}_{r}\le 0.13\%$.
- $L/t=5$: $0.04\le {\epsilon}_{r}\le 0.36\%$.
- $L/t=10$: $0.01\le {\epsilon}_{r}\le 0.44\%$.
- $L/t=20$: $0.01\le {\epsilon}_{r}\le 0.28\%$.

## 5. Conclusions

- The present finite element model incorporated the rule of mixtures to evaluate the effective mechanical and thermal properties of the FG constituents, where the volume distribution of the ceramic was considered by means of the power law.
- The behavior of the present finite element model was checked by a comparison with the literature and simulations in a finite element commercial code, with the findings showing that the aforementioned results are close to the present ones and behave in a similar manner. Maximum axial displacements and transverse deflections are now available for a comparison with studies that have dealt with the thermal–structural problem presented here.
- In the thermal–structural analysis of FG beams subject to the boundary conditions considered in this article, the higher relative errors were obtained when short beams (e.g., $L/h=3$) were modeled.
- In addition, we found that normal stresses predicted by the present finite element model were in good agreement with those obtained using plane elements.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FGM | Functionally graded material |

CBT | Classical beam theory |

FSDT | First-order shear deformation theory |

TSDT | Third-order shear deformation theory |

HSDT | Higher-order shear deformation theory |

FGB | Functionally graded beams |

FG | Functionally graded |

L | Beam length |

b | Beam width |

t | Beam thickness |

${T}_{top}$, ${T}_{bot}$ | Top and bottom surface temperatures |

${q}_{0}$ | Uniform distributed load |

x, y, z | Rectangular coordinate variables |

P | Material property |

${P}_{top}$ | Top constituent property |

${P}_{bot}$ | Bottom constituent property |

${V}_{top}$ | Volume distribution of the top constituent |

n | Power law exponent |

$K\left(z\right)$ | Thermal conductivity |

${K}_{top}$, ${K}_{bot}$ | Thermal conductivity of top and bottom constituents |

i | Index of the sum |

$\eta $ | Number of terms used in the series for the approximation |

${u}^{0}$, ${w}^{0}$, ${\varphi}^{0}$ | Axial displacement, transverse displacement, and rotation of a |

point located at the centroidal axis x | |

${\left(\xb7\right)}^{M}$ | Quantity related to mechanical effects |

${\left(\xb7\right)}^{T}$ | Quantity related to thermal effects |

${\epsilon}_{11}$, ${\gamma}_{13}$ | Axial and transverse strains |

$\alpha \left(z\right)$ | Thermal expansion coefficient |

${T}_{ref}$ | Temperature of reference |

${\sigma}_{11}$, ${\gamma}_{13}$ | Normal and shear stresses |

$E\left(z\right)$, $G\left(z\right)$ | Young’s and shear moduli |

$\nu \left(z\right)$ | Poisson’s ratio |

$\delta {W}_{I}$, $\delta {W}_{E}$ | Internal and external virtual works |

$\mathbf{f}$ | Vector of external forces |

${h}_{e}$ | One-dimensional domain |

$\psi $ | Interpolation functions |

${\mathbf{K}}^{\mathbf{e}}$ | Element’s stiffness matrix |

${\mathbf{K}}^{ij}$ | Submatrices of element’s stiffness matrix |

${\mathbf{F}}_{\mathbf{T}}^{\mathbf{e}}$ | Element’s thermal force vector |

${\mathbf{F}}_{\mathbf{M}}^{\mathbf{e}}$ | Element’s force vector |

Al | Aluminum |

Al_{2}O_{3} | Alumina |

C-F | Clamped-free boundary conditions |

$\Delta T$ | Temperature difference between top surface temperature and reference temperature |

S-R | Simply supported boundary conditions |

ZnO_{2} | Zirconia |

Present | Numerical results of the present model |

Plane | Numerical results of the plane model |

Exact | Exact solution results |

${\epsilon}_{r}$ | Relative error |

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**Figure 1.**FG beam subjected to a distributed load with different temperatures of the top and bottom surfaces.

**Figure 2.**Transverse deflection of various FG beams subjected to thermal and distributed loads where ${q}_{0}=100$ N/m (${T}_{top}=400$ °C, ${T}_{bot}=300$ °C, and ${T}_{ref}=300$ °C) under (

**a**) C-F and (

**b**) S-R boundary conditions.

**Figure 3.**Transverse deflection of FG beams $(n=1)$ subjected to various $\Delta T$ and a distributed load ${q}_{0}=-{10}^{4}$ N/m (${T}_{bot}=20$ °C and ${T}_{ref}=0\xb0\mathrm{C}$) under (

**a**) C-F and (

**b**) S-R boundary conditions.

**Figure 4.**Variation of the normal stress ${\sigma}_{xx}$ of a C-F FG beam subjected to a mechanical and thermal load for $n=1$ and (

**a**) $L/h=5$; (

**b**) $L/h=20$ (${q}_{0}=-{10}^{4}$ N/m, ${T}_{bot}=20$ °C, and ${T}_{ref}=0\xb0\mathrm{C}$).

**Figure 5.**Variation of the normal stress ${\sigma}_{xx}$ of an S-R FG beam subjected to a mechanical and thermal load for $n=1$ and (

**a**) $L/h=5$; (

**b**) $L/h=20$ (${q}_{0}=-{10}^{4}$ N/m, ${T}_{bot}=20$ °C, and ${T}_{ref}=0\xb0\mathrm{C}$).

**Figure 6.**Normal stress (${\sigma}_{xx}$) of various FG beams subjected to thermal and distributed loads where ${q}_{0}=-{10}^{4}$ N/m (${T}_{top}=120$ °C, ${T}_{bot}=20$ °C, and ${T}_{ref}=0$ °C) under (

**a**) C-F and (

**b**) S-R boundary conditions.

Material | E (GPa) | $\mathit{\nu}$ | K (W/m°C) | $\mathit{\alpha}$ (1/°C) |
---|---|---|---|---|

Aluminum (Al) | 70 | 0.3 | 204 | $23\times {10}^{-6}$ |

Alumina (Al_{2}O_{3}) | 380 | 0.3 | 10.4 | $7.4\times {10}^{-6}$ |

**Table 2.**Maximum displacements for a clamped-free FG beam subjected to only thermal load $(L/t=5$, ${T}_{top}=400$ °C, ${T}_{bot}={T}_{ref}=300$ °C).

n = 0.5 | n = 1 | n = 5 | ||||
---|---|---|---|---|---|---|

$\mathit{\eta}$ | ${\mathit{u}}^{\mathbf{0}}\times {\mathbf{10}}^{-\mathbf{3}}$ | ${\mathit{w}}^{\mathbf{0}}\times {\mathbf{10}}^{-\mathbf{3}}$ | ${\mathit{u}}^{\mathbf{0}}\times {\mathbf{10}}^{-\mathbf{3}}$ | ${\mathit{w}}^{\mathbf{0}}\times {\mathbf{10}}^{-\mathbf{3}}$ | ${\mathit{u}}^{\mathbf{0}}\times {\mathbf{10}}^{-\mathbf{3}}$ | ${\mathit{w}}^{\mathbf{0}}\times {\mathbf{10}}^{-\mathbf{3}}$ |

10 | 1.7247 | $-9.2646$ | 1.9362 | $-8.9328$ | 3.3252 | $-10.4942$ |

25 | 1.4931 | $-8.7550$ | 1.7070 | $-8.3815$ | 3.1461 | $-10.0095$ |

50 | 1.4283 | $-8.5021$ | 1.6477 | $-8.1640$ | 3.0990 | $-9.8696$ |

100 | 1.4162 | $-8.4439$ | 1.6369 | $-8.1186$ | 3.0906 | $-9.8436$ |

200 | 1.4156 | $-8.4410$ | 1.6365 | $-8.1165$ | 3.0902 | $-9.8425$ |

400 | 1.4156 | $-8.4410$ | 1.6365 | $-8.1165$ | 3.0902 | $-9.8425$ |

Material | E (GPa) | $\mathit{\nu}$ | C-F | S-R |
---|---|---|---|---|

Aluminum (Al) | 70 | 0.3 | Bottom | Top |

Zirconia (ZnO_{2}) | 200 | 0.3 | Top | Bottom |

**Table 4.**Maximum dimensionless transverse deflection of a C-F FG beam subjected to a uniform load distribution $\left(L/t=4\right)$.

Work | n = 0 | n = 0.2 | n = 1 | n = 5 | n = 10 |
---|---|---|---|---|---|

Present | 46.51490 | 54.00955 | 74.33885 | 94.71622 | 102.49537 |

Plane | 46.57096 | 53.96973 | 74.24512 | 94.62760 | 102.51855 |

Vo et al. [19] | 46.51500 | 54.01125 | 74.33875 | 94.71625 | 102.49625 |

**Table 5.**Maximum dimensionless transverse deflection of a C-F FG beam subjected to a uniform load distribution $\left(L/t=16\right)$.

Work | n = 0 | n = 0.2 | n = 1 | n = 5 | n = 10 |
---|---|---|---|---|---|

Present | 43.92621 | 51.13280 | 70.50375 | 88.53483 | 95.67928 |

Plane | 43.86304 | 51.15649 | 70.39235 | 88.44528 | 95.62302 |

Vo et al. [19] | 43.92625 | 51.13375 | 70.50250 | 88.53375 | 95.67875 |

**Table 6.**Maximum dimensionless transverse deflection of an S-R FG beam subjected to a uniform load distribution $\left(L/t=4\right)$.

Work | $\mathbf{n}=0$ | $\mathbf{n}=0.5$ | $\mathbf{n}=1$ | $\mathbf{n}=5$ | $\mathbf{n}=10$ |
---|---|---|---|---|---|

Present | 15.04884 | 9.53850 | 8.36849 | 6.50747 | 6.00623 |

Plane | 15.05273 | 9.54069 | 8.29951 | 6.49140 | 6.02041 |

Şimşek [29] | 15.04922 | 9.53958 | 8.36862 | 6.50755 | - |

Vo et al. [19] | 15.04948 | 9.53828 | 8.36823 | 6.50742 | - |

**Table 7.**Maximum dimensionless transverse deflection of an S-R FG beam subjected to a uniform load distribution $\left(L/t=16\right)$.

Work | $\mathbf{n}=0$ | $\mathbf{n}=0.5$ | $\mathbf{n}=1$ | $\mathbf{n}=5$ | $\mathbf{n}=10$ |
---|---|---|---|---|---|

Present | 13.14767 | 8.34089 | 7.38270 | 5.78671 | 5.31468 |

Plane | 13.05415 | 8.28457 | 7.32718 | 5.74005 | 5.27186 |

Şimşek [29] | 13.14779 | 8.34180 | 7.38268 | 5.78672 | - |

Vo et al. [19] | 13.14714 | 8.34063 | 7.38255 | 5.78672 | - |

**Table 8.**Maximum transverse deflection for C-F isotropic beams subjected to a thermal load for various $L/t$ ratios.

Model | L/t | ||
---|---|---|---|

3 | 5 | 20 | |

Present | −0.010350 | −0.028750 | −0.46000 |

Plane | −0.010552 | −0.029037 | −0.46027 |

Exact | −0.010350 | −0.028750 | −0.46000 |

**Table 9.**Maximum transverse deflection for S-R isotropic beams subjected to a thermal load for various $L/t$ ratios.

Model | L/t | ||
---|---|---|---|

3 | 5 | 20 | |

Present | 0.0025875 | 0.0071875 | 0.115 |

Plane | 0.0025765 | 0.0071092 | 0.11327 |

Exact | 0.0025875 | 0.0071875 | 0.115 |

**Table 10.**Maximum displacements of a C-F FG beam subjected to thermal load and distributed load ${q}_{0}=100$ N/m (${T}_{top}=400$ °C, ${T}_{bot}=300$ °C, and ${T}_{ref}=300$ °C).

$\mathbf{n}=0.5$ | $\mathbf{n}=1$ | $\mathbf{n}=5$ | $\mathbf{n}=10$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Model | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | |

$L/t=3$ | Present | 0.8497 | $-3.0410$ | 0.9822 | $-2.9236$ | 1.8544 | $-3.5354$ | 2.2992 | $-4.7209$ |

Plane | 0.8514 | $-3.0958$ | 0.9955 | $-2.9825$ | 1.9406 | $-3.6603$ | 2.4064 | $-4.9210$ | |

$L/t=5$ | Present | 1.4162 | $-8.4435$ | 1.6370 | $-8.1181$ | 3.0906 | $-9.8428$ | 3.8312 | $-13.1500$ |

Plane | 1.4182 | $-8.5175$ | 1.6506 | $-8.1941$ | 3.1780 | $-10.0020$ | 3.9411 | $-13.4360$ | |

$L/t=10$ | Present | 2.8325 | $-33.7633$ | 3.2740 | $-32.4618$ | 6.1814 | $-39.4007$ | 7.6611 | $-52.6523$ |

Plane | 2.8356 | $-33.8830$ | 3.2889 | $-32.5950$ | 6.2720 | $-39.7390$ | 7.7786 | $-53.3280$ | |

$L/t=20$ | Present | 5.6653 | $-134.9740$ | 6.5487 | $-129.7477$ | 12.3642 | $-157.4974$ | 15.3221 | $-210.5132$ |

Plane | 5.6725 | $-134.9600$ | 6.5690 | $-129.8300$ | 12.4650 | $-158.3800$ | 15.4580 | $-212.4600$ |

**Table 11.**Maximum displacements of an S-R FG beam subjected to thermal load and distributed load ${q}_{0}=100$ N/m (${T}_{top}=400$ °C, ${T}_{bot}=300$ °C, and ${T}_{ref}=300$ °C).

$\mathbf{n}=0.5$ | $\mathbf{n}=1$ | $\mathbf{n}=5$ | $\mathbf{n}=10$ | ||||||
---|---|---|---|---|---|---|---|---|---|

Model | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | |

$L/t=3$ | Present | 0.8496 | $0.7618$ | 0.9822 | $0.7320$ | 1.8544 | $0.8743$ | 2.3005 | $1.1647$ |

Plane | 0.8299 | $0.7606$ | 0.9806 | $0.7324$ | 1.9665 | $0.8841$ | 2.4383 | $1.1980$ | |

$L/t=5$ | Present | 1.4161 | $2.1125$ | 1.6369 | $2.0308$ | 3.0906 | $2.4514$ | 3.8324 | $3.2722$ |

Plane | 1.3967 | $2.1124$ | 1.6356 | $2.0342$ | 3.2037 | $2.4735$ | 3.9729 | $3.3271$ | |

$L/t=10$ | Present | 2.8323 | $8.4445$ | 3.2738 | $8.1194$ | 6.1811 | $9.8449$ | 7.6621 | $13.1523$ |

Plane | 2.8139 | $8.4458$ | 3.2736 | $8.1329$ | 6.2972 | $9.9199$ | 7.8097 | $13.3030$ | |

$L/t=20$ | Present | 5.6646 | $33.7796$ | 6.5473 | $32.4830$ | 12.3613 | $39.4332$ | 15.3207 | $52.6880$ |

Plane | 5.6500 | $33.7210$ | 6.5522 | $32.4700$ | 12.4870 | $39.6530$ | 15.4860 | $53.1430$ |

**Table 12.**Maximum displacements of a C-F FG beam $(n=1)$ subjected to various $\Delta T$ and a distributed load ${q}_{0}=-{10}^{4}$ N/m (${T}_{bot}=20$ °C and ${T}_{ref}=0\xb0\mathrm{C}$).

$\mathbf{\Delta}\mathit{T}=100\xb0\mathbf{C}$ | $\mathbf{\Delta}\mathit{T}=200\xb0\mathbf{C}$ | $\mathbf{\Delta}\mathit{T}=300\xb0\mathbf{C}$ | $\mathbf{\Delta}\mathit{T}=400\xb0\mathbf{C}$ | ||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{L}/\mathit{h}$ | Model | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ |

3 | Present | 1.8938 | $-1.5267$ | 2.8760 | $-4.4503$ | 3.8582 | $-7.3739$ | 4.8403 | $-10.2976$ |

Plane | 1.9195 | $-1.5620$ | 2.9150 | $-4.5445$ | 3.9105 | $-7.5270$ | 4.9060 | $-10.5100$ | |

5 | Present | 3.1554 | $-4.2698$ | 4.7924 | $-12.3883$ | 6.4293 | $-20.5069$ | 8.0662 | $-28.6255$ |

Plane | 3.1814 | $-4.2973$ | 4.8320 | $-12.4920$ | 6.4826 | $-20.6870$ | 8.1332 | $-28.8810$ | |

10 | Present | 6.3018 | $-17.6686$ | 9.5756 | $-50.1384$ | 12.8495 | $-82.6082$ | 16.1234 | $-115.0780$ |

Plane | 6.3288 | $-17.7010$ | 9.6176 | $-50.3030$ | 12.9060 | $-82.9060$ | 16.1950 | $-115.5100$ | |

20 | Present | 12.5308 | $-80.1733$ | 19.0785 | $-210.0480$ | 25.6263 | $-339.9227$ | 32.1740 | $-469.7974$ |

Plane | 12.5620 | $-80.1610$ | 19.1300 | $-210.1200$ | 25.6980 | $-340.0800$ | 32.2660 | $-470.0300$ |

**Table 13.**Maximum displacements of an S-R FG beam $(n=1)$ subjected to various $\Delta T$ and a distributed load ${q}_{0}=-{10}^{4}$ N/m (${T}_{bot}=20$ °C and ${T}_{ref}=0\xb0\mathrm{C}$).

$\mathbf{\Delta}\mathit{T}=100\xb0\mathbf{C}$ | $\mathbf{\Delta}\mathit{T}=200\xb0\mathbf{C}$ | $\mathbf{\Delta}\mathit{T}=300\xb0\mathbf{C}$ | $\mathbf{\Delta}\mathit{T}=400\xb0\mathbf{C}$ | ||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{L}/\mathit{h}$ | Model | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ | ${\mathit{u}}^{0}\times {10}^{-3}$ | ${\mathit{w}}^{0}\times {10}^{-3}$ |

3 | Present | 1.8943 | $0.3802$ | 2.8765 | $1.1122$ | 3.8587 | $1.8443$ | 4.8408 | $2.5763$ |

Plane | 1.8927 | $0.3807$ | 2.8732 | $1.1131$ | 3.8538 | $1.8455$ | 4.8343 | $2.5779$ | |

5 | Present | 3.1577 | $1.0502$ | 4.7946 | $3.0809$ | 6.4316 | $5.1117$ | 8.0685 | $7.1425$ |

Plane | 3.1563 | $1.0540$ | 4.7919 | $3.0882$ | 6.4276 | $5.1224$ | 8.0632 | $7.1565$ | |

10 | Present | 6.3199 | $4.1343$ | 9.5938 | $12.2529$ | 12.8677 | $20.3715$ | 16.1416 | $28.4900$ |

Plane | 6.3194 | $4.1525$ | 9.5931 | $12.2840$ | 12.8670 | $20.4170$ | 16.1400 | $28.5490$ | |

20 | Present | 12.6762 | $15.5429$ | 19.2240 | $48.0127$ | 25.7718 | $80.4825$ | 32.3195 | $112.9523$ |

Plane | 12.6790 | $15.5870$ | 19.2310 | $48.0440$ | 25.7840 | $80.5010$ | 32.3370 | $112.9600$ |

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## Share and Cite

**MDPI and ACS Style**

Valencia Murillo, C.E.; Gutierrez Rivera, M.E.; Celaya Garcia, L.D.
Thermal–Structural Linear Static Analysis of Functionally Graded Beams Using Reddy Beam Theory. *Math. Comput. Appl.* **2023**, *28*, 84.
https://doi.org/10.3390/mca28040084

**AMA Style**

Valencia Murillo CE, Gutierrez Rivera ME, Celaya Garcia LD.
Thermal–Structural Linear Static Analysis of Functionally Graded Beams Using Reddy Beam Theory. *Mathematical and Computational Applications*. 2023; 28(4):84.
https://doi.org/10.3390/mca28040084

**Chicago/Turabian Style**

Valencia Murillo, Carlos Enrique, Miguel Ernesto Gutierrez Rivera, and Luis David Celaya Garcia.
2023. "Thermal–Structural Linear Static Analysis of Functionally Graded Beams Using Reddy Beam Theory" *Mathematical and Computational Applications* 28, no. 4: 84.
https://doi.org/10.3390/mca28040084