Thermoelastic Behaviors of Temperature-Dependent Multilayer Arches under Thermomechanical Loadings

Abstract

This work presents analytical solutions for thermoelastic behaviors of multilayer arches with temperature-dependent (TD) thermomechanical properties under thermomechanical loadings. The temperature is varied across the thickness of the arch. Firstly, an arched-slice model is developed, which divides every layer of the arch into numerous hypothetical arched slices with uniform thermomechanical properties. Based on the model, the nonlinear heat conduction equations across the thickness of the arch are solved using the iteration approach, and then the thermoelastic equations obtained from the two-dimensional thermoelasticity theory are solved using the state-space approach and transfer-matrix approach. The present solutions are compared with those obtained using the finite element method and the Euler–Bernoulli theory (EBT). It is found that the error of the EBT increases when the angle of the arch increases or the length-to-thickness ratio decreases. Finally, numerical examples are conducted to analyze the effects of surface temperature and TD thermomechanical properties on the temperature, displacement, and stress distributions of a sandwich arch. The results show that the temperature dependency of thermomechanical properties is a key parameter in predicting the thermoelastic behaviors of the arch in a high-temperature environment.

1. Introduction

Multilayer arches have been widely applied in various engineering structures, such as bridges [1,2,3] and storage tanks [4,5]. The thermomechanical analysis of the multilayer arches has attracted considerable interest, given that the structures may work in severe environments. For example, the multilayer thermal protection system of space vehicles will suffer aerodynamic heating during hypersonic flying, which makes the thermomechanical analysis important for the design [6]. In addition to inducing thermal stresses, the temperature change in the arches also changes the thermomechanical properties [7,8]. These effects could weaken the load-carrying capacity of the arches, and even cause structural failure. Such a problem needs a thorough study.

Several theories [9,10] for the mechanical analysis of arches have been reported, which are commonly based on the straight beam theories [11,12,13,14]. Among these theories, the most basic and popular one is the Euler–Bernoulli theory (EBT), with many successful applications [15,16,17,18]. Based on the EBT and state-space approach, the thermal vibration of cross-ply laminated arches was studied by Khdeir [19]. According to the EBT and using Green’s function method, Rezaiee-Pajand et al. [20] studied the deformations of arches under thermomechanical loadings. The results showed that the in-plane and out-of-plane displacements can be induced through the lateral temperature distribution. Vargas et al. [21] proposed a method for the static analysis of cross-ply laminated arches under thermomechanical loadings. The EBT was applied to obtain the in-plane stress distributions, while the Airy stress function was applied to obtain the out-of-plane ones. According to the EBT and von Kármán hypothesis, Keibolahi et al. [16] studied the nonlinear vibration of a shallow arch under rapid heating. The Ritz method was applied to discretize the motion equation, and the Newton–Raphson method was applied to obtain the solution.

As a result of the neglect of shear deformation effects, the EBT is only suitable for the analysis of thin arches, but has considerable errors for the analysis of thick arches. Thus, diverse refined theories have been proposed, such as the first-order theory [22,23,24] and higher-order theories [25,26,27,28]. Additionally, the exact thermoelasticity theory [29,30] can also be used to solve the arch problems. Using the thermoelasticity theory, Qian et al. [31] studied the static behavior of laminated arches under thermal loading. The exact thermoelastic solutions were presented by using the Fourier series expansion method.

The above studies were carried out based on the hypothesis of temperature-independent (TI) thermomechanical properties. The hypothesis may be tenable when the arch is under a limited temperature range. However, temperature dependency of thermomechanical properties must be included for a wide temperature range, otherwise considerable errors will arise [32,33,34,35]. Based on the EBT, the thermal bending, buckling, and vibration behaviors of temperature-dependent (TD) functionally graded (FG) arches were studied [36,37,38,39,40]. According to the first-order theory, Javani et al. [41] studied the thermally induced vibration of FG shallow arches considering the TD thermomechanical properties. The generalized differential quadrature method was applied to solve the transient heat conduction equation and the motion equation. Using the third-order arch theory and two-step perturbation technique, Babaei et al. [42,43] performed the thermal bending and vibration analysis for TD FG shallow arches. The results indicated that the mechanical performance of the arch was greatly affected by the temperature dependency of thermomechanical properties. Based on a refined tube theory proposed by Zhang and Fu [44], the nonlinear bending behavior of TD shallow curved tubes was studied using the two-step perturbation technique [45,46].

The literature report indicates the lack of analytical solutions for multilayer arches considering TD thermomechanical properties. Motivated by this fact, we propose an arched-slice model to study the thermoelastic behaviors of TD multilayer arches under thermomechanical loadings. Asymptotic analytical solutions of temperature, displacements, and stresses are obtained, and the effects of material and load parameters on the thermoelastic behaviors are discussed.

2. Problem Statement and Arched-Slice Model

A simply supported multilayer circular arch is shown in Figure 1. In the initial stress-free state, the arch is under a uniform temperature. We consider the arch is heated from the inside and outside surfaces and subjected to a radial load Q(φ) on the outside surface. The thermomechanical properties of every layer are TD.

2.1. Basic Assumptions

The study on the multilayer arch is based on the following assumptions:

i.

The inside and outside surface temperatures are uniformly distributed on the surfaces and the two lateral surfaces are adiabatic;

ii.

Each layer is made of a homogenous isotropic material with uniform thickness;

iii.

Each layer is perfectly bonded with the adjacent layer;

iv.

The analysis is within the framework of linear elasticity and small strains.

2.2. Arched-Slice Model

Based on the first assumption, the temperature only varies across the thickness of the arch, hence the thermomechanical properties are also r-dependent. For such a problem, it is impractical to exactly solve the heat conduction equations and thermoelasticity equations due to their r-dependent coefficients. To facilitate asymptotic solutions, we propose an arched-slice model as shown in Figure 2. In this model, every layer is divided into several hypothetical arched slices. When every slice is sufficiently thin, the r-dependent coefficients can be approximated as constant ones by letting $r={\overline{r}}_i$ (see Figure 3).

3. Temperature Solution

Based on the arched-slice model, an iteration approach will be applied to solve the nonlinear heat conduction equations for the TD case (i.e., the thermomechanical properties of the arch are TD). Before introducing the iteration procedure, we first give an exact solution of the heat conduction equations for the TI case (i.e., the thermomechanical properties are assumed to be TI).

3.1. Temperature Solution for the TI Case

The temperature field in the TI case is governed by the following.

(i) Heat conduction of the ith (i = 1, 2, …, q) slice across the r-direction,

$$\frac{d{T}_i}{dr}+r\frac{d^2{T}_i}{d{r}^2}=0$$

(1)

(ii) Temperature and flux at the interface between the ith (i = 1, 2, …, q − 1) and (i + 1)th slices,

$${\left\{\begin{array}{c}{T}_i\\ {\overline{k}}_i\frac{d{T}_i}{dr}\end{array}\right\}}_{r=r_i}={\left\{\begin{array}{c}{T}_{i+1}\\ {\overline{k}}_{i+1}\frac{d{T}_{i+1}}{dr}\end{array}\right\}}_{r=r_i}$$

(2)

(iii) Inside and outside surface temperatures,

$$\left\{\begin{array}{c}{T}_1(r_0)\\ T_q(r_q)\end{array}\right\}=\left\{\begin{array}{c}{T}_{in}\\ T_{out}\end{array}\right\}$$

(3)

Using Equations (1)–(3) yields the following temperature solution:

$$T_i=\frac{A_i+\frac{{\overline{k}}_1}{{\overline{k}}_i}\ln r}{A_q+\frac{{\overline{k}}_1}{{\overline{k}}_q}\ln r_q}\left(T_{out}-T_{in}\right)+T_{in}$$

(4)

where

$$A_{\xi }={\overline{k}}_1{\displaystyle \sum _{j=2}^{\xi }\left(\frac{\ln r_{j-1}}{{\overline{k}}_{j-1}}-\frac{\ln r_{j-1}}{{\overline{k}}_j}\right)}-\ln r_0,\xi =i,q$$

(5)

3.2. Temperature Solution for the TD Case

According to the exact solution Equation (4), we further use the iteration approach to solve the heat conduction problem for the TD case. To implement the iteration approach, an initial solution should be assumed. Here, we assume that the temperature is initially a linear function of r. The iteration procedure is shown in Figure 4.

4. Displacement and Stress Solutions

In Section 3, we obtained the temperature solution, which can be used for the displacement and stress analyses in this section.

4.1. Basic Equations

Based on the two-dimensional thermoelasticity theory, the deformed state of the arch is governed by the following [47].

(i) Equilibrium equation of the ith (i = 1, 2, …, q) slice,

$$\frac{\partial {\sigma }_r^i}{\partial r}+\frac{1}{r}\frac{\partial {\tau }_{\phi r}^i}{\partial \phi }+\frac{{\sigma }_r^i-{\sigma }_{\phi }^i}{r}=0,\frac{1}{r}\frac{\partial {\sigma }_{\phi }^i}{\partial \phi }+\frac{\partial {\tau }_{\phi r}^i}{\partial r}+\frac{2{\tau }_{\phi r}^i}{r}=0$$

(6)

(ii) Strain–displacement relation,

$${\epsilon }_{\phi }^i=\frac{u_r^i}{r}+\frac{1}{r}\frac{\partial u_{\phi }^i}{\partial \phi },{\epsilon }_r^i=\frac{\partial u_r^i}{\partial r},{\gamma }_{\phi r}^i=\frac{1}{r}\frac{\partial u_r^i}{\partial \phi }+\frac{\partial u_{\phi }^i}{\partial r}-\frac{u_{\phi }^i}{r}$$

(7)

(iii) Stress–strain relation in the plane-stress condition,

$$\left\{\begin{array}{c}{\sigma }_{\phi }^i\\ {\sigma }_r^i\\ {\tau }_{\phi r}^i\end{array}\right\}=\left[\begin{array}{ccc}\frac{{\overline{E}}_i}{1-{\overline{\mu }}_i^2}& \frac{{\overline{E}}_i{\overline{\mu }}_i}{1-{\overline{\mu }}_i^2}& 0\\ \frac{{\overline{E}}_i{\overline{\mu }}_i}{1-{\overline{\mu }}_i^2}& \frac{{\overline{E}}_i}{1-{\overline{\mu }}_i^2}& 0\\ 0& 0& \frac{{\overline{E}}_i}{2(1+{\overline{\mu }}_i)}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_{\phi }^i\\ {\epsilon }_r^i\\ {\gamma }_{\phi r}^i\end{array}\right\}-\left\{\begin{array}{c}{t}_i\\ t_i\\ 0\end{array}\right\}$$

(8)

where $t_i={\overline{E}}_i{\overline{\alpha }}_i({\overline{T}}_i-T_0)/(1-{\overline{\mu }}_i)$.

(iv) Displacements and stresses at the interface between the ith (i = 1, 2, …, q − 1) and (i + 1)th slices,

$${\left\{\begin{array}{c}{u}_{\phi }^i\\ u_r^i\\ {\tau }_{\phi r}^i\\ {\sigma }_r^i\end{array}\right\}}_{r=r_i}={\left\{\begin{array}{c}{u}_{\phi }^{i+1}\\ u_r^{i+1}\\ {\tau }_{\phi r}^{i+1}\\ {\sigma }_r^{i+1}\end{array}\right\}}_{r=r_i}$$

(9)

(v) Inside and outside surface stresses,

$${\left\{\begin{array}{c}{\sigma }_r^1\\ {\tau }_{\phi r}^1\end{array}\right\}}_{r=r_0}=\mathbf{0},{\left\{\begin{array}{c}{\sigma }_r^q\\ {\tau }_{\phi r}^q\end{array}\right\}}_{r=r_q}=\left\{\begin{array}{c}-Q(\phi )\\ 0\end{array}\right\}$$

(10)

(vi) Simply supported boundary conditions,

$${\left\{\begin{array}{c}{u}_r^i\\ {\sigma }_{\phi }^i\end{array}\right\}}_{\phi =0,\theta }=\mathbf{0}$$

(11)

Applying the state-space approach [48] to Equations (6)–(8) yields

$$\frac{\partial }{\partial r}\left\{\begin{array}{c}{u}_{\phi }^i\\ {\sigma }_r^i\\ {\tau }_{\phi r}^i\\ u_r^i\end{array}\right\}=\left[\begin{array}{cccc}\frac{1}{r}& 0& \frac{2(1+{\overline{\mu }}_i)}{{\overline{E}}_i}& -\frac{1}{r}\frac{\partial }{\partial \phi }\\ \frac{{\overline{E}}_i}{r^2}\frac{\partial }{\partial \phi }& \frac{{\overline{\mu }}_i-1}{r}& -\frac{1}{r}\frac{\partial }{\partial \phi }& \frac{{\overline{E}}_i}{r^2}\\ -\frac{{\overline{E}}_i}{r^2}\frac{{\partial }^2}{\partial {\phi }^2}& -\frac{{\overline{\mu }}_i}{r}\frac{\partial }{\partial \phi }& -\frac{2}{r}& -\frac{{\overline{E}}_i}{r^2}\frac{\partial }{\partial \phi }\\ -\frac{{\overline{\mu }}_i}{r}\frac{\partial }{\partial \phi }& \frac{1-{\overline{\mu }}_i{}^2}{{\overline{E}}_i}& 0& -\frac{{\overline{\mu }}_i}{r}\end{array}\right]\left\{\begin{array}{c}{u}_{\phi }^i\\ {\sigma }_r^i\\ {\tau }_{\phi r}^i\\ u_r^i\end{array}\right\}+\left\{\begin{array}{c}0\\ \frac{{\overline{\mu }}_i-1}{r}{t}_i\\ -\frac{{\overline{\mu }}_i-1}{r}\frac{\partial t_i}{\partial \phi }\\ \frac{1-{\overline{\mu }}_i{}^2}{{\overline{E}}_i}{t}_i\end{array}\right\}$$

(12)

The induced stress ${\sigma }_{\phi }^i$ can be obtained as

$${\sigma }_{\phi }^i={\overline{\mu }}_i\left({\sigma }_r^i+t_i\right)+\frac{{\overline{E}}_i}{r}{u}_r^i+\frac{{\overline{E}}_i}{r}\frac{\partial u_{\phi }^i}{\partial \phi }-t_i$$

(13)

4.2. General Solution to the Thermoelasticity Equations

Given that the coefficient matrix of Equation (12) has the variable r, it is impractical to exactly solve the equation. Here, we can replace the variable coefficient matrix with a constant one by setting $r={\overline{r}}_i$. It should be mentioned that the error induced by the replacement is negligible due to the thinness of every slice. Under this circumstance, Equations (12) and (13) can be approximated as

$$\frac{\partial }{\partial r}\left\{\begin{array}{c}{u}_{\phi }^i\\ {\sigma }_r^i\\ {\tau }_{\phi r}^i\\ u_r^i\end{array}\right\}=\left[\begin{array}{cccc}\frac{1}{{\overline{r}}_i}& 0& \frac{2(1+{\overline{\mu }}_i)}{{\overline{E}}_i}& -\frac{1}{{\overline{r}}_i}\frac{\partial }{\partial \phi }\\ \frac{{\overline{E}}_i}{{\overline{r}}_i{}^2}\frac{\partial }{\partial \phi }& \frac{{\overline{\mu }}_i-1}{{\overline{r}}_i}& -\frac{1}{{\overline{r}}_i}\frac{\partial }{\partial \phi }& \frac{{\overline{E}}_i}{{\overline{r}}_i{}^2}\\ -\frac{{\overline{E}}_i}{{\overline{r}}_i{}^2}\frac{{\partial }^2}{\partial {\phi }^2}& -\frac{{\overline{\mu }}_i}{{\overline{r}}_i}\frac{\partial }{\partial \phi }& -\frac{2}{{\overline{r}}_i}& -\frac{{\overline{E}}_i}{{\overline{r}}_i{}^2}\frac{\partial }{\partial \phi }\\ -\frac{{\overline{\mu }}_i}{{\overline{r}}_i}\frac{\partial }{\partial \phi }& \frac{1-{\overline{\mu }}_i^2}{{\overline{E}}_i}& 0& -\frac{{\overline{\mu }}_i}{{\overline{r}}_i}\end{array}\right]\left\{\begin{array}{c}{u}_{\phi }^i\\ {\sigma }_r^i\\ {\tau }_{\phi r}^i\\ u_r^i\end{array}\right\}+\left\{\begin{array}{c}0\\ \frac{{\overline{\mu }}_i-1}{{\overline{r}}_i}{t}_i\\ -\frac{{\overline{\mu }}_i-1}{{\overline{r}}_i}\frac{\partial t_i}{\partial \phi }\\ \frac{1-{\overline{\mu }}_i{}^2}{{\overline{E}}_i}{t}_i\end{array}\right\}$$

(14)

$${\sigma }_{\phi }^i={\overline{\mu }}_i\left({\sigma }_r^i+t_i\right)+\frac{{\overline{E}}_i}{{\overline{r}}_i}{u}_r^i+\frac{{\overline{E}}_i}{{\overline{r}}_i}\frac{\partial u_{\phi }^i}{\partial \phi }-t_i$$

(15)

To satisfy the simply supported conditions in Equation (11), we assume the solutions of Equation (14) as

$$\left\{\begin{array}{c}{u}_{\phi }^i\\ u_r^i\\ {\tau }_{\phi r}^i\\ {\sigma }_r^i\end{array}\right\}={\displaystyle \sum _{m=1}^{\infty }\left\{\begin{array}{c}{U}_m^i(r)\cos ({\beta }_m\phi )\\ W_m^i(r)\sin ({\beta }_m\phi )\\ {\Gamma }_m^i(r)\cos ({\beta }_m\phi )\\ R_m^i(r)\sin ({\beta }_m\phi )\end{array}\right\}}$$

(16)

where ${\beta }_m=\frac{m\pi }{\theta }$ and $U_m^i(r)$, $W_m^i(r)$, ${\Gamma }_m^i(r)$, and $R_m^i(r)$ are unknowns.

The stress $t_i$ in Equation (14) can be expanded as

$$t_i(\phi )={\displaystyle \sum _{r=1}^{\infty }{t}_m^i\sin ({\beta }_m\phi )}$$

(17)

where $t_m^i=\frac{2}{\theta }{\displaystyle {\int }_0^{\theta }{t}_i\sin ({\beta }_m\phi )d\phi }$.

Using Equations (16) and (17), the partial differential equation shown in Equation (14) can be reduced to an ordinary differential one as follows:

$$\frac{d{\delta }_i(r)}{dr}=M_i{\delta }_i(r)+N_i$$

(18)

where

$${\delta }_i(r)=\left\{\begin{array}{c}{U}_m^i(r)\\ R_m^i(r)\\ {\Gamma }_m^i(r)\\ W_m^i(r)\end{array}\right\},M_i=\left[\begin{array}{cccc}\frac{1}{{\overline{r}}_i}& 0& \frac{2(1+{\overline{\mu }}_i)}{{\overline{E}}_i}& -\frac{1}{{\overline{r}}_i}{\beta }_m\\ -\frac{{\overline{E}}_i}{{\overline{r}}_i^2}{\beta }_m& \frac{{\overline{\mu }}_i-1}{{\overline{r}}_i}& \frac{1}{{\overline{r}}_i}{\beta }_m& \frac{{\overline{E}}_i}{{\overline{r}}_i^2}\\ \frac{{\overline{E}}_i}{{\overline{r}}_i^2}{\beta }_m^2& -\frac{{\overline{\mu }}_i}{{\overline{r}}_i}{\beta }_m& -\frac{2}{{\overline{r}}_i}& -\frac{{\overline{E}}_i}{{\overline{r}}_i^2}{\beta }_m\\ \frac{{\overline{\mu }}_i}{{\overline{r}}_i}{\beta }_m& \frac{1-{\overline{\mu }}_i{}^2}{{\overline{E}}_i}& 0& -\frac{{\overline{\mu }}_i}{{\overline{r}}_i}\end{array}\right],N_i=\left\{\begin{array}{c}0\\ \frac{{\overline{\mu }}_i-1}{{\overline{r}}_i}{t}_m^i\\ \frac{1-{\overline{\mu }}_i}{{\overline{r}}_i}{\beta }_m{t}_m^i\\ \frac{1-{\overline{\mu }}_i{}^2}{{\overline{E}}_i}{t}_m^i\end{array}\right\}$$

(19)

Similarly, substituting Equations (16) and (17) into Equation (15) gives

$${\sigma }_{\phi }^i={\displaystyle \sum _{m=1}^{\infty }\left[{\overline{\mu }}_i{R}_m^i(r)+({\overline{\mu }}_i-1)t_m^i+\frac{{\overline{E}}_i}{{\overline{r}}_i}{W}_m^i(r)-\frac{{\overline{E}}_i}{{\overline{r}}_i}{\beta }_m{U}_m^i(r)\right]\sin ({\beta }_m\phi )}$$

(20)

Based on the matrix theory, the general solution to Equation (18) is [48]

$${\delta }_i(r)=A_i(r-r_{i-1}){\delta }_i(r_{i-1})+B_i(r-r_{i-1}),r\in [r_{i-1},r_i]$$

(21)

where

$$A_i(r-r_{i-1})=\exp \left[M_i(r-r_{i-1})\right],B_i(r-r_{i-1})=\exp \left[M_i(r-r_{i-1})\right]M_i^{-1}{N}_i-M_i^{-1}{N}_i$$

(22)

4.3. Transfer-Matrix Approach

The radial load Q(φ) can be expanded as

$$Q(\phi )={\displaystyle \sum _{m=1}^{\infty }{Q}_m\sin ({\beta }_m\phi )}$$

(23)

where $Q_m=\frac{2}{\theta }{\displaystyle {\int }_0^{\theta }Q(\phi )\sin ({\beta }_m\phi )d\phi }$.

Substituting Equations (16) and (23) into the boundary conditions in Equations (9) and (10) yields

$${\mathit{\delta }}_{i+1}(r_i)={\delta }_i(r_i)$$

(24)

$${\delta }_1(r_0)={\left\{\begin{array}{cccc}{U}_m^1(r_0)& 0& 0& W_m^1(r_0)\end{array}\right\}}^T,{\delta }_q(r_q)={\left\{\begin{array}{cccc}{U}_m^q(r_q)& -Q_m& 0& W_m^q(r_q)\end{array}\right\}}^T$$

(25)

Substituting $r=r_i$ into Equation (21) yields

$${\delta }_i(r_i)=A_i(h_i){\delta }_i(r_{i-1})+B_i(h_i)$$

(26)

Based on the continuity condition Equation (24) at all the interfaces and using the transfer-matrix approach, the following relations can be obtained:

$${\delta }_i(r_i)={\displaystyle \prod _{s=i}^1\left[A_s(h_s)\right]}{\delta }_1(r_0)+{\displaystyle \sum _{s=1}^{i-1}\left\{{\displaystyle \prod _{j=i}^{s+1}\left[A_j(h_j)\right]}{B}_s(h_s)\right\}}+B_i(h_i),i=1,2,\dots ,q$$

(27)

Incorporating the boundary condition Equation (25) into Equation (27) yields

$${\left\{\begin{array}{cccc}{U}_m^q(r_q)& -Q_m& 0& W_m^q(r_q)\end{array}\right\}}^T=S{\left\{\begin{array}{cccc}{U}_m^1(r_0)& 0& 0& W_m^1(r_0)\end{array}\right\}}^T+\overline{S}$$

(28)

where

$$S=\left[\begin{array}{ccc}{S}_{11}& \cdots & S_{14}\\ ⋮& \ddots & ⋮\\ S_{41}& \cdots & S_{44}\end{array}\right]={\displaystyle \prod _{i=q}^1{A}_i(h_i)},\overline{S}={\left\{\begin{array}{ccc}{\overline{S}}_1& \cdots & {\overline{S}}_4\end{array}\right\}}^T={\displaystyle \sum _{i=1}^{q-1}\left\{{\displaystyle \prod _{j=q}^{i+1}\left[A_j(h_j)\right]}{B}_i(h_i)\right\}}+B_q(h_q)$$

(29)

Decomposing Equation (28) gives the solution of $U_m^1(r_0)$ and $W_m^1(r_0)$ as follows:

$$\left\{\begin{array}{c}{U}_m^1(r_0)\\ W_m^1(r_0)\end{array}\right\}=-{\left[\begin{array}{cc}{S}_{21}& S_{24}\\ S_{31}& S_{34}\end{array}\right]}^{-1}\left\{\begin{array}{c}{Q}_m+{\overline{S}}_2\\ {\overline{S}}_3\end{array}\right\}$$

(30)

Substituting Equation (30) into Equation (25) yields ${\delta }_1(r_0)$. Incorporating ${\delta }_1(r_0)$ into Equation (27) and using the continuity condition Equation (24) yields ${\delta }_i(r_{i-1})$. Incorporating ${\delta }_i(r_{i-1})$ into Equation (21) gives ${\delta }_i(r)$ for every slice. Finally, the displacements $u_{\phi }^i$ and $u_r^i$ and stresses ${\sigma }_{\phi }^i$, ${\sigma }_r^i$ and ${\tau }_{\phi r}^i$ are obtained from Equations (16) and (20).

The present method focuses on arches with simply supported ends. The method can also be developed to deal with other support conditions. For example, the clamped end can be transformed into a simply supported one by adding the unknown longitudinal surface forces, which can be finally determined by the zero displacement condition at the end [49].

5. Numerical Results and Discussion

Numerical examples were conducted to study the thermoelastic behaviors of a steel-concrete-steel sandwich arch.

Table 1. Young’s moduli of steel and concrete at different temperatures [50,51].

T (°C)	20	100	150	200	300	400	500	600	700
Es (GPa)	210	210	-	189	168	147	126	65.1	27.3
Ec (GPa)	30	30	30	-	-	-	-	-	0

Note: Linear interpolation is applied to obtain Young’s moduli at other temperatures.

lists the Young’s moduli of steel and concrete at different temperatures [50,51].

Table 2. Temperature-dependent coefficients for steel and concrete [50,51].

P(T)	P2	P1	P0
ks (W/m°C)	−3.33 × 10−2	0	54
kc (W/m°C)	5.7 × 10−7	−3.36 × 10−3	1.36
αs (°C−1)	0	4 × 10−9	1.208 × 10−5
αc (°C−1)	1.4 × 10−11	2.8 × 10−10	6.0056 × 10−6
μs	0	0	0.3
μc	0	0	0.2

shows the thermal conductivities, thermal expansion coefficients, and Poisson’s ratios, which are all expressed by quadratic polynomials as $P(T)=P_2{T}^2+P_{1}T+P_0$. Unless otherwise stated, some of the geometry and load parameters are fixed at $H_1=H_3=0.1H$, Q(φ) = 5000 N/m, $T_{in}=T_0=20°\mathrm{C}$, while the outside surface temperature $T_{out}$, thickness H, angle θ, mean radius $r_m$ ($r_m=0.5(r_0+r_q)$), and mean length L ($L=\theta r_m$) are the variables. Given that the present method is based on the arched-slice model, we assumed that the arch was divided into q arched slices with equal thickness.

5.1. Validation of the Solutions

The convergence and accuracy of the present method were studied. Firstly, we considered the sandwich arch (θ = 0.5π rad, $r_m$ = 1.1 m, H = 0.2 m) heated at $T_{out}=200 °\mathrm{C}$.

Table 3. Convergence and comparison study of the temperature solutions (unit: °C).

Positions	FE Solutions	Present Solutions
			s = 1	s = 2	s = 3	s = 4
r = 1.05 m	53.660	q = 10	54.184	53.714	53.723	53.723
		q = 20	54.133	53.652	53.661	53.661
		q = 50	54.135	53.654	53.663	53.664
		q = 100	54.133	53.652	53.661	53.661
		q = 200	54.133	53.652	53.661	53.661
r = 1.1 m	109.212	q = 10	110.006	109.214	109.214	109.214
		q = 20	110.006	109.214	109.214	109.214
		q = 50	110.006	109.214	109.214	109.214
		q = 100	110.006	109.214	109.214	109.214
		q = 200	110.006	109.214	109.214	109.214
r = 1.15 m	165.390	q = 10	165.940	165.465	165.456	165.456
		q = 20	165.888	165.400	165.391	165.391
		q = 50	165.890	165.403	165.394	165.394
		q = 100	165.888	165.400	165.391	165.391
		q = 200	165.888	165.400	165.391	165.391

and

Table 4. Convergence and comparison study of the displacement and stress solutions.

Variables	FE Solutions	Present Solutions
			m = 5	m = 25	m = 45	m = 65	m = 85
$u_{\phi }^r$(mm)	2.15	q = 10	2.17	2.15	2.14	2.14	2.14
		q = 50	2.18	2.15	2.15	2.15	2.15
		q = 100	2.18	2.15	2.15	2.15	2.15
		q = 200	2.18	2.15	2.15	2.15	2.15
		q = 400	2.18	2.15	2.15	2.15	2.15
		q = 600	2.18	2.15	2.15	2.15	2.15
$u_r^m$(mm)	5.68	q = 10	5.68	5.67	5.67	5.67	5.67
		q = 50	5.69	5.68	5.68	5.68	5.68
		q = 100	5.69	5.68	5.68	5.68	5.68
		q = 200	5.69	5.68	5.68	5.68	5.68
		q = 400	5.69	5.68	5.68	5.68	5.68
		q = 600	5.69	5.68	5.68	5.68	5.68
${\sigma }_{\phi }^m$(MPa)	12.7	q = 10	10.3	10.2	10.2	10.2	10.2
		q = 50	12.5	12.2	12.3	12.3	12.3
		q = 100	12.8	12.5	12.5	12.5	12.5
		q = 200	12.9	12.6	12.6	12.6	12.6
		q = 400	13.0	12.7	12.7	12.7	12.7
		q = 600	13.0	12.7	12.7	12.7	12.7

show the convergence results for the temperature, displacement, and stress solutions, as well as the comparison with the finite element (FE) solutions. In Table 4, $u_{\phi }^r$ denotes $u_{\phi }$ at φ = 0, $r=r_m$; $u_r^m$ denotes $u_r$ at φ = 0.5θ, $r=r_m$; and ${\sigma }_{\phi }^m$ denotes ${\sigma }_{\phi }$ at φ = 0.5θ, $r=r_m$. The FE analysis was conducted using the commercial package ABAQUS. The four-node heat element, DC2D4, and the four-node plane stress element, CPS4R, were applied to model the arch. Given the symmetry, we only modelled half of the arch, as shown in Figure 5. The FE solutions were obtained by dividing the face and core layers with 4 and 16 elements along the r-direction, respectively, and 50 elements along the φ-direction. It can be seen from Table 3 and Table 4 that the present solutions converge quickly with the increase of the slice number q, iterative step s, and half wave number m. The FE solutions match well with the present ones.

The sandwich arch can also be studied using the EBT [19]. Here, the deflections $u_r^m$ of the arch (Q(φ) = 0, $T_{out}=400°\mathrm{C}$) obtained using EBT and thermoelasticity theory are compared. Figure 6a shows the effects of the length-to-thickness ratio L/H on the comparison results when the angle and mean radius are fixed at θ = 0.3π rad and $r_m$ = 10 m, respectively; Figure 6b displays the effects of the angle θ on the comparative results when L/H = 10 and $r_m$ = 10 m; and Figure 6c displays the effects of the curvature $1/r_m$ on the comparative results when L/H = 10 and θ = 0.3π rad. We can see that the EBT solutions match well with the present ones for thin and shallow arches, but the error of the EBT increases when L/H decreases or θ increases; the curvature $1/r_m$ almost does not affect the accuracy of the EBT solutions.

5.2. Parametric Study

We considered the sandwich arch (θ = 0.75π, $r_m$ = 1.1 m, H = 0.2 m) heated with $T_{out}$ = 100 °C, 200 °C, and 300 °C. The temperature, displacement, and stress distributions of the arch in the TD and TI cases are both considered for comparison. Note that the reference temperature, $T_0$ = 20 °C, was used to determine the thermomechanical properties for the TI case.

5.2.1. Temperature Distribution

Figure 7 shows the temperature distributions across the thickness for the TD and TI cases. We found that T changes rapidly in the concrete layer, but is almost invariable in the steel layer, given that $k_c$ is much smaller than $k_s$. By increasing $T_{out}$, the difference of T in the concrete layer between the TD and TI cases increases. To further illustrate the effects of $T_{out}$ and temperature dependency of thermomechanical properties on the temperature field of the arch, we show the relation between $T_{out}$ and T at $r=r_m$, considering both the TD and TI cases, in Figure 8. We can see that T nonlinearly changes with $T_{out}$ in the TD case, but linearly changes with $T_{out}$ in the TI case. When $T_{out}$ = 300 °C, the difference in T between the two cases is 6.38%.

5.2.2. Displacement and Stress Distributions

Given that the arch considered in this work is under combined thermal and mechanical loadings, here we separately study the mechanical responses induced by the two kinds of loadings. Note that whether the arch is under thermal loading or mechanical loading, the TD and TI cases are considered to show the effects of temperature dependency of thermomechanical properties on the induced mechanical responses.

Firstly, the displacements and stresses induced by thermal loading were studied. Figure 9 shows the distributions of $u_{\phi }^T$ at φ = 0 and $u_r^T$ and ${\sigma }_{\phi }^T$ at φ = 0.5θ for the TD and TI cases, where the superscript T indicates the thermal loading. It is seen that, by increasing $T_{out}$, the difference of displacement and stress distributions between the two cases increases. Figure 9a,b show that, for any fixed $T_{out}$, the TD case has a larger deformation compared with the TI case, because the high-temperature environment can degrade the stiffness of the arch when considering the temperature dependency of thermomechanical properties. Figure 9c shows that, because of the mismatch of thermomechanical properties between the face and core layers, ${\sigma }_{\phi }^T$ is discontinuous at the interfaces.

Figure 10 shows $u_r^T$ and ${\sigma }_{\phi }^T$ at φ = 0.5θ and $r=r_q$ vs. $T_{out}$, considering both the TD and TI cases. We can see that $u_r^T$ and ${\sigma }_{\phi }^T$ nonlinearly change with $T_{out}$ in the TD case, but linearly change with $T_{out}$ in the TI case. The phenomenon is similar to that of temperature, as shown in Figure 8. It was also found that $u_r^T$ in the TD case is always larger than that in the TI case for any fixed $T_{out}$. However, ${\sigma }_{\phi }^T$ in the TD case is not always larger than that in the TI case with the increase of $T_{out}$. When $T_{out}=300°\mathrm{C}$, the differences between the two cases are 7.96% and 4.20% for $u_r^T$ and ${\sigma }_{\phi }^T$, respectively. By combining Figure 8 and Figure 10, it is concluded that the temperature dependency of thermomechanical properties is a key parameter in predicting the thermoelastic behaviors of the arch in high-temperature environments.

Secondly, the displacements and stresses induced by mechanical loading were studied. Figure 11 shows the distributions of $u_r^M$ and ${\sigma }_{\phi }^M$ at φ = 0.5θ and ${\tau }_{\phi r}^M$ at φ = 0 for the TD and TI cases, where the superscript M indicates the mechanical loading. It is shown that when $T_{out}=100°\mathrm{C}$, the TD and TI cases have the same displacement and stress distributions, given that the thermomechanical properties (Young’s modulus and Poisson’s ratio) of the constituents remain constant when $T\le 100°\mathrm{C}$ (see Table 1 and Table 2). It is also noticed that in the TD case, $u_r^M$ increases with $T_{out}$, because the stiffness of the arch is degraded with $T_{out}$; however, the distributions of ${\sigma }_{\phi }^M$ and ${\tau }_{\phi r}^M$ are almost unchanged with the increase of $T_{out}$. Furthermore, Figure 9 and Figure 11 show that the deformations and stresses induced by thermal loading are much bigger than those induced by mechanical loading. Hence, thermal loading has a major influence on the mechanical behaviors of the arch in this study.

6. Conclusions

An arched-slice model is proposed to study the thermoelastic behaviors of TD multilayer arches under thermomechanical loadings. Based on the model, the temperature solution is obtained by applying the iteration approach to the nonlinear heat conduction equations, and then the displacement and stress solutions are obtained by applying the state-space approach and transfer-matrix approach to the two-dimensional thermoelasticity equations. The convergence and accuracy of the present solutions were studied. The effects of surface temperature and TD thermomechanical properties on the temperature, displacement, and stress distributions of a sandwich arch are studied. The following findings were obtained.

i.

The present solutions converge quickly with high accuracy. By comparing the present solutions with those predicted by the EBT, it was found that for thin and shallow arches, the deflection predicted by the EBT agrees well with that predicted by the present thermoelasticity theory. However, the error of the EBT increases when the angle θ increases or the length-to-thickness ratio L/H decreases.

ii.

The temperature, displacement, and stress distributions nonlinearly change with the surface temperature in the TD case, but linearly change in the TI case.

iii.

By separately studying the mechanical responses of the arch induced by thermal loading and mechanical loading, two main effects of temperature can be revealed, directly inducing thermal stresses and deformations and affecting the responses induced by mechanical loading.

iv.

When the arch is subjected to thermal loading, the temperature dependence of thermomechanical properties has a much greater impact on the stress solution than on the temperature and displacement solutions.

v.

Regardless of whether the arch is subjected to thermal loading or mechanical loading, the induced deformations in the TD case are always greater than those in the TI case.