Analysis of laminated beams with temperature-dependent material properties subjected to thermal and mechanical loads

Composite Structures 227 (2019) 111304

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Analysis of laminated beams with temperature-dependent material properties subjected to thermal and mechanical loads

T

⁎

Zhong Zhang, Ding Zhou , Hai Fang, Jiandong Zhang, Xuehong Li College of Civil Engineering, Nanjing Tech University, Nanjing 211816, PR China

A R T I C LE I N FO

A B S T R A C T

Keywords: Laminated beam Temperature-dependent material property Thermo-elasticity theory Iterative algorithm Transfer-matrix method

This work focuses on the temperature, displacement and stress analysis of simply-supported laminated beams with temperature-dependent material properties subjected to thermal and mechanical loads. The temperature is considered to be constant along the length of the beam, however, varies along the thickness. A thin-layer model is introduced for analysis: every layer of the beam is divided into finite sublayers, each of which is approximately assumed to have uniform material properties. Based on Fourier’s law, the temperature field is obtained by an iterative algorithm. Based on the thermo-elasticity theory, the two-dimensional displacement and stress solutions are obtained by using the transfer-matrix method. The comparative study shows that the results from the two-dimensional finite element method agree well with the present ones; however, the results from the classical beam theory have considerable errors, especially for thick beams. Finally a sandwich beam is taken as an example. The analysis reveals that the temperature not only produces deformations and stresses itself, but also affects the deformations and stresses induced by mechanical loads.

1. Introduction The excellent mechanical property and environmental sustainability of composite laminated beams have led to their wide use in modern engineering [1–3]. The mechanical performance of these structures in high temperature environment has attracted significant attention. In general, temperature variation will induce internal stresses due to different thermo-elastic properties between contiguous layers, and reduce the stiffness due to softening of the beams. These effects could weaken the load-carrying capacity of the beams, and even cause structural failure. Considerable efforts have been devoted to the analysis of beams based on various theories, which have been reviewed by a lot of researchers [4–7]. These theories are mainly divided into three categories, namely, classical beam theory (CBT), first-order shear deformation theory (FSDT, also known as Timoshenko beam theory), and higher-order shear deformation theories (HSDTs). The CBT based on the Euler-Bernoulli hypothesis is popular in engineering due to its simplicity. Timoshenko [8] is perhaps the first one to study the thermo-elastic behavior of laminated beam. In his work, the bending analysis of a bimetal strip subjected to uniform thermal loads was proposed based on the CBT. An analytical model for the flexural vibrations of thin EulerBernoulli beam resonators subjected to thermal and mechanical loads

⁎

was presented by Sharma and Kaur [9]. The governing equation was solved using the Laplace transform technique and Adomian decomposition method. According to the CBT and modified couple stress theory, Mohandes and Ghasemi [10] studied the vibration and bending behaviors of micro/nanolaminated beam subjected to thermal loads. The governing equation derived from Hamilton’s principle was numerically solved using the generalized differential quadrature method. Ebrahimi and Barati [11] studied the buckling behavior of curved Euler-Bernoulli FG beams subjected to different temperature profiles. The CBT neglects the transverse shear deformation, which makes it less accurate for thick beams. In order to improve the accuracy of the CBT, the FSDT was proposed with a hypothesis that the transverse shear strain is invariable along the thickness direction. The FSDT gives a better prediction of deflections and stresses for both thin and moderately thick beams and hence has been widely applied into thermomechanical problems by many researchers [12–19]. However, the hypothesis of constant transverse shear strain is still not the real case as derived from the two-dimensional (2-D) elasticity theory. Therefore, it was necessary to improve the theories for a more accurate prediction. This was achieved by the HSDTs, in which the transverse displacement along the thickness direction is described using various higher-order functions [7]. By introducing a higher-order zigzag theory, Kapuria et al. [20] studied the thermal stresses of laminated beams subjected to

Corresponding author. E-mail address: [email protected] (D. Zhou).

https://doi.org/10.1016/j.compstruct.2019.111304 Received 9 May 2019; Received in revised form 20 July 2019; Accepted 31 July 2019 Available online 07 August 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.

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thermal loads. Li and Qiao [21] studied the thermal postbuckling of laminated beams resting on two-parameter elastic foundations using Reddy’s higher-order theory. Zhang and Fu [22] proposed a higherorder model for tubes, in which the displacement distribution was described in the Leurent series form. Based on this refined model, bending, buckling, and vibration of FG tubes with temperature-dependent material properties were studied [23,24]. Ghalami-Choobar et al. [25] gave the static analysis of laminated beams under thermal and mechanical loads by introducing a unified zig-zag theory, in which many beam theories were included via some shape functions such as sinusoidal and parabolic functions. Some beam problems have also been solved using the exact elasticity theory. Compared with the solutions from the classical theory and refined theories, the solutions from the elasticity theory are generally more accurate since no hypotheses introduced. Eslami et al. [26] reviewed the thermo-elasticity theory and presented its application in some practical problems. Sankar and Tzeng [27] presented an analytical solution for predicting the thermal stress distribution in FG beams. Xu and Zhou [28] presented the thermo-elastic analysis for beams of changeable thickness under thermal and mechanical loads. The exact elasticity solutions for laminated beams under uniform thermo-load and mechanical loads were studied by Zhang et al. [29]. The literature review indicates that not much attention has been paid to the temperature-dependent effects on the thermal and mechanical performance of laminated beams. The objective of this work is to build an analytical model for considering the effects of high temperature on the temperature, displacement, and stress distributions in laminated beams with temperature-dependent material properties. In this work, the displacement and stress solutions are obtained based on the exact 2-D thermo-elasticity theory, which renounces any transverse shear deformation hypothesis compared with the classical theory and various shear deformation theories. In the numerical examples, a sandwich beam is considered. The thermal load and the mechanical load are separately studied to show the effects of temperature on structure responses.

Fig. 2. Partition of an arbitrary layer along the thickness direction.

can be approximately regarded as uniform. 2.1. Temperature-independent state The thermal conductivity ki of each sublayer is uniform on the basis of the thin-layer model, but it is still temperature-dependent. To obtain the temperature field for the temperature-dependent state, we first carry out the exact temperature solution for the temperature-independent state by considering ki to be constant. Using the local coordinate yi (see Fig. 2), the equations governing the temperature distribution are: (1) 1-D steady-state heat equation for the ith sublayer [30],

d ⎛ dTi ⎞ ⎜k i ⎟ = 0 dyi ⎝ dyi ⎠

(1)

where Ti is the temperature and ki is temperature-independent. (2) Compatibility of the temperature and thermal flux between contiguous sublayers,

Ti (hi ) = Ti + 1 (0), ki

dTi dyi

= ki + 1 yi = hi

dTi + 1 dyi + 1

, (i = 1, 2, … p ̂ − 1) yi + 1 = 0

(2) 2. Temperature solution

(3) Top and bottom surface temperatures,

As shown in Fig. 1, a simply-supported laminated beam of length L p and thickness H (H = ∑i = 1 Hi ) comprises several well bonded layers, each of which is made of an isotropic linear elastic material with temperature-dependent properties. A global coordinate system is built with the axes along the length and thickness directions denoted by x and y, respectively. The beam is subjected a distributed load Q(x) and is heated from a uniform temperature T0 . Finally, the temperatures on the top and bottom surfaces are denoted by Tt and Tb , respectively. The temperature is assumed to be invariable along the x-direction, then the heat conduction problem becomes a 1-D one governed by the coordinate y. However, variable temperature along the thickness direction leads to variable coefficients in the governing equations owing to the temperature-dependent material properties. A thin-layer model (see Fig. 2) is introduced to transform the equations into those with invariable coefficients. In this model, every layer of the beam is equally divided into q sublayers (the total sublayer number is p ̂ = pq ), each with a very small thickness hi = Hj / q (j = 1, 2, … , p, and i = q(j-1) + 1, q(j-1) + 2, … , qj). In this case, the material properties of each sublayer

T1 (0) = Tb, Tp ̂ (h p )̂ = Tt

(3)

The general solution of Eq. (1) is

Ti = ai yi + bi

(4)

where ai and bi are the unknowns. Substituting Eq. (4) into Eq. (2) gives the relations between the ith sublayer and the bottom sublayer: i−1

ai =

k1 k a1, bi = a1 ∑ 1 hj + b1 ki k j=1 j

(5)

Taking i = p ̂ and substituting Eqs. (4) and (5) into Eq. (3) gives

a1 =

Tt − Tb p̂

∑ j=1

k1 h kj j

, b1 = Tb (6)

Substituting Eq. (6) into Eq. (5) obtains ai and bi for the ith sublayer. Therefore, the temperature solution for the temperature-independent state is derived by substituting Eq. (5) into Eq. (4). 2.2. Temperature-dependent state The thermal conductivity ki in Eq. (1) actually varies with temperature, in which case the exact temperature solution is hard to obtain. To deal with this problem an iterative algorithm is proposed, in which the initial approximate temperature Tini is linear along the yi -direction. In the jth step of the algorithm, the thermal conductivity of each sublayer is constant and determined by the temperature on the mid-plane of the sublayer, which can be obtained from the (j − 1)th step. The

Fig. 1. Geometry, coordinate and load condition of the laminated beam. 2

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Therefore, the temperature increment ΔTi ̂ of the ith sublayer is

ΔTi ̂ = Ti ̂ − T0

(8)

The thermal expansion coefficient, Young’s modulus, and Poisson’s ratio of the ith sublayer, which can be determined by the uniform temperature Ti ,̂ are denoted by αi , Ei , and μi , respectively. With the plane stress condition, the equations governing the deformed state are: (1) Equilibrium equations with neglecting volume forces, i i ∂τxy ∂σyi ∂τxy ∂σxi + = 0, + =0 ∂x ∂yi ∂yi ∂x

where

σxi

and

σyi

(9)

are the normal stresses;

i τxy

is the shear stress.

(2) Geometrical relations,

εxi =

∂ui i ∂v i ∂ui ∂v , ε y = i , γxy = + i ∂x ∂yi ∂yi ∂x

(10)

where ui and vi are the displacements; i γxy is the shear strain.

εxi

and

ε yi

are the normal strains;

(3) Thermo-elastic constitutive relations,

σxi =

Ei 1 − μi2

(εxi + μi ε yi ) − ti, σyi = i τxy =

Ei 1 − μi2

(ε yi + μi εxi ) − ti,

Ei γi 2(1 + μi ) xy

(11)

where

ti = Fig. 3. Flowchart for showing the iteration of temperature solution.

(12)

(4) Simply-supported boundary conditions,

σxi = 0, vi = 0, at x = 0, L

temperature solution in each step can be obtained in the same way as shown in the Section 2.1. A flowchart for the iteration of the temperature solution is given in Fig. 3.

(13)

(5) Compatibility of the displacements and stresses between contiguous sublayers,

3. Displacement and stress solutions

ui (x , hi ) = ui + 1 (x , 0), vi (x , hi ) = vi + 1 (x , 0), i i+1 σyi (x , hi ) = σyi+ 1 (x , 0), τxy (x , hi ) = τxy (x , 0)

3.1. Thermo-elasticity equation

(14)

(6) Top and bottom surface stresses,

In the thin-layer model, the thickness of each sublayer is very small, hence the temperature Ti ̂ (i = 1, 2, …, p )̂ of each sublayer can be approximately regarded as uniformly distributed and equivalent in magnitude to the actual temperature at the mid-plane of the sublayer (see Fig. 4), i.e.

h Ti ̂ = Ti ⎛ i ⎞ ⎝2⎠

Ei αi ΔTi ̂ 1 − μi

1 σyp ̂ (x , h p )̂ = −Q (x ), σy1 (x , 0) = 0, τxyp ̂ (x , h p )̂ = τxy (x , 0) = 0

(15)

Substituting the first of Eq. (11) into the first of Eq. (13) and using Eq. (10), we can rewrite the stress boundary condition as

∂v Ei ⎛ ∂ui + μi i ⎞⎟ = ti, at x = 0, L ⎜ ∂yi ⎠ 1 − μi2 ⎝ ∂x

(7)

Fig. 4. Discretization of the continuous temperature along the thickness direction. 3

(16)

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dRi (yi ) = Si Ri (yi ) + Ti dyi

Eq. (16) is valid at the two lateral ends of the ith sublayer, hence ti is considered as the longitudinal surface forces. Therefore, the displacements induced by the longitudinal thermal load and by the longitudinal surface forces are identical. In this case σxi can be expressed by

σxi = σ~xi + H (x ) ti + H (x − L) ti − ti

where i 0 S4i − ϕm ⎤ ⎧Um (yi ) ⎫ ⎡ 0 0 ⎧ ⎫ ⎪ ⎥ ⎢ 0 i 0 0 0 ϕm ⎪ Ym (yi ) ⎪ ⎪ ⎪ ⎪ ⎥ ⎢ = , Ri (yi ) = , Si = T 2ti − 2(−1)mti i i 2 i ⎢ S2 ϕm S1 ϕm 0 ⎨ Xmi (yi ) ⎬ ⎨ ⎬ 0 ⎥ L ⎥ ⎢ ⎪ i ⎪ ⎪ ⎪ i i 0 ⎪ ⎪ ⎩ ⎭ 0 0 ⎥ ⎢ ⎦ ⎣− S1 ϕm S3 ⎩ Vm (yi ) ⎭ (28)

(17)

where the last item − ti is identical to that given in Eq. (11); σ~xi takes zero value at the two ends of the ith sublayer; H (x ) ti and H (x − L) ti are the longitudinal surface forces acted at x = 0 and L, respectively. H(x) is the unit pulse function with the following definition:

1, x = 0 H (x ) = ⎧ , 0≤x≤L ⎨ ⎩ 0, x ≠ 0

The solution of Eq. (27) is (18)

Ri (yi ) = Ai (yi ) Ri (0) + Bi (yi )

Substituting Eq. (17) into Eq. (9), we can rewrite the equilibrium equations as i i ∂τxy ∂σyi ∂τxy ∂σ~xi + = δ (x ) ti − δ (x − L) ti, + =0 ∂x ∂yi ∂yi ∂x

Ai (yi ) = exp(Si yi ), Bi (yi ) = [exp(Si yi ) − I ] Si−1 Ti

∞, x = 0 ∞, x = L dH (x ) dH (x − L) , δ (x − L) = =⎧ =⎧ ⎨ ⎨ dx dx ⎩ 0, x ≠ 0 ⎩ 0, x ≠ L (20)

exp(Si yi ) = c1i (yi ) I + c2i (yi ) Si + c3i (yi ) Si2 + c4i (yi ) Si3

(k = 1, 2, 3, 4) are dependent on the eigenvalues of the where matrix Si . According to the first of Eq. (28), the eigenvalues of Si are

κ1 = κ3 = ϕm , κ2 = κ 4 = −ϕm

i ⎧ c1 (yi ) ⎫ ⎡ 1 ⎪ ⎪ c2i (yi ) ⎪ ⎪ ⎢1 =⎢ ⎨ c3i (yi ) ⎬ ⎢ 0 ⎪ i ⎪ ⎢ ⎪ c4 (yi ) ⎪ 0 ⎩ ⎭ ⎣

Combining Eqs. (10), (19), and (21) leads to the following governing equation:

u 0 S4i − β ⎤ ⎧ ui ⎫ 0 ⎧ ii ⎫ ⎡ 0 ⎧ ⎫ ⎢ 0 σ 0 0 − β 0 ⎥ ⎪ σyi ⎪ ⎪ ⎪ ∂ ⎪ y⎪ ⎥ ⎢ + = i ⎬ i ⎬ ⎨ δ (x ) ti − δ (x − L) ti ⎬ 0 ⎥ ⎨ τxy ∂yi ⎨ τxy ⎢− S2i β 2 S1i β 0 ⎪ ⎪ ⎢ i ⎪ ⎥⎪ ⎪ ⎪ S3(i) ti ⎭ 0 0 ⎦ ⎩ vi ⎭ ⎩ S3i ⎩ vi ⎭ ⎣ S1 β

−1

κ13 ⎤ 2 κ2 κ 2 κ 23 ⎥ ⎥ 1 2κ1 3κ12 ⎥ ⎥ 1 2κ2 3κ 22 ⎦

⎧ exp(κ1 yi ) ⎫ ⎪ exp(κ2 yi ) ⎪ ⎨ yi exp(κ1 yi ) ⎬ ⎪ ⎪ ⎩ yi exp(κ2 yi ) ⎭

Expanding the normal stress ti in Eq. (25) and the applied load Q(x) gives (23)

∞

∞

∑ tmi sin(ϕm x ), Q (x ) = ∑ qm sin(ϕm x )

ti (x ) =

m=1

3.2. Displacement and stress in a sublayer

∞

1 2 ∑ cos(ϕm x ) + L , δ (x − L) = L m=1

∞

∑ m=1

tmi =

1 (−1)mcos(ϕm x ) + L (24)

∞ ∞ ui = ∑m = 1 Umi (yi ) cos(ϕm x ), vi = ∑m = 1 Vmi (yi ) sin(ϕm x ) ∞ ∞ i i i τxy = ∑m = 1 Xm (yi ) cos(ϕm x ), σy = ∑m = 1 Ymi (yi ) sin(ϕm x ) − ti

Substituting ui and express σ~xi as

σ~xi = −

(34)

2 L

∫0

L

ti sin(ϕm x ) dx , qm =

2 L

∫0

L

Q (x ) sin(ϕm x ) dx

(35)

Applying Eq. (25) to Eq. (14) and using the first of Eq. (34), we can rewrite the compatibility conditions as

Ri + 1 (0) = Ri (hi ) + Ri, i + 1

where ϕm = mπ / L . The displacement and stress solutions are assumed to be

σyi

m=1

where

Expanding the delta functions in Eq. (22) gives

2 δ (x ) = L

(33)

3.3. Displacement and stress in the laminated beam

where

Ei

κ1 κ12

Substituting Eq. (33) back into Eq. (31) obtains the matrix exponential exp(Si yi ) , and then Ai (yi ) and Bi (yi ) can be calculated from Eq. (30). Finally the vector Ri (yi ) is obtained from Eq. (29). (22)

S1i = −μi , S2i = Ei, S3i =

(32)

Eq. (32) shows that the matrix Si has repeated eigenvalues, in which case cki (yi ) are given by

(21)

2(1 + μi ) ∂ , S4i = ,β= Ei ∂x

(31)

cki (yi )

Ei Ei Ei i = = (εxi + μi ε yi ), σyi = (ε yi + μi εxi ) − ti, τxy γi 2(1 + μi ) xy 1 − μi2 1 − μi2

1 − μi2

(30)

In Eq. (30), I is a 4 × 4 identity matrix. To analytically calculate Ai (yi ) and Bi (yi ) , we can expand the matrix exponential exp(Si yi ) into the following polynomial based on the Cayley-Hamilton theorem:

(19)

In Eq. (19), δ (x ) ti − δ (x − L) ti can be treated as a longitudinal volume force, then the longitudinal normal stress is σ~xi . Hence Eq. (11) can be rewritten as

σ~xi

(29)

where

where δ(x) and δ(x − L) are the delta functions defined as

δ (x ) = −

(27)

(36)

where T

Ri, i + 1 = { 0 tmi+ 1 − tmi 0 0 } (25)

(37)

Applying Eq. (25) to Eq. (15) and using Eq. (34), we can rewrite the top and bottom surface conditions as

in Eq. (25) into the first of Eq. (21), one can

Xm1 (0) = 0, Ym1 (0) = tm1 , Xmp ̂ (h p )̂ = 0, Ymp ̂ (h p )̂ = tmp ̂ − qm

(38)

∞

∑ m=1

[S2i ϕm Umi (yi ) + S1i Ymi (yi )] sin(ϕm x )

Taking yi = hi and yi + 1 = hi + 1 into Eq. (29), we have (26)

Ri (hi ) = Ai (hi ) Ri (0) + Bi (hi ), Ri + 1 (hi + 1)

Accordingly, applying Eq. (26) to Eq. (17) obtains the stress σxi . It can be observed that the solutions of vi and σxi exactly meet Eq. (13). Introducing Eqs. (24) and (25) into Eq. (22) yields

= Ai + 1 (hi + 1) Ri + 1 (0) + Bi + 1 (hi + 1)

(39)

Combining Eqs. (36) and (39) one can obtain the relation between 4

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supported one by adding the unknown longitudinal surface forces, which can be finally determined by the zero displacement condition at the end [31].

Table 1 Temperature-dependent Young’s moduli of the carbon steel and concrete [32,33]. Temperature

Es / Es0

Ec / Ec0

20 100 150 200 300 400 500 600 700 800 900 1000 1100 1200

1.000 1.000 – 0.900 0.800 0.700 0.600 0.310 0.130 0.090 0.0675 0.0450 0.0225 0

1.000 1.000 1.000 – – – – – 0 – – – – –

4. Convergence and comparison study To demonstrate the procedure we take a two-layer thick beam as an example, of which the geometric parameters are L = 2 m , H1 = 0.1 m , and H2 = 0.4 m . The upper layer are made of concrete while the lower layer carbon steel. The temperature-dependent properties of the two materials are [32,33] The carbon steel:

ks = 54 − 3.33 × 10−2T W/(m°C), 20° C≤ T < 800°C, αs = 1.208 × 10−5 + 4 × 10−9T °C−1, 20 ° C≤ T < 750 °C, μs = 0.3

(46)

The concrete: Note: Es and Ec denote the temperature-dependent Young’s moduli of the carbon steel and concrete, respectively; Es0 = 210 GPa and Ec0 = 30 GPa is the Young’s moduli of the carbon steel and concrete at T0 = 20 °C , respectively. Linear interpolation can be employed to obtain the Young’s moduli of the two materials at arbitrary temperature.

kc = 1.36 − 1.36 × 10−3T + 5.7 × 10−7T 2 W/(m°C), 20° C≤ T ≤ 1200°C, α c = 6.0056 × 10−6 + 2.8 × 10−10T + 1.4 × 10−11T 2 °C−1, 20 ° C≤ T ≤ 805 °C, μc = 0.2

contiguous sublayers as

(47) Table 1 shows the relations between temperature and Young’s moduli for the two materials. The top and bottom surface temperatures steadily rise to Tt = 200 °C and Tb = 150 °C from T0 = 20 °C , respectively. The applied load is Q(x) = 5000 N/m.

Ri + 1 (hi + 1) = Ai + 1 (hi + 1) Ai (hi ) Ri (0) + Ai + 1 (hi + 1)[Bi (hi ) + Ri, i + 1] + Bi + 1 (hi + 1) (40) By analogy, the relation between the ith sublayer and the bottom sublayer is

4.1. Convergence study

Ri (hi ) i−1

k+1

⎧ ⎫ = ∑ ∏ [Aj (hj )][Bk (hk ) + Rk, k + 1] + ⎨ j=i ⎬ k=1 ⎩ ⎭ (i = 1, 2, … p)̂ i−2

Ri (0) =

∏ [Ak (hk )] R1 (0) + Bi (hi),

k+1

∑⎧∏

[Aj (hj )][Bk (hk ) + Rk, k + 1]

⎨ j=i−1 ⎩ ^) + Bi − 1 (hi − 1) + Ri − 1, i , (i = 2, 3, …p k=1

We first study the convergence of the temperature solution by gradually increasing the sublayer number q. Table 2 lists the temperature solutions at five different point. The data indicate that the present temperature solutions converge quickly as q increases. The numerical results rounded off to three decimal places keep invariable when q ≥ 50. Even only using ten sublayers still guarantees sufficient accuracy. Then we study the convergence of the displacement and stress solutions by truncating the series terms m up to M, i.e. m = 1, 2, …, M. Table 3 gives the displacements ul , vm and stress σm for different sublayer numbers q and different truncated terms M, in which ul denotes u at x = 0, y = 0.5H, vm denotes v at x = 0.5L, y = 0.5H, and σm denotes σx at x = 0.5L, y = 0.5H. The data show a good convergence of the present solution as q and M increase. The displacements and stresses for M = 45 and q = 100 can guarantee satisfied accuracy, hence the infinite series terms and the sublayer numbers are, respectively, fixed at M = 45 and q = 100 in the following computations unless stated.

1 k=i

(41)

⎫ + ⎬ ⎭

1

∏

[Ak (hk )] R1 (0)

k=i−1

(42)

Substituting Eq. (38) into Eq. (41) yields p̂ {Um (h p )̂

T T tmp ̂ − qm 0 Vmp ̂ (h p )̂ } = C {Um1 (0) tm1 0 Vm1 (0) } + C¯

(43)

where

⎡ C11 ⎢C21 C=⎢ C ⎢ 31 ⎣C41

C12 C22 C32 C42 T

C¯ = {C¯1 C¯2 C¯3 C¯ 4 } =

C13 C23 C33 C43

p ̂− 1 ∑i = 1

C14 ⎤ C24 ⎥ 1 = ∏i = p ̂ Ai (hi ), C34 ⎥ ⎥ C44 ⎦

{

i+1 ∏ j = p ̂ [Aj (hj )][Bi (hi )

4.2. Comparison study

+ Ri, i + 1]

+ B p ̂ (h p )̂ Eliminating

}

As is generally known, the finite element (FE) method can provide reliable results with high computational cost. The accuracy of the FE solution can be verified by comparison with the present solution.

(44)

Ump ̂ (h p )̂

and

Vmp ̂ (h p )̂

from Eq. (43), one has

1 p̂ 1 ⎧Um (0) ⎫ = ⎡C21 C24 ⎤ ⎧tm − qm − C¯2 − tm C22 ⎫ 1 C C ⎥ ¯ ⎨Vm1 (0) ⎬ ⎢ ⎨ ⎬ 31 34 − C3 − tm C32 ⎦ ⎩ ⎩ ⎭ ⎣ ⎭

Table 2 Convergence study of the temperature (°C).

−1

(45)

Sublayer number

Solving Eq. (45) obtains R1 (0) ; substituting R1 (0) into Eq. (42) ob̂ substituting Ri (0) into Eq. (29) obtains tains Ri (0) (i = 1, 2, … , p ); Ri (yi ) . Finally, the displacements and stresses are solved by applying Ri (yi ) to Eqs. (17), (25), and (26). The present analysis focuses on beams with simply-supported ends. The analysis can also be developed to deal with other support conditions. For example, the clamped end can be transformed into a simply-

q=1 q=5 q = 10 q = 50 q = 100

5

Position y = 0.1H

y = 0.3H

y = 0.5H

y = 0.7H

y = 0.9H

150.152 150.144 150.144 150.144 150.144

156.516 156.378 156.369 156.367 156.367

168.940 168.642 168.639 168.637 168.637

181.364 181.068 181.065 181.062 181.062

193.788 193.659 193.649 193.647 193.647

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Table 3 Convergence study of the displacement and stress components. Solution

Sublayer number

Truncated term number M=5

M = 15

M = 25

M = 35

M = 45

ul (mm)

q=1 q=5 q = 10 q = 50 q = 100 q = 300

−1.063 −1.076 −1.076 −1.077 −1.077 −1.077

−1.109 −1.120 −1.120 −1.120 −1.120 −1.120

−1.121 −1.131 −1.131 −1.132 −1.132 −1.132

−1.126 −1.136 −1.136 −1.136 −1.137 −1.137

−1.129 −1.139 −1.139 −1.139 −1.139 −1.139

vm (mm)

q=1 q=5 q = 10 q = 50 q = 100 q = 300

−0.907 −0.817 −0.813 −0.812 −0.812 −0.812

−0.908 −0.818 −0.815 −0.814 −0.814 −0.814

−0.908 −0.818 −0.815 −0.814 −0.814 −0.814

−0.908 −0.818 −0.815 −0.814 −0.814 −0.814

−0.908 −0.818 −0.815 −0.814 −0.814 −0.814

σm (MPa)

q=1 q=5 q = 10 q = 50 q = 100 q = 300

5.390 8.317 7.731 7.495 7.435 7.435

5.887 8.664 8.150 7.944 7.891 7.891

5.912 8.733 8.190 7.970 7.914 7.914

5.908 8.703 8.176 7.964 7.910 7.910

5.908 8.719 8.183 7.966 7.911 7.911

Fig. 5. Comparison of the mid-span deflection among the present solution vpresent , the 1-D Euler-Bernoulli solution v1 - D , and the FE solution v FE for the FG beam.

as the ratio L/H decreases. Table 4 gives the comparative results of the temperatures, displacements and stresses at five different points for the before-mentioned twolayer beam. For x = 0.25L, the values of all the displacement and stress components are non-zero. The FE solution is obtained using the software ABAQUS. The data indicate that the FE solution matches well with the present solution and the relative errors are all less than 2%. In the present model, variable temperature along the thickness direction leads to variable material properties which are the distinctive feature of FG materials [27]. Therefore, the present elasticity solution can be used to verify the accuracy of the solution of FG beams based on the classical theory. Here, the deflection of a simply-supported FG beam is calculated based on the CBT [27] and the present elasticity theory. Some parameters of the FG beam are [27]

5. Numerical examples Consider a sandwich beam with the top and bottom layers made of the carbon steel and the core made of the concrete. Some parameters are L = 2 m, H = 0.2 m (H1=0.04 m, H2 =0.14 m, and H3 = 0.02 m), Q (x) = 5 kN/m, T0 =20 °C, and Tb = 50 °C, while the top surface temperature Tt is variable. In the following analysis, the temperature, displacement, and stress solutions are obtained on the basis of two states: (i) the material properties are in temperature-dependent (TD) state; (ii) the material properties are in temperature-independent (TID) state, i.e. they are constants and taken as those at T0 = 20 °C. 5.1. Temperature field

E (y ) = Eb exp(λE y ), α (y ) = αb exp(λ α y ), ΔT (y ) = ΔTb exp(λT y ) Eb = 1 GPa, αb = 1 × 10−4 °C−1, ΔTb = 100 °C, λT = λE = 230 m−1, λ α = 0, H = 0.01 m

The temperature field is studied. Fig. 6 plots the temperature distribution along the thickness direction for the sandwich beam subjected to three different top surface temperatures Tt = 100 °C, 200 °C, and 300 °C. It can be observed that the temperature distribution in every layer is linear in the TID state but is nonlinear in the TD state, since the thermal conductivity is constant in the TID state but is variable with temperature in the TD state. The temperature T almost has no change in the carbon steel layers compared with that in the concrete layer, since the thermal conductivity of the carbon steel is much bigger than that of the concrete. Fig. 7 plots the temperature T at y = 0.5H versus the top surface temperature Tt . It is seen that with the increase of Tt , the temperature T is linearly increased in the TID state but is nonlinearly increased in the TD state; the discrepancy between the two states is increased as Tt rises.

(48)

where λT , λE , and λ α are constants denote the gradation of the temperature, Young’s modulus, and Poisson’s ratio, respectively; the symbol b denotes the quantities on the bottom surface. The length L is variable. For the plane stress analysis, the Poisson’s ratio μ is not required in the CBT but is essential in the present elasticity theory, hence we take μ = 0.3 in this example. Fig. 5 shows the comparison of the mid-span deflection (v at x = 0.5L, y = 0.5H) among the present solution vpresent , the 1-D solution v1 - D [27] based on the CBT, and the FE solution v FE . It is found that the present solution agrees well with the FE solution; the present solution and the 1-D solution show a good consistency for slender beams, however the discrepancy enlarges gradually Table 4 Comparison study between the present solution and the FE solution at x = 0.25L. Position

Method

T (°C)

u (mm)

v (mm)

σx (MPa)

σy (MPa)

τxy (MPa)

y = 0.1H

Present FE Present FE Present FE Present FE Present FE

150.144 150.144 156.367 156.366 168.637 168.636 181.062 181.063 193.647 193.648

−0.783 −0.783 −0.698 −0.698 −0.613 −0.613 −0.531 −0.530 −0.448 −0.448

−0.814 −0.815 −0.691 −0.692 −0.603 −0.603 −0.501 −0.501 −0.388 −0.388

−16.248 −16.237 15.186 15.192 7.020 7.023 0.216 0.212 −6.188 −6.201

0.355 0.353 1.213 1.208 1.340 1.343 0.808 0.810 0.124 0.126

0.999 0.983 0.671 0.670 −0.206 −0.210 −0.840 −0.835 −0.505 −0.502

y = 0.3H y = 0.5H y = 0.7H y = 0.9H

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Fig. 6. Temperature distribution along the thickness direction for the sandwich beam subjected to different top surface temperatures.

Fig. 7. Temperature at y = 0.5H versus the top surface temperature for the sandwich beam.

This can be attributed to the growing difference of the thermal conductivities between the two states with increasing temperature. 5.2. Displacement and stress fields To clearly show the effects of high temperature on the mechanical behavior of beams, the responses of the sandwich beam are divided into two parts: part T explains the responses of the beam only subjected to the thermal load, which can be achieved by setting the mechanical load Q(x) to be zero; part Q explains the responses of beam only subjected to the mechanical load (but the effects of temperature on the material properties are still considered), which can be achieved by setting the thermal expansion coefficients αs and α c to be zero. Thus for these two parts the displacements and stresses are expressed by T Q Q Q Q Q {u v σx σy τxy } = {uT vT σxT σyT τxy } + {u v σx σy τxy }

Fig. 8. Distributions of displacement and stress components along the thickness direction for the sandwich beam subjected to the thermal load: (a) displacement uT at x = 0; (b) displacement vT at x = 0.5L; (c) stress σxT at x = 0.5L.

and (b) that the deformation in the TD state is bigger than that in the TID state when Tt is fixed, since the stiffness of the beam is decreased with the increase of temperature when considering the temperature dependence of material properties. It is found from Fig. 8(c) that σxT has a sudden change at the interface between contiguous layers, since the material properties of contiguous layers are different. In the TID state σxT is increased with the increase of Tt ; however, in the TD state σxT may

(49)

T Q Q Q Q Q where {uT vT σxT σyT τxy } and {u v σx σy τxy } are the displacement and stress components of parts T and Q, respectively. Firstly the displacements and stresses of part T are studied. Fig. 8 shows the distributions of uT at x = 0, vT at x = 0.5L, and σxT at x = 0.5L for the sandwich beam subjected to different top surface temperatures Tt = 100 °C, 200 °C, and 300 °C. It is seen from Fig. 8(a)

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Fig. 9. Displacement and stress components at x = 0.5L, y = 0.9H− versus the top surface temperature for the sandwich beam subjected to the thermal load: (a) displacement vT ; (b) stress σxT .

not be monotonously increased with the increase of Tt (see σxT at x = 0.5L, y = 0.9H−, where the superscript – denotes the lower surface of the interface between the top and middle layers). To further show the relation we plots the curves of vT and σxT at x = 0.5L, y = 0.9H− with respect to Tt for the TD and TID states, as shown in Fig. 9. It is seen that in the TID state, vT and σxT are linearly increased as Tt rises; in the TD state, vT is nonlinearly increased as Tt rises, while σxT is first increased and then decreased. The discrepancies of vT and σxT between the two states are both increased as Tt rises. Then the displacements and stresses of part Q are studied. Fig. 10 Q at x = 0 plots the distributions of vQ at x = 0.5L, σxQ at x = 0.5L, and τxy for the sandwich beam. Fig. 10(a) shows that in the TD state, the deformation of the beam is increased with the increase of Tt , since the stiffness of the beam is decreased as temperature rises. Fig. 10(b) and Q are (c) indicate that in the TD state, the distributions of σxQ and τxy almost invariable when Tt changes. It is noticed from Fig. 10(a), (b), and Q in the TID state are identical (c) that the distributions of vQ , σxQ , and τxy to those in the TD state at Tt = 100 °C. This can be attributed to the fact that the Young’s moduli and Poisson’s ratios of the carbon steel and concrete are invariable when temperature is not more than 100 °C (see Table 1 and Eqs. (46) and (47)). Combining Figs. 8 and 10 it can be observed that the displacements and stresses induced by the thermal load are much greater than those induced by the mechanical load.

Fig. 10. Distributions of displacement and stress components along the thickness direction for the sandwich beam subjected to the mechanical load, considering the effects of temperature on the material properties: (a) displacement Q vQ at x = 0.5L; (b) stress σxQ at x = 0.5L; (c) stress τxy at x = 0.

6. Conclusions Temperature, displacement and stress analysis of simply-supported laminated beams with temperature-dependent material properties subjected to thermal and mechanical loads is carried out. The temperature field is obtained from Fourier’s law by an iterative algorithm,

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and the displacement and stress solutions are obtained from the 2-D thermo-elasticity theory using the transfer-matrix method. The effects of the surface temperature on the temperature, displacement and stress fields in a sandwich beam are studied. Some important findings are listed as:

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1) The results obtained from the present 2-D elasticity theory and the 1-D classical beam theory show a good consistency for thin beams; however, the discrepancy is significant for thick beams. 2) The temperature is linearly distributed in every layer of the beam when the material properties are temperature-independent; however, it is nonlinearly distributed when the material properties are temperature-dependent. 3) The temperature, displacements and stresses linearly vary with the surface temperature when the material properties are temperatureindependent; however, nonlinearly vary when the material properties are temperature-dependent. The discrepancy between the temperature-dependent state and temperature-independent state is increased as the surface temperature rises. 4) The mechanical responses of the beam induced by thermal load and mechanical load are studied separately. The results reveal that the effects of temperature on the beam are mainly divided into two aspects: (i) producing deformations and stresses itself; (ii) affecting the deformations and stresses induced by the mechanical load. 5) Compared with the stresses induced by mechanical load, the deformations induced by the same load are more affected by the temperature. 7. Data availability The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is financially supported by the National Key Basic Research Program of China (Grant No. 2012CB026205), the National Natural Science Foundation of China (Grant No. 51608264; 51778288), and the Transportation Science and Technology Project of Jiangsu Province (Grant No. 2014Y01). Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.compstruct.2019.111304. References [1] Yan JB, Liew JYR, Zhang MH, Sohel KMA. Experimental and analytical study on ultimate strength behavior of steel–concrete–steel sandwich composite beam structures. Mater Struct 2015;48:1523–44. https://doi.org/10.1617/s11527-0140252-4. [2] Zhang L, Liu W, Wang L, Ling Z. Mechanical behavior and damage monitoring of pultruded wood-cored GFRP sandwich components. Compos Struct 2019;215:502–20. https://doi.org/10.1016/j.compstruct.2019.02.084. [3] Triantafyllou GG, Rousakis TC, Karabinis AI. Corroded RC beams patch repaired and strengthened in flexure with fiber-reinforced polymer laminates. Compos Part B Eng 2017;112:125–36. https://doi.org/10.1016/j.compositesb.2016.12.032. [4] Aydogdu M. A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Phys E 2009;41:1651–5. https://doi.org/10.1016/j.physe. 2009.05.014. [5] Hetnarski RB, Eslami MR. Thermal stresses – advanced theory and applications.

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