Elastic Solutions for Orthotropic Laminated Beams Under Temperature Variations and Loads
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摘要: 由多种工程材料组成的复合材料层合结构因其卓越的可设计性和优良的力学性能，在建筑工程、航天航空、汽车工业等领域有着广泛的应用. 该文研究了温度环境中正交各向异性固支层合梁的热力学行为，并基于热弹性理论推导出了热应力和位移的精确解. 该方法适用于荷载和变温环境共同作用下，任意厚度和层数的正交各向异性固支层合梁. 首先，引入单位脉冲函数和Dirac函数，将固支边界等效为简支边界和一组纵向边界反力. 其次，以位移和应力作为状态变量，并联合基本方程建立状态空间方程，利用Fourier级数将状态空间方程进行简化. 然后，根据相邻层界面处位移和应力的连续性关系，推导出层合梁顶层和底层之间的位移和应力关系. 最后，利用层合结构上下表面应力和位移的边界条件，最终确定正交各向异性固支层合梁内任意一点的位移和应力. 收敛性和对比分析表明了该方法的有效性和准确性. 同时，探究了温度环境和长厚比对正交各向异性固支层合梁内位移和应力分布的影响.Abstract: Composite laminated structures, composed of multiple engineering materials, have wide-range applications in building engineering, aerospace, automobile industry and other fields, owing to the exceptional designability and mechanical performance. The thermodynamic behaviors of clamped orthotropic laminated beams were investigated in temperature variations and the exact solutions for thermal stresses and displacements were derived based on the theory of thermoelasticity. This method is applicable to laminated beams with arbitrary thickness and number of layers, under both loads and variable temperatures. Firstly, the clamped support was equivalently transformed into a simple support boundary and a pair of transverse boundary reaction forces through introduction of the unit impulse function and the Dirac function. Additionally, the state equation was formulated with the displacement and stress as state variables and the fundamental equations combined. The Fourier series was employed to simplify the state space equation. The relationships of displacements and stresses between the top and bottom layers of the laminated beam were sequentially derived based on the continuities of displacements and stresses at the interfaces of the adjacent layers. Ultimately, the displacement and stress at any point in the orthotropic laminated beam were determined by means of the stress and displacement boundary conditions at the upper and lower surfaces of the structure. Convergence and comparison analyses demonstrate the effectiveness and accuracy of the proposed method. Furthermore, the effects of the temperature and the length-to-thickness ratio on the distributions of displacements and stresses in the orthotropic laminated beams, were discussed in detail.
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图 1 正交各向异性固支层合梁结构示意图
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 1. Diagram of the clamped orthotropic laminated beam
图 2 固支边界的等效几何模型
Figure 2. The equivalent geometric model for the laminated beam with clamped supports
图 4 在不同温度环境中，层合梁在x=0.8 m处，u, v, σx和τxy沿厚度方向的分布
Figure 4. Distributions of u, v, σx and τxy along the thickness direction at x=0.8 m for the laminated beam in different temperature environments
图 5 温度环境ΔT=40 ℃时，层间应力σx在y=0.2 m和0.6 m处沿x方向的分布
Figure 5. Distributions of σx for ΔT=40 ℃ at y=0.2 m and 0.6 m along the x direction
图 6 不同长厚比的层合梁在x=L/10处，u, v, σx和τxy沿厚度方向的分布
Figure 6. Distributions of u, v, σx and τxy along the thickness direction at x=L/10 for the laminated beam with different length-to-thickness ratios
material E1/GPa E2/GPa G12/GPa μ12 α1/(10-5 ℃-1) α2/(10-5 ℃-1) steel 200 200 76.9 0.30 1.2 1.2 concrete 30 30 12.5 0.20 0.7 0.7 timber 12.7 0.98 0.8 0.37 0.31 2.36 
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表 2 位移和应力收敛性分析
Table 2. Numerical solutions of displacements and stresses
n1/n2/n3 M x=2 m, y=0.05 m x=6 m, y=0.3 m u/mm v/mm τxy/MPa σx/MPa u/mm v/mm τxy/MPa σx/MPa 3/8/3 15 0.002 16 -0.624 -0.030 1 -219 0.040 5 -0.113 0.005 29 -3.70 20 0.002 21 -0.621 -0.031 3 -212 0.038 8 -0.112 0.005 15 -4.09 25 0.002 22 -0.622 -0.033 3 -218 0.037 7 -0.105 0.005 16 -3.95 30 0.002 22 -0.622 -0.033 3 -218 0.037 7 -0.105 0.005 15 -3.94 35 0.002 22 -0.621 -0.033 3 -216 0.037 7 -0.105 0.005 15 -3.95 4/10/4 15 0.002 18 -0.623 -0.030 7 -219 0.040 9 -0.106 0.005 25 -3.70 20 0.001 99 -0.620 -0.031 7 -212 0.039 3 -0.103 0.005 66 -4.03 25 0.002 12 -0.622 -0.033 3 -218 0.037 7 -0.105 0.005 16 -3.95 30 0.002 22 -0.622 -0.033 3 -218 0.037 7 -0.105 0.005 15 -3.95 35 0.002 22 -0.622 -0.033 3 -218 0.037 7 -0.105 0.005 15 -3.95 5/12/5 15 0.002 18 -0.624 -0.030 3 -219 0.040 7 -0.108 0.005 62 -3.70 20 0.002 00 -0.620 -0.033 1 -212 0.039 1 -0.106 0.005 22 -4.19 25 0.002 22 -0.622 -0.033 3 -218 0.038 9 -0.105 0.005 15 -3.95 30 0.002 22 -0.622 -0.033 3 -218 0.037 7 -0.105 0.005 15 -3.95 35 0.002 22 -0.622 -0.033 3 -218 0.037 7 -0.105 0.005 15 -3.95
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表 3 本文方法与有限元解的比较
Table 3. Comparison of the present method with the FE solution
position method u/mm v/mm σx/MPa τxy/MPa x=1 m y=0 m present -0.006 90 -0.693 -209 0 FE -0.006 92 -0.688 -209 0 y=0.25 m present 0.034 70 -0.215 -4.13 0.011 0 FE 0.034 80 -0.215 -4.10 0.011 0 y=0.4 m present 0.037 30 0.112 -4.05 0.010 8 FE 0.037 80 0.113 -4.11 0.010 3 y=0.65 m present -0.035 80 0.620 -210 -0.906 0 FE -0.035 40 0.617 -214 -0.905 0
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