GREEN’S FUNCTION METHOD FOR STATIC AND DYNAMIC ANALYSIS OF FUNCTIONALLY GRADED BEAMS
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摘要: 功能梯度梁静动态响应的数值分析方法一般局限于有限元法，存在有限元法的固有缺点，有必要发展新的数值求解方法。将功能梯度梁静力分析的控制微分方程转化为与匀质材料梁静力分析控制微分方程相一致的形式，并利用匀质材料梁静力问题的格林函数，开展功能梯度梁的静力分析。在此基础上，进一步推导获得功能梯度梁的柔度矩阵，据此建立功能梯度梁的运动方程，开展功能梯度梁的动力特性分析和动力响应分析。数值算例表明，采用格林函数法可以高效准确地分析功能梯度梁的静力响应与动力行为，验证了方法的计算精度与效率。Abstract: The numerical analysis of static and dynamic problems of functionally graded material beams is usually conducted with the finite element method and suffers from the inherent deficiencies of the method. The governing equations for static analysis of functionally graded material beams are transformed to the forms for static analysis of isotropic material beams, and the Green’s functions corresponding to isotropic material beams are used for static analysis of functionally graded material beams. On this basis, the compliance matrix of a functionally graded material beam is further derived, and the equation of motion is established and solved for the dynamic problem of the functionally graded material beam. Numerical examples show that the proposed method can analyze the static and dynamic problems of functionally graded material beams with high efficiency and accuracy, indicating the feasibility of the present approach.
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图 1 与功能梯度梁等效的匀质梁
Figure 1. A uniform beam equivalently to the functionally graded beam
图 3 功能梯度梁跨中点C的轴向与横向位移
Figure 3. Axial and transverse displacement of node C at midspan of the functionally graded beam
图 4 功能梯度梁固定端点A的轴力和弯矩
Figure 4. Axial force and bending moment of node A at fixed end of the functionally graded beam
图 5 功能梯度梁前三阶横向自由振动固有频率
Figure 5. The first three natural frequencies of functionally graded beam’s transverse free vibration
图 7 功能梯度梁跨中点C横向位移、速度和加速度
Figure 7. Transverse displacement, velocity, acceleration of node C at midspan of the functionally graded beam
表 1 功能梯度梁跨中点C的轴向与横向位移
Table 1. Axial and transverse displacement of node C at midspan of the functionally graded beam
位移 方法 子段数n，离散单元数Ne 4 10 20 50 100 轴向位移/
(×10−5 m)解析解 − − 1.328 − − 有限元法 1.314 1.326 1.327 1.328 1.328 格林函数法
(d1=d2=0.2 m)1.349 1.331 1.329 1.328 1.328 格林函数法
(d1=d2=0.4 m)1.349 1.331 1.329 1.328 1.328 格林函数法
(d1=0.2 m, d2=0.2 m)1.349 1.331 1.329 1.328 1.328 横向位移/
(×10−3 m)解析解 − − −2.447 − − 有限元法 −2.403 −2.439 −2.445 −2.447 −2.447 格林函数法
(d1=d2=0.2 m)−2.489 −2.451 −2.448 −2.447 −2.447 格林函数法
(d1=d2=0.4 m)−2.489 −2.451 −2.448 −2.447 −2.447 格林函数法
(d1=0.2 m, d2=0.2 m)−2.489 −2.451 −2.448 −2.447 −2.447 
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表 2 功能梯度梁固定端点A的轴力与弯矩
Table 2. Axial force and bending moment of node A at fixed end of the functionally graded beam
内力 方法 子段数n，离散单元数Ne 4 10 20 50 100 轴力/kN 解析解 − − 3.166 − − 有限元法 3.181 3.168 3.166 3.166 3.166 格林函数法
(d1=d2=0.2 m)3.066 3.149 3.161 3.165 3.165 格林函数法
(d1=d2=0.4 m)3.066 3.149 3.161 3.165 3.165 格林函数法
(d1=0.2 m, d2=0.2 m)3.066 3.149 3.161 3.165 3.165 弯矩/(kN·m) 解析解 − − −0.625 − − 有限元法 −0.634 −0.627 −0.626 −0.625 −0.625 格林函数法
(d1=d2=0.2 m)−0.677 −0.634 −0.628 −0.626 −0.625 格林函数法
(d1=d2=0.4 m)−0.677 −0.634 −0.628 −0.626 −0.625 格林函数法
(d1=0.2 m, d2=0.2 m)−0.677 −0.634 −0.628 −0.626 −0.625
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表 3 功能梯度梁前三阶横向自由振动固有频率
Table 3. The first three natural frequencies of functionally graded beam’s transverse free vibration
频率 方法 子段数n，离散单元数Ne 10 20 30 40 50 200 第一阶频率/
(×103 rad/s)有限元法 1.320 1.323 1.324 1.324 1.324 1.324 格林函数法
(m=4)1.322 1.323 1.323 1.323 1.323 1.323 格林函数法
(m=6)1.323 1.324 1.324 1.324 1.324 1.324 格林函数法
(m=8)1.323 1.324 1.324 1.324 1.324 1.324 格林函数法
(m=10)1.322 1.324 1.324 1.324 1.324 1.324 第二阶频率/
(×103 rad/s)有限元法 4.275 4.307 4.312 4.314 4.315 4.317 格林函数法
(m=4)4.298 4.303 4.303 4.304 4.304 4.304 格林函数法
(m=6)4.308 4.314 4.315 4.315 4.316 4.316 格林函数法
(m=8)4.308 4.315 4.316 4.317 4.317 4.317 格林函数法
(m=10)4.309 4.315 4.316 4.317 4.317 4.317 第三阶频率/
(×103 rad/s)有限元法 8.853 8.992 9.018 9.027 9.031 9.038 格林函数法
(m=4)8.885 8.897 8.898 8.899 8.899 8.899 格林函数法
(m=6)8.989 8.995 8.994 8.994 8.994 8.993 格林函数法
(m=8)9.011 9.032 9.034 9.035 9.035 9.036 格林函数法
(m=10)9.010 9.033 9.036 9.037 9.037 9.038
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