ADAPTIVE MESH REFINEMENT ANALYSIS OF FINITE ELEMENT METHOD FOR ELASTIC BUCKLING OF CRACKED CIRCULARLY CURVED BEAMS
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摘要: 该文建立圆弧形曲梁裂纹的截面损伤缺陷比拟方案，实现裂纹大小(深度)、位置、数目的模拟。引入变截面Euler-Bernoulli梁的h型有限元网格自适应分析方法，求解含裂纹损伤圆弧曲梁弹性屈曲问题，得到优化的网格和满足预设误差限的高精度屈曲荷载和屈曲模态解答。数值算例表明该算法中网格非均匀加密可适应裂纹损伤引起的屈曲模态变化，应用于各类曲梁夹角和裂纹损伤分布工况下的弹性屈曲研究，定量分析了裂纹损伤程度对圆弧曲梁的屈曲荷载和屈曲模态的影响，检验了该文算法的精确性和可靠性。Abstract: The scheme for the cross-section damage defects in circularly curved beam is established to simulate the magnitude (depth), location and number of the cracks. The h-version adaptive finite element method for non-uniform Euler-Bernoulli beam is introduced to solve the elastic buckling of circularly curved beam with cracks. Using the proposed method, the final optimized meshes and high-precision buckling loads and modes meeting the preset error tolerance can be obtained. Numerical examples show that the non-uniform mesh refinement can adapt to the change of buckling mode induced by crack damage, which is applied to the elastic buckling analysis for some typical kinds of subtended angles and crack damage distribution conditions of circularly curved beams. Furthermore, the influence of crack damage on the buckling load and mode of circular curved beams is quantitatively analyzed, and the accuracy and reliability of the proposed algorithm are verified.
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图 2 含裂纹损伤圆弧曲梁截面损伤和加载示意图
Figure 2. Diagram of cross-section damage defect and loading for circularly curved beam with crack damage
图 5 不同裂纹损伤位置下圆弧曲梁弹性屈曲
Figure 5. Elastic buckling of circularly curved beam with different locations of crack damage
图 6 不同裂纹损伤大小下圆弧曲梁弹性屈曲
Figure 6. Elastic buckling of circularly curved beam with different magnitudes of crack damage
图 8 多裂纹损伤下网格分布和屈曲模态差
Figure 8. Mesh distribution and buckling modes of circularly curved beam with multiple cracks
表 1 圆弧曲梁根据径厚比和夹角分类
Table 1. Categories of circularly curved beams according to ratio of radius and thickness and subtended angle
按径厚比$R/h$ 厚曲梁 中厚曲梁 薄曲梁 $R/h < 40$ $R/h = 40$ $R/h > 40$ 按曲梁夹角θ0 浅梁 中等深度梁 深度梁 超深度梁 $\theta _0^{} < 40_{}^ \circ $ $\theta _0^{} = 40_{}^ \circ $ $40_{}^ \circ < \theta _0^{} \leqslant 180_{}^ \circ $ $\theta _0^{} > 180_{}^ \circ $ 
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表 3 两端简支曲梁不同夹角下屈曲荷载值
Table 3. Buckling loads of of curved beam with hinged–hinged supports under different subtended angles
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表 4 不同裂纹损伤位置下曲梁屈曲荷载值
Table 4. Buckling loads of curved beam with crack damage at different locations
损伤位置$\theta _{\rm{c}}^{}$ 屈曲荷载$\bar q_{\rm{c}}^{}$ 损伤位置$\theta _{\rm{c}}^{}$ 屈曲荷载$\bar q_{\rm{c}}^{}$ 0 0.999 999 π/3 0.937 817 π/12 0.994 679 5π/12 0.924 499 π/6 0.978 168 π/2 0.919 641 π/4 0.958 574 − −
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表 5 不同裂纹损伤大小下曲梁屈曲荷载值
Table 5. Buckling loads of curved beam with crack under different magnitudes
损伤大小$h_{\rm{c}}^{}/h$ 屈曲荷载$\bar q_{\rm{c}}^{}$ 0.1 0.995 226 0.2 0.988 219 0.3 0.976 756 0.4 0.957 391 0.5 0.919 641
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表 6 多裂纹损伤不同位置下曲梁屈曲荷载值
Table 6. Buckling loads of curved beam with multiple cracks at different locations
工况 第1条裂纹位置 第2条裂纹位置 第3条裂纹位置 屈曲荷载$\bar q_{\rm{c}}^{}$ I π/4 (I1) π/2 (I2) 3π/4 (I3) 0.856 826 II π/4 (II1) π/2 (II2) π/12 (II3) 0.881 116 III 5π/12 (III1) π/2 (III2) 7π/12 (III3) 0.800 923
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