ELASTIC BUCKLING OF THIN PLATES WITH RECTANGULAR HOLES UNDER COMPRESSION, BENDING, SHEAR AND COMBINED BENDING SHEAR
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摘要: 该文分析、开发、验证和总结了单个或多个矩形孔对四边简支板在弯曲、压缩、剪切单独受力以及弯剪复合受力时的临界弹性屈曲应力的影响以及其近似闭合形式表达式。表达式的形式基于经典板稳定理论近似，并通过采用ABAQUS壳体有限元的参数研究得到开发和验证。表达式可作为壳体有限元特征屈曲分析的一种方便的替代方法。有限元参数研究表明，孔会产生独特的屈曲模式，孔间距改变会减少或增加板的临界弹性屈曲应力，且间隔足够大时多孔板屈曲应力可用单孔板计算公式代替。经验证的闭合形式表达式及其相关限制旨在足够通用，以适应工程实践中常见的孔尺寸和间距范围。Abstract: Closed-form expressions considering the influence of single or multiple rectangular holes on the critical elastic buckling stress of plates under bending, compression, shearing and combined bending shear are developed, validated and summarized. The expression forms are based on classical plate stability approximations, and are developed and verified by parametric studies employing shell finite elements. The expressions serve as a convenient alternative to shell finite element eigen-buckling analysis. The finite element parameter studies show that holes may produce unique buckling modes, and the change of hole spacing may reduce or increase the critical elastic buckling stress of the plate. When the spacing between holes is long enough, the buckling stress equation of single-hole plate can be used for multiple-hole plate. The validated closed-form expressions and their associated limitations are intended to be versatile enough to accommodate the range of hole sizes and spacings commonly found in engineering practice.
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Key words:
- steel structure /
- web opening /
- elastic buckling /
- ABAQUS /
- prediction method /
- interaction
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图 4 单孔对均匀压缩四边简支板屈曲应力 σcr的影响(h=300)
Figure 4. Influence of a single hole on
σcr for uniformly compressed simply-supported plates on four sides (h=300) 
图 5 短均匀压缩四边简支板的屈曲模态(h0/h=0.6, l/l0=4)
Figure 5. Buckling modes for short uniformly compressed simply-supported plates on four sides (h0/h=0.6, l/l0=4)
图 6 开单孔长均匀压缩四边简支板的屈曲模态
Figure 6. Buckling modes for long uniformly compressed simply-supported plates on four sides with a single hole
图 7 多孔对长均匀受压简支板屈曲应力 σcr的影响
Figure 7. Influence of multiple holes on σcr for long uniformly compressed simply-supported plates
图 8 开多孔均匀压缩简支板的屈曲模态
Figure 8. Buckling modes for long uniformly compressed simply-supported plates with multiple holes
图 9 均匀压缩板的ABAQUS屈曲应力 σcr变化
Figure 9. ABAQUS σcr variations of uniformly compressed plates
图 10 净截面到总截面的弹性屈曲应力转换的截面图
注：σcr2,net为开孔段上方未加劲板边缘纵向应力
Figure 10. Diagram of a section of plate demonstrating the elastic bucking stress conversion from the net to the gross section
图 12 单孔对纯弯简支板屈曲应力 σcr的影响(h=300)
Figure 12. Influence of a single hole on σcr for bending simply-supported plates (h=300)
图 13 开单孔长纯弯四边简支板的屈曲模态
Figure 13. Buckling modes for long bending simply-supported plates on four sides with a single hole
图 14 多孔对长纯弯简支板屈曲应力 σcr的影响
Figure 14. Influence of multiple holes on σcr for long bending simply-supported plates
图 16 净截面到总截面的弹性屈曲应力转换的截面图
注：σcr2,net为开孔段上方未加劲板外侧边缘纵向应力； $\sigma_{{\rm{cr}}2, {\rm{net}}}' $为开孔段上方未加劲板内侧边缘纵向应力
Figure 16. Diagram of a section of plate demonstrating the elastic bucking stress conversion from the net to the gross section
图 18 纯弯板拟合公式与模型结果对比
Figure 18. Comparison of fitting equation with modeling results of bending plates
图 20 单孔对纯剪四边简支板屈曲应力σcr的影响(h=300)
Figure 20. Influence of a single hole on σcr for shear simply-supported plates on four sides (h=300)
图 21 开单孔长纯剪四边简支板的屈曲模态
Figure 21. Buckling modes for long shear simply-supported plates on four sides with a single hole
图 23 多孔对长纯剪简支板屈曲应力σcr的影响
Figure 23. Influence of multiple holes on σcr for long shear simply-supported plates
图 24 纯剪板拟合公式与模型结果对比
Figure 24. Comparison of fitting equation with modeling results of shear plates
图 26 TONG文献与模型的相关关系曲线对比
Figure 26. Comparison of correlation curve between TONG’s equation and modeling results
图 27 板宽对相关关系曲线的影响(h=300)
Figure 27. Influence of plate width on correlation curve (h=300)
图 28 板长对相关关系曲线的影响(h=300)
Figure 28. Influence of plate length on correlation curve (h=300)
图 29 多孔对长弯剪简支板屈曲应力σcr的影响
Figure 29. Influence of multiple holes on σcr for long combined bending and shear simply-supported plates
表 1 不同全局尺寸因子下的特征值
Table 1. Eigenvalues under different global dimension factors
全局尺寸因子/mm 纯压 纯弯 纯剪 2.5 1768.0 10 509 2386.9 5.0 1769.4 10 527 2390.3 10.0 1774.1 10 595 2403.2 15.0 1780.4 10 689 2423.7 20.0 1786.2 10 800 2444.3 
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表 2 PHAM文献公式与模型对比
Table 2. Comparison of PHAM's equation with modeling results
方孔参数 文献公式 文献模型 本文模型 l0/l kv/kss kv/kss 相对误差/(%) kv/kss 相对误差/(%) 0.00 1.000 0.9987 0.1 0.9987 0.1 0.10 0.875 0.8819 −0.8 0.8824 −0.8 0.20 0.660 0.6680 −1.2 0.6684 −1.3 0.30 0.469 0.4771 −1.7 0.4772 −1.7 0.40 0.326 0.3347 −2.6 0.3344 −2.5 0.50 0.225 0.2361 −4.7 0.2353 −4.4 0.60 0.166 0.1694 −2.0 0.1686 −1.5 0.70 0.120 0.1240 −3.2 0.1236 −2.9 0.80 0.080 0.0926 −13.6 0.0921 −13.1 注：模型与文献完全一致，方板宽度200 mm，厚度1.5 mm。
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表 3 弯曲应力和剪切应力局部屈曲相关关系拟合公式的指数系数βb
Table 3. Coefficient βb in the fitting equation of the local buckling correlation between bending stress and shear stress
开孔宽度
h0/mm指数系数βb l0=
15 mml0=
30 mml0=
45 mml0=
60 mml0=
75 mml0=
90 mml0=
105 mml0=
120 mml0=
135 mml0=
150 mml0=
165 mml0=
180 mml0=
195 mml0=
210 mm240 1.701 1.648 1.601 1.555 1.510 1.469 1.432 1.402 1.378 1.362 1.356 1.364 1.392 1.451 270 1.589 1.557 1.529 1.499 1.467 1.436 1.407 1.382 1.363 1.351 1.348 1.357 1.384 1.436 300 1.499 1.480 1.465 1.447 1.427 1.405 1.383 1.364 1.349 1.340 1.340 1.350 1.374 1.419 330 1.429 1.417 1.410 1.401 1.389 1.374 1.359 1.346 1.335 1.330 1.331 1.341 1.362 1.399 360 1.374 1.366 1.363 1.360 1.353 1.345 1.335 1.327 1.320 1.317 1.319 1.329 1.347 1.376 390 1.332 1.325 1.324 1.323 1.321 1.317 1.312 1.307 1.304 1.303 1.306 1.315 1.329 1.353 420 1.298 1.291 1.291 1.292 1.292 1.292 1.290 1.288 1.287 1.288 1.291 1.299 1.311 1.329 450 1.271 1.264 1.264 1.266 1.267 1.269 1.269 1.269 1.270 1.272 1.275 1.282 1.291 1.304 480 1.249 1.242 1.241 1.243 1.246 1.248 1.250 1.251 1.253 1.256 1.260 1.265 1.272 1.281
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表 4 弯曲应力和剪切应力局部屈曲相关关系拟合公式的指数βt
Table 4. Coefficient βt in the fitting equation of the local buckling correlation between bending stress and shear stress
开孔宽度
h0/ mm指数系数βb l0=
15 mml0=
30 mml0=
45 mml0=
60 mml0=
75 mml0=
90 mml0=
105 mml0=
120 mml0=
135 mml0=
150 mml0=
165 mml0=
180 mml0=
195 mml0=
210 mm240 1.882 1.859 1.849 1.848 1.854 1.863 1.876 1.893 1.912 1.934 1.953 1.963 1.949 1.890 270 1.839 1.819 1.812 1.814 1.821 1.832 1.845 1.859 1.873 1.884 1.888 1.878 1.842 1.763 300 1.803 1.785 1.778 1.779 1.785 1.793 1.804 1.813 1.819 1.820 1.812 1.787 1.737 1.648 330 1.773 1.755 1.746 1.744 1.747 1.752 1.757 1.760 1.760 1.752 1.733 1.699 1.642 1.549 360 1.746 1.728 1.718 1.712 1.711 1.711 1.710 1.708 1.700 1.685 1.659 1.619 1.558 1.467 390 1.721 1.703 1.691 1.683 1.677 1.672 1.666 1.657 1.644 1.623 1.592 1.548 1.486 1.399 420 1.697 1.680 1.666 1.654 1.645 1.635 1.624 1.611 1.592 1.567 1.533 1.487 1.426 1.344 450 1.674 1.657 1.642 1.628 1.615 1.601 1.586 1.568 1.546 1.517 1.481 1.435 1.376 1.299 480 1.652 1.635 1.619 1.603 1.587 1.570 1.552 1.531 1.505 1.475 1.437 1.391 1.335 1.263 注：拟合值可以有±0.05的调整区间。
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