DIRECT PREDICTION OF MAXIMUM DEFLECTION FOR PLASTICALLY DEFORMED STRUCTURES UNDER INTENSE DYNAMIC PULSE
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摘要: 经过多年的研究, 由中国学者提出和研发的膜力因子法和饱和分析方法已被证明是分析和预测冲击、爆炸等强动载荷作用下梁、板等结构件的塑性大变形行为的有力工具. 在这两套理论工具相结合所获得的一系列最新成果的基础上, 文章提出一种对梁和板在强脉冲作用下的最大挠度的直接预测方法. 考虑了膜力和弯矩相互作用的准确屈服条件, 同时假定位移场近似地按照与准静态破损机构相似的模态发生变化, 该方法直接从膜力因子的表达式出发, 依据外载作的功与塑性耗散相等的能量条件, 只需要求解初等方程就可以简单明晰地得到梁和板在矩形脉冲作用下的最大挠度, 极大地简化了数学推导. 与同时考虑准确屈服条件和瞬态响应阶段的完全解以及具有上下界的模态解相比, 这一方法能够同样准确但更简单地计入膜力对结构大变形承载能力的效应, 为工程设计提供比完全解更简明、比模态解更精准的梁和板最大塑性变形的估算公式; 再同改进的脉冲等效技术相结合, 这种直接预测方法有望进一步拓展到更复杂的结构件, 获得广泛的工程应用.Abstract: After years of research, the membrane factor method (MFM) and saturation analysis (SA) method proposed and developed by Chinese scholars have been proven to be effective powerful tools in analyzing and predicting the large plastic deformation behavior of structural members such as beams and plates under intense dynamic loading such as impact and explosion. Based on recent results obtained by the combination of these two sets of theoretical tools, this paper proposes a direct prediction of deflection (DPD) method to predict the maximum (saturated) deflection of beams and plates subjected to intense loading pulses. This method does not rely on the governing equations of the structure; rather, it only needs to establish elementary equations based on the balance of internal and external work, whilst the former can be directly integrated from the expressions of relevant membrane factors. While the interaction between bending moment and membrane force (i.e., exact yield locus) is considered, the predictions on the maximum deflection can be simply obtained by solving the elementary equations, thus greatly simplifying the mathematical derivation. Compared with the complete solution, which considers both the exact yield criterion and the transient response phase, as well as the upper and lower bounds resulted from modal solution, the proposed DPD method can more simply yet still accurately account for the effect of membrane force on the load-carrying capacity of the structure in large deformation. Consequently, this DPD method can provide a series of calculation formulae on the maximum plastic deflection of beams and plates, which are more concise than complete solutions, more accurate than modal solutions, and easier for the use in engineering design. Combined with a refined pulse equivalency technique, this DPD method is expected to be further extended to other structures under general pulse loading and achieve a wide range of engineering applications.
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图 2 弯矩与轴力的交互作用屈服面及与之关联的流动法则
Figure 2. Interactive yield locus between bending moment and axial force and its associated flow rule
图 3 边界为简支可移和简支不可移的圆板的膜力因子fn 随板中点的无量纲挠度的变化 , 同时也代表了圆板的承载能力在大挠度变形过程中的变化
Figure 3. Variation of membrane factor fn of circular plates (with movable simply supported and immovable simply supported boundaries) with the dimensionless central deflection; it also represents the variation of the load-carrying capacity of circular plates during large deformation
图 4 均布矩形脉冲载荷作用下的固支方板
Figure 4. Fully clamped square plate under uniformly distributed rectangular pulse loading
图 5 均布脉冲载荷下方板的固定变形模态
Figure 5. Time-independent deformation mode of a square plate under uniformly distributed pulse loading
图 6 载荷−挠度曲线及内外功平衡示意图
Figure 6. Load-deflection curve and a sketch on internal and external power balance
图 7 固支梁的无量纲最大挠度随载荷幅值的变化
Figure 7. Dimensionless maximum deflection varies with dimensionless pulse amplitude for fully clamped beams
图 8 简支梁的无量纲最大挠度随载荷幅值的变化
Figure 8. Dimensionless maximum deflection varies with dimensionless pulse amplitude for simply supported beams
图 9 固支方板的无量纲最大挠度随载荷幅值的变化
Figure 9. Dimensionless maximum deflection varies with dimensionless pulse amplitude for fully clamped square plates
表 1 分析梁和板塑性大变形的几类方法
Table 1. Comparison of several methods for analyzing large plastic deformation of beams and plates
Method Saturation analysis (SA) Membrane factor method (MFM) Finite element method (FEM) Material mode Rigid-perfectly plastic (R-PP) Rigid-perfectly plastic (R-PP) Elastic-perfectly plastic (E-PP) or Elastic-plastic Applicable objects Dynamic large deformation Quasi-static or dynamic large deformation Quasi-static or dynamic large deformation Deformation mode Modal deformation field Transient and modal deformation field Transient deformation field Yield surface Circumscribed and inscribed square limit surfaces Exact limit surface Exact limit surface Main results The upper and lower bounds of the maximum
deflection and its dependence on
external loading parametersMaximum deflection and its dependence on external loading parameters Maximum and final deflection, but each
set of loading parameters needs to be calculated
one by one
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表 2 梁和板的膜力因子一览
Table 2. Membrane factors for beams and plates
Structure and boundary conditions Membrane factor ${f_n}$ Reference Finite length beam, simply supported, axial immovable $\left\{ \begin{gathered} 1 + 4{\delta ^2}\;\;\;\;(\delta \leqslant 1/2) \\ 4\delta \;\;\;\;\;\;\;\;\;{\kern 1 pt} {\kern 1 pt} (\delta \geqslant 1/2) \\ \end{gathered} \right.$ Chen et al.[10] Finite length beam, fully clamped, axial immovable $\left\{ \begin{gathered} 1 + {\delta ^2}\;\;\;{\kern 1 pt} {\kern 1 pt} \;\;(\delta \leqslant 1) \\ 2\delta \;\;\;\;\;\;\;\;\;{\kern 1 pt} {\kern 1 pt} (\delta \geqslant 1) \\ \end{gathered} \right.$ Chen et al.[10] Infinite beam, local deformation $\left\{ \begin{gathered} 1 + {\delta ^2}\;\;\;{\kern 1 pt} {\kern 1 pt} \;\;(\delta \leqslant 1) \\ 2\delta \;\;\;\;\;\;\;\;\;{\kern 1 pt} {\kern 1 pt} (\delta \geqslant 1) \\ \end{gathered} \right.$ Yu et al.[6] Circular plate, simply supported, radially movable $\left\{ \begin{gathered} 1 + \frac{1}{3}{\delta ^2}\;\;\;{\kern 1 pt} {\kern 1 pt} \;\;(\delta \leqslant 1) \\ \delta + \frac{1}{{3\delta }}\;\;\;\;\;{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} (\delta \geqslant 1) \\ \end{gathered} \right.$ Calladine[11]
, Kondo et al.[12]Circular plate, simply supported, radially immovable $\left\{ \begin{gathered} 1 + \frac{4}{3}{\delta ^2}\;\;\;{\kern 1 pt} {\kern 1 pt} \;\;\;{\kern 1 pt} {\kern 1 pt} (\delta \leqslant 1/2) \\ 2\delta + \frac{1}{{6\delta }}\;\;\;\;\;{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} (\delta \geqslant 1/2) \\ \end{gathered} \right.$ Yu et al.[7] Circular plate, fully clamped $ \left\{ \begin{gathered} 1 + \frac{1}{2}{\delta ^2}\;\;\;{\kern 1 pt} {\kern 1 pt} \;\;(\delta \leqslant 1) \\ \delta + \frac{1}{{2\delta }}\;\;\;\;\;{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} (\delta \geqslant 1) \\ \end{gathered} \right. $ Yu et al.[7] Square plate, simply supported $\left\{ \begin{gathered} 1 + \frac{4}{3}{\delta ^2}\;\;\;{\kern 1 pt} {\kern 1 pt} \;\;\;{\kern 1 pt} {\kern 1 pt} (\delta \leqslant 1/2) \\ 2\delta + \frac{1}{{6\delta }}\;\;\;\;\;{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} (\delta \geqslant 1/2) \\ \end{gathered} \right.$ Chen et al.[8] Square plate, fully clamped $ \left\{ \begin{gathered} 1 + \frac{1}{2}{\delta ^2}\;\;\;{\kern 1 pt} {\kern 1 pt} \;\;(\delta \leqslant 1) \\ \delta + \frac{1}{{2\delta }}\;\;\;\;\;{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} (\delta \geqslant 1) \\ \end{gathered} \right. $ Chen et al.[8]
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表 3 按式(14)及式(16)或式(18)计算得到的固支梁的最大挠度
Table 3. The maximum deflection of fully clamped beam calculated by Eq. (14) and Eq. (16) or Eq. (18)
$\lambda $ 1 1.2 4/3 1.5 2 4 6 10 ${\delta _s}$
(14)&(16)0 0.775 1.000 1.229 1.817 3.915 5.944 9.967 ${\delta _s}$
(14)&(18)0 0.775 1.083 1.278 1.833 3.917 5.944 9.967
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