A fast equivalent-isotropic-plate model for stiffened circular plates under pulse loading
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摘要: 加筋板在爆炸与冲击防护中应用广泛，而其动力响应的快速求解一直是工程中关注的重点。对于径向均匀加筋的圆板，基于刚度叠加思想，提出了一种将其等效为各向同性平板的方法，用于分析其在脉冲载荷下弹性阶段的动力响应。结合理论推导与数值方法，显式地给出了简洁的等效平板厚度公式。经验证，提出的等效方法建立了加筋圆板与均质圆板间的内在联系，适用于多种加筋尺寸、材料及载荷形式。等效圆板与加筋圆板的最大挠度偏差不超过6%，低阶振动频率偏差不超过10%。相比于直接对加筋圆板进行计算，等效分析方法大大提高了求解效率，且保证了很高的计算精度，在冲击响应预测和结构优化等工程应用中具有重要意义。Abstract: Stiffened panels are widely used in the explosion and impact protection, thus a fast and accurate method for solving their dynamic response is highly desired in engineering. Based on the idea of stiffness superposition, a novel equivalent-isotropic-plate method is proposed in this paper to convert the radial and uniformly stiffened circular plate into an isotropic flat plate, so as to analyze its dynamic response in the elastic stage under uniform pulse loading. Since obtaining the dynamic response of an isotropic plate is mature and convenient, the equivalent analysis can overcome the computational difficulty of anisotropy in direct modeling, thus greatly improving the solving efficiency. Through the linear superposition of the plate and stiffener dynamic equations, a concise formula of the equivalent plate thickness is derived explicitly. The equivalent parameter in the formula is obtained with the assistance of simulation and numerical fitting, which directly measures the strengthening effect of the stiffeners on the plate. Employing the equivalent-isotropic-plate model, the overall dynamic response of a stiffened circular plate can be represented by that of an equivalent isotropic plate with acceptable accuracy, especially for low-order vibrations and center deflections. It is verified that the equivalent method can be successfully applied to a variety of stiffening types, materials, and load forms. The deviation of the maximum deflection response of the equivalent flat plate from that of the original stiffened circular plate does not exceed 6%, and the deviation of the response frequency does not exceed 10%. This completely meets the engineering requirements. The equivalent-isotropic-plate model verifies the feasibility of isotropic equivalence, and reveals the intrinsic connection between the radial stiffened circular plate and the homogeneous circular plate, which is of great significance in engineering applications such as response prediction and structural optimization.
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Key words:
- stiffened plate /
- circular plate /
- dynamic response analysis /
- equivalent method /
- pulse loading
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图 2 加强肋方向s1和s2的坐标变换
Figure 2. The coordinate transformation of the directions s1 and s2 of the stiffeners
图 3 LS-DYNA中十字形加筋圆板的壳单元网格划分
Figure 3. The mesh of cross-stiffened circular plate using shell element in LS-DYNA
图 4 不同厚度圆板在线性衰减脉冲（压力峰值50 kPa，衰减时间2 ms）作用下的最大响应挠度及拟合曲线
Figure 4. The maximum deflections and fitting curve of circular flat plates with different thicknesses under linear decaying pulse with the peak pressure of 50 kPa and the decay time of 2 ms
图 5 3种不同十字形加筋圆板的
$h_{\mathrm{s}}^2 $ 和$h_{\mathrm{e}}^2 $ 的关系Figure 5. The relations between
$h_{\mathrm{s}}^2 $ and$h_{\mathrm{e}}^2 $ for three different cross-stiffened circular plates
图 6 不同n、β设置下加筋圆板的等效参数K
Figure 6. The equivalent parameter K of the stiffened circular plates with different n and β
图 7 等效参数K(n, β)拟合函数图
Figure 7. The fitting function graphs of equivalent parameter K(n, β)
图 8 3种不同加筋圆板及其等效平板、等质量修正板、等质量分布修正板的中心挠度-时间曲线对比
Figure 8. Comparisons of the central deflection-time curves among three different stiffened circular plates and the corresponding equivalent plates, equal mass revised plates, equal mass distribution revised plates
图 9 不同时刻n4β0.07加筋圆板(左)与对应等效平板(右)的挠度分布云图
Figure 9. The deflection contour maps of the stiffened circular plate n4β0.07 (left) and the corresponding equivalent plate (right) at different times
图 10 不同n、β设置下加筋圆板及其等效平板（采用式(16)或式(19)）在同等线性衰减脉冲载荷下的响应情况对比
Figure 10. Comparisons of the dynamic response of the stiffened circular plates and the corresponding equivalent plates (by Eq.(16) or Eq. (19)) with different n and β under the same linear decaying pulse
图 12 不同线性衰减脉冲压力下的n4β0.07加筋圆板与对应等效平板的最大挠度
Figure 12. The maximum deflections of the stiffened circular plate n4β0.07 and the corresponding equivalent plate under the triangle decaying pulse with different pressure
图 13 矩形脉冲（RP）、线性衰减脉冲（LDP）和等腰三角脉冲（ITP）下的n4β0.07加筋圆板与对应等效平板的中心挠度-时间曲线
Figure 13. The central deflection-time curves of the stiffened circular plate n4β0.07 and the corresponding equivalent plate under the rectangle pulse (RP), linear decaying pulse (LDP), and isosceles triangular pulse (ITP)
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