Homogeneous Generalized Yield Function for Frame Members with Box Section
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摘要: 为了克服传统广义屈服函数的局限性，建立了适用于平面和空间框架结构，且对截面几何尺寸不敏感的箱型构件通用齐次广义屈服函数，提出了箱型框架极限承载力分析的高效线弹性迭代方法. 首先，通过对比分析遴选出对截面几何参数不敏感的箱型截面广义屈服函数作为基准；然后，根据全面试验法确定了拟合配点，并通过回归分析建立了考虑轴力和双向弯矩联合作用的箱型截面通用齐次广义屈服函数；最后，结合弹性模量缩减策略，提出了箱型截面框架结构极限承载力分析的高效线弹性迭代方法. 研究结果表明：建立的箱型截面通用齐次广义屈服函数不仅克服了普通广义屈服函数计算结果不稳定的缺陷，而且克服了现存齐次广义屈服函数存在的对截面几何尺寸敏感且不适用于空间结构等问题；建立的结构极限承载力分析的方法较传统的弹塑性增量分析法相对误差不超过3%，且计算时间不足传统分析法的10%，均表明本文建立方法的精确性和适用性.Abstract: In order to overcome limitations of the traditional generalized yield function (GYF), a homogeneous generalized yield function (HGYF) is proposed for box sections, which is insensitive to sectional geometry of box sections and applicable for plane and spatial frame structures; and a linear-elastic iterative method with higher efficiency is presented for ultimate bearing capacity of framed structures. Firstly, different generalized yield functions (GYF) are investigated and the suitable one is selected as insensitive to the geometric parameters of box sections. Then a set of fitting points are determined according to the comprehensive test method, based on which an HGYF was developed by regression analysis with a wide range of application for box sections. Finally, a linear-elastic iterative method is presented with high efficiency for the ultimate bearing capacity of frames with box section on the basis of elastic modulus reduction method (EMRM). Numerical examples show that the proposed HGYF achieves satisfying stable results with a wide range of application for plane and spatial frames, overcoming not only the instability of the traditional GYF, but also the sensitivity of the existing HGYF to geometry of sections which is unsuitable for spatial frames. The relative error of the established method is less than 3% compared with the traditional elasto-plastic incremental analysis method, and the calculation time is less than 10% that of the traditional analysis method, which shows the accuracy and efficiency of this method.
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图 4 基于fZ和
${\bar f_4}$ 的EMRM迭代过程Figure 4. Iterative process of the EMRM based on
${f_{\rm{Z}}}\;{\rm{\& }}\;{\bar f_4}$ 
图 6 基于fZ和
${\bar f_4}$ 的EMRM迭代计算过程Figure 6. Iterative process of the EMRM based on
${f_{\rm{Z}}}\;{\rm{\& }}\;{\bar f_4}$ 表 1 残差均方差
Table 1. Residual mean-square deviation
${{\bar f}_2}$ ${{\bar f}_3}$ ${{\bar f}_4}$ 0.068 0.057 0.042 
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表 3 不同方法计算结果的收敛性分析
Table 3. Convergences of the results from different methods
计算方法 文献 参数 构件离散单元数 2 3 4 6 8 10 12 EMRM 本文HGYF 极限承载力/(kN•m−1) 60.53 59.27 60.09 59.65 相对差/% 2.08 1.34 0.73 计算时间/s 7.4 11.2 13.7 19.6 文献[15] HGYF 极限承载力/（kN•m−1） 58.65 57.29 57.99 57.58 相对差/% 2.32 1.22 0.71 计算时间/s 13.3 15.9 19.0 20.6 EPIA 极限承载力/（kN•m−1） 75.76 69.47 66.48 63.83 62.56 61.74 61.37 相对差/% 8.30 4.30 3.99 1.99 1.31 0.60 计算时间/s 12.0 17.2 18.8 24.9 37.4 43.2 48.6
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表 4 荷载初值对结构极限承载力的影响
Table 4. Influence of the initial load on ultimate bearing capacity
计算方法 文献 荷载初值/
（kN•m−1）极限荷载/
（kN•m−1）计算
时间/s相对
误差/%EPIA 61.74 43.2 EMRM GYF[6] 30 93.36 15.3 51.21 60 56.95 12.8 7.76 120 27.47 11.8 55.51 GYF[7] 30 80.12 9.7 29.77 60 51.10 8.9 17.23 120 28.97 9.3 53.08 GYF[8] 30 64.08 17.7 3.79 60 37.27 15.6 39.63 120 18.81 14.9 69.53 本文HGYF 30 60.09 17.3 2.67 60 60.09 17.6 2.67 120 60.09 16.5 2.67
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表 5 几何参数
Table 5. Geometry parameters
$\beta $ 截面参数/m 梁 柱 $\gamma = 5$ $\gamma = 10$ $\gamma = 5$ $\gamma = 10$ $ 1.0$ b 0.10 0.10 0.20 0.20 h 0.10 0.10 0.20 0.20 t1 0.02 0.01 0.04 0.02 t2 0.02 0.01 0.04 0.02 $ 1.5$ b 0.10 0.10 0.20 0.20 h 0.15 0.15 0.30 0.30 t1 0.03 0.015 0.06 0.03 t2 0.01 0.01 0.04 0.02 $2.0$ b 0.10 0.10 0.20 0.20 h 0.20 0.20 0.40 0.40 t1 0.04 0.02 0.08 0.04 t2 0.02 0.01 0.04 0.02
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表 6
$\gamma $ 和$\beta $ 对计算结果的影响Table 6. Influence of
$\gamma $ and$\beta $ on the results计算方法 文献 $\beta $ $\gamma $ 极限荷载/
（kN•m−1）相对误差/% EPIA 1.0 5 31.44 10 19.49 1.5 5 70.19 10 42.25 2.0 5 137.96 10 81.02 EMRM 本文
HGYF1.0 5 30.67 2.45 10 19.07 2.15 1.5 5 68.72 2.09 10 41.44 1.92 2.0 5 134.91 2.21 10 79.24 2.20 文献[17]
HGYF1.0 5 30.21 3.91 10 18.75 3.80 1.5 5 66.54 5.20 10 40.06 5.18 2.0 5 127.02 7.93 10 74.46 8.10
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表 7 几何尺寸
Table 7. Geometry parameters
m 构件 b h t1 t2 梁 0.1 0.15 0.025 0.015 柱 0.2 0.30 0.050 0.030
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表 8 内力组合对结构极限承载力的影响
Table 8. Influence of internal force combination on ultimate bearing capacity
参数 EPIA EMRM HGYF[17] 本文HGYF 极限荷载/（kN•m−1） 28.94 151.37 27.85 计算时间/s 554.2 2.6 50.9 相对误差/% 423.05 3.77
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表 9 荷载初值对结构极限承载力的影响
Table 9. Influence of the initial load on ultimate bearing capacity
计算方法 文献来源 荷载初值/（kN•m−1） 极限荷载/（kN•m−1） 计算时间/s 相对误差/% EPIA 50 28.94 554.2 EMRM GYF[8] 10 44.44 30.2 53.56 30 20.68 27.7 28.54 50 14.11 28.8 51.24 GYF[9] 10 42.87 31.3 48.12 30 19.63 32.9 32.17 50 12.39 32.9 57.19 GYF[10] 10 34.66 37.3 19.77 30 19.17 36.1 33.76 50 11.26 35.2 61.09 本文HGYF 10 27.85 50.9 3.77 30 27.85 50.8 3.77 50 27.85 50.9 3.77
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表 10
$\beta $ 和$\gamma $ 对结果的影响Table 10. Influence of
$\beta $ and$\gamma $ on the results计算方法 $\beta $ $\gamma $ 极限荷载/
（kN•m−1）相对误差/% EPIA 1.0 5 16.95 10 10.53 1.5 5 34.26 10 21.21 2.0 5 60.62 10 37.69 EMRM 1.0 5 16.73 1.30 10 10.40 1.23 1.5 5 33.49 2.25 10 20.74 2.24 2.0 5 59.27 2.23 10 36.87 2.18
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