In-plane stability analysis of frame-type concrete strut for foundation pit engineering
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摘要: 基坑规范未明确混凝土框架支撑平面稳定的计算长度取值，钢结构、混凝土规范及相关研究的框架柱计算长度系数公式亦未考虑支撑长度及两端固定边界条件的影响。本文根据受力模型将框架支撑按巨型格构柱进行系统平面稳定分析，通过缀板式格构柱稳定屈曲精确理论解，考虑框架支撑单肢作用、支撑长度及两端固定的影响，推导出支撑计算长度系数公式。用能量法推导了固端边界条件下，考虑钢立柱侧向刚度影响的支撑平面计算长度系数公式。公式计算结果与有限元特征值屈曲分析结果接近，精度满足工程设计要求。Abstract: The excavation specifications do not specify the calculation length value for the stability of the concrete frame support plane. Currently, the calculation length coefficient formula for frame columns in steel structures, concrete specifications, and related research does not consider the influence of support length and fixed boundary conditions at both ends. Based on the load-bearing model, a systematic in-plane stability analysis of the frame bracing system is conducted by treating the bracing members as mega-lattice columns. Utilizing the exact theoretical solution for the stability buckling of battened lattice columns, and accounting for the individual member action, the effective length of the bracing, and the fixed-end boundary conditions, a formula for the effective length factor of the bracing is derived. Furthermore, the energy method is employed to deduce the in-plane effective length factor formula for bracing members with fixed-end boundary conditions, incorporating the lateral stiffness of the steel columns. The calculated results demonstrate a high degree of agreement with those obtained from finite element eigenvalue buckling analysis, confirming that the proposed formulas achieve sufficient accuracy for practical engineering design applications.
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图 2 某工程对撑布置及简图
Figure 2. Layout and schematic diagram of the support system in a certain project
图 5 框架支撑有限元计算模型及结果
Figure 5. Finite element calculation model and results of frame support
图 6 多杆支撑公式与SAP2000计算长度系数比较
Figure 6. Comparison of multi-bar support formula and SAP2000 calculation length coefficient
图 7 考虑钢立柱作用的支撑模型（单位：mm）
Figure 7. Support model considering the effect of steel columns (Unit: mm)
图 8 弯曲梁与剪力梁并联计算简图
Figure 8. Simplified diagram for parallel calculation of bending beam and shear beam
图 9 两端固定布置多道弹性支座压杆模型
Figure 9. Model of a compression bar with multiple elastic supports arranged at both ends
图 11 二端固定临界荷载与$ Kl\mathrm{_k} $关系图
Figure 11. Relationship between fixed critical load at both ends and $ Kl\mathrm{_k} $
图 12 k=1.0，n=7 SAP2000屈曲分析结果
Figure 12. Buckling analysis results of SAP2000 with k=1.0 and n=7
表 1 公式与SAP2000计算长度系数对比表
Table 1. Comparison table of formulas and SAP2000 calculated length coefficients
层数$ n $ 2 3 4 5 6 7 8 9 10 $ {\mu }_{\text{SAP}2000} $ 0.99 1.26 1.49 1.66 1.78 1.87 1.94 1.99 2.02 35386 20339 16049 14222 $ {\mu }_{公式} $ 0.91 1.23 1.47 1.64 1.76 1.85 1.91 1.96 2.0 $ {\mu }_{\text{s}} $ 2.17 
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表 2 μ值计算结果对比（$ {K}_{1}=1.0\;\text{kN}/\text{mm} $， $ G=2.64 $）
Table 2. Calculation Results of μ ($ {K}_{1}=1.0\text{ kN}/\text{mm} $ and $ G=2.64 $)
长度$ {j}={n} $ 3 4 5 6 7 8 9 10 $ \mu {}_{公式} $ 1.09 1.16 1.23 1.27 1.27 1.31 1.32 1.32 模态$ m $ 1 1 2 2 3 3 3 4 $ \mu_{\mathrm{S\text{A}P}2000} $ 1.10 1.15 1.23 1.27 1.28 1.30 1.31 1.32 模态$ m $ 1 1 2 2 3 3 3 4
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表 3 μ值计算结果对比（$ K_2=0.5\ \text{k}\mathrm{N}/\text{mm} $， $ G=1.32 $）
Table 3. Calculation Results of μ ($ K_2=0.5\text{ k}\mathrm{N}/\text{mm} $ and $ G=1.32 $)
长度${j}={n} $ 3 4 5 6 7 8 9 10 $ \mu {}_{公式} $ 1.16 1.27 1.29 1.37 1.41 1.41 1.44 1.46 模态$ m $ 1 1 2 2 2 2 3 3 $ \mu_{\mathrm{S\text{A}P}2000} $ 1.17 1.28 1.3 1.38 1.41 1.42 1.44 1.46 模态$ m $ 1 1 1 2 2 2 3 3
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