Static Pushover Analysis of Frame Structure Based on Force Analogy Method
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摘要: 传统静力推覆分析方法求解结构非线性变形需对结构整体刚度矩阵进行实时地合成与分解，该过程将占用大量计算资源.基于拟力法的纤维梁有限元分析方法进行静力推覆分析，在迭代求解结构非线性变形时，首先对弹性刚度矩阵进行分解，计算出侧向荷载作用下的弹性位移；然后通过反复调用弹性刚度矩阵的分解结果与弹性位移，减少回代计算量；最后采用算法时间复杂度理论定量对比了该方法与传统方法的计算效率，通过一榀八层钢筋混凝土框架结构数值算例，分析比较了两种方法的计算结果与算法时间复杂度. 结果表明：两种方法顶点位移-基底剪力曲线基本吻合，层间位移角与楼层之间的关系曲线也基本一致，两者的最大误差出现在第3层，为3.72%，与传统方法相比，基于拟力法的静力推覆分析方法算法时间复杂度降低了80%，计算效率至少是传统方法的5倍.Abstract: In traditional pushover analysis, solving nonlinear deformation need to update and decompose global stiffness matrix in real time, which costs many computing resources. The static pushover analysis for nonlinear fiber beam element is conducted on the basis of the force analogy method (FAM). Firstly, the elastic displacement under the lateral load was calculated by the factorization of elastic stiffness matrix before the iteration of nonlinear deformation. Secondly, the computational cost of back iteration was decreased by utilizing the factorization result of elastic stiffness matrix and elastic displacement during iterations. Finally, the algorithm complexity theory was utilized to evaluate the efficiency of the proposed algorithm and a classical method. The computational results and algorithm time complexity of the two methods were compared through a numerical results of 2D eight-floor frame structure. Results show that with two methods the vertex displacement-base shear curve basically coincides as well as the story drift curve, and the maximum error of story drift is on the 3rd floor with 3.72%. Compared with the traditional method, the algorithm complexity of the proposed method is decreased about 80%, and its computing efficiency is increased at least five times.
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图 1 基于拟力法的静力推覆分析流程
Figure 1. Flow chart for static pushover analysis based on force analogy method
图 5 塑性自由度数与顶点位移关系
Figure 5. Relationship between number of plastic degrees and roof displacement
表 1 时间复杂度
Table 1. Time complexity
计算步骤 计算项目 时间复杂度 1 $\left( {\Delta { {{X} }_1} } \right)_1^{\left( j \right)} = \Delta \lambda _1^{\left( j \right)}{ {{U} }_{\rm{F} } }$ $n$ 2 $\Delta { {{u} }_0} = \left( { {{K} }_{ {\rm{pr} } }^{\rm{T} } } \right)_1^{\left( j \right)}{\left( {\Delta { {{X} }_1} } \right)^{\left( j \right)} }$ $2\alpha d - d$ 3 $\left( {{{{K}}_{{\rm{pl}}}}} \right)_1^{\left( j \right)} = \left( {{{{K}}_{{\rm{rr}}}}} \right)_1^{\left( j \right)} - \left( {{{K}}_{{\rm{pr}}}^{\rm{T}}} \right)_1^{\left( j \right)}{{K}}_{\rm{e}}^{ - 1}\left( {{{{K}}_{{\rm{pr}}}}} \right)_1^{\left( j \right)}$ $\beta d/2$ 4 $\Delta {{{u}}_1} = \left( {{{K}}_{{\rm{pl}}}^{ - 1}} \right)_1^{\left( j \right)}\Delta {{{u}}_0}$ $m_{\rm{d}}^2d + 8{m_{\rm{d}}}d + d$ 5 $\Delta {{{u}}_2} = \left( {{{{K}}_{{\rm{pr}}}}} \right)_1^{\left( j \right)}\Delta {{{u}}_1}$ $2\alpha d - d$ 6 $\left( {\Delta { {{X} }_2} } \right)_1^{\left( j \right)} = {{K} }_{\rm{e} }^{ - 1}\Delta { {{u} }_2}$ $2mn - n - {m^2}/2 + 1/2$ 7 $\left( {\Delta {{X} } } \right)_1^{\left( j \right)} = \left( {\Delta { {{X} }_1} } \right)_1^{\left( j \right)} + \left( {\Delta { {{X} }_2} } \right)_1^{\left( j \right)}$ $n$ 
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表 2 截面配筋
Table 2. Parameters of sections
楼层 梁 中柱 边柱 截面的宽，高/mm 配筋/mm2 截面的宽，高/mm 配筋/mm2 截面的宽，高/mm 配筋/mm2 1 $300,\;600$ 6874 $700 ,\;700$ 8758 $700 ,\;700$ 11235 2 $300 ,\;600$ 6874 $700 ,\;700$ 5024 $700 ,\;700$ 5024 3 $300 ,\;600$ 6874 $700 ,\;700$ 5024 $700 ,\;700$ 5024 4 $300 ,\;600$ 6874 $700 ,\;700$ 5024 $700 ,\;700$ 5024 5 $300 ,\;600$ 4688 $600 ,\;600$ 3786 $600 ,\;600$ 3786 6 $300 ,\;600$ 4688 $600 ,\;600$ 3786 $600 ,\;600$ 3786 7 $300 ,\;600$ 4688 $600 ,\;600$ 3786 $600 ,\;600$ 3786 8 $300 ,\;600$ 4688 $600 ,\;600$ 3786 $600 ,\;600$ 3786 注：配筋指截面中钢筋截面面积.
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