吉林大学学报(工学版) ›› 2025, Vol. 55 ›› Issue (7): 2333-2342.doi: 10.13229/j.cnki.jdxbgxb.20231088

• 交通运输工程·土木工程 • 上一篇    

铰接杆系结构静力分析的力法基本方程统一形式

张沛1,2(),冯健2,3,周继凯1(),尚志兵1   

  1. 1.河海大学 土木与交通学院，南京 210098
    2.东南大学 混凝土及预应力混凝土结构教育部重点实验室，南京 211189
    3.东南大学 土木工程学院，南京 211189
  • 收稿日期:2023-10-23 出版日期:2025-07-01 发布日期:2025-09-12
  • 通讯作者: 周继凯 E-mail:zhangpei250131@163.com;zhoujikaihhu@hotmail.com
  • 作者简介:张沛（1983-），男，副教授，博士. 研究方向：新型预应力空间结构. E-mail： zhangpei250131@163.com
  • 基金资助:
    国家自然科学基金项目(51878147);中国博士后科学基金项目(2020M671319);江苏省博士后科研项目(2020Z317);东南大学混凝土及预应力混凝土结构教育部重点实验室开放课题项目(CPCSME2022-01)

Unified form of basic equations of force method for static analysis of pin-bar assemblies

Pei ZHANG1,2(),Jian FENG2,3,Ji-kai ZHOU1(),Zhi-bing SHANG1   

  1. 1.College of Civil and Transportation Engineering，Hohai University，Nanjing 210098，China
    2.Key Laboratory of Concrete and Pre-stressed Concrete Structures of Ministry of Education，Southeast University，Nanjing 211189，China
    3.School of Civil Engineering，Southeast University，Nanjing 211189，China
  • Received:2023-10-23 Online:2025-07-01 Published:2025-09-12
  • Contact: Ji-kai ZHOU E-mail:zhangpei250131@163.com;zhoujikaihhu@hotmail.com

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摘要：

以力法分析为基础，从平衡方程增量形式出发，考虑初始内力的影响，结合变形协调条件和本构关系，推导了小变形、线弹性条件下铰接杆系结构静力分析基本方程。运用线性代数及广义逆矩阵理论，实现内力增量和坐标增量的解耦，最终得到的力法基本方程组由广义平衡方程和广义协调方程两部分组成，二者的系数矩阵互为转置，形式上与静定动定结构力法分析基本方程一致。当研究对象为静定动定结构时，该方程组退化为传统力法基本方程，故可将其视为传统力法考虑初始内力刚化效应的直接推广，适用于各类铰接杆系结构的小变形、线弹性静力分析，其计算精度随预应力水平及结构刚度的提高而增大。

关键词: 铰接杆系结构, 静力分析, 初始内力, 力法, 广义逆

Abstract:

On the basis of linear elasticity hypothesis and force method， an analytic theory for static analysis of pin-bar assemblies is developed from the incremental form of equilibrium equations， where the effect of initial internal forces， the compatibility equations and constitutive equations are taken into account. Then the displacements and the increments of axial force are decoupled by using linear algebra and Moore-Penrose generalized inverse theory. The basic formulas proposed finally consists of two parts — generalized equilibrium equations and generalized compatibility equations， both of which have square coefficient matrices of full rank being transposed with each other. In other words， they are formally consistent with the basic equations using in traditional force method， and will degenerate into the latter ones in dealing with the statically and kinematically determinate structures. Therefore， the proposed theory can be regarded as an extended version of the traditional force method considering the stiffening effect of initial internal forces， which is applicable to any pin-bar assemblies with small deformation and linear elasticity static structural analysis. Its calculation accuracy is increasing with the increment of prestress level and structural stiffness.

Key words: pin-bar assemblies, static analysis, initial internal force, force method, generalized inverse

中图分类号: 

  • TU323

图1

3索段平面铰接体系"

表1

算例1单元内力 (N)"

外荷载单元号初始内力 n内力增量δ n
矩阵位移法矩阵力法

本文

方法

W=30 N①67.082 09.431 107.621 2
②6010.113 108.198 5
③67.082 08.927 307.044 6
W=3000 N①6 708.2259.777 80256.075 8
②6 000259.930 30255.766 9
③6 708.2207.046 30201.453 9

表2

算例1节点位移 (mm)"

外荷载节点号位移δ x矩阵位移法矩阵力法本文方法
W=30 N节点1X向分量-5.163 6-5.160 2-5.193 0
Y向分量-12.331 6-12.040 4-11.808 7
节点2X向分量-5.082 1-5.160 2-5.121 5
Y向分量-10.869 5-10.320 3-10.089 6
W=3 000 N节点1X向分量-6.009 3-5.160 2-6.000 0
Y向分量-4.697 4-12.040 4-4.781 7
节点2X向分量-3.751 8-5.160 2-3.771 1
Y向分量-3.116 0-10.320 3-3.153 2

图2

矩阵力法计算模型"

图3

12单元预应力索杆结构"

图4

16单元预应力索杆结构（新增拉索索长可控）"

表3

算例2单元内力 (N)"

工况号单元编号初始内力 n内力增量δ n
矩阵位移法矩阵力法本文方法
工况1①~④1 311.347.196 245.592 246.253 6
⑤~⑧2 389.5-9.605 4-10.242 9-10.390 0
⑨~?-2 853.3-120.965 2-118.327 6-119.707 8
工况2①~④1 311.331.876 8030.325 1
⑤~⑧2 389.5-5.826 20-6.755 3
⑨~?-2 853.3-68.379 90-65.954 2
?~?062.003 3061.136 7

表4

算例2节点位移 (mm)"

工况号节点编号位移δ x矩阵位移法矩阵力法本文方法
工况1节点1X向分量4.296 84.326 54.318 9
Y向分量-5.279 6-5.380 1-5.387 7
Z向分量-3.440 9-3.486 3-3.494 6
节点2X向分量5.279 65.380 15.387 7
Y向分量4.296 84.326 54.318 9
Z向分量-3.440 9-3.486 3-3.494 6
节点3X向分量-4.296 8-4.326 5-4.318 9
Y向分量5.279 65.380 15.387 7
Z向分量-3.440 9-3.486 3-3.494 6
节点4X向分量-5.279 6-5.380 1-5.387 7
Y向分量-4.296 8-4.326 5-4.318 9
Z向分量-3.440 9-3.486 3-3.494 6
工况2节点1X向分量5.115 19.111 45.077 2
Y向分量-5.713 8-9.111 4-5.778 0
Z向分量-3.611 3-5.466 8-3.634 8
节点2X向分量5.713 89.111 45.778 0
Y向分量5.115 19.111 45.077 2
Z向分量-3.611 3-5.466 8-3.634 8
节点3X向分量-5.115 1-9.111 4-5.077 2
Y向分量5.713 89.111 45.778 0
Z向分量-3.611 3-5.466 8-3.634 8
节点4X向分量-5.713 8-9.111 4-5.778 0
Y向分量-5.115 1-9.111 4-5.077 2
Z向分量-3.611 3-5.466 8-3.634 8

图5

单元①~④内力曲线"

图6

单元⑤~⑧内力曲线"

图7

单元⑨~?内力曲线"

图8

单元?~?内力曲线"

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