Journal of Jilin University(Engineering and Technology Edition) ›› 2025, Vol. 55 ›› Issue (7): 2333-2342.doi: 10.13229/j.cnki.jdxbgxb.20231088

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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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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

CLC Number: 

  • TU323

Fig.1

A planar pin-jointed system with 3 cables"

Table 1

Internal forces of element in example 1"

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

Table 2

Displacements of nodes in example 1"

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

Fig.2

Computational model used in matrix force method"

Fig.3

A prestressed cable-strut structure with 12 members"

Fig.4

A prestressed cable-strut structure with 16 members（lengths of four additional members are controllable）"

Table 3

Internal forces of element in example 2"

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

Table 4

Displacements of nodes in example 2"

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

Fig.5

Force curves of ①~④ members"

Fig.6

Force curves of ⑤~⑧ members"

Fig.7

Force curves of ⑨~? members"

Fig.8

Force curves of ?~? members"

[1] Pellegrino S. Analysis of prestressed mechanisms[J]. International Journal of Solids and Structures, 1990, 26(12): 1329-1350.
[2] 张沛, 冯健. 张拉整体结构的稳定性判定及刚度分析[J]. 土木工程学报, 2013, 46(10): 48-57.
Zhang Pei, Feng Jian. Stability criterion and stiffness analysis of tensegrity structures[J]. China Civil Engineering Journal, 2013, 46(10): 48-57.
[3] 王艳蒙, 刘贺平, 罗阿妮. 张拉整体基本单元几何稳定构型分析[J]. 哈尔滨工程大学学报, 2022, 43(2): 268-273.
Wang Yan-meng, Liu He-ping, Luo A-ni. Analysis of geometrically stable configurations of the basic tensegrity unit[J]. Journal of Harbin Engineering University, 2022, 43(2): 268-273.
[4] 张沛, 冯健, 周继凯. 含冗余拉索的棱柱型张拉整体结构成形过程分析[J]. 东南大学学报:自然科学版, 2022, 52(5): 848-855.
Zhang Pei, Feng Jian, Zhou Ji-kai. Analysis on forming process of prismatic tensegrity structures with redundant cables[J]. Journal of Southeast University (Natural Science Edition), 2022, 52(5): 848-855.
[5] 陈耀, 冯健, 马瑞君. 对称型动不定杆系结构的可动性判定准则[J]. 建筑结构学报, 2015, 36(6): 101-107.
Chen Yao, Feng Jian, Ma Rui-jun. A unified criterion for movability of kinematically indeterminate frameworks with symmetry[J]. Journal of Building Structures, 2015, 36(6): 101-107.
[6] 冯晓东, 戴冠鸥, 杨伟家, 等. 基于向量式结构力学的张拉整体结构可展性能研究[J]. 西安建筑科技大学学报:自然科学版, 2021, 53(3): 350-357.
Feng Xiao-dong, Dai Guan-ou, Yang Wei-jia, et al. Research on developable property of tensegrity structures based on vector mechanics of structures[J]. Journal of Xi'an University of Architecture & Technology(Natural Science Edition), 2021, 53(3): 350-357.
[7] 马烁, 袁行飞, 杨柳. 预应力索杆结构模型相似理论及其验证[J]. 建筑结构学报, 2021, 42(7): 95-103.
Ma Shuo, Yuan Xing-fei, Yang Liu. Validation of similitude laws for prestressed cable-strut structures[J]. Journal of Building Structures, 2021, 42(7): 95-103.
[8] Guest S. The stiffness of prestressed frameworks: a unifying approach[J]. International Journal of Solids and Structures. 2006, 43(3-4): 842-854.
[9] 龙驭球, 包世华, 袁驷. 结构力学Ⅱ——专题教程[M]. 4版. 北京: 高等教育出版社, 2018.
[10] 张沛, 冯健. 一种求解索杆张力结构整体自应力模态的能量方法[J]. 土木工程学报, 2013, 46(6): 62-68.
Zhang Pei, Feng Jian. An energy method for overall self-stress modes calculation of cable-strut tension structures[J]. China Civil Engineering Journal, 2013, 46(6): 62-68.
[11] Zhang P, Kawaguchi K, Feng J. Prismatic tensegrity structures with additional cables: Integral symmetric states of self-stress and cable-controlled reconfiguration procedure[J]. International Journal of Solids and Structures, 2014, 51(25-26): 4294-4306.
[12] Pellegrino S. Structural computations with the singular value decomposition of the equilibrium matrix[J]. International Journal of Solids and Structures, 1993, 30(21): 3025-3035.
[13] 涂桂刚, 崔昌禹, 江红, 等. 基于广义逆矩阵理论的杆系结构形态创构方法[J]. 土木工程学报, 2020, 53(5): 25-31.
Tu Gui-gang, Cui Chang-yu, Jiang Hong, et al. A structural morphogenesis method for frame structure based on generalized inverse matrix theory[J]. China Civil Engineering Journal, 2020, 53(5): 25-31.
[14] 薛素铎, 鲁建, 李雄彦, 等. 跳格布置对环形交叉索桁结构静动力性能的影响[J]. 吉林大学学报:工学版, 2019, 50(5): 1687-1697.
Xue Su-duo, Lu Jian, Li Xiong-yan, et al. Influence of grid-jumping arrangement on static and dynamic performance of annular crossed cable-truss structure[J]. Journal of Jilin University(Engineering and Technology Edition), 2019, 50(5): 1687-1697.
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