﻿ 基于无应力状态控制法的钢桁梁桥起拱研究
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 哈尔滨工程大学学报  2018, Vol. 39 Issue (12): 1941-1946  DOI: 10.11990/jheu.201706034 0

### 引用本文

DAN Qilian. Pre-camber setting of truss bridge based unstressed state control method[J]. Journal of Harbin Engineering University, 2018, 39(12), 1941-1946. DOI: 10.11990/jheu.201706034.

### 文章历史

Pre-camber setting of truss bridge based unstressed state control method
DAN Qilian
School of Civil Engineering, Southwest Jiaotong University, Chengdu 610031, China
Abstract: This paper focuses on the method of setting pre-camber for steel truss girder bridges with a known pre-camber curve. The relative elevation between two adjacent joints is taken as the input variable to deduce the mathematical expressions of the length variation of each member based on the mechanical equilibrium equation of a structure formed in stages. Considering the pre-camber setting characteristics of through steel truss girder bridges, the length variation formula of each member is further developed in three cases:only the length of upper chord member bar is changed; the lengths of the upper chord and diagonal wed member bar are changed; and the lengths of the upper chords, diagonal wed, and vertical member bar are changed simultaneously. An example study was carried out to illustrate applications of the method. The analysis results show that the unstressed pre-camber setting method is an absolutely geometrical method, which will not generate stress in members and supports during the pre-camber setting. The numerical results are in accordance with the design values and the reference value, which illustrates that the pre-camber setting used in this study is reliable. The method is clear and reliable and deserves to be developed for applications in the other types of bridges.
Keywords: steel truss girder bridge    pre-camber curve    relative elevation    plant pre-camber    unstressed pre-camber setting method

1 位移荷载起拱法

 $\left[ {\begin{array}{*{20}{c}} {{F_{x1}}}\\ {{F_{y1}}}\\ {{F_{x2}}}\\ {{F_{y2}}}\\ {{F_{x3}}}\\ {{F_{y3}}}\\ {{F_{x4}}}\\ {{F_{y4}}} \end{array}} \right] = \mathit{\boldsymbol{K}}\left[ {\begin{array}{*{20}{c}} {{u_1}}\\ {{v_1}}\\ {{u_2}}\\ {{v_2}}\\ {{u_3}}\\ {{v_3}}\\ {{u_4}}\\ {{v_4}} \end{array}} \right]$ (1)

 ${u_1} = {v_1} = {u_2} = {v_2} = 0$ (2)

 $\left\{ \begin{array}{l} {F_{x3}} = 0\\ {F_{y3}} = 0\\ {F_{x4}} = 0 \end{array} \right.$ (3)

 $\left\{ \begin{array}{l} {g_1}{u_3} + {g_2}{c_2}\left( {{u_3}{c_2} + {v_3}{s_2}} \right) = 0\\ {g_2}{s_2}\left( {{u_3}{c_2} + {v_3}{s_2}} \right) + {g_3}\left( {{v_3} - {v_4}} \right) = 0\\ {g_4}{u_4} = 0 \end{array} \right.$ (4)

2 无应力状态起拱法 2.1 起拱原理

 Download: 图 2 无应力状态起拱法示意图 Fig. 2 Schematic diagram of the unstressed pre-camber setting method

 $\left( {\sum\limits_{i = 1}^4 {{\mathit{\boldsymbol{k}}_i}} } \right)\mathit{\boldsymbol{\delta = P}} + \sum\limits_{i = 1}^4 {{\mathit{\boldsymbol{P}}_{0i}}}$ (5)

 ${\mathit{\boldsymbol{k}}_i} = \left[ {\begin{array}{*{20}{c}} {{g_i}c_i^2}&{{g_i}{c_i}{s_i}}&{ - {g_i}c_i^2}&{ - {g_i}{c_i}{s_i}}\\ {{g_i}{c_i}{s_i}}&{{g_i}s_i^2}&{ - {g_i}{c_i}{s_i}}&{ - {g_i}s_i^2}\\ { - {g_i}c_i^2}&{ - {g_i}{c_i}{s_i}}&{{g_i}c_i^2}&{{g_i}{c_i}{s_i}}\\ { - {g_i}{c_i}{s_i}}&{ - {g_i}s_i^2}&{{g_i}{c_i}{s_i}}&{{g_i}s_i^2} \end{array}} \right]$ (6)
 $\mathit{\boldsymbol{\delta }} = {\left[ {\begin{array}{*{20}{c}} {{u_1}}&{{v_1}}&{{u_2}}&{{v_2}}&{{u_3}}&{{v_3}}&{{u_4}}&{{v_4}} \end{array}} \right]^{\rm{T}}}$ (7)
 ${\mathit{\boldsymbol{P}}_{0i}} = {\left[ {\begin{array}{*{20}{c}} {{g_i}{c_i}\Delta {L_i}}&{{g_i}{s_i}\Delta {L_i}}&{ - {g_i}{c_i}\Delta {L_i}}&{ - {g_i}{s_i}\Delta {L_i}} \end{array}} \right]^{\rm{T}}}$ (8)

 $\begin{array}{*{20}{c}} {\left[ {\begin{array}{*{20}{c}} {{g_1}c_1^2 + {g_2}c_2^2 + {g_3}c_3^2}&{{g_1}{c_1}{s_1} + {g_2}{c_2}{s_2} + {g_3}{c_3}{s_3}}&{ - {g_3}c_3^2}&{ - {g_3}{c_3}{s_3}}\\ {{g_1}{c_1}{s_1} + {g_2}{c_2}{s_2} + {g_3}{c_3}{s_3}}&{{g_1}c_1^2 + {g_2}c_2^2 + {g_3}c_3^2}&{ - {g_3}{c_3}{s_3}}&{ - {g_3}s_3^2}\\ { - {g_3}c_3^2}&{ - {g_3}{c_3}{s_3}}&{{g_3}c_3^2 - {g_4}c_4^2}&{{g_3}{c_3}{s_3} + {g_4}{c_4}{s_4}}\\ { - {g_3}{c_3}{s_3}}&{ - {g_3}s_3^2}&{{g_3}{c_3}{s_3} + {g_4}{c_4}{s_4}}&{{g_3}s_3^2 - {g_4}s_4^2} \end{array}} \right].}\\ {\left[ {\begin{array}{*{20}{c}} {{u_3}}\\ {{v_3}}\\ {{u_4}}\\ {{v_4}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} { - {g_1}{c_1}\Delta {L_1} - {g_2}{c_2}\Delta {L_2} - {g_3}{c_3}\Delta {L_3}}\\ { - {g_1}{s_1}\Delta {L_1} - {g_2}{s_2}\Delta {L_2} - {g_3}{s_3}\Delta {L_3}}\\ {{g_3}{c_3}\Delta {L_3} - {g_4}{c_4}\Delta {L_4}}\\ {{g_3}{s_3}\Delta {L_3} - {g_4}{s_4}\Delta {L_4}} \end{array}} \right]} \end{array}$ (9)

 $\left[ {\begin{array}{*{20}{c}} { - {c_1}}&{ - {s_1}}&0&0\\ { - {c_2}}&{ - {s_2}}&0&0\\ { - {c_3}}&{ - {s_3}}&{{c_3}}&{{s_3}}\\ 0&0&{ - {c_4}}&{{s_4}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{u_3}}\\ {{v_3}}\\ {{u_4}}\\ {{v_4}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {\Delta {L_1}}\\ {\Delta {L_2}}\\ {\Delta {L_3}}\\ {\Delta {L_4}} \end{array}} \right]$ (10)

 $\left\{ \begin{array}{l} \Delta {L_1} = - {u_3}{c_3} - {v_3}{s_3}\\ \Delta {L_2} = - {u_3}{c_2} - {v_3}{s_2}\\ \Delta {L_3} = \left( {{u_4} - {u_3}} \right){c_3} + \left( {{v_4} - {v_3}} \right){s_3}\\ \Delta {L_4} = - {u_4}{c_4} - {v_4}{s_4} \end{array} \right.$ (11)

2.2 杆件伸缩

2.2.1 仅伸缩上弦杆

 Download: 图 3 伸缩上弦杆示意图 Fig. 3 Schematic diagram of the length change in upper chords
 $\Delta {L_2} = \Delta {L_3} = \Delta {L_4} = 0$ (12)

v4为已知输入，由图中几何关系知，在上弦杆的伸缩变化过程中，阴影部分形状将不发生变化，它仅发生刚体转动，转角用α表示。联立式(11)~(12)可得上弦杆伸缩量。

v4已知及ΔL4=-u4c4-v4s4=0得

 ${u_4} = - {v_4}{t_4}$ (13)

 ${u_3}{c_3} + {v_3}{s_3} = {u_4}{c_3} + {v_4}{s_3}$ (14)

 ${u_3}{c_2} + {v_3}{s_2} = 0$ (15)

 $\Delta {L_1} = \frac{{\left( {{c_1}{s_2} - {s_1}{c_2}} \right)\left( {{s_3} - {c_3}{t_4}} \right)}}{{{c_2}{s_3} - {s_2}{c_3}}}{v_4}$ (16)

 $\alpha = \frac{{{v_4}}}{{{L_4}}}$ (17)
2.2.2 同时伸缩上弦杆和斜腹杆

 Download: 图 4 同时伸缩上弦杆和斜腹杆 Fig. 4 Length change in both upper chords and diagonal web
 $\Delta {L_3} = \Delta {L_4} = 0$ (18)

v4已知及ΔL4=-u4c4-v4s4=0可得式(13)。

 $\left\{ \begin{array}{l} {v_3} = {v_4} + {L_3}\left[ {\sin \left( {{\theta _3} - \beta } \right) - {s_3}} \right]\\ {u_3} = {u_4} + {L_3}\left[ {\cos \left( {{\theta _3} - \beta } \right) - {c_3}} \right] \end{array} \right.$ (19)

 $\left\{ \begin{array}{l} \Delta {L_1} = \left( {{c_1}{t_4} - {s_1}} \right){v_4} - {L_3}\left[ {\cos \left( {{\theta _3} - \beta } \right) + } \right.\\ \;\;\;\;\;\;\;\;\left. {\sin \left( {{\theta _3} - \beta } \right) - {c_1}{c_3} - {s_1}{s_3}} \right]\\ \Delta {L_2} = \left( {{c_2}{t_4} - {s_2}} \right){v_4} - {L_3}\left[ {\cos \left( {{\theta _3} - \beta } \right) + } \right.\\ \;\;\;\;\;\;\;\;\left. {\sin \left( {{\theta _3} - \beta } \right) - {c_2}{c_3} - {s_2}{s_3}} \right] \end{array} \right.$ (20)
2.2.3 同时伸缩上弦杆、竖杆和斜腹杆。

 $\left\{ \begin{array}{l} \Delta {L_1} = - {u_3}{c_3} - {v_3}{s_3}\\ \Delta {L_2} = - {u_3}{c_2} - {v_3}{s_2}\\ \Delta {L_3} = \left( {{s_3} - {c_3}{t_4}} \right){v_4} - {u_3}{c_3} - {v_3}{s_3} \end{array} \right.$ (21)

2.3 两种方法比较

1) 两种方法均是以单个节间桁段为研究对象，建立起拱模型，并且都是以节间下弦节点的相对预拱度为输入，杆件的伸缩量作为输出，从而达到起拱目的方法。

2) 荷载位移起拱法将相对预拱度作为强迫位移，根据杆端内力之和与外荷载相等建立关系式，推导出杆件伸缩量的表达式，表达式中含有力学参数；无应力状态起拱法由无应力状态法力学平衡条件直接建立杆件伸缩量与节点位移的关系，表达式中不含物理力学参数，不会产生起拱附加力。

3) 文献[9]在推导位移荷载起拱法时，仅针对竖杆为竖直且弦杆水平的情况进行推导，所以仅适用对此类平面N形钢桁梁的起拱；而无应力状态起拱法在推导时并未限定各杆件的具体倾角，适用于任何平面N形钢桁梁。

3 数值验证

 Download: 图 5 榕江特大桥主桁结构立面图 Fig. 5 The elevation of the main truss of the Rongjiang Bridge

 Download: 图 6 榕江特大桥厂设预拱度曲线(mm) Fig. 6 The pre-camber curve of the Rongjiang Bridge (mm)

4 结论

1) 直接以无应力状态法力学平衡方程得出杆件伸缩量数学表达式，式中不包含任何物理力学参数，为纯几何起拱方法，起拱过程不会产生起拱附加力。

2) 根据下承式钢桁梁桥预拱度设置特点，进一步得出了只伸缩上弦杆；同时伸缩上弦杆和斜腹杆；同时伸缩上弦杆、竖杆和斜腹杆三种情况下的杆件伸缩公式。可针对具体部位进行不同杆件伸缩，对桁梁进行无应力起拱。

3) 数值计算结果与设计值和参考文献计算值吻合，说明采用本文方法进行预拱度设置的可靠性。

4) 与荷载位移起拱法相比，二者的计算原理不同，但二者对N形桁架的起拱效果是一致的。

 [1] 中铁大桥勘测设计院有限公司. TB 10002.2-2005, 铁路桥梁钢结构设计规范(附条文说明)[S].北京: 中国铁道出版社, 2005. China Railway Bridge Survey and Design Institute Co., Ltd. TB 10002.2-2005, Code for design on steel structure of railway bridge[S]. Beijing: China Railway Publishing House, 2005. (0) [2] 中华人民共和国交通运输部. JTG D64-2015, 公路钢结构桥梁设计规范[S].北京: 人民交通出版社, 2015. People's Republic of China Ministry of Transport. JTG D64-2015, Specifications for design of highway steel bridge[S]. Beijing: People's Transportation Press, 2015. (0) [3] 冯海舟, 熊健民. 大跨度钢桁梁斜拉桥预拱度分析[J]. 北方交通, 2013(3): 57-60. FENG Haizhou, XIONG Jianmin. Analysis on the camber of large-span steel truss cable-stayed bridge[J]. Northern Communications, 2013(3): 57-60. DOI:10.3969/j.issn.1673-6052.2013.03.019 (0) [4] 冯沛. 大跨度铁路连续钢桁梁桥预拱度设置研究[J]. 铁道标准设计, 2016, 60(4): 62-64. FENG Pei. Study on camber-setting of large span railway steel truss bridge[J]. Railway standard design, 2016, 60(4): 62-64. (0) [5] 陈小佳, 崔太雷, 封仁博. 基于几何正装法的N式钢桁梁桥预拱度设置研究[J]. 铁道建筑, 2017(1): 72-75. CHEN Xiaojia, CUI Tailei, FENG Renbo. Study on pre-camber setting for n-type steel truss bridge based on geometric-forward-installation method[J]. Railway engineering, 2017(1): 72-75. (0) [6] 胡步毛, 艾宗良, 袁明, 等. 基于非线性规划实现钢桁连续梁预拱度[J]. 铁道工程学报, 2010, 27(4): 49-52. HU Bumao, AI Zongliang, YUAN Ming, et al. Use nonlinear programming to achieve camber of steel truss continuous bridge[J]. Journal of railway engineering society, 2010, 27(4): 49-52. DOI:10.3969/j.issn.1006-2106.2010.04.011 (0) [7] 李佳莉, 张谢东, 陈卫东, 等. 基于多目标规划的连续钢桁梁预拱度设置研究[J]. 武汉理工大学学报(交通科学与工程版), 2016, 40(2): 360-364. LI Jiali, ZHANG Xiedong, CHEN Weidong, et al. Multi-objective programming based study on pre-camber setting of steel truss continuous girder bridge[J]. Journal of Wuhan University of Technology (transportation science & engineering), 2016, 40(2): 360-364. DOI:10.3963/j.issn.2095-3844.2016.02.032 (0) [8] 向律楷, 鄢勇, 袁明, 等. 钢桁梁预拱度设置方法研究[J]. 四川建筑, 2015, 35(1): 150-153. XIANG Lyukai, YAN Yong, YUAN Ming, et al. Study on pre-camber setting of truss bridge[J]. Sichuan architecture, 2015, 35(1): 150-153. DOI:10.3969/j.issn.1007-8983.2015.01.054 (0) [9] 蔡禄荣, 王荣辉, 王钰. 大跨度柏式钢桁梁桥厂制预拱度设置研究[J]. 铁道学报, 2013, 35(4): 96-101. CAI Lurong, WANG Ronghui, WANG Yu. Study on plant precamber setting of large-span n type steel truss bridge[J]. Journal of the China railway society, 2013, 35(4): 96-101. DOI:10.3969/j.issn.1001-8360.2013.04.015 (0) [10] 苑仁安, 秦顺全, 王帆. 分阶段成形杆系结构几何非线性平衡方程[J]. 桥梁建设, 2014, 44(2): 50-55. YUAN Renan, QIN Shunquan, WANG Fan. Equilibrium equation for geometric nonlinearity of frame structure formed in stages[J]. Bridge construction, 2014, 44(2): 50-55. (0) [11] 许磊平, 秦顺全, 马润平. 基于平面壳单元的分阶段成形结构平衡方程[J]. 西南交通大学学报, 2013, 48(5): 857-862. XU Leiping, QIN Shunquan, MA Runping. Equilibrium equation derivation of structures formed by stages based on plane shell element[J]. Journal of Southwest Jiaotong University, 2013, 48(5): 857-862. DOI:10.3969/j.issn.0258-2724.2013.05.012 (0)