一类考虑翘曲畸变的空间梁单元位移模式

张尧; 董军; 李国华; 王秀芳

doi:10.6052/j.issn.1000-4750.2024.02.S021

一类考虑翘曲畸变的空间梁单元位移模式

张尧^1,,
董军^2, ,,
李国华^2,,
王秀芳^2,

1.
交通运输部公路科学研究院，北京 100088
2.
北京建筑大学工程结构与新材料北京市高等学校工程研究中心，北京 100044

基金项目:

国家自然科学基金项目(51808025)；北京市自然科学基金项目(8202012)；长江学者和创新团队发展计划资助项目(IRT_17R06)；北京建筑大学北京未来城市设计高精尖创新中心项目(UDC2019021424)；北京建筑大学金字塔人才培养工程项目(JDYC20200329)

详细信息

作者简介:
张　尧(1996−)，男，四川人，硕士生，主要从事桥梁与隧道工程、有限元程序设计研究(E-mail: 792641209@qq.com)

李国华(1974−)，女，山东人，讲师，博士，主要从事结构力学、结构工程抗震等研究(E-mail: liguohua@bucea.edu.cn)

王秀芳(1984−)，女，内蒙古人，副教授，博士，主要从事材料力学、古建筑保护等研究(E-mail: wangxiufang@bucea.edu.cn)

通讯作者:
董　军(1967−)，男，山东人，教授，博士，主要从事桥梁与隧道工程、结构工程及计算力学研究(E-mail: jdongcg@bucea.edu.cn)

中图分类号: TU31
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出版历程
- 收稿日期: 2024-02-21
- 修回日期: 2025-01-02
- 网络出版日期: 2025-02-18
- 刊出日期: 2025-06-24

A KIND OF DISPLACEMENT MODE OF SPATIAL BEAM ELEMENT CONSIDERING THE EFFECT OF WARPING AND DISTORTION

1.
Research Institute of Highway Ministry of Transport, Beijing 100088, China
2.
Higher Institution Engineering Research Center of Structural Engineering and New Materials, Beijing University of Civil Engineering and Architecture, Beijing 100044, China

摘要

摘要:
基于弹性理论和扭转几何关系，推导了单结点6自由度、低次、具备翘曲畸变计算能力的空间梁单元位移模式，并丰富了“两结点双线性插值梁单元(Bm_ZY)”的空间模型。进一步通过自编的ZQFEM有限元程序建立了约束扭转模型，研究了截面形状、壁厚对翘曲畸变的影响。数值算例验证了该梁单元位移模式的正确性，同时数值算例分析表明：翘曲、畸变效应，会显著增大截面的扭转刚度；Midas Civil的计算结果并不连续。
- Python /
- 有限元分析 /
- 空间梁单元 /
- 翘曲 /
- 畸变 /
- 扭转
Abstract:
Based on the elastic theory and on the torsional geometry relationship of a constrained beam, deduced is a constrained torsional model of a single-node 6-degree-of-freedom, and of low-order beam element with warpage and distortion calculation ability, and enriched is the space model of "two-node bilinear interpolation beam element (Bm_ZY)" . Furthermore, the constrained torsional model was established by ZQFEM software, and the influence of section shape and wall thickness on warpage and distortion was studied. The numerical examples verify the correctness of the displacement mode of the beam element, and the numerical analysis also shows that the warpage and distortion effects will significantly increase the torsional stiffness of the cross section of the beam, and that the results of Midas Civil calculations are not continuous.
- Python /
- finite element analysis /
- space beam element /
- warp /
- distortion /
- torsion

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图 1 梁单元坐标系

Figure 1. Beam element coordinate system

下载: 全尺寸图片幻灯片

图 2 空心截面扭转角与壁厚曲线图

Figure 2. A Curve of torsional Angle and wall thickness of hollow section

下载: 全尺寸图片幻灯片

1 矩形截面积分示意图

1. Integral diagram of rectangular section

下载: 全尺寸图片幻灯片

表 1 实心截面扭转角计算结果

Table 1 Calculation results of torsional angle of solid belly section

截面形状	Midas Civil_翘曲	Midas Civil_非翘曲	Python_初等梁^[14]	《材料力学》^[6]	Bm_ZY
实心圆形 (相对误差)	127 323 954.474 (+0.00%)	127 323 954.474 (+0.00%)	127 323 954.474 (+0.00%)	127 323 954.474 (+0.00%)	61 941 383.257 (−51.35%)
实心矩形 (相对误差)	9 121 250.417 (+14.02%)	10 288 856.640 (+28.61%)	8 000 000.000 (+0.00%)	8 000 000.000 (+0.00%)	3 182 634.608 (−60.22%)
实心正方形 (相对误差)	79 038 701.805 (+5.38%)	88 888 888.889 (+18.51%)	75 000 000.000 (+0.00%)	75 000 000.000 (+0.00%)	32 142 857.143 (−57.14%)

下载: 导出CSV

1 名词解释

1 Explanation of nouns

符号	含义	符号	含义	符号	含义	符号	含义
γ_yz	局部坐标系yoz剪应变	θ_yz	局部坐标系yoz面转角	“,”(下角标)	偏导简化符号	r	截面半径(外轮廓)
γ_zx	局部坐标系xoz剪应变	θ_zx	局部坐标系xoz面转角	A	梁单元截面积	h	截面高(外轮廓)
γ_xy	局部坐标系xoy剪应变	θ_xy	局部坐标系xoy面转角	L	梁单元长度	b	截面宽(外轮廓)
ε_x	局部坐标系x轴向应变	u	局部坐标系x轴向位移	µ	泊松比	E	弹性模量
ε_y	局部坐标系y轴向应变	v	局部坐标系y切向位移	U₂	弹性常数^[15]	U₁	弹性常数^[15]
ε_z	局部坐标系z轴向应变	w	局部坐标系z切向位移	D	本构矩阵	B	几何矩阵
k_s	剪切修正系数(剪切系数)	k_A	变截面参数	q	位移列阵	N	形函数矩阵
ε	应变矩阵	−	−	−	−	−	−

下载: 导出CSV

2 积分表

2 Table of integration

i-j-k	η_i-j-k
1-0-0	$\dfrac{{{b^2}{k_1} + {h^2}{k_2}}}{8}\dfrac{1}{{{h^2}}}$
2-0-0	$\dfrac{{{b^3}[\ln (k{k_1} + {k_1}) + {k_1}k{k_1}] + {h^3}[\ln (k{k_2} + {k_2}) + {k_2}k{k_2}]}}{{48}}\dfrac{1}{{{h^3}}}$
2-2-0	$\dfrac{{{b^3}[ - \ln (k{k_1} + {k_1}) + {k_1}k{k_1}] + 2{h^3}\ln (k{k_2} + {k_2})}}{{48}}\dfrac{1}{{{h^3}}}$
2-0-2	$\dfrac{{2{b^3}\ln (k{k_1} + {k_1}) + {h^3}[ - \ln (k{k_2} + {k_2}) + {k_2}k{k_2}]}}{{48}}\dfrac{1}{{{h^3}}}$
3-0-0	$\dfrac{{{b^4}(k{k_1^3}/3 + {k_1}) + {h^4}(k{k_2^3}/3 + {k_2})}}{{64}}\dfrac{1}{{{h^4}}}$
3-2-0	$\dfrac{{{b^4}k{k_1^3}/3 + {h^4}{k_2}}}{{64}}\dfrac{1}{{{h^4}}}$
3-0-2	$\dfrac{{{b^4}{k_1} + {h^4}k{k_2^3}/3}}{{64}}\dfrac{1}{{{h^4}}}$
3-2-2	$\dfrac{{{b^4}({k_1} - \arctan {k_1}) + {h^4}({k_2} + \arctan {k_1})}}{{64}}\dfrac{1}{{{h^4}}}$
3-4-0	$\dfrac{{{b^4}({k_1^3}/3 - {k_1} + \arctan {k_1}) + {h^4}(\pi /2 - \arctan {k_1})}}{{64}}\dfrac{1}{{{h^4}}}$
3-0-4	$\dfrac{{{b^4}\arctan {k_1} + {h^4}({k_2^3}/3 - {k_2} - \arctan {k_1})}}{{64}}\dfrac{1}{{{h^4}}}$
4-2-0	$\dfrac{{{b^5}[ - 5\ln (k{k_1} + {k_1}) + 3{k_1}k{k_1} + 2k{k_1^3}{k_1}] + 4{h^5}[\ln (k{k_2} + {k_2}) + {k_2}k{k_2}]}}{{1024}}\dfrac{1}{{{h^5}}}$
4-0-2	$\dfrac{{4{b^5}[\ln (k{k_1} + {k_1}) + {k_1}k{k_1}] + {h^5}[ - 5\ln (k{k_2} + {k_2}) + 3{k_2}k{k_2} + 2k{k_2^3}{k_2}]}}{{1024}}\dfrac{1}{{{h^5}}}$
4-4-0	$\dfrac{{{b^5}[3\ln (k{k_1} + {k_1}) + 5{k_1^3}k{k_1} - 3{k_1}k{k_1^3}] + 8{h^5}\ln (k{k_2} + {k_2})}}{{1024}}\dfrac{1}{{{h^5}}}$
4-0-4	$\dfrac{{8{b^5}\ln (k{k_1} + {k_1}) + {h^5}[3\ln (k{k_2} + {k_2}) + 5{k_2^3}k{k_2} - 3{k_2}k{k_2^3}]}}{{1024}}\dfrac{1}{{{h^5}}}$
5-2-0	$\dfrac{{{b^6}(k{k_1^5}/5 + {k_1^3}/3) + {h^6}(k{k_2^3}/3 + {k_2})}}{{320}}\dfrac{1}{{{h^6}}}$
5-0-2	$\dfrac{{{b^6}(k{k_1^3}/3 + {k_1}) + {h^6}(k{k_2^5}/5 + {k_2^3}/3)}}{{320}}\dfrac{1}{{{h^6}}}$

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参考文献(17)

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一类考虑翘曲畸变的空间梁单元位移模式

通讯作者: 董 军(1967−)，男，山东人，教授，博士，主要从事桥梁与隧道工程、结构工程及计算力学研究(E-mail: jdongcg@bucea.edu.cn)

计量

出版历程

A KIND OF DISPLACEMENT MODE OF SPATIAL BEAM ELEMENT CONSIDERING THE EFFECT OF WARPING AND DISTORTION

计量

出版历程

目录

通讯作者:
董　军(1967−)，男，山东人，教授，博士，主要从事桥梁与隧道工程、结构工程及计算力学研究(E-mail: jdongcg@bucea.edu.cn)