﻿ 变截面波形钢腹板箱梁剪应力计算理论
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 同济大学学报(自然科学版)  2019, Vol. 47 Issue (4): 475-481.  DOI: 10.11908/j.issn.0253-374x.2019.04.004 0

### 引用本文

LIU Chao, HUANG Yuhao, GAO Zhan. Calculation Theory of Shear Stress in Variable Section Box Girder with Corrugated Steel Webs[J]. Journal of Tongji University (Natural Science), 2019, 47(4): 475-481. DOI: 10.11908/j.issn.0253-374x.2019.04.004

### 文章历史

Calculation Theory of Shear Stress in Variable Section Box Girder with Corrugated Steel Webs
LIU Chao , HUANG Yuhao , GAO Zhan
College of Civil Engineering, Tongji University, Shanghai 200092, China
Abstract: Under the plane-section assumption and the premise that no relative slip exists, a calculation theory of shear stress in the variable depth and width box sections with corrugated steel webs based on elastic beam elements is given. The calculation theory proposes that shear, moment and axial force will all produce shear stress, and the latter two can produce shear stress only in the case of variable cross section. For further analysis, a finite element model is established, and a comparison among the traditional calculation theory, the calculation theory and the results of the finite element model is conducted. The calculation theory is proven to be reliable.
Key words: corrugated steel webs    calculation theory of shear stress    shear stress distribution    sections of variable width and depth    finite element analysis

1 剪应力计算方法 1.1 计算基本假定

1.2 波形钢腹板变截面计算理论

 图 1 波形钢腹板箱梁截面和截面微元区段 Fig.1 Box girder section with corrugated steel webs and microelement sector of section
 $\tau=\frac{\mathrm{d} F}{\mathrm{d} x} \frac{1}{b(y)}$ (1)

 $F=F_{N}+F_{M}$ (2)
 $F_{N}=\int_{0}^{y_{0}} \sigma_{y N} b(y) \mathrm{d} y=\frac{N}{A} A_{y_{0}}$ (3)
 $F_{M}=\int_{0}^{y_{0}} \sigma_{y M} b(y) \mathrm{d} y=\frac{M}{I} \int_{0}^{y_{0}}\left(y_{\mathrm{c}}-y\right) b(y) \mathrm{d} y$ (4)

 $\begin{array}{c}{\int_{0}^{y_{0}}\left(y_{c}-y\right) b(y) \mathrm{d} y=y_{c} \int_{0}^{y_{0}} b(y) \mathrm{d} y-} \\ {\int_{0}^{y_{0}} yb(y) \mathrm{d} y=\left(y_{\mathrm{c}}-y^{*}\right) A_{y_{0}}=S_{y_{0}}}\end{array}$ (5)

 $F=\frac{N}{A} A_{y_{0}}+\frac{M}{I} S_{y_{0}}$ (6)

 $N \tan \alpha \mathrm{d} x+Q \mathrm{d} x-\frac{1}{2} q(\mathrm{d} x)^{2}=\mathrm{d} M$ (7)
 图 2 轴向微元区段 Fig.2 Axial segmental infinitesimal element

 $\frac{\mathrm{d} M}{\mathrm{d} x}=N \tan \alpha+Q$ (8)

 $\begin{array}{l} \tau = \frac{1}{{b(y)}}\frac{{{\rm{d}}F}}{{{\rm{d}}x}} = \frac{1}{{b(y)}}\frac{{{\rm{d}}\left( {\frac{N}{A}{A_{{y_0}}} + \frac{M}{I}{S_{{y_0}}}} \right)}}{{{\rm{d}}x}} = \\ \;\;\;\;\;\frac{1}{{b(y)}}\left( {\frac{{\frac{{{\rm{d}}\left( {N{A_{{y_0}}}} \right)}}{{{\rm{d}}x}}A - \frac{{{\rm{d}}A}}{{{\rm{d}}x}}\left( {N{A_{{y_0}}}} \right)}}{{{A^2}}} + } \right.\\ \;\;\;\;\;\left. {\frac{{\frac{{{\rm{d}}\left( {M{S_{{y_0}}}} \right)}}{{{\rm{d}}x}}I - \frac{{{\rm{d}}I}}{{{\rm{d}}x}}\left( {M{S_{{y_0}}}} \right)}}{{{I^2}}}} \right) = \\ \;\;\;\;\;\frac{1}{{b(y)}}\left( {\frac{{{A_{{y_0}}}}}{A}\frac{{{\rm{d}}N}}{{{\rm{d}}x}} + \frac{N}{A}\frac{{{\rm{d}}{A_{{y_0}}}}}{{{\rm{d}}x}} - \frac{{N{A_{{y_0}}}}}{{{A^2}}}\frac{{{\rm{d}}A}}{{{\rm{d}}x}} + \frac{M}{I}\frac{{{\rm{d}}{S_{{y_0}}}}}{{{\rm{d}}x}} + } \right.\\ \;\;\;\;\;\left. {\frac{{{S_{{y_0}}}}}{I}\frac{{{\rm{d}}M}}{{{\rm{d}}x}} - \frac{{M{S_{{y_0}}}}}{{{I^2}}}\frac{{{\rm{d}}I}}{{{\rm{d}}x}}} \right) = \frac{M}{{Ib(y)}}\left( {\frac{{{\rm{d}}{S_{{y_0}}}}}{{{\rm{d}}x}} - } \right.\\ \;\;\;\;\;\frac{{{S_{{y_0}}}}}{I}\frac{{{\rm{d}}I}}{{{\rm{d}}x}}) + \frac{{Q{S_{{y_0}}}}}{{Ib(y)}} + \frac{N}{{b(y)}}\left( {\frac{{{S_{{y_0}}}\tan \alpha }}{I} + } \right.\\ \;\;\;\;\;\frac{1}{A}\frac{{{\rm{d}}{A_{{y_0}}}}}{{{\rm{d}}x}} - \frac{{{A_{{y_0}}}}}{{{A^2}}}\frac{{{\rm{d}}A}}{{{\rm{d}}x}}) + \frac{{{\rm{d}}N}}{{{\rm{d}}x}}\frac{{{A_{{y_0}}}}}{{b(y)A}} \end{array}$ (9)

 $\begin{array}{l} \tau = \frac{M}{{Ib(y)}}\left( {\frac{{{\rm{d}}{S_{{y_0}}}}}{{{\rm{d}}x}} - \frac{{{S_{{y_0}}}}}{I}\frac{{{\rm{d}}I}}{{{\rm{d}}x}}} \right) + \frac{{Q{S_{{y_b}}}}}{{Ib(y)}} + \\ \;\;\;\frac{N}{{b(y)}}\left( {\frac{{{S_{{y_0}}}\tan \alpha }}{I} + \frac{1}{A}\frac{{{\rm{d}}{A_{{y_0}}}}}{{{\rm{d}}x}} - \frac{{{A_{{y_0}}}}}{{{A^2}}}\frac{{{\rm{d}}A}}{{{\rm{d}}x}}} \right) \end{array}$ (10)

 $\tau=\frac{Q S_{y_{0}}}{I b(y)}$ (11)

 $\tau=\tau_{M}+\tau_{Q}+\tau_{N}$ (12)

 $A=b_{1} t_{1}+b_{2} t_{2}$ (13)
 $\begin{array}{c}{I=\frac{1}{12}\left(b_{1} t_{1}^{3}+b_{2} t_{2}^{3}\right)+b_{1} t_{1}\left(y_{\mathrm{c}}-\frac{t_{1}}{2}\right)^{2}+} \\ {b_{2} t_{2}\left(h-y_{\mathrm{c}}-\frac{t_{2}}{2}\right)^{2}}\end{array}$ (14)
 $y_{\mathrm{c}}=\frac{b_{1} \frac{t_{1}^{2}}{2}+b_{2} t_{2}\left(h-\frac{t_{2}}{2}\right)}{A}$ (15)

 $\frac{\mathrm{d} A}{\mathrm{d} x}=b_{1} \frac{\mathrm{d} t_{1}}{\mathrm{d} x}+t_{1} \frac{\mathrm{d} b_{1}}{\mathrm{d} x}+b_{2} \frac{\mathrm{d} t_{2}}{\mathrm{d} x}+t_{2} \frac{\mathrm{d} b_{2}}{\mathrm{d} x}$ (16)

 \begin{aligned} \frac{\mathrm{d} I}{\mathrm{d} x}=& k_{1} \frac{\mathrm{d} b_{1}}{\mathrm{d} x}+k_{2} \frac{\mathrm{d} b_{2}}{\mathrm{d} x}+k_{3} \frac{\mathrm{d} t_{1}}{\mathrm{d} x}+k_{4} \frac{\mathrm{d} t_{2}}{\mathrm{d} x}+\\ & k_{5} \frac{\mathrm{d} y_{\mathrm{c}}}{\mathrm{d} x}+k_{6} \frac{\mathrm{d} h}{\mathrm{d} x} \end{aligned} (17)

 $\begin{array}{*{20}{c}} {{k_1} = t_1^3 + {{\left( {{y_{\rm{c}}} - \frac{{{t_1}}}{2}} \right)}^2}{t_1},{k_2} = t_2^3 + {{\left( {h - {y_{\rm{c}}} - \frac{{{t_2}}}{2}} \right)}^2}{t_2},}\\ {{k_3} = \frac{{{b_1}t_1^2}}{4} + {b_1}{{\left( {{y_{\rm{c}}} - \frac{{{t_1}}}{2}} \right)}^2} - {b_1}{t_1}\left( {{y_{\rm{c}}} - \frac{{{t_1}}}{2}} \right),}\\ {{k_4} = \frac{{{b_2}t_2^2}}{4} + {b_2}{{\left( {h - {y_{\rm{c}}} - \frac{{{t_2}}}{2}} \right)}^2} - {b_2}{t_2}\left( {h - {y_{\rm{c}}} - \frac{{{t_2}}}{2}} \right),}\\ {{k_5} = 2{b_1}{t_1}\left( {{y_{\rm{c}}} - \frac{{{t_1}}}{2}} \right) - 2{b_2}{t_2}\left( {h - {y_{\rm{c}}} - \frac{{{t_2}}}{2}} \right),}\\ {{k_6} = 2{b_2}{t_2}\left( {h - {y_{\rm{c}}} - \frac{{{t_2}}}{2}} \right)} \end{array}$ (18)

(1) y0处于上翼缘板内(0 < y0 < t1)

 $\begin{array}{*{20}{c}} {{S_{{y_0}}} = {b_1}{y_0}\left( {{y_{\rm{c}}} - \frac{{{y_0}}}{2}} \right),{A_{{y_0}}} = {b_1}{y_0},\frac{{{\rm{d}}{A_{{y_0}}}}}{{{\rm{d}}x}} = \frac{{{\rm{d}}{b_1}}}{{{\rm{d}}x}}{y_0},}\\ {\frac{{{\rm{d}}{S_{{y_0}}}}}{{{\rm{d}}x}} = {y_0}\left( {{y_{\rm{c}}} - \frac{{{y_0}}}{2}} \right)\frac{{{\rm{d}}{b_1}}}{{{\rm{d}}x}} + {b_1}{y_0}\frac{{{\rm{d}}{y_{\rm{c}}}}}{{{\rm{d}}x}}} \end{array}$ (19)

(2) y0处于腹板内(t1y0ht2)

 $\begin{array}{c}{S_{y_{0}}=b_{1} t_{1}\left(y_{c}-\frac{t_{1}}{2}\right), A_{y_{0}}=b_{1} t_{1}, \frac{\mathrm{d} A_{y_{0}}}{\mathrm{d} x}=\frac{\mathrm{d} b_{1}}{\mathrm{d} x} t_{1}+\frac{\mathrm{d} t_{1}}{\mathrm{d} x} b_{1}} ,\\ {\frac{\mathrm{d} S_{y_{0}}}{\mathrm{d} x}=t_{1}\left(y_{\mathrm{c}}-\frac{t_{1}}{2}\right) \frac{\mathrm{d} b_{1}}{\mathrm{d} x}+b_{1} t_{1} \frac{\mathrm{d} y_{\mathrm{c}}}{\mathrm{d} x}+} \\ {\left(y_{\mathrm{c}} b_{1}-b_{1}\right) \frac{\mathrm{d} t_{1}}{\mathrm{d} x}}\end{array}$ (20)

(3) y0处于下翼缘板内(ht2 < y0 < h)

 $\begin{array}{l} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{A_{{y_0}}} = {b_1}{t_1} + {b_2}\left( {{y_0} + {t_2} - h} \right),\\ \frac{{{\rm{d}}{A_{{y_0}}}}}{{{\rm{d}}x}} = {t_1}\frac{{{\rm{d}}{b_1}}}{{{\rm{d}}x}} + {b_1}\frac{{{\rm{d}}{t_1}}}{{{\rm{d}}x}} + \left( {{y_0} + {t_2} - h} \right)\frac{{{\rm{d}}{b_2}}}{{{\rm{d}}x}} + {b_2}\frac{{{\rm{d}}{t_2}}}{{{\rm{d}}x}} - {b_2}\frac{{{\rm{d}}h}}{{{\rm{d}}x}},\\ {S_{{y_0}}} = {b_1}{t_1}\left( {{y_{\rm{c}}} - \frac{{{t_1}}}{2}} \right) - {b_2}\left( {y + {t_2} - h} \right)\frac{{\left( {h - {t_2} + {y_0} - 2{y_{\rm{c}}}} \right)}}{2}\\ \frac{{{\rm{d}}{S_{{y_0}}}}}{{{\rm{d}}x}} = {\left. {\frac{{{\rm{d}}{S_{{y_0}}}}}{{{\rm{d}}x}}} \right|_{\left( {{y_0} = {t_1}} \right)}} + \frac{1}{2}\left( {{y_0} + {t_2} - h} \right)\left( {h - {t_2} + {y_0} - } \right.\\ \;\;\;\;\;\;\;\;\;\;\;2{y_{\rm{c}}})\frac{{{\rm{d}}{b_2}}}{{{\rm{d}}x}} + {b_2}\left( {\left( {h - {y_{\rm{c}}} - {t_2}} \right)\frac{{{\rm{d}}{t_2}}}{{{\rm{d}}x}} - } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left( {h - {y_{\rm{c}}} - {t_2}} \right)\frac{{{\rm{d}}h}}{{{\rm{d}}x}} - \left( {{y_0} + {t_2} - h} \right)\frac{{{\rm{d}}{y_{\rm{c}}}}}{{{\rm{d}}x}}) \end{array}$ (21)

2 有限元模型

 图 3 有限元模型 Fig.3 Finite element model
 图 4 箱梁截面及波形钢腹板 Fig.4 Box girder section and corrugated steel web

3 结果分析 3.1 有限元正确性验证

 $\delta_{M}=\int_{0}^{l} \frac{M_{0}(x) M_{F}(x)}{E I(x)} \mathrm{d} x$ (22)

 $U_{\mathrm{s}}=\int_{0}^{l} \frac{Q^{2}}{2 A G} \mathrm{d} x=\int_{0}^{l} \frac{\tau^{2}}{2 G} \mathrm{d} x \mathrm{d} y \mathrm{d} z$ (23)

 $\delta_{Q}=\frac{\partial U_{\mathrm{s}}}{\partial Q}$ (24)

 图 5 各种挠度计算方法对比 Fig.5 Comparison of various deflection calculation methods
3.2 整体应力分析

 图 6 悬臂梁剪应力云图 Fig.6 Shear stress nephogram of cantilever girder
3.3 截面剪应力分析

 图 7 截面剪应力分布 Fig.7 Sectional shear stress distribution

3.4 顶、底板剪力分配

 $\rho_{\rm{w}}=\frac{\int \tau_{\rm{w}} \mathrm{d} A}{Q}$ (20)

 图 8 剪力分配比例 Fig.8 Distribution ratio of shear force

3.5 剪力、轴力、弯矩剪应力分布

 图 9 剪应力分配比例 Fig.9 Distribution proportion of shear stress

4 结论

(1) 提出了一种针对变宽变高波形钢腹板桥梁的剪应力计算方法，认为剪力、弯矩和轴力均会产生剪应力，并且后两者仅仅在变截面时产生剪应力.本文方法计算结果与有限元模型计算结果吻合程度高，可为波形钢腹板桥梁的抗剪设计提供依据.

(2) 针对变截面波形钢腹板桥梁，传统剪应力计算理论适用性较差.在腹板断面上，弯矩和轴力产生的剪应力方向与剪力剪应力方向相反，并且在弯矩较大时，弯矩和轴力产生的剪应力与剪力剪应力的比值可达0.6以上，从而导致实际剪应力数值小于剪力剪应力，这也是传统剪应力计算理论误差较大的原因.

(3) 波形钢腹板截面顶板剪应力很小，底板剪应力相对顶板而言较大，主要来源于底板受到的压应力的竖向分力.在无弯矩或弯矩很小区域认为仅由腹板承剪是可以接受的，但当弯矩较大时，顶、底板承剪比例能达到50%以上，因此应考虑顶、底板的承剪能力.传统计算理论、计算模型和规范认为剪力全部由钢腹板承担而不考虑混凝土顶、底板作用，容易造成波形钢腹板浪费，顶、底混凝土板不足的情况.

(4) 对单箱双室波形钢腹板桥梁进行有限元分析，发现腹板间存在剪应力分配，中腹板剪应力较小，边腹板剪应力较大.本文提出的计算理论对边腹板吻合程度高，在一定程度上提高了结构安全性.

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