﻿ 既有管道与内衬叠合界面受力性能及计算方法
 哈尔滨工业大学学报  2020, Vol. 52 Issue (11): 167-174  DOI: 10.11918/201904203 0

### 引用本文

ZHAO Yahong, MA Baosong, ZHANG Haifeng, HE Chunliang, SHI Guopeng. Mechanical behavior and calculation method of interface between host pipeline and lining[J]. Journal of Harbin Institute of Technology, 2020, 52(11): 167-174. DOI: 10.11918/201904203.

### 文章历史

1. 中国地质大学 工程学院，武汉 430074;
2. 中山大学 土木工程学院，广东 珠海 519082

Mechanical behavior and calculation method of interface between host pipeline and lining
ZHAO Yahong1, MA Baosong2, ZHANG Haifeng1, HE Chunliang1, SHI Guopeng1
1. Faculty of Engineering, China University of Geosciences, Wuhan 430074, China;
2. School of Civil Engineering, Sun Yat-sen University, Zhuhai 519082, Guangdong, China
Abstract: To understand the repair effect of trenchless centrifugal spraying method on defective pipeline and reveal the failure mechanism of the repaired interface and "pipeline-lining" system, a composite curved beam model was established. Analytical formulae of the cross section stress and the interface stress of the composite curved beam with different materials were derived by using the method of variable cross section, and the accuracy of the formulae was verified through the Three Edge Bearing Test (TEBT) of the repaired pipeline under concentrated load. The criterion for judging the coordinated deformation of the "pipe-lining" system was given—when the resistance of the interface was greater than the load, a composite structure was formed; otherwise, a unitized structure was formed. Through the analysis of the parameter study, a simplified interface shear stress formula was obtained based on the ratio β of lining thickness to host wall thickness, the ratio η of elastic modulus, and the diameter of the existing pipe D. The accuracy error between the proposed model and the mechanical model was less than 1%. Finally, the design method of lining repair for existing pipelines with different failure modes was given.
Keywords: reinforced concrete pipe    composite curved beam    co-deformation    design method    lining thickness

1 叠合曲梁应力模型

1.1 截面正应力

 图 1 叠合曲梁应力分析 Fig. 1 Stress analysis of composite curved beam

 $\left\{ {\begin{array}{*{20}{l}} {{\sigma _{{\rm{sa}}}} = \frac{{N{E_{\rm{a}}}}}{{{E_{\rm{a}}}{A_{\rm{a}}} + {E_{\rm{b}}}{A_{\rm{b}}}}} - \frac{{M{E_{\rm{a}}}}}{{({E_{\rm{a}}}{A_{\rm{a}}} + {E_{\rm{b}}}{A_{\rm{b}}})r}} + \frac{{M{E_{\rm{a}}}}}{{{E_{\rm{a}}}{J_{{\rm{za}}}} + {E_{\rm{b}}}{J_{{\rm{zb}}}}}} \cdot \frac{y}{{1 - y/r}},{y^\prime } - {h_{\rm{a}}} - {h_{\rm{b}}} \le y \le {y^\prime } - {h_{\rm{b}}},}\\ {{\sigma _{{\rm{sb}}}} = \frac{{N{E_{\rm{b}}}}}{{{E_{\rm{a}}}{A_{\rm{a}}} + {E_{\rm{b}}}{A_{\rm{b}}}}} - \frac{{M{E_{\rm{b}}}}}{{({E_{\rm{a}}}{A_{\rm{a}}} + {E_{\rm{b}}}{A_{\rm{b}}})r}} + \frac{{M{E_{\rm{b}}}}}{{{E_{\rm{a}}}{J_{{\rm{za}}}} + {E_{\rm{b}}}{J_{{\rm{zb}}}}}} \cdot \frac{y}{{1 - y/r}},{y^\prime } - {h_{\rm{b}}} \le y \le {y^\prime }.} \end{array}} \right.$ (1)

1.2 界面径向应力

 $\begin{array}{*{20}{l}} {{\sigma _{\rm{r}}} = \frac{1}{{b(r - y)}}\left[ {\left( {\frac{{N{E_{\rm{b}}}}}{{{E_{\rm{a}}}{A_{\rm{a}}} + {E_{\rm{b}}}{A_{\rm{b}}}}} - \frac{{M{E_{\rm{b}}}}}{{({E_{\rm{a}}}{A_{\rm{a}}} + {E_{\rm{b}}}{A_{\rm{b}}})r}}} \right){{\bar A}_{\rm{b}}} + } \right.}\\ {\left. {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{{M{E_{\rm{b}}}}}{{{E_{\rm{a}}}{J_{{\rm{za}}}} + {E_{\rm{b}}}{J_{{\rm{zb}}}}}} \cdot {{\bar K}_{{\rm{zb}}}}} \right],{y^\prime } - {h_{\rm{b}}} \le y \le {y^\prime }.} \end{array}$ (2)

y=y′-hb, 即为叠合材料界面处的径向应力.当上下材料相同时，式(2)退化为弯曲单梁的界面径向应力计算公式.

1.3 界面剪切应力

 ${\tau ^\prime } = \frac{{rV}}{{b(r - y)}}\left( {\frac{{{E_{\rm{b}}}{{\bar K}_{{\rm{zb}}}}}}{{{E_{\rm{a}}}{J_{{\rm{za}}}} + {E_{\rm{b}}}{J_{{\rm{zb}}}}}} - \frac{1}{r} \cdot \frac{{3E_{\rm{a}}^2h_{\rm{a}}^3 + 4E_{\rm{a}}^2h_{\rm{a}}^2{h_{\rm{b}}} + 3{E_{\rm{a}}}{E_{\rm{b}}}h_{\rm{a}}^2{h_{\rm{b}}} + 5{E_{\rm{a}}}{E_{\rm{b}}}{h_{\rm{a}}}h_{\rm{b}}^2 + E_{\rm{b}}^2h_{\rm{b}}^3}}{{{E_{\rm{a}}}{h_{\rm{a}}}{h_{\rm{b}}}({E_{\rm{a}}}{h_{\rm{a}}} + {E_{\rm{b}}}{h_{\rm{b}}}){k_1}}}} \right).$ (3)

 $\overline {{\tau ^\prime }} = \frac{{3V}}{{2b}}\frac{{{{({E_{\rm{a}}}h_{\rm{a}}^2 + 2{E_{\rm{a}}}{h_{\rm{a}}}{h_{\rm{b}}} + {E_{\rm{b}}}h_{\rm{b}}^2)}^2} - 4{{({E_{\rm{a}}}{h_{\rm{a}}} + {E_{\rm{b}}}{h_{\rm{b}}})}^2} \cdot {y^2}}}{{{E_{\rm{b}}}{h_{\rm{a}}}h_{\rm{b}}^3({E_{\rm{a}}}{h_{\rm{a}}} + {E_{\rm{b}}}{h_{\rm{b}}}){k_1}}},$ (4)

y=y′-hb, 所得值即为下端界面的剪切应力τ′, 叠合梁剪切应力界面处存在突变，这是由于假定剪切应力沿界面的宽度方向均匀分布引起的.根据力的平衡原理，界面的实际剪切应力应为式(3)、(4)乘相应的模量比.

2 叠合曲梁受力模型判断标准 2.1 界面黏结张拉强度

2.2 界面抗剪强度

GB50010—2015《混凝土结构设计规范》给出不配箍筋叠合板叠合面的抗剪强度标准值取0.6 MPa; JTG TJ22—2008《公路桥梁加固设计规范》中规定配置箍筋的叠合面，混凝土纯剪切强度设计值取抗压强度设计值的0.12倍; 对于不配置钢筋的叠合面，混凝土纯剪切强度设计值取0.45 MPa.

2.3 叠合曲梁协调变形条件

 $K \cdot S < R.$ (5)

3 试验验证及参数研究

3.1 1 000 mm管道修复模型验证

Shi等[5]的试验研究中，既有管道为DN1 000 mm加筋混凝土管材，混凝土抗拉强度为4.41 MPa，弹性模量Ea为31 950 MPa，考虑到管材的腐蚀缺陷，原壁厚由100 mm减至82 mm; 修复用砂浆的抗拉强度为2.93 MPa，弹性模量Eb=8 900 MPa，修复后内衬壁厚为50 mm.试验中测得的3次破坏荷载分别为105.1，117.2及131.9 kN/m，平均值为118.1 kN/m.

 图 2 管道-内衬结构尺寸及受力模式(mm) Fig. 2 Pipe-lining structure dimensions and force patterns(mm)
 ${M = \frac{{PR}}{\pi } \cdot \left( {1 - \frac{\pi }{2}{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha } \right),}$ (6)
 ${V = \frac{P}{2}{\rm{cos}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha ,}$ (7)
 ${N = \frac{P}{2}{\rm{sin}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \alpha .}$ (8)

 图 3 叠合界面随荷载变化的失效过程 Fig. 3 Failure process of composite interface with increasing load P
 图 4 修复后的管道试件在荷载达到Pc前脱离[5] Fig. 4 Repaired pipe sample breaks away before the crack load reaches Pc[5]

3.2 材料弹性模量对界面应力的影响

 图 5 界面应力与材料模量比间的关系 Fig. 5 Relation between interface stress and material modulus ratio

 图 6 叠合曲梁模型简化误差 Fig. 6 Simplified error of composite curved beam model
3.3 管道直径对界面剪切应力的影响

3.2的研究表明，界面剪切应力是内衬壁厚设计的控制因素，而界面径向应力条件往往都能满足.因此，下文仅研究不同参数对界面剪切应力的影响.图 7为不同直径管道在修复后的界面剪切应力分布情况，图中给出了4种β(内衬厚度与既有管道壁厚比值)的界面剪切应力-管径曲线.

 图 7 几种β值对应的界面剪切应力-直径关系(η=1.19) Fig. 7 Relation between interface shear stress and pipe diameter with different β(η=1.19)

3.4 内衬厚度对界面剪切应力的影响

 图 8 几种不同管径对应的τ-β变化关系 Fig. 8 Relation between interface shear stress and β with different diameters

3.5 界面剪切应力的简化公式

 $\tau = \frac{{DV}}{{b(D - 2y)}} \cdot {\kappa _\tau } \cdot {D^{ - 0.911}}.$ (9)

4 内衬厚度设计方法

4.1 管道的破坏模式

 图 9 三边承载试验中管顶荷载与管径变形量关系 Fig. 9 Relation between pipe top load and pipe diameter deformation in TEBT
4.2 内衬厚度设计方法

 图 10 内衬与既有管道形成叠合结构受力模式 Fig. 10 Patterns of composite structure between lining and host pipeline

 $\left\{ {\begin{array}{*{20}{l}} {\tau = \frac{V}{{b(D - 2y)}} \cdot {\kappa _\tau } \cdot {D^{0.089}} < [\tau ],}\\ {{\sigma _r} = \frac{1}{{b(r - y)}}({\kappa _{{\sigma _1}}}N \pm {\kappa _{{\sigma _2}}}M) < [{\sigma _{\rm{r}}}].} \end{array}} \right.$ (10)

 ${\sigma _{\rm{z}}} = \frac{N}{{\bar A}} \pm \frac{{My}}{{\bar I}} < [{\sigma _{\rm{z}}}].$ (11)

 图 11 “管道-内衬”体系内衬壁厚设计示意 Fig. 11 Design of lining wall thickness in"pipe-lining" system
5 结论

1) 内衬层与既有管道协调变形的条件是界面的径向张拉应力与黏结张拉强度、界面剪切应力与抗剪强度的大小关系.当抗力大于荷载时，两者协调变形，形成叠合结构，否则形成复合结构.

2) 通常界面的抗剪条件为不同受力模型的主要判断标准.界面的剪切应力可简化为与旧管管径D、内衬厚度与既有管道壁厚比值β及材料的弹性模量比η等参数相关的函数.

3) 当修复材料与既有管道材料的弹性模量比值较大时(如用玻纤树脂及碳纤维塑料内衬修复既有管道)，必须大幅度增加两种材料界面的抗剪强度，以保证“管道-内衬”体系协调变形.

4) 界面叠合作用会随着外荷载的增加而逐步失效，为了兼顾工程安全性及经济性，应以管道正常使用极限状态，根据界面剪切应力公式设计修复壁厚.最理想的状态是界面脱离均发生在荷载达到Pc之后.

5) 内衬壁厚设计时应进行剩余强度评价，并选取管顶、管底和起拱线等危险点进行设计.应同时考虑协调变形条件及材料的性能，若协调变形条件无法满足，则只能按照复合结构模型进行壁厚设计.

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