工程科学与技术   2019, Vol. 51 Issue (2): 71-77

1. 长沙理工大学 土木工程学院，湖南 长沙 410114;
2. 湖南科技大学 土木工程学院，湖南 湘潭 411201

Calculation Method of Rock-socketed Depth for Rock-socketed Piles Based on Bedrock Horizontal Failure
YIN Pingbao1, YANG Ying1, LEI Yong2, HE Wei1, ZHANG Jianren1
1. School of Civil Eng., Changsha Univ. of Sci. & Technol., Changsha 410114, China;
2. College of Civil Eng., Hunan Univ. of Sci. and Technol., Xiangtan 411201, China
Abstract: In order to calculate the rock-socketed depth of rock-socketed piles, the failure characteristics of rock outside piles and the horizontal force and bending moment on the top of bedrock should be taken into account. According to horizontal ultimate bearing characteristics of rock-socketed pile, a simplified analysis model of rock-socketed depth calculation was established. Based on the Hoek–Brown strength criterion and static balance principle, a theoretical calculation formula of rock-socketed depth of rock-socketed piles was deduced with the consideration of horizontal force and bending moment at the top of bedrock. Comparison and analysis of an engineering example showed that values of rock-socketed depth calculated by theoretical formulas are closer to the method of codes. And then the influences of horizontal load, pile diameter, rock uniaxial compressive strength and rock mass rating indexRMR on rock-socketed depths were discussed. The results showed that the rock-socketed depth increases linearly with the increase of horizontal force, non-linearly with the increase of bending moment, and decreases non-linearly with the increase of pile diameter, rock uniaxial compressive strength and rock mass rating index RMR. At the same condition, when pile diameter increases from 1.0 m to 2.0 m and 3.0 m, rock-socketed depth decreases by 32% and 44%, respectively. When rock uniaxial compressive strength increases from 15 MPa to 30, 45 and 60 MPa, rock-socketed depth reduces by 44.4%, 59.7% and 67.6%, respectively. When the value of rock mass rating index RMR increases from 30 to 45, 60 and 75, rock-socketed depth reduces by 48.9%, 72.3% and 84.2%, respectively. Rock mass rating index RMR has a significant effect on rock-socketed depth. If the value of rock mass rating index RMR is more than 85, the rock-socketed depth will less than 1.0 m. When determining the optimum rock-socketed depth of actual rock socketed piles, the strength and quality of rocks and pile diameters should be considered comprehensively.
Key words: pile foundation    rock-socketed depth    horizontal load    Hoek–Brown strength criterion    ultimate bearing capacity

1 计算模型及破坏准则 1.1 简化计算模型建立

 图1 嵌岩深度计算模型 Fig. 1 Calculation model for rock-socketed depth

1）嵌岩桩由于水平力作用而产生水平转动，嵌岩段桩侧岩石受到桩前法向正应力和桩侧切向剪应力共同作用。假定嵌岩段桩侧岩石水平抗力自上而下随深度呈现线性变化，如图1所示，且满足如下关系：

 $p = - \eta {\textit{z}} + \kappa$ (1)

2）以圆形桩为例，如图2所示，在水平荷载作用下，基岩顶面处桩侧法向应力 $\sigma$ 和切向剪应力 $\tau$ 分别为：

 图2 桩侧法向应力及切向剪应力分布 Fig. 2 Distribution of normal stress and shear stress outside piles

 $\sigma = {\sigma _{\rm{m}}}\cos \,\alpha$ (2)
 $\tau {\rm{ = }}{\tau _{\rm{m}}}\sin\;2\alpha$ (3)

3）根据图12中的几何关系，可知基岩顶面处桩侧岩石水平极限抗力是由桩侧总法向应力 $p$ n和总水平摩阻力 $p_\tau$ 两部分组成，其计算式为：

 ${p_{\rm{n}}}{\rm{ = 2}}\int_0^{\textstyle\frac{{\text{π}} }{2}} {\frac{d}{2}{\sigma _{\rm{m}}}} {\cos ^2}\;\alpha {\rm{d}}\alpha {\rm{ = }}\frac{{\text{π}} }{4}d{\sigma _{\rm{m}}}$ (4)
 ${p_{\rm{\tau }}}{\rm{ = 2}}\int_0^{\textstyle\frac{{\text{π}} }{2}} {\frac{d}{2}{\tau _{\rm{m}}}} \sin (2\alpha )\cos \,\alpha {\rm{d}}\alpha = \frac{2}{3}d{\tau _{\rm{m}}}$ (5)

 ${p_{\rm{u}}} = {p_{\rm{n}}} + {p_{\rm{\tau }}} = d\left( {\frac{{\text{π}} }{4}{\sigma _{\rm{m}}}{\rm{ + }}\frac{2}{3}{\tau _{\rm{m}}}} \right)$ (6)

1.2 Hoek–Brown破坏准则

 图3 岩石的强度准则和Mohr应力圆 Fig. 3 Strength criterion for rock and Mohr stress circles

 ${\tau _{\rm{s}}} = g(\sigma )$ (7)

 $\frac{{{\sigma _1} - {\sigma _3}}}{2} = \frac{{{\tau _{\rm{s}}}}}{{\cos \,\beta }}$ (8)
 $\frac{{{\sigma _1}{\rm{ + }}{\sigma _3}}}{2} = \sigma {\rm{ + }}{\tau _{\rm{s}}}\tan \,\beta$ (9)

Hoek–Brown破坏准则对应的大、小主应力之间的关系为[1415]

 ${\left( {{\sigma _1}{\rm{ - }}{\sigma _3}} \right)^2} = {\sigma _3}{\sigma _{\rm{c}}}m + \sigma _{\rm{c}}^2s$ (10)

 $m = {m_0}\exp \frac{{RMR - 100}}{a}$ (11)
 $s = \exp \frac{{RMR - 100}}{b}$ (12)

 ${\left( {q + \frac{{m{\sigma _{\rm{c}}}}}{8}} \right)^2} = \frac{m}{4}{\sigma _{\rm{c}}}p + \left( {\frac{{{m^2}}}{{64}} + \frac{s}{4}} \right)\sigma _{\rm{c}}^2$ (13)

$\lambda = m{\sigma _{\rm{c}}}/8$ $\xi = 8s/{m^2}$ ，则式（13）可进一步无量纲化为：

 ${(\bar q{\rm{ + }}1)^2} = 2(\bar p + \xi ){\rm{ + }}1$ (14)

 $\sin \,\rho = \frac{{{\rm{d}}\bar q}}{{{\rm{d}}\bar p}} = \frac{1}{{1 + \bar q}}$ (15)

 $\bar \tau = \frac{\tau }{\lambda } = \frac{{1 - \sin\, \rho }}{{\tan \rho }}$ (16)
 $\bar \sigma = \frac{\sigma }{\lambda } + \xi = {\left( {\frac{{1 - \sin \,\rho }}{{\sin\, \rho }}} \right)^2}\frac{{\left( {1 + 2\sin \,\rho } \right)}}{2}$ (17)

 $2{\sin ^3}\,\rho - (2\bar \sigma + 3){\sin ^2}\,\rho + 1 = 0$ (18)

 $\rho {\rm{ = }}\frac{{\text{π}} }{2} - \sqrt[\scriptstyle 4]{{\frac{{8\bar \sigma }}{3}}}$ (19)

 $\bar \tau \approx 1.25{\bar \sigma ^{0.75}}$ (20)

 图4 Hoek–Brown破坏准则的拟合曲线 Fig. 4 Fitting curves of Hoek–Brown strength criterion

 $\frac{\tau }{\lambda } = 1.25{\left( {\frac{\sigma }{\lambda } + \xi } \right)^{0.75}}$ (21)

2 基桩嵌岩深度计算 2.1 岩石水平极限抗力计算

 ${\sigma _1}{\rm{ = }}{\sigma _{\rm{m}}}$ (22)
 ${\sigma _3}{\rm{ = }}{\sigma _{\rm{v}}} = \sum\limits_{i = 1}^n {{l_i}} {\gamma _{{\rm{si}}}}$ (23)

 ${\sigma _{\rm{m}}} = {\sigma _{\rm{v}}} + \sqrt {{\sigma _{\rm{c}}}m{\sigma _{\rm{v}}} + \sigma _{\rm{c}}^2s}$ (24)

$\alpha$ =π/4时，桩侧切向剪应力 $\tau$ m最大，根据式（2）可得桩侧岩石的法向应力为：

 $\sigma = {\sigma _{\rm{m}}}\cos \frac{{\text{π}} }{4}{\rm{ = }}\frac{{\sqrt 2 }}{2}{\sigma _{\rm{m}}}{\rm{ = }}\frac{{\sqrt 2 }}{2}\left( {{\sigma _{\rm{v}}} + \sqrt {{\sigma _{\rm{c}}}m{\sigma _{\rm{v}}} + \sigma _{\rm{c}}^2s} } \right)$ (25)

 \begin{aligned}[b] {\tau _{\rm{m}}} &= 1.{\rm{25}}\lambda {\left( {\frac{\sigma }{\lambda } + \xi } \right)^{0.75}} = \\ &\,\,\,\,\,\,\,\,\,\,1.25\lambda {\left\{ {\frac{{\sqrt 2 }}{{2\lambda }}\left[ {{\sigma _{\rm{v}}} + \sqrt {{\sigma _{\rm{c}}}m{\sigma _{\rm{v}}} + \sigma _{\rm{c}}^2s} } \right] + \xi } \right\}^{0.75}} \end{aligned}\!\!\!\!\!\! (26)

 \begin{aligned}[b]{p_{\rm{u}}} = & d\Bigg\{\frac{{\text{π}} }{4}\left( {{\sigma _{\rm{v}}} + \sqrt {{\sigma _{\rm{c}}}m{\sigma _{\rm{v}}} + \sigma _{\rm{c}}^2s} } \right) + \Bigg. \\ & \left.\frac{{5\lambda }}{6}{{\left[ {\frac{{\sqrt 2 }}{{2\lambda }}\left( {{\sigma _{\rm{v}}} + \sqrt {{\sigma _{\rm{c}}}m{\sigma _{\rm{v}}} + \sigma _{\rm{c}}^2s} } \right) + \xi } \right]}^{0.75}} \right\}\end{aligned} (27)
2.2 嵌岩深度计算

 $H - \int_0^{\textstyle\frac{\kappa }{\eta }} {( - \eta x + \kappa )} {\rm{d}}x + \int_{\textstyle\frac{\kappa }{\eta }}^{{h_{\rm{r}}}} {(\eta x - \kappa )} {\rm{d}}x = 0$ (28)

 ${M_{\rm{H}}} - \int_0^{\textstyle\frac{\kappa }{\eta }} {x( - \eta x + \kappa )} {\rm{d}}x + \int_{\textstyle\frac{\kappa }{\eta }}^{{h_{\rm{r}}}} {x(\eta x - \kappa )} {\rm{d}}x = 0$ (29)

 $\kappa {h_{\rm{r}}}^2 - 4H{h_{\rm{r}}} - 6{M_{\rm{H}}} = 0$ (30)

 ${\left. p \right|_{{\textit{z}} = 0}} = \kappa = {p_{\rm{u}}}$ (31)

 ${h_{\rm{r}}} = \frac{{2H}}{{{p_{\rm{u}}}}} + \sqrt {\frac{{4{H^2}}}{{{p_{\rm u}^2}}} + \frac{{6{M_{\rm{H}}}}}{{{p_{\rm{u}}}}}}$ (32)
3 算例验证

4 影响因素分析

4.1 水平力H和桩径d的影响

 图5 不同水平荷载H下hr–MH关系曲线 Fig. 5 hr–MH curves with different horizontal load H

 图6 不同桩径d下hr–MH关系曲线 Fig. 6 hr–MH curves with different pile diameter d

4.2 ${\sigma}_{\bf c}$ ${{RMR}}$ 的影响

 图7 不同 ${\sigma _{\bf c}}$ 和RMR下hr–H关系曲线 Fig. 7 hr–MH curves with different ${\sigma _{\bf c}}$ and RMR

 图8 不同 ${\sigma _{\bf c}}$ 和MH下hr–RMR关系曲线 Fig. 8 hr–RMR curves with different ${\sigma _{\bf c}}$ and MH

5 结　论

1）基桩嵌岩深度 $h$ r随基岩顶面处水平力 $H$ 增加近似呈现线性关系增大，随基岩顶面处弯矩 $M_{\rm H}$ 增加呈现非线性关系增大，而随桩径 $d$ 增加呈现非线性关系减小。与桩径 $d$ =1.0 m相比，桩径 $d$ 每增大1.0 m最小嵌岩深度 $h$ r约分别减小32%和44%。

2）基桩嵌岩深度 $h$ r随岩石单轴抗压强度 ${\sigma _{\rm{c}}}$ 和岩体地质力学分类指标 $RMR$ 增大呈现非线性关系减小。其他条件相同时，与 ${\sigma _{\rm{c}}}$ =15 MPa相比，岩石单轴抗压强度指标 ${\sigma _{\rm{c}}}$ 每增加15 MPa，嵌岩深度 $h$ r分别减小44.4%、59.7%及67.6%。与 $RMR$ =30相比，岩体地质力学分类指标 $RMR$ 每增加15，嵌岩深度 $h$ r分别减小48.9%、72.3%及84.2%。

3）其他条件相同时，岩体地质力学分类指标 $RMR$ 对嵌岩深度 $h$ r的影响比岩石单轴抗压强度 ${\sigma _{\rm{c}}}$ 更为显著，但当 $RMR$ 大于85时，基桩嵌岩深度 $h$ r尚不足1.0 m，且受岩石强度指标 ${\sigma _{\rm{c}}}$ 的影响甚小。实际工程设计时，应兼顾岩石的强度、质量及桩径等方面确定嵌岩桩的最佳嵌岩深度。

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