This paper proposes a semi-analytical solution for the ultimate bearing capacity q_u of both smooth and rough circular shallow foundations on rock mass. Specifically, a three-dimensional (3D) Hoek–Brown (HB) is adopted, in conjunction with equilibrium equations under axisymmetric conditions, to derive the governing equations. The method of characteristics is utilized to solve the stress and failure characteristics mesh to determine the q_u. The proposed solution is verified by using it to analyze test foundations. Comparison with an HB criterion-based solution is performed to highlight the importance of 3D strength. Furthermore, parametric studies are performed to investigate the effects of rock mass properties (intact rock constant $m_{\mathrm{i}}$, geological strength index GSI, intact rock unconfined compressive strength σ_c) and foundation diameter (B) on the q_u, failure surface size, and vertical stress distribution on the foundation base. The results indicate that ignoring the 3D strength and the rock mass weight would lead to underestimation of q_u. Besides, the ultimate bearing capacity factor $N_{\sigma }$ (ratio of q_u to σ_c) increases with $m_{\mathrm{i}}$, GSI and B but decreases with ${\sigma }_{\mathrm{c}}$. The failure surface size is significantly affected by $m_{\mathrm{i}}$, GSI, B, ${\sigma }_{\mathrm{c}}$ and rock mass unit weight. The stress distribution on the foundation base has higher variance (higher possibility of stress concentration) at smaller ${\sigma }_{\mathrm{c}}$, GSI, and larger B, rock mass unit weight.

中文翻译：

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考虑三维强度的光滑与粗糙圆形岩石地基极限承载力半解析解

本文提出了岩体上光滑和粗糙圆形浅基础的极限承载力q _u的半解析解。具体来说，采用三维（3D）Hoek-Brown（HB），结合轴对称条件下的平衡方程，推导控制方程。利用特征法对应力和失效特征网格进行求解，确定q _u。通过使用它来分析测试基础来验证所提出的解决方案。与基于 HB 准则的解决方案进行比较，以强调 3D 强度的重要性。此外，还进行了参数研究来研究岩体特性（完整岩石常数）的影响${米}_{\mathrm{我}}$、地质强度指数GSI、完整岩石无侧限抗压强度σc )和q _u上的基础直径( B ₎、破坏面尺寸以及基础上的竖向应力分布。结果表明，忽略3D强度和岩体重量会导致q _u的低估。此外，极限承载力系数${氮}_{\sigma }$（ q _u与σ _c之比）随${米}_{\mathrm{我}}$，GSI 和B但随${\sigma }_{\mathrm{C}}$。破坏面尺寸受以下因素影响显着${米}_{\mathrm{我}}$, GSI, B ,${\sigma }_{\mathrm{C}}$和岩体单位重量。地基应力分布在较小的情况下具有较高的方差（应力集中的可能性较高）${\sigma }_{\mathrm{C}}$，GSI，以及较大的B，岩体单位重量。