Abstract The self-weight of freestanding natural arches imparts stresses on the lintel and abutments, setting the conditions for rock fracture and erosion. Rock is weak in tension, therefore an ideal arch form is one that minimizes tensile stresses and supports the weight of the lintel in compression, e.g. an inverted catenary. However, rock mass structure, including bedding, cross-bedding and other discontinuities, often imparts strong control on the geometry of arches, leading to forms that are less favorable from a stress perspective. Here we analyze a suite of nineteen arch models created from ground- and drone-based photogrammetry to assess the three-dimensional static stress field under gravitational loading. The models represent a range of arch lengths from 4 to 88 m, as well as a variety of forms, and use material properties previously calibrated from dynamic analysis of ambient vibrations. Our results demonstrate that arches shaped like beams have relatively high tensile stresses and a nearly symmetrical statistical distribution of principal stresses, while those with convex forms have comparably lower tensile stresses and statistical distributions favoring compressive stresses. In-situ observations of tensile cracks frequently correspond to the location of predicted tensile stresses in our models. We calculated the ratio of mean principal stresses for each arch, which is theoretically 1 for a flat prismatic beam and approaches infinity for an ideal inverted catenary. We found several arches with mean principal stress ratio around 1 and that values increased with lintel convexity. These results indicate that while self-sculpture might attempt to create ideal stress-based forms, discontinuities can control lintel geometry and natural arches evolve with forms that may be less favorable for long-term stability. The mean principal stress ratio is a simple metric to classify and compare arches, which may be useful for assessment of arch stability supporting conservation and hazard analyses.

中文翻译：

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梁与悬链线之间：几何形状对天然岩拱重力应力和稳定性的影响

摘要 独立式天然拱的自重对门楣和桥台施加应力，为岩石破裂和侵蚀创造条件。岩石抗拉力弱，因此理想的拱形形式是将拉应力降至最低并在压缩时支撑过梁重量的形式，例如倒悬链。然而，岩体结构，包括层理、交错层理和其他不连续性，通常对拱的几何形状有很强的控制，导致从应力角度来看不太有利的形式。在这里，我们分析了一套由基于地面和无人机的摄影测量创建的 19 拱模型，以评估重力加载下的三维静态应力场。这些模型代表了从 4 到 88 m 的一系列拱长度，以及各种形式，并使用先前通过环境振动动态分析校准的材料属性。我们的结果表明，梁形状的拱具有相对较高的拉伸应力和几乎对称的主应力统计分布，而凸形拱具有相对较低的拉伸应力和有利于压缩应力的统计分布。拉伸裂纹的原位观察通常对应于我们模型中预测拉伸应力的位置。我们计算了每个拱的平均主应力的比率，理论上，对于扁平棱柱梁，该比率为 1，而对于理想的倒悬链则接近无穷大。我们发现了几个平均主应力比约为 1 的拱形，并且这些值随着门楣的凸度而增加。这些结果表明，虽然自我雕塑可能会尝试创建理想的基于应力的形式，但不连续性可以控制门楣的几何形状，自然拱门的形式可能不太利于长期稳定性。平均主应力比是对拱进行分类和比较的简单度量，可用于评估支持保护和危险分析的拱稳定性。