砌体梁分析(Voussoir Beam Analysis)[纪念钱鸣高院士]
1 引言
2 采矿顶住砌体梁分析
2.1 破坏模式
砌体梁的破坏模式有三种：第一种是在柱墩出垂直滑动引起的剪切破坏，其计算原理与前两种方法相同；第二种是由于重力作用引起的弯曲破坏，这种破坏模式与弹性分析的弯曲类似，但梁会发生断裂；第三种是由于压缩产生的局部粉碎性破坏。砌体梁分析作了如下假设：
(1) 顶柱不能承受拉应力，当发生弯曲时，主要通过形成抛物线压缩拱来实现自支撑；
(2) 砌体梁的变形发生在柱墩的滑动之前，抗滑稳定性在压力拱形成之后决定;
(3) 不考虑由原岩应力和/或开挖的几何形状引起的初始侧向应力，假定梁最初无应力；
(4) 横切结构的角度小于节理面假设的最小摩擦角。
2.2 输入参数
当使用Mohr-Coulomb强度准则进行刚性或弹性分析时，不需要原岩的单轴抗压强度UCS，但进行Voussoir分析，由于需要使用UCS计算局部挤压破坏的安全系数，因此无论采用哪一种强度准则，都必须输入UCS值。此外，如果使用Hoek-Brown准则，则需要输入岩石的弹性模量和泊松比。
2.3 计算指标
砌体梁分析有三个计算指标：
(1) 剪切安全系数
剪切安全系数的计算方法如下式所示:
其中
(2) 压缩安全系数
压缩安全系数FOS=UCS/产生的最大水平压应力
(3) 拱穿透概率
拱穿透概率(Arch Snap-Thru Probability)是砌体的主要破坏模式，计算使用的是Diederichs(1999)建议的算法[Diederichs, M. S. and P. K. Kaiser (1999). "Tensile strength and abutment relaxation as failure control mechanisms in underground excavations." 。
在Voussoir分析中计算的屈曲参数(buckling parameter)代表了由Voussoir分析迭代程序确定的在给定的几何形状和岩体模量下不稳定的拱形结构的百分比。Hutchinson等人(1996)确定出屈曲参数=35%作为超过屋顶应被视为不稳定的极限。这个35%的设计极限恰好对应于跨中挠度的变形梁，即为梁厚度的10%。因此，拱的稳定性也可以通过监测中跨的位移来评估，相对于无弯曲的状态。上述输入数据的计算结果如下：
可以看出，在剪切作用下破坏的失败的概率为0.6%，压缩作用下破坏的概率为0%，拱穿透下的破坏概率很低。如果逐渐减小矿柱的厚度，破坏概率逐渐增加，当矿柱厚度达到0.5m时，三种破坏模式同时达到，致使顶柱塌落。
2.4 砌体梁理论的挑战
随着数值模拟技术的发展，传统的砌体梁理论受到了严重挑战，特别是基于离散块体理论的技术能够更深入地了解顶柱的破坏机理，Hatzor(1998)[The stability of a laminated Voussoir beam: Back analysis of a historic roof collapse using DDA]使用DDA方法模拟了砌体梁；Oliveira(2014)[Revisiting the applicability of voussoir beam theory for tunnel design in Sydney]使用UDEC模拟了砌体梁；Abousleiman(2020)[Expanding Understanding of the Voussoir Beam Analog in Flat Roof Excavations Using the Discrete Element Method]也使用离散元模拟了砌体梁，下面总结这方面最新的研究动态。
(2017) Subsidence Prediction of Overburden Strata and Surface Based on the Voussoir Beam Structure Theory [82]
(2012) Deformation mechanisms and stability analysis of undermined sedimentary rocks in the shallow subsurface
(1998) The Stability of a Laminated Voussoir Beam - Back Analysis of a Historic Roof Collapse Using DDA
(2014) Revisiting The Applicability Of Voussoir Beam Theory For Tunnel Design In Sydney
(1999) Stability of large excavations in laminated hard rock masses the voussoir analogue revisited
(1999) Tensile strength and abutment relaxation as failure control mechanisms in underground excavations
(1996) Analysis and design of an underground hard rock Voussoir beam roof
(2017) Voussoir beam model for lower strong roof strata movement in longwall mining – Case study
(2021) Expanding application of the voussoir beam analog to horizontally bedded and passively bolted flat-roof excavations using the discrete element method
(2007) Modeling of a Multiply Jointed Voussoir Beam in the Centrifuge
(1999) The Voussoir Beam Reaction Curve
(2019) Failure mechanism and fracture aperture characteristics of hard thick main roof based on voussoir beam structure in longwall coal mining
(2009) Narrabri Coal Mine Stage 2 Longwall Project Mine Subsidence Predictions and Impact Assessment
(1998) Numerical Evaluation of the Response in Bending of An Underground Hard Rock Voussoir Beam Roof
(1987) Back-analysis of roof conditions in the Great Northern Seam, Newcastle Coal Measures, Australia, using voussoir beam theory
(1996) Cablebolting in Underground Mines
(2018) Μulti-jointed stratified hard rock roof analysis and design
(1998) The Influence of Joint Spacing on the Stability of a Laminated Voussoir Beamback Analysis of a Roof Failure Using DDA
(1991) Failure of a linear Voussoir arch a laboratory and numerical study
(2002) Risk considerations for crown pillar stability assessment for mine closure planning
(2020) Strategies for surface crown pillar design using numerical modelling – A case study
(2011) Stability Analysis of Shallow Tunnel of Norra Länken [5]
