A semi-analytical approach is proposed for addressing the minimum thrust and minimum thickness analysis of spherical masonry domes under their self-weight. The classical differential equilibrium equations of axially symmetric shells are resorted to, calling for the exact computation of self-weight and center of mass of an infinitesimal dome voussoir. The minimum thrust analysis is accomplished by determining a statically admissible stress state in the dome, that is associated to a kinematically admissible settlement mechanism of minimum thrust. Such a stress state entails vanishing hoop stresses in the lower part of the dome, where the opening of meridional cracks occurs. An original numerical scheme is proposed for automatically computing a distribution of hoop stresses in the dome cap, producing a statically admissible stress state. A suitable modification of the semi-analytical procedure for the minimum thrust analysis is then adopted to perform the minimum thickness analysis. Numerical results are presented for spherical domes with parameterized oculus and embrace angles, complemented by the comparison with alternative equilibrium models, and with approximate treatments of the dome self-weight adopted in the literature. The merit of the proposed procedure is finally proved in application to a real case. As an iconic result, a refined estimate of the minimum thickness of a hemispherical dome is derived, resulting in 0.04284 times the radius of the mid-surface.

提出了一种半解析方法来解决自重作用下球形砖石穹顶的最小推力和最小厚度分析。求助于经典的轴对称壳的微分平衡方程，要求精确计算无穷小圆顶拱顶的自重和质心。最小推力分析是通过确定圆顶中的静态允许应力状态来实现的，该状态与最小推力的运动学上允许的沉降机制相关。这种应力状态会在圆顶的下部发生环向裂纹的消失，从而消除环向应力。提出了一种原始数值方案，用于自动计算圆顶帽中的环向应力分布，从而产生静态允许的应力状态。然后采用最小推力分析的半分析程序的适当修改，以执行最小厚度分析。给出了带有参数化的眼球和包角的球形穹顶的数值结果，并与其他平衡模型进行了比较，并对文献中采用的穹顶自重进行了近似处理。所提出程序的优点最终在实际案例中得到了证明。作为标志性的结果，得出了半球形圆顶的最小厚度的精确估计，得出中间表面半径的0.04284倍。给出了带有参数化的眼球和包角的球形穹顶的数值结果，并与其他平衡模型进行了比较，并对文献中采用的穹顶自重进行了近似处理。所提出程序的优点最终在实际案例中得到了证明。作为标志性的结果，得出了半球形圆顶的最小厚度的精确估计，得出中间表面半径的0.04284倍。给出了带有参数化的眼球和包角的球形穹顶的数值结果，并与其他平衡模型进行了比较，并对文献中采用的穹顶自重进行了近似处理。所提出程序的优点最终在实际案例中得到了证明。作为标志性的结果，得出了半球形圆顶的最小厚度的精确估计，得出中间表面半径的0.04284倍。