莫尔圆极点法原理及应用

Principle and application of pole point method of Mohr's circle

莫尔圆在岩土强度理论、应力路径、挡土压力及地基承载力分析等方面得到了广泛应用,也常用来解释试验数据和预测岩土变形。极点是莫尔圆上一个特殊的点,对于给定的应力状态,具有唯一性;用作图法,借助极点可以方便、快速求得复杂条件下的应力状态(应力大小和方向),避免使用繁复的数值计算,这是极点法最突出的优势。莫尔圆上极点的位置与应力方向密切相关,岩土力学中规定正应力以压为正,则必须设定正确的剪应力正方向。基于微元隔离体力的平衡条件,证明了必须规定剪应力绕微元体逆时针为正,才能确定正确的极点位置及求得问题的正确解答。采用反证法,证明了极点的唯一性;并给出了极点法的可靠性证明。最后,介绍了极点法求解复杂应力问题及确定应力不连续面位置的优越性。

Mohr＇s circle is a geometric representation of the two-dimensional stress state and is very useful to perform quick and efficient estimations. It is also popularly used in geotechnical fields such as soil strength, stress path, earth pressure and bearing capacity. It is often used to interpret the test data, to analyze complex geotechnical problems, and to predict soil behaviors. The pole point on Mohr＇s circle is a point so special that it can help to readily find stresses on any specified plane by using diagram instead of complicated computation. However, the orientation of the pole point on MoM＇s circle is closely related to the directions of stresses. In soil mechanics, conventionally, because the compressive normal stress is positive, the positive shear stress should be appropriately defined for using the pole point method correctly. Based on the equilibrium of the isolated element, the positive shear stress will cause a counterclockwise rotation of the infinitesimal element. In addition, the uniqueness of the pole point is verified by using the proof by the contradiction. The validity of the pole point method is testified by the corresponding theoretical method. It is concluded that the pole point method is used much more easily than the theoretical method. Finally, two relatively complex examples are given by using the pole point method to determine the stress state and the discontinuity in the undrained soils, respectively.