岩土工程分析四川大学,GEOTECHNICAL ANALYSIS Stress at a Point in 3D,岩土工程分析四川大学,Stresses at a Point in 3D,岩土工程分析四川大学,Stresses at a Point in 3D,Moment Equilibrium gives,Hence, there are only six independent stress components:,岩土工程分析四川大学,Stresses on a general plane,Assuming there is one such plane, where the shear stresses are zero and only a normal stress exists.,Equilibrium of forces leads to:,岩土工程分析四川大学,Stresses on a general plane,Where:,There are three real roots for , which are: 1 2 3,岩土工程分析四川大学,Stress Invariants,The state of stress at a point in 3 Dimensions is uniquely defined by the three principal stresses,The stress invariants reduce to:,岩土工程分析四川大学,Deviatoric Stresses,岩土工程分析四川大学,Deviatoric Stresses,岩土工程分析四川大学,Determination of the principal stresses,岩土工程分析四川大学,Determination of the principal stresses,The solutions are:,岩土工程分析四川大学,Worked Example,Determine: a) The stress Invariants I1 , I2 , I3 b) The principal stresses 1 , 2 , 3,岩土工程分析四川大学,Worked Example Solution,岩土工程分析四川大学,Worked Example Solution,Note: 1 2 3,岩土工程分析四川大学,Equations of Equilibrium,Taking into account the stress change with coordinates. Consider all forces acting in the x direction, we can obtain:,(Where X is the body force in the x direction).,岩土工程分析四川大学,Equations of Equilibrium,Similarly, if we consider the forces in the y and z directions, we will obtain similar equations:,Where Y and Z are the body force in the y and z directions, respectively.,岩土工程分析四川大学,Exe.,(1) For the stress state presented by the stress tensor,Determine: a) The stress Invariants I1 , I2 , I3 b) The principal stresses 1 , 2 , 3,(2) Prove the Equations of Equilibrium,Where X,Y and Z are the body forces in the x,y and z directions respectively.,岩土工程分析四川大学,