岩土工程分析——四川大学1.ppt

Nottingham Centre for Geomechanics GEOTECHNICAL ANALYSIS Stress at a Point in 3D Stresses at a Point in 3D z x y ?zz ?yz ?yy ?yx ?zy ?zx ?xy ?xz ?xx Stresses at a Point in 3D Moment Equilibrium gives Hence, there are only six independent stress components: Stresses on a general plane z x y ? Assuming there is one such plane, where the shear stresses are zero and only a normal stress exists. Equilibrium of forces leads to: Stresses on a general plane Where: There are three real roots for ?, which are: ?1 ?2 ?3 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: Deviatoric Stresses Deviatoric Stresses Determination of the principal stresses Determination of the principal stresses The solutions are: Worked Example Determine: a) The stress Invariants I1 , I2 , I3 b) The principal stresses ?1 , ?2 , ?3 Worked Example Solution Worked Example Solution Note: ?1 ?2 ?3 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). ?yx ?zx ?x z y x dz dx dy 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. 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. Nottingham Centre for Geomechanics