岩土工程分析——四川大学2.ppt

Nottingham Centre for Geomechanics GEOTECHNICAL ANALYSIS Stress at a Point in 2D σyy σyx σxy σxx y 0 x The state of stress is defined by: Moment Equilibrium gives σxy = σyx Hence, there are only three independent stress components: σxx, σyy and σxy = σyx Stresses at a Point in 2D Transformations of Stresses y x σyy σxy σyx σxx σss σst s A O B θ t Consider a wedge of material with unit thickness as shown above Determine Transformations of Stresses All components of force acting in the s direction are to be summed and equated to zero: So we will have: y x σyy σxy σyx σxx σss σst s A O B θ t By considering the force equilibrium in the t direction, we will obtain: The above two equations can be rewritten as: y x σyy σxy σyx σxx σss σst s A O B θ t The stresses in the t direction can be easily shown to be: y x σyy σxy σyx σxx σss σst s A O B θ t Principal Stresses Principal stresses act on a plane where zero shear stresses are acting on Therefore, let σts = 0 In other words, when ? is defined by the above expression, the plane AB will be a principal plane with zero shear stresses. The two principal stresses are: Where: σ1 σ2 Major principal stress Minor principal stress Graphical representation σyy σyx σxy σxx y 0 x σts σtt σss σst s θ t Graphical representation A 2θ A B B ?1 0 Shear stress Normal stress ?2 Coordinates of point A: Coordinates of point B: From A’, B’: Equation of Equilibrium Taking into account the variation of stresses with position, we can write down the stresses acting in the x direction. Sum all components of force in the x direction and let it equal to zero, we have: Similarly, by considering all the stresses acting in the y direction, we can obtain: Exe 1. Derive the following equation . σxy = σyx . 2. Derive the Equation of Equilibrium in the Y direction. Nottingham Centre for Geomechanics