1. 1/61/6 M.Chrzanowski: Strength of Materials SM1-08: Continuum Mechanics: Stress distribution CONTINUUM MECHANICS (STRESS DISTRIBUTION)

2. 2/62/6 M.Chrzanowski: Strength of Materials SM1-08: Continuum Mechanics: Stress distribution State of stress Stress distribution Stress vector

3. 3/63/6 M.Chrzanowski: Strength of Materials SM1-08: Continuum Mechanics: Stress distribution x2x2 x1x1 x3x3 Volume V Surface S Volume V 0 Surface S 0 Stress vector Volumetric force GGO theorem Surface traction (loading)

4. 4/64/6 M.Chrzanowski: Strength of Materials SM1-08: Continuum Mechanics: Stress distribution On the body surface stress vector has to be balanced by the traction vector Stress on the body surface Coordinates of vector normal to the surface This equation states statics boundary conditions to comply with the solution of the equation: This equation (Navier equation) reflects internal equilibrium and has to be fulfilled in any point of the body (structure).

5. 5/65/6 M.Chrzanowski: Strength of Materials SM1-08: Continuum Mechanics: Stress distribution We have to deal with the set of 3 linear partial differential equations. Navier equation in coordintes reads: There are 6 unknown functions which have to fulfil static boundary conditions (SBC): We need more equations to determine all 6 functions of stress distribution. To attain it we have to consider deformation of the body.

6. 6/66/6 M.Chrzanowski: Strength of Materials SM1-08: Continuum Mechanics: Stress distribution Comments 1.Equation is derived from one of two equilibrium equations, i.e. that the sum of forces acting over the body has to vanish. 2.The other equilibrium equation – sum of the moments equals zero – yield already assumed symmetry of stress matrix, σ ij = σ ji 3.Navier equation is the special case of the motion equation i.e. uniform motion (no inertia forces involved). The inertia effects can be included by adding d’Alambert forces to the right hand side of Navier equation.

7. 7/67/6 M.Chrzanowski: Strength of Materials SM1-08: Continuum Mechanics: Stress distribution  stop