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CONTINUUM MECHANICS ( STRESS DISTRIBUTION )PowerPoint Presentation

CONTINUUM MECHANICS ( STRESS DISTRIBUTION )

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CONTINUUM MECHANICS ( STRESS DISTRIBUTION )

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CONTINUUM MECHANICS(STRESS DISTRIBUTION)

Stress vector

State of stress

Stress distribution

Surface traction(loading)

Volume V

SurfaceS

Stress vector

x3

Volume V0

SurfaceS0

Volumetric force

x2

x1

GGO theorem

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).

Navier equation

in coordintes reads:

We have to deal with the set of 3 linear partial differential equations.

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.

Comments

- Equation is derived from one of two
equilibrium equations, i.e. that the sum of forces acting over the body has to vanish.

- The other equilibrium equation – sum of the moments equals zero – yield already assumed symmetry of stress matrix, σij= σji

- 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.

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