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Mechanical - Finite Element Analysis - Dynamic Analysis Using Element Method

Equation of Motion

   Posted On :  25.09.2016 12:13 pm

There is two types of equation of motion Longitudinal vibration of beam or axial vibration of a rod Transverse vibration of a beam

EQUATION OF MOTION

 

There is two types of equation of motion

 

Longitudinal vibration of beam or axial vibration of a rod

Transverse vibration of a beam


 

where NiNxi and Nyi are shape functions. This is an incompatible element! The stiffness matrix is still of the form

 

k =   BTEBdV ,

where B is the strain-displacement matrix, and E the stress- strain matrix.

 

Minding Plate Elements:

 

4-Node Quadrilateral

8-Node Quadrilateral

Three independent fields.

Deflection w(x,y) is linear for Q4, and quadratic for Q8.

 

 

Discrete Kirchhoff Element:

 

Triangular plate element (not available in ANSYS). Start with a 6-node riangular element,


Total DOF = 9  (DKT Element).

 

Incompatible w(x,y);  convergence is faster (w is cubic along each edge) and it is efficient.

 

Test Problem:

 

 


L/t = 10,   = 0.3

ANSYS 4-node quadrilateral plate element.

ANSYS Result for wc

 

Mesh           wc (  PL2/D)

2        2                           0.00593

4        4                           0.00598

8        8                           0.00574

16      16                         0.00565

          :                            :

Exact Solution                         0.00560

 

Question:Converges from “above”? Contradiction to what we learnt about the nature of the FEA solution?

Reason:  This is an incompatible element ( See comments on p. 177).

 

 

Shells and Shell Elements

 

Shells  Thin structures witch span over curved surfaces.

 


 

Example:

 

Sea shell, egg shell (the wonder of the nature); Containers, pipes, tanks; Car bodies;

 

Roofs, buildings (the Superdome), etc. Forces in shells:

 

Membrane forces + Bending Moments

 

(cf.  plates:  bending only)


Example: A Cylindrical Container.


 

Shell Theory:

Thin shell theory

 

Shell theories are the most complicated ones to formulate and analyze in mechanics (Russian’s contributions).

 

Engineering Craftsmanship Demand strong analytical skill

 

Shell Elements:


cf.: bar + simple beam element => general beam element.

DOF at each node:


 

Based on shell theori es;

Most general shell el ements (flat shell and plate elements are subsets);

 

Complicated in form ulation.

 

Test Cases:

 




 


Tags : Mechanical - Finite Element Analysis - Dynamic Analysis Using Element Method
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