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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 *Ni*, *Nxi* and *Nyi* are shape functions. This is an incompatible element! The stiffness matrix is still of the form

**k = B**T**EB***dV*** **,

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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