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 = BTEBdV ,
where B is the strain-displacement matrix, and E the stress- strain matrix.
Minding Plate Elements:
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.
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.
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.
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
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.