UNIT III FINITE ELEMENT
METHOD
INTRODUCTION
• General Methods of the Finite Element
Analysis
• 1. Force Method – Internal forces are
considered as the unknowns of the problem.
• 2. Displacement or stiffness method –
Displacements of the nodes are considered
• as the unknowns of the problem
Discretisation of a structure
• In order to represent a half-hole model in a wave
digital modelling context, a decomposition of the
instantaneous variables ( and ) into wave variables is
required. Taking a three-port modelling approach (as
described in, and applying to the network in the
modelling structure depicted in results. Because the
main bore is modelled as a digital waveguide,
both and must equal the main bore
characteristic impedance . The scattering equations of
the three-port junction that models the wave
interaction at the intersection between the main bore
and the tonehole are:
Displacement functions
•
Displacement function for the CST element
.1.Select the element type which infers the displacement function is
specified
2. Discretize the component
3. Define a constitutive relationship (stress – strain relationship)
4. Assemble the element stiffness matrix
5. Assemble the component, or global stiffness matrix
6. Solve Solve the system of equations for unknonwn nodal
displacements the system of equations for unknonwn nodal
displacements
7. Solve for Element Strains and Stresses
TRUSS ELEMENTS
It is made up of several bars, riveted or welded
together. The following
assumptions are made while finding the forces
in a truss,
(a) All members are pin joints, (b) The truss is
loaded only at the joints, (c) The
self – weight of the members is neglected unless
stated
• Stiffness Matrix [K] for a truss element
Beam element
• A beam element is defined as a long, slender
member that is subjected to vertical loads and
moments, which produce vertical
displacements and rotations. The degrees of
freedom for a beam element are a vertical
displacement and a rotation at each node, as
opposed to only an horizontal displacement at
each node for a truss element.
Plane stress analysis
• which includes problems such as plates with
holes, fillets, or other changes in geometry
that are loaded in their plane resulting in local
stress concentrations.
Plane strain analysis
• which includes problems such as a long g g
underground box culvert subjected to a
• uniform load acting constantly over its length
or a long
• cylindrical control rod subjected to a load that
remains constant over the rod length
TRUSS STRUCTURES
80 kN
60 kN
80 kN
C
60 kN
D
A
C
D
B
A
1
2
B
Primary structure
Truss Structures
(a) Remove the redundant member (say
AB) and make the structure
•
a primary determinate structure
•
The condition for stability and
indeterminacy is: r+m>=<2j,
•
Since, m = 6, r = 3, j = 4, (r + m =) 3 + 6 > (2j
=) 2*4 or 9 > 8  i = 1
b)Find deformation ABO along AB:
ABO = (F0uABL)/AE
F0 = Force in member of the primary structure
due to applied load
uAB= Forces in members due to unit force
applied along AB
Truss Structures
(c) Determine deformation along AB due to unit load applied along
AB:
d) Apply compatibility condition along AB:
ABO+fAB,ABFAB=0
d) Hence determine FAB
(e) Determine the individual member forces in a particular member
CE by
FCE = FCE0 + uCE FAB
where FCE0 = force in CE due to applied loads on primary structure
(=F0), and uCE = force in CE due to unit force applied along AB (= uAB)