Course
Year
: S0912 - Introduction to Finite Element Method
: 2010
BASIC CONCEPT OF FEM
Session 3
COURSE 3
Content:
• Modelling of element and structure
• Stiffness of element
• Load – displacement relationship
• Boundary condition
• Global and Local coordinate
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MODELLING OF ELEMENT AND STRUCTURE
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MODELLING OF ELEMENT AND STRUCTURE
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MODELLING OF ELEMENT AND STRUCTURE
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MODELLING OF ELEMENT AND STRUCTURE
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MODELLING OF ELEMENT AND STRUCTURE
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STIFFNESS OF ELEMENT
A typical member stiffness relation has the following general form:
Qm = km.qm + Qom
(1)
where
m = member number m.
Qm = vector of member's characteristic forces, which are unknown internal forces.
km = member stiffness matrix which characterises the member's resistance against
deformations.
qm = vector of member's characteristic displacements or deformations.
Qom = vector of member's characteristic forces caused by external effects (such as
known forces and temperature changes) applied to the member while qm = 0.
If qm are member deformations rather than absolute displacements, then Qm are
independent member forces, and in such case (1) can be inverted to yield the so-called
member flexibility matrix, which is used in the flexibility method.
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STIFFNESS OF ELEMENT
For a system with many members interconnected at points called nodes, the members' stiffness
relations such as Eq.(1) can be integrated by making use of the following observations:
• The member deformations qm can be expressed in terms of system nodal displacements r in
order to ensure compatibility between members. This implies that r will be the primary unknowns.
• The member forces Qm help to the keep the nodes in equilibrium under the nodal forces R. This
implies that the right-hand-side of (1) will be integrated into the right-hand-side of the following
nodal equilibrium equations for the entire system:
R = Kr + Ro
where
R = vector of nodal forces, representing external forces applied to the system's nodes.
K = system stiffness matrix, which is established by assembling the members' stiffness
matrices
km
r = vector of system's nodal displacements that can define all possible deformed
configurations of
the system subject to arbitrary nodal forces R.
o
R = vector of equivalent nodal forces, representing all external effects other than the nodal
forces
which are already included in the preceding nodal force vector R. This vector is
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LOAD – DISPLACEMENT RELATIONSHIP
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LOAD – DISPLACEMENT RELATIONSHIP
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BOUNDARY CONDITION
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BOUNDARY CONDITION
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GLOBAL & LOCAL COORDINATE
• Rectangular Cartesian global
reference coordinate system{Y1 ,Y2 ,Y3 }
• Orthogonal curvilinear coordinate
system to describe geometry and
deformation {1 ,2 ,3}
• Curvilinear local finite element
coordinates {1 ,2 ,3}
• Locally orthonormal body coordinates
define material symmetry and
structure, {X1 , X 2 , X 3} related to the
finite element coordinates by a rotation
about the (1 ,2 ) -normal axis through
the "fiber angle" , 1
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From Costa et al, J Biomech Eng 1996;118:452-463
GLOBAL & LOCAL COORDINATE
Z
Global axes
Y
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X
GLOBAL & LOCAL COORDINATE
Element nodal forces, element stiffness and displacements in local coordinates
f  k u
Element nodal forces, element stiffness and displacements in global coordinates
F  K U ,
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K   R k R  , R   R 


1
BASIC STEPS OF THE FINITE-ELEMENT METHOD (FEM)
1. Establish strong formulation
– Partial differential equation
2. Establish weak formulation
– Multiply with arbitrary field and integrate over element
3. Discretize over space
– Mesh generation
4. Select shape and weight functions
5. Compute element stiffness matrix
– Local and global system
6. Assemble global system stiffness matrix
7. Apply nodal boundary conditions
– temperature/flux/forces/forced displacements
8. Solve global system of equations
– Solve for nodal values of the primary variables
(displacements/temperature)
9. Compute temperature/stresses/strains etc. within the element
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– Using nodal values and shape functions