finite element method-the direct stiffness method
TRANSCRIPT
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Finite
Element
MethodBy the Direct Stiffness Method (DSM)Engr Y. K. Galadima
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General Procedure Pre-processing
1. Idealisation
, ,
, ,
,
,
,
,
2
31 4
Idealisation simply
means creating a
mathematical
model of the
physical systemby making
necessary
assumptions
1 2 3
NOTE
The subscript notations adopted in this
presentation are slightly different from those
used in the lecture notes
This presentation may contain error
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General Procedure Pre-processing
2. Discretisation (decomposition) (2.4.2.2)
a) Disconnection (2.4.2.2.1)
2
3
1 4
1 2 3
y
The local or element
axes are denoted by
and
The local coordinate
system is selected
such that the
axis aligned with the
longitudinal axis ofthe element
The global
coordinate axes are
denoted by and
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General Procedure
Pre-processing2. Discretisation (decomposition) (2.4.2.2)
b) Localisation/isolation (2.4.2.2.2)
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Pre-processing2. Discretisation (decomposition) (2.4.2.2)
b) Localisation/isolation (2.4.2.2.2)
We will use this
element as a genericelement to derive the
stiffness equations for
the truss elements
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General Procedure
Pre-processing2. Discretisation (decomposition) (2.4.2.2)
c) Derivation of Member Stiffness Equations (2.4.2.2.3)
=
= cos 90
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General ProcedurePre-processing
2. Discretisation (decomposition) (2.4.2.2)c) Derivation of Member Stiffness Equations (2.4.2.2.3)
No shear for truss elements, hence
=
00 0
00 0
0
0 0
0
0 0
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General Procedure
Pre-processing2. Discretisation (decomposition) (2.4.2.2)c) Derivation of Member Stiffness Equations (2.4.2.2.3)
Now the task is to find expressionsfor the INFLUENCECOEFFICIENTS , ,
and
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General ProcedurePre-processing
2. Discretisation (decomposition) (2.4.2.2)
c) Derivation of Member Stiffness Equations (2.4.2.2.3)
=
, =
=
, =
Hence
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General Procedure
Pre-processing2. Discretisation (decomposition) (2.4.2.2)c) Derivation of Member Stiffness Equations (2.4.2.2.3)
=
1 0
0 0
1 0
0 01 0
0 0
1 0
0 0
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General Procedure
Pre-processing3. Globalisation (2.4.2.3)
,
,
,
,
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General Procedure
Pre-processing3. Globalisation: Force fieldexpressing the joint forces wrt their components in theglobal coordinate gives
= cos + sin
= s i n + cos
And
= cos + sin
= s i n + cos
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General ProcedurePre-processing
3. Globalisation: Force field
=
0 0
0 00 0
0 0
Or in short notation
=
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General Procedure
Pre-processing3. Globalisation: Displacement field
=
0 0
0 0
0 00 0
Or in short notation
=
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General Procedure
Pre-processing
3. Globalisation: substituting theexpressions for and will give
=
Or
=
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General Procedure
Pre-processing
3. Globalisation: if we write therelationship btw the nodal forces in theglobal coordinate system and their
corresponding displacement as =
Then
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General Procedure
Pre-processing
3. Globalisation: comparing the last twoexpression shows that
= Thus
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General Procedure
Pre-processing3. Globalisation
=
0 0
0 00 0
0 0
1 0
0 0
1 0
0 01 0
0 0
1 0
0 0
0 0
0 00 0
0 0
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General Procedure
Pre-processing3. Globalisation
=
Is the global stiffness matrix
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General Procedure
Pre-processing3. Globalisation: thus the global stiffness matrix forthe element becomes
=
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General Procedure
Pre-processing3. Globalisation: therefore, for the
element, the global stiffness equation is
=
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I think we should take a break here
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Questions
???