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# PPT: Force Method Notes | EduRev

## GATE : PPT: Force Method Notes | EduRev

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Module III Module III
Direct stiffness method
• Introduction – element stiffness matrix – rotation transformation
Direct stiffness method
matrix – transformation of displacement  and load vectors and
stiffness matrix – equivalent nodal forces and load vectors –
assembly of stiffness matrix and load vector – determination of assembly of stiffness matrix and load vector determination of
nodal displacement and element forces – analysis of plane truss
beam and plane frame (with numerical examples) – analysis of
grid space frame (without numerical examples) grid – space frame (without numerical examples)
Page 2

Module III Module III
Direct stiffness method
• Introduction – element stiffness matrix – rotation transformation
Direct stiffness method
matrix – transformation of displacement  and load vectors and
stiffness matrix – equivalent nodal forces and load vectors –
assembly of stiffness matrix and load vector – determination of assembly of stiffness matrix and load vector determination of
nodal displacement and element forces – analysis of plane truss
beam and plane frame (with numerical examples) – analysis of
grid space frame (without numerical examples) grid – space frame (without numerical examples)
Introduction
• The formalised stiffness method involves evaluating the
displacement transformation matrix C
MJ
correctly p y
• Generation of matrix C
MJ
is not suitable for computer
programming
H th l ti f di t tiff thd • Hence the evolution of direct stiffness method
Page 3

Module III Module III
Direct stiffness method
• Introduction – element stiffness matrix – rotation transformation
Direct stiffness method
matrix – transformation of displacement  and load vectors and
stiffness matrix – equivalent nodal forces and load vectors –
assembly of stiffness matrix and load vector – determination of assembly of stiffness matrix and load vector determination of
nodal displacement and element forces – analysis of plane truss
beam and plane frame (with numerical examples) – analysis of
grid space frame (without numerical examples) grid – space frame (without numerical examples)
Introduction
• The formalised stiffness method involves evaluating the
displacement transformation matrix C
MJ
correctly p y
• Generation of matrix C
MJ
is not suitable for computer
programming
H th l ti f di t tiff thd • Hence the evolution of direct stiffness method
Direct stiffness method
• We need to simplify the assembling process of S
J
, the
J
assembled structure stiffness matrix
• The key to this is to use member stiffness matrices for actions The key to this is to use member stiffness matrices for actions
and displacements at BOTH ends of each member
If b di l  d ih f  • If member displacements are expressed with reference to
global co-ordinates, the process of assembling S
J
simple simple
Page 4

Module III Module III
Direct stiffness method
• Introduction – element stiffness matrix – rotation transformation
Direct stiffness method
matrix – transformation of displacement  and load vectors and
stiffness matrix – equivalent nodal forces and load vectors –
assembly of stiffness matrix and load vector – determination of assembly of stiffness matrix and load vector determination of
nodal displacement and element forces – analysis of plane truss
beam and plane frame (with numerical examples) – analysis of
grid space frame (without numerical examples) grid – space frame (without numerical examples)
Introduction
• The formalised stiffness method involves evaluating the
displacement transformation matrix C
MJ
correctly p y
• Generation of matrix C
MJ
is not suitable for computer
programming
H th l ti f di t tiff thd • Hence the evolution of direct stiffness method
Direct stiffness method
• We need to simplify the assembling process of S
J
, the
J
assembled structure stiffness matrix
• The key to this is to use member stiffness matrices for actions The key to this is to use member stiffness matrices for actions
and displacements at BOTH ends of each member
If b di l  d ih f  • If member displacements are expressed with reference to
global co-ordinates, the process of assembling S
J
simple simple
Member oriented axes (local coordinates)
d t t i t d  ( l b l di t ) and structure oriented axes (global coordinates)
L d
x
y
L
Local axes
L d L
L d
sin L d ?
L d?
x
Y Y Y Y
cos L d?
?
Global axes
y
Global axes Global axes
?
X X X X
Global axes
Page 5

Module III Module III
Direct stiffness method
• Introduction – element stiffness matrix – rotation transformation
Direct stiffness method
matrix – transformation of displacement  and load vectors and
stiffness matrix – equivalent nodal forces and load vectors –
assembly of stiffness matrix and load vector – determination of assembly of stiffness matrix and load vector determination of
nodal displacement and element forces – analysis of plane truss
beam and plane frame (with numerical examples) – analysis of
grid space frame (without numerical examples) grid – space frame (without numerical examples)
Introduction
• The formalised stiffness method involves evaluating the
displacement transformation matrix C
MJ
correctly p y
• Generation of matrix C
MJ
is not suitable for computer
programming
H th l ti f di t tiff thd • Hence the evolution of direct stiffness method
Direct stiffness method
• We need to simplify the assembling process of S
J
, the
J
assembled structure stiffness matrix
• The key to this is to use member stiffness matrices for actions The key to this is to use member stiffness matrices for actions
and displacements at BOTH ends of each member
If b di l  d ih f  • If member displacements are expressed with reference to
global co-ordinates, the process of assembling S
J
simple simple
Member oriented axes (local coordinates)
d t t i t d  ( l b l di t ) and structure oriented axes (global coordinates)
L d
x
y
L
Local axes
L d L
L d
sin L d ?
L d?
x
Y Y Y Y
cos L d?
?
Global axes
y
Global axes Global axes
?
X X X X
Global axes
1.  Plane truss member
Stiffness coefficients in local coordinates
1
3
2
4
y
0
0
Degrees of freedom
1
??
??
Unit displacement
x
EA
L
EA
L
??
??
corr. to DOF 1
00
EA EA
? ?
-
??
[]
00
0000
00
M
LL
S
EA EA
??
? ?
? ?
? ?
-
??
=
Member stiffness matrix in local
coordinates
0000
L L
??
?
?
? ?
?
?
```
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## Structural Analysis

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