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 Page 1


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
can be made 
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
can be made 
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
can be made 
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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