Direct Stiffness Method:Direct Stiffness Method:
Plane Frame
Plane Frame Analysis
All the members lie in the same plane.
Members are interconnected by rigid or pin joints.
The internal stress resultants at a cross-section of member
consist of bending moment, shear force and an axial force.
The significant deformations in the plane frame are only
flexural and axial.
Stiffness matrix of the member is derived in its local co-
ordinate axes and then it is transformed to global co-ordinate
system.
Members are oriented in different directions and hence before
forming the global stiffness matrix it is necessary to refer all
the member stiffness matrices to the same set of axes.
This is achieved by transformation of forces and
displacements to global co-ordinate system.
Plane Frame, Member Stiffness Matrix
The frame members have six degrees of freedom
*au
a b
*au
*
aw *a
*
bu
*bw
*b
=
*
*
*
*
*
b
b
b
a
a
a
ab
w
u
w
*r
b b
Plane Frame, Member Stiffness Matrix
The forceforceforceforce displacementdisplacementdisplacementdisplacement relationshiprelationshiprelationshiprelationship can be written:
**
aab uX
*abM
*baM
*abX
*abZ
*baX
*baZ
a b
==
=
*
*
*
*
*
*
*
*
*
*
b
b
b
a
a
a
ababab
ab
ab
ab
ab
ab
ab
ab
w
u
w
u
M
Z
X
M
Z
X
**** krkR
member vector of secondarysecondarysecondarysecondary local forcesforcesforcesforces
member vector of local joint displacements
member local stiffness member local stiffness member local stiffness member local stiffness matrixmatrixmatrixmatrix
*Rab
*rab*k ab
Plane Frame, Member Stiffness Matrix
Member locallocallocallocal stiffness matrix
l
EA
l
EA0000
0000
l
EA
l
EA
FixedFixedFixedFixed FixedFixedFixedFixed connectionconnectionconnectionconnection FixedFixedFixedFixed HingedHingedHingedHinged connectionconnectionconnectionconnection
=
l
EI
l
EI
l
EI
l
EIl
EI
l
EI
l
EI
l
EIl
EA
l
EAl
EI
l
EI
l
EI
l
EIl
EI
l
EI
l
EI
l
EIll
ab
460
260
6120
6120
0000
260
460
6120
6120
22
2323
22
2323
*k
=
000000
03
033
0
0000
03
033
0
03
033
0
0000
323
22
323
l
EI
l
EI
l
EIl
EA
l
EAl
EI
l
EI
l
EIl
EI
l
EI
l
EIll
ab*k
HingedHingedHingedHinged FixedFixedFixedFixed connectionconnectionconnectionconnection HingedHingedHingedHinged HingedHingedHingedHinged connectionconnectionconnectionconnection
=
l
EI
l
EI
l
EIl
EI
l
EI
l
EIl
EA
l
EA
l
EI
l
EI
l
EIl
EA
l
EA
ab
3300
30
3300
30
0000
000000
3300
30
0000
22
233
233
*k
=
000000
000000
0000
000000
000000
0000
l
EA
l
EA
l
EA
l
EA
ab*k
HingedHingedHingedHinged FixedFixedFixedFixed connectionconnectionconnectionconnection HingedHingedHingedHinged HingedHingedHingedHinged connectionconnectionconnectionconnection
Plane Frame, Member Stiffness Matrix
Member globalglobalglobalglobal stiffness matrix kab
abababab TkTk* = T abababab TkTk =
=
0cossin000
0sincos000
000100
0000cossin
0000sincos
abab
abab
abab
ab
T
x
z
a
b
Tab transformation matrix
100000
0cossin000 abab z angle of angle of angle of angle of
transformation transformation transformation transformation
b
Plane Frame, Member vector of primary forces
Member vector of primary locallocallocallocal forces is corresponding
to the fixedfixedfixedfixed endendendend rererereactionactionactionaction due to external load.
The fixed end reactions were derived by forceforceforceforce methodmethodmethodmethod The fixed end reactions were derived by forceforceforceforce methodmethodmethodmethod
(as well as secondary forces for member stiffness matrix).
a b*abX
*baX
=*
*
*
ab
ab
ab
M
Z
X
*R
a b*abZ
*abM *
baZ
*baM
=
*
*
*
ab
ab
ab
abab
M
Z
X*R
Plane Frame, Member vector of primary forces
a) Continuous load
Member vector of primary locallocallocallocal forces
1q 2qa) Continuous load
1n2n
Member connection
+
+
20/)37(
6/)2(
21
21
lqq
lnn
+
+
40/)916(
6/)2(
21
21
lqq
lnn
+
+
40/)411(
6/)2(
21
21
lqq
lnn
( )
+
+
6/2
6/)2(
21
21
lqq
lnn
*
*
ab
ab
Z
X
+
+
+
+
60/)32(
20/)73(
6/)2(
60/)23(
221
21
21
221
21
lqq
lqq
lnn
lqq
+
+
+
0
40/)114(
6/)2(
120/)78(
21
21
221
lqq
lnn
lqq
+
+
+
120/)87(
40/)169(
6/)2(
0
221
21
21
lqq
lqq
lnn
( )
+
+
0
6/2
6/)2(
0
21
21
21
lqq
lnn
=
*
*
*
*
ba
ba
ba
ab
ab
M
Z
X
M
Z
*abR
Plane Frame, Member vector of primary forces
a) Loading by force
Member vector of primary locallocallocallocal forces
F
ba) Loading by force
a ba b
Member connection
( )
322 2/)3(
/
lblbF
lbF
z
x
/
/
lbF
lbF
z
x
( )
2/)3(
/
32 lblbF
lbF
z
x
+
32 /)2(
/
lalbF
lbF
z
x
*
*
ab
ab
Z
X( )
( )
( ) ( )
+
2
32
2/
2/)3(
/
0
2/)3(
lalabF
lalaF
laF
lblbF
z
z
x
z
0
/
/
0
laF
laF
z
x
z( )
( ) ( )
( )
+
0
2/)3(
/
2/
2/)3(
322
2
lalaF
laF
lblabF
lblbF
z
x
z
z
+
+
22
32
22
/
/)2(
/
/
/)2(
lbaF
lblaF
laF
labF
lalbF
z
z
x
z
z
=
*
*
*
*
ba
ba
ba
ab
ab
M
Z
X
M
Z
*abR
Plane Frame, Member vector of primary forces
a) Loading by bending moment
Member vector of primary locallocallocallocal forces
Ma ba) Loading by bending moment a ba b
Member connection
3/6
0
lMab ( ) ( )
2/3
0
322 lblM ( ) ( )
322 2/3
0
lalM
/
0
lM
*
*
ab
ab
Z
X
2
3
2
/)32(
/6
0
/)32(
/6
lalMa
lMab
lblMb
lMab ( ) ( )( )
( ) ( )
0
2/3
0
2/)3(
2/3
322
222
lblM
lblM
lblM ( ) ( )
( ) ( )( )
222
322
2/)3(
2/3
0
0
2/3
lalM
lalM
lalM
0
/
0
0
/
lM
lM
=
*
*
*
*
ba
ba
ba
ab
ab
M
Z
X
M
Z
*abR
Plane Frame, Member vector of primary forces
Member vector of primary globalglobalglobalglobal forces
*Tababab RTR = ababab RTR =
=
0cossin000
0sincos000
000100
0000cossin
0000sincos
abab
abab
abab
ab
T
x
z
a
b
100000
0cossin000 abab z angle of angle of angle of angle of
transformation transformation transformation transformation
b
Plane Frame, The load displacement equation
After establishing the globalglobalglobalglobal stiffnessstiffnessstiffnessstiffness matrixmatrixmatrixmatrix and loadloadloadload
vectorvectorvectorvector, the load displacement relationship can be
written:written:
Global stiffness matrix K is established by the
localization of member global stiffness matrixes kab Global load vector F is taken as difference between
vector of joint loads S and vector of primary global
FKrFrK == 1
v
