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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 *
aϕ*
bu
*bw
*bϕ
=
*
*
*
*
*
b
b
b
a
a
a
ab
w
u
w
ϕ
ϕ*r
bϕ bϕ
Plane Frame, Member Stiffness Matrix
� The forceforceforceforce displacementdisplacementdisplacementdisplacement relationshiprelationshiprelationshiprelationship can be written:
**ˆ
aab uX∧
*abM
∧*baM∧
*abX
∧*abZ
∧*baX
∧*baZ
a b
⋅=⋅=
=
*
*
*
*
*
*
*
*
*
*
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
b
b
b
a
a
a
ababab
ab
ab
ab
ab
ab
ab
ab
w
u
w
u
M
Z
X
M
Z
X
ϕ
ϕ**** 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
*Rabˆ
*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 * ⋅⋅= Tabababab TkTk ⋅⋅=
−
−
=
0cossin000
0sincos000
000100
0000cossin
0000sincos
abab
abab
abab
ab
γγγγ
γγγγ
T
x
zγ γ γ γ
a
b
Tab … transformation matrix
100000
0cossin000 abab γγ zγ γ γ γ … … … … 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
αF
ba) Loading by force αa b
a 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
γγγγ
γγγγ
T
x
z
a
b
100000
0cossin000 abab γγ zγ γ γ γ … … … … 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
vector of joint loads S and vector of primary global
forces
� Vector of primary global forces is established by the
localization of member vectors of primary global forces
RSF −=R
R
abR
Plane Frame, Member vector of known forces
� Member vector of joint displacements
a
a
w
u
� Member vector of secondary global forces
=
b
b
b
a
a
ab
w
u
w
ϕ
ϕr
uX̂
⋅=⋅=
=
b
b
b
a
a
a
ababab
ab
ab
ab
ab
ab
ab
ab
w
u
w
u
M
Z
X
M
Z
X
ϕ
ϕkrkR
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
Plane Frame, Member vector of known forces
� Member vector of known globalglobalglobalglobal forces
ab
ab
ab
ab
ab
ab
Z
X
Z
X
Z
Xˆ
ˆ
� Member vector of known locallocallocallocal forces
=
+
=+=
ba
ba
ba
ab
ab
ab
ab
ab
ab
ab
ab
ab
ab
ab
ab
ababab
M
Z
X
M
Z
M
Z
X
M
Z
M
Z
X
M
Z
ˆ
ˆ
ˆ
ˆ
ˆ
R̂RR
*ab XX
=
⋅=⋅=
*
*
*
*
*
*
*
ba
ba
ba
ab
ab
ab
ba
ba
ba
ab
ab
ab
abababab
M
Z
X
M
Z
X
M
Z
X
M
Z
X
TRTR
a
b
q = 5,0 kN/m
cdef 2 4
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
5
3
F1 = 8 kN
cdef 2 4
1
3
8
A1=A3= 0,52 m2
I1=I 3= 0,0062 m4
A2=A4= 0,26 m2
I2=I 4= 0,0031m4
4 104
F2 = 5,3 a
b6
I2=I 4= 0,0031m
E = 21 GPa
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
Degrees of freedom:
� Frame is kinematically indeterminate to 5th degree.
= e
e
e
u
w
u
ϕr
d
d
w
u
Code number:
� Non-zero code number is assigned code number is assigned code number is assigned code number is assigned to each unknown.
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
4
3
2
1
= e
e
e
u
w
u
ϕr
(1 2 3) (4 5 0)(0 0 0)
5
4
d
e
w
u
(0 0 0)
(0 0 0)
Member parameters:
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
Member E [kPa] A I l cos sin Code numbers
1 21000000 0,52 0,0062 8 0 -1 0 0 0 1 2 32 21000000 0,26 0,0031 10 1 0 1 2 3 4 5 02 21000000 0,26 0,0031 10 1 0 1 2 3 4 5 03 21000000 0,52 0,0062 12 0 1 4 5 0 0 0 04 21000000 0,26 0,0031 6 1 0 4 5 0 0 0 0
(1 2 3)(4 5 0)
(0 0 0) Member connection:
1. ae … hinged – fixed
2. ed … fixed – hinged
(0 0 0)
(0 0 0)
2. ed … fixed – hinged
3. db … hinged – fixed
4. dc … hinged - fixed
Loading:
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
Mem
ber
1
n1
Mem
ber
2
n1
Mem
ber
3
n1
Mem
ber
4
n1
n2 n2 n2 n2
q1 q1 5,00 q1 q1 5,00q2 q2 5,00 q2 q2 0,00Fx
* Fx* Fx
* Fx*
Fz* Fz
* Fz* 8,00 5,30 Fz
*M
emb
er
Mem
ber
2
Mem
ber
3
Mem
ber
Fz* Fz
* Fz* 8,00 5,30 Fz
*
a* a* a* 5,00 8,00 a*
b* b* b* 7,00 4,00 b*
M M M Ma* a* a* a*
b* b* b* b*
1q 2q
1n2n
FzF
a ba b
xF
M
a ba b
Member local stiffness matrix:
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
Fixed - Fixed Fixed - Hinged Hinged - Fixed Hinged - Hinged
Mem
ber
1 1365000 0 0 -1365000 0 0 1365000 0 0 -1365000 0 0 1365000 0 0 -1365000 0 0 1365000 0 0 -1365000 0 0
0 3052 -12206 0 -3052 -12206 0 763 -6103 0 -763 0 0 763 0 0 -763 -6103 0 0 0 0 0 0
48825
Mem
ber
0 -12206 65100 0 12206 32550 0 -6103 48825 0 6103 0 0 0 0 0 0 0 0 0 0 0 0 0
-1365000 0 0 1365000 0 0 -1365000 0 0 1365000 0 0 -1365000 0 0 1365000 0 0 -1365000 0 0 1365000 0 0
0 -3052 12206 0 3052 12206 0 -763 6103 0 763 0 0 -763 0 0 763 6103 0 0 0 0 0 0
0 -12206 32550 0 12206 65100 0 0 0 0 0 0 0 -6103 0 0 6103 48825 0 0 0 0 0 0
Fixed - Fixed Fixed - Hinged Hinged - Fixed Hinged - Hinged
Mem
ber
2 546000 0 0 -546000 0 0 546000 0 0 -546000 0 0 546000 0 0 -546000 0 0 546000 0 0 -546000 0 0
0 781 -3906 0 -781 -3906 0 195 -1953 0 -195 0 0 195 0 0 -195 -1953 0 0 0 0 0 0
0 -3906 26040 0 3906 13020 0 -1953 19530 0 1953 0 0 0 0 0 0 0 0 0 0 0 0 0
-546000 0 0 546000 0 0 -546000 0 0 546000 0 0 -546000 0 0 546000 0 0 -546000 0 0 546000 0 0
0 -781 3906 0 781 3906 0 -195 1953 0 195 0 0 -195 0 0 195 1953 0 0 0 0 0 0
0 -3906 13020 0 3906 26040 0 0 0 0 0 0 0 -1953 0 0 1953 19530 0 0 0 0 0 0
Fixed - Fixed Fixed - Hinged Hinged - Fixed Hinged - Hinged
Mem
ber
3 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0
0 904 -5425 0 -904 -5425 0 226 -2713 0 -226 0 0 226 0 0 -226 -2713 0 0 0 0 0 0
0 -5425 43400 0 5425 21700 0 -2713 32550 0 2713 0 0 0 0 0 0 0 0 0 0 0 0 0
-910000 0 0 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0 910000 0 0
Mem
ber
-910000 0 0 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0 910000 0 0
0 -904 5425 0 904 5425 0 -226 2713 0 226 0 0 -226 0 0 226 2713 0 0 0 0 0 0
0 -5425 21700 0 5425 43400 0 0 0 0 0 0 0 -2713 0 0 2713 32550 0 0 0 0 0 0
Fixed - Fixed Fixed - Hinged Hinged - Fixed Hinged - Hinged
Mem
ber
4 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0
0 3617 -10850 0 -3617 -10850 0 904 -5425 0 -904 0 0 904 0 0 -904 -5425 0 0 0 0 0 0
0 -10850 43400 0 10850 21700 0 -5425 32550 0 5425 0 0 0 0 0 0 0 0 0 0 0 0 0
-910000 0 0 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0 910000 0 0 -910000 0 0 910000 0 0
0 -3617 10850 0 3617 10850 0 -904 5425 0 904 0 0 -904 0 0 904 5425 0 0 0 0 0 0
0 -10850 21700 0 10850 43400 0 0 0 0 0 0 0 -5425 0 0 5425 32550 0 0 0 0 0 0
Member vector of primary local forces:
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
F - F F - H H - F H - H
Mem
ber
1 0,00 0,00 0,00 0,000,00 0,00 0,00 0,000,00 0,00 0,00 0,000,00 0,00 0,00 0,00
Mem
ber
0,00 0,00 0,00 0,000,00 0,00 0,00 0,000,00 0,00 0,00 0,00
F - F F - H H - F H - H
Mem
ber
2 0,00 0,00 0,00 0,00-25,00 -31,25 -18,75 -25,0041,67 62,50 0,00 0,000,00 0,00 0,00 0,00
-25,00 -18,75 -31,25 -25,00-41,67 0,00 -62,50 0,00
F - F F - H H - F H - H
3 0,00 0,00 0,00 0,00
Mem
ber
3 0,00 0,00 0,00 0,00-6,36 -8,76 -4,07 -6,4318,32 27,89 0,00 0,000,00 0,00 0,00 0,00-6,94 -4,54 -9,23 -6,87-19,14 0,00 -28,31 0,00
F - F F - H H - F H - H
Mem
ber
4 0,00 0,00 0,00 0,00-10,50 -12,00 -8,25 -10,009,00 12,00 0,00 0,000,00 0,00 0,00 0,00-4,50 -3,00 -6,75 -5,00-6,00 0,00 -10,50 0,00
Member transformation matrix:
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
0,00 -1,00 0,00 0,00 0,00 0,00 0,00 1,00 0,00 0,00 0,00 0,00
Mem
ber
1
0,00 -1,00 0,00 0,00 0,00 0,001,00 0,00 0,00 0,00 0,00 0,000,00 0,00 1,00 0,00 0,00 0,000,00 0,00 0,00 0,00 -1,00 0,000,00 0,00 0,00 1,00 0,00 0,000,00 0,00 0,00 0,00 0,00 1,00
Mem
ber
2
1,00 0,00 0,00 0,00 0,00 0,000,00 1,00 0,00 0,00 0,00 0,000,00 0,00 1,00 0,00 0,00 0,000,00 0,00 0,00 1,00 0,00 0,000,00 0,00 0,00 0,00 1,00 0,00
Mem
ber
3
0,00 1,00 0,00 0,00 0,00 0,00-1,00 0,00 0,00 0,00 0,00 0,000,00 0,00 1,00 0,00 0,00 0,000,00 0,00 0,00 0,00 1,00 0,000,00 0,00 0,00 -1,00 0,00 0,000,00 0,00 0,00 0,00 0,00 1,00
Mem
ber
4
1,00 0,00 0,00 0,00 0,00 0,000,00 1,00 0,00 0,00 0,00 0,000,00 0,00 1,00 0,00 0,00 0,000,00 0,00 0,00 1,00 0,00 0,00
Mem
ber
2
0,00 0,00 0,00 0,00 1,00 0,000,00 0,00 0,00 0,00 0,00 1,00
Mem
ber
4
0,00 0,00 0,00 0,00 1,00 0,000,00 0,00 0,00 0,00 0,00 1,00
Member global stiffness matrix: TkTk ⋅⋅= *T
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
0 0 0 1 2 3 code no. 4 5 0 0 0 0 code no.0 0 0 1 2 3 code no.
Mem
ber
1
763 0 0 -763 0 -6103 0
0 1365000 0 0-1365000 0 0
0 0 0 0 0 0 0
-763 0 0 763 0 6103 1
0-1365000 0 0 1365000 0 2
-6103 0 0 6103 0 48825 3
1 2 3 4 5 0 code no.
Mem
ber
2
546000 0 0 -546000 0 0 1
0 195 -1953 0 -195 0 2
0 -1953 19530 0 1953 0 3
-546000 0 0 546000 0 0 4
4 5 0 0 0 0 code no.
Mem
ber
3
226 0 0 -226 0 2713 4
0 910000 0 0 -910000 0 5
0 0 0 0 0 0 0
-226 0 0 226 0 -2713 0
0 -910000 0 0 910000 0 0
2713 0 0 -2713 0 32550 0
4 5 0 0 0 0 code no.
Mem
ber
4
910000 0 0 -910000 0 0 4
0 904 0 0 -904 -5425 5
0 0 0 0 0 0 0
-910000 0 0 910000 0 0 0
Mem
ber
2
-546000 0 0 546000 0 0 4
0 -195 1953 0 195 0 5
0 0 0 0 0 0 0
Mem
ber
4
-910000 0 0 910000 0 0 0
0 -904 0 0 904 5425 0
0 -5425 0 0 5425 32550 0
Member vector of primary global forces
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
*Tababab RTR ⋅=
code no.code no.
code no.
Mem
ber
1
0,00 0
0,00 0
0,00 0
0,00 1
0,00 2
0,00 3
code no.
0,00 1
-31,25 2
code no.
Mem
ber
3
4,07 4
0,00 5
0,00 0
9,23 0
0,00 0
-28,31 0
code no.
0,00 4
5
Mem
ber
2 -31,25 2
62,50 3
0,00 4
-18,75 5
0,00 0
Mem
ber
4 -8,25 5
0,00 0
0,00 0
-6,75 0
-10,50 0
Global stiffness matrix and vector of primary forces (partial
calculation): ““““Localization” Localization” Localization” Localization” according to the code number
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
1 2 3 4 5
1 763 0 6103 0,00 1
Me
mb
er
1 1 763 0 6103 0,00 1
2 0 1365000 0 0,00 2
3 6103 0 48825 0,00 3
4 4
5 5
1 2 3 4 5
Me
mb
er
2 1 546000 0 0 -546000 0 0,00 1
2 0 195 -1953 0 -195 -31,25 2
3 0 -1953 19530 0 1953 62,50 3
4 -546000 0 0 546000 0 0,00 4
5 0 -195 1953 0 195 -18,75 5
1 2 3 4 51 2 3 4 5
Me
mb
er
3 1 1
2 2
3 3
4 226 0 4,07 4
5 0 910000 0,00 5
1 2 3 4 5
Me
mb
er
4 1 1
2 2
3 3
4 910000 0 0,00 4
5 0 904 -8,25 5
Global stiffness matrix (summation of partial calculations)
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
K 1 2 3 4 5
1 546763 0 6103 -546000 0
Global load vector:
1 546763 0 6103 -546000 02 0 1365195 -1953 0 -1953 6103 -1953 68355 0 19534 -546000 0 0 1456226 05 0 -195 1953 0 911099
R S F = S - R 00F R S F = S - R0,00 1 1 0,00 1
-31,25 2 20,00 2 51,25 2
62,50 3 40,00 3 -22,50 3
4,07 4 4 -4,07 4
-27,00 5 5 27,00 5
=
⋅⋅
⋅
=
=
0
0
40
20
0
0
02
44
4
0
q
q
F
F
M
F
F
zd
xd
e
ze
xe
S
The load displacement equation:
FKrFrK 1 ⋅=⇒=⋅ −
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
2
1
000037,0
000001,0
e
e
w
u
Member vector of joint displacements:
� Creating according to the code number
5
4
3
2
000030,0
000002,0
000329,0
000037,0
−−=
=
d
d
e
e
w
u
w
ϕr
code no. Member1 code no. Member2 code no. Member3 code no. Member4
0 0,000000 1 0,000001 4 -0,000002 4 -0,000002
0 0,000000 2 0,000037 5 0,000030 5 0,000030
0 0,000000 3 -0,000329 0 0,000000 0 0,000000
1 0,000001 4 -0,000002 0 0,000000 0 0,000000
2 0,000037 5 0,000030 0 0,000000 0 0,000000
3 -0,000329 0 0,000000 0 0,000000 0 0,000000
Member vector of secondary global forces:
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
code no. Member1 code no. Member2 code no. Member3 code no. Member4
0 2,01 1 2,01 4 0,00 4 -2,07
Member vector of known global forces:
0 2,01 1 2,01 4 0,00 4 -2,070 -50,61 2 0,64 5 27,62 5 0,030 0,00 3 -6,44 0 0,00 0 0,001 -2,01 4 -2,01 0 0,00 0 2,072 50,61 5 -0,64 0 -27,62 0 -0,033 -16,06 0 0,00 0 -0,01 0 -0,16
Member1 Member2 Member3 Member4code no. Member1 code no. Member2 code no. Member3 code no. Member4
0 2,01 1 2,01 4 4,07 4 -2,070 -50,61 2 -30,61 5 27,62 5 -8,220 0,00 3 56,06 0 0,00 0 0,001 -2,01 4 -2,01 0 9,23 0 2,072 50,61 5 -19,39 0 -27,62 0 -6,783 -16,06 0 0,00 0 -28,31 0 -10,66
Member vector of known local forces:
Direct Stiffness Method: Plane Frame
Example Example Example Example 1111
Member1 Member2 Member3 Member4
50,61 2,01 27,62 -2,072,01 -30,61 -4,07 -8,220,00 56,06 0,00 0,00
Diagrams of internal forces:Diagrams of internal forces:Diagrams of internal forces:Diagrams of internal forces:
0,00 56,06 0,00 0,00-50,61 -2,01 -27,62 2,07-2,01 -19,39 -9,23 -6,78
-16,06 0,00 -28,31 -10,66
