Advanced Methods of Structural Analysis

11.7 Analysis of Redundant Frames 409

k 2 D

EI 2

l 2

Œ3D

2 EI

10

Œ3D

EI

5

Œ3 ;

k 3 D

EI 3

l 3

Œ3D

EI

3

Œ3D

EI

5

Œ5 :

For whole structure the stiffness matrix in local coordinates is

kQD

2

4

k 1 00

0 k 2 0

00 k 3

3

(^5) DEI
5
2
6
6
4
4200
2400
0030
0005
3
7
7
5
Matrix procedures.For whole structure the stiffness matrix in global coordinates
K D AkAQ
T
D

0111
0:2 0:2 0 0:333
EI
5
2
6
6
4
4200
2400
0030
0005
3
7
7
5 
2
6
6
4
0 0:2
1 0:2
10
1 0:333
3
7
7
5
D EI

2:4 0:093
0:093 0:207
The entries of this matrix are unit reactions of the displacement method in canonical
form (Example 8.2).
The determinant of this matrix is detKD0:48815, so the inverse matrix
K^1 D
1
EI

0:4241 0:1905
0:1905 4:9165
:
The matrix resolving equationKZEDPEallows us to find the vector of unknown
displacements
ZED
Z 1 .rad/
Z 2 .m/

DK^1 PE D
1
EI

0:4241 0:1905
0:1905 4:9165

11:193
5

D
1
EI
3:7944
22:452

These values present the angle of rotation of the rigid joint and linear displacement
of the cross-bar; they have been obtained previously by the displacement method
(Example 8.2).