Advanced Methods of Structural Analysis

11.7 Analysis of Redundant Frames 405

moments are zero. Therefore, the vector offixed-end moments (vector of internal
forces of the first state) at sections 1–3 on the basis of theMP^0 andS-ediagrams

becomesSE 1 D
000

̆T
.
The static matrix is constructed on the basis of theZ-PandS-ediagrams.
Figure11.26f shows the free body diagram for joint 1 subjected to possible mo-
mentP 1 and two unknown internal forces in vicinity of joint 1 (bending moments
S 2 andS 3 /. Equilibrium condition isP 1 DS 2 CS 3.
It is obvious that

P 2 D

S 1

h



S 2

h

:

Thus, the static matrix becomes

AD



011

1=h 1=h 0

Stiffness matrix for each finite element and stiffness matrixkQfor all structure in
local coordinates are

k 1 D

EI 1

l 1



42

24

D

EI

h



42

24

;k 2 D

EI 2

l 2

Œ3D

EI

l

Œ3

Thus, the stiffness matrices of the structure in the local coordinates

kQD



k 1 0

0 k 2

D

EI

h

2

4

420

240

003

3

5

Matrix procedures:Intermediate matrix complex

kAQ TDEI

h

2

4

420

240

003

3

(^5) 
2
4
0 1=h
1 1=h
10
3
(^5) DEI
h
2
4
2 6=h
4 6=h
30
3
5
Stiffness matrix for whole structure in global coordinates
KDAkAQ
T
D

011
1= h 1= h 0
EI
h
2
4
2 6= h
4 6= h
30
3
5
„ ƒ‚ ...
kAQT
D
EI
h

7 6= h
6=h^2 12 = h^2
For 2  2 matrix, we can use the following useful relationship: if