Advanced Methods of Structural Analysis

11.6 Analysis of Continuous Beams 401

The stiffness matrix of all structure in global coordinates and inverse stiffness
matrix are

KDAkAQ

T

D



1100

0011

EI

l

2

6

6

4

30

42

24

03

3

7

7

5 D

EI

l



72

27

;

K^1 D

l

45 EI



7  2

 27

To construct the matricesPandS 1 we need to consider the unit moving load in all
spans separately. Next for each loading we need to construct the bending moment
diagram in primary system of displacement method and show the corresponding
Joint-load diagram.

LoadPD 1 in the first span The bending moment diagram in the primary system
and equivalent joint-load diagram are shown in Fig.11.25b.
On the basis of the joint-load andZ-Pdiagrams, the vectorPof the external joint
loads for any positionuof unit loadPin the first span is

PEDl

0:5u



1 u^2



0



On the basis of theMP^0 andS-ediagrams the vectorS 1 of unknown internal forces
S 1 S 4 in the first state for any positionPbecomes

SE 1 Dl

2 6 6 6 6 6

0:5u



1 u^2



0

0

0

3 7 7 7 7 7

Now we can determine the entries of both of these matrices when moving loadPis
placed at the sections 2 and 4 and performcorresponding matrix procedures.

PD 1 at the section 2

(uD0:333; D0:667/

PD 1 at the section 4

(uD0:667; D0:333/

PEDl

&

0:1480

0

'

;SE 1 D

2

(^66)
(^66)
(^66)
M 1
M 2
M 3
M 4
3
(^77)
(^77)
(^77)
1
Dl
2
(^66)
(^66)
(^66)
0:1480
0
0
0
3
(^77)
(^77)
(^77)
PEDl
&
0:1851
0
'
;SE 1 D
2
(^66)
(^66)
(^66)
M 1
M 2
M 3
M 4
3
(^77)
(^77)
(^77)
1
Dl
2
(^66)
(^66)
(^66)
0:1851
0
0
0
3
(^77)
(^77)
(^77)
(continued)