Advanced Methods of Structural Analysis

11.6 Analysis of Continuous Beams 399

The final vector of required bending moments (kNm) is

ESDSE 1 CkAQ TZDEI 0

2 6 6 6 6 6 6 6 6

0:00667

0:00333

0:03

0:03

0:015

3 7 7 7 7 7 7 7 7

CEI 0

2 6 6 6 6 6 6 6 6

0:00313

0:00627

0:0204

0:0126

0:0024

3 7 7 7 7 7 7 7 7

DEI 0

2 6 6 6 6 6 6 6 6

0:0098

0:0096

0:0096

0:0174

0:0174

3 7 7 7 7 7 7 7 7

Corresponding final bending moment diagram is presented in Fig.11.24f.

Example 11.5.Design diagram of the uniform three-span continuous beam is pre-
sented in Fig.11.25a. Construct the influence lines for bending moments at the
supportsBandC(sections 6 and 12, respectively).

Solution.Each span of the beam is divided in equal portions and specified sections
are numerated (0–18). Next we needto show the displacement-load (Z-P)andS-e
diagrams (Fig.11.25a). Unknown momentsS 1 ,S 2 arise at supportBandS 3 ,S 4 at
supportC.

Static matrix TheZ-PandS-ediagrams allow us to constructing the following
equilibrium equations:

P 1 DS 1 CS 2

P 2 DS 3 CS 4

so the static matrix of the structure is

A.24/D



1100

0011

Stiffness matrix Stiffness matrices for each finite element are

k 1 D

EI

l

Œ3 ; k 2 D

EI

l



42

24

; k 3 D

EI

l

Œ3 :

Stiffness matrix of all structure in local coordinates and intermediate complex

kAQ Tare

kQDEI

l

2

6

6

4

3000

0420

0240

0003

3

7

7

5 ;

kAQ TDEI

l

2

6

6

4

3000

0420

0240

0003

3

7

7

5 

2

6

6

4

10

10

01

01

3

7

7

5 D

EI

l

2

6

6

4

30

42

24

03

3

7

7

5