Advanced Methods of Structural Analysis

11.6 Analysis of Continuous Beams 395

Stiffness matrices for left and right spans are

k 1 D

3 EI 1

l 1

Œ1D

3 EI 1

8

Œ1D

3 EI 1

40

Œ5 ;

k 2 D

3 EI 2

l 2

Œ1D

3 EI 1

10

Œ1D

3 EI 1

40

Œ4 :

Internal stiffness matrix for all beam in local coordinates is

kQD



k 1 0

0k 2

D

3 EI

40



50

04

Stiffness matrix for all structure in global coordinates and its inverse are

KDAkAQ

T

D

11

̆



3 EI

40



50

04



1

1



D

27

40 EI

!K^1 D

40 EI

27

Displacement of the joint 1

ZEDK^1 EPD^40 EI

27

4:16D

6:163

EI

Vector of internal forces of the second state

SE 2 DkAQ TZED^3 EI

40



50

04



1

1



6:163

EI

D

2:3111

1:8489



Final internal forces

ESfinDES 1 CSE 2 D

16

20:16



C

2:3111

1:8489



D

18:31

18:31



The negative sign means that momentS 2 , according toS-ediagram, is directed in

opposite direction, so external fibers right at the joint 1 are located above the neutral

line. Final bending moment diagram is shown in Fig.11.23g.

Equilibrium condition

P

M 1 D 0 for joint 1 is satisfied. The bending moment

at pointkmay be calculated considering second span as simply supported beam

subjected to forcePand moment 18.31 kNm counterclockwise.

Notes:

1.Analysis of this beam has been performed early by the displacement method
(Chap. 8 ) so the reader has an opportunity to compare analysis of the same beam
by different methods.