Advanced Methods of Structural Analysis

11.6 Analysis of Continuous Beams 403

They are

EPDl

u^2

u^2 



;SE 1 Dl

2 6 6 6 6 6

0

u^2

u^2 

0

3 7 7 7 7 7

After that we will determine the entries of these both matrices when moving loadP
is placed at the sections 8 and 10 and perform corresponding matrix procedures.

Load PD 1 in the third spanThis case can be considered elementary by tak-
ing into account the symmetry of the beam and loading in the first span. The final
influence lines for moments at the supportsBandCare shown in Fig.11.25e.
If loadPD 1 is placed at sections with odd numbers1; 3; : : :then ordinates of
influence linesMBandMCmay be computed by similar way.

Discussion

Dimension of each submatrix depends on the type of element. For truss member and
bending fixed-pinned member, each submatrix is a scalar. For fixed-fixed member,
a stiffness matrix is.22/; for this element the relationships between the bending
moments at the ends and angular displacements at the ends according TableA.4is
described by (11.7), so

kff D

EI

l



42

24

:

It is obvious that it is possible toexpandthe number of unknown forces at the
ends of the element and consider the displacements at the ends. For example, we
can consider four components for unknown forces (not only the bending moments,
but the reactions also) and four components for end displacements (not only the
angular displacements, but the lineardisplacements too). According TableA.4,the
force–displacement relationships can be presented in the following form

2 6 6 6 6 6

R 1

M 1

R 2

M 2

3 7 7 7 7 7

D

2 EI

l

2

6

6

4

6=l^2 3=l 6=l^2 3=l

3=l 2 3=l 1

6=l^2 3=l 6=l^2 3=l

3=l 1 3=l 2

3

7

7

5

„ ƒ‚ ...

kff



2 6 6 6 6 6

 1

' 1

 2

' 2

3 7 7 7 7 7

In this case we need to use for analysis the.44/– stiffness matrix. Similarly for
pinned-fixed member the stiffness matrix becomes

kpf D

3 EI

l

2

4

1=l^2 1=l^2 1=l

1=l^2 1=l^2 1=l

1=l 1=l 1

3

(^5) :