Advanced Methods of Structural Analysis

11.6 Analysis of Continuous Beams 393

Notes:

1.In different textbooks the algorithm above (or slightly modified algorithm) is
referred differently. They are the matrix displacement method (do not confuse
with the displacement method in matrix form), finite element method, stiffness
method. All of them realize the one general idea: presentation of the structure as a
set of separate elements with necessary demands about consistent of deformation.

2.The entries of the stiffness matrixKDAkAQ

T
present the unit reactions of the
displacement method in canonical form.
3.Generally speaking, a vector of internal unknown forcesSmay be constructed by
different forms. In this textbook, the vectorSis presented in thesimplest form.
This vector contains onlynonzerobending moments (for beams and frames)
for each separate member at the ends. This choice of the vectorSleads to the
very compact stiffness matrix. Indeed, for fixed-pinned beam, this matrix con-
tains only one entry, for fixed-fixed standard member, this matrix has dimension
.22/. Certainly, we can expand the vector stateSincluding, for example, the
shear and axial force. However, this leads to the stiffness matrix with expanded
dimensions. Even if the final result would contain more complete information,
observing over all matrices is difficult. Therefore, we limited our consideration
of the MSM only forsimplestpresentation of vectorS. This leads to the sim-
ple and vivid of intermediate and final results, and significant simplification of
numerical procedures.

11.6 Analysis of Continuous Beams

This section presents a detailed analysis of statically indeterminate continuous

beams subjected to different types of exposures.

Design diagram of the uniform two-span beam subjected to fixed load is shown

in Fig.11.23a. This structure may be presented as a set of two finite elements: they

areA-1 and 1-B. To transmit the given load to the joint load, we need to calculate

the fixed end moments at support 1 (Fig.11.23b). They are equal to

M1A^0 D

ql 12

8

D

q 82

8

D16 .kNm/andM1B^0 D

Pl 2

2





1 ^2



D

12  10

2

0:4



1 0:4^2



D20:16 .kNm/;

so the equivalent moment20:16 16 D4:16acts clockwise (Fig.11.23c) and cor-

responding joint-load diagram (first state) is shown in Fig.11.23d.

The beam has one unknown angular displacement at support 1 and corresponding

one possible external joint load. The displacement-load (Z-P) diagram is presented

in Fig.11.23e. Having the joint-load andZ-Pdiagrams, we can construct the vector