8.5 Special Types of Exposures 299
The equilibrium condition of the horizontal element is used to calculateR 2 s
X
XD 0 !R 2 sD0:00336EI:The canonical equations become
2:4Z 1 C0:093Z 2 C0:0076D0;
0:093Z 1 C0:207Z 2 0:00336D0:The roots of these equations are
Z 1 D0:386 10 ^2 rad;
Z 2 D1:797 10 ^2 m:The bending moment diagram is constructed by the following expression using the
principle of superposition
MsDMN 1 Z 1 CMN 2 Z 2 CMs^0 ;whereMs^0 DMaCMbCMis the bending moment in the primary system caused
by the settlements of the support. The corresponding computation is presented in
Ta b l e8.5.
Ta b l e 8. 5 Calculation of bending moments
Points MN 1 MN 1 Z 1 MN 2 MN 2 Z 2 MaCMBCM' Ms
1 0:4 0:155 C0:24 0:431 1:04 0:454
3 C0:8 0:309 0:24 0:431 C0:64 0:100
4 1:0 0:386 0:333 0:598 0:0 0:212
5 0:0 0:0 0:0 0:0 0:0 0:0
6 0:6 0:232 0:0 0:0 0:12 0:112
8 0:0 0:0 0:0 0:0 0:0 0:0
Factor EI 10 ^2 EI EI 10 ^2 EI 10 ^2 EI 10 ^2 EI
signs of
bending moments
+ −+
−The final bending moment diagram due tothe settlements of the support is pre-
sented in Fig.8.11f. This same diagram was obtained previously by the force method
(Example 7.3).
Discussion. In case of the structure subjected to external loading, the internal force
distribution depends only on the relative stiffness of the elements, while in case of
the settlements of support the distribution of internal forces depends on both the
relative and the absolute stiffness of the elements.
In the case of settlements of a support, the calculation of free terms is an ele-
mentary procedure using the displacement method, while by the force method free
terms are more difficult to calculate. Therefore, for settlement of support problems
the displacement method is preferable to the force method.