Advanced Methods of Structural Analysis

290 8 The Displacement Method

The canonical equations of the displacement method become:

2:4Z 1 C0:093Z 2 

11:1933

EI

D0;

0:093Z 1 C0:207Z 2 

5

EI

D0: (a)

The roots of these equations present the primary unknowns and they are equal to

Z 1 D

3:794

EI

.rad/; Z 2 D

22:450

EI

.m/:

The final bending moment diagram is constructed by a formula using the principle
of superposition:

MPDMN 1 Z 1 CMN 2 Z 2 CMP^0 :

The corresponding calculations for the specified points (Fig.8.8c) are presented
in Table8.3. The signs of the bending moments in the unit conditions are conven-
tional.

Ta b l e 8. 3 Calculation of bending moments
Points MN 1 MN 1 Z 1 MN 2 MN 2 Z 2 MP^0 M(kN m)
1 0:4 1:5176 C0:24 C5:388 C4:1667 C8:037
2 C0:2 C0:7588 0:0 0:0 2:0833 1:324
3 C0:8 C3:0352 0:24 5:388 C4:1667 C1:814
4 1:0 3:794 0:333 7:476 0:0 11:27
5 0:0 0:0 0:0 0:0 0:0 0:0
6 0:6 2:2764 0:0 0:0 C15:36 C13:084
7 0:36 1:3658 0:0 0:0 9:984 11:349
8 0:0 0:0 0:0 0:0 0:0 0:0
Factor EI EI

6

3

4

signs of

bending moments

+ −

+

−

The final bending moment diagramMis presented in Fig.8.8g.Thesamedia-
gram was obtained by the force method (Example 7.2).
Construction of shear and axial force diagrams, as well as computation of the
reactions is described in detail in Example 7.2.

Summary. The presentation of the equation of the displacement method in canoni-
cal form is conveniently organized and also prescribes a well-defined algorithm for
the analysis of complex structures. Special parts of structural analysis can be carried
out more easily with the canonical form ofthe displacement method. These include
the construction of influence lines (Chap. 10 ), Stability (Chap. 13), and Vibration
(Chap. 14).