Advanced Methods of Structural Analysis

250 7 The Force Method

RN'1,RN'2are reactions in direction of'-displacement due to the same primary

unknowns

The bending moment diagrams, unit displacementsıik, and reactionsRNi1RNi2in
the primary system caused by the unit primary unknownsX 1 andX 2 are shown in
Fig.7.18c.
The expressions (c) in an expanded form are

1sD.RNa1aCRNb1bCRN'1'/D.1aC 0 bC 10 '/

D.10:02C 0 0:01C 10 0:01/D0:12m; (7.4)

2sD.RNa2aCRNb2bCRN'2'/D.0aC 1 bC 8 '/

D.00:02C 1 0:01C 8 0:01/D0:09m:

Canonical equations become

666:67X 1 C275X 2 0:12EID0;

275X 1 C170:67X 2 0:09EID0: (7.5)

The roots of these equations are

X 1 D1:119 10 ^4 EI;

X 2 D7:076 10 ^4 EI: (7.6)

The bending moment diagram is constructed using the superposition principle ac-
cording to (7.15)
MDMN 1 X 1 CMN 2 X 2 : (7.7)

The corresponding calculations are presented in Table7.12.

Ta b l e 7. 1 2 Calculation of bending moments

Points MN 1 MN 1 X 1 MN 2 MN 2 X 2 M

1  10 1.119 8:0 5:6608 4:5418

3  10 1.119 3:0 2:1228 1:0038

4 0.0 0.0 3:0 2:1228 2:1228

5 0.0 0.0 0.0 0.0 0.0

6  10 1.119 0.0 0.0 1.1190

8 0.0 0.0 0.0 0.0 0.0

Factor 10 ^3 EI 10 ^3 EI 10 ^3 EI

signs of

bending moments

+ −

+

−

The resulting bending moment diagram is presented in Fig.7.18d.

Static verification A free body diagram for rigid joint of the frame is shown
in Fig.7.18d. The extended fibers are shown by dotted lines. Equilibrium of the
rigid joint X

MD.2:12281:00381:119/ 10 ^3 EID0: