Advanced Methods of Structural Analysis

280 8 The Displacement Method

2.According to the reciprocal reactionstheorem, the secondary reactions satisfy
the symmetry conditionrikDrki, i.e.,r 12 Dr 21.
3.The dimensions of unit reactionrikare determined by the following rule: the
dimension ofreaction(force or moment) at indexiis divided by the dimension
of thedisplacement(linear or angular) at indexk. In our caseŒr 11 D kN mrad,
Œr 12 DkN mm DkN,Œr 21 DkNradDkN,Œr 22 DkNm.

8.2.4 Procedure for Analysis.......................................

For analysis of statically indeterminate continuous beams and frames by the dis-

placement method in canonical form the following procedure is suggested:

1.Define the degree of kinematical indeterminacy and construct the primary sys-

tem of the displacement method

2.Formulate the canonical equations of the displacement method (8.4)

3.Apply successively unit displacementsZ 1 D 1 ,Z 2 D 1;:::; Zn D 1 to

the primary structure. Construct the corresponding bending moment diagrams

MN 1 ;MN 2 ;:::;MNnusing TablesA.3–A.8

4.Calculate the main and secondary unit reactionsrik

5.Construct the bending moment diagramMP^0 due to the applied load in the pri-

mary system and calculate the free termsRiPof the canonical equations

6.Solve the system of equations with respect to unknown displacementsZ 1 ,

Z 2 ;:::;Zn

7.Construct the bending moment diagrams by formula

MPDMN 1 Z 1 CMN 2 Z 2 CCMNnZnCMP^0 : (8.5)

The termMNnZnrepresents a bending momentdiagram due to actual displace-

mentZn.ThetermMP^0 represents a bending moment diagram in the primary

system due to actual load

8.Compute the shear forces using the Schwedler theorem considering each mem-

ber due to the given loads and end bending moments and construct the corre-

sponding shear diagram

9.Compute the axial forces from the consideration of the equilibrium of joints of

the frame and construct the corresponding axial force diagram

10.Calculate reactions of supports and check them using the equilibrium condi-

tions for an entire structure as a whole or for any separated part

Let us show the application of this algorithm to the analysis of a uniform contin-

uous two-span beamA-1-B(Fig.8.6a). According to (8.1), this continuous beam

is kinematically indeterminate to the first degree. Indeed, supportAprevents linear

displacement of the joints and there is one rigid joint at support 1, therefore there is

only angular displacement at this support. Thus, the primary unknown is the angular