Advanced Methods of Structural Analysis

318 9 Mixed Method

r 210

D



kN m

kN

DŒm:

Equilibrium condition of induced constraint 2 from bendingmoment diagramM 2

leads to coefficient of the displacement method

r 22 D 1 EIC0:5EID1:5EI.kN m/rad/:

Theorem of reciprocal displacements and reactions leads toı^012 Dr 210 D^12 m=rad:

The unit ofı^0 ikis defined by following rule: the unit ofdisplacement,which

corresponds to the indexi, should be divided by unit of factor, which corresponds

to the indexk. In our case we get

ı 210

DŒm=rad.

For calculation of free terms of canonical equations we need to construct the

bending moment diagram in the primary system due to distributed load q; this

diagram is presented in Fig.9.2e. Specific ordinates ofMP^0 diagram are follows:

M 2  3 DM 3  2 Dql^2 =12D.6 82 /=12D 32 kN m;MCDql^2 =24D 16 kN m:

From this diagram we getR2PD32 .kN m/and1PD 0.

Canonical equations of the mixed method becomes:

864

EI

X 1 C12Z 2 D0;

12X 1 C1:5EIZ 2  32 D0:

Roots of these equations areX 1 D0:266kN;Z 2 D19:2=EI.rad/

9.2.3 Computation of Internal Forces..............................

Resulting bending moments acting at different cross sections of the frame are cal-

culated by formula

MPDM 1 X 1 CM 2 Z 2 CMP^0 : (9.3)

Corresponding calculation for specified points is presented in Table9.1. Resulting

bending moment diagram is presented in Fig.9.2f.

Verification.The static and kinematical verifications may be considered. They are:

1.Free body diagram for joint 2 is presented in Fig.9.2f. Direction of the moments
is shown according to location of the extended fibers, which are shown by dotted
lines. Equilibrium equation

P
M 2 D3:2C9:612:8D 0 is satisfied.
2.Displacement in the direction of eliminated constraintAin the original sys-
tem must be zero. This displacement may be computed by multiplication of two
bending moment diagrams: one of them is the resultant bending moment diagram
MPfor the entire given structure and second is bending moment diagram caused
by theX 1 D 1 inany primary system of the force method. One version of the