Advanced Methods of Structural Analysis

274 8 The Displacement Method

number of rigid joints is 2, hence the degree of kinematical indeterminacy of the en-
tire structure equals 3: the angular displacements of rigid joints and the horizontal
displacement of the crossbar.
In case (b), the number of rigid joints equals 3 and the number of independent
linear displacements,ndD 2 , so the degree of kinematical indeterminacy equals 5.
Case (c) presents a multistorey frame. JointCshows that memberCDis con-
nected with vertical memberAEby a hinge, while the memberACEdoes not have
a hinge at pointC. To construct the hinged scheme, it is necessary to introduce
hinges at jointsB,D, and jointCfor memberACE(note that hingeCincludes
the original hinge at jointCand the introduced hinge). For the given structure, the
number of rigid joints is 3 (jointsB,D,andCfor memberACE) and the number
of independent linear joint displacementsis 2, so the total number of unknowns by
the displacement method is 5.
Case (d) also presents a multistorey frame, in which the crossbars are absolutely
rigid. Since the joints cannot rotate, the number of unknown angles of rotation is
zero and the number of independent linear joint displacements is 2.

8.1.2 Primary System and Primary Unknowns....................

The primary system of the displacement method is obtained from the given one by
introducingadditional constraints to prevent rotation of all rigid joints and all in-
dependent displacements of various joints. These introduced constraints are shown
by the shaded squares and the double lines, respectively. Primary systems for some
structures are shown in Fig.8.3.
Primary unknownsZi.iD1;2;:::;n/representdisplacementsof introduced
constraints (angles of rotations and/or linear displacements of various joints of a
frame). The number of primary unknowns,n, for each structure equals to the de-
grees of its kinematical indeterminacy.

2

1

1

3

2

4

6

5

3

1 2

(^14)
(^23)
5
ab
cd
Fig. 8.3 Design diagrams and primary systems of the displacement method