272 8 The Displacement Method8.1.1 Kinematical Indeterminacy..................................
In the case of the force method, the unknowns are forces at the redundant con-
straints. Knowing these forces we can find the distribution of internal forces and
after that, displacements at any point of a structure.
The fundamental approach in the displacement method is the opposite: initially
we calculate thedisplacementsat the ends of the members and then the internal
forces in the members. Thus, the primary unknowns in the displacement method are
thedisplacements.
Analysis of a structure by the displacement method is based on the following
assumptions:1.The deformations of the members caused by axial and shear forces can be
neglected.
2.The difference between the length of the deformable element and its initial length
can be neglected.
Analysis of any statically indeterminate structure by the displacement method be-
gins with determining the degree ofkinematicalindeterminacy. Generally, the
degree ofkinematicalindeterminacynof a structure is determined by the formula
nDnrCnd; (8.1)wherenris the number of unknown angles of rotation of the rigid joints of a structure
andndis the number of independent lineardisplacements of the joints.
Note that, in general, the degrees ofkinematicaland statical indeterminacy are
not equal.
To calculate the number of linear displacements,nd, we need to introduce the
concept of the “hinged system or scheme.” A hinged system is obtained from the
original structure by introducing hinges at all rigid joints and supports while con-
sidering all members of the hinged scheme to be absolutely rigid. The degree of
mobility of a hinged system is determined by the number of additional members,
which would transform a hinged system into a geometrically unchangeable struc-
ture. The degree of mobility in turn determines the number of independent linear
joint displacements,nd.
Let us consider the structure shown in Fig.8.1a; the elastic curve is shown by
a dotted line. Since axial deformations are neglected and support 1 isunmovable,
the joints of the entire structure have no horizontal displacements. Since the number
of rigid joints equals two, thennrD 2. To show the hinged scheme, we introduce
hinges at all rigid joints; this scheme represents a geometrically unchangeable struc-
ture. Indeed, the structure 1-2-3 presents a rigid disc and joint 4 is connected with
this rigid disc and with the ground at support 5. SondD 0 and the total degree of
kinematical indeterminacy equals two.
Due to the action of the external load, rigid joints 2 and 4 rotate by angles' 2 and
' 4 , respectively. These angular displacements,' 2 and' 4 , determine completely the
deformable shape of the structure and represent the unknowns of the displacement
method.