Advanced Methods of Structural Analysis

11.1 Basic Idea and Concepts 371

X

Y

1

A

P

x 3

y 3

2

3

x 1

y 1 x 2

y 2

D

4

a

Fig. 11.2 Local and global coordinate systems

The members 1, 2, 3, 4 met at the jointA. Internal forcesNiof the elements
1–4 does not coincide with given external loadP. Axial deformationiof each
element does not coincide with vertical displacementof jointA. Therefore, we
need to distinguish between the internal and external forces, deflection of separate
members and displacements of joints of the structure. For this purpose, we introduce
two coordinate systems.Each elementof the structure, correspondinginternalforce
anddeformationmay be referred to thelocalcoordinate systemx-y. The origin
of this system coincides with initial point of this element and one of the axes is
directed along the element itself as shown for members 1–3; for member 4, the
local coordinates are not shown. Thestructure in whole,theexternalforce, and
displacementsof the joints are referenced toglobalcoordinatesX-Y.
The difference between the local displacementsi of each element and the
global vertical displacementof jointAcan be seen from the follows relationships:

x 1 D; y 1 D 0 I

x 2 D0; y 2 DI

x 3 Dsin ̨; y 3 Dcos ̨;:::

11.1.3 Displacements of Joints and Degrees of Freedom

Once a structure is presented as a set of finite elements, we need to identify the
possible displacements of the ends of eachmember. In fact these displacements
present unknowns of the displacement method (angular displacements of the rigid
joints and their independent linear displacements). These displacements are called
thepossibleangular and linear displacements of the joints. The termpossiblemeans
that in given structure such displacement is possible, but not necessarily utilized.
For example, in case of two span continuous beam, a section at the middle support
generally rotates; however, if the beam and its loading are symmetrical then this
angle of rotation is zero. In summary, wecan see that any possible displacement of
the joint is a displacement inglobalcoordinates.
Degree of kinematical indeterminacynof a structure is determined by the for-
mulanD nrCnd,wherenris a number of unknown angles of rotation of the
rigid joints of a structure andndis a number of independent linear displacements of
joints. Parameterndefines the degrees of freedom of a structure.