Advanced Methods of Structural Analysis

1.4 Degrees of Freedom 11

Equilibrium conditions leads to the following results:

JointA:

X

YD 0 W RAN 1 sin'D 0 !N 1 D

RA

sin'

D

Pb

.aCb/sin'

;

JointC:

X

YD 0 W N 4 CN 1 sin'D 0 !N 4 DN 1 sin'D

Pb

aCb

;

JointD:

X

YD 0 W N 4 D0:

We received for internal forceN 4 two contradictory (or inconsistent) results. Why

this happens? To answer on this question let us consider the very important concept

“degrees of freedom.”

1.4 Degrees of Freedom

A number of independent parameters, which define configuration of a system with-

out deformation of its members is called the degree of freedom. The number of

degrees of freedom for any structure may be calculated by the Chebushev’ formula

WD3D2HS 0 ; (1.1)

whereD,H,andS 0 are the number of rigid discs, simple hinges, and constraints

of supports, respectively.

For trusses the degrees of freedom may be calculated by formula

WD2JSS 0 ; (1.2)

whereJandSare the number of joints and members of the truss, respectively.

Special cases.There are three special cases possible.

1.W>0. The system is geometrically changeable and cannot be used in engineer-
ing practice.
2.W D 0. The system has the necessary number of elements and constraints to
be geometrically unchangeable structure. However, the system still can be in-
appropriate for engineering structure. Therefore, this case requires additional
structural analysis to check if the formation of the structure and arrangement of
elements and constraints is correct. This must be done according to rules, which
are considered above. For example, let us consider systems, which are presented
in Fig.1.13a–c.