Advanced Methods of Structural Analysis

3.2 The Generation of Statically Determinate Trusses 47

3.2 The Generation of Statically Determinate Trusses...................

The trusses according to their generation are subdivided into simple, compound, and

complex ones.

3.2.1 Simple Trusses................................................

Simple trusses are formed on the basis of a hinge-connected triangle with any addi-

tional joint attached by means of two additional members. The relationship between

the number of bars (S) and joints (J) for simple statically determined and geomet-

rically unchangeable trusses is expressed as

SD2J3: (3.1)

This formula can be derived using two principally different approaches.

1.The unchangeability condition of a truss.A simple truss has an initial triangle
disc and each new joint is attached to the previous rigid disk using two bars
(Fig.1.4c). The total number of bars isSD 3 C2.J3/. In this formula, the
first number represents the three bars used for the initial rigid disc. The 3 within
the brackets arises from the three joints of the initial triangle, soJ 3 represents
the number of joints attached to the initial triangle. The coefficient 2 stems from
the number of bars associated with each additional joint. Therefore, the total
number of bars is given by (3.1).
2.The statically determinacy condition of a truss.For each joint, two equilibrium
equations must be satisfied:

P

X D0;

P
Y D 0. The number of unknown
internal forces is equal to the total number of membersS. The total number
of unknowns, including the reactions of the supports, isSC 3. The number of
equilibrium equations equals2J. Therefore, a truss is statically determinate if
SD2J 3.
Thus, the two different approaches both lead to the same (3.1) for the kinematical
analysis of a simple truss. Therefore, if asimpletruss is statically determinate, its
geometry must be unchangeable and vice versa. Note, however, that (3.1), while
necessary, is an insufficient condition on its own to determine that a truss is geomet-
rically unchangeable. It is indeed possible to devise a structure which satisfies (3.1)
yet connects the bars in such a way that the structure is geometrically changeable.
For more examples see Chap. 1.
If the number of barsS>2J 3 , then the system is statically indeterminate;
this case will be considered later in Part 2. If the number of barsS<2J 3 ,
then the system is geometrically changeable and cannot be used as an engineering
structure.