(zero degree of freedom)
STABILITY AND DETERMINACY
After reading this lesson the student will be able to
1. Differentiate between various structural forms such as beams, plane truss,
space truss, plane frame, space frame, arches, cables, plates and shells.
2. State and use conditions of static equilibrium.
3. Calculate the degree of static and kinematic indeterminacy of a given structure
such as beams, truss and frames.
4. Differentiate between stable and unstable structure.
1.
Stability and Determinacy of Structures:
Structural stability must be judged by the number and arrangement of
the members and the connection of the
structure. They are determined by inspection or by formula. For
convenience, we shall deal with the general stability and determinacy of
beam trusses and rigid frames in separate sections
1.1.
Stability and Determinacy of Beams.
A criterion may be established for the statical stability and determinacy
of beams.
Let :
(r) = no. of reactions
(c) = The total no. of equations of conditions.
(Where: c=1 for an internal hinge, c=2 for an internal roller and c=0 for
beams without internal connection)
(c + 3) = The total no. of the equilibrium equations.
Support Reactions
1)
Roller: One unknown element.
(2 degrees of freedom)
2) Link or strut: One unknown element.
The beam is set to be:
(2 degrees of freedom)

Unstable if (r < c+3 )

Determinate if stable if (r=c+3)
3) Hinge: Two unknown elements.
(1 degree of freedom)
4) Fixed: Three unknown elements.
 Indeterminate if stable if (r > c +3 )
The degree of indeterminacy (m) can be obtained by:
m = r – (c+3)
Further illustrations are given below

Unstable if( b + r < 2j )

Determinate if stable if (b+r = 2j )
 Indeterminate if stable if (b+r > 2j )
The degree of indeterminacy (m) can be obtained by:
m =( b+r) - (2j)
1.2. Stability and Determinacy of Trusses.
A truss is composed of a number of bars connected at their ends by a number of
pinned joints so as to form a network, usually a series of triangles and mounted
of a number of supports. Each bar of a truss is a two force member; hence each
represents one unknown elements for the entire system is counted by the
number of bars (internal) plus the number of independent reaction elements
(external). Thus, if we let b denote the number of bars and r the number of
reaction components, the total number of unknown elements of the entire
system is b+r. Now if the truss in in equilibrium, every isolated portion must
likewise be in equilibrium. For truss having j joints, the entire system may be
separated into j free bodies in which each joint yields two equilibrium equations,
∑ 𝐹𝑥 = 0 and ∑ 𝐹𝑦 = 0 for the concurrent force system acting on it.We may thus
establish criteria for the statical stability of and the determinacy of a truss by
counting the total unknowns and the total equations.
The truss is set to be:
Further illustrations are given below
1.3. Stability and Determinacy of Frames.
A rigid frame is built of beams and columns connected rigidly such as the one
shown in figure a. The stability and determinacy of a rigid frame may also be
investigated by comparing the number of unknowns( internal unknowns and
reactions unknowns) with the number of equations of statics available for
their solution. Like a truss, a rigid frame may be separated into a number of
free bodies of joints as shown in figure b, which requires that every member
of frame be taken apart. As discussed, there are usually three unknowns
magnitude (N.V, M) existing in a cut section of a member. However, if these
quantities are known at one section of a member, similar quantities for any
other section of the same member can be determined. Hence, there are only
three independent, internal unknown elements for each member in frame.
We may thus establish criteria for the statical stability of and the determinacy of
a truss by counting the total unknowns and the total equations.
Let :
(b) = no. of frame members
(r) = no. of reactions
(j) = no. of joints.
(c) = The total no. of equations of conditions.
(Where: c=1 for an internal hinge, c=2 for an internal roller and c=0 for
beams without internal connection)
(c = no. of members connected at joint – 1)
The frame is set to be:
 Unstable if (3b + r < 3j + c )
 Determinate if stable if (3b + r = 3j + c )
 Indeterminate if stable if (3b + r > 3j + c )
The degree of indeterminacy (m) can be obtained by:
m = (3b+r) – (3j+c)



Unstable if (U<E)
Determinate if stable if (U=E)
Indeterminate if stable if (U>E)
The degree of indeterminacy (m) can be obtained by:
m=U-E
1.4.
Stability and Determinacy of Composite Structures.
(E) = no. of equilibrium equations
(U) = no. of unknowns
The structure is set to be: