How to calculate degree of static Indeterminacy and kinematic indeterminacy. Reference used : Structural analysis by R C Hibbler

1. TYPES OF THE STRUCTURE
MASS STRUCTURES
Mass structures are those which resist Load
by virtue of their weight.
Framed structures
A framed structures resist applied Load by
virtue of its geometry
Basic elements of framed structures
Rods, beams, columns, plate or slabs

2. Plane frame
A structure having all its member in one plane is called plane frame

3. Space frame
A structure having a member in three dimensions is called space frame

4. Structures
Structure is an assemblage of
number of components like slab,
beams, columns and foundations
which remain in equilibrium
Any structure is designed for the
stress resultants of bending moment,
shear force, deflection, torsional
stresses, and axial stresses.

5. If these moments, shears and stresses
are evaluated at various critical sections,
then based on these, the proportioning
can be done.
Evaluation of these stresses, moments
and forces and plotting them for that
structural component is known as
Structural analysis.
Determination of dimensions for these components of these
stresses and proportioning is known as design.

6. CLASSIFICATION OF STRUCTURES
Statically Determinate structure
• The structure for which the reactions at the supports & internal
forces in the members can be found out by the conditions of
static equilibrium, is called a statically determinate structure.
•Example of determinate structures are:
simply supported beams, cantilever beams, single and double
overhanging beams, three hinged arches , etc.
•There are three basic conditions of static equilibrium:
ΣH = 0
ΣV = 0
ΣM = 0

7. Statically indeterminate structure
• The structure for which the reactions at the supports & the internal forces in the
members can not be found out by the conditions of static equilibrium, is called statically
indeterminate structure.
• Examples of indeterminate structures are: fixed beams, continuous beams, fixed arches,
two hinged arches, portals, multistoried frames , etc.
• If equations of static equilibrium are not sufficient to determine all the unknown
reactions (vertical, horizontal & moment reactions) acting on the structure, it is called
externally indeterminate structure or externally redundant structure.
• If equations of static equilibrium are not sufficient to determine all the internal forces
and moments in the member of the structure, even though all the external forces acting
on the structure are known, is called internally indeterminate structure or internally
redundant structure.

8. For a coplanar structure there are at
most three equilibrium equations for
each part, so that if there is a total of n
parts and r force and moment reaction
components, we have
In particular, if a structure is statically
indeterminate, the additional equations
needed to solve for the unknown
reactions are obtained by relating the
applied loads and reactions to the
displacement or slope at different points
on the structure
These equations, which are referred to
as compatibility equations, must be
equal in number to the degree of
indeterminacy of the structure.
Compatibility equations involve the
geometric and physical properties of
the structure

11. Classify each of the beams shown in Figs. a through d as statically determinate or
statically indeterminate.
If statically indeterminate, report the number of degrees of indeterminacy. The beams
are subjected to external loadings that are assumed to be known and can act anywhere
on the beams.

13. Classify each of the pin-connected structures shown in Figs. a through d as statically determinate or statically
indeterminate.
If statically indeterminate, report the number of degrees of indeterminacy. The structures are subjected to arbitrary
external loadings that are assumed to be known and can act anywhere on the structures.

15. Classify each of the frames shown in Figs. 2–22a through 2–22c as statically determinate or statically
indeterminate. If statically indeterminate, report the number of degrees of indeterminacy. The frames are
subjected to external loadings that are assumed to be known and can act anywhere on the frames.

17. To ensure the equilibrium of a structure or its
members, it is not only necessary to satisfy the
equations of equilibrium, but the members must
also be properly held or constrained by their
supports regardless of how the structure is loaded.
Two situations may occur where the conditions for
proper constraint have not been met.
Stability.
a) Partial Constraints.
Instability can occur if a structure or one of its
members has fewer reactive forces than equations of
equilibrium that must be satisfied. The structure
then becomes only partially constrained.
Here the equation ΣFx = 0, will not be satisfied for the loading conditions,
and therefore the member will be unstable.

18. b) Improper Constraints.
In some cases there may be as many unknown
forces as there are equations of equilibrium;
however, instability or movement of a structure or
its members can develop because of improper
constraining by the supports. This can occur if all
the support reactions are concurrent at a point
From the free-body diagram of the beam it is seen
that the summation of moments about point O will
not be equal to zero (Pd 0); thus rotation about
point O will take place.

19. Another way in which improper constraining leads to
instability occurs when the reactive forces are all
parallel. An example of this case is shown in Fig.
Here when an inclined force P is applied, the
summation of forces in the horizontal direction will
not equal zero.
In general, then, a structure will be geometrically
unstable—that is, it will move slightly or collapse
•if there are fewer reactive forces than equations of
equilibrium; or
•if there are enough reactions, instability will occur if
the lines of action of the reactive forces intersect at a
common point or are parallel to one another.

20. If the structure is unstable, it does not matter if it is
statically determinate or indeterminate. In all cases
such types of structures must be avoided in practice.

21. Classify each of the structures in Figs.a through d as stable or unstable. The structures are subjected
to arbitrary external loads that are assumed to be known.

23. kinematic indeterminacy
Degrees of Freedom.
When a structure is loaded, specified points on it, called nodes, will
undergo unknown displacements.
These displacements are referred to as the degrees of freedom for the
structure, and in the displacement method of analysis it is important to
specify these degrees of freedom since they become the unknowns
when the method is applied.
The number of these unknowns is referred to as the degree in which
the structure is kinematically indeterminate.

24. To determine the kinematic indeterminacy we can imagine
the
structure to consist of a series of members connected to
nodes, which are usually located at joints, supports, at the
ends of a member, or where the members have a sudden
change in cross section.
In three dimensions, each node on a frame or beam
can have at most three linear displacements and
three rotational displacements; and
in two dimensions, each node can have at most two
linear displacements and one rotational
displacement.
For example any load P applied to the beam will cause node A only to rotate (neglecting
axial deformation), while node B is completely restricted from moving. Hence the beam
has only one unknown degree of freedom, θA, and is therefore kinematically
indeterminate to the first degree.

25. The beam in Fig.b has nodes at A, B, and C, and so
has four degrees of freedom, designated by the
rotational displacements θA, θB, θC, and the vertical
displacement ∆C.
It is kinematically indeterminate to the fourth
degree.
Consider now the frame in Fig. c.
Again, if we neglect axial deformation of the
members, an arbitrary loading P applied to the
frame can cause nodes B and C to rotate, and these
nodes can be displaced horizontally by an equal
amount.
The frame therefore has three degrees of freedom,
θB, θC, ∆ B, and thus it is kinematically indeterminate
to the third degree.