**Statically Indeterminate Structures**

Statically indeterminate structures are those which can not be solved by using the equations of static equilibrium alone. Additional compatibility equations are required to solve indeterminate structures e.g. fixed ended beam, continuous beam, propped cantilever etc.

It is very important to understand the behaviour of different types of supports. In general every hinged support will have two reaction components (along x and y axes) . Roller support will have one reaction in the direction perpendicular to the surface on which the rollers are kept. Fixed support will have three reactions ; two in x and y directions and the third is moment about z-axis at the support.

Let's consider a beam which has roller supports at A and B and
a hinged support at C. Its free body diagram is given below which
shows that the given beam has 4 support
reactions (C_{x }along x-axis and A_{y}, B_{y}, C_{y }
along y-axis).

Every member of a structure will provide three equations of static
equilibrium (ΣF_{x} = 0, ΣF_{y} = 0 and ΣM_{z}
= 0).

(i) If the number of unknown reaction components are equal to the
number of equations of equilibrium, the structure is known as
**statically determinate**.

(ii) If the unknowns are more than the number of equations of
equilibrium the structure is known as **statically
indeterminate**. The difference between unknown reaction
components and the equations of equilibrium is known as **degree of
indeterminacy**. Therefore the beam shown in the
above figure is statically
indeterminate of degree one as it has 4 unknown reactions and only 3
equations of static equilibrium equilibrium..

(iii) If the unknown reaction
components are less than the number of equilibrium equation, the
structure is known as **unstable**. A structure is also
defined as unstable if all the reaction components are concurrent or
parallel (even if reactions are equal or more than the number of
equations). This is evident from the beam shown in the above figure
that if we change the hinged support at C to a roller support, all
the reactions will be vertical and hence parallel to each other and
this beam will not be able to support any horizontal force applied
on it.

**Methods for Solving Indeterminate Structures**

Most of the structures being used in construction are indeterminate in nature. Force method of analysis or displacement method of analysis could be used to solve for indeterminate structures.

Some of the common methods to solve for indeterminate structures are;

Method of consistent deformation (compatibility method also known as flexibility method),

Moment distribution method, (Watch our **Video Lecture on Moment distribution method for solving indeterminate structures** This video is presenting different steps of moment distribution method with the help of an illustrated solved example.)

three-moment theorem

,Energy theorems

,Stiffness method etc.

You can also use our
**
Moment Distribution Calculator**
for solving indeterminate beams

Use our
**
Fixed Beam
Calculator **to
determine fixed-end moments for different loading cases.

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