Degree of Redundancy | Static and Kinematic Indetermiancy

غير معرف أغسطس 15, 2017
In order to solve an indeterminate structure, it is necessary to simplify the structure to a simple system of simply supported beam or to a respective cantilever action system. This is done by cutting sufficient supports and the structural members. Thus the number of redundants is equal to the number of forces and the moments that is required in order to restore the continuity.



Types of Indeterminacy


A structural system can be described by employing two types of indeterminacies. They are:

  1. Static Indeterminacy
  2. Kinematic Indeterminacy

Static Indeterminacy


The static indeterminacy refers to the number of forces that have to be released to transform a structure into a stable and statically determinate system.

A statically indeterminate structure cannot be solved by the set of equilibrium equation alone. There comes the need of additional set of equations that are called as condition of compatibility.Below table shows the equation to determine the static indeterminacy of different structural system. The respective designations used are:

ns =Degree of Indeterminacy
m= Number of Members
j = number of joints including the support
r = number of reactions
t = number of released like hinges = mr - 1 ; Where mr = Number of elements meeting at that hinge

Table -1

System
Type Of Structure
Degree Of Static Indeterminacy
ns

2D Truss

Pin-Jointed Plane Structure

m + r – 2j
3D Truss
Pin-Jointed Space Structure
m +r – 3j
2D Beam & Frames
Rigid-Jointed Plane Structure
3(m – j ) + r – t
3D Beams & Frames
Rigid-Jointed Space Structure
6(m – j) + r – t

Kinetic Indeteminacy


This type of indeterminacy refers to the number of independent components of joint displacements with respect to the specified set of axes. These displacements formed at the nodal points will completely describe the response of the structure to any kind of loading condition.


If all the joint displacements are restrained, the structure can be made kinematically determinate.

The degree of kinematic indeterminacy can be defined as the number of unrestrained components of the joint displacements. This value is considered as mentioned below:

  1.  Any joint in space will have 6 degrees of freedom which are 6 independent components of displacements i.e. three rotations and three translations.
  2. Any joint in plane frame will have 3 degrees of freedom i.e. two translations and one rotation.
The degree of Kinematic Indeterminacy can be given as,
Ik = NJ - C

Here,
 N = number of Degrees of Freedom
 J = Number of Joints
C= Number of Restraints Against displacement giving rise to reaction components
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