It is the number of independent components of joint displacements with respect to a specified set of axes. Kinematic indeterminacy also refered as degrees of freedom for a structure, is the number of unrestrained component of joint displacements. An elastic body is called a structure when a set of loads are applied the elastic body deforms and also the system is set up against the deformation.
Kinematic indeterminacy of a structure
If the displacement components of the joints of a skeletal structure can not determined by compatibility equations alone, it is called as kinematically indeterminate structure., i.e., number of unknown displacement components is greater than the number of compatibility equations.
Equations of Compatibility
The number of equations is equal to the number of constraints, which is present due to the support conditions and other factors such as the inextensibility of the members.
Independent displacement components
The value of independent displacement components depends on the type of joint. The list is given below in tabular form:
|Type of Joint||Degree of Freedom|
|Pin joint of a plane frame||2 Translations|
|Pin joint of a space frame||3 Translations|
|Rigid joint of a plane frame||1 rotation and 2 translations|
|Rigid joint of a space frame||3 rotations and 3 translations|
Based on the type of support also the number of degree of freedom varies, i.e
|Type of support||D.O.F.|
|Pinned supports||1 rotation|
|Roller support||2 (1 rotation and 1 translation in x direction.)|
|Free end||3 (1 rotation and 2 translations (x and y directions))|
|Vertical guided roller||1 translation in y direction|
|Horizontal guided roller||1 translation in x direction|
Formulation of Kinematic indeterminacy
For rigid jointed plane surface is given by :
Dk = NJ – C
- J is the number of degrees of freedom of each joint.
- J is the number of joints.
- C is the number of reaction components (r), if the members are extensible.
For example for a rigid joint plane surface considering axial strains of members, Dk = 3j-r., but when we neglect axial strains Dk = 3j – (m + r).
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