It is the number of independent components of joint displacements with respect to a set of axes. Kinematic indeterminacy also referred 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 components of the joints of a structure can not be determined by compatibility eqn. alone, it is called as kinematically indeterminate structure., i.e., number of unknown displacement components is greater than the number of equations.
Equations of Compatibility
The number of eqn. is equal to the number of unknowns, which is present due to the support conditions and other factors such as the in flexible of the members.
Independent displacement components
The value of independent 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. |
| Fixed support | 0 |
| 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
For rigid jointed plane surface is given by :
Dk = NJ – C
Where, N
- J is the number of degrees of freedom of each joint.
- J is the number of joints.
- C is the number of reaction (r) if the members are flexible.
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).
Do check this post to know about structure.
Conclusions
This blog describes indeterminacy as a situation in which the number of unknowns in a system exceeds the number of equations available to solve for them.
Key learnings of this post are as follows:
- Types: The different types of indeterminacy and how we calculate using the degrees of freedom are discussed.
- Importance: The importance of considering indeterminacy when designing mechanical systems, as it can impact the stability of the system.
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