What is the difference between static and kinematic indeterminacy of structures?

Static indeterminacy and kinematic indeterminacy are terms used in structural engineering to describe different aspects of the stability and behavior of a structure under various loading conditions.

1. Static Indeterminacy:

   • Definition: Static indeterminacy refers to the condition where a structure has more unknown support reactions or internal forces than the number of available equilibrium equations.
   • Equations of Equilibrium: In static equilibrium, a structure must satisfy the equations of equilibrium (e.g., ΣF = 0 and ΣM = 0). The number of unknowns (reactions and internal forces) must be equal to or less than the number of equations for a structure to be statically determinate.
   • Consequences: If a structure is statically indeterminate, the application of equilibrium equations alone is insufficient to determine all the unknown forces and reactions. Additional compatibility equations or deformation considerations are needed for analysis.

2. Kinematic Indeterminacy:

   • Definition: Kinematic indeterminacy refers to the condition where a structure restricts not only the external loads but also the possible deformations or movements it can undergo.
   • Degrees of Freedom: In the context of kinematic indeterminacy, "degrees of freedom" refers to the number of independent ways a structure can deform or move. If a structure is kinematically indeterminate, it restricts more degrees of freedom than the external loads can induce.
   • Consequences: The additional constraints on deformations lead to internal forces and displacements that cannot be determined by considering external loads alone. Techniques like the flexibility method are used to address kinematic indeterminacy by considering deformations directly.

Relationship:

• A statically determinate structure can still be kinematically indeterminate and vice versa.
• In a statically determinate structure, the number of unknowns (reactions and internal forces) can be determined using equilibrium equations.
• In a kinematically determinate structure, the deformations and movements can be completely determined based on the external loads and support conditions.
• Statically indeterminate structures require additional methods beyond equilibrium equations, while kinematically indeterminate structures may need consideration of deformation compatibility.

Example:

• A cantilever beam with a fixed support at one end and a roller support at the other is statically determinate (as there are three reactions that can be determined using equilibrium equations) but kinematically indeterminate (as the roller support allows vertical movement, creating an additional degree of freedom that cannot be determined solely by considering external loads).

Understanding both static and kinematic indeterminacy is essential in the analysis and design of structures, as it influences the methods used for structural analysis and the determination of unknown forces and deformations.