Short Answer:

Static determinacy is a condition in which all the reactions and internal forces in a structure or mechanical system can be determined only by using the equations of static equilibrium. In such a system, the number of unknown forces equals the number of equilibrium equations available.

In simple words, a structure or body is said to be statically determinate when it can be completely analyzed using the basic equilibrium equations without needing any additional relations, such as deformation or material properties. Examples include a simply supported beam or a three-hinged arch.

Detailed Explanation :

Static Determinacy

Static determinacy is an important concept in engineering mechanics and structural analysis. It helps engineers identify whether the forces and reactions in a structure can be found using only the conditions of equilibrium, or if additional information (like deflection or deformation) is required.

In a statically determinate structure, the reactions and internal forces can be determined solely by applying the equations of static equilibrium. This means that the structure’s stability and load distribution can be completely analyzed using simple equilibrium laws.

In contrast, when the number of unknown reactions exceeds the available equilibrium equations, the structure becomes statically indeterminate, and additional compatibility equations are needed for analysis.

Definition

Static determinacy can be defined as:

“A structure or body is said to be statically determinate if all the unknown reactions and internal forces can be determined completely by applying the equations of static equilibrium only.”

If additional equations or conditions are needed (such as deflection or strain compatibility), the structure is statically indeterminate.

Conditions of Static Determinacy

The basic conditions of static equilibrium for a rigid body are:

• In two-dimensional (2D) systems:

These are three independent equations.

• In three-dimensional (3D) systems:

These are six independent equations.

If the number of unknown reactions equals the number of available equilibrium equations, the structure is statically determinate.

If there are fewer unknowns, the structure is unstable, and if there are more unknowns, the structure is statically indeterminate.

Mathematical Condition for Static Determinacy

1. For 2D Structures:
   Let,
   = number of unknown reaction components,
   = number of equilibrium equations (which is 3 for 2D).

Then the condition for static determinacy is:

1. • If , the structure is unstable.
   • If , the structure is statically determinate.
   • If , the structure is statically indeterminate.
2. For 3D Structures:
   The condition becomes:

because there are six independent equilibrium equations in space.

Examples of Statically Determinate Structures

1. Simply Supported Beam:
   • Has one pin support and one roller support.
   • Pin provides two reactions (horizontal and vertical).
   • Roller provides one reaction (vertical).
   • Total reactions = 3, which equals the number of equilibrium equations.
   • Hence, the beam is statically determinate.
2. Three-Hinged Arch:
   • Contains three hinges and can be analyzed using three equilibrium equations.
   • Internal and external reactions can be determined by static laws.
3. Truss with Proper Member Arrangement:
   • For a perfect truss: , where
     = number of members,
     = number of reactions,
     = number of joints.
   • When this equation is satisfied, the truss is statically determinate.

Physical Meaning

In a statically determinate structure:

• The reactions depend only on geometry and loading, not on material properties or deformation.
• The structure is stiff and stable.
• It can be easily analyzed using basic equations of equilibrium.
• Any removal or failure of a support or member causes the structure to become unstable.

This makes statically determinate systems simple to design and calculate but less flexible in accommodating deformation or redundancy.

Importance of Static Determinacy

1. Simplified Analysis:
   The reactions and internal forces can be determined using only equilibrium equations without complex mathematical methods.
2. Economical Design:
   Since no redundant supports exist, material and manufacturing costs can be minimized.
3. Predictable Behavior:
   The structure’s response to loads is straightforward and easy to understand.
4. Educational Value:
   Statically determinate problems are used as the foundation for learning structural mechanics.

Limitations of Static Determinacy

1. Lack of Redundancy:
   If any support or member fails, the entire structure collapses because there are no alternative load paths.
2. Limited Practical Use:
   Most real-world structures are statically indeterminate for better strength and load distribution.
3. Sensitivity to Support Settlements:
   Small deformations or settlements can cause large changes in internal forces since the structure cannot adjust to them.
4. Vibration Issues:
   Since it is not redundant, vibration absorption and damping capacity are limited.

Comparison with Statically Indeterminate Structures

Aspect	Statically Determinate	Statically Indeterminate
Unknowns	Equal to equilibrium equations	More than equilibrium equations
Analysis Method	Using equilibrium equations only	Requires compatibility and deformation analysis
Redundancy	No redundancy	Redundant supports or members present
Stability	Just stable	More stable due to redundancy
Examples	Simply supported beam, perfect truss	Fixed beam, continuous beam

Practical Applications

1. Bridges: Simply supported beams are often designed as statically determinate for easy analysis.
2. Frames and Machines: Early mechanical designs often use determinate supports for predictable load distribution.
3. Cranes and Lifting Equipment: Use determinate designs for stability and simplicity.
4. Trusses: Perfect trusses in roofs and towers are often statically determinate for efficient load transfer.
5. Foundations: Properly designed foundations rely on determinate support conditions to prevent uneven loading.

Conclusion

Static determinacy refers to a condition where all unknown reactions and internal forces can be determined using only the equilibrium equations. It provides a simple yet powerful method for analyzing structures and mechanical systems. When the number of unknowns equals the number of equilibrium equations, the system is statically determinate. These systems are stable, predictable, and easy to analyze but lack redundancy and flexibility. Understanding static determinacy is fundamental in mechanical and structural engineering design for ensuring both efficiency and stability.