Short Answer:

A statically indeterminate bar is a bar or structural member in which the number of unknown reactions or internal forces is more than the number of available independent equilibrium equations. In such cases, static equilibrium equations alone are not enough to find all the unknowns. Additional equations from material deformation and compatibility conditions are required.

In simple words, a statically indeterminate bar cannot be solved only by using static laws of equilibrium. The deformation or elongation of the bar must also be considered to find the forces, stresses, or reactions in the member.

Detailed Explanation :

Statically Indeterminate Bar

A statically indeterminate bar is a structural member that has more supports or constraints than necessary for maintaining equilibrium. This condition makes it impossible to determine all the internal forces and reactions using only the three static equilibrium equations—∑Fx = 0, ∑Fy = 0, and ∑M = 0. Therefore, additional relations are required, which come from the material’s deformation (elastic behavior) and geometric compatibility.

In simple terms, if a bar has more constraints than needed, it is over-restrained, and such a bar is known as a statically indeterminate bar. The analysis of this type of bar involves both statics and elasticity.

Explanation with Example

Let us consider a uniform bar that is fixed at both ends and subjected to a load at some point along its length. In this case, both ends of the bar provide reactions. However, only one reaction can be found directly using the equation of equilibrium because the bar has two unknown reactions (at two supports) but only one equation from static equilibrium. Therefore, the bar is said to be statically indeterminate to degree one.

This means one additional condition, such as the compatibility of deformation, is needed to solve for the remaining unknown. The compatibility condition ensures that the total deformation of the bar matches the boundary conditions (no relative movement at fixed ends).

Degree of Indeterminacy

The degree of indeterminacy refers to the number of additional equations required beyond the static equilibrium equations to solve the problem completely.
For a bar:

• Statically determinate bar: The number of unknowns equals the number of equilibrium equations.
• Statically indeterminate bar: The number of unknowns exceeds the number of equilibrium equations.

If there is one extra unknown, the system is said to be indeterminate to degree one; if two extra unknowns, then indeterminate to degree two, and so on.

Conditions for Equilibrium and Compatibility

To solve a statically indeterminate bar, two types of conditions are used:

1. Equilibrium Conditions:
   These are the basic static equations like

which ensure that the body is in a stable and balanced state.

1. Compatibility Conditions:
   These conditions state that the deformation of connected parts must be consistent. For example, if a bar is fixed at both ends, the total elongation between the fixed supports must be zero.

By combining both equilibrium and compatibility conditions, we can find all the unknown reactions and internal stresses.

Formula and Analysis

For a statically indeterminate bar, the stress and strain relationships are found using Hooke’s Law and the deformation formula for an axial member:

where:

•  = change in length,
•  = axial load,
•  = length of the bar,
•  = cross-sectional area,
•  = modulus of elasticity.

In an indeterminate bar, several sections may experience different internal forces, and the total deformation must satisfy the compatibility condition. The combined use of these relationships allows us to solve for the unknown reactions.

Practical Example

Imagine a steel bar rigidly fixed at both ends and subjected to a temperature rise. Since both ends are fixed, the bar cannot expand freely, resulting in thermal stresses. This is another case of a statically indeterminate condition because the stresses developed depend not only on static equilibrium but also on thermal expansion and constraint conditions.

Similarly, in compound or composite bars made of different materials joined together, the stresses depend on how each material deforms under the same load. These also fall under statically indeterminate systems.

Advantages and Disadvantages

Advantages:

• Provides greater rigidity and strength.
• Distributes loads more evenly among supports.
• Offers better safety in case one support fails.

Disadvantages:

• Analysis is more complex.
• Temperature changes or settlement of supports can induce unwanted stresses.
• Difficult to construct and maintain due to over-restraint.

Conclusion

A statically indeterminate bar is one that cannot be solved using only static equilibrium equations because of extra supports or constraints. It requires additional conditions from material deformation and compatibility to determine all internal forces and reactions. These bars provide higher rigidity and strength but are more complex to analyze. Understanding statically indeterminate systems is important for designing safe and efficient mechanical and structural components.