# Statically Determinate and Indeterminate Trusses

**Trusses** are commonly used in construction to support roofs, bridges, and other structures. They consist of a framework of interconnected bars or beams that are designed to resist external loads. Trusses can be classified into two types based on Determinacy: statically determinate and indeterminate. This article will explore the differences between these two types of trusses and their basic principles.

🔗Types of Trusses, Their Characteristics and Applications

🔗Perfect, Imperfect, Deficient and Redundant Truss

## What are Statically Determinate Trusses?

In the context of engineering mechanics, the term "statically determinate" has a specific meaning. A truss is said to be statically determinate if all of the forces acting on it can be calculated using the equations of static equilibrium alone. This means that the support reaction forces and internal forces in the members of the truss can be determined without the need for additional equations. Thus, static determinacy simplifies the process of analyzing trusses and makes them easier to design and build.

When it comes to truss analysis and design, one of the most critical aspects is determining the force in each of its members. The primary objective is to ensure that each member can withstand the applied loads without failure. For planar trusses to be considered statically determinate, it is important to note that the sum of the number of members and the number of support reactions cannot exceed twice the number of joints present in the structure.

Newton's Laws are relevant not just to the structure as a whole, but also to every joint or node in the truss. For any node subjected to external forces or loads to remain stationary, certain criteria must be met. The sum of all forces (both horizontal and vertical) acting on the node and the sum of all moments acting about the node must be equal to zero. Analyzing these conditions at each node makes it possible to determine the magnitude of both compression and tension forces.

In order to effectively solve problems related to Statically Determinate Trusses, several assumptions must be considered. For example, it is generally assumed that members will only be subjected to axial forces, meaning that shear force and bending moment can be ignored. Self-weight is another factor that is typically disregarded in such scenarios. Additionally, it is often assumed that all members are linear and that all joints are smooth and frictionless. Finally, all loads and reactions are assumed to act directly or indirectly at joints only.

## What is Statically Indeterminate Truss?

When the reaction forces and internal forces in the truss members cannot be determined through equilibrium equations, the structure is referred to as a statically indeterminate structure. Newton's Laws alone are insufficient to determine the member forces in such structures. In statically indeterminate structures, the number of unknown forces is greater than the number of equilibrium equations available to solve them.

In these trusses, the internal forces cannot be calculated using the equations of statics alone. Instead, they require additional equations, such as compatibility equations or force-displacement equations, to determine the internal forces.

Two-dimensional structures can be analyzed using three equilibrium equations. Two methods can be used when dealing with statically indeterminate structures: the Force method and the Displacement method. Redundant forces are considered unknowns in the Force method, while the Displacement method considers displacements as unknowns.

## Degree of Static Indeterminacy

The degree of static indeterminacy of a truss can be calculated using the formula

where m represents the total number of members, r_{e} refers to the total external reactions, and j is the total number of joints.

D_{S} = m+r_{e} – 2j

Where m represents the total number of members, r_{e} refers to the total external reactions and j is the total number of joints.

A truss is considered determinate if the degree of static indeterminacy, as determined by the equation, is equal to zero, D_{S} = 0

If D_{S} is greater than zero, the truss structure is considered indeterminate. D_{S} > 0

Because a statically determinate truss cannot have more members than what is required for stability, it is not a fail-safe structure. Therefore, if one member fails, the entire truss can collapse. To ensure public safety, adding redundant members to the truss structure is often necessary.