Structural Stability and Influence Lines Analysis

Structural Stability

Stability depends on the capacity of the support. Ensure that the system is a structure, not a mechanism. Structures become unstable due to inadequate external support or loading.

Determining Stability

To determine stability, imagine a rigid body in a 2D coordinate system where each point is able to translate and rotate. We calculate the degree of freedom (DOF) using the following formula:

DOF = 3Rb - R - 2H

Conditions for Structural Instability

• Parallel reactions
• Concurrent forces

Influence Lines (IL)

An Influence Line (IL) is a graphic representation of the relationship between a component force, restraint reaction, or displacement value and the position of a unit force. There are two primary methods for development: the equilibrium equation method and the Mueller-Breslau principle.

IL Development Methods

• Equilibrium Equations: This method is based on the equations of static equilibrium. The IL graph is plotted using equations for the requested internal forces.
• Mueller-Breslau Principle: The ordinate value of the IL is proportional to the ordinates of the deflection from the structure. This is obtained by removing the restraint corresponding to the function from the structure and introducing a force that causes a displacement in the positive direction.

Calculating Internal Forces via IL

1. Concentrated Load: The value of the force is calculated as the product of the concentrated load and the corresponding ordinate of the IL.
2. Uniform Distributed Loads: Calculated via the area under the IL curve.
3. Applied Moment: The value of the force is calculated as the product of the moment and the tangent of the angle between the IL and the axis.

Moment Envelope

The moment envelope is a graph showing the variation of the minimum and maximum values for a function along a member due to all possible loading conditions.

Methods of Truss Calculation

• Method of Joints: Only axial forces appear in each joint. We use static equilibrium. Calculations are only possible in joints where one or two unknown forces appear in intersected bars.
• Method of Sections: Based on passing an imaginary cutting plane through the truss to divide it into two sections. There are only three equations of equilibrium available.
• Henneberg's Method: Allows for the calculation of forces in truss structures where other methods are difficult to apply. This method depends on moving one bar within the structure from one position to another.