Axial Loads

Saint Venant's Principle:

It states that, "A system of forces applied to some segment of a solid body in equilibrium, produce stresses in the body that rapidly diminishes with increasing distance from the segment".

Elastic Deformation of an Axially Loaded Member:

To evaluate formula for deformation, consider diagram.

Statically Indeterminate System:

Many problems in which internal forces cannot be determined from statics alone. Problems of this type are called Statically Indeterminate.

Steps for solution are discussed ahead:

1. Draw free body diagram (F.B.D.).
2. Equations of equilibrium of forces and moments.
3. Equations describing geometry of deformations or Compatibility of Displacements.
4. Relationships between Load-deformation or Stress-Strain for the material.

Principle of Superposition:

If system is statically indeterminate if it is held by more supports than are required to maintain its equilibrium (which creates more number of unknowns). We designate one of reactions as a Redundant to eliminate corresponding support.

Principle of Superposition states that: "Stress or displacements in complex loading are obtained by subdividing into components and later algebraically summing the stress or displacements caused by each load".

Conditions must be satisfied:

• Loading must be linearly related to stress or displacement that is to be determined.
• Loading must not significantly change the original geometry or configuration of member.

There are two types of analysis which are discussed ahead:

1. First Method of Analysis:

There are three steps for the First method of analysis for Indeterminate problem. Steps are given below:

1. Equilibrium equation.
2. Compatibility equation.
3. Stress-Strain relation or Deformation equation.

2. Flexibility or Force Method Analysis:

There are three steps for the Force method analysis for Indeterminate problem. Steps are given below:

1. Compatibility equation.
2. Stress-Strain relation or Deformation equation.
3. Equilibrium equation.

To solve statically indeterminate systems, consider following diagram:

Thermal Effects:

External loads are not the only source of stresses and strains in a structure. Thermal effects are commonly found in both mechanical and structural analysis (rise of stress and strain) that are open to atmosphere.

• Changes in temperature, produce expansion or contraction of structural materials, resulting in thermal strain and stresses.
• Thermal effects are more important in design of statically indeterminate structures than statically determinate ones. 

References:

• Material from Class Lectures + Book named Engineering Mechanics of Materials by R.C. Hibbeler (10th Edition) + my knowledge. 
• Pics and GIF from Google Images.  
• Videos from YouTube.