What is castiglious I and II theorem and it's applications 2021

Castiglious I theorem:-

If the material is linearly elastic,supports are unyielding and temperature is constant,Then the first partial derivative of total strain energy with respect to the diflection/slope will give the load/moment at that point.

    

Derivative of total strain energy w.r.t diflection

Derivative of total strain energy w.r.t slope

Where:-

U= strain energy

P= load

M= moment

∆= diflection

 θ   = slope

Assumptions:-

1. Material is linearly elastic

2.supports are unyielding  

3.temperature is constant 

Castiglious II theorem:-

The The first partial derivative of the total strain energy w.r.t. the load/ moment will give the diflection/ slope  first partial derivative of the total strain energy w.r.t. the load/ moment will give the diflection/ slope 

Derivative of total strain energy w.r.t load

Derivative of total strain energy w.r.t moment