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 diflection


Derivative of total strain energy w.r.t slope
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


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