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

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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
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Kaolinite, illite, montmorilonite, How the clay minerals kaolinite, illite and montmorillonite are formed by schematic diagram.

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 Kaolinite:- Kaolinite is the most common minerals of the kaolinite group of clay minerals. Its basic structural unit consists of gibbsite  sheet with aluminium atoms at the center. It is joined to silica sheet through the unbalanced oxygen atoms at the apexes of silica sheet. The total thickness of the basic structural unit is about 7 angstrom. The basic structural unit of kaolinite minerals is symbolised as shown in figure. Kaolinite minerals The basic structural units of kaolinite minerals are joined together by hydrogen bond. It develops between the oxygen atoms of silica sheet and the hydroxyls of gibbsite sheet. Since the hydrogen bond is fairly strong, it is extremely difficult to separate the layers, and hence, kaolinite minerals is relatively stable. Moreover, water cannot penetrate through the layers of the structural units of kaolinite minerals. Kaolinite shows relatively little swell on wetting. China clay is almost pure Kaolinite. Montmorillonite Montmorillonite is the mos
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Bettie's law (Rayleigh theorem),What is Bettie's law? 2021

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  Bettie's law (Rayleigh      theorem):- When material is linearly elastic, supports are unyielding and temperature is constant Then the virtual work done by P-system of forces in going through displacement caused by Q-system of forces is equal to the virtual work done by Q-system of forces in going through displacement caused by P-system of forces. Limitations :- 1.material is linearly elastic 2.temperature is constant 3.supports are unyielding Example:- Special cases for Bettie's law:- Case -1 :- If theta-1 be the rotational displacement in the direction of M1 caused by  M2 and theta-2 be the rotational displacement in the direction of M1 caused by M1.
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