What is slope diflection method, slope diflection method equation 2021

 What is slope diflection method?

Here the joint displacement ( θ , Δ ) are taken as unknown and the joints moments are derived by the force displacement rotation which is called as slope diflection method.

Note:- 

In this method axial force and shear force effects are neglected and only Bending moment effects considered.

Sign convention:-

(I) fixed end moment:- clockwise (+) , anticlockwise (-)

(ii) rotation:- same above

(iii) settlement:- 
clockwise positive when rotation is clockwise

Anticlockwise negative when rotation is anticlockwise

Slope diflection equation:-

Considering a continuous beam where A , B are the intermediate supports. The final end moments developed at A and B (Mab, Mba) will be due to the effects of

(i) effects of loading  (M̅ab)
(ii) rotation of joint A  (θa ).   (M̅ab1)
(iii) rotation of joint B (θb)  (M̅ab2)
(iv) effects of settlement of supports   (M̅ab3)

By the superposition theorem,the final end moments will be the summetion off all the effects combined.

Mab = M̅ab + M̅ab1 + M̅ab2 + M̅ab3

Mab = M̅ab + 4EIθa/L  + 2EIθb/L  + 
 (−6EIΔ/L²)

Mab = M̅ab + 2EI(2θa + θb − 3Δ/L)

Mba = M̅ba + 2EI/L(2θb + θa − 3Δ/L)


It is called slope diflection equation


  1. Bettie's law के बारे मैं जानने के लिए:- click here
  2. Three moment theorem के बारे मैं जानने के लिए:- click here
  3. Unit load method के बारे मैं जानने के लिए:- click here

Comments

Post a Comment

Popular posts from this blog

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

Image
 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 i...
Read more

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

Image
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
Read more

Bettie's law (Rayleigh theorem),What is Bettie's law? 2021

Image
  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.
Read more