Learning stand by GkG

Stiffness: -  It is difained as the moment required at a joint in a member to produced unit rotation of that joint. It is dinoted by " K "                 M = kθ Unit rotation.  θ = 1                M = K Case.1-- when far end is fixed          K = 4EI/L Case.2-- when far end is hinged      K = 3EI/L Case.3-- when far end is free         K = 0 Relative stiffness:-                 R.S  =  K/4E Case.1-- Far end is fixed .          K = 4EI/L       ,         R.S = I/L Case.2-- far end is hinged.      K = 3EI/L       ,        R.S = 3I/4L            D.F. = (R.S member)/(R.S joint) R.S = Relative stifness D.F = distribution ...

  Super position theorem:- According to this theorem the resultent stress function for multiple loading is equal to the sum of the effect of the individual loading. It is valid for beams and frames and both the determinate and indeterminate structure. Assumptions of Superposition theorem:- Hook's law valid Supports are unyielding 3.Temperature is constant   Example:- Superposition theorem exam ple           gkgautam6036.blogspot.com

  Minimum potential energy theorem:- The minimum potential energy theorem state that the partial derivative of the potential energy w.r.t redundant reaction is zero.                                       ∂U/∂R = 0  R =  Redundant reaction ( number of support reaction - equilibrium equation ) If a structure is loaded and there is any redundant reaction then the true value of that redundant reaction can be determined if the potential energy of the structure is minimum.     gkgautam6036.blogspot.com अगर आप किसी भी topic से related nots चाहते हैं तो कमेंट करे।

These topics cover in this page (strain energy से related सब टॉपिक के लिए भी इसे ही लीख सकते हैं )  Strain energy for Axial loading , Strain energy for Bending moment Strain energy for castiglious theorem What is strain energy:-                                                                      When a gradual load is applied on an elastic body , then the body is strained and work is done on the body which is stored in the form of internal energy. This internal work done by the body in resisting the straining is called strain energy.  Strain energy For Axial loading :-       Strain energy For Axial loading diagram                             U = P× ΔL/2 According to hook's law.            ...

  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 m...

  What is three moment theorem?       Three moment theorem express the relationship between the moment at three successive supports. The supports "moments" can be determined by the Three moment equation. It is mostly suitable for continuous beam.  According to this theorem the supports moments Ma, Mb, Mc are evaluated by using the equation.         Three moment theorem equation                 Where:- a1 = area of free Bending moment diagram of span AB a1 = area of free Bending moment diagram of span BC X1 = distance of centroid of Bending moment diagram (AB) from side A (left) X1 = distance of centroid of Bending moment diagram (BC) from side C (right) For example:- Numericals page 1 Numericals page 2 Numericals page 3 Numericals page 5        gkgautam6036.blogspot.com         

  Unit load method:- It is forced method. It is used for determinate and indeterminate structure.It is a modification of castiglious II theorem. In this method we apply a virtual load/moment of unit magnitude in the direction of deflection/rotation and at that particular point. Unit load method formula Where:- M - Bending moment due to the external applied load m1 - Bending moment due to the unit load applied at that point of deflection is asked m2 - Bending point due to unit moment applied at that point of rotation is asked             gkgautam6036.blogspot.com

  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.

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