Target GATE & ESE 2022 (Strength of Material)

 👉Elongation in prismatic bar 

                        △ = ү.L^2/2E

                        △ = WL/2AE

Here, △ = elongation in prismatic bar

           L = length of bar

           W = self weight of the bar = ρ.A.L

          E = young's modulus of elasticity

          ү = Unit weight of material

👉 Elongation in conical bar

               ∆ = ү.L^2/6E

👉 Elongation in conical bar due to self weight is one third of elongation in prismatic bar due to self weight. 

👉 Important Relationship

    E = 2N (1+μ)

   E = 3K(1-2μ)

   μ = (3K-2N)/(6K+2N)

   E = 9KN/3K+N

where,

        N = Modulus of rigidity/Shear Modulus

        K = bulk modulus

        E = Modulus of elasticity/ Elastic modulus

        µ = Poisson's ratio ( 0 to 0.50)

👉   Modulus of Resilience is energy stored upto elastic limit per unit volume.

👉       △Sudden = 2.△static

👉 strain is the fundamental behaviour but stress is a derived concept because strain can measured with some instrument  and is a fundamental quantity however stress can only be derived , it can not be measured.

👉 If loading of nth degree, then shear force diagram is of (n+1) degree and bending moment diagram  is of (n+2) degree

👉For bending moment M to be maximum

     dM/dx =0 and we know dM/dx  = Shear force (V)

    Bending moment is maximum at the section where shear force is zero or changes sign.

👉        dM/dx = V       (Slope of Bending moment = shear force)

             dV/dx  = w        (Slope of shear force = loading intensity)

             M = ∫ V dx

             V  = ∫ w dx

             d2M/dx2 = dV/dx  = w

   M = Bending moment

   V = shear force

   w = loading intensity

 👉Point of contra-flexure is the point where bending moment changes it's sign

👉        Real Beam             Conjugate Beam

        Hinged Support              Hinged Support

        Free Support                  Fixed support

        Fixed support                  Free support

        Internal roller                  Internal hinge

        Internal hinge                  Internal roller

   

                                                Happy Learning

                             

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