Solid Mechanics

 👉Plane having principal stress has no tangential stress.

👉Mutually perpendicular planes are 180 degree apart in Mohr circle and have numerically equal value of shear stress.

👉Maximum shear stress = radius of Mohr's circle

👉         Theory of Failure                                               Also Known as  

       Maximum principal stress theory             Rankine theory, Lame's theory, Maximum stress theory

        Maximum principal strain theory                             St. Venant theory

           Maximum shear stress theory                   Tresca theory, Guest theory, Columb theory

          Maximum strain energy theory                              Beltrami Haigh theory

      Maximum shear strain energy theory                 Distortion energy theory, Von-Mises theory

👉 Bending Equation 

                                  M/I = σ/y  = E/R

where, M = Bending Moment

           σ  = Bending stress

           I =  Moment of inertia

          y = Distance of the fibre from Neutral axis

         R = Radius of curvature

         E = young's modulus of elasticity

👉 Torsion Equation:

                      T/Ip = τ/r = τmax/R  = N.θ/l

👉 Strain energy per unit volume = 1/2 x shear stress x shear strain

👉 Strain energy in torsion = 1/2 x T x θ

👉  For Brittle materials, Normal stress theory gives best result

👉 For ductile material maximum shear stress theory is applicable.

                                                                Happy Learning

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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