Objective: To learn how one can draw shear

Objective: To learn how one can draw shear face diagram (SFD) and bending moment diagram (BMD) using graphical method or summation method. 1 Dr. Ali H. Al-Gadhib w(x)x w(x)

k(x) (a) M V O M+M N

A V+V x Free-body diagram of segment x. (b) Cross-Sectional area of segment The idea is to find relationship between load w(x) and shear V(x) and to find relationship between shear V(x)

and moment M(x) using equilibrium equations. Dr. Ali H. Al-Gadhib 2 Load Shear/Moment Relationship w(x) w(x)x k(x) M

V +Fy = 0: M+M O V+V N

A x Free-body diagram of segment x. Cross-Sectional area of segment V w(x) x (V + V) = 0 V = + w(x) x dV

w x dx Slop of shear diagram at each point = - Distributed load intensity at each point +MO = 0: - V x M + w(x) x [k(x)] + (M + M) = 0 M = V x + w(x) k(x)2 dM V dx

Slop of moment diagram at each point = Shear at each point Dr. Ali H. Al-Gadhib 3 4 Dr. Ali H. Al-Gadhib Summary dV w ( x ) dx

dM V( x ) dx VB = VA + area of load between A & B. MB = MA + area of shear between A & B. A VA

B Area of load between A & B VA VB Area of shear between A & B MA MA

MB If load +ve, then shape of the shear is positive. 5 If shear is ve, then shape of the shear is negative. Dr. Ali H. Al-Gadhib Draw the shear and moment diagrams for the beam. 6

Dr. Ali H. Al-Gadhib 7 Dr. Ali H. Al-Gadhib Draw the shear and moment diagrams for the beam. 8 Dr. Ali H. Al-Gadhib 9 Dr. Ali H. Al-Gadhib

Draw the shear and moment diagrams for the beam. 10 Dr. Ali H. Al-Gadhib 11 Dr. Ali H. Al-Gadhib Draw the shear and moment diagrams for the overhanging beam.

12 Dr. Ali H. Al-Gadhib 13 Dr. Ali H. Al-Gadhib 14 Dr. Ali H. Al-Gadhib

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