1. CHAPTER #3 SHEAR FORCE & BENDING MOMENT
Introduction
Types of beam and load
Shear force and bending Moment
Relation between Shear force and Bending Moment

2. INTRODUCTION Devoted to the analysis and the design of beams
Beams – usually long, straight prismatic members
In most cases – load are perpendicular to the axis of the beam
Transverse loading causes only bending (M) and shear (V) in beam

3. Types of Load and Beam The transverse loading of beam may consist of
Concentrated loads, P1, P2, unit (N)
Distributed loads, w, unit (N/m)

4. Types of Load and Beam
Beams are classified to the way they are supported
Several types of beams are shown below
L shown in various parts in figure is called ‘span’

5. Determination of Max stress in beam

6. SHEAR & BENDING MOMENT DIAGRAMS
Shear Force (SF) diagram – The Shear Force (V) plotted against distance x Measured from end of the beam
Bending moment (BM) diagram – Bending moment (BM) plotted against distance x Measured from end of the beam

7. DETERMINATIONS OF SF & & BM
The Shear & bending moment diagram will be obtained by determining the values of V and M at selected points of the beam

8. DETERMINATIONS OF SF & & BM
The Shear V & bending moment M at a given point of a beam are said to be positive when the internal forces and couples acting on each portion of the beam are directed as shown in figure below
The shear at any given point of a beam is positive when the external forces (loads and reactions) acting on the beam tend to shear off the beam at that point as indicated in figure below

9. DETERMINATIONS OF SF & & BM
The bending moment at any given point of a beam is positive when the external forces (loads and reactions) acting on the beam tend to bend the beam at that point as indicated in figure below

18. Relation between Shear force and Bending Moment
When a beam carries more than 2 or 3 concentrated load or when its carries distributed loads, the earlier methods is quite cumbersome
The constructions of SFD and BMD is much easier if certain relations existing among LOAD, SHEAR & BENDING MOMENT
There are 2 relations here:-
Relations between load and Shear
Relations between Shear and Bending Moment

19. Relations between load and Shear
Let us consider a simply supported beam AB carrying distributed load w per unit length in figure below
Let C and C’ be two points of the beam at a distance Δx from each other
The shear and bending moment at C will be denoted as V and M respectively; and will be assumed positive, and
The shear and bending moment at C’ will be denoted as V+ ΔV and M + ΔM respectively

20. Relations between load and Shear (cont.)
Writing the sum of the vertical components of the forces acting on the F.B. CC’ is zero
Dividing both members of the equation by Δx then letting the Δx approach zero, we obtain

21. Relations between load and Shear (cont.)
The previous equation indicates that, for a beam loaded as figure, the slope dV/dx of the shear curve is negative; the numerical value of the slope at any point is equal to the load per unit length at that point
Integrating the equation between point C and D, we write

22. Relations between Shear and Bending moment
Writing the sum of the moment about C’ is zero, we have
Dividing both members of the eq. by Δx and then letting Δx approach zero we obtain

23. Relations between Shear and Bending moment (cont.)
The equation indicates that, the slope dM/dx of the bending moment curve is equal to the value of the shear
This is true at any point where a shear has a well-defined value i.e. at any point where no concentrated load is applied.
It also show that V = 0 at points where M is Maximum
This property facilitates the determination of the points where the beam is likely to fail under bending
Integrate eq. between point C and D, we write

24. Relations between Shear and Bending moment (cont.)
The area under the shear curve should be considered positive where the shear is positive and vice versa
The equation is valid even when concentrated loads are applied between C and D, as long as the shear curve has been correctly drawn.
The eq. cease to be valid, however if a couple is applied at a point between C and D.