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

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

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)

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’

Determination of Max stress in beam

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

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

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

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

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

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

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

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

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

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

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.