CHAPTER #3CHAPTER #3SHEAR FORCE & BENDING SHEAR FORCE & BENDING

MOMENTMOMENT Introduction Introduction Types of beam and loadTypes of beam and load Shear force and bending MomentShear force and bending Moment Relation between Shear force and Bending Relation between Shear force and Bending

MomentMoment

INTRODUCTIONINTRODUCTION

Devoted to the analysis and the design of beams Devoted to the analysis and the design of beams Beams – usually long, straight prismatic membersBeams – usually long, straight prismatic members In most cases – load are perpendicular to the axis of the In most cases – load are perpendicular to the axis of the

beam beam Transverse loading causes only bending (M) and shear Transverse loading causes only bending (M) and shear

(V) in beam(V) in beam

Types of Load and BeamTypes of Load and Beam

The transverse loading of beam may consist ofThe transverse loading of beam may consist of Concentrated loads, P1, P2, unit (N)Concentrated loads, P1, P2, unit (N) Distributed loads, w, unit (N/m) Distributed loads, w, unit (N/m)

Types of Load and BeamTypes of Load and Beam

Beams are classified to the way they are supportedBeams are classified to the way they are supported Several types of beams are shown belowSeveral types of beams are shown below L shown in various parts in figure is called ‘span’L shown in various parts in figure is called ‘span’

Determination of Max stress in Determination of Max stress in beambeam

I

cMm

3

2

12

16

1

bhI

bhS

S

Mm

SHEAR & BENDING MOMENT DIAGRAMSSHEAR & BENDING MOMENT DIAGRAMS

Shear Force (SF) diagramShear Force (SF) diagram – The – The Shear Force (V) plotted against Shear Force (V) plotted against distance x Measured from end of distance x Measured from end of the beamthe beam

Bending moment (BM) diagram Bending moment (BM) diagram – – Bending moment (BM) plotted Bending moment (BM) plotted against distance x Measured against distance x Measured from end of the beamfrom end of the beam

DETERMINATIONS OF SF & & BMDETERMINATIONS OF SF & & BM

The Shear & bending moment The Shear & bending moment diagram will be obtained by diagram will be obtained by determining the values of V determining the values of V and M at selected points of and M at selected points of the beamthe beam

DETERMINATIONS OF SF & & BMDETERMINATIONS OF SF & & BM

The Shear V & bending moment M at a given point of a beam are said 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 to be positive when the internal forces and couples acting on each portion of the beam are directed as shown in figure belowportion of the beam are directed as shown in figure below

The shear at any given point of a beam is positive when the external 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 forces (loads and reactions) acting on the beam tend to shear off the beam at that point as indicated in figure belowbeam at that point as indicated in figure below

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

Relation between Shear force and Relation between Shear force and Bending MomentBending Moment

When a beam carries more than 2 or 3 concentrated load or when its carries distributed loads, the earlier methods is quite cumbersomeWhen 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 MOMENTThe constructions of SFD and BMD is much easier if certain relations existing among LOAD, SHEAR & BENDING MOMENT

There are 2 relations here:-There are 2 relations here:- Relations between load and ShearRelations between load and Shear

Relations between Shear and Bending MomentRelations between Shear and Bending Moment

Relations between load and ShearRelations between load and Shear

Let us consider a simply supported beam AB carrying distributed load w per unit length in figure belowLet 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 Let C and C’ be two points of the beam at a distance ΔΔx from each otherx from each other

The shear and bending moment at C will be denoted as V and M respectively; and will be assumed positive, andThe 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+ The shear and bending moment at C’ will be denoted as V+ ΔΔV and M + V and M + ΔΔM respectivelyM respectively

Relations between load and Shear (cont.)Relations between load and Shear (cont.) Writing the sum of the vertical components Writing the sum of the vertical components

of the forces acting on the F.B. CC’ is zero of the forces acting on the F.B. CC’ is zero

xwV

xwVVV

0

Dividing both members of the equation by Dividing both members of the equation by ΔΔx x

then letting the then letting the ΔΔx approach zero, we obtain x approach zero, we obtain

wdx

dV

The previous equation indicates that, for a beam loaded as figure, the slope 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 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 any point is equal to the load per unit length at that point

Integrating the equation between point C and D, we writeIntegrating the equation between point C and D, we write

)( DandCbetweencurveloadunderareaVV

dxwVV

CD

x

x

CD

D

C

Relations between load and Shear (cont.)Relations between load and Shear (cont.)

Relations between Shear and Bending momentRelations between Shear and Bending moment

Writing the sum of the moment about C’ is Writing the sum of the moment about C’ is

zero, we havezero, we have

2)(2

1

0)2(

xwxVM

xxwxVMMM

Dividing both members of the eq. by Dividing both members of the eq. by ΔΔx and x and

then letting then letting ΔΔx approach zero we obtainx approach zero we obtain

Vdx

dM

The equation indicates that, the slope dM/dx of the bending moment curve is equal to the value of the shearThe 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.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 MaximumIt 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 bendingThis property facilitates the determination of the points where the beam is likely to fail under bending

Integrate eq. between point C and D, we writeIntegrate eq. between point C and D, we write

)DandCbetweencurveshearunderareaMM

dxVMM

CD

x

x

CD

D

C

Relations between Shear and Bending moment (cont.)Relations between Shear and Bending moment (cont.)

The area under the shear curve should be considered positive where the shear is positive and The area under the shear curve should be considered positive where the shear is positive and

vice versavice versa

The equation is valid even when concentrated loads are applied between C and D, as long as the The equation is valid even when concentrated loads are applied between C and D, as long as the

shear curve has been correctly drawn.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.The eq. cease to be valid, however if a couple is applied at a point between C and D.

)DandCbetweencurveshearunderareaMM

dxVMM

CD

x

x

CD

D

C

Relations between Shear and Bending moment (cont.)Relations between Shear and Bending moment (cont.)