BTE 1013ENGINEERING SCIENCES

8. SHEAR FORCE AND BENDING MOMENT

NAZARIN B. NORDINnazarin@icam.edu.my

What you will learn:

CHAPTER 8SHEAR FORCE & BENDING MOMENT

Introduction Types of beam and loadShear force and Bending MomentRelation 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

I

cMm I

cMm

3

2

12

16

1

bhI

bhS

S

Mm

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

xwV

xwVVV

0

Dividing both members of the equation by

Δx then letting the Δx approach zero, we

obtain

wdx

dV

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

)( DandCbetweencurveloadunderareaVV

dxwVV

CD

x

x

CD

D

C

Relations between load and Shear (cont.)

Relations between Shear and Bending Moment

Writing the sum of the moment about C’ is

zero, we have

2)(2

1

0)2(

xwxVM

xxwxVMMM

Dividing both members of the eq. by Δx and

then letting Δx 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 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

)DandCbetweencurveshearunderareaMM

dxVMM

CD

x

x

CD

D

C

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.

)DandCbetweencurveshearunderareaMM

dxVMM

CD

x

x

CD

D

C

Relations between Shear and Bending Moment (cont.)

QUESTION 1

If the beam carries loads at the positions shown in figure, what are the reactive forces at the supports? The weight of the beam may be neglected.

QUESTION 2

If the beam carries loads at the positions shown in figure, what are the reactive forces at the beam? The weight of the beam may be neglected.

QUESTION 3

Determine the shear force and bending moment at points 3.5m and 8.0m from the right-hand end of the beam. (neglect the weight of the beam)

QUESTION 4A beam of length 5.0m and neglect the weight rests on supports at each end and a concentrated load of 255N is applied at its midpoint. Determine the shear force and bending moment at distances from the right-hand end of the beam of a) 1.5mb) 2.4mc) Draw the shear force and bending moment diagram.

QUESTION 5A cantilever has a length of 2m and a concentrated load of 8kN is applied to its free end. Determine the shear force and bending moment at distances of a) 0.5m b) 1.0mc) Draw the shear force and bending moment diagram.(neglect the weight of the beam.

QUESTION 6A beam of length 5.5m supports at each end and a concentrated load of 135N is applied at 2.5m from the left hand end. Determine the shear force and bending moment at distances of;a) 0.8m b) 1.2mc) Draw the shear force and bending moment diagram.(neglect the weight of the beam)

QUIZ

A beam of length 1m supports at each end and a concentrated load of 1.5N is applied at the centre. Determine the shear force and bending moment at distances of;a) 0.25m b) 0.65mc) Draw the shear force and bending moment diagram.(neglect the weight of the beam)

THANK YOU