3_shear force bending moment_sm

7/29/2019 3_Shear Force Bending Moment_SM

1/28

CHAPTER #3

SHEAR FORCE & BENDINGMOMENT

Introduction

Types of beam and load

Shear force and bending Moment

Relation between Shear force andBending Moment


2/28

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


3/28

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)


4/28

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


5/28

Determination of Max stress in

beam

I

cM

m

3

2

12

16

1

bhI

bhS

S

M

m


6/28

SHEAR & BENDING MOMENT DIAGRAMS

Shear Force (SF) diagram TheShear Force (V) plotted againstdistance x Measured from end ofthe beam

Bending moment (BM) diagramBending moment (BM) plottedagainst distance x Measuredfrom end of the beam


7/28

DETERMINATIONS OF SF & & BM

The Shear & bending momentdiagram will be obtained bydetermining the values of Vand M at selected points ofthe beam


8/28


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

The shear at any given point of a beam is positive when the externalforces (loads and reactions) acting on the beam tend to shear off the

beam at that point as indicated in figure below


9/28


The bending moment at any given point of a beam is positive when theexternal forces (loads and reactions) acting on the beam tend to bendthe beam at that point as indicated in figure below


10/28


11/28


12/28


13/28


14/28


15/28


16/28


17/28


18/28

Relation between Shear force andBending 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


19/28

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 x 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+ V and

M +M respectively


20/28

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

x then letting thex approach zero, we

obtain

wdx

dV


21/28

The previous equation indicates that, for a beam loaded as figure,

the slope dV/dx of the shear curve is negative; the numerical value ofthe 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.)


22/28

Relations between Shear and Bendingmoment

Writing the sum of the moment about C iszero, we have

2)(

2

1

0)2

(

xwxVM

xxwxVMMM

Dividing both members of the eq. byx and

then lettingx approach zero we obtain

Vdx

dM


23/28

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 Bendingmoment (cont.)


24/28

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 Bendingmoment (cont.)


25/28


26/28


27/28


28/28

3_shear force bending moment_sm

Documents