chapter 2 -shear force and bending moment

8/12/2019 Chapter 2 -Shear Force and Bending Moment

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Universiti Tun Hussein Onn Malaysia

(UTHM)

Engineering Mechanics :

SOLID MECHANICS I

Shear Force and Bending Moment


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Todays Objectives:

Students will be able to:1. determine stress in members caused by bending

2. discuss how to establish shear and moment diagrams for a beam or shaft

3. determine the largest shear and moment in a member, and specify where

they occur

Topics:

BeamShear force and Bending Moment

DiagramGraphical Method for

Constructing Shear and Moment

Diagrams

CHAPTER 2 : SHEAR FORCE AND

BENDING MOMENT


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2.0 APPLICATION


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As shown, failure of this table occurred at the brace support on its right

side. The bending moment diagram for the table loading would indicate

this to be the point of maximum internal moment


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2.1 BEAM


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INTRODUCTION

Beamsstructural members that are slender and

supporting transverse loads at various points alongthe longitudinal axis of the member.

Transverse loadings of beams are classified asconcentratedloads or distributedloads.

Applied loads result in internal forces consistingof a shear force (from the shear stress

distribution) and a bending couple (from the

normal stress distribution).

In order to design a beam, it is necessary todetermine the maximum shear and moment in the

beam which is express Vand Mas functions of

arbitrary positionxalong axis.

These functions can be represented by graphs calledshear and moment diagrams

Engineers need to know the variationof shear anmoment along the beam to know where to reinforce

it


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Classification of Beam Support


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2.2 SHEAR AND MOMENT

DIAGRAMS


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Shear and Bending Moment

Sign conventions for shear forces Vand V

and bending couplesMand M

Shear and bending-moment functions must be determined for each regionof

the beam betweenany two discontinuities of loading


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Procedure of Analysis

Support reactionsDetermine all reactive forces and couple moments acting on beamResolve all forces into components acting perpendicular and parallel to beams axisShear and moment functionsSpecify separate coordinates x having an origin at beams left end, and extendingto regions of beam between concentrated forces and/or couple moments, or where there

is no discontinuity of distributed loadingSection beam perpendicular to its axis at each distancexDraw free-body diagram of one segmentMake sure Vand Mare shown acting in positive sense, according to sign conventionsum forces perpendicular to beams axis to get shearSum moments about the sectioned end of segment to get moment

Plot shear diagram (Vvs.x) and moment diagram (Mvs.x)If numerical values are positive, values are plotted above axis, otherwise, negative

values are plotted below axis

It is convenient to show the shear and moment diagrams directly below the free-body

diagram

Shear and moment Diagram


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Example 1

Draw the shear and moment diagrams for the beam shown below.


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Example 2

Draw the shear and moment diagrams for the beam shown below.


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Example 3

Draw the shear and moment diagrams for beam shown below.


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Example:4


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3.3 GRAPICAL METHOD FORCONSTRUCTING SHEAR

AND MOMENT DIAGRAMS


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Relation between Distributed Load,

Shear Force and Bending Moment

xwV

xwVVVFy

0:0

D

C

x

xCD dxwVV

wdx

dV

Relationship between load and shear:

2

2

1

02

:0

xwxVM

xxwxVMMMMC

D

C

x

x

CD dxVMM

Vdx

dM

Relationship between shear and bending moment:


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dV

dx= w(x)

dM

dx= V

Slope ofshear diagram

at each point

Slope of momentdiagram at each

point

=

distributed loadintensity at

each point

= shear ateach point

SUMMARY :

V= w(x) dx

= -F

M= V(x) dx

= MoChange in

shear

Change in

moment

= area under

distributed loading

= area under shear

diagram


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EXAMPLES OF DIAGRAM :


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Procedure of Analysis

Support reactionsDetermine support reactions and resolve forces acting on the beam intocomponents that are perpendicular and parallel to beams axis

Shear diagram

Establish VandxaxesPlot known values of shear at two ends of the beamSince dV/dx = w, slope of the shear diagram at any point is equal to the (-ve)intensity of the distributed loading at that point

To find numerical value of shear at a point, use method of sections andequation of equilibrium or by using V= w(x) dx, i.e., change in the shear

between any two points is equal to (-ve) area under the load diagram between

the two pointsSince w(x) must be integrated to obtain V, then if w(x)is a curve of degree

n, V(x)will be a curve of degree n+1


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Procedure of Analysis (Cont.d)

Moment diagramEstablish Mandxaxes and plot known values of the moment at the ends of

the beam

Since dM/dx = V, slope of the moment diagram at any point is equal to theshear at the point

At point where shear is zero, dM/dx = 0and therefore this will be a point ofmaximum or minimum momentIf numerical value of moment is to be determined at the point, use method ofsections and equation of equilibrium, or by using M= V(x) dx, i.e., changein moment between any two pts is equal to area under shear diagram between

the two pts

Since V(x) must be integrated to obtain M, then if V(x) is a curve of degree n,M(x) will be a curve of degree n+1


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Example 5

Draw the shear and moment diagrams for the beam in figure

below.


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Example 7


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Example 8


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Example 9


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Example 10


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chapter 2 -shear force and bending moment

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