chapter 2 -shear force and bending moment
TRANSCRIPT
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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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