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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams Beams structural members designed to support
loadings perpendicular to their axes Beams straight long bars with constant cross-
sectional areas
A simply supported beam is pinned at one endand roller supported at
the other
A cantilevered beam isfixed at one end and free
at the other
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams For actual design of a beam, apply
- Internal shear force V and the bending momentM analysis
-
- Appropriate engineering code to determinebeams required cross-sectional area
Variations of V and M obtained by the method ofsections
Graphical variations of V and M are termed as
shear diagram and bending moment diagram
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams Internal shear and bending moment
functions generally discontinuous, or theirslopes will be discontinuous at pointswhere a distributed load changes or where
concentrated forces or couple momentsare applied
Functions must be applied for each
segment of the beam located between anytwo discontinuities of loadings
Internal normal force will not beconsidered
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Load applied to a beam actperpendicular to the beams axis andhence produce only an internal shear
For design purpose, the beamsresistance to shear, and particularly to
bending, is more important than itsability to resist a normal force
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and DiagramsSign Convention To define a positive and negative shear
force and bending moment acting on the
Positive directions are denoted by aninternal shear force that causes clockwise
rotation of the member on which it actsand by an internal moment that causescompression or pushing on the upper part
of the member
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Sign ConventionA positive moment
would tend to bend the
member if it wereelastic, concave upwards
Loadings opposite to theabove are considerednegative
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Procedure for AnalysisSupport Reactions
Determine all the reactive forces and
couple moments acting on the beam Resolve them into components acting
perpendicular or parallel to the beams
axis
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and DiagramsProcedure for Analysis
Shear and Moment Reactions
Specify separate coordinates x having an origin
of the beams between concentrated force and/orcouple moments or where there is no continuityof distributed loadings
Section the beam perpendicular to its axis ateach distance x and draw the FBD of one of thesegments
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Procedure for Analysis
Shear and Moment Reactions
V and M are shown acting in their positive sense
he shear V is obtained by summing the forcesperpendicular to the beams axis
The moment M is obtained by summing
moments about the sectioned end of thesegment
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and DiagramsProcedure for Analysis
Shear and Moment Diagrams
Plot the shear diagram (V versus x) and the
If computed values of the functions describing Vand M are positive, the values are plotted abovethe x axis, whereas negative values are plottedbelow the x axis
Convenient to plot the shear and the bending
moment diagrams below the FBD of the beam
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Example 7.7
Draw the shear and bending moments
diagrams for the shaft. The support at A is a
thrust bearing and the support at C is ajournal bearing.
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Solution
Support Reactions
FBD of the shaft
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Solution
kNVFy 5.2;0 ==+
mx ..;==
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Solution
kNV
VkNkNFy
5.2
055.2;0
=
==+
mkNxM
xkNmxkNMM
.)5.210(
0)(5.2)2(5;0
=
=+=
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and DiagramsSolution
Shear diagram internal shear force is always
positive within the shaft AB
Just to the right of B, the shearforce changes sign and remainsat constant value for segmentBC
Moment diagram
Starts at zero, increases linearlyto B and therefore decreases tozero
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and DiagramsSolution
Graph of shear and momentdiagrams is discontinuous at
ie, A, B, C
All loading discontinuous aremathematical, arising from
the idealization of aconcentrated force andcouple moment
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Example 7.8
Draw the shear and bending diagrams for
the beam.
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Solution
Support Reactions
FBD of the beam
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and DiagramsSolution
Distributed loading acting on thissegment has an intensity of 2/3 x at
force after the segment is isolated asa FBD
For magnitude of the
resultant force,
(x)(2/3 x) = 1/3 x2
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Solution
Resultant force acts through the centroid of thedistributed loading area, 1/3 x from the right
1
mkN
x
xM
xx
xMM
kNx
V
xy
.99
0933
1;0
39
3;
3
2
2
=
=
+=
=
==
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7.2 Shear and MomentEquations and Diagrams7.2 Shear and Moment
Equations and Diagrams
Solution
For point of zero shear,
xV 09
3
==
For maximum moment,
( ) ( )
mkN
mkNM
mx
.12.3
.9
20.520.59
20.5
3
max
=
=
=