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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