NAZARIN B. NORDIN nazarin@icam.edu.my

tobascothwackΠολεοδομικά Έργα

15 Νοε 2013 (πριν από 3 χρόνια και 4 μήνες)

88 εμφανίσεις

NAZARIN B. NORDIN

nazarin@icam.edu.my

What you will learn:




CHAPTER 8

SHEAR FORCE & BENDING MOMENT


Introduction


Types of beam and load


Shear force and Bending Moment


Relation 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
c
M
m


3
2
12
1
6
1
bh
I
bh
S


S
M
m


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
Δ
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+
Δ
V and
M +
Δ
M 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



x
w
V
x
w
V
V
V









0

Dividing both members of the equation by
Δ
x then letting the
Δ
x approach zero, we
obtain

w
dx
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

)
(
D
and
C
between
curve
load
under
area
V
V
dx
w
V
V
C
D
x
x
C
D
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
(
x
w
x
V
M
x
x
w
x
V
M
M
M















Dividing both members of the eq. by
Δ
x and
then letting
Δ
x approach zero we obtain

V
dx
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

)
D
and
C
between
curve
shear
under
area
M
M
dx
V
M
M
C
D
x
x
C
D
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.

)
D
and
C
between
curve
shear
under
area
M
M
dx
V
M
M
C
D
x
x
C
D
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 4

A 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)
1.5m

b)
2.4m

c)
Draw the shear force and bending moment diagram.

QUESTION 5

A 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)
1.0m

c)
Draw the shear force and bending moment diagram.

(neglect the weight of the beam.

QUESTION 6

A 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)
1.2m

c)
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)
0.65m

c)
Draw the shear force and bending moment diagram.

(neglect the weight of the beam)