ME2113-1 DEFLECTION AND BENDING STRESSES IN BEAMS

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15 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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ME
2
11
3
-
1



DEFLECTION AND BENDING STRESSES IN BEAMS


(EA
-
02
-
21)













SEMESTER 3


201
3
/20
1
4

















NATIONAL UNIVERSITY OF SINGAPORE

DEPARTMENT OF MECHANICAL ENGINEERING


INTRODUCTION


Beams are one of the commonest components encount
ered in structures. In most cases, they
are transversely loaded and thus undergo bending. Bending induces stresses in a beam and
causes it to deflect. In the design and analysis of beam
-
type structures, the stresses and
deflections generated must be examin
ed to ascertain that they are within acceptable limits.
Simple beam theory provides expressions for the calculation of these quantities. This
experiment provides the means to investigate how beam theory can be applied in practice.



OBJECTIVE AND SCOPE


To

study the application of beam theory in practice, by subjecting a cantilever to various point
loads and examining the resulting stresses and deflection. From these values, to also
determine the Young’s modulus and Poisson’s ratio of the beam material. The

magnitudes
and signs of the strains and stresses at two locations along the beam are investigated in terms
of their relation to each other and in accordance with beam theory.



THEORY


Figure 1 shows a cantilever of cross
-
section
b
x
h,

subjected to a dow
nward point load
P

at a
distance
L

from the built
-
in
-
end.




Fig. 1 Cantilever subjected to a point load



The bending moment
M
xz

at the point
x

from the built
-
in
-
end is given by:



M
xz

=
-
P(L
-

x)

(1)
The longitudinal normal stress

xx

is related to t
he bending moment by:


y
I
M
z
xz
xx




(2)


where

the y
-
coordinate of the point of interest is with respect to the horizontal mid
-
plane of
the beam.


I
bh
z

1
12
3

is the second moment of area of the beam cross
-
section about the z
-
axis.


For example, at the lower surface, y =
-
h/2 ; hence the longitudinal stress there is:


)
2
(
)
2
/
(
h
I
M
z
xz
h
y
xx







(3)


Normal strain is related to normal stress by Hooke’s law:


)
(
1
zz
yy
xx
xx
E








(4a)


)
(
1
yy
xx
zz
zz
E








(4b)

where
E

is Young’s mod
ulus and


is Poisson’s ratio.


For a transversely
-
loaded cantilever,

yy

=

zz

= 0. Hence, the stress
-
strain relationship
reduces to:


xx
xx
xx
xx
E
E






or


(5)

and

xx
xx
zz
E












(6)


The vertical deflection
v

of the point at
x = L
where the l
oad
P

is applied is given by:


z
L
EI
PL
3
3


v

(7)


EXPERIMENTAL EQUIPMENT

With reference to Fig. 1, the experimental equipment comprises:


1.

A cantilever
aluminium

beam (b =25.
6
mm, h =
6.06
mm) with a load
-
carrying hanger
at a distance
L =250mm
fro
m the built
-
in
-
end.


2.

6

x
250g

weighs to act as transverse loads on the beam.


3.

A dial
-
gauge to measure the vertical deflection of the loading point at
x =

L
.


4.

Two pairs of strain gauges mounted on: (1) the upper surface of the beam at a distance
d
1

= 50mm from the clamped end. (2) the underside of the beam at a distance
d
2

=
150mm from the clamped end; At each location, there are two strain gauges, one
aligned longitudinally in the
x
-
direction and the other in the horizontal transverse
z
-
direction.


5.

A switching box to which wires from the four strain gauges are connected to four of
its channels. This switching box facilities selection of the gauge which a reading is to
be taken from.


6.

A strain meter connected to the switching box. This gives th
e value of strain measured
by the selected strain gauge.


7.

A gripper.


EXPERIMENTAL PROCEDURE

Part I


1.1

Identify the strain quantity (i.e.

xx1

,

zz1

,

xx2

or

zz2

) measured by each gauge and the
channel to which it is connected on the switching box.


1.2

Wi
th no loads on the hanger, set the dial gauge reading to zero.


1.3

Select a channel on the switching box for the first set of strain gauge readings and zero
the strain meter.


1.4

Load the beam in
six

steps of
2
5
0

g weight increments and record the deflection of
the
load point and the reading of the strain gauge selected [note whether the strain readings
are positive (tensile) or negative (compressive)]. Record the deflection and strain
readings upon unloading as a check.


1.5

Select the other channels in turn, zero t
he dial gauge if necessary and repeat the
loading/unloading process together with the recording of deflection and strain readings.


1.6

Tabulate all the readings and plot the following:


Graph 1
-

Load
P

against vertical deflection
v

. From the slope of this
graph and
Eqn. (7), calculate Young’s modulus
E

of the beam material.



Graph 2
-


zz1

against

xx1

and

zz2

against

xx2
.

From the slope of the lines,
calculate Poisson’s ratio using Eqn. (6).



Graph3
-

Load
P

against

xx1
. Calculate the slope of this g
raph.



Graph 4
-

Determine, using Eqn. (1), the bending moment
magnitudes

at the two
strain gauge locations,
x = d
1

and
x = d
2

,
for each of the
six

applied
loads. Then, use Eqn. (2) or (3) to calculate the theoretical
magnitude

of the longitudinal stress
es (

xx1

and

xx2
) at the beam surface (
y = h
/2)
for the
six

loads. Use these four pairs of stress values to plot four lines
showing the theoretical variation of maximum longitudinal stress with
location along the beam (i.e.

xx

against
x

).




From the
ma
gnitudes

of the longitudinal strain readings (

xx1

and

xx2
)
corresponding to the
six

values of applied load, calculate the
corresponding experimental stress magnitudes (

xx1

and

xx2
) using the
value of
E

derived from Graph 1 and Eqn. (5). Plot these expe
rimental
points on the same graph with the
six

lines showing the theoretical
variation of maximum stress with beam location.


Part II



Attach the gripper to the end of the beam (Fig. 2) and grip the gripper as tight as possible.

** DO NOT PULL DOWN
ON THE
CANTILEVER.




Record the highest reading

xx1

of strainmeter, and evaluate your handgrip force.



Fig 2.

DISCUSSION

1.

Comment on the signs of the strains (

xx1,


zz1,

xx2
and

zz2
) with respect to the location
and orientation of the strain gauges and h
ow the beam is loaded.

2.

With reference to Graph 4, comment on the slopes of the
six

theoretical lines and also
on how stress varies with beam location.

3.

Comment on the accuracy of your handgrip force.



Sltoh/Nov.1999

(Amended 4 Jul 2013)