EK305: MECHANICS OF MATERIALS LAB#4 BEAM BENDING TEST

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Nov 15, 2013 (3 years and 8 months ago)

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EK305: MECHANICS OF
MATERIALS

LAB#4

BEAM BENDING TEST





PRELAB:

1. Obtain the equation for the deflection of
a beam in a cantilever configuration with a
point load applied at the non
-
fixed end.
Assume the load to be P, length of the
beam to be L, its tra
nsverse moment of
inertia to be I and modulus of Elasticity to
be E. P is applied downwards.

2. What is the deflection equation of the beam
when the load is applied at the center of the
beam instead of at the end? (You may simply
quote standard equations f
rom the appendix of
your book for question 2 alone).



OBJECTIVE:

To observe the effect of material,
loading position and boundary conditions on the
bending and deflection of slender beams.



THEORY:

The differential equation governing the bending
deflec
tions of thin elastic beams with constants
E and I is






E I v’’= M(x)






(1)

where

E


= Young’s modulus of material

I


= Moment of inertia of the beam
about the axis of bending

V


=
Transverse beam deflection

M(x) = Internal moment



Integr
ating and applying appropriate boundary
conditions result in the solution to the above
differential equation.

For brittle material like glass, it is often useful to
define a parameter called the bending modulus.
It is essentially the elasticity modulus, t
hough it
is found out using beam deflection in the
transverse direction rather than a direct tensile
test. This number is found to differ from the
Young’s modulus determined using tensile tests.
Tensile tests for brittle material require
sophisticated supp
orts so as not to crush the
specimen. The theoretical expression is




E
b

= (L
3

/(4bh
3
)) (P/ δ)





(2)

where




E
b

= Bending modulus




L = Length of the sample between
supports




P = Load applied at x = L/2




b = breadth




h = thickness




δ

= deflection of sample at x =
L/2













PROCEDURE:

SAFETY: As this lab involves breaking glass
samples, SAFETY GOGGLES MUST BE
WORN IN ORDER TO PREVENT GLASS
SHARDS FROM ACCIDENTALLY FALLING
INTO ONE’S EYES.



There are two parts to this experimen
t. The first
part involves determining the Bending modulus
of glass.

1.

Mount the glass sample between the two
cylindrical supports.

2.

Make sure the loading is in the center of the
glass sample.

3.

Run the experiment while taking data
simultaneously.

4.

Stop the
experiment once the glass specimen
breaks.

5.

Repeat the experiment for 5 samples to get
adequate data.



The second part involves cantilever
configuration of metallic beams.

1.

Mount one end between vices.

2.

Locate the load at the unsupported end of the
canti
lever (or at the center, depending on the
experiment performed).

3.

Trace out the current curve of the cantilever.

4.

Apply a predefined load. Note that the load
should be large enough to produce perceptible
deflection.

5.

Trace out the curve with the load appli
ed.

6.

Release the load.

7.

Repeat the experiment with two different
material and two different load configurations.



PROCESSING DATA:

There are two different data that you need to
process.

For glass, you need to:

1.

Plot the displacement (x
-
axis) versus load
(y
-
axis). Plot these as points; do not join the
points.

2.

Draw a regression line between the load and
displacement. The slope of this line is the
bending modulus. Average this slope over the 5
samples that you are processing data for.

3.

Compare it with the t
heoretical predictions of
equation 2.



For the metallic samples, you need to:

1.

Plot the values of cantilever beam displacement
(y
-
axis) versus the distance measured along the
beam (x
-
axis).

2.


On the same plot, draw the curve to the
theoretical equation o
btained as solution to the
differential equation. NOTE THAT THE
DEFLECTION CURVE WILL BE
DIFFERENT FOR DIFFERENT LOADING
CONDITIONS.

3.

Draw relevant conclusions from these curves.





QUESTIONS:

Derive E
b

from equation 1 applied to the
three
-
point configur
ation used for the glass
sample. Use E
b

in place of E in equation 1.
Assume that you are applying the load at the
center of the specimen, and that the moment of
inertia of the sample about the transverse axis is
bh
3
/12. (Hint: There are two boundary
condi
tions. One states that the displacement in
the y direction at the supports is zero. The other
is that the slopes match at x = L/2).