Lab 7 Exercise: Beam Bending Under 3-Point Loading

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MECH207
4/18/18
Lab 7 Exercise: Beam Bending Under 3-Point Loading
Bend testing (also flexural testing) is commonly performed to measure the flexural strength and
modulus of all types of materials and products. Here we apply the load perpendicular to the longitudinal
axis of the specimen to compare beams of different x-sectional geometries. This is important because
for a given application one must choose a material and a beam geometry such that it does not bend
significantly when supporting the loads required for the application. This experiment uses a universal
testing machine and a three point bend fixture to bend the beam and acquire load vs. deflection data
needed to calculate the properties of the beam.
Load (N)
Position (mm)
y
h
Z
b
Test Procedure:
1. Produce beams of different x-sectional geometry for comparison. Measure the sample
geometry carefully, including span length (L) between supports and calculate moment of
inertia about z.
2. Adjust the support span length determined by your sample size (here L will be 28cm).
3. Place the test beam on the 3 point bend fixture and begin the test, collecting load vs.
deflection data.
4. End the test after bending to 5% strain or until the sample breaks.
5. Answer the Questions below.
MECH207
4/18/18
Questions to Answer:
1) Under 3-point bending, what is the maximum shear force (V) and bending moment (M) for a
point load applied in the middle of the beam? Show free body diagrams and equilibrium
equations (solve for all reactions). Draw shear force and bending moment diagrams for V and
M (be sure to label max/min values as a function of P and L)! (20 pts)
2) Calculate the moment of inertia (Iz) and section modulus (S) for the two simple beam
configurations below (measure the width and thickness of each board for composite beam).
Then calculate Iz and S for YOUR beam –be sure to show all steps using the parallel axis
theorem (see lecture notes and Ch 10)! (20 pts)
Vertical
Orientation
y
h
Z
Horizontal
Orientation
b
y
Z
h
b
Your Beam
?
MECH207
4/18/18
3) From the class data collected, fill out Chart below (Pmax, dmax, and m for ALL Beams;
remaining columns fill out just for the 1 horizontal, 2 vertical, and YOUR beam. (20 pts)
Beam
Flexural
Strength
Pmax (N)
dmax
(m)
m
(slope,
N/m)
Vmax
(N)
Mmax
(Nm)
ymax
(m)
Iz
(m4)
S
(m3)
sxmax
(MPa)
tmax
(MPa)
1horizontal
2vertical
Beam 3
Beam 4
Beam 5
Beam 6
4) From the tabulated data above, determine which has the best properties in terms of flexural
strength (Pmax) and flexural stiffness proportional to slope m. Did the result surprise you? How
did the results correlate with S? (10 pts)
5) According to the flexure formula in Chapter 5, the normal stress is given by:
My
where M is bending moment, y is distance along specimen height, and I is
I
moment of inertia. When/where is this a maximum (on the beam’s length and x-section)?
Compare the stress withstood by each beam. (10 pts)
?x ? ?
ef
(m/m)
MECH207
4/18/18
VQ
where V is the shear force, Q is the first moment and b is
Ib
the width of the beam at the centroid. When/where is this a maximum (on the beam’s length
and x-section)? Calculate tmax for your beam and the horizontal and vertical beams. (10 pts)
6) Max shear is given by: ? ?
7) Calculate the maximum flexural strain for each (prior to failure) using the formula:
6?h
where d is deflection at the beam center (prior to failure) and L is the span
L2
length of the beam supports and h is the height of the beam. Which beam deformed the most?
Rank them from highest to lowest. (10 pts)
?f ?
L3 m
*Bonus: Calculate Young’s modulus E using the formula: E ?
. Does this vary widely for
48I
each of the beams? Does this match literature values for this type of wood? Why might there
be variation or discrepancy? (Is you’re beam purely wood or other material?)

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