Activity 2.1.

2 Beam Deflection
Introduction
Engineers must look for better ways to build structures. Less material typically
means that structures will be lighter and less expensive. Knowing the moment of
inertia for different shapes is an important consideration for engineers as they strive
to make designs lighter and less expensive.
Equipment
● 1- 2x4 (preferably straight, free of knots and imperfections)
● Dial calipers or a ruler with 1/32 divisions
● 2 - 1 foot lengths of 2x4 for use as supports
● Tape measure
● Permanent marker
● Floor scale
● Cinder block (Concrete Masonry Unit)
Procedure
You will determine the weight of one of your classmates using nothing more than a
standard 2x4 and a measuring device. This activity will provide you with a better
understanding of Moment of Inertia and how it can be used to determine the strength
of beams.

Preliminary lab calculations to determine beam Modulus of Elasticity

1. Calculate beam Moment of Inertia

B – width of the beam (in.) Vertical Orientation Horizontal Orientation

I = (1.5)(3.5)^3 / 12 = I =(3.5)(1.5)^3 /12 =
h – height of the beam (in.) 5.359375 in^4 0.984375 in^4

I – Moment of Inertia (in.4)

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Principles Of Engineering Activity 2.1.2 Beam Deflection – Page 1
Position the beam as shown below.

2. Measure the span between the supports. Record your measurement below.

Total Span (L) = _______96___in.

3. Measure the distance between the floor and the bottom of the beam.

Pre-Loading Distance (DPL) = ______8.5____in. (measured from middle)

4. Position a volunteer (V1) to stand carefully on the middle of the beam. Have a
person on either side of the beam to help support the volunteer. Measure the
distance between the floor and the bottom of the beam.

Applied Load Distance (DAL) = ______7.2_____in. (measured from middle)

5. Calculate the maximum beam deflection ( MAX ).

MAX = DPL - DAL

MAX = ______1.3____ in.

6. Determine the weight of volunteer (V1) using the classroom floor scale.

Volunteer weight (F) ______150______ lb

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Principles Of Engineering Activity 2.1.2 Beam Deflection – Page 2
 7. Calculate your beam’s Modulus of Elasticity (it is important to know that each
beam will have its own specific Modulus of Elasticity) by rearranging the equation
for beam maximum deflection to isolate (E). Show all work.
Rearrange the 𝐹𝐿
3
𝐸=
equation 48𝐼⍙𝑀𝐴𝑋

to solve in terms of E
E= (150lbs.)(96 in.)^3 / (48(1.3in)( 0.984375 in^4))
Substitute known
values

E=132710400lb-in^3/(61.425in^5)
Simplify

E=2160527.473psi
Solve

Note: An object’s Modulus of Elasticity is a material-based property and stays the

same regardless of orientation.

Calculate volunteer (V2) weight

8. Position the beam as shown below.

9. Measure the span between the supports. Record your measurement below.

Total Span (L) = _____96_____in.

10. Measure the distance between the floor and the bottom of the beam.

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Principles Of Engineering Activity 2.1.2 Beam Deflection – Page 3
 Pre-Loading Distance (DPL) = _____9.5_____in. (measured from top)

11. Position a second volunteer (V2) to stand carefully in the middle of the beam.
Have a person on either side of the beam to help support the volunteer. Measure
the distance between the floor and the bottom of the beam.

Applied Load Distance (DAL) = _______8____in. (measured from top)

12. Calculate the maximum beam deflection ( MAX ).

MAX = DPL - DAL

MAX = ____1.5______ in.

13. Calculate volunteer (V2) weight by rearranging the equation for maximum
deflection to isolate (F). Show all work.
Rearrange the 48𝐸𝐼⍙𝑀𝐴𝑋
𝐹=
equation 𝐿
3

to solve in terms of F
F=48(2160527.473psi)(0.984375in^4)(1.5in)/(96in^3)
Substitute known
values

F=153127384.6lb-in^3/884736in^3
Simplify

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Principles Of Engineering Activity 2.1.2 Beam Deflection – Page 4
 F=173.08lb
Solve

Determining Beam Deflection

14. Using the information you collected and calculated in steps 1 – 14, calculate the
max deflection of the beam if volunteer (V2) is positioned to stand on the beam in
a vertical orientation.

Substitute known △ 3
173.0769231𝑙𝑏*(96𝑖𝑛)
𝑀𝐴𝑋=
values 48*2160527.473𝑝𝑠𝑖*5.359375𝑖𝑛
4

Simplify 153127384.6𝑙𝑏*𝑖𝑛
3
△𝑀𝐴𝑋 = 2
55579692.3𝑙𝑏*𝑖𝑛

Solve △𝑀𝐴𝑋 =. 27551𝑖𝑛

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Principles Of Engineering Activity 2.1.2 Beam Deflection – Page 5
 15. Verify your calculated max deflection answer and work to your instructor by
having volunteer (V2) carefully stand in the middle of the beam. Place a person
on either side of the beam to help support the volunteer. Measure the distance
between the floor and the bottom of the beam.
Calculated deflection: ___.276__________
Measured deflection: _____11.25-10.75=.5________
Instructor signature: ___________N/A________________ Date: ____1/9/15____

Practice Problem

16. Complete the chart below by calculating the cross-sectional area, Moment of
Inertia, and beam deflection, given a load of 250 lbf, a Modulus of Elasticity of
1,510,000 psi, and a span of 12 ft. Show all work in your engineering notebook.

Beam A B C D E F
Common
2x6 2x6 2x8 2x8 2x10 2x10
Name

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Principles Of Engineering Activity 2.1.2 Beam Deflection – Page 6
 Actual
Dimensions 1.5 x 5.5 1.5 x 5.5 1.5 x 7.25 1.5 x 7.25 1.5 x 9.25 1.5 x 9.25
(in.)
Vertical or
Horizontal
Orientation
Cross-Secti
onal Area 8.25 8.25 10.875 10.875 13.875 13.875
(in.2)
Moment of
Inertia 20.80 1.56 47.63 2.04 98.93 2.60
(in.4)
Beam
Deflection .50 6.66 .22 5.05 .10 3.96
(in.)

Conclusion
1. Using Excel, create a Deflection vs. Moment of Inertia graph. What is the
relationship between moment of inertia and beam deflection?

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Principles Of Engineering Activity 2.1.2 Beam Deflection – Page 7
Moment of inertia and beam deflection are inversely proportional meaning that their
product is constant (the graph will form a hyperbola)

2. How could you increase the Moment of Inertia (I) of a beam without increasing its
cross-sectional area?
Keep the area constant while increasing the height and decreasing width. Height
and width should always be inversely proportional.

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Principles Of Engineering Activity 2.1.2 Beam Deflection – Page 8