This document describes the design and construction of a demonstrator to show equivalent stiffness in beams made of different configurations of two materials. The authors designed five beam configurations with equal stiffness but varying cross-sectional geometries using acrylic and HDPE. They developed equations to calculate the required dimensions and 3D printed the beams. Testing showed the beams deflected similarly under an applied load, validating the equivalent stiffness concept. Challenges in the non-ideal material properties and tolerances were discussed.

1. Two-Material Beam Equivalent Stiffness
Demonstrator
by
Jared Bruton and Michael Morgan
ME - CE 501 Final Project
Dr. Brian Jensen
Spring Term 2015
June 18, 2015

2. 1 Problem Explanation
The purpose of this project was to explore multi-material beam stiffness through designing a
demonstrator to show equivalent stiffness in beams of different configurations of materials and
cross-sectional geometry. As beams of multiple materials appear in real-world engineering
problems quite frequently, a knowledge of how to analyze them is useful. A visualization
tool that shows these concepts which are discussed in class and practiced in homework will
be a valuable aid in helping students gain a more intuitive perspective on the topic.
2 Problem Approach
We chose to demonstrate the concept by constructing multiple beams with common stiffness
and visually different cross sections using various combinations of two materials. As stiffness
is not directly visible, we elected to demonstrate stiffness with deflection. To do this, we
constrained each cantilevered beam’s design so that under an equal tip load P, with equal
length L, and, as mentioned previously, with equal stiffness (EiIi) the beams would deflect
a common distance δ, as shown in Figure 5.
P
δOriginal beam
Deflected Beam
L
Figure 1: Boundary conditions and constraints for all beams.
2.1 Cross Section Configurations
Five configurations shown in Figure 2 were considered to show multiple ways of achieving
the same stiffness with two-material beams. In the following configurations we will refer
to parameters h and b which are constant across all of the beams. In order to provide a
comparison of the geometry required to result in equal stiffness between the two materials
individually, Beam 1 and Beam 5 were made solely of Material B and Material A, respectively.
Beam 1 has height 2h and is considered the baseline beam as from its geometry we calculated
the desired stiffness S for all of the beams. Beam 2 is a combination of both materials, each
with height h and the upper portion (Material A) with scaled width b2. Beam 3 mimics
a sandwich structure with equal heights h3 of Material B on each side of the Material A
section, which has height h. Finally, Beam 4 has equal heights of each material.
1

3. Beam 1 Beam 2 Beam 3 Beam 5Beam 4
2h
h
h
Material A - Acrylic
Material B - HDPE
h
h4
h4
b b
b
b
b
b2
h3
h5
Figure 2: Five configurations of two-material beams, all with equivalent stiffness. Orange values
are calculated, grey values are fixed. Stiffness S is calculated from the geometry of
Beam 1.
2.2 Governing Equations
As outlined in class, we began with the small-deflection equation, shown in Equation 1.
δ =
PL3
3EI
(1)
We adapted the equation to be suitable for two-material beams by replacing EI with
(EiIi), for each material i in the beam. Throughout the remainder of this work we will
refer to the stiffness term (EiIi) as S. The governing deflection equation becomes as
follows.
δ =
PL3
3 (EiIi)
=
PL3
3S
(2)
In this equation, I is the polar moment of inertia about the neutral axis. The neutral
axis can be calculated as shown in Equation 3.
yna =
(Ei ¯yIi)
(EiAi)
(3)
Now, as we focus in on the stiffness, it is clear that S is a function of only material
properties and cross-sectional geometries of the beam. It also depends on the neutral axis
which itself is determined by those two factors. However, in Beam 1, Beam 3, and Beam 5,
the symmetric cross-sections mean that the neutral axis is at the center of the cross-section
and therefore independent of the material properties. This makes the calculation of unknown
dimensions (as shown in Figure 2) for these beams trivial. In Beam 2 and Beam 4 though,
the neutral axis is affected by the material properties, rendering these derivations much more
lengthy. All derivations can be found in Appendix A: Calculations.
Additionally, in order to avoid yielding in the first use (and consequently, the only use),
the stress also was of interest to us. This is given by:
σ =
−MyE
(EiIi)
(4)
2

4. where M is the applied moment, E is the modulus of the material of interest, and y is the
distance from the point of interest to the neutral axis.
2.3 Material Selection
Based on the equations given in the previous section, we developed a spreadsheet to para-
metrically calculate the cross-sectional geometry of each beam based on the properties of
the chosen materials and the calculated stiffness of Beam 1. In this spreadsheet, we also
included the max stress in each material of each beam, the safety factor based on each of
those (using max principle stress theory), deflection calculations, a tolerance analysis for
material thickness, and plots of the cross-section dimensions for visualization purposes. This
spreadsheet can be seen in Figure 3. This tool allowed us to easily change the materials of
the beams and automatically view the updated data.
Figure 3: A screenshot of the spreadsheet we developed to design the beams.
With the spreadsheet created, we began to explore different material combinations. We
determined that, in order to work effectively as a demonstration tool, we wanted a visually
significant difference in the thickness of the beams. However, when we adjusted the materials
to give a large difference, the top segment of Beam 2 became paper thin and impossible to
fabricate. We learned that both the width of that segment and the difference in thickness of
the other beams related directly to the ratio of the Young’s Moduli of the two materials used,
EB/EA, where EA > EB. We further determined that the values of both moduli needed to
be fairly low to allow a substantial deflection. By visual inspection of the geometry graphs
on the spreadsheet, we determined the ideal modulus ratio to be between 0.2 and 0.3.
3

5. From this point we created a table comparing the ratios of the moduli of candidate
materials. This table is shown in Figure 4. Along both the top and the left side is the same
list of materials and their respective moduli. The table calculates each ratio and highlights
anything in the desired range (between 0.2 and 0.3). After sifting through the valid options,
we identified the most feasible and inexpensive combination to be acrylic and high density
polyethylene (HDPE). According to the values we found online, these two materials had
a modulus ratio of 0.25 and would result in significant deflection as well as an acceptable
difference in the thicknesses of the beams.
Figure 4: A screenshot of the material combination table.
2.4 Hardware Design and Construction
As a structure to hold the beams, we designed a single fixture to closely approximate a
cantilever connection for all five beams. See Figure 5 for side view of boundary conditions
and construction method.
This frame was also constructed such that applying the force load P would not cause
the frame to flip. We did this by constructing it of a sufficiently heavy material and with
a footprint large enough that the load applied would not create a moment to cause this
problem. This was necessary to maintain ease-of-use when showing the demonstrator to an
audience.
Beam thicknesses were calculated as discussed in Sections 2.2, 2.3, and Appendix A:
Calculations. The beams were all manufactured using a Shopbot CNC router to machine
the beams to their specified thicknesses. After they were machined to be within tolerances,
composite beams were adhered using Loctite plastic-bonding superglue.
4

6. P
δOriginal beam
Deflected Beam
L
Plywood Faceplate
Bolted connection to frame
Figure 5: Construction and boundary conditions of cantilever beams. Beams were bolted to a
frame (left side of graphic) and a close-fitting plywood faceplate was fitted around
the beam to eliminate vertical translation and closely approximate a cantilevered end
condition.
3 Final Design and Results
The final theoretical design consists of the five beam configurations with length L = 200
mm. The load equals 1.23 lbs. Material A is acrylic, Material B is HDPE; rather than using
the values we found on the internet, the moduli we used in our calculations were physically
measured from our material (discussed more in Section 4). Based on these numbers and the
calculated beam geometries, the deflection should equal 17.5 mm and the lowest safety factor
against failure is 2.7 (well above our minimum of 2.0). With tolerances of plus or minus .125
mm (.005 inches), the largest error in deflection due to the worst-case tolerance stack-up is
approximately 19% (20.8 mm) and the lowest safety factor came down to only 2.5.
As can be seen in Figure 6, our final fixture was constructed successfully. It also performed
as desired when the load was applied, with no instability problems. All five beams were
manufactured and behave very closely to our predictions. Applying the 1.23 lb force to the
end of all 5 beams resulted in deflections close to the calculated value of 17.5 mm. See Table
1 and Table 2 for comparison of target and measured deflections. Each beam performed
very well considering all of the tolerances and sources of error involved. We believe that our
median percent error of 2.9% was well within our realistic goals for the beams.
(a) Constructed fixture with all five
beams.
(b) Reverse side of fixture, with can-
tilever bolts.
Figure 6: Constructed fixture and beams.
5

7. Table 1: Comparison of calculated and actual (measured) deflection results.
Beam 1 Beam 2 Beam 3 Beam 4 Beam 5
Calculated deflection (mm) 17.5 17.5 17.5 17.5 17.5
Measured deflection (mm) 18 17 19 18 21
Percent Error (%) 2.9 2.9 8.6 2.9 20
Table 2: Mean, Median, and Standard Deviation of results
Target Value Mean Value Median Value Standard Deviation
Deflection (mm) 17.5 18.6 18.0 1.4
Percent Error (%) 0 7.4 2.9 6.7
4 Challenges Faced and Discussion
We encountered numerous challenges during the course of this project. Several are high-
lighted and our approach for overcoming them detailed below.
• Non-ideal material properties - When selecting our two materials, we were looking
for an E/E ratio between 0.2 and 0.3. We used online material properties databases
such as MATWEB to compare different combinations of materials before choosing
to use HDPE and Acrylic and ordering our stock materials. We suspected that the
material properties would not match our ideal values, so we decided to physically test
our materials upon arrival to identify their Young’s Moduli. We found average values
for the moduli by using Equation 5 and a deflection test in which we measured the
deflection after clamping the beams at various lengths and applying various amounts
of weight. A comparison of values found online and measured values is presented in
Table 3 along with their respective E/E ratios. By calculating the actual values, we
were able to account for this challenge before constructing any of our beams. We re-ran
our calculations to find the new dimensions required for common-stiffness beams, and
although our final beam thicknesses are not as drastically different as we desired due
to the higher E/E ratio, our results were not significantly impacted.
E =
PL3
3Iδ
(5)
Table 3: Measured versus Ideal Young’s Modulus for Acrylic and HDPE
Online Measured
Acrylic E = 3.2GPa E = 2.5GPa
HDPE E = 0.8GPa E = 1.1GPa
E/E Ratio 0.25 0.43
6

8. • Tolerances of manufacturing equipment - The Shopbot CNC router we used did not
have a very accurate depth measurement. Consequently, we had to fix our beam to
the table, perform a conservative pass to get the thickness close to our desired value,
remove the beam, measure the actual thickness, and make repeat adjustments to get
to the desired value. It took us about four times for each beam to get within our
desired final tolerance, and in all cases the machine did not hit the desired thickness
on the first pass. If we were to re-do the project, we would potentially use a CNC mill
or manual mill to speed up the process and get beter tolerances on our manufactured
beams.
• Residual stresses due to machining - We used a bandsaw to cut the HDPE bar to
provide stock materials for our beams. Upon removal of the beam from the saw, we
noticed a permanent curvature in the once-straight rod. See Figure 7 for a photo of
the beam after cutting. To reduce the curvature, we glued the portions of the beam
that were affixed to acrylic such that the curvature was flattened out and close to
zero. However, this likely contributed to error in the deflection of the beam. Moving
forward, we would consider annealing the beam or loading it such that it creeps back
to being straight after cutting it with the saw.
Figure 7: HDPE after cutting on the band saw. Note positive curvature.
5 Conclusion
In conclusion, we successfully built a demonstrator to show the concept of equivalent stiffness
in two-material beams. Not only did we learn much about multiple material beams, but we
learned a great deal about the importance of manufacturing accuracy with our beams, and
also much about the difficulty of manufacturing a design to match ideal values. Our final
beams have stiffness values within about 10% on all but one beam. We are pleased with our
demonstrator and are confident that it can be used to effectively demonstrate the concept
of multi-material beams.
7

9. 6 Appendix A: Calculations
6.1 Calculating unknown dimensions for each beam
6.1.1 Beam 1
Values of b and h were chosen for Beam 1 based on available materials and engineering
judgement.
6.1.2 Beam 2
Measured from a zero datum at the bottom of the beams,
yna =
EA
h
2
bBh + Eb
3h
2
bA
EAbAh + EBbBh
=
EBhbB + 3EAhbA
2(EAbA + EBbB)
(6)
Now, let
N =
EBbB + 3EAbA
EBbB + EAbA
(7)
so
yna = N(
h
2
) (8)
and the stiffness, S = (EiIi), with I about the neutral axis, is
S = EA
bAh3
12
+ bAh(
3h
2
− N
h
2
)2
+ EB
bBh3
12
+ bBh(
h
2
− N
h
2
)2
(9)
We used an online calculator to solve for bA, but the result was ridiculously long, so we
used the built-in Excel Solver to calculate bA.
6.1.3 Beam 3
By symmetry, yna is at the midpoint of the beam. Using the parallel axis theorem,
S = EA
bh3
A
12
+ 2EB
bh3
B
12
+ bhB
hA + hB
2
2
(10)
Solving for hB,
hB = 3
−EAbh3
A + EBbh3
A + 12S
8bEB
−
hA
2
(11)
8

10. 6.1.4 Beam 4
Measured from a zero datum at the bottom of the beams,
yna =
Eb
h
2
hb + EA
3h
2
hb
2hb(EA + EB)
=
EBh + 3EAh
2(EA + EB)
(12)
Now, let
M =
EB + 3EA
EB + EA
(13)
so
yna = M
h
2
(14)
and the stiffness, S = (EiIi), with I about the neutral axis, is
S =
EAbh3
12
+ EAbh
3h
2
− M
h
2
2
+
EBbh3
12
+ EBbh
h
2
− M
h
2
2
(15)
Solving for h yields
h = 3
12S
b(3EAM2 − 18EAM + 28EA + 3EBM2 − 6EBM + 4EB
(16)
6.1.5 Beam 5
Due to symmetry, yna is at the center of the beam.
S = EA
bh3
12
(17)
Solving for h,
h = 3
12S
bEA
(18)
9