AEROSPACE 305W STRUCTURES & DYNAMICS LABORATORY

Laboratory Experiment #1

Bending of Selectively Reinforced Beams

January 21, 2015

Kyle Ellingson

Lab Partners: Keagan Boyce, Abigail Brandtmeier, Mitansh Shah, Darren Slotnick

Course Instructor: Dr. Stephen Conlon,

Lab TA: Andrew Goodyear

Abstract

This laboratory experiment compares the effectiveness of both aluminum and composite I-beam

reinforcement methods to increase the stiffness and strength of a beam. By measuring the end

displacement and near-wall strains of an unreinforced aluminum beam, an aluminum-aluminum

reinforced beam, and an aluminum-composite reinforced beam under identical loading patterns with

an LVDT and strain gauges respectively, it is concluded that both reinforcement methods are

effective in reducing strains and displacements for an end-loaded cantilevered I-beam and H-beam. It

should be noted that the composite reinforcement method is significantly more effective in achieving

these goals and much more efficient for aerospace beam bending applications. This conclusion stems

from the composite material’s considerably higher modulus of elasticity and lower density relative to

the aluminum in tested specimens.

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Table 1: Bending Stiffness of Experimental Specimens

I. Introduction

The purpose of this experiment was to study the flexural behavior of beams using both an analytical

and experimental approach in order to explore the benefits of using composite materials for

reinforcement of a beam. A beam is defined as a bar that is capable of carrying loads applied

transversely to its longest dimension. For an efficient aerospace design, one needs to account for the

maximum stresses that a system will be under. Additionally, the goal is to use the minimal amount of

material that allows the design to satisfy the performance requirements including: weight, stability,

strength, and cost. A thorough understanding of design constraints and requirements is crucial to

forming an efficient design. In this experiment, stiffness and strength are closely examined in order

to minimize deflections and internal strains. Specifically, this lab utilizes the study of static flexural

behavior of cantilevered I-beams to determine the relationship of structural deformation to applied

loads and the effectiveness of using high modulus composite materials to selectively reinforce

beams. Beam bending theory gives the following governing equations upon which theoretical values

for displacement and strain have been obtained for the specific loading pattern used in the

experiment:

𝑤(𝑥) =𝑃

𝐸𝐼[

𝑥3

6−

𝐿𝑥2

2] and 𝜀(𝑥) = −

𝑧𝑃

𝐸𝐼[𝑥 − 𝐿]

Where E is the modulus of elasticity; I is the area moment of inertia; P is the loading force; x is the

distance of the relevant sensor to the rigid wall; L is the length of the beam. The use of reinforcing

doublers on I-beams has a significant effect on the flexural rigidity of a beam. Flexural rigidity, or

bending stiffness, is equal to the product of the modulus of elasticity and the area moment of inertia.

The bending stiffness values for the specimens used in the experiment are given below in Table 1.

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II. Experimental Procedure

Figure 1 illustrates the experimental apparatus. A structural beam was clamped to a stiff test fixture

at one end to simulate a cantilevered condition. At the other end, a hanging weight supplied a known

external load. The deformation of the beam was measured using a strain gage mounted on the upper

surface near the root; a LVDT was used to measure the transverse displacement at one point along

the beam. The strain was read using a strain box and recorded with LabView.

Three specimens were used in this experiment: 1) an aluminum irregular T-beam; 2) an aluminum

irregular T-beam reinforced with aluminum caps; 3) an aluminum irregular T-beam reinforced with

graphite/epoxy caps. Figure 2 and Table 1 together summarize some relevant characteristics of the

specimens. The gage factor for the strain gage was 2.125, and its nominal resistance was 350 ohms.

The base aluminum beam material had a Young's modulus of 10.0 x 106 psi, a yield stress of 40 ksi,

and a density of 0.10 pounds per cubic inch. The composite had a longitudinal Young's modulus of

19.8 x 106 psi and a density of 0.0506 pounds per cubic inch.

Figure 2: Important Specimen Dimensions

Figure 1: The Experimental Apparatus

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Once all sensors were connected, on, and zeroed, the sensor distance measurements (LS, LD, LF) were

recorded. The sensor distance measurements recorded for each specimen loading arrangement are

noted below in Table 3.

With the LabView data recording program running, five data points were recorded at each of the load

conditions. The forward loading conditions ranged from 0.0lb to 5.0lb in increments of 0.5lb. Then

each specimen was backward loaded from 5.0lb to 0.0lb in 1.0lb increments. To correct recorded

data, existing strain in gauges prior to loading was subtracted from measured strain data. Once this

data was recorded and saved, the specimen was unclamped and removed. This process was carefully

repeated for each remaining specimen.

Table 2: Relevant Specimen Properties

Table 3: Apparatus Sensor Distance Measurements

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Table 4: Theoretical Displacement and Strain Values for I-Beams

Table 5: Averaged Measured Displacement and Strain Values for I-Beams

% 𝑬𝒓𝒓𝒐𝒓 =|𝑬𝒙𝒑. −𝑻𝒉𝒆𝒐𝒓. |

𝑻𝒉𝒆𝒐𝒓.∗ 𝟏𝟎𝟎

III. Results

Both the theoretical and averaged experimental data of the displacements and strains are presented

below in Tables 4 and 5.

Table 6 shows the percentage error of displacement and strain between the theoretical data and the

experimental data (of the forward loading).

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Table 6: Percentage Error of Displacement and Strain Data

Figure 3: Displacement vs. Force of Al, - Specimen

Figure 4: Displacement vs. Force of Al, Al Specimen

Figures 3, 4, and 5 show the discrepancies between the theoretical and experimental data for

displacement.

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Figure 5: Displacement vs. Force of Al, C Specimen

Figure 6: Strain vs. Force of Al, - Specimen

Figure 7: Strain vs. Force of Al, Al Specimen

Figures 6, 7, and 8 show the discrepancies between the theoretical and experimental data for strain.

(μS)

(μS)

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Figure 8: Strain vs. Force of Al, C Specimen

IV. Discussion

On comparing the theoretical and experimental results for beam displacement, there is a clear

inconsistency. The actual behavior of the unreinforced aluminum beam most closely resembles the

predicted behavior compared to the other specimens upon inspection of Figures 3, 4, and 5. Table 6

reports displacement percentage error values ranging from 11.0% to 63.4% for the unreinforced

aluminum beam. The displacement errors range from 144.2% to 190.6% and 229.1% to 276.6% for

the aluminum-aluminum reinforced beam and the aluminum-composite reinforced beam

respectively.

On comparing the prediction accuracy for strain, there is a relatively smaller range of inaccuracy.

Similar to displacement predictions, the experimental behavior of the unreinforced aluminum beam

most closely resembles the predicted strain values throughout the course of loading. Figures 6, 7, and

8 provide a graphical representation of the theoretical and experimental strain behaviors for each

specimen. Referring to Table 6, the unreinforced aluminum beam strain behavior hosted a range of

percentage error of only 7.2% to 14.8%. The percentage error for the strain behaviors of the

aluminum-aluminum reinforced beam and the aluminum-composite reinforced beam ranged from

42.8% to 55.6% and 72.9% to 85.7% respectively.

Discrepancies between the predicted displacement and strain behaviors and the experimentally

determined values arise from both systematic and precision errors that occurred throughout the

experiment. Though efforts to limit both types of errors were taken, they have a considerable impact

on the accuracy of the predicted beam behaviors regardless. Possible systematic or bias errors come

(μS)

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from the experimental apparatus equipment used. The sensor distance measurements may have been

skewed by an inadequate measuring device (measuring tape) and an inability to read measurements

from an ideal point of view. Also, the strain gauges used during the experiment had accumulated a

level of residual strain from extensive prior use that, though thought to be accounted for, may have

contributed to the inconsistencies in strain behavior. Precision or random errors that may have

occurred stem from the values of Young’s modulus used for both the aluminum and composite

materials during the formation of the behavior predictions. These generally accepted values of E are

obtained as an average of many past experiments and therefore lend themselves to a certain level of

inaccuracy because the materials the specimens were composed of are not ideal and do contain

inconsistencies on a microscopic level. The observation that the highest percentage error values were

associated with the aluminum, composite reinforced beam could be due, in part, to the imperfect

lamination of the doublers with the aluminum. The assumption of ideal material behaviors that were

used in the derivation of theoretical strain and displacement values is therefore violated. As shown in

Table 5, the strain and displacement values corresponding the same load weight for forward and

backward loading differ. This phenomenon is known as hysteresis and occurs because real materials

act as nonlinear springs under loading and unloading. The dependence of the output of the system is

not only based on its current input, but also on the history and internal state of each specific

specimen. Lastly, another assumption used in the derivation of theoretical values for both strain and

displacement that was violated during the experiment was that the cantilevered beam was not

technically attached to a rigid wall, but rather to a clamp that closely resembles an ideal rigid wall

connection.

On inspection of the percentage error ranges in Table 6, it is evident that the experimental strain

results correlate more closely with the theoretical behavior predicted. As previously stated, the

assumptions the experiment violated change the actual boundary conditions of the beam under which

the predictions were derived. It seems that the displacement behavior of the end of the beam is more

sensitive to these boundary conditions than the strain behavior near the rigid connection.

Both the theoretical and experimental data suggest that reinforcement of I-beams is highly effective

in increasing the stiffness of beams and beam applications. Using values from Table 5, at the highest

loading condition of 5.0lb, the aluminum-aluminum reinforced beam and the aluminum composite

reinforced beam reduced displacements relative to the unreinforced aluminum beam by 34.7% and

46.8% respectively. The strength of the beam was also increased significantly with the addition of

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both aluminum and composite reinforcements. Using experimental results from Table 5 again, it is

determined that the strains at the highest loading condition relative to that of the unreinforced

aluminum beam decreased by 35.8% and 49.6% respectively. While both methods of reinforcement

are evidently effective in increasing the stiffness and strength of a beam, it is apparent that the

composite reinforcement is significantly more effective. This result arises from the fact that the

composite used in the experiment has a significantly higher modulus of elasticity than the aluminum

used. Additionally, the use of composites for reinforcement has the added benefit of a more

lightweight structure because it has a much lower density than aluminum.

It should be noted that Figures 3 through 8 contain experimental results of each specimen being

loaded in an H configuration rather than the standard I configuration. Upon evaluation, these

measurements taken are not particularly useful as a beam loaded in this condition is not practical.

The strain gauge was located in on the horizontal web of the H-beam which does not obtain the

maximum strains that should occur under loading. Figures 6, 7, and 8 show a minimal range of

strains induced throughout loading. While the H-beam configuration increases the area moment of

inertia, a beam under this loading would certainly undergo localized column buckling of its vertical

flanges resulting in catastrophic failure rather than local failure before considerable beam

displacement occurred.

V. Conclusions and Recommendations:

In this experiment, the stiffness and strength qualities of three different beam specimens were

analyzed through the collection of displacement and strain values over the course of identical loading

patterns. The three specimens used were an unreinforced aluminum beam, an aluminum-aluminum

reinforced beam, and an aluminum-composite reinforced beam. Using basic beam bending theory

and MATLAB, theoretical displacement and strain behaviors were calculated for each specific

specimen. With strong efforts to model the loading condition and boundary condition assumptions

used in the derived theoretical behavior models in the experimental apparatus, the experimental data

was compiled and graphed through the use of LabView application software and MS Excel.

Inconsistencies between the theoretical and experimental behaviors of each beam are accounted for

by both systematic and precision errors that occur both in the experiment and predictive modeling.

Results show significant improvements in stiffness and strength through the use of both aluminum

and composite reinforcement. Of the two methods, it is concluded that composite reinforcement is

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more effective in aerospace beam applications as the combination of a composite’s low density and

high modulus of elasticity can provide a system with substantially improved strength and stiffness

while reducing the overall weight of a system.

For a more conclusive and accurate repetition of this experiment, there are several procedural

changes that can be made. By collecting more instance of strain and displacement at varying sensor

locations along the beam, one can more confidently and completely improve the understanding of

complete beam loading behaviors. Another recommendation would be to test more specimens

reinforced with varying composites to better understand the role of composite properties in beam

reinforcement. To increase the precision and accuracy experimental data, it is recommended to use

more precise measuring devices for the gauge and sensor distance values. Lastly, for a more idealistic

behavior of loaded I-beam specimens, it is recommended to use specimens with limited prior loading

to reduce the effects of hysteresis and to use new strain gauges with each test to limit the effect of

gauge creep.

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Appendix: MATLAB Script Used For Theoretical Data

clear

clc

E=10e6;

L=8.75;

I_1=0.009234;

x_1=0.1875;

d_1=0.1875;

I_2=0.021845;

x_2=0.0625;

d_2=8;

I_3=0.0341371;

x_3=0.0625;

d_3=7.625;

z_1=0.239;

z_2=0.284;

z_3=0.277;

EI_3=341371;

format long;

for p=[0:0.5:5];

w_1=(1/(E*I_1))*(p*(d_1)*(d_1)*(d_1)/6-p*L*(d_1)*(d_1)/2);

end

for p=[0:0.5:5];

w_2=(1/(E*I_2))*(p*(d_2)*(d_2)*(d_2)/6-p*L*(d_2)*(d_2)/2);

end for p=[0:0.5:5];

w_3=-1*(1/(EI_3))*(p*(d_3)*(d_3)*(d_3)/6-p*L*(d_3)*(d_3)/2);

end

for p=[0:0.5:5];

epsilon_1=-z_1/(E*I_1)*(p*x_1-p*L);

end

for p=[0:0.5:5];

epsilon_2=-z_2/(E*I_2)*(p*x_2-p*L);

end

for p=[0:0.5:5];

epsilon_3=-z_3/(EI_3)*(p*x_3-p*L);

end