MECH 312 Final Assignment Fall 2015
You have a choice of a last exam or a project for 100 points. The exam would occur on December 11 during class time. The project report is due the same day.
The object of this assignment is to model and verify the motion of a double pendulum like the ones shown below. You can suggest another. The equations have been derived and solved before so I urge you to do a bit of researching.

Steps:
1. Form a team of three students. Try to choose one team member who you have not worked with before.
2. Choose a 2 degree of freedom case.
3. Derive the equations of motion. Research your case.
4. Solve for the natural frequencies and mode shapes.
5. Build one. Keep it simple and inexpensive, something you can easily make at home.
6. Use your equations to specifically model the behavior of your system.
a. Use the actual masses and lengths to calculate the natural frequencies and mode shapes
b. Give your pendulum an initial displacement in the shape of your mode shape 1, calculate and plot the response
c. Same as “b” with mode shape 2
d. Give both masses an initial displacement, calculate and plot the responses.
7. Repeat 6b through 6d experimentally to approximately verify the first and second natural frequencies, mode shapes, and motion of the system.
8. Prepare a report in article format. Be sure that you have:
a. Abstract
b. Introduction
c. Theoretical Development
d. Results
e. Discussion
f. Citations
Example Report

Concrete Design – Lab Project 2

Names
Abstract
Four concrete beams with varying degrees of reinforcement were tested and the nominal and cracking moments were observed at failure. Each beam had dimensions of 6”x10”x96”. Beam 1 had one #3 bar and no stirrups. Beams 2 and 3 had two #4 bars. Beam 4 had two #5 bars. Beams 3 and 4 had #3 stirrups spaced at 4”. The observed nominal and cracking moments for each beam were found to be greater than the theoretical moments. The observed cracking moments were much closer to the theoretical cracking moments than the observed and theoretical nominal moments. Beams 1 and 2 were without stirrups and failed due to shear cracking. The presence of stirrups prevented failure due to shear forces in beams 3 and 4. These beams failed due to concrete crushing in the compression zone.
Introduction
Four concrete beams were under-reinforced with different amounts of reinforcement, so the behavior of the beams could be studied, as they were loaded to failure. The behavior of the beams with the presence of stirrups was compared to the beams without stirrups. The observed nominal and cracking moments were compared to the theoretical moments.
The main function of stirrups in concrete beams is to prevent failure due to shear forces (McCormac, 2014). Without stirrups the beams will develop diagonal cracks along the length of the beam and fail due to shear forces. With stirrups the beams will fail when the concrete in the compression zone is crushed. A beam that has a balanced steel ratio is one for which the tensile steel will theoretically just reach its yield point at the same time the extreme compression concrete fibers attain a strain equal to 0.003 (McCormac, 2014). The text provides guidelines and design methods to achieve ductile beam behavior. It also describes methods for calculating nominal and cracking moments in the beams when loaded to failure.
The calculated cracking moment capacities in reinforced concrete members seem to be quite accurate (McCormac, 2014). For nominal moments, the percent difference of the observed and theoretical values decreases as the maximum force is increased. For cracking moments, the percent difference between the observed and theoretical values increases as the maximum force is increased.
The theoretical nominal and cracking moments of the four beams were calculated and compared to the observed values in this study. The behavior of the beams at failure without stirrups was compared to the behavior of the beams at failure with stirrups.
Experimental Methods
To observe the added strength due to presence of stirrups, four beams were made, each having dimensions of 6”x10”x96”. Beam 1 was made with 1 #3 bar and no stirrups. Beam 2 had 2 #4 bars and no stirrups. Beam 3 had 2 #4 bars and was made with stirrups placed roughly every 4”. The stirrups were #3 bars. Lastly, Beam 4 was made with 2 #5 bars and with stirrups. To make each beam, concrete was poured into pre-made forms and compacted with a steel rod every one-third interval as the depth increased.
After pouring each beam, a trowel was used to flatten the wet concrete surface of the beam. The four beams were allowed to reach a fully cured condition. The 28th day is the accepted length of time for the concrete to cure to maximum strength. In the lab each beam was loaded to failure with a hydraulically operated machine, called a Riehle hydraulic press. All were placed and loaded as simply supported beams and loaded to failure. A load cell was used to measure the resisting force of the beam. Peak forces and the value of the force at the appearance of the first flexure crack were observed and recorded.
Theoretical Discussion
Below are the pertinent equations used in calculations for the nominal and cracking moments of the test beam:
The theoretical cracking moment may be calculated using expression (1) from McCormac (2009).
(1)
where
(1a)
Ig=Gross moment of inertia of the cross-section
(1b)
yt= Distance from the centroid to the bottom of the cross-section
(1c)
fr= Tensile Strength
fc’= Compressive strength of the bottom layer of the beam
b= thickness of beam
h= height of beam
The theoretical nominal moment may be determined using expression (2) from McCormac (2009).
(2)
where
fy= yield strength
?= reinforcement ratio
F= strength reduction factor

The observed moments were determined using expression (3).
(3)
where
M= observed cracking or nominal moment
w= the weight per unit length (60#/ft for this case)
L= distance between simple supports
Results
Table 1 shows the top and bottom strengths of the beam retrieved from the cylinder samples of the concrete mixture. Table 1 also shows the loads applied when the first crack appeared in each beam, and the maximum load applied to each beam. Table 2 shows the observed and calculated nominal moments, as well as the observed and calculated cracking moments. In addition, the percent difference for each beam was calculated and tabulated as seen in Table 3. The observed nominal moments were determined using the top layer strength of the beams in Table 1. The observed cracking moments were determined using the bottom layer strength of the beams in Table 1. Figure 1 shows a graph comparing the results of the observed and calculated nominal strengths and cracking moments of each beam. Figures 2 and 3 show photos of beams 2 and 4 at failure. The cracks in beam 2 are propagated out due to lack of stirrups. The cracks in beam 4 are relatively vertical because the stirrups prevented the propagation of cracking.
Table 1 Top and Bottom Beam Strengths and Loads Applied to Beams
Beam # fc' Top
(ksi) fc' Bottom
(ksi) Force @ Cracking
Pcr (Kips) Maximum Force
Pmax (Kips)
Beam 1 4.960 4.720 2.150 3.300
Beam 2 6.160 6.940 3.100 11.500
Beam 3 5.760 5.250 2.300 11.380
Beam 4 6.320 5.120 3.350 12.900
Table 2 Observed and Calculated Nominal Strengths and Cracking Moments
Beam # Mn Observed
(Kips in) Mn Calculated (Kips in) MCR Observed (Kips in) MCR Calculated (Kips in)
Beam 1 84.96 37.43 57.36 51.53
Beam 2 281.76 160.83 80.16 62.48
Beam 3 278.88 157.85 60.96 54.34
Beam 4 315.36 210.51 86.16 53.67
Table 3 Percent Difference of Observed and Calculated Nominal and Cracking Moments
Beam # Nominal Moment, Mn Cracking Moment, Mcr
Observed
(Kips in) Calculated
(Kips in) Percent Difference Observed
(Kips in) Calculated
(Kips in) Percent Difference
Beam 1 84.96 37.43 126.98% 57.36 51.53 11.31%
Beam 2 281.76 160.83 75.19% 80.16 62.48 28.30%
Beam 3 278.88 157.85 76.67% 60.96 54.34 12.18%
Beam 4 315.36 210.51 49.81% 86.16 53.67 60.54%
Figure 1 Graph of Observed and Calculated Beam Results

Figure 2 Beams 2 at Failure

Figure 3 Beams 4 at Failure
Technical Discussion
By calculating the theoretical moments, the accuracy of the observed nominal and cracking moments can be determined. In all beam cases, the observed nominal and cracking moment were greater than the calculated or theoretical moments. Looking at the progression of beams, it is determined that beams containing stirrups (beams 3 and 4) had fairly high nominal and cracking moments compared to the beams that contained no stirrups (beams 1 and 2).
Another element of the test results is the correlation between percent difference and beam number. For the nominal moments, the percent difference between observed and calculated values decreased as beam number and maximum force increased. Conversely, for the cracking moment, the percent difference between the observed and calculated values slightly increased as the maximum force increased. First, in order to understand the basis of this observation, it is noted that the observed moment calculations were collected when the first flexural crack became noticeable and because of this, error in moment calculations become relevant. Precision in this case, is determined by the time at which moment readings are taken in relation to the actual time of flexural cracking. This is what causes the percent difference in observed and theoretical moment calculations. If flexural cracking and the time it takes to become noticeable correlate to the type of beam being tested, in the cases presented in this laboratory test, beams with stirrups as well as beams without stirrups do give insight to the beam’s projected moment capacities and cracking properties.
When identifying the cracking properties of the tested beams in this laboratory test, it was determined that beams 3 and 4, for the most part, prevented diagonal tension cracking along the length of the beam. These beams failed due to concrete crushing in the compression zone, given the main function of stirrups in concrete beams is to prevent failure due to shear forces. Contrariwise, beams 1 and 2 had noticeable diagonal cracks along the length of the beam and failed due to shear forces.
Conclusion
It was determined that the addition of stirrups in concrete beams increased the beams resistance to failure due to shear cracking. Additionally the beams with stirrups failed due to compressive forces, in the compressive zone, which was expected. The beams that did not contain stirrups failed because of shear forces which caused diagonal cracking along the length of the beams. The computations for the cracking moments of the beams gave a reasonable prediction of the actual beam values, while the computed nominal moments for the beams weren’t as accurate. As beam number and maximum force increased, the percent difference between the observed and calculated nominal moments decreased greatly. The percent difference between the observed and calculated cracking moments followed a different pattern, and increased, as the beam number and maximum force increased.
Literature Cited
McCormac, J. C., “Design of Reinforced Concrete 9th edition,” Wiley, New York, 2014, print.
Appendix
Table 4 illustrates the given data in Table 1 converted into units applicable for the calculations on the test beams
Table 4 Top and Bottom Beam Strengths and Loads Applied to Beams (Different Units)
fc' Top
(psi) fc' Bottom
(psi) Force @ Cracking
Pcr (lbs) Maximum Force
Pmax (lbs)
4960 4720 2150 3300
6160 6940 3100 11500
5760 5250 2300 11380
6320 5120 3350 12900
Table 5 shows the computations of the gross moment of inertia, the tensile strength, and the distance from the centroid to the bottom of the cross-section.
Table 5 Gross Moment of Inertia, Tensile Strength and centroid distance of each beam
Beam # Ig fr yt
Beam 1 500 515.3 5
Beam 2 500 624.8 5
Beam 3 500 543.4 5
Beam 4 500 536.7 5
Table 6 shows the areas for each of the beams’ steel reinforcement.
Table 6 Steel Reinforcement Areas
Beam 1 (1) #3 Bars (in2)= 0.11
Beam 2 & 3 (2) #4 Bars (in2)= 0.20
Beam 4 (2) #5 Bars (in2)= 0.31
Table 7 shows the yield strengths and reinforcement ratios for each beam.
Table 7 Yield Strengths and Reinforcement Ratios
fy (ksi) ?Actual ?min ?max
Beam 1 55 0.002619048 0.0036 0.0200
Beam 2 67.5 0.00952381 0.0030 0.0239
Beam 3 67.5 0.00952381 0.0030 0.0230
Beam 4 60 0.014761905 0.0033 0.0240
Figure 4 shows a snapshot of the calculations performed to get the nominal and cracking moments for the test beams.
Figure 4 Nominal and Cracking Strength Calculations
Figure 5 shows a snapshot of the calculations performed to get the nominal and cracking moments for the test beams continued.
Figure 5 Nominal and Cracking Strength Calculations Continued