12-100 Introduction to Civil and Environmental Engineering
Project 3
Bridge over Troubled Waters
Final report
Group 11
Klein, Andrew J.
Strømme, Torstein J.
Jang, Yoon Jae
Abedian, Alexander A.
Luo, Siyun
McHugh, Alexander G.
Contents
This report consists of two main parts: One part concerning the plank problem, and one part
concerning the cardboard bridge. Most of this report is focused on the bridge problem, which
also was the major component of the project. As our bridge did not withstand the final testing,
this report also includes a portion on those bridges that did pass. This is found within the
cardboard bridge -section of the report.
Executive summary
done
Introduction
Problems addressed
Role of team members
done
done
done
The Plank Problem, results
Our prediction
Observations
done
done
done
The Cardboard Bridge, results
Description of structure
Challenges during construction
Results and observations
done
done
done
done
Bridges that passed
Bridge A: A covered flat bridge (Group 1)
Bridge B: A covered arch bridge (Group 3)
Bridge C: A massive beam (Group 5)
done
done
done
done
Discussion of differences
done
Recommendations
done
Summary
in progress
Conclusion
done
more on weak ends?
can use revision
Alex A
Executive summary
Project “Bridge over troubled water” focused on two major tasks: Calculation regarding the
provided plank and constructing a cardboard structure with budget as well as resource
constraints. The goal of the plank problem was to calculate and predict the failure probability as
the testing subject that weighs approximately 180 pounds reaches the midspan of the plank
provided by the client. Moreover, Client required the position along the plank that has 50% of
failure probability for the testing subject. Same testing subject was used for the second task:
constructing cardboard structure.
The goal of the structure was to build a bridge that would withstand the weight of the testing
subject as it crosses the structure. Unlike the plank problem, the construction of the bridge had
to be approached with thorough discussion and plan as the client requested specific conditions
to be satisfied. First, the constructed structure has to be 60 inches long that would span 48
inches. Furthermore, the client requested the structure to be built within requested time
interval with limited resource and budget of $20. All the demanded conditions were
successfully satisfied, yet the structure failed to provide proper support and collapsed. The
main reason of the failure was that the structure was not balanced in terms of weight
distribution making it to be fragile on both ends.
Introduction
Project 3, Bridge over Troubled Water, is a project that requires group discussions and
coordinated effort in the application of construction engineering principles. The project is
divided into two separate major tasks; the plank problem, and the cardboard bridge
construction. Though different, both these tasks require an understanding of the maximum
bending moment for a material, that is, how much weight a material can hold over a certain
stretch without breaking. A general understanding of how different factors affect this maximum
bending moment is crucial in solving the tasks successfully.
Problems addressed
This report will focus on the differences between predicted and observed performance. In case
of the plank problem, the observation itself will be a trivial affair, whereas the observation of
the failing bridge raises a number of questions we need to address, such as: why it failed, what
made other bridges pass, and how we can make a better bridge next time.
Role of team members
Klein, Andrew J.: Assembly of PowerPoint presentation, writing on the plank problem
Strømme, Torstein J.: Taking photos, compiling and reviewing final report, writing small pieces
Jang, Yoon Jae: Writing executive summary
Abedian, Alexander A.: Writing summary and conclusion
Luo, Siyun: Writing survey of additional cardboard structures
McHugh, Alexander G.: Writing on the cardboard bridge
Aside from these individual efforts, the group as a whole have had discussions and executed
collaborative decision-making throughout the project.
The Plank Problem
For the plank problem the group was required to measure
the breaking loads and maximum bending moments of six
planks made of the same wood material. Using the data
collected the group was to make a statistical analysis to
predict the likelihood of a plank (of the same material as the
test planks) breaking with a particular weight at any given
point and the position along the plank at which there is a
50% chance of failure under the weight of Professor Irving
Oppenheim, approximately 180 lbs.
Our prediction
Photo 1
After testing the plank material, it was determined that the failure stress was equal to 8533.61
lb/in2 and that the standard deviation was 2277.16 lb/in2 (Table 1). The process of retrieving
these numbers is thoroughly described in the preliminary report.
Table 1: Sample Statistical Analysis
Sample mean failure stress (σfailure) [lb/in2]
8533.61
Sample std. dev. failure stress [lb/in2]
2277.16
Using the measured dimensions of the 1”x3” plank, we used equation 1 and 2 to determine
Mmax for the plank (Table 2).
Equation 1
S = bd2/6 [in3]
Equation 2
(σfailure) = Mmax/S [lb/in2]
Table 2: Mean and Std. Dev. Moment Capacity, 1”x3” Beam
Estimated Mmax [lb*in]
1684.10
Estimated Mmax std. dev. [lb*in]
411.61
The next step was to calculate the point along the plank at which there would be a 50%
probability of failure. We labeled this point “x.” In order to make the calculation we made the
assumption that the beam experienced no net torque about points A or B and no net moment
forces at any point. Using the assumptions we were able to set up a system of equations that
allowed us solve for the position x at which the sum of the moments about the cut and the sum
of the torques about point A equals zero (Equations 3 and 4 as well as Figure 1 and Table 5). In
the calculations the moment force found in Table 2 and the given force of 180 lb were used.
Equation 3
Tnet at A = Tccw- Tcw = 0
= Fb*48” – 180 lbs.*x = 0
Figure 1
Equation 4
Σmcut = 0
M-Fa*x = 0
Table 3: 50% Failure Probability Position, 1”x3” Beam
50% Failure Probability Position x (in.)
12.73
This number was sufficient to represent the 50% probability position because the
distribution was assumed to be normal and there should be no assumed deviation from the
“average” plank with characteristics the same as the mean calculations.
A different approach had to be taken to determine the probability of failure at midspan. For these calculation we knew that the position x at which the 180 lb. force would be
applied was 24”, as this is half the length of the chasm. We did not know the sample M Midpoint
(figure A6). Due to the symmetry of the system, we knew that the balancing normal forces at A
and B were each one half of 180 lb, or 90 lb. Using equation 4 we solved for the MMidpoint
necessary to sever the beam. Plugging sample mean into equation 5 along with the σstd. dev. and
estimated M yielded a Z-score which was interpreted by using a Z-score table. The value
obtained from the table was the probability of failure at midpoint for our 1”x3” plank under a
180 lb weight.
Equation 5
Table 6: Z-score and Probability of Failure at Midpoint, 1”x3” Beam
Z-score
1.156
Associated Probability
87.7%
Observations
When Irving Oppenheim actually walked on the plank, it showed that our calculated estimate
was fairly accurate. While he may not have made it exactly 12 inches before the plank broke, he
did not make it halfway across the plank, which would have been 24”. We estimate the actual
distance before the break to be about 10 inches.
The Cardboard Bridge
Part of this project comprises of the construction of a
cardboard bridge of length 60” that can withstand the
weight of a 180 lb professor walking across the bridge.
The cost of the bridge construction should stay within a
Photo 2
budget limit of $20.00. A basic understanding of
engineering theories and creativity were employed in creating a list of alternative ideas for the
cardboard bridge. The bridge constructed turned out to break at the very first step of the
professor, due to insufficient support under the top layer of the bridge.
Description of structure
Figure 3
The chosen bridge design was a rectangular prism built
around two cardboard cross braces, the girders, that each
would cover the entire span necessary. These two pieces
would stand vertically, to maximize depth, thus be able to
carry heavy loads, and would traverse the diagonals of the
bridge, crossing in the middle (see Figure 2). Furthermore,
these crossbeams featured small tabs along their top and
bottom (see figure 3). On top of and beneath the cross
braces laid horizontal cardboard pieces to hold the
Figure 2
Photo 3
structure together and provide an even surface for walking (see photo 3). The aforementioned
tabs were inserted into slits cut in the top and bottom faces and folded down. This support
mechanism was thought to provide adequate lateral support while saving supplies. The
remaining cardboard would be used to help distribute the weight over more of the girders,
while also making sure an object wouldn’t fall “through” the bridge for lack of support. The
complete internal structure is seen in Photo 2.
This was design was chosen for its assumed stability and great depth. Assessing the formula for
failure stress, it seemed advantageous to maximize the depth of the structure as opposed to
the width, for the depth contributed heavily to the theoretical strength. It was felt that the
diagonal main supports would not only provide high moment forces against any loads, but
would resist warping of the structure, also. With all remaining material, other minor braces [see
figure B5] would be cut to shape and inserted at even intervals to create a bond between the
main cross braces and also
Challenges during construction
Two main problems were encountered during the construction and planning of our bridge. This
first was a material problem. The plank appeared weaker than expected when a light test load
was applied. It was also tilted slightly when freestanding and when under load. These problems
were addressed by binding the midsection with tape to make it denser (see Photo 2). Two more
supporting pieces were placed on each side, as well as on the middle of the plank. A total of
eight supporting pieces were added to the bridge. Tape was applied to the bottom of the bridge
in multiple layers in order to make it sturdier and to resist the tensional forces that would be
applied to the bottom piece.
The second main problem was the difficulty in cutting the cardboard with the scissors. We
simply cut harder and more precisely to compensate for the difficulty. The precision was hard
to come by, which was the reason for the bridge being slightly different from the design.
Results and observations
The full structural potential of the bridge was not fully assessed, for it collapsed under the load
within a few inches of the starting ledge. It was observed that the structure warped, twisting
the main support beams and contorting the secondary supports as well. Cardboard is naturally
stronger when force is applied parallel to the plane and weaker when force is applied directly
perpendicular to the face, thus when these vertical beams deformed the structure collapsed,
becoming extremely weak and tearing. It was as if the load
was supported by a few interlocked yet flat strips of
cardboard, which was not sufficient.
The warping phenomenon was exaggerated by the fact that
the main support braces were cut, then interlocked, making
them weaker and dependent on one another. The
interlocking mechanism was certainly weaker than other
methods of adhesion, such as wood glue or packing tape.
There was also a dearth of supports at the terminal ends of
the bridge, meaning that while the center may have been
sufficient to carry the load, the ends would give out in any
case.
Photo 4
These problems could be addressed a number of ways. It is likely that the height of the
structure, while mathematically advantageous, contributed to the flexibility and thus weakness
of the bridge. Because the suppleness of the cardboard is exaggerated over greater distances,
the supports would be generally weaker but more rigid had they been shorter. This resizing
would yield a second advantage, however; allowing more cardboard to be used in other areas.
The surplus could be used to reinforce, both vertically and horizontally, the terminal ends and
the main support braces. Furthermore, the tabbed construction system was relatively weak as
well; the 1” strips could not hold the beams sufficiently to the horizontal faces when under
significant horizontal pressure. Tape, with a very high tensional resistance, would likely prove
more adapt at the job.
Bridges that passed
All in all, only 3 out of a total of 11 bridges built passed the test i.e. they were able to support a
person of weight 180 lbs walking across the bridge. Below is a description for each of the three
bridges that passed, observations on their performance, as well as an analysis of how and why
the engineering designs of these bridges worked.
The three successful bridges employed vastly different concepts of engineering design.
Nevertheless, all three ideas worked, for the strength and rigidity adduced to the structure by
the design was sufficient for supporting the instructor’s weight.
Bridge A: A covered flat bridge (Group 1)
Group 1 built a covered bridge in the form of a
plank. The bridge is thin, compared to the other
bridges built. The interior design of the bridge
resembles a criss-cross pattern as viewed from
the top of the bridge (Figure 4).
Photo 5
Bridge A was, surprisingly, able to support the weight of the instructor on the
bridge for a substantially long period of time, contrary to first impressions of the
slimness and strength of the bridge. The instructor could comfortably stand on
the bridge.
Bridge A was successful because multiple pieces of cardboard were placed in
between the top and bottom cardboard pieces of the bridge (as can be seen
from Figure 4). Also, the small size of the plank meant that there was more
cardboard for Group 1 to place in between the top and bottom of the bridge.
Figure 4
Bridge B: A covered Brown Truss arch bridge (Group 3)
Group 3 built a covered bridge in the form of an arch, with
an interior design similar to that of the Brown Truss
(Figure 5), where members in the bridge are laid out in a
criss-cross pattern as viewed from the side of the bridge.
Photo 6
The Brown truss is also known as an economically viable
bridge; in other words, it uses the least amount of materials, which is a plus, considering the
limited amount of cardboard provided for this project.
Bridge B was successful in supporting the weight of the instructor walking across the bridge.
Towards the end, bridge B did buckle slightly under pressure from the top. Nevertheless, the
instructor had an easy time walking over the bridge, for the bridge was wide.
Bridge B was successful because the criss-crossing of the members placed the members under
constant compression. Also, the shape of the arch forces the members to support each other.
This places the members in further compression, such that the more weight placed on the
bridge, the stronger the structure became. As a result, the additional compression to the
members from the weight of the person walking on the bridge is minimal. This allows the
members to withstand the pressure placed on the bridge by the weight of the person walking
on the bridge.
Figure 5
Bridge C: A massive beam (Group 5)
Group 5 built a bridge in the form of a beam
spanning from one side to another, with stabilizing
supports attached to each end of the beam. Strips
of cardboard were cut out and taped together to
Photo 7
form the beam. The beam is then placed on its side
over the opening, with the corrugated side of the cardboard facing upwards.
Bridge C was successful in supporting the weight of the instructor walking across the bridge,
although the instructor had to balance while walking due to the narrowness of the bridge.
Bridge C was successful because of the way the cardboard strips are placed across the opening.
The basic concept applied in the design of this bridge is this: pressure applied in the middle of a
strip of cardboard placed horizontally over the opening is less rigid and therefore more likely to
bend than if the strip of cardboard were placed vertically over the opening. Taping multiple
strips of cardboard pieces together amplifies the strength and rigidity of vertically placed
cardboard, resulting in a beam that can withstand a great amount of pressure placed on it. Also,
to prevent the beam from swaying side to side should the instructor misplace his weight,
supports are attached to each end of the beam, so that the beam will maintain rotational
equilibrium while the instructor walks on the bridge.
Discussion of differences
Of the discussed bridges, our design is most similar to bridge A. The major difference from that
bridge to ours is that bridge A has evenly distributed support under the cover, whereas our
support is heavily focused on the center of the bridge. Our bridge was also deeper, so we had
less cardboard to distribute under the cover, leading to insufficient support at the ends.
The two other discussed bridges also provided the support at the ends that our bridge lacked,
preventing such failures that our bridge was subject to.
Lessons learned and future recommendations
The primary lesson learned was the functionality of the bridge as a unit; no one piece can exist
on its own, and the individual pieces must help support each other without losing their
individual strength. A second critical lesson learned was that of precision; due to the
awkwardness of the scissors used and the resulting inaccuracy, many pieces were not exactly as
specified in the plans, while the edges were frayed and thus weakened. Glue and tape would
not bond as strongly to these jagged joints, either. Sharper, more exacting tools, such as a boxcutter knife or open knife, would have yielded cleaner cuts, at the risk of safety. Further
precision would be gained from utilizing all the time available for construction; of the three 2hour periods granted for construction, the bridge was built in two sittings, likely at the cost of
some craftsmanship.
A final lesson to be learned from this project was simplicity in size and in concept. Some of the
strongest bridges in the world are suspension bridges, but given the materials presented in this
project, such complex designs would be unfeasible. Similarly, many successful designs were
seemingly simple, but took advantage of the materials’ strengths. These groups created
appropriately sized structures, as well, with just enough surface area to for the load to lie
comfortably, and no more. With the small size, and resulting excess of material, the successful
bridges were exceptionally dense and thus exceptionally strong.
As for any future bridges, they should be constructed in a way such that the slit principle is
unnecessary. And if the principle is to be used, it should be made sure that the slits are not so
tight that they make the structure twist in any direction, causing unstable structures. And most
importantly, it must be made sure that the bridge can support its assigned weight over the
entire stretch, not only in the middle.
Summary
Our final bridge design was what we thought to be the best bridge design option. We had two
cross-sections that we squeezed together with tape in the center to create more support there.
Unfortunately, this design failed under the Professor's weight. It was crushed within the first
step he took.
how we chose ...how we constructed ...what happened
Conclusion
Where he stepped was virtually unsupported. This, in retrospect, was not a smart idea. Even if
the Professor had stepped farther out, he still would have collapsed the bridge. There was one
sufficiently supported section of the bridge and that was in the dead center of the bridge. If we
were given the chance to change our bridge, we would put supports evenly distributed
throughout the span of the bridge. There would be no gaping holes under the top layer of
cardboard. It would be filled with vertically placed cardboard beams. This would significantly
reduce the chance of collapse. With the knowledge that we now have, there is no question we
can now construct a much more stable and all around stronger bridge.