ALEXANDER CUBELLIS

The Fractal Stage and the Iterative Process:

 

Our group implemented the use of an optimization algorithm that would allow us to quickly determine acceptable ranges of values for various factors that would influence the structural integrity of the bridge. Namely our group designed our bridge with the following dfx’s in mind:

 

Design for Moment: Metric (mm⁴), Higher is better—increasing the moment of inertia will allow for the bridge to withstand the rotational force on the beam.

Design for Constructability:  Metric components to piece together, less is better—allowing for the bridge to be constructed in a shorter period of time and have a lesser likelihood of failing due to poor design.

Design to Withstand Shear Failure:  Metric (Force in N), Lower is better—decreasing the likelihood that the bridge will fail in shear at the supports—the location of the highest shear in the beam. To mediate this factor our group tried to use as many diaphragms as possible given the limited material available as the diaphragms enable the shearing load to be dissipated into the web of the beam).

 

Constraints:

• Size of the mat-board (restricts the dimensionality of the bridge with specific restrictions on the minimum length and width of the bridge)

• Quantity of contact cement available for construction (restricts the number of diaphragms that can be designed). As the number of diaphragms increase, so does the quantity of contact cement that can be used.

 

Quantitative Analysis and Reasoning:

The taller and wider the beam becomes so does the I (moment of inertia). However since the I is proportional to h³ and only b¹ to maximize material usage, the beam was designed to be taller than wider. Additionally, since I is inversely proportional to the stress and the angle of curvature as illustrated by Navier’s equation, and theta = M/EI, our bridge was designed to have as high an I-value as possible [1].

 

These physical relationships are summarized in the algorithm to the left. 

 

Key Design Decisions:

1. Implementation of a Pi-Beam design: This design creates a shape with the highest moment of inertia since it allows the user to make the web as tall as possible. As a result our group decided not to use a central column as to eliminate the difficulty in construction and thus improving the constructability of the design.

2. Use of additional shear reinforcement at the supports: Our group decided to allocate extra material specifically for designing the shear reinforcement at the supports as this was the expected location of shear failure.

 

The Dynamic Components: The Effect of Time:

Given the time constraints (each of the group members also had to submit a Praxis assignment the weekend before the bridge assignment was due) our group was forced to perform the entire iterative process in the span of approximately 6-7 hours (the assignment was due the next day). As a result, our group designed a beam that was theoretically constructible given the loading conditions and the material available.  However, our group failed to consider the conditions needed so that the contact cement would adhere appropriately to give the bridge its rigidity. The contact cement requires a flat surface wide enough so that when the glue bonds it will be able to appropriately resist shear stress. Since the width of the mat-board is only 1.27 cm, this does not allow for optimal drying conditions.

 

Since the bridge design was deemed to be no longer viable, our group was forced to then completely alter the design. As a result our group removed one of the flange layers so that the extra area could be used on the web to create the flaps. This then allowed for the glue to bond accordingly.

 

Our Final Design is shown at left.

 

Reflection:

On testing day our bridge was designed to withstand a load of 803 N. When the bridge was actually put to the test it was able to safely support the 400 N load applied by the train but failed prematurely at only 753 N under the point load testing.

 

Images of the where the bridge failed can be viewed in the slideshow above.

 

Interestingly, as is shown in the video to the left, our bridge failed in tension at the support where the point load was applied. Originally, our group did not design for the bridge to fail in tension since tensile failure occurs at 16 MPa and therefore is much less critical than the 4 MPa sheer stress of the matboard. Therefore our group was puzzled as to why this design failed in this way.

Upon analyzing the location where the beam failed in more detail, there is evidence that the failure occured as a result of the glue applied at the flange. Our group was so concerned about making sure that there was ample surface area to apply the glue that we did not pay attention to the method by which the glue was applied. Our group briefly interviewed Professor Evan Bentz, the on-site professor who analyzed the failure of our bridge design. He was quite intrigued by the failure and commented that tensile failure for this type of bridge design was unusual since the tensile strength of the mat-board is significantly stronger than the shear capacity and a bridge of this design should have failed in shear before failing in tension. He mentioned that the failure was “likely caused to due to uneven cement application and as a result caused an uneven distribution of E in the beam. Therefore as the compressive stress applied to the flange increased the adhered portion of the flange and the web began to pull apart resulting in the tensile failure.” The application of the contact cement resulted in the formation of a weak boundary later (WBL) vertically along the interface between the web and the diaphragm and this is what distributed the stress into the web as opposed to the diaphragm. [4]

Comparison to the Winning Design:
The highest load held by any of the bridge designs was just over 2000 N. This bridge was so well constructed that it did not fail under shear stress or due to a design flaw as was common amongst the other designs but actually under the crushing of the material. Their design was quite different from ours in that the group decided to use the central column and three lateral box girders along the length of the beam.  This design was very effective as it allowed for even dissipation of stress and reduced the mid-span deflection of the beam. It is interesting to note that the winning group had a lower I value since the height and cross sectional area of their beam was smaller than ours. However, it appears that the most important design decision was to include the column so as to allow for stress dissipation and to minimize deflection at mid-span.