##### Fish Fertilize Trees

By Maya C. Lemaire

### 2018

### 2018

### Investigating Truss Structured Bridges: Which type is most tolerant?

Abstract

Whether it be the sturdy Baltimore truss, the reliable Bailey truss, the aesthetic Bowstring truss, the basic Pratt truss or the classic Warren truss, all truss bridges have one similar property: they transfer the load to a much wider area through the use of a truss as its main element. A truss is simply a structure of connected elements to form triangular units. Therefore, it is easy to imagine that a relatively complex structure like a truss bridge would react to a load respective of its design.

Hence, when I set out to investigate the truss bridge which was the best, I needed to come up with a definition for 'best' so I decided that, in terms of this research article, the best bridge will be the bridge that puts the bars under the least stress under a load of 5000 kN.

I found that the Parker bridge was the best out of the three I investigated (the Warren truss, Pratt truss, and Parker truss). It had an average stress of 10394182.37 Pascals in tension and 14393741.06 Pascals in compression with the cross-sectional area of each bar being 0.25168m2 due to the maximum material amount being 30 meters cubed per bridge.

Introduction

Truss bridges are one of the most popular and recognizable bridges on the planet. From the bridge over the River Kwai in Kanchanaburi - Thailand to the Hart bridge in Florida – USA, a great majority of people will be able to name a reputable truss bridge. There are also numerous types of truss bridges, each distinct and individual from any other type. In the middle of the first of my two weeks at the Oxford Scholastica Academy, I chose my project that was to last the majority of the course:

Investigate how different types of truss bridges react to the same force acting upon the center of the bridge.

This involved researching different truss bridge designs and their respective strength and using various aspects of mathematics to calculate the forces acting upon the bridge. I started by designing three different truss bridges of the most divergent types, namely the Warren, Pratt, and Parker truss bridges. Next, I calculated the forces acting upon them. I then had to convert the forces to stress in order to find the bridge that would be the most effective. However, I had to limit the amount of materials I had to work with 30m3 of material in order to maintain a fair test. In addition to this, each bridge also would have to be 30m long so that the resulting moment would be the same. In this article, I will reveal and explain my findings on the truss structure that is most effective and the material it should be made out of.

Literature Review

In 2017, a team of students at the Al-Mansour University College in Iraq designed and analyzed a truss bridge as partial fulfillment of the requirements for a B.Sc. in Civil Engineering (link in references). However, this research paper will be focusing on multiple designs of truss bridges and how they all react to the same load rather than focusing on formulating and analyzing one design of a truss bridge like the team at Al- Mansour University College did.

In 2008, Baylor University's Faculty of Engineering and Computer Science produced a presentation on the fundamentals of calculating forces of tension and compression on a truss structure (link in references).

Whilst this research article does involve a lot of calculations regarding forces and vectors, what sets this article aside from Baylor University's presentation is the fact that this article is more about exploring different truss structures and actually goes on to convert the forces to stress for a more of an in-depth analysis of which truss structure is the best.

Method

The independent variable (the variable that was changed) of my investigation was the structure of the bridge, and the dependent variable (the variable measured) was the forces acting upon the bars and the average stresses (under compression and tension). The control variables include the amount of material from which to make the bridge, the length of the bridge (30 meters) and the position and weight of the force applied (5000kN in the center of each bridge).

First, I was given a general equation which the bridge should have to follow in order to have a higher chance of success. The general equation was:

B + R - 2J = 0

Where:

B - the total amount of bars used to construct the bridge.

R - the total amount of reaction forces acting upon the bridge. R is usually three because on most bridges, there is a fixed joint on one end and a roller joint on the other and since the fixed joint has two reaction forces (one acting vertically and the other horizontally) and the roller joint has only one reaction force acting vertically (because there is no more horizontal reaction force due to the roller joint’s ability to move horizontally), the total amount of reaction forces sums up to three.

Therefore:

B + R - 2J = 0

Can be rearranged to:

R = 2J – B

And since R ideally retains a value of 3:

3 = 2J – B

Below are diagrams of the bridges I designed: *note that all bridges are 30 meters in length.

The Warren truss bridge I designed has an ideal number of bars and joints because when we plug it into the equation, we get:

R = (2 x 13) – 23 R = 26 – 23 R=3

The Parker truss bridge I designed also has an ideal number of bars and joints because:

R = (2 x 12) – 21 R = 24 – 21 R=3

Furthermore, the Pratt truss bridge I designed also has an ideal number of bars and joints because:

R = (2 x 12) – 21 R = 24 – 21 R=3

Statics Calculations

In this section of the article, a demonstration will be provided of the mathematics used in order to find the forces acting upon each joint and bar.

It is explicit that the bridge is symmetrical, hence, we only need to solve for one half. Since we know that the moment around joint 1 is equal to the force applied at the center (W) multiplied by the length of the element connecting the force and the joint (L), we get:

Hence the moment at both points 1 and 2 is 50 kN.

We know that the reaction forces in the x and y directions sum to 0 since the bridge doesn’t move. Hence:

Results

In this section, I will present the forces on each element of each truss bridge I designed, and I will present the thickness of each element according to the amount of material I set out to use.

However, we have to keep in mind that the minimum forces are 0 due to the fact that this is a redundant structure. This means that it has more supports than necessary as a failsafe if one of the beams or joints fail. Elements 13 and 17 have no forces acting upon them due to the fact that the load is taken, instead, by elements 7,18 and 21,12 respectively.

The Pratt bridge is also a redundant structure due to the fact that elements 8 and 12 do not bear any load. Instead, they are supported by elements 7,18 and 21,17 respectively. However, should any one of these beams fail, elements 8 and 12 will bear the respective loads as a replacement.

Discussion

As seen from the bar chart above, the Warren truss had the highest stress out of all of the bridge types. I think that this is due to the fact that elements 3 and 4 have little support and so they are forced to bear most of the 5000kN load. Also, element 21 bore a jaw dropping 32534944.61 pascals of stress in compression which was mainly due to the fact that it was situated directly overhead the load and was attached to 2 other elements which were attached to the point of action of the load. In terms of average stress in compression, the Pratt bridge had a relatively high stress (17774495.09N) which is caused by the inward pointing diagonal supports pulling on the upper horizontal elements. Although the Parker truss also had similar supports, the upwards sloping elements along the top of the bridge counteract the compressive force.

I think that it was the Warren’s simple structure that had no supports that let it down. As seen in the other two designs, they had some form of support beams that point towards the center, the Warren however, did not. On the other hand, in a more realistic situation, the load would not be applied to the center of the bridge and would be more evenly spread in intervals.

This makes the Parker truss bridge the best choice out of the three because of its relatively low stress under tension and compression. With the lowest maximum stresses and the lowest average stress under compression, the Parker bridge is proved to be the most efficient structure at distributing the 5000kN load which I think is owed to the unique arrangement of the elements at the top of the bridge. Not only does it counteract the compressive force caused by the load and the diagonal supports, but it also allows for elements 13 and 17 to be zero force members, making the structure redundant.

Conclusion

Below is a table of results from my investigation:

As seen from the table above, by the definition of ‘best’ presented at the start of the article, I conclude that the best bridge is the Parker truss bridge.

It has the lowest:

-Maximum stress under tension - 16224344.41 Pascals

-Average stress under tension - 10394182.37 Pascals

-Maximum stress under compression - 21273009.38 Pascals

And the Parker bridge also had the second lowest average stress under compression (14393741.06 Pascals).

Recommendations for future recreations of this investigation

- Have a more evenly distributed load along the bridge to make the investigation more realistic.

- Use a 3D model of the truss bridges to conduct the experiment on for a more realistic investigation.

- Investigate more truss structures (Baltimore truss, Bailey truss, Bowstring truss... etc.).

- Investigate materials that could be used (now that we know the stress, we only need to study Young’s modulus for an idea of the materials that could be used)

Bibliography

This article was written with courtesy of the following websites used solely for research purposes:

- Unknown, (2019). Truss Bridge - Types, History, Facts and Design. History of Bridges. [Online]. Available: http://www.historyofbridges.com/facts-about-bridges/truss- bridge/

- Unknown, (2019). Truss Bridge. Wikiwand. [Online]. Available: https://www.wikiwand.com/en/Truss_bridge

- Qasim. O.A Ph.D, (2019). Analysis and Design of Steel Truss Bridges. ResearchGate. [Online]. Available:

https://www.researchgate.net/profile/Ola_Qasim/publication/323693087_Analysis_and_Design_of_Steel_Truss_Bridges/link s/5aa5bf100f7e9badd9ab5c52/Analysis-and-Design-of-Steel- Truss-Bridges.pdf?origin=publication_detail

- Grady, W.M, (2008). Analysis of Structures. Baylor University. [Online]. Available: http://web.ecs.baylor.edu/faculty/grady/_29_trusses.pdf

Designs and simulations were done on https://platform.skyciv.com/

Written by Mark Chan

Year 11 Student at Bangkok Patana School

Published in the Oxford Scholastica Journal of Engineering