# Introduction

Figure 1.1: Cables can hold tension but not compression

Cables can be made of practically any material, such as wool or any other natural or artificial fibers, but in most engineering applications cables will be made of steel.

Cables have a few properties which are worth noting. First, while it is possible to pull on something with a cable or to have a cable take a tensional load, cables cannot push on something or take compressional loads. Another property of cables is a consequence of Newton's third law[2], every action has an equal and opposite reaction. In the case of cables this means that if a tensional load is applied at one end of the cable that same force will be experienced throughout the length of the cable, all the way to the opposite end.

# Benefits

Figure 2.1: Tension of Cable

One benefit of cable structures is that the tension forces in the cables can be used as points of references in the analysis of stuctures. For example, Figure 2.1 involves a weight (w) which causes tension in the cable. This weight pushes the cable down. The cable is now in tension and forms an angle with the horizontal. Given this weight and the angles that the cable forms with the horizontal, the tension in the cables can be calculated with simple vector addition. If this cable was in compression, it would take more than the weight to determine anything within the structure.[3]

Figure 2.2: Guy Wire Example

Stability is another benefit of tensile structures. The tension in the cables cause the resulting structure to be stable. This is commonly demonstrated in tents. As tents, or any structure of that nature, are held by guy wires, which are connected to a grounded point located at the centre of the structure, loads are free to act anywhere on the structure. This is because the guy wires are initially in tension. One can also say the guy wires are in pre-tension because of the fact that the guy wires are in stationary position until a load is being applied to it.[4]

Another benefit of cables structures is that compared to reinforced concrete, cables weigh significantly less[7] and they can generally hold materials such as Kevlar vinyl[8] and other 'soft materials'. These soft materials act like a sheet or blanket being pulled out at each corner, tension in the material opens it up and gives it structure. Cables supply the required tension to keep these ‘soft materials’ open and rigid.

# History

From ancient Roman canopies and rope bridges, to modern day tents and suspension bridges cable structures, also known as tensile structures, are applications of "tension-only" members.[4] It was not known that steel cables, which first appeared during the industrial revolution,[4] could effectively be used in structures until the late nineteenth century, when Vladimir Shukhov of Nizhny Novgorod, Russia constructed the first tensile steel shell.[9] Although Vladimir's creation was a roof instalment, this opened doors for today's engineers to explore what is possible with cable structures.

The technology behind cable structures has advanced since the time of Vladimir Shokhov, allowing for bigger and more ambitious projects to be undertaken. An organization by the name of Geiger engineers have built numerous tensile structures and roofs over the past 30 years, from the Canada Harbour Place Roof Replacement to the more recent B.C. Place Stadium Revitalization which added a new retractable roof.[10] Another organization that has pushed the envelope of tensile structures is Eide Industries, who have focused their efforts towards using tensile structures for architectural purposes.[11] Eide Industries' tensile structures have a unique aesthetic to them whic would not be possible in conventional structures. As well fabrics are now being added to these types of structures giving rise to numerous benefits such as making them more cost efficient, making them more environmentally friendly, increasing their life span, making them easier to install and many more.[12]

# Types of Cable Structures

## Cable-Stayed Bridges

Figure 4.1: Simplified illustration of a Cable Stayed Bridge

In our day-to-day life we come across a variety of different bridges. Some of them have cables as a vital part of their structure. One of the examples of such type of bridges are Cable-Stayed Bridges. These bridges are very common when it comes to pedestrian bridges, highway bridges and bridges for pipelines[13]. According to W.Podolny and J. Scalzi's book "Construction and Design of Cable Stayed Bridges", it was during the 1950's that these types of bridges gained popularity with Germany being on the forefront who constructed several Cable stayed bridges across river Rhine. Few of the main reasons for the development of these types of bridges was their low cost of construction, the speed of erection and the fact that they had the potential to cover a relatively longer spans[14].

Figure 4.2: Deflection of Cable Stayed Bridge

W.Podolny and J.Scalzi define a modern day cable stayed-bridge as " A bridge which consists of a superstructure of steel or reinforced concrete members that is supported at one or more points by cables extending from one or more towers."[15]. The cables transfer their tensile load to the towers as shown in the Figure [4.1]. This load is then transferred to the main column on which the tower is constructed. The most common type of materials used for the superstructure of these kind of bridges are either concrete or steel. And each of them have their advantages and disadvantages.

## Suspension Bridges

Figure 4.3: Simplified illustration of a Suspension bridge

Suspension bridges are one of the most beautiful civil engineering structures in the world. It is a beautiful combination of ropes, steel and concrete. The earliest known occurrence of a suspension bridge was a bridge built across the Indus River near the Swat in A.D. 400. The origins of the suspension bridges can be traced back to the warm countries of South-East Asia due to their availability of creepers, vines and other trailing plants [16]. In a suspension bridge there is a suspender cable which runs the entire length of the bridge and is supported by two or more towers. From this suspender cable, vertical or radiating rods or suspension cables are suspended which hold up the deck of the bridge. As illustrated in the Figure [4.3]. According to Pugsley's book "The Theory of Suspension Bridges", suspension bridges only captured the western people's interest on a large scale after the introduction of wrought iron [17]. The Brooklyn Bridge is considered one of the biggest successes in the civil engineering field. Its successful construction gave rise to a lot of excitement amongst the engineers all over the world and the American engineers came to be recognized as the experts in the construction of suspension bridges[18]. The design theory which was developed by Moiseiff and Steinman was used in the construction of all the suspension bridges.

# Example Question

As was previously mentioned cables can only take load in tension, this must be taken into account when analyzing cable structures. For example, it can be seen by inspection that the simple cable structure depicted in figure 5.1 is statically indeterminate with the vertical reactions of cables A ($A_{y}$) and B ($B_{y}$) as well as the vertical and horizontal reactions at pin C ($C_{y}$ and $C_{x}$) being unknown. The solution can be simplified by implementing the need for cables A and B to be in tension for them to supply a reaction.

Figure 5.1: A simple cable structure

First examine the collapse mechanism of the structure if the two cables were to be removed. As can be seen in figure 5.2 the beam would rotate clockwise about the pin C. To counteract this rotation cable B would have to supply a force upward, translating to tension in the cable, and cable A would have to supply a downward force, translating to compression in the cable. Since cables are unable to take load in compression the reaction force of the cable A will be zero.

Figure 5.2: The collapse mechanism of the structure with its cable supports removed

Now that the system has been simplified to three unknowns it only remains to use the equations of static equilibrium to solve for them, the free body diagram of the system can be seen in figure 3.3. By taking the sum of the forces in the x direction it can be seen that $\sum F_{x}= C_{x}=0$. To solve for the remaining two reactions start by taking the moment about the pin C.

Figure 5.3: The free body diagram of the structure with the support reaction at A removed

$\sum M_{C} = 0$

$0 = (B_{y} \cdot 4m) - (100kN \cdot 2m)$

$B_{y} = \frac{200kN \cdot m}{4m$

$B_{y} = 50kN$

Finally, to solve for the remaining reaction simply take the sum of the forces in the y direction

$\sum F_{y} = 0$

$0 = B_{y} + C_{y} - 100kN$

$C_{y} = 100kN - 50kN$

$C_{y} = 50kN$

3 Dimensional Cable Supporters Example

Consider the example above illustrating a suspended load carried by three cable supports. The load is assumed to be a chandelier weighing 60 lbs. It can, however, be a part of a steel deck due to the wide variety of applications of cable supporters. The analysis and calculation principles remain the same nonetheless. For the object to be in equilibrium, the sum of the forces in all directions must be equal to zero. Since the example provides the distances in the x, y, and z axes between the end of each supporter (where it is connected to the wall) and the origin, where the load is connected, one can establish a unit vector for every cable. A unit vector represents the force components on every axis that a cable support is experiencing when carrying the load. After establishing a unit vector for every cable, the components of every axis are summed separately and then solved as part of a three equation-three unknown system. Within the force components, i refers to the x axis, j to the y, and k to the z.

We begin the analysis with the first cable on the left (Cable A). The cable is in the X-Z axes (4 feet in the x direction and 3 feet in the z direction), hence it is categorized as 4i + 3k.

The next cable (Cable B) is 5 feet right, 6 feet up, and 4 feet south of the origin (out of the page), hence characterized by -5i + 4j + 6k.

The third cable is (Cable C) is 5 feet right, 6 feet up, and 5 feet north of the origin (into the page), hence characterized by -5i - 5j + 6k.

The load is represented by -k.

Now that we have the direction of each cable, we can multiply the force by the unit vector to separate the components.

Cable A:

Unit Vector: (4i + 3k) / √(4^2 + 3^2) = 0.8i + 0.6k Force: 0.8Fa (i) + 0.6Fa (k)

Cable B:

Unit Vector:(-5i + 4j + 6k) / √(5^2 + 4^2 + 6^2) = -0.57i + 0.46j + 0.68k Force: -0.57Fb (i) + 0.46Fb (j) + 0.68Fb (k)

Cable C:

Unit Vector: (-5i - 5j + 6k) / √(5^2 + 5^2 + 6^2) = -0.54i -0.54j + 0.65k Force: -0.54Fc (i) - 0.54Fc (j) + 0.65Fc (k)

Next, we can proceed to sum the forces about the x, y, and axes independently.

Sum X-axis: 0.8Fa - 0.57Fb - 0.54Fc = 0

Sum Y-axis: 0.46Fb - 0.54Fc = 0

Sum Z-axis: 0.6Fa + 0.68Fb + 0.65Fc - 60 = 0

Solving the equations above nets the following result:

Fa = 38.5 lbs Fb = 29.9 lbs Fc = 25.5 lbs

Note the similarity between the forces in cables B and C as they are symmetrical in the x and z axes.

Understanding the analysis of cable supports is an important skill for structural engineers to hone, and the example above is an excellent representation of one of many systems and applications of cable supports.

# References

1. Bell, John. "Facultatea De Construcții Din Timișoara." Facultatea De Construcții Din Timișoara. N.p., n.d. Web. 24 Nov. 2013. <http://www.ct.upt.ro/users/AurelStratan/bsd%5Cc05-bsd.pdf>.
2. "Newton's Third Law of Motion." The Physics Classroom. ComPADRE, n.d. Web. 25 Nov. 2013. <http://www.physicsclassroom.com/class/newtlaws/u2l4a.cfm>.
3. T. Henderson. (1996). Tension in Cables. [Online]. Available: http://www.physicsclassroom.com/class/vectors/u3l3c.cfm
4. . P. Gossen. (2004, November). Design With Cables. [Online]. Available: www.structuremag.org/OldArchives/2004/november/DesignWithCables.pdf
5. Lamb, Robert, and Michael Morrissey. "How Bridges Work." HowStuffWorks. N.p., n.d. Web. 24 Nov. 2013. <http://www.howstuffworks.com/engineering/civil/bridge7.htm>.
6. "The Golden Gate Bridge." British Columbia Institute of Technology, n.d. Web. 24 Nov. 2013. <http://commons.bcit.ca/civil/edufacts/golden_gate.html>.
7. Buchholdt, H. A. "Design Consideration." An Introduction to Cable Roof Structures. Cambridge [Cambridgeshire: Cambridge UP, 1985. 277. Print.
8. "How Kevlar Fabrics Are Used." Kevlar Fabric. N.p., n.d. Web. 25 Nov. 2013. <http://www.kevlarfabric.com/how-kevlar-fabrics-are-used>.