The tectonics of tunnelling and bridging non connected territories James Ho.
Contents
Introduction to beams and bridges..........................................................................02 Principals of tunnels and bridges explained............................................................03 - 05 Tunnelling vs bridging and what architectural features can these offer?.................06 The tunnel.................................................................................................................07 - 09 The bridge................................................................................................................09 Beam bridges.....................................10 - 12 Arch bridges.......................................12 - 13 Truss bridges......................................13 - 14 Hanging bridges................................14 - 17 Case studies............................................................................................................17 Laerdal-Aurland tunnel, Norway.........17 Alamillo bridge, Seville.......................18 Sydney Harbour bridge, Australia......19 Architectural features..........................20
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The tectonics of tunnelling and bridging non-connected territories Tunnelling vs. Bridging and what architectural features can these offer?
pals that make them icons for their ability to elude gravity to answer the question of “Tunnelling vs. Bridging and what architectural features can these offer?”
these two seem to pierce or hurdle their obstacles. I will be introducing several principals and laws of physics to ease the explanation of the physics of the tunnel and the bridge such as; free force diagrams (natural forces and exterior forces), Isaac Newton’s laws in particularly Newton’s laws of motion and resonance & frequency.
2
F
Free Force Diagram is a simple illustrative drawing of arrows showing the forces acting on an object. The object is usually illustrated as a dot or square and the arrows show a force, therefore the
Kg
F
Natural forces are the forces exerted by the actual object under gravity i.e. Its own weight. F
FIG 1. Free force diagram
F
∞
F
F
External forces are the forces acting on it that are not resultant from it self i.e. Friction or wind.
states that if an object is set in motion then that object will continue to be in motion until and adverse force acts upon it.
FIG
F
Kg
Kg
Kg
Kg
f
F states that the force of a moving object is proportional to its mass and acceleration hence the formula F=m.a
f FIG 3. Newton´s second law
3
FIG 4. Newton´s third law
states that every force has an equal and opposite reaction force.
y
x
Resonance occurs when a force is applied to an object and causes it to vibrate, every object has a natural frequency ing) and this is the maximum capacity of displacement / vibration of the object. Stable line, no energy transfer, no force acting and no frequency.
FIG 5. Stationary line
y
x
Picture the example of a singer hitting a high pitch note that causes a glass to break. This means that the singer must provide a note at an equivalent frequency as the glasses natural frequency causing this glass to resonate at its maximum capacity causing it to break. When a force is applied, energy is transferred and a frequency along the line is established.
FIG 6. A wave motion
4
y
x If the force applied is increased so does the energy dissipated along the line creating a larger amplitude and frequency. Reaching the natural frequency will mean that the amplitude and frequency will be the largest achievable.
FIG 7. A wave with high amplitudes
y
x it is channelled back and forth along the line, which is how the a bridge will behave under duress, this is know as a wave or stationary wave.
FIG 8. A standing wave
5
Tunnelling vs. Bridging and what architectural features can these offer? We know that every successful stationary system of mechanics balances and equates the numerous forces acting on it, making it a body of forces in equilibrium – a system of forces that are in harmony or balanced. This is what the organs of a bridge and tunnel do, by applying the forces in the precise direction and quantity. To do this, various systems of structures within the main structural logic are applied and combined to produce this equilibrium. I will be cataloguing and explaining these systems for both the tunnel and the bridge. Tunnels and bridges share several of the same principles to keep a load or structure in position or suspended to allow us to travel through or over mountains, valleys and even oceans.
that both of these structures need to provide; same for both the tunnel and bridge.
Required features from a tunnel & bridge. - Firm foundations. - Strong structure. - Effective function & use.
and effective function and use. These three are the
Firm Foundations The foundations are the vertical structures that are often attached to elements that extrude out from them, most importantly they channel all the forces to the ground and they are the strongest element of any structure. When it comes to foundations the most important factor is the soil in which they will sit. There are many number of soil types and combinations, but they can generally be divided into three composites; water, air and solid particles.
use a large surface area to distribute the weight evenly to prevent the foundation it self from sinking into the soil by spreading out the weight so the force exerted on the ground is smaller per square unit of measure. use long narrow columns that travel deeper into the soil than Spread Foundations reaching the bedrock in soils of less tion between them and the soil. A strong structure The main structures used in tunnelling and bridging are: and suspension structures. These can be combined to meet the needs of the purpose, use and the environment in which they may sit. The materials used are important as they can provide good aesthetics, function, and physical qualities but above all for the properties that they can provide when they are subjected to different forces and physical conditions such as pulling (tension), pushing (compression) and sliding (shear). Materials used are timber, concrete, iron, steel and more recently aluminium, plastics and the combination of these. Tension the stress in a line made taught and the best material to use in tension is steel. Compression two forces acting towards each other, literally squashing a material together, compressing the particles of the material. Concrete is a material that works best in compression. Wide and short materials are strong in compression and long thin materials are good in tension. Bricks and stone are short and wide relative to their height making them an appropriate material choice for arches; these are some of the oldest forms of bridging. is a sliding action like the one of a scissor but in this case instead of using this force to cut it used to hold an object in a sta6
tionary position. Like a stone been pushed on a surface, the shear force acts perpendicular to the natural force of the brick which is gravity pushing the brick down. So by increasing the size of the brick will increase the weigh, friction and shear force stopping it from sliding.
Required features from a tunnel & bridge.
Effective function & use
- Firm foundations. - Strong structure.
Both structures are intricate, arduous and expensive to construct, so it is vital that the adequate structures are chosen and applied in an orderly fashion to produce its intended result effectively.
- Effective function & use.
forces to create objects that seem to deceive gravity. I will now look at a tunnel and bridge individually to analyse how each one applies these techniques.
The tunnel vs the bridge - The tunnel considering tunnel excavations. The variations depend on there composition of the three main elements in the soil; water, air and solid particles. a composition of water, air, solid particles and weak rock.
The tunnel. - Soil survey & exploration. - Excavation.
a composition of small amounts of water and air combined with solid particles of soft rock; shale, chalk, and sandstone usually compressed together. soils with high water content, air, clay and sand, usually submerge in water. The procedure of tunnelling is the same for all types, ronmental control. The only difference lays in the use and purpose of the tunnel, many tunnels are used for a limited time such as ore extraction so the aesthetics, materials and the general design aspect is of less importance and safety been the most important aspect always. Tunnels are also important for human transport and every day use for longer periods of time which brings in aspects of aesthetics and comfort playing a larger role.
- Soil removal. - Ground support. - Environmental control.
Soil survey and exploration Geological surveys and drifts – exploration tunnels, will give an approximation to the soil type and condition. Soils are very unpredictable and the only way to be certain of the soil conditions is to bore or drill ahead of the actual dig, a common procedure used today. These techniques have originated from the oil industry and in recent years Japanese engineers have provided techniques that allow to identifying dangerous soil conditions in advance. Excavation Various machines and equipment sizes carry out the process of excavation. The use of equipment depends greatly on the size of the bore and soil conditions. Smaller diameter tunnels can be excavated with hydraulic and power drills handled by hand and on a larger scale huge excavation perforated using the appropriate equipment and then removed. But every now and then segments of extremely hard rock are met during the excavation and this is when specialist in pyrotechnics are called in to break down this hard rock with the use of explosives. 7
As soils are so unpredictable in many cases a combination of all of these techniques are used for the same tunnel. There are two main ways of advancing when boring; full face boring and
The tunnel.
.
- Soil survey & exploration. - Excavation. - Soil removal. - Ground support. - Environmental control. FIG 9. Full-face boring
Full face boring is when soil removal advances equally as it moves forward, uniformly perpendicular to the dig direction.
FIG 10. Heading and bench boring
is when excavation takes place in segments at different heights creating steps. A technique used in soils that are known to be structurally weak. Soil removal
environments. The ability to excavate lays in the machinery used today in the tunnel industry especially in the robust drill heads and bore heads that can perforate even the hardest rock. But the next most important aspect is the ability for these machines to remove the broken rock on to a conveyor system which then carries the materials out of the tunnel by the use of conveyor belts, carts and trucks to allow room to work more effectively. More modern equipment and advances in science have allowed tunnel boring to be precise down to the millimetre by the use of GPS and lasers guided drills like the ones used in for the construction of the channel tunnel crossing (1988-1994) between England and France. Ground Support This is the installation of structures to hold the tunnel up and in shape, the shape of the tunnel plays a big role in its structural properties and I will look into that later on. Engineers can calculate the time that passes from the moment that the tunnel is bored to the time 8
that supports need to be erected and this is calculated from the type of soil and ground conditions. This is called the – stand up time, The tunnel. - Soil survey & exploration. - Soil removal. - Ground support. - Environmental control.
FIG 11. Force distribution on arch shape
Concrete is the favourable material when it comes to tunnel construction for its impermeable features, ability to withstand heat and above all its ability to support large amounts of pressure.
FIG 12. Cylindrical tunnel
FIG 13. Horseshoe tunnel
Tunnels are generally always in shape as this is the strongest form that holds its shape under high pressures. Cylindrical tunnel shapes can be altered but always resemble a cylindrical shape like the tunnel shape. Environmental control -
In this environment it is important to have a fresh air supply and in many cases the air cycling system is the largest mechanical feature on site running the whole length of the tunnel, which also has to remove deadly gases and harmful particles in the air especially after ery and even special body suits to protect against high temperatures and abrasions. The tunnel vs the bridge - The Bridge Bridges are one of the oldest forms of engineering and important links of communication. Bridges have a spacial relationship with the landscape and context in which they sit. Apart from creating a link they form part of our country sides and cities and are true expre9
sive sculptures of architectural tectonics. Types of Bridges.
Bridges can be categorised into four structural types or categories; The beam bridge A beam bridge is the simplest tectonic type for constructing a bridge, simple and very effective. These bridges use towers imbedded in foundations as a secure base to span outwards in opposite directions to counterbalance their weight, a typically steel truss structure.
- Arched bridge. - Truss bridge. - Hanging bridge.
The beams projecting away from the towers are usually at an angle, this helps transfer the forces to the tower foundations. The smaller the angle to the vertical the easier the loads and forces are channelled to the ground. Like a person holding a bag of shopping out at arms reach perpendicular to his or her chest. By lowering your arms and creating an acute angle towards your feet the forces travel easier to the ground the same would happen if you went the opposite way and created an acute angle towards your head, the forces would travel easier through your arm and down to the ground.
θ
Kg
θ
Kg
Kg
FIG 14. Resolving horizontally & vertically
So by placing the cantilevers at an angle they help transfer the forces horizontally to the vertical foundation beam and down the foundations. All the forces combine at the concrete foot of the counter lever towers and are then transferred to the ground, this is only possible due to the ability of concrete foundations to support great loads as we know now that concrete functions best when in compression. cally designed to carry bending moments. The anatomy of beam bridges can be divided into three main elements: b
a/ The beams run the length of the bridge and in cross-section to ensure they stay in the correct shape. b/ The c/ The
a
is the main section of the bridge that the user will use, the part where you will drive or walk over. keep everything up. c
FIG 15. Cross section through a beam bridge
10
The simplest example of a beam bridge is a – this is a beam that spans over a short distance and is supported at either end. A tree trunk set across a stream horizontally is a simple example of a supported beam bridge. Types of Bridges.
- Arched bridge. - Truss bridge. - Hanging bridge. FIG 16. Supported beam bridge
The most common beam bridges today are made from concrete. *We will assume that all our examples are uniform and weight distribution is equal through out. R1
R2
N
FIG 17. Free force diagram of a
A free force diagram will show the forces acting on this supported beam bridge. You have three forces acting – of the beam spanning across a threshold and two forces acting at either end of the beam which is the reaction forces of the beam supported on the both sides.
F
F
F FIG 18. Free force diagram of an
Now imagine the same structure but with an extra foundation tower in the middle. This will allow a greater length in the bridge span and this process can be repeated to increase the length of the bridge, this is called an Imagine a force that will act as a car crossing the bridge in the middle of the beam at a single instance, this will create a bending motion in the beam like a u-shape, this is called sagging. 11
Hogging Types of Bridges. - Beam bridge.
Sagging
- Truss bridge. - Hanging bridge.
FIG 19. Diagrams of Hogging & Sagging
Picture the same car travelling across the bridge and this time with the extra support, sagging will produced an opposite u-shaped were the tower is positioned, this is called . The arch bridge An arch acts with the same principal of the beams, vertical and horizontal components set at obtuse angles (to the ground). The forces are then transferred easier through the angle of incident; the joint between beam and foundation tower, allowing a smooth channel for the forces to be returned to the ground. The arch bridge has numerous elements to its anatomy – and .
FIG 20. Voussoirs of a bridge
FIG 21.
of a bridge
The voussoirs or the arch rib is the curved underbelly of the bridge that holds all the v-shaped stone boulders that creates the arch shape of the underpass. The are the triangular shaped corners above the arch and bellow the deck.
FIG 22. Centrig of a bridge
The are the vertical structures that are often attached to elements that extrude out from it and most importantly channel forces to the ground, as coverd previously. The centrig 12
The most common arched bridges are stone bridges, which are well known feature of the Spanish countryside. The Romans made self the stone slab which is normally heavy and wide, which is one of the principals to provide a low centre of gravity. This makes the stone slab stable and secure especially under adverse forces such as; wind, vibration and the load that a bridge will be carrying. This principal of stone masonry is overlaid and built up to create the desired bridge.
Types of Bridges.
is so effective that just the stones them self’s can be used as long as they are shaped and placed correctly, this is a skilled job and how dry stonewalls are made.
- Truss bridge.
- Beam bridge.
- Hanging bridge.
FIG 23. The structural principal of an arch bridge
The key is to keep a low centre of gravity and by interlocking the stone slabs to reduce weak points from forces acting in multiple directions. This way all forces are channelled to the ground from the very top along the voussoirs. So picture an arch connected to a foundation tower made from stone slabs in this manner. The arch transfers its load laterally to the foundation tower this then transfers the forces down to the centre of its structure and securely to the ground. This principal is sound and acts as a template that can be repeated to give the desired length of a bridge. fers the load of the foundation towers, maintaining and following the shape of the arch. The truss bridge
Types of Bridges.
The simplest truss bridge structure is a king post truss bridge, which is simply a triangle, like a scalene triangle in proportion. Picture
- Beam bridge. - Arched bridge.
close the triangular shape.
- Hanging bridge.
FIG 24. King post truss bridge
There are many variations in the king post truss even though they all follow the same structural principal with shorter or longer beams.
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Types of Bridges. - Beam bridge. - Arched bridge.
- Hanging bridge. FIG 25. Truss bridge variations
Trusses are structures that can carry heavy loads wile been; small in volume and can be shaped in order to over come any obstacle. They are also mounted and dismantled easily as they can be made as assembly parts making the construction process very easy. The individual parts are bolted, clamped or pinned. The joinery is a key part of a bridge structure especially a truss bridge as it is mainly formed by many smaller elements. All the forces will be distributed through all the parts of the bridge structure but this will only apply if all the parts stay together. In steel bridges the most common joinery used is a bolt & nut or welded, how ever before World War II rivets were used which are cylindrical steel pieces with a steel head on the one end and in large structures theses were inserted wile still hot and the other end of the cylinder was hammered in. Timber bridges use a timber cylindrical shapes that are wedged through two over lapping holes. However the most common joinery used today is welding. This allows a better adhesion between materials, is better resistant to weathering and allows a better distribution of adhered surface over a larger surface area. An important factor is the quality of the weld and this lays entirely on the skill of the welder. Hanging Bridge There are two forms of hanging bridges; suspension and
Types of Bridges. - Beam bridge.
bridges.
As for any bridge the vital elements for the construction of a successful bridge are; towers, bridge deck, foundations and for a hanging bridge the following are also needed;
- Arched bridge. - Truss bridge.
FIG 26. Suspended lines
The structural logic behind a in the line, creating a curved shape. To reduce the tension in the line and the depth of the u-shape of the curve, props are placed at intervals along the line creating a wave like pattern in the. A stiff deck is then literally suspended from a secondary set of cables that extend vertically from the curved set of lines. 14
Types of Bridges. - Beam bridge. - Arched bridge. - Truss bridge.
FIG 27. Suspension bridge
One of the main differences between suspension bridges and
bridges are the anchor points.
The suspension bridge carries the tension along the bridges’ length which means the tension is anchored at each end of the line at two key anchor points, meaning that if either of these points fail then the whole system will fail. Therefore anchorages are very important in suspension bridges and large weights have to be attached at these points to resist the pulling force of the resultant forces of the bridge. Cable-stayed bridges have numerous anchor points along the length of the deck, spreading the total tension in the line along the bridge at numerous points and with no main line in tension running the length of the deck. The tension in the line is directly proportional to the total dead load of the bridge and indirectly proportional to the parabola of the line. Meaning that the longer the length and heavier the bridge is the more tension in the lines and the greater the sag.
FIG 28 a. Cable stayed bridge
The structural logic in the cable-stayed bridge is similar to the suspension bridge but instead of having lines spanning from tower to tower in a u-shaped the lines are taught and run from the tower to the deck directly.
Santiago Calatrava´s work in The Netherlands and Israel. 15
Types of Bridges. - Beam bridge. - Arched bridge. - Truss bridge.
FIG 28 b. Cable stayed bridge
Now lets look at how bridges behave under external forces that make them unsafe, the structures and resultant actions taken to over come these complications in bridge design. The type of wave motion which is most hazardous to bridges and how these waves a created.
Anti-node Node
FIG 29. Variations in wave behaviour
The motion created from the resultant forces acting on a bridge; vehicles, nature (winds) etc provide the bridge with, energy - the points a wave motion is created from this wave and will travel backwards and forwards along the same axial line (a standing wave) creating nodes and anti-nodes which depending on the frequency created by these external forces, will be larger or smaller.
the bridge will sway at its maximum capacity and this will result in the possible failure of the bridge. Suspension and cable stayed bridges are particularly susceptible to this as the wires provide less stiffness than a solid structure would. There are several ways to prevent this resonance, by around the bridge.
- altering the way air travels
Dampening is a process in which materials are applied to absorb the energy of these external forces resulting in lessening the kinetic energy and so reducing the amplitude of the wave motion. 16
I will now look at some examples of existing bridges to show how some of these structural principals are used and can form effective structures that do hurdle over waterways and punch through mountains to achieve a link between two non-connected territories. I will be analysing the following tunnel and bridges; the Laerdal-Aurland tunnel in Sogn og Fjordane, Norway, the Alamillo bridge in Seville, Spain and the Sydney harbour bridge in Sydney, Australia. Laerdal-Aurland tunnel
The longest tunnel in the world is located two hundred kilometres north east of Bergen, roughly in the middle of the country between only route available by land. Tunnelling is a favourable civil construction type that is common in Norway due to its rough terrain and heavy snow conditions especially in the mountainous regions of the country. The tunnel extends through an entire segment of a mountain range between Laerdal and Aurland a successful achievement after more hard and course rock composites.
at both ends and worked towards the centre, wile at the same time a two kilometre secondary tunnel was dug from a side valley in the centre of the dig allowing to simultaneously dig from the extremities to its core and from the core towards the extremities with the use of guided laser beams. Like with any other tunnel, environmental control and ground support are key aspects for a sound structure under extreme duress and in this particular tunnel these aspects were heightened considering the vastness of this project. New and extreme measures had to be taken to ensure the safety of the motorist that would use this passage to cross from one side of in-tunnel treatment plant housed in its own cavern nine and a half kilometres from Aurland. This plant draws fresh air from its openings twenty-four and a half thousand metres from each other and expels it at an opening at its centre.
FIG 30. Laerdal-Aurland tunnel, Norway.
Considering the length of this tunnel security measures had to be implemented into its design to ensure open access and the ability for the largest vehicle to turn around within the tunnel in an emergency situation. The entire journey is divided into four sections transi-
concrete horseshoe structure resembles the mountains rocky conditions. All of these measures have been taken to break down the twenty-minute journey to avoid extended periods of monotonous travel. These are some of the safety precautions taken in addition to the standard use of; photo-monitoring, emergency telephones, fail safe
FIG 32 a. Laerdal-Aurland tunnel interior
FIG 32 b. Laerdal-Aurland tunnel interior
FIG 32 c. Laerdal-Aurland tunnel interior
FIG 31. Tunnel path
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Alamillo bridge The Alamillo bridge of Seville is a cable-stayed bride located in the northern part of the city. This bridge connects the main city of Seville to the island of la Cartuja an area brought to life thanks to the link that this bridge has provided. Designed by Spanish architect, engineer and sculptor, Santiago Calatrava.
side of the mast, a mast grounded by huge spread foundations.
FIG 33. Alamillo bridge, Seville, Andalusia, Spain.
FIG 34. Elevation view from the south
A mast made from a compound of steel and concrete that stands one hundred and forty two metres in the air and two hundred and example of equilibrium, how the weight of the mast equates the pulling force of the entire deck all through twenty-six steel cables in tension. An elaborate example of forces in harmony, as though it was a giant harp. FIG 35 a. Tensile wires of a cable stayed bridge
The cables are attached to a hexagonal box girder which is the central horizontal support running the length of the deck. Which also
and others when constructing in this new strand of the cable stayed bridge typology.
FIG 35 b. Dampening system
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Sydney Harbour Bridge
But the main reason why I chose this bridge is because it combines the structural principals of two types of bridge structures; the arch and the truss, perhaps one of the reasons why it is the longest steel truss span arched bridge in the world. Built between 1922 and
It is located in the hart of Sydney and nicknamed the “coat hanger”, connecting Sydney’s world famous harbour with the northern of seawater and one thousand one hundred and forty nine metres in total length grounding its extremities safely on both sides of this active and scenic bay.
FIG 36. Sydney Harbour bridge, Sydney, Australia
FIG 37. Bridge entrance and toll
FIG 38. Elevation view from the East
FIG 39. Concrete pillars faced with granite
The bridge triggered a demand for a more aesthetic focus on bridge design especially for its time as all other bridges in the past were designed to be purely functional and utilitarian. This bridge set president for another demand from bridges and that was aesthetics and so the steel work remained as it was which has its charm just by it self but the vertical structural supports pulling the span of high-strength concrete foundations drive deep into the soil using spread foundations twelve metres in depth anchoring the numerous trusses to the ground. The steel arch length was constructed from each shore and met in the middle and the cross-sectional elements were hoisted in place from barges. The trusses and other steel elements were fastened together using ten-centimetre long rivets weighing three and a half kilograms each. This all came together on February 1932 when the bridge was successfully load-tested with ninety-six steam locomotives.
but also a landmark for Sydney’s waterfront.
FIG 40. Construction of the deck
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The intention of this essay is to answer the question, Tunnelling vs. Bridging and what architectural features can these offer? Under the research topic of, “The tectonics of tunnelling and bridging non connected territories”. Required features from a tunnel & bridge.
and bridges. Free force diagrams to show the forces acting on a body, Newton’s laws of motion to establish an understanding of how forces behave and to elaborate on the principal of equilibrium and resonance & frequency that shows how a bridge behaves when subjected to multiple forces.
structure and effective function & use. From the tunnel I learned that tunnels are constructed purely based on a concave shape primarily derived from the shape of a cylinder. The shape allows the forces acting on it to be distributed along the arch and so eliminating weak points in the form.
- Firm foundations. - Strong structure. - Effective function & use.
The tunnel. - Soil survey & exploration. - Excavation.
, which determines the boring type, machinery used and the safety of the environment. Excavation, the process of perforating the rock using, power drills, explosives and Moles or T.B.M´s (tunnel, boring, machines) and the type of excavation, full face boring or . the removal of the soil on a conveyor systems and most importantly the effective and continuous removal of it as this directly affects the pace of excavation. environmental control, which establishes an appropriate and safe living environment.
construction process of any structure.
- Soil removal. - Ground support. - Environmental control.
Types of Bridges.
Bridges have shown me that there is a bridge type for each site especially to overcome a type of environment or obstacle. This is the
- Beam bridge.
and
- Arched bridge.
The objective of a bridge and tunnel is to establish a physical connection between two territories divided by an obstruction whether it may be a mountain, river, valley, etc. Successfully creating an important aspect within a society, communication. After that aesthetics come into play making our tunnels and bridges into monuments within our cities and landscapes because of their ability to overcome the physics of nature.
- Truss bridge.
From the four structural types of bridges all are used in architecture with the exception of the hanging structural type, but could be implemented as well. An example of bridge structures used in architecture are the connection between two rooms in a building, the walls are the structural supports like the foundation towers of a bridge, the ceiling spans from wall to wall just like the deck of a bridge would, successfully used depends on the resultant volume created in this connection. Were as in the case of a bridge or tunnel the volume plays a small role and the actual link between two territories being the most important aspect. Yet they both used, beams, arches, trusses and even suspension & tension systems. Concluding that the same structure type is used but the difference lays in the product & use. The structural type used in architecture depends on the form it creates between spaces, focusing on the aesthetics and comfort of the space and the structural type used in tunnelling and bridging depends on the actual connection between two spaces, primarily to be able to create a longer projection of the deck with the least supports possible.
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Illustrations.
FIG 1. Free force diagram, James Ho, p3
FIG 26. Suspended lines, James Ho, p14
FIG
FIG 27. Suspension bridge, James Ho, p15
FIG 3. Newton´s second law, James Ho, p3
FIG 28 a. Cable stayed bridge, James Ho, p15
FIG 4. Newton´s third law, James Ho, p4
FIG 28 b. Cable stayed bridge, James Ho, p16
FIG 5. Stationary line, James Ho, p4
FIG 29. Variations in wave behaviour, James Ho, p16
FIG 6. A wave motion, James Ho, P4
FIG 30. Map of Oslo showing tunnel location, https://maps.google.es, p17
FIG 7. A wave with high amplitudes, James Ho, p5
FIG 31. Tunnel path, https://maps.google.es, p17
FIG 8. A standing wave, James Ho, p5
FIG 32 a. Laerdal-Aurland tunnel interior, http://www.worldbiggest.net/world-longest-road-tunnel/, 11.01.2013, p17
FIG 9. Full-face boring, James Ho, p8 FIG 32 b. Laerdal-Aurland tunnel interior, http://www.worldbiggest.net/world-longest-road-tunnel/, 11.01.2013, p17
FIG 10. Heading and bench boring, James Ho, p8 FIG 11. Force distribution on arch shape, James Ho, p9
FIG 32 c. Laerdal-Aurland tunnel interior, http://www.worldbiggest.net/world-longest-road-tunnel/, 11.01.2013, p17
FIG 12. Cylindrical tunnel, James Ho, p9 FIG 33. Alamillo bridge, Seville, Andalusia, https://maps.google.es, 11.01.2013, p18 FIG 13. Horseshoe tunnel, James Ho, p9 FIG 34. Elevation view from the south, http://www.calatrava.com/#/Selected%20works/Architecture/ Seville?mode, 11.01.2013, p18
FIG 14. Resolving horizontally & vertically, James Ho, p10 FIG 15. Cross section through a beam bridge, James Ho, p10
FIG 35 a. Tensile wires of a cable stayed bridge, http://www.bristol.ac.uk/civilengineering/bridges/Pages/ NotableBridges/Alamillo.html, 11.01.2013, p18
FIG 16. Supported beam bridge, James Ho, p11 FIG 17. Free force diagram of a
James Ho, p11
FIG 18. Free force diagram of an FIG 19. Diagrams of Hogging &
James Ho, p11 James Ho, p12
FIG 35 b. Dampening system, http://www.bristol.ac.uk/civilengineering/bridges/Pages/NotableBridges/ Alamillo.html, 11.01.2013, p18 FIG 36. Sydney Harbour bridge, Sydney, Australia, https://maps.google.es, 11.01.2013, p19 FIG
FIG 20. Voussoirs of a bridge, James Ho, p12 FIG 21.
of a bridge, James Ho, p12
FIG 22. Centrig of a bridge, James Ho, p12
FIG bridge603x320.jpg, 11.01.2013, p19 FIG
-
FIG 23. The structural principal of an arch bridge, James Ho, p13 FIG 40. Construction of the deck, http://www.harbourbridge.com.au, 11.01.2013, p19 FIG 24. King post truss bridge, James Ho, p13 FIG 25. Truss bridge variations, James Ho, p14
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