Principle Of Superposition - Understanding Structural Concepts

The principle of superposition according to Williams and Todd 2000 ―states that the total effects of two different inputs to a systems equal to the sum of their effects when applied separately.‖ Thus, if a beam deflects by a distance ∆₁ under a load A and a distance ∆₂ under a load B, then the beam‘s deflection under a load (A+B) will be ∆1+∆2.

∆1

Fig 1.1: A Beam of length l with a vertical load A applied at l/2

∆2

Fig 1.2: Same Beam as in 1.1 but with a vertical load B applied at same position as 1.1

A+B

∆1+∆2

Fig 1.3: Same Beam as in 1.1 with a vertical load A+B applied at l/2

Assumptions of Super-position principle

1. The material is elastic and elastic limit is not exceeded during loading

2. The geometry of the material does not change or change in geometry is substantially small

Plate 1: A simply supported beam unloaded Plate2: Deflection of beam loaded A

Plate3: Deflection of beam with load A+B References

Williams M.S. & Todd J.D., (2000) Structures theory and analysis, Palgrave Macmillan

1.20 Why a Roly-Poly Toy does not fall...????

Amrit Pal Singh

Concept: Stable Equilibrium by lowering the Centre of Gravity

Definitions:-

Equilibrium- It is defined as the state of rest of a body, during which the net effect of forces on the body is zero i.e. F=0.

Three states of equilibrium:

a. Stable Equilibrium- A body is in stable equilibrium when the mass is concentrated at the bottom thus lowering the position of center of gravity. So when the body is tilted, the centre of gravity gets raised but the body moves back to its stable equilibrium position by making the centre of gravity as low as possible. Example – Roly ploy toy

b. Unstable Equilibrium – A body is in unstable equilibrium when the mass is concentrated at the top, when a small force is applied on such a body it makes the body unstable and the body ultimately falls to make the centre of gravity as low as possible. Example – a person standing on one foot.

c. Neutral equilibrium- – A body is in neutral equilibrium when the body is placed in such a position so that when a force acts on the body, centre of gravity is neither raised nor lowered.

Example – A dumbbell lying on ground along its longer dimension

Centre of gravity – A rigid body is made up of different members each having a specific weight.

Center of gravity of a body is a point where we assume the weight of all the members of the body to be concentrated.

Answer to Why…???? - Roly poly toy is generally hollow and spherical at the bottom with the entire weight concentrated at the bottom so the centre of gravity is in the middle of spherical section at the bottom as shown by red circle. At equilibrium centre of gravity is exactly above the point of contact with the ground as shown by the red line. ―When the toy is tilted the center of gravity rises from the green line to the orange line‖, and the center of gravity is no longer over the point of contact with the ground as clarified by the yellow line. . This produces a righting moment which returns the toy to the equilibrium position. . Such a Roly poly toys is not homogenous as its density varies across the body. An object like this has only one stable and unstable point so no matter how you move the toy it will return to its original position.

Based on this example we can explain why it is so important for a Structure to have a broad base with its centre of gravity as low as possible and within the base of its support. If the centre of gravity of the structure lies outside the base of the support it may be unstable and it may fail but if centre of gravity is within the base of support and if a force is applied on the structure the structure will move around a fixed point at the base and retain its original position after some time therefore the position of the centre of gravity is crucial in a structure.

Conclusion - If a system has its centre of gravity concentrated nearer the base of the support, it is in stable equilibrium, small disturbances to the system may lead to temporary changes in the position of the system but ultimately the system returns to its original position. So when the roly-poly toy is tilted at an angle it wobbles for some time and then attains its original position.

References

1.21 “Cable supported structure”analysis BINGNAN SHEN

―Cable supported structure‖is a clear definition for both structural members and features―Cable‖ means the arched tension cables and ―supported‖denotes the effect of

compression struts，while ―structure‖ represents the upper layer in the whole structure. Cable supported structures are defined as cable supported plane structure，separate cable supported spatial structure and inseparable supported spatial structure.

Fig.1 Cable supported portal frame Practical example:

Structure is composed by the rods and lasso composition. The lasso offer tension force so that the structure internal force get balance and the structure remain stable. The internal force path shows in the picture below.

Chord-tension beams example:

Use common ruler as beams, both ends can be thought of as hinge support.

Only use common ruler to support and put a load of 2.5kg at the middle of beam.

It is clearly that the deflection is obvious.

Use cloth, ruler and plastic box constitute a chord-tension beam. As shown below:

Put a load of 4kg at the middle of beam.

Deflection of the structure is very small. Compared with the formal simply supported beams, stiffness improves a lot.

Load flow path of chord-tension beams:

The bottom lasso sustain tension force, largely reduced the bending moment of the curved upper beams.

Assume the area of the uprights component is A1, the area of lower component is A2, the height of chord-tension beam is h. The second moment of area of the uprights component is I1. The second moment of area of lower component is I2

Then we can get the second moment of area of the whole structure:

Conclusion:

1. A string of the structure of the upper structure has certain stiffness, this makes the structure of the construction and the joint structure than the flexible structure will greatly simplified.

2. The forces path on the system structure is reasonable. High strength pressed lasso the introduction of the upper structure and lower cable-strut formed part of the whole, and work together; A type of the cable tension through the inner structure make the generation and load the opposite displacement, partly offset by the load.

3. In theory have to maximize the use of structure material characteristics, use of the least build large span steel structure of the building, etc.

1.22 Stress Distribution in Real Life

Vlamakis Emmanouil

Concept: Stress Distribution

Structures: Simple column with footing, the using of snowshoes and a push-pin

Introduction:

A common problem in Engineer‘s life but also in human‘s day to day life is the load enforcement in to non-cohesive ground. Particularly the stresses that are developed from a body from its weight or from the load which is acting on it may be produce disequilibrium or settlement in non-cohesive ground.

Practical Example 1: Simple column with footing

For F = 10KN

σ

₁

= F/A

₁

= 10KN/m

and σ

₂

= F/A

₂

= 100,000KN/m

σ

= 10,000 σ

As it is illustrated in figure (1a-b) above, for a simple column foundation there are two cases, one with a large surface at the bottom of the column (figure 1a) and another without any extra surface at the bottom (figure1b).

Consequently, according to figure 1a, we recognise that the applied force in the column is distributed in the whole area. As a result it is produced a stress distribution. On the other hand in figure 1b it is obvious that the column without footing produces high stress concentration which is 10 thousand times bigger than the first case.

Practical Example 2: The using of snowshoes.

This is an example with the human body when someone wants to play with snow.

How it is possible to walk on it?

Figure 2

As it is observed from figure above without snowshoes it is difficult to move because our feet settle in to the snow. This is the same problem as the previous.

The snowshoe produces a stress distribution along its surface; as a result the surface of the snow which is come in contact with the snowshoe can resist easer the weight of the body.

Practical Example 2: Push-pin example

Figure 3

In this example we can observe the meaning of stress distribution by applying the same force. As we can see in figure 3 the finger applies force to the push-pin and the pin applies force to the board but the push-pin react from the resistance of the board so it is also applies the same force to our finger. As a result we have the same force but on the opposite site. The area of the head of the push-pin is A

and the area of the nib is A

but we know that A

=400* A

.

Hence, σ

finger

=F/ A

and σ

board

=F/A

by dividing these two equations we have,

σ

finger

/σ

board

= A

/A

and by applying the previous assumptions we finally have,

σ

board

= 400* σ

finger

Consequently the stress which is produced to the finger is 400 times smaller than the stress which is produced to the board. Thus this is the reason that the pin is fixed in to the board.

Conclusion:

In the conclusion the main concept of these examples are to understand how we can distribute the applied stress in order to avoid the settlement and also the disequilibrium. This is obvious from the simple examples above that by increasing the area we generate lower stress to the ground. Hence more stable structures.

References:

 www.structuralconcepts.org

 www.digitalschool.minedu.gov.gr/modules/ebook/show.

 www.google.com

1.23 Using Global Buckling to Erect a Camping Tent Keiran Murphy

A structural member that is loaded axially in compression is often referred to as a compression member and are the main members of modern day structures. A compression member is usually vertical and is known as a column.

A column is said to be in stable equilibrium when it is under load and returns to its straight form even after a lateral force is applied and removed (See Figure 1.1).

There comes a point however, when this compression load is increased and the column doesn‘t return to its original form after lateral forces are applied. This load is known as its critical or buckling load.

A state of instability is reached when the column load is increased so much that uncontrollable deflection occurs in the lateral direction due to such application (See Figure 1.2). A short column will fail by direct compression, whereas a longer column under axial or eccentric loading will tend to reach this state of instability more easily resulting in it buckling / bending. This is due to their slender nature and so the load carrying capacity of a column depends on a ratio of length to cross-sectional area (often referred to as its slenderness ratio).

A formula derived by Euler to describe the maximum axial load a slender column can carry before buckling is shown below:

Where

F = maximum or critical load on column E = modulus of elasticity

I = area moment of inertia

L = unsupported length of column K = column effective length factor (value depends on the conditions of end support of the column See Figure 1.3)

Fig. 1.1

Using the formula above to calculate the maximum load a column can withstand before buckling, is a vital stage within the design process. This is because if a state of instability where to occur within a structure, it could potentially be a life threatening situation and so global buckling is designed to be avoided at all costs.

However, there are some applications in everyday life where this structural concept of global buckling is very useful. One of these applications is outdoor camping and the use of a tent.

The theory behind this concept of erecting a camping tent works totally opposite to that used to prevent structural elements failing due to bending collapse. Tent poles are designed to be highly elastic and so can cope with huge amounts of eccentric loading past their critical load point without failing. This gives the tent huge diversity for the user without the need for them to worry about the safety of the product.

Camping tents are also designed with slenderness ratio in mind. Once connected all together, the length of such is extremely greater in comparison to its cross sectional area, hence the force required to bend the pole is minimal (in accordance to Euler‘s formula).

The smaller the value of EI, which is determined by the materials properties and dimensions, and the larger the length, L, will result in a smaller driving force required to buckle to section. This would give greater ease to the consumer, saving both time and work effort to erect the tent. A greater value of L also provides a larger living environment by creating a much wider circumference of arc (larger dome).

By testing this theory of global buckling with a camping tent will prove just how easy and effective the structural concept actually is. The photographic demonstration can be seen below:

1 Compression force is applied

Stages 1 and 2 from above explaining the demonstration can now be repeated using the actual shell of a tent to show how the theory becomes reality.

Using the theory of global buckling to erect camping tents has proven to be extremely useful, as not only is it a very simple idea in theory, but one that is exceptionally practical, functional and

ingenious at the same time. The tent poles lightweight material properties and ease of construction methods make it an essential item on any camping holiday to create a perfect home from home. With very little effort, knowledge or experience needed to use, yet having huge benefits along with a great satisfaction of comfort due to its robustness and safety factors, the camping tent is an excellent example of how structural concepts are used in everyday lives.

References:

1. BETZ, Professor Joseph AIAA, American Institute of Aeronautics and Astronautics Column Buckling 1 Force applied at either end

is not great enough, so pole

2 Greater driving force is applied causing the pole to buckle, raising the shell so that a dome is formed, and a new „home from home‟ is created.

1.24 Bamboo bionic structure application in the buildings

TANG li

1. Introduction

Bamboo widely used in China for many years, people use in life such as load-bearing; Building materials (

figure 1, 2

), because bamboo has good toughness and it was broad at the base also each segment gradually become thin with the increase of height, its like a ladder shape equal strength structural. (

Figure 3

)

Figure 1 Figure 2

Figure 3

2. Analysis

From the view of mechanics, the each segment such as

Horizontal resistance to twist box, and can improve Horizontal

resistance to squash and shearing ability so in the wind lord resistance on each paragraph bending deformation basically the same ability. The characteristics of thick at the bottom then top fine was also adapted to the bottom had stronger bending

moment than top.(

figure 4

)

Figure 4

The bamboo’s structure is a good mechanics model, people

quoted this kind of structure use in high-rise building design, such as “Taipei 101” (

^{figure 5}

) and “Jin Mao Tower ” (

^{figure 6}

)

Figure 5 Figure 6

These buildings are above 400 meters, in the design of high

building, people met many problem and the main thing is that the strong winds caused the sway of the building, especially in the typhoon area but use the bamboo structure can solve this

problem, because the “bamboo joint” structure like ring to truss and outrigger together effect on building, they will greatly

enhance the structure’s stiffness and reduce lateral displacement.

3. Example

This is a simple structure do not have the “bamboo joint” (

figure7

) and another simple structure have “bamboo joint” (

^{figure 8}

) in the same loads of performance.

Figure 7 Figure 8

In the obvious contrast we can found the “bamboo joint” in the role of high buildings is very important.

4. Conclusion

The sum up, bamboo’s structure has good toughness and

stiffness because it has bamboo joint. In the high building, outrigger and the ring to truss is building’s “bamboo joint”.

Bamboo bionic structure use in modern buildings not only for beautiful but also practical.

5. References

http://www.docin.com/p-225863664.html http://www.structuralconcepts.org

http://ztzx.forestry.gov.cn/

1.25 Post-Tensioned Concrete Concept Răzvan SENCU

Pre-stressing is a method for overcoming concrete poor tensile strength (Wikipedia.com). This method implies some tendons, namely strands or high strength steel cables, which are the tensioned elements while they induce compression stresses to the concrete either by bonding or by ends anchorages.(Sami Khan, 1995). As the name sais ―pre‖ means before. However, there are known two main techniques, one being the pre-tensioning and the second post-tensioning. The difference between is that for the first case the cables are tensioned before concrete pouring and released once the concrete become cure, while for the second method usually the components are precast and tensioned by the tendons once they had been mounted in situ.

In other words, the pre-stressed concrete is considered to be a combination between concrete and steel, similar to ordinary reinforced concrete, that comes to form a ―resisting couple of forces‖ that will counterbalance the ―external applied bending‖. (David Childs, 2010)

―Post-tensioning is a technique of pre-loading the concrete‖ in such a way that the ―tensile stresses that are induced by dead and live loads‖ will be eliminated or considerable reduced. (Sami Khan, 1995) The main advantage for using pre-stressing is that the concrete beams can span longer with reduced cross section.

Figure 1. Infinity Bridge across River Tees (free photos Flickr.com)

A small research was carried out on The Infinity Bridge. The following picture shows the load transfer path of the bridge. It is fairly obvious that the twin-arches give not only vertical reactions but also horizontal thrusts.

Figure 2. Load transfer path (modified picture from http://geometrygym.blogspot.com/)

A key feature of the design of The Infinity Bridge is that the horizontal reactions of the arches are not taken by the middle pier. They discharge on the lateral strands (see figure 1 right photo) which were provided to tie the bridge‘s concrete deck. In this elegant manner they achieve both the post-tensioning of the deck and the stability of the arches at the same time.

The above images explain the structural concept of the Infinity Bridge‘s post-tensioned concrete deck.

Let us assume that we have a simple idealized arch with no initial loading as in the first image. At the both ends there is connected one wire each, different in colour with little helmets on their opposite ends. It can be observed in second image that once the arch is loaded will give lateral movement of

both ends. Having the two wires crossing each other, the distance between the little helmets will decrease, therefore it can be concluded that any member between this two helmets will be in

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