Structure

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A truss.

A structure is any physical body that is intended to sustain loads. [1] For example an arch, artery_, a bicycle, a building, a dome, a pneumatic tire, a post and lintel system, or a truss.

Contents

1   Function

2   Properties

A structure that recovers its original shape when the load causing the deflection is removed is called elastic. Many solids including rubber, wood, and even steel are elastic. [2] Structures that remain distorted when the load is removed are called plastic. [2] For example, clay and gold.

3   Behavior

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Simplified model of distortion of interatomic bonds under mechanical strain.

When any structure deflects under load, the material from which it is made is itself stretched or contracted, internally, throughout all of its parts and in due proportion, down to a molecular scale. When we deform a stick or a steel spring, the atoms and molecules of which the material is made have to move further apart, or else squash close together, when the material as a whole is stretched or compressed. The chemical bonds which join the atoms to each other, and so hold the solid together, are strong and stiff. So when the material as a whole is stretched or compressed, this can only be done by stretching or compressing many millions of strong chemical bonds which vigorously resisted being deformed. Thus, these bonds produce the required large forces of reaction. [2]

4   History

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Filippo Brunelleschi's dome of the Florence Cathedral (1436).

The intellectual basis of the rules of medieval craftsmen was not very different from that of a cookbook. By relying on traditional proportions, it is sometimes safe to scale up masonry from small churches to large cathedrals; for other kinds of structures this way of doing things will not work. This is the reason why for a very long time the size of the largest ships remained virtually constant though buildings got bigger and bigger. [1]

The effective beginnings of the serious study of structures may be said to be due to the persecution and obscurantism of the Inquisition_. [1] In 1633, Galileo Galilei fell foul of the Church on account of his revolutionary astronomical discoveries, which were considered to threaten the very bases of religious and civil authority. He was most firmly headed off astronomy and, after his famous recantation, he was allowed to retire to his villa at Arctri, near Florence_. Living there, virtually under house-arrest, he took up the study of the strength of materials at almost seventy. [1]

Galileo was allowed to correspond with scholars in various parts of Europe. Among his many surviving letters, there are several about structure, and his correspondence with Marin Mersenne, a Jesuit priest in France, seemed to have been particularly fruitful. [1]

Edme Marriote, a priest in the wine country, spent most of his life working on the laws of terrestrial mechanics and on the strength of rods in tension and in bending. Under Louis XIV he helped to found the `French Academy of Sciences`_, and was in favor with both the Church and State. By Mariotte's time, the whole subject of the behavior of materials and structures under loads was beginning to be called the science of elasticity. [1]

The achievement of understanding that if a structure is to resist a load (and not break or move), it can only do so by pushing back at it with an equal and opposite force is credited to Robert Hooke. [2] By 1676, Hooke saw clearly that not only must solids resist weights or other mechanical loads by pushing back at them, but also that every kind of solid changes its shape when a mechanical force is applied to it (by stretching or contracting itself) and that it is this change of shape which enables the solid to do the pushing back. Thus, when we hang a brick from the end of a piece of string, the string gets longer, and it is just this stretching which enables the string to pull upwards on the brick and so prevent it from falling. All materials and structures deflect when loaded, although to greatly varying extents, and unless this deflection is too large for the purposes of the structure, it is not in any way a fault but rather an essential character without which no structure would be able to work. [2]

Hooke tested a variety of objects made from various materials and having various geometrical forms such as springs and wires and beams. Having hung a succession of weights upon them and measured the resulting deflects, he showed that the deflection in any given structure was proportional to the load ("Hooke's Law'), that is, a load of 200 pounds would cause twice as much deflection as a load of 100 pounds. Hooke published his experiments in 1679 in a paper called "De potentia restitutiva or of a spring". The paper contained the famous statement "ut tension sic vis" ("as the extension, so the force"), known since as "Hooke's law". [2]

Throughout the eighteenth century, little real progress was made in the study of elasticity. The form in which Hooke's originally proposed Hooke's law was of limited use to engineers since the deflection of a structure is affected both by its size and geometrical shape, and those two effects needs to be quantified for them to be useful. Further, a bitter enmity existed between `Issac Newton`_ and Hooke, and Newton, who lived on for twenty-five years after Hooke died, devoted a good deal of this time to denigrating Hooke's memory and the importance of applied science. [2]

Apart from the prejudices of the eighteenth century, the main reason why the science of elasticity got stuck for so long was that the few scientists who did study it tried to deal with forces and deflects by considering the structure as the whole rather than by analyzing the forces and extensions which could be shown to exist at any given point within the material. All through the eighteenth century and well into the nineteen, clever men such Leonhard Euler and `Thomas Young`_ attempted to solve what now seem to be straightforward problems through incredible intellectual contortions. [3]

Augstin Cauchy put forward the ideas of stress and strain_ in a generalized form in a paper to the French Academy of Sciences in 1822. [3] This enabled the science of elasticity to become a practical tool for engineers rather than a hunting-ground for philosophers.

Cauchy perceived that this idea of stress can be used not only to predict when a material will break but also to describe the state at any point inside a solid in a much more general way.

5   References

[1](1, 2, 3, 4, 5, 6) J.E. Gordon. 1978. Structures. Chapter 1: The structure in our lives.
[2](1, 2, 3, 4, 5, 6, 7) J.E. Gordon. 1978. Structures. Chapter 2: Why structures carry loads.
[3](1, 2, 3, 4) J.E. Gordon. 1978. Structures. Chapter 3: The invention of stress and strain.

[3]

The concept of the elastic conditions at a specified point inside a material is the concept of stress and strain.

Stress in a solid is like pressure in a liquid or gas, except that while pressure acts in three directions, pressure acts only in one. Stress is a measure of how hard the molecules which make up a material are being pushed together or pulled apart as a result of external forces.

The stress in any direction at a given point in a material is simply the force which happens to be acting in that direction at that point, divided by the area on which the force acts. If we call the stress at a certain point s, then:

stress = s = force / area = P / A

where P = force and A is the area over which the force can be considered as acting.

For example, if we hang a 5kg brick from the end of a piece of string that has a cross-section of 2 square mm, then the stress in the string will be 2.5 kg of force per square millimeter. [3]

Stress can be expressed in any units of force dived by and units of area, however it it typically described in pound per square inch (p.s.i.) in English-speaking countries and kilograms per square centimeter (kgf/cm^2) elsewhere.

The stress in a material is a condition which exists at a point and it is not especially associated with any particular cross-sectional area.

Stress tells us how hard, with how much force, the atoms at any point in a solid are being pulled apart.

Strain tells us how far they are being pulled apart. Thus if a rod which has original length L is caused to stretch by an amount l by the action of a force on it, then the strain in a rod e will be l / L. For example, if the original length of a string is 2 meters and the weight of a brick causes it to stretch by 1 centimeter, then the strain in the string is (1 / 200) cm or 0.5% m. Engineering strains are usually quite small, so engineers often express strains as percentages to reduce the opportunity for confusion with decimal points.

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When we want to test a material we generally make a "test-piece" from it. The shapes vary a good deal, but usually they have a steam on which measurements can be made, and two thickened ends by which they can be attached to the testing machine.

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The stress in the steam of the test-piece is obtained by dividing the load recorded at each stage by the area of its cross-section. The strain of the material is obtained by measuring the extension of the steam by means of a sensitive device called an extensometer. With equipments of this kind, it is generally easy to measure the stress and strain which occur within a specimen of a material as we increase the load upon it. The relationship between stress and strain for that material is given by the graph between them which is called a stress-strain diagram.

When we plot the stress-strain diagrams for metals and a number of other common solids, we find that at least for moderate stresses the graph is a straight line. When this is so we speak of the material as "obeying Hooke's law" or sometimes of a "Hookean material".

We also find that the slope of the straight part of the graph varies greatly for different materials.