## Introduction

Statics is a branch of Mechanics that studies the forces and their effects in rigid bodies in balance. Just like humans, objects and buildings need a skeleton too. This is known as the structure. Bridges, amusement parks, chairs, and many other objects that surround us, have structures, if they didn’t they wouldn’t be able to stand. Structures are really important parts of buildings and objects.

If the structure collapses, the consequences can be really awful. Even in objects that look simple, there is an analysis of forces acting on it. Structures need to be rigid, resistant, and steady. Even in the most creative buildings, architects, and engineers responsible for them must have had to take into account the form and structure before carrying out its construction. A great example of this is the Eiffel Tower. Its engineer’s Gustave Eiffel. He was a French engineer and constructor, whose most famous constructions are the Statue of Liberty and the Eiffel Tower.

## Gustav Eiffel

The Eiffel Tower was the winning entry in a competition for a ‘centerpiece’ for the Paris Exposition of 1889. The design by an engineer named Gustav Eiffel was selected from over three hundred entries for its striking design and for its economical structure which displayed the French prowess in metal construction. The Eiffel Tower is an iron tower built on the Champ de Mars beside the Seine River in Paris. The tower has become a global icon of France and is one of the most recognizable structures in the world.

The imposing tower, with its 6300 tons of iron forged in 18000 pieces united by 2500000 rivets, has a height of 300 meters. It was built between the years 1887 and 1889 for the Universal Exposition of 1889 en Paris, France. It was built in over two years and was the highest building in the world for over 40 years, until the construction in 1930 of the Chrysler Building in New York. In 2003, the Tower celebrated the fact that it has welcomed over 200 million visitors in 114 years. Royalty, stars, tourists, international celebrities, all form part of the history of one of Paris’s most astounding jewels.

As with the Pyramids, the Leaning Tower of Pisa, the Acropolis, the Coliseum, and the Statue of Liberty, the Eiffel Tower stirs the curiosity and evokes admiration. Alexandre Gustave Eiffel Born in Dijon, Cote-d’Or, France on December 15, 1832. An engineering innovator and a major contributor to 19th-century engineering. Eiffel graduated at the Ecole Centrale des Arts et Manufactures. He was awarded a double degree in literature and science. He was hired by a Paris construction firm specializing in railway equipment, becoming chief of research for the General Railway Equipment Co.

Eiffel later became the one who reshaped the face of Paris in the 19th century. Most of Eiffel’s bridges or other structures are still in use. He got a place in history by the famous Eiffel Tower, Paris landmark, that tapering, modular black wrought iron structure rising almost 1000 feet near the Seine River. Eiffel was proud of his tower, but he felt that it cast a long shadow over his career and prevented his public and professional appreciation of his larger talents as an engineer and researcher. I ought to be jealous of the tower; it is much more famous than I am,” he once said. “People seem to think it is my only work, whereas I have done other things, after all. ” Some of his works to his credit include some of the longest-span iron bridges, the internal bracing for the Statue of Liberty, modular buildings. He also contributes to research into aerodynamics, meteorology, and telecommunications. Eiffel was described by chroniclers as “conceiving his structures as a complete whole rather than in two mental phases, one relating to form and the other to matter … e thought in materials. ” He eschewed secrecy and routinely permitted publication of blueprints and calculation of his completed works. The great architect Le Corbusier said of Eiffel, “His calculations were always inspired by an admirable instinct for proportion; his goal was elegance. ” Eiffel conceives a principle that helps him to construct the Eiffel tower, after that he uses the same method in different things this was called the Gustav Eiffel construction principle. This principle is: Suppose for an instant that each side is equipped with simple trusses forming a shell of a structure resistant to strong winds named P’, P”, P”‘, PIV.

## Facts about Eiffel Tower

Eiffel Tower We already told about some of the histories of the Eiffel tower, and this investigation only focus on structural analysis to help us to understand the application of statics on a daily basis.

Here are some facts of the Eiffel tower some are taken from the official page of the tower:

- The Eiffel Tower, a symbol of innovative technique at the end of the 19th century, has maintained its universal image.
- The most spectacular transformations are also those that are the most visible: it has changed color six times in its lifetime and its lighting effects have been designed at different moments to decorate the tower for a day or for longer periods of time.
- The developments that have been added to the different levels such as the various pavilions and constructions on the first and second floors.
- Also the roofing of the galleries and shelters providing refuge from bad weather conditions including the landing stages, covered walkways, etc. as well as the numerous technical and service quarters and the spaces located directly over or under these levels.
- The elevator and staircases – construction or replacement of – and the elevator platform could also fall into this same category of developments, except in cases where they have become integrated parts of the structure.
- In fact, to obtain the 300 meters, the Tower is basically composed of two elements – a base, which is a sort of bar stool, very sturdy, standing on 4 main pillars that are bonded and extended with a much lighter batter at the smaller level that constitutes the second floor, – a tower firmly attached a top.
- The greatest difficulty in erecting it was the bonding of the four main pillars on the first floor.
- The erection of the pillars – auto-stable – above the first floor was less difficult. As for the tower, it was erected with even less difficult, apart from working at heights.

The parts used to construct the Tower All of the iron came from the factories of Mr. Dupont and Mr. Found, blacksmiths located in Pompey (Meurthe-et-Moselle), who were represented in Paris by their director Mr. A. Pregre The rivets came from Mr. Letroyeur and Mr. Bouvard in Paris. The quality was that of boiler or locomotive rivets. Initial height: 312 m (to the top of the flagpole). Current height (including antennas): 324 m. The deepest foundations: (North and West) lie 15 meters underground. In each of these foundations, four pillars of masonry are built, which bear the four uprights of each leg of the Tower, known as rafters. Total weight: 10,100 tons. Weight of the iron structure: 7,300 tons. Space between the pillars: 1st platform: 4,415 m2 2nd inner platform: 1,430 m2 rd inner platform: 250 m2. Height of the platforms: 1st platform: 57 m 2nd inner platform: 115 m 3rd inner platform: 276 m. A number of steps in the East pillar to the top: 1,665. The number of rivets (total): 2,500,000. Weight of paint used: 60 tons for each repainting campaign. The number of elevators: From the ground to the second floor: 5 (one in the East pillar, one in the West pillar, one in the North pillar, one private elevator in the South pillar for the “Jules Verne” restaurant and one good elevator in the South pillar). From the second floor to the top: two sets of two duo lifts. Elevator speed: 2 meters/second. Passenger flow and capacity of the elevators

North pillar: 920 persons/hour East pillar: 650 persons/hour West pillar: 650 persons/hour Duolifts: 1,140 persons/hour Jules Verne: 10 persons/ascent Goods elevator in the South pillar: 30 persons or 4 tons/ascent. The number of persons working in the Tower: SNTE: 280 Restaurants: 240 Souvenirs: 50 Other: around 50. Number of antennas: 120 The Tower is made exclusively of wrought iron, although steel was available at the time. Eiffel chose iron because steel was more expensive since it was a new development, and also because he was experienced and confident in iron design.

## Structural Analysis

First, we need to take the geometry of the Eiffel tower.

At the time it was built, many people were shocked by its daring shape, but Eiffel and his engineers understood the importance of wind forces and knew that if they were going to build such a tall structure, they had to be certain it would withstand the wind. “Not to what phenomenon did I give primary concern in designing the Tower? It was wind resistance. Well then! I hold that the curvature of the monument’s four outer edges, which is as mathematical calculation dictated it should be(…) will give a great impression of strength and beauty, for it will reveal to the eyes of the observer the boldness of the design as a whole” -translated from the French newspaper Le Temps of 14 February 1887.

As we see the form of the Eiffel tower is not a perfect parabola, instead is a result of mathematical equations and graphical results that lead to the true shape of the tower. As we discussed Gustav was worried about the wind factor. Eiffel’s engineers used graphical results to calculate the strength needed to support its tremendous weight, as well as empirical evidence to account for the effects of wind. During years mathematicians and architects tried to find an equation that suits the shape of the tower. An equation was created by French Eiffel Tower aficionado Christophe Chouard, who posted it on his Web site and challenged engineers and mathematicians worldwide to find its solution. This is the equation:

At any height on the Tower, the moment of the weight of the higher part of the Tower, up to the top, is equal to the moment of the strongest wind on this same part. Writing the differential equation of this equilibrium allows us to find the “harmonious equation” that describes the shape of the Tower. Writing the equation: Let A be a point on the edge of the Tower. Let x be the distance between the top of the Tower and A. Let P(x) be the weight of the part of the Tower above A, up to the top of the Tower. Let f(x) be the half-width of the Tower at A. The moment of the weight of the Tower relative to point A is equal to P(x)·f(x). Let us consider a slice of the Eiffel Tower located at a distance t from the top of the Tower, its thickness being equal to it.

Viewed from the top, this slice looks like a square whose width is 2·f(t). The forces applying on this slice are 1. its weight dP(t), which is proportional to its volume: dP(t)=k·4·f(t)2·dt 2. the horizontal wind dV(t), which is proportional to the slice’s surface to the wind: dV(t)=K·2·f(t)·dt Writing the differential equation: The absolute value of the moment of dP(t) relative to point A is equal to dP(t)·f(x). The absolute value of the moment of dV(t) relative to point A is equal to dV(t)·(x-t). These 2 moments relative to point A should be equal: Let a be. The function, which gives the width of the Eiffel Tower as a function of the distance from the top, is a solution to the following equation:

Weidman of the CU-Boulder mechanical engineering department found one solution – a downward-facing parabola, but it has the wrong curvature. In 2003 Weidman was introduced to Professor Iosif Pinellas, an expert in mathematical analysis. Calculations by Pinellas showed that all existing solutions to Chouard’s equation must be either parabola-like or “explode to infinity” at the top of the tower. Weidman began to read more about the life of Eiffel and his construction efforts. He tracked down a copy of a communication from Eiffel to the French Society of Civil Engineers dated March 30, 1885, the document affirmed that Eiffel planned to counterbalance wind pressure with the tension between the tower’s construction elements. “Eiffel discovered this form of construction produces no load in the diagonal truss elements commonly used to counteract the bending moment, or torque, of the wind, and hence those truss members could be eliminated,” Weidman said. “This allows for a reduction of the tower weight and reduces the surface area exposed to the wind. ” Based on the information, Weidman derived a new equation for the skyline profile – one that “embraces Eiffel’s deep concern for the effects of wind-loading on the tower,” he said. He found an exact solution of the equation in the form of an exponential function that closely matches the shape of the tower’s upper half. Plotting the actual shape of the tower reveals two separate exponential sections that are hooked together. Since Eiffel did not seem confident in estimating the wind torque on the tower, the “overdesigned” the bottom section.

The Eiffel tower, notes Weidman, “is a structural form molded by the wind. ” “Eiffel was worried about the wind throughout his building career,” The mathematical calculations dictating the shape of the Tower have been examined in a recent article in the American Journal of Physics by Joseph Gallant. The force of the wind (dF) produces a torque around the bottom left corner of the tower which is countered by the force of the Tower’s weight (dW) By balancing the maximum torque generated by the wind with the torque generated by the Tower’s own weight, has derived an equation which describes the shape of the Tower: Where f(x) is the half? width of the Tower at height, f0 is the half? width of the Tower at the ground and w(x) is the maximum wind pressure the Tower can withstand at a height. Eiffel, experienced in designing open lattice structures, allowed for a large safety margin by designing the Tower to withstand wind pressures of 4 kN/m2. The fastest winds recorded at the Tower reached a speed of 214 km/h in 1999 and would have produced pressures of just 2. 28 kN/m2. The structural analysis is based on a previous structural analysis made by David P. Billington. Therefore the images and calculations made are copyright from the author and note that this is only for investigation purposes.

The first step is to idealize the tower, this for practical proposes, therefore the structural analysis of the real tower is too complex for the investigation. There are four observation decks. The widths of the Tower corresponding to each of these heights have been calculated from the equations for a parabola, which is an idealization of the Tower’s shape; its true shape we have already discussed. With these two equations, we can calculate the height, width, and angle of the tower. The basic structures of the Tower are lattice-work columns at each of the four corners of the Tower but are lightweight columns. Such composite structural elements require detailed calculations in analysis, so the composite columns are idealized by single solid cross-section columns, each with an area of 800 square inches. We are also going to simplify it making the analysis in two dimensions. So we are going to combine the two front and the two back columns into two of double thickness.

This results in an area of 1600 square inches for each of the simplified supports. Overall internal forces are found from an analysis centered on the tower axis. Internal forces in the individual elements, which act on the curved element axes, are found from the overall forces. When the tower axis is used, the connections between the columns will be idealized as continuous. They are actually continuous above the second platform, but below this point, they are made only by the lower platform and the ground. Now for the analysis, we have to find the external and internal loads that affect the structure of the tower. The weight of the Tower is known to be 18,800 kips.

This weight is not distributed uniformly along with the height of the Tower; there is more material at the base than near the top. For our purposes, the weight will be divided among the three portions defined earlier. For the live load as we call it the weight of people and machinery on the platforms. This load is not uniform and constant but for a practical purpose, an estimate of the live load is set to 50 pounds per square foot. This live load is calculated over the two lower platforms where it is most significant. The first platform is 240 square feet and the second platform is 110 square feet.

This load acts through the centroid of the Tower, which is located 257 feet above the ground. The wind pressure on the Tower is stronger near the top than at the bottom, but the wind force is fairly uniform because the Tower is tapered. This analysis will use the assumption that the wind is uniform load acting all along with the Tower and only acting in one direction, in this case, the left side of the tower, but a conservatively high force is used to simulate high wind speeds.

The centroid of this force is halfway up the Tower so P, the idealized point wind load, acts at this point. Now we can find the reactions of the tower. The overall reactions at the base of the Tower are easily found from the wind and gravity loads. Overall vertical and horizontal reactions will develop to balance the respective loads. A moment reaction will also develop to balance the horizontal load applied through its centroid a distance l/2 from the support. The reactions at the base of each column are necessary to find the internal forces in the individual columns.

Each column will logically develop half of the horizontal and vertical reactions found for the entire structure. So we divide V and H into two. Each colum supports 11,140kN vertical and 1,280kN horizontal. The wind force will create a higher vertical reaction in the leeward support and a lower vertical reaction in the windward support because the wind alone would create compression in the leeward support and tension in the windward support. The internal forces in the columns are found using the reactions, the loads, and the principles of equilibrium. The simplest internal forces are the axial ones, which result from the vertical loads and reactions. Forces from Vertical Loads

The axial force in the columns is the inclined force, N, shown acting along the axis of one column. This force is equal to the combination of a vertical and a horizontal component. The gravity loads and reactions are vertical and create the vertical component; the horizontal component develops because the vertical forces are carried in an inclined column. These three forces create a ‘force polygon’ where, drawn head to tail, the scaled components are equal to the axial force, and three forms a closed shape. The magnitudes of the horizontal components and the axial force can be found from the vertical force and the angle of the column axis with trigonometry.

The overall bending moment from the horizontal wind load will produce tension (T) in one column and compression (C) in the other. The values of the bending moment all along the height of the Tower must be known to find the resulting tension and compression forces in the columns.

This calculation uses the bending moments found around the tower axis to find axial forces that act along the column axes. At the base of the Tower the moment force is M = 1,260,000 ft-k (the reaction force), and the width, d, is 328 feet. Therefore, Where T = +3,850 k and C = -3,850 k because compression forces are designated as negative. Both of these forces are vertical reactions to the wind. The axial forces in the inclined elements resulting from the wind load are found from the vertical reactions and the horizontal reactions to the wind found earlier. On the leeward side, where the wind creates a vertical compression reaction, Vc and H contribute part of their forces along the axis of the column.

The width of the base is equally capable of resisting the bending moment at the base as the width at the second platform is capable of resisting the bending moment at that point. Total Axial Forces Now that the axial forces from both vertical and horizontal loads have been calculated, the total axial forces in the column supports are found by simply adding the axial force found from the two loads. Compression forces are negative and tension forces are positive. This is shown in the first diagram, below. The second diagram shows the corresponding forces on the second platform. At this point (the second platform) the windward column experiences tension forces rather than the compression found near the base. This creates no structural problems because wrought iron resists tension just as well as compression.

The explanations used and the estimates found in this analysis are all valid, and accurate, but still, the analysis must be viewed as a simplification of the actual situation. A brief review of the assumptions used in this analysis will emphasize its simplicity; the actual Tower’s geometry has been idealized in a number of critical ways by using solid columns and a two-dimensional model, and the loads, both dead and wind, were simplified.

## Conclusion

Statics is a branch of Mechanics that studies the forces and their effects in rigid bodies in balance. Statics are present everywhere, every structure that we see are subject to the principles of statics. Like in the investigation the Eiffel tower is subject to reactions, and bending moments, so this where statics come in.

Thanks to the principles and laws of static they can see how much the Eiffel tower can hold, and what the Eiffel tower need to fight the bending moment, and its own weight. Every standing structure needs the principles of statics, and even in our daily life, we interact with statics, like a door. We see that we require statics in every aspect of our life.

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