1.
Engineering
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The term Engineering is derived from the Latin ingenium, meaning cleverness and ingeniare, meaning to contrive, devise. Engineering has existed since ancient times as humans devised fundamental inventions such as the wedge, lever, wheel, each of these inventions is essentially consistent with the modern definition of engineering. The term engineering is derived from the engineer, which itself dates back to 1390 when an engineer originally referred to a constructor of military engines. In this context, now obsolete, a referred to a military machine. Notable examples of the obsolete usage which have survived to the present day are military engineering corps, the word engine itself is of even older origin, ultimately deriving from the Latin ingenium, meaning innate quality, especially mental power, hence a clever invention. The earliest civil engineer known by name is Imhotep, as one of the officials of the Pharaoh, Djosèr, he probably designed and supervised the construction of the Pyramid of Djoser at Saqqara in Egypt around 2630–2611 BC. Ancient Greece developed machines in both civilian and military domains, the Antikythera mechanism, the first known mechanical computer, and the mechanical inventions of Archimedes are examples of early mechanical engineering. In the Middle Ages, the trebuchet was developed, the first steam engine was built in 1698 by Thomas Savery. The development of this gave rise to the Industrial Revolution in the coming decades. With the rise of engineering as a profession in the 18th century, similarly, in addition to military and civil engineering, the fields then known as the mechanic arts became incorporated into engineering. The inventions of Thomas Newcomen and the Scottish engineer James Watt gave rise to mechanical engineering. The development of specialized machines and machine tools during the revolution led to the rapid growth of mechanical engineering both in its birthplace Britain and abroad. John Smeaton was the first self-proclaimed civil engineer and is regarded as the father of civil engineering. He was an English civil engineer responsible for the design of bridges, canals, harbours and he was also a capable mechanical engineer and an eminent physicist. Smeaton designed the third Eddystone Lighthouse where he pioneered the use of hydraulic lime and his lighthouse remained in use until 1877 and was dismantled and partially rebuilt at Plymouth Hoe where it is known as Smeatons Tower. The United States census of 1850 listed the occupation of engineer for the first time with a count of 2,000, there were fewer than 50 engineering graduates in the U. S. before 1865. In 1870 there were a dozen U. S. mechanical engineering graduates, in 1890 there were 6,000 engineers in civil, mining, mechanical and electrical. There was no chair of applied mechanism and applied mechanics established at Cambridge until 1875, the theoretical work of James Maxwell and Heinrich Hertz in the late 19th century gave rise to the field of electronics
2.
Force
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In physics, a force is any interaction that, when unopposed, will change the motion of an object. In other words, a force can cause an object with mass to change its velocity, force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity and it is measured in the SI unit of newtons and represented by the symbol F. The original form of Newtons second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. In an extended body, each part usually applies forces on the adjacent parts, such internal mechanical stresses cause no accelation of that body as the forces balance one another. Pressure, the distribution of small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate. Stress usually causes deformation of materials, or flow in fluids. In part this was due to an understanding of the sometimes non-obvious force of friction. A fundamental error was the belief that a force is required to maintain motion, most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved-on for nearly three hundred years, the Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known, in order of decreasing strength, they are, strong, electromagnetic, weak, high-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction. Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was famous for formulating a treatment of buoyant forces inherent in fluids. Aristotle provided a discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotles view, the sphere contained four elements that come to rest at different natural places therein. Aristotle believed that objects on Earth, those composed mostly of the elements earth and water, to be in their natural place on the ground. He distinguished between the tendency of objects to find their natural place, which led to natural motion, and unnatural or forced motion
3.
Beam (structure)
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A beam is a structural element that primarily resists loads applied laterally to the beams axis. Its mode of deflection is primarily by bending, the loads applied to the beam result in reaction forces at the beams support points. The total effect of all the acting on the beam is to produce shear forces and bending moments within the beam. Beams are characterized by their manner of support, profile, length, historically beams were squared timbers but are also metal, stone, or combinations of wood and metal such as a flitch beam. Beams generally carry vertical gravitational forces but can also be used to carry horizontal loads, the loads carried by a beam are transferred to columns, walls, or girders, which then transfer the force to adjacent structural compression members. In light frame construction joists may rest on beams, in carpentry a beam is called a plate as in a sill plate or wall plate, beam as in a summer beam or dragon beam. In engineering, beams are of types, Simply supported - a beam supported on the ends which are free to rotate and have no moment resistance. Fixed - a beam supported on both ends and restrained from rotation, over hanging - a simple beam extending beyond its support on one end. Double overhanging - a simple beam with both ends extending beyond its supports on both ends, continuous - a beam extending over more than two supports. Cantilever - a projecting beam fixed only at one end, trussed - a beam strengthened by adding a cable or rod to form a truss. In the beam equation I is used to represent the moment of area. It is commonly known as the moment of inertia, and is the sum, about the axis, of dA*r^2, where r is the distance from the neutral axis. Therefore, it not just how much area the beam section has overall. The greater I is, the stiffer the beam in bending, internally, beams experience compressive, tensile and shear stresses as a result of the loads applied to them. Above the supports, the beam is exposed to shear stress, there are some reinforced concrete beams in which the concrete is entirely in compression with tensile forces taken by steel tendons. These beams are known as prestressed concrete beams, and are fabricated to produce a more than the expected tension under loading conditions. High strength steel tendons are stretched while the beam is cast over them, then, when the concrete has cured, the tendons are slowly released and the beam is immediately under eccentric axial loads. This eccentric loading creates a moment, and, in turn
4.
Virtual work
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Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement will be different for different displacements, among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the according to the principle of least action. The work of a force on a particle along a displacement is known as the virtual work. The principle of work had always been used in some form since antiquity in the study of statics. It was used by the Greeks, medieval Arabs and Latins, working with Leibnizian concepts, Johann Bernoulli systematized the virtual work principle and made explicit the concept of infinitesimal displacement. He was able to solve problems for both bodies as well as fluids. Bernoullis version of virtual work law appeared in his letter to Pierre Varignon in 1715 and this formulation of the principle is today known as the principle of virtual velocities and is commonly considered as the prototype of the contemporary virtual work principles. In 1743 DAlembert published his Traite de Dynamique where he applied the principle of work, based on the Bernoullis work. His idea was to convert a dynamical problem into static problem by introducing inertial force, consider a point particle that moves along a path which is described by a function r from point A, where r, to point B, where r. It is possible that the moves from A to B along a nearby path described by r + δr. The variation δr satisfies the requirement δr = δr =0, the components of the variation, δr1, δr2 and δr3, are called virtual displacements. This can be generalized to a mechanical system defined by the generalized coordinates qi. In which case, the variation of the qi is defined by the virtual displacements δqi. Virtual work is the work done by the applied forces. When considering forces applied to a body in equilibrium, the principle of least action requires the virtual work of these forces to be zero. Consider a particle P that moves from a point A to a point B along a trajectory r and it is important to notice that the value of the work W depends on the trajectory r. Suppose the force F is the same as F, the variation of the work δW associated with this nearby path, known as the virtual work, can be computed to be δ W = W ¯ − W = ∫ t 0 t 1 d t
5.
Direct integration of a beam
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Direct integration is a structural analysis method for measuring internal shear, internal moment, rotation, and deflection of a beam. These constants are determined by using either the forces at supports, for internal shear and moment, the constants can be found by analyzing the beams free body diagram. For rotation and displacement, the constants are found using conditions dependent on the type of supports, for a cantilever beam, the fixed support has zero rotation and zero displacement. For a beam supported by a pin and roller, both the supports have zero displacement, take the beam shown at right supported by a fixed pin at the left and a roller at the right. There are no applied moments, the weight is a constant 10 kN, taking x as the distance from the pin, w =10 Integrating, V = − ∫ w d x = −10 x + C1 where C1 represents the applied loads. For a more realistic situation, such as a load of 1 kN and an EI value of 5,000 kN·m². Note that for the rotation θ the units are divided by meters. This is because rotation is given as a slope, the displacement divided by the horizontal change. Mechanics Materials, sixth edition, Pearson Prentice Hall,2005, beam Deflection by Double Integration Method
6.
Macaulay's method
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Macaulay’s method is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. Use of Macaulay’s technique is convenient for cases of discontinuous and/or discrete loading. Typically partial uniformly distributed loads and uniformly varying loads over the span, the first English language description of the method was by Macaulay. The actual approach appears to have developed by Clebsch in 1862. This equation is simpler than the beam equation and can be integrated twice to find w if the value of M as a function of x is known. Using these integration rules makes the calculation of the deflection of Euler-Bernoulli beams simple in situations where there are multiple point loads, the Macaulay method predates more sophisticated concepts such as Dirac delta functions and step functions but achieves the same outcomes for beam problems. An illustration of the Macaulay method considers a simply supported beam with a single eccentric concentrated load as shown in the adjacent figure, the first step is to find M. e. The Macaulay brackets help as a reminder that the quantity on the right is zero when considering points with x < a, as w =0 at x =0, C2 =0. Assuming that this happens for x < a we have P b x 22 L − P b 6 L =0 or x = ±1 /23 Clearly x <0 cannot be a solution. Even when the load is as near as 0. 05L from the support, hence in most of the cases the estimation of maximum deflection may be made fairly accurately with reasonable margin of error by working out deflection at the centre
7.
Direct stiffness method
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It is a matrix method that makes use of the members stiffness relations for computing member forces and displacements in structures. The direct stiffness method is the most common implementation of the finite element method, in applying the method, the system must be modeled as a set of simpler, idealized elements interconnected at the nodes. The material stiffness properties of elements are then, through matrix mathematics. The structure’s unknown displacements and forces can then be determined by solving this equation, the direct stiffness method forms the basis for most commercial and free source finite element software. The direct stiffness method originated in the field of aerospace, researchers looked at various approaches for analysis of complex airplane frames. These included elasticity theory, energy principles in structural mechanics, flexibility method and it was through analysis of these methods that the direct stiffness method emerged as an efficient method ideally suited for computer implementation. Between 1934 and 1938 A. R. Collar and W. J. Duncan published the first papers with the representation, aeroelastic research continued through World War II but publication restrictions from 1938 to 1947 make this work difficult to trace. Finally, on Nov.61959, M. J. Turner, head of Boeing’s Structural Dynamics Unit, a typical member stiffness relation has the following general form, Q m = k m q m + Q o m where m = member number m. Q m = vector of members characteristic forces, which are unknown internal forces, K m = member stiffness matrix which characterizes the members resistance against deformations. Q m = vector of members characteristic displacements or deformations, Q o m = vector of members characteristic forces caused by external effects applied to the member while q m =0. This implies that r will be the primary unknowns, the member forces Q m help to the keep the nodes in equilibrium under the nodal forces R. K = system stiffness matrix, which is established by assembling the members stiffness matrices k m. This vector is established by assembling the members Q o m, the system stiffness matrix K is square since the vectors R and r have the same size. In addition, it is symmetric because k m is symmetric, where q m can be found from r by compatibility consideration. It is common to have Eq. in a form where q m and Q o m are, respectively, in such case, K and R o can be obtained by direct summation of the members matrices k m and Q o m. The method is known as the direct stiffness method. The advantages and disadvantages of the stiffness method are compared and discussed in the flexibility method article. The first step using the direct stiffness method is to identify the individual elements which make up the structure. Once the elements are identified, the structure is disconnected at the nodes, each element is then analyzed individually to develop member stiffness equations
8.
Plate theory
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In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions, the typical thickness to width ratio of a plate structure is less than 0.1. A plate theory takes advantage of this disparity in length scale to reduce the full three-dimensional solid mechanics problem to a two-dimensional problem, the aim of plate theory is to calculate the deformation and stresses in a plate subjected to loads. Of the numerous theories that have been developed since the late 19th century. These are the Kirchhoff–Love theory of plates The Mindlin–Reissner theory of plates Note, the Kirchhoff–Love theory is an extension of Euler–Bernoulli beam theory to thin plates. The theory was developed in 1888 by Love using assumptions proposed by Kirchhoff and it is assumed that a mid-surface plane can be used to represent the three-dimensional plate in two-dimensional form. If φ α are the angles of rotation of the normal to the mid-surface, If the rotations of the normals to the mid-surface are in the range of 10° to 15°, the strain-displacement relations can be approximated using the von Kármán strains. The equilibrium equations for the plate can be derived from the principle of virtual work, the quantities σ α β are the stresses. It is more convenient to work with the stress and moment results that enter the equilibrium equations and these are related to the displacements by = and = −. The moments corresponding to these stresses are = −2 h 3 E3 The displacements u 10 and u 20 are zero under pure bending conditions, in index notation, w,11110 +2 w,12120 + w,22220 =0. In index notation, w,11110 +2 w,12120 + w,22220 = − q D and in direct notation In cylindrical coordinates, for an orthotropic plate =11 − ν12 ν21. Therefore, =2 h 1 − ν12 ν21, the dynamic theory of plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes. The governing equations simplify considerably for isotropic and homogeneous plates for which the in-plane deformations can be neglected and have the form D = − q −2 ρ h ∂2 w 0 ∂ t 2, where D is the bending stiffness of the plate. For a uniform plate of thickness 2 h, D, =2 h 3 E3, in direct notation Note, the Einstein summation convention of summing on repeated indices is used below. In the theory of plates, or theory of Raymond Mindlin and Eric Reissner. However, the strain is constant across the thickness of the plate. This cannot be accurate since the stress is known to be parabolic even for simple plate geometries. To account for the inaccuracy in the strain, a shear correction factor is applied so that the correct amount of internal energy is predicted by the theory