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59-567: In the context to structural analysis, a structure refers to a body or system of connected parts used to support a load. Important examples related to Civil Engineering include buildings, bridges, and towers; and in other branches of engineering, ship and aircraft frames, tanks, pressure vessels, mechanical systems, and electrical supporting structures are important. To design a structure, an engineer must account for its safety, aesthetics, and serviceability, while considering economic and environmental constraints. Other branches of engineering work on

118-415: A hierarchical organization , but hierarchy makes it easier for a listener to understand and remember the music. In analogy to linguistic terminology, motifs and phrases can be combined to make complete musical ideas such as sentences and phrases . A larger form is known as the period . One such form that was widely used between 1600 and 1900 has two phrases, an antecedent and a consequent , with

177-427: A hierarchy (a cascade of one-to-many relationships), a network featuring many-to-many links , or a lattice featuring connections between components that are neighbors in space. Buildings , aircraft , skeletons , anthills , beaver dams , bridges and salt domes are all examples of load -bearing structures. The results of construction are divided into buildings and non-building structures , and make up

236-417: A pointer that links them together in a particular order. Out of these any number of other data structures can be created such as stacks , queues , trees and hash tables . In solving a problem, a data structure is generally an integral part of the algorithm . In modern programming style, algorithms and data structures are encapsulated together in an abstract data type . Software architecture

295-470: A pressure vessel , plates, shells, and three-dimensional solids. Commercial computer software for structural analysis typically uses matrix finite-element analysis, which can be further classified into two main approaches: the displacement or stiffness method and the force or flexibility method . The stiffness method is the most popular by far thanks to its ease of implementation as well as of formulation for advanced applications. The finite-element technology

354-443: A combination of wood and metal such as a flitch beam . Beams primarily carry vertical gravitational forces , but they are also used to carry horizontal loads such as those due to earthquake or wind, or in tension to resist rafter thrust ( tie beam ) or compression ( collar beam ). The loads carried by a beam are transferred to columns , walls , or girders , then to adjacent structural compression members , and eventually to

413-409: A connecting rod, a truss, a beam, or a column, but also a cable, an arch, a cavity or channel, and even an angle, a surface structure, or a frame. Once the dimensional requirement for a structure have been defined, it becomes necessary to determine the loads the structure must support. Structural design, therefore begins with specifying loads that act on the structure. The design loading for a structure

472-403: A continuous system such as a plate or shell is modeled as a discrete system with a finite number of elements interconnected at finite number of nodes and the overall stiffness is the result of the addition of the stiffness of the various elements. The behaviour of individual elements is characterized by the element's stiffness (or flexibility) relation. The assemblage of the various stiffness's into

531-506: A crystal have a structure that involves repetition of a basic unit called a unit cell . The atoms can be modeled as points on a lattice , and one can explore the effect of symmetry operations that include rotations about a point, reflections about a symmetry planes, and translations (movements of all the points by the same amount). Each crystal has a finite group, called the space group , of such operations that map it onto itself; there are 230 possible space groups. By Neumann's law ,

590-404: A half cadence in the middle and a full cadence at the end providing punctuation. On a larger scale are single-movement forms such as the sonata form and the contrapuntal form , and multi-movement forms such as the symphony . A social structure is a pattern of relationships. They are social organizations of individuals in various life situations. Structures are applicable to people in how

649-863: A master stiffness matrix that represents the entire structure leads to the system's stiffness or flexibility relation. To establish the stiffness (or flexibility) of a particular element, we can use the mechanics of materials approach for simple one-dimensional bar elements, and the elasticity approach for more complex two- and three-dimensional elements. The analytical and computational development are best effected throughout by means of matrix algebra , solving partial differential equations . Early applications of matrix methods were applied to articulated frameworks with truss, beam and column elements; later and more advanced matrix methods, referred to as " finite element analysis ", model an entire structure with one-, two-, and three-dimensional elements and can be used for articulated systems together with continuous systems such as

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708-457: A multilevel hierarchy of structures employing biominerals and proteins , at the bottom of which are collagen fibrils . In biology , one of the properties of life is its highly ordered structure, which can be observed at multiple levels such as in cells , tissues , organs , and organisms . In another context, structure can also observed in macromolecules , particularly proteins and nucleic acids . The function of these molecules

767-461: A redundant structure is needed so that if a component fails it has backups. A high redundancy is an essential part of the design of several systems in the Space Shuttle . As a branch of philosophy, logic is concerned with distinguishing good arguments from poor ones. A chief concern is with the structure of arguments. An argument consists of one or more premises from which a conclusion

826-434: A society is as a system organized by a characteristic pattern of relationships. This is known as the social organization of the group. Sociologists have studied the changing structure of these groups. Structure and agency are two confronted theories about human behaviour. The debate surrounding the influence of structure and agency on human thought is one of the central issues in sociology. In this context, agency refers to

885-451: A structural engineer must determine information such as structural loads , geometry , support conditions, and material properties. The results of such an analysis typically include support reactions, stresses and displacements . This information is then compared to criteria that indicate the conditions of failure. Advanced structural analysis may examine dynamic response , stability and non-linear behavior. There are three approaches to

944-471: A variety of diagrams called structural formulas . Lewis structures use a dot notation to represent the valence electrons for an atom; these are the electrons that determine the role of the atom in chemical reactions. Bonds between atoms can be represented by lines with one line for each pair of electrons that is shared. In a simplified version of such a diagram, called a skeletal formula , only carbon-carbon bonds and functional groups are shown. Atoms in

1003-404: A wide variety of non-building structures . A structural system is the combination of structural elements and their materials. It is important for a structural engineer to be able to classify a structure by either its form or its function, by recognizing the various elements composing that structure. The structural elements guiding the systemic forces through the materials are not only such as

1062-408: Is inferred . The steps in this inference can be expressed in a formal way and their structure analyzed. Two basic types of inference are deduction and induction . In a valid deduction, the conclusion necessarily follows from the premises, regardless of whether they are true or not. An invalid deduction contains some error in the analysis. An inductive argument claims that if the premises are true,

1121-573: Is a box (a square shell); the most efficient shape for bending in any direction, however, is a cylindrical shell or tube. For unidirectional bending, the Ɪ-beam or wide flange beam is superior. Efficiency means that for the same cross sectional area (volume of beam per length) subjected to the same loading conditions, the beam deflects less. Other shapes, like L-beam (angles), C (channels) , T-beam and double-T or tubes, are also used in construction when there are special requirements. This system provides horizontal bracing for small trenches, ensuring

1180-664: Is actually a numerical method for solving differential equations generated by theories of mechanics such as elasticity theory and strength of materials. However, the finite-element method depends heavily on the processing power of computers and is more applicable to structures of arbitrary size and complexity. Regardless of approach, the formulation is based on the same three fundamental relations: equilibrium , constitutive , and compatibility . The solutions are approximate when any of these relations are only approximately satisfied, or only an approximation of reality. Each method has noteworthy limitations. The method of mechanics of materials

1239-434: Is determined by their shape as well as their composition, and their structure has multiple levels. Protein structure has a four-level hierarchy. The primary structure is the sequence of amino acids that make it up. It has a peptide backbone made up of a repeated sequence of a nitrogen and two carbon atoms. The secondary structure consists of repeated patterns determined by hydrogen bonding . The two basic types are

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1298-467: Is limited to very simple structural elements under relatively simple loading conditions. The structural elements and loading conditions allowed, however, are sufficient to solve many useful engineering problems. The theory of elasticity allows the solution of structural elements of general geometry under general loading conditions, in principle. Analytical solution, however, is limited to relatively simple cases. The solution of elasticity problems also requires

1357-488: Is much higher than that for solid cross sections such a rod or bar. In this way, stiff beams can be achieved with minimum weight. Thin walled beams are particularly useful when the material is a composite laminate . Pioneer work on composite laminate thin walled beams was done by Librescu . The torsional stiffness of a beam is greatly influenced by its cross sectional shape. For open sections, such as I sections, warping deflections occur which, if restrained, greatly increase

1416-469: Is now sophisticated enough to handle just about any system as long as sufficient computing power is available. Its applicability includes, but is not limited to, linear and non-linear analysis, solid and fluid interactions, materials that are isotropic, orthotropic, or anisotropic, and external effects that are static, dynamic, and environmental factors. This, however, does not imply that the computed solution will automatically be reliable because much depends on

1475-404: Is often specified in building codes . There are two types of codes: general building codes and design codes, engineers must satisfy all of the code's requirements in order for the structure to remain reliable. There are two types of loads that structure engineering must encounter in the design. The first type of loads are dead loads that consist of the weights of the various structural members and

1534-405: Is related linearly to strain, that the material (but not the structure) behaves identically regardless of direction of the applied load, that all deformations are small, and that beams are long relative to their depth. As with any simplifying assumption in engineering, the more the model strays from reality, the less useful (and more dangerous) the result. There are 2 commonly used methods to find

1593-489: Is the specific choices made between possible alternatives within a framework. For example, a framework might require a database and the architecture would specify the type and manufacturer of the database. The structure of software is the way in which it is partitioned into interrelated components. A key structural issue is minimizing dependencies between these components. This makes it possible to change one component without requiring changes in others. The purpose of structure

1652-420: Is to optimise for (brevity, readability, traceability, isolation and encapsulation, maintainability, extensibility, performance and efficiency), examples being: language choice , code , functions , libraries , builds , system evolution , or diagrams for flow logic and design . Structural elements reflect the requirements of the application: for example, if the system requires a high fault tolerance, then

1711-421: Is zero. Therefore, the magnitude and direction of the reaction forces can be calculated. This type of method uses the force balance in the x and y directions at each of the joints in the truss structure. At A, At D, At C, Although the forces in each of the truss elements are found, it is a good practice to verify the results by completing the remaining force balances. At B, This method can be used when

1770-531: The deflection of beams include "method of virtual work " and the "slope deflection method". Engineers are interested in determining deflections because the beam may be in direct contact with a brittle material such as glass . Beam deflections are also minimized for aesthetic reasons. A visibly sagging beam, even if structurally safe, is unsightly and to be avoided. A stiffer beam (high modulus of elasticity and/or one of higher second moment of area ) creates less deflection. Mathematical methods for determining

1829-648: The infrastructure of a human society. Built structures are broadly divided by their varying design approaches and standards, into categories including building structures, architectural structures , civil engineering structures and mechanical structures. The effects of loads on physical structures are determined through structural analysis , which is one of the tasks of structural engineering . The structural elements can be classified as one-dimensional ( ropes , struts , beams , arches ), two-dimensional ( membranes , plates, slab , shells , vaults ), or three-dimensional (solid masses). Three-dimensional elements were

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1888-444: The parallel axis theorem and the fact that most of the material is away from the neutral axis , the second moment of area of the beam increases, which in turn increases the stiffness. An Ɪ-beam is only the most efficient shape in one direction of bending: up and down looking at the profile as an 'Ɪ'. If the beam is bent side to side, it functions as an 'H', where it is less efficient. The most efficient shape for both directions in 2D

1947-525: The superposition principle to analyze a member undergoing combined loading. Solutions for special cases exist for common structures such as thin-walled pressure vessels. For the analysis of entire systems, this approach can be used in conjunction with statics, giving rise to the method of sections and method of joints for truss analysis, moment distribution method for small rigid frames, and portal frame and cantilever method for large rigid frames. Except for moment distribution, which came into use in

2006-415: The α-helix and the β-pleated sheet . The tertiary structure is a back and forth bending of the polypeptide chain, and the quaternary structure is the way that tertiary units come together and interact. Structural biology is concerned with biomolecular structure of macromolecules. Chemical structure refers to both molecular geometry and electronic structure. The structure can be represented by

2065-417: The 1930s, these methods were developed in their current forms in the second half of the nineteenth century. They are still used for small structures and for preliminary design of large structures. The solutions are based on linear isotropic infinitesimal elasticity and Euler–Bernoulli beam theory. In other words, they contain the assumptions (among others) that the materials in question are elastic, that stress

2124-439: The analysis: the mechanics of materials approach (also known as strength of materials), the elasticity theory approach (which is actually a special case of the more general field of continuum mechanics ), and the finite element approach. The first two make use of analytical formulations which apply mostly simple linear elastic models, leading to closed-form solutions, and can often be solved by hand. The finite element approach

2183-472: The basis for structural analysis. This is usually done using numerical approximation techniques. The most commonly used numerical approximation in structural analysis is the Finite Element Method . The finite element method approximates a structure as an assembly of elements or components with various forms of connection between them and each element of which has an associated stiffness. Thus,

2242-424: The beam forces (internal forces of the beam and the forces that are imposed on the beam support) include the " moment distribution method ", the force or flexibility method and the direct stiffness method . Most beams in reinforced concrete buildings have rectangular cross sections, but a more efficient cross section for a beam is an Ɪ- or H-shaped section which is typically seen in steel construction. Because of

2301-415: 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 compression more than the expected tension under loading conditions. High strength steel tendons are stretched while the beam is cast over them. Then, when

2360-416: The beams are horizontal and carry vertical loads. However, any structure may contain beams, such as automobile frames, aircraft components, machine frames, and other mechanical or structural systems. Any structural element , in any orientation, that primarily resists loads applied laterally across the element's axis is a beam. Historically a beam is a squared timber, but may also be made of metal, stone, or

2419-411: The bottom to enclose an arc of larger radius in tension. This is known as sagging ; while a configuration with the top in tension, for example over a support, is known as hogging . The axis of the beam retaining its original length, generally halfway between the top and bottom, is under neither compression nor tension, and defines the neutral axis (dotted line in the beam figure). Above the supports,

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2478-692: The conclusion is likely. Beam (structure) A beam is a structural element that primarily resists loads applied laterally across the beam's axis (an element designed to carry a load pushing parallel to its axis would be a strut or column). Its mode of deflection is primarily by bending , as loads produce reaction forces at the beam's support points and internal bending moments , shear , stresses , strains , and deflections . Beams are characterized by their manner of support, profile (shape of cross-section), equilibrium conditions, length, and material. Beams are traditionally descriptions of building or civil engineering structural elements, where

2537-472: The concrete has cured, the tendons are slowly released and the beam is immediately under eccentric axial loads. This eccentric loading creates an internal moment, and, in turn, increases the moment-carrying capacity of the beam. Prestressed beams are commonly used on highway bridges. The primary tool for structural analysis of beams is the Euler–Bernoulli beam equation . This equation accurately describes

2596-555: The elastic behaviour of slender beams where the cross sectional dimensions are small compared to the length of the beam. For beams that are not slender a different theory needs to be adopted to account for the deformation due to shear forces and, in dynamic cases, the rotary inertia. The beam formulation adopted here is that of Timoshenko and comparative examples can be found in NAFEMS Benchmark Challenge Number 7. Other mathematical methods for determining

2655-421: The equations of linear elasticity . The equations of elasticity are a system of 15 partial differential equations. Due to the nature of the mathematics involved, analytical solutions may only be produced for relatively simple geometries. For complex geometries, a numerical solution method such as the finite element method is necessary. It is common practice to use approximate solutions of differential equations as

2714-412: The ground. In light frame construction , joists may rest on beams. In engineering, beams are of several types: In the beam equation , the variable I represents the second moment of area or moment of inertia : it is the sum, along the axis, of dA · r , where r is the distance from the neutral axis and dA is a small patch of area. It measures not only the total area of the beam section, but

2773-661: The individual human capacity to act independently and make free choices. Structure here refers to factors such as social class , religion , gender , ethnicity , customs, etc. that seem to limit or influence individual opportunities. In computer science , a data structure is a way of organizing information in a computer so that it can be used efficiently. Data structures are built out of two basic types: An array has an index that can be used for immediate access to any data item (some programming languages require array size to be initialized ). A linked list can be reorganized, grown or shrunk, but its elements must be accessed with

2832-996: The main option available to early structures such as Chichen Itza . A one-dimensional element has one dimension much larger than the other two, so the other dimensions can be neglected in calculations; however, the ratio of the smaller dimensions and the composition can determine the flexural and compressive stiffness of the element. Two-dimensional elements with a thin third dimension have little of either but can resist biaxial traction. The structure elements are combined in structural systems . The majority of everyday load-bearing structures are section-active structures like frames, which are primarily composed of one-dimensional (bending) structures. Other types are Vector-active structures such as trusses , surface-active structures such as shells and folded plates, form-active structures such as cable or membrane structures, and hybrid structures. Load-bearing biological structures such as bones, teeth, shells, and tendons derive their strength from

2891-492: The model and the reliability of the data input. Structure A structure is an arrangement and organization of interrelated elements in a material object or system , or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as biological organisms , minerals and chemicals . Abstract structures include data structures in computer science and musical form . Types of structure include

2950-536: The moment balance, which gives a maximum of 3 equations to find a maximum of 3 unknown truss element forces through which this cut is made. Find the forces FAB, FBD and FCD in the above example The truss elements forces in the remaining members can be found by using the above method with a section passing through the remaining members. Elasticity methods are available generally for an elastic solid of any shape. Individual members such as beams, columns, shafts, plates and shells may be modeled. The solutions are derived from

3009-491: The restriction that there is always some numerical error. Effective and reliable use of this method requires a solid understanding of its limitations. The simplest of the three methods here discussed, the mechanics of materials method is available for simple structural members subject to specific loadings such as axially loaded bars, prismatic beams in a state of pure bending , and circular shafts subject to torsion. The solutions can under certain conditions be superimposed using

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3068-555: The secure installation of utilities. It's specifically designed to work in conjunction with steel trench sheets. A thin walled beam is a very useful type of beam (structure). The cross section of thin walled beams is made up from thin panels connected among themselves to create closed or open cross sections of a beam (structure). Typical closed sections include round, square, and rectangular tubes. Open sections include I-beams, T-beams, L-beams, and so on. Thin walled beams exist because their bending stiffness per unit cross sectional area

3127-584: The solution of a system of partial differential equations, which is considerably more mathematically demanding than the solution of mechanics of materials problems, which require at most the solution of an ordinary differential equation. The finite element method is perhaps the most restrictive and most useful at the same time. This method itself relies upon other structural theories (such as the other two discussed here) for equations to solve. It does, however, make it generally possible to solve these equations, even with highly complex geometry and loading conditions, with

3186-438: The square of each patch's distance from the axis. A larger value of I indicates a stiffer beam, more resistant to bending. Loads on a beam induce internal compressive , tensile and shear stresses (assuming no torsion or axial loading). Typically, under gravity loads, the beam bends into a slightly circular arc, with its original length compressed at the top to form an arc of smaller radius, while correspondingly stretched at

3245-557: The symmetry of a crystal determines what physical properties, including piezoelectricity and ferromagnetism , the crystal can have. A large part of numerical analysis involves identifying and interpreting the structure of musical works. Structure can be found at the level of part of a work, the entire work, or a group of works. Elements of music such as pitch , duration and timbre combine into small elements like motifs and phrases , and these in turn combine in larger structures. Not all music (for example, that of John Cage ) has

3304-401: The truss element forces of only a few members are to be found. This method is used by introducing a single straight line cutting through the member whose force has to be calculated. However this method has a limit in that the cutting line can pass through a maximum of only 3 members of the truss structure. This restriction is because this method uses the force balances in the x and y direction and

3363-413: The truss element forces, namely the method of joints and the method of sections. Below is an example that is solved using both of these methods. The first diagram below is the presented problem for which the truss element forces have to be found. The second diagram is the loading diagram and contains the reaction forces from the joints. Since there is a pin joint at A, it will have 2 reaction forces. One in

3422-531: The weights of any objects that are permanently attached to the structure. For example, columns, beams, girders, the floor slab, roofing, walls, windows, plumbing, electrical fixtures, and other miscellaneous attachments. The second type of loads are live loads which vary in their magnitude and location. There are many different types of live loads like building loads, highway bridge loads, railroad bridge loads, impact loads, wind loads, snow loads, earthquake loads, and other natural loads. To perform an accurate analysis

3481-406: The x direction and the other in the y direction. At point B, there is a roller joint and hence only 1 reaction force in the y direction. Assuming these forces to be in their respective positive directions (if they are not in the positive directions, the value will be negative). Since the system is in static equilibrium, the sum of forces in any direction is zero and the sum of moments about any point

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