Structural analysis comprises the set of physical laws and mathematics required to study and predict the behavior of structures. The common subjects of structural analysis are buildings, bridges, aircraft, ships, and any other engineering artifacts whose integrity is judged largely based upon its ability to withstand loads. It incorporates the fields of mechanics, dynamics, and the many failure theories. From a theoretical perspective the primary goal of structural analysis is the computation of deformations, internal forces, and stresses. However, in practice, structural analysis can be viewed more abstractly as a method to drive the engineering design process or prove the soundness of a design without a dependence on directly testing it. In engineering mechanics, deformation is a change in shape due to an applied force. ... In physics, force is an influence that may cause a body to accelerate. ... Stress is the internal distribution of force per unit area that balances and reacts to external loads applied to a body. ...

To perform an accurate analysis a structural engineer must determine such information as structural loads, geometry, support conditions, and materials 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 behaviour. This article or section does not adequately cite its references or sources. ... Structural elements are used in structural analysis to simplify the structure which is to be analysed. ... Stress is the internal distribution of force per unit area that balances and reacts to external loads applied to a body. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ...

There are three approaches to the analysis: the mechanics of materials approach (also know 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 formulation leading to closed-form solutions. The third, actually a numerical method for solving differential equations, is very widely used for structural analysis. The equations solved by the finite element method are generated by theories of mechanics such as elasticity theory and strength of materials. Analytical formulations apply mostly to simple linear elastic models and can often be accomplished by hand. 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. Strength of materials is materials science applied to the study of engineering materials and their mechanical behavior in general (such as stress, deformation, strain and stress-strain relations). ... 3-D elasticity is one of three methods of structural analysis. ... Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... Finite element analysis (FEA) or finite element method (FEM) is a numerical technique for solution of boundary-value problems. ... A standard definition of mechanical equilibrium is: A system is in mechanical equilibrium when the sum of the forces, and torque, on each particle of the system is zero. ... The term compatibility has the following meanings: In telecommunication, the capability of two or more items or components of equipment or material to exist or function in the same system or environment without mutual interference. ...

Each method has noteworthy limitations. The method of mechanics of materials 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 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 the restriction that there is always some numerical error. Effective and reliable use of this method requires a solid understanding of its limitations.

Strength of materials methods (classical methods)

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 the superposition principle to analyze a member undergoing combined loading. Solutions for special cases exist for common structures such as thin-walled pressure vessels. A statically determinate beam, bending under an evenly distributed load. ... In linear algebra, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. ...

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 for small rigid frames, and portal frame and cantilever method for large rigid frames. Except for moment distribution, which came into use in 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. In architecture and structural engineering, a truss is a static structure consisting of straight slender members inter-connected at joints into triangular units. ... Hardy Cross, 1885-1959, born in Nansemond County, Virginia, was an engineer and the developer of the moment distribution method. ... Face The 1930s (years from 1930–1939) were described as an abrupt shift to more radical and conservative lifestyles, as countries were struggling to find a solution to the Great Depression, also known in Europe as the World Depression. ...

The solutions are based on linear isotropic infinitessimal elasticity and Euler-Bernoulli beam theory. In other words, they contain the assumptions (among others) that the materials in question are elastic, that stress 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. In engineering mechanics, deformation is a change in shape due to an applied force. ...

Elasticity methods

Elasticity methods are available for 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 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. // Linear elasticity The linear theory of elasticity models the macroscopic mechanical properties of solids assuming small deformations. ...

Many of the developments in the mechanics of materials and elasticity approaches have been expounded or initiated by Stephen Timoshenko. Stephen Timoshenko Stephen P. Timoshenko or Stepan Prokofyevich Timoshenko (Ukrainian: , Russian: , December 23, 1878 – May 29, 1972), is reputed to be the father of modern engineering mechanics. ...

Finite element methods

Finite element method models a structure as an assembly of elements or components with various forms of connection between them. Thus, 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. The behaviour of individual elements is characterised by the element's stiffness or flexibility relation, which altogether leads to the system's stiffness or flexibility relation. To establish the element's stiffness or flexibility relation, 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. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

Early application of matrix methods were for 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 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, by far, more popular thanks to its ease of implementation as well as of formulation for advanced applications. The finite-element technology 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 the model and the reliability of the data input. Finite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. ... Steel Pressure Vessel A pressure vessel is a closed, rigid container designed to hold gases or liquids at a pressure different from the ambient pressure. ... As one of the methods of structural analysis, the Direct Stiffness Method (DSM), also known as the displacement method or matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. ... In structural engineering, the flexibility method is the classical consistent deformation method for computing member forces and displacements in structural systems. ...

Time-line

• 1452-1519 Leonardo da Vinci made many contributions.
• 1638: Galileo Galilei published the book "Two New Sciences" in which he examined the failure of simple structures.
• 1660: Hooke's law by Robert Hooke.
• 1687: Issac Newton published "Philosophiae Naturalis Principia Mathematica" which contains the Newton's laws of motion.
• 1750: Euler-Bernoulli beam equation.
• 1667-1748: Daniel Bernoulli introduced the principle of virtual work.
• 1707-1783: Leonhard Euler developed the theory of buckling of columns.
• 1826: Claude-Louis Navier published a treatise on the elastic bahaviors of structures.
• 1873: Carlo Alberto Castigliano presented his dissertation "Intorno ai sistemi elastici", which contains his theorem for computing displacement as partial derivative of the strain energy. This theorem includes the method of least work as a special case.
• 1936: Hardy Cross' publication of the moment distribution method which was later recognized as a form of the relaxation method applicable to the problem of flow in pipe-network.
• 1941: Alexander Hrennikoff submitted his D.Sc thesis in MIT on the discretization of plane elasticity problems using a lattice framework.
• 1942: R. Courant divided a domain into finite subregions.
• 1956: J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp's paper on the "Stiffness and Deflection of Complex Structures". This paper introduces the name "finite-element method" and is widely recognized as the first comprehensive treatment of the method as it is known today.

The Mona Lisa Leonardo di ser Piero da Vinci (April 15, 1452 – May 2, 1519) was an Italian polymath: scientist, mathematician, engineer, inventor, anatomist, painter, sculptor, architect, musician, and writer. ... KDFSAJFKASJDKFJASDKLJFDKLASJFLKJASKLFJLAKSJFLKSJALFKJSKLJFto the Sun-centered solar system which Galileo supported. ... Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ... Robert Hooke, FRS (July 18, 1635 – March 3, 1703) was an English polymath who played an important role in the scientific revolution, through both experimental and theoretical work. ... Sir Isaac Newton in Knellers portrait of 1689. ... Newtons own copy of his Principia, with handwritten corrections for the second edition. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... The elementary Euler-Bernoulli beam theory is a simplification of the linear isotropic theory of elasticity which allows quick calculation of the load-carrying capacity and deflection of common structural elements called beams. ... Daniel Bernoulli Daniel Bernoulli (Groningen, February 8, 1700 – Basel, March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland. ... A force F, which may be real (actual) or imaginary (fictitious), acting on a particle is said to do virtual work when the particle is imagined to undergo a real or imaginary displacement component D in the direction of the force. ... Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... In engineering, buckling is a failure mode characterised by a sudden failure of a structural member that is subjected to high compressive stresses where the actual compressive stresses at failure are smaller than the ultimate compressive stresses that the material is capable of withstanding. ... Claude-Louis Navier (born Claude Louis Marie Henri Navier on February 10, 1785 in Dijon, died August 21, 1836 in Paris) was a French engineer and physicist. ... Carlos Alberto Castigliano (9 November 1847 – 25 October 1884) was an Italian mathematician and physicist known for Castiglianos method for determining displacements in a linear-elastic system based on the partial derivative of strain energy. ... Castiglianos method, named for Carlo Alberto Castigliano, is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the strain energy. ... Hardy Cross, 1885-1959, born in Nansemond County, Virginia, was an engineer and the developer of the moment distribution method. ... Alexander Hrennikoff (1896–December 31, 1984) was a Russian-Canadian Civil Engineer, a founder of the Finite Element Method. ... The Massachusetts Institute of Technology (MIT) is a private, coeducational research university located in Cambridge, Massachusetts. ... Richard Courant (born January 8, 1888 at Lublinitz, today Poland, died January 27, 1972 at New York/USA) was a German and American mathematician. ...

References

• ASME Eyewitness Series: Finite Element Method (FEM) - Historical Timeline

• A Historical Outline of Matrix Structural Analysis: A Play in Three Acts

• Probabilistic assessment of structures using Monte Carlo simulation by Jan Hlavacek