Structural analysis
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Structural analysis is a branch of Solid Mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and their components. In contrast to theory of elasticity, the models used in structure analysis are often differential equations in one spatial variable. Structures subject to this type of
analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
include all that must withstand loads, such as buildings, bridges, aircraft and ships. Structural analysis uses ideas from
applied mechanics Applied mechanics is the branch of science concerned with the motion of any substance that can be experienced or perceived by humans without the help of instruments. In short, when mechanics concepts surpass being theoretical and are applied and e ...
, materials science and
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
to compute a structure's
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
s, internal
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s,
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
es, support reactions, velocity, accelerations, and
stability Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Linear stability ** Lyapunov stability ** Orbital stability ** Structural sta ...
. The results of the analysis are used to verify a structure's fitness for use, often precluding
physical test A physical test is a qualitative or quantitative procedure that consists of determination of one or more characteristics of a given product, process or service according to a specified procedure. ASTM E 1301, Standard Guide for Proficiency Testing ...
s. Structural analysis is thus a key part of the engineering design of structures."Science Direct: Structural Analysis"


Structures and loads

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 Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewage ...
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 Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
work on a wide variety of
non-building structure A nonbuilding structure, also referred to simply as a structure, refers to any body or system of connected parts used to support a load that was not designed for continuous human occupancy. The term is used by architects, structural engin ...
s.


Classification of 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 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.


Loads

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 is often specified in
building code A building code (also building control or building regulations) is a set of rules that specify the standards for constructed objects such as buildings and non-building structures. Buildings must conform to the code to obtain planning permiss ...
s. 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 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.


Analytical methods

To perform an accurate analysis a structural engineer must determine information such as
structural load A structural load or structural action is a force, deformation, or acceleration applied to structural elements. A load causes stress, deformation, and displacement in a structure. Structural analysis, a discipline in engineering, analyzes the ef ...
s,
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, 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 Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be periodic, such as the motion of a pendulum—or random, such ...
,
stability Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Linear stability ** Lyapunov stability ** Orbital stability ** Structural sta ...
and non-linear behavior. There are three approaches to the analysis: the mechanics of materials approach (also known as strength of materials), the
elasticity theory In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are ...
approach (which is actually a special case of the more general field of
continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
), and the
finite element The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat t ...
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 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.


Limitations

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 Pure bending ( Theory of simple bending) is a condition of stress where a bending moment is applied to a beam without the simultaneous presence of axial, shear, or torsional forces. Pure bending occurs only under a constant bending moment (M) sinc ...
, 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. 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 A truss is an assembly of ''members'' such as beams, connected by ''nodes'', that creates a rigid structure. In engineering, a truss is a structure that "consists of two-force members only, where the members are organized so that the assembl ...
analysis,
moment distribution method The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal. The method only accounts for flexural effects and ignores axia ...
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. 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 is related linearly to strain, that the material (but not the structure) behaves identically regardless of direction of the applied load, that all
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
s 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.


Example

There are 2 commonly used methods to find 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 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 is zero. Therefore, the magnitude and direction of the reaction forces can be calculated. :\sum M_A=0=-10*1+2*R_B \Rightarrow R_B=5 :\sum F_y=0=R_+R_B-10 \Rightarrow R_=5 :\sum F_x=0=R_


Method of joints

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, :\sum F_y=0=R_+F_\sin(60)=5+F_\frac \Rightarrow F_=-\frac :\sum F_x=0=R_+F_\cos(60)+F_=0-\frac\frac+F_ \Rightarrow F_=\frac At D, :\sum F_y=0=-10-F_\sin(60)-F_\sin(60)=-10-\left(-\frac\right)\frac-F_\frac \Rightarrow F_=-\frac :\sum F_x=0=-F_\cos(60)+F_\cos(60)+F_=-\frac\frac+\frac\frac+F_ \Rightarrow F_=0 At C, :\sum F_y=0=-F_ \Rightarrow F_=0 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. :\sum F_x=-F_=-0=0 \Rightarrow verified At B, :\sum F_y=R_B+F_\sin(60)+F_=5+\left(-\frac\right)\frac+0=0 \Rightarrow verified :\sum F_x=-F_-F_\cos(60)=\frac-\frac\frac=0 \Rightarrow verified


Method of sections

This method can be used when 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 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


=Method 1: Ignore the right side

= : :\sum M_D=0=-5*1+\sqrt*F_ \Rightarrow F_=\frac :\sum F_y=0=R_-F_\sin(60)-10=5-F_\frac-10 \Rightarrow F_=-\frac :\sum F_x=0=F_+F_\cos(60)+F_=\frac-\frac\frac+F_ \Rightarrow F_=0


=Method 2: Ignore the left side

= : :\sum M_B=0=\sqrt*F_ \Rightarrow F_=0 :\sum F_y=0=F_\sin(60)+R_B=F_\frac+5 \Rightarrow F_=-\frac :\sum F_x=0=-F_-F_\cos(60)-F_=-F_-\left(-\frac\right)\frac-0 \Rightarrow F_=\frac 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

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 the equations of
linear elasticity Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mec ...
. 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.


Methods using numerical approximation

It is common practice to use approximate solutions of differential equations as 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 (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
. 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, 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 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 In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
, solving
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s. 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 The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
", 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 As one of the methods of structural analysis, the direct stiffness method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. It is a ' ...
and the force or
flexibility method In structural engineering, the flexibility method, also called the method of consistent deformations, is the traditional method for computing member forces and displacements in structural systems. Its modern version formulated in terms of the me ...
. 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 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.


Timeline

*1452–1519
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
made many contributions *1638:
Galileo Galilei Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
published the book "
Two New Sciences The ''Discourses and Mathematical Demonstrations Relating to Two New Sciences'' ( it, Discorsi e dimostrazioni matematiche intorno a due nuove scienze ) published in 1638 was Galileo Galilei's final book and a scientific testament covering muc ...
" in which he examined the failure of simple structures *1660:
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
by
Robert Hooke Robert Hooke FRS (; 18 July 16353 March 1703) was an English polymath active as a scientist, natural philosopher and architect, who is credited to be one of two scientists to discover microorganisms in 1665 using a compound microscope that ...
*1687:
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
published "'' Philosophiae Naturalis Principia Mathematica''" which contains the
Newton's laws of motion Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: # A body remains at rest, or in moti ...
*1750: Euler–Bernoulli beam equation *1700–1782: Daniel Bernoulli introduced the principle of
virtual work In mechanics, 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 is different for ...
*1707–1783:
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
developed the theory of
buckling In structural engineering, buckling is the sudden change in shape ( deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a ...
of columns *1826:
Claude-Louis Navier Claude-Louis Navier (born Claude Louis Marie Henri Navier; ; 10 February 1785 – 21 August 1836) was a French mechanical engineer, affiliated with the French government, and a physicist who specialized in continuum mechanics. The Navier–Stok ...
published a treatise on the elastic behaviors of structures *1873:
Carlo Alberto Castigliano Carlo Alberto Castigliano (9 November 1847, in Asti – 25 October 1884, in Milan) was an Italian mathematician and physicist known for Castigliano's method for determining displacements in a linear-elastic system based on the partial deriva ...
presented his dissertation "''Intorno ai sistemi elastici''", which contains
his theorem His or HIS may refer to: Computing * Hightech Information System, a Hong Kong graphics card company * Honeywell Information Systems * Hybrid intelligent system * Microsoft Host Integration Server Education * Hangzhou International School, i ...
for computing displacement as partial derivative of the strain energy. This theorem includes the method of 'least work' as a special case *1878-1972
Stephen Timoshenko Stepan Prokofyevich Timoshenko (russian: Степан Прокофьевич Тимошенко, p=sʲtʲɪˈpan prɐˈkofʲjɪvʲɪtɕ tʲɪmɐˈʂɛnkə; uk, Степан Прокопович Тимошенко, Stepan Prokopovych Tymoshenko; ...
father of modern
Applied mechanics Applied mechanics is the branch of science concerned with the motion of any substance that can be experienced or perceived by humans without the help of instruments. In short, when mechanics concepts surpass being theoretical and are applied and e ...
including the Timoshenko–Ehrenfest beam theory *1936:
Hardy Cross Hardy Cross (1885–1959) was an American structural engineer and the developer of the moment distribution method for structural analysis of statically indeterminate structures. The method was in general use from c. 1935 until c. 1960 when it was ...
' 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 Alexander Pavlovich Hrennikoff (russian: Александр Павлович Хренников; 11 November 1896 — 31 December 1984) was a Russian-Canadian structural engineer, a founder of the Finite Element Method. Biography Alexander was b ...
submitted his D.Sc. thesis in
MIT The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...
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" introduces the name "finite-element method" and is widely recognized as the first comprehensive treatment of the method as it is known today


See also

*
Limit state design Limit State Design (LSD), also known as Load And Resistance Factor Design (LRFD), refers to a design method used in structural engineering. A limit state is a condition of a structure beyond which it no longer fulfills the relevant design criteria ...
*
Structural engineering theory Structural engineering depends upon a detailed knowledge of loads, physics and materials to understand and predict how structures support and resist self-weight and imposed loads. To apply the knowledge successfully structural engineers will need ...
*
Structural integrity and failure Structural integrity and failure is an aspect of engineering that deals with the ability of a structure to support a designed structural load (weight, force, etc.) without breaking and includes the study of past structural failures in order to ...
* Stress–strain analysis *
von Mises yield criterion The maximum distortion criterion (also von Mises yield criterion) states that yielding of a ductile material begins when the second invariant of deviatoric stress J_2 reaches a critical value. It is a part of plasticity theory that mostly applie ...
* Probabilistic Assessment of Structures * Structural testing


References

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