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Structural analysis is the determination of the effects of loads on physical structures and their components. Structures subject to this type of analysis include all

that must withstand loads, such as buildings, bridges, vehicles, machinery, furniture, attire, soil strata, prostheses and biological tissue. Structural analysisincorporates the fields of applied mechanics, materials science and applied mathematics to compute a structure's deformations, internal forces, stresses, supportreactions, accelerations, and stability. The results of the analysis are used to verify a structure's fitness for use, often saving physical tests. Structural analysis isthus a key part of the engineering design of structures.

Contents

1 Structures and Loads

1.1 Classification of structures

1.2 Loads

2 Analytical methods

3 Strength of materials methods (classical methods)

3.1 Example

3.1.1 Method of Joints

3.1.2 Method of Sections

3.1.2.1 Method 1: Ignore the right side

3.1.2.2 Method 2: Ignore the left side

4 Elasticity methods

5 Methods using numerical approximation

6 Timeline

7 See also

8 References

9 External links

Structures and Loads[edit]

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 areimportant. In order to design a structure, one must serve a specified function for public use, the engineer must account for its safety, aesthetics, and serviceability,

while taking into consideration economic and environmental constraints. Other branches of engineering work on a wide variety of nonbuilding structures.

Classification of structures[edit]

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 thematerials 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[edit]

Once the dimensional requirement for a structure have been defined, it becomes necessary to determine the loads the structure must support. In order to design a

structure, it is therefore necessary to first specify the loads that act on it. The design loading for a structure is often specified in building codes. There are twotypes of codes: general building codes and design codes, engineer must satisfy all the codes requirements for a reliable structure.

There are two types of loads that structure engineering must encounter in the design. First type of load is called 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. Second type of load is 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[edit]

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 behavior. There are three approaches to 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 to simple linear elastic

models, lead to closed-form solutions, and can often be solved by hand. The finite element approach is actually a numerical method for solving differentialequations 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 ofelasticity 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 twodiscussed 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)[edit]

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 besuperimposed 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 trussanalysis, 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 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 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.

Example[edit]

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 issolved using both of these methods. The first diagram below is the presented problem for which we need to find the truss element forces. 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, we have a roller joint and hence we

only have 1 reaction force in the y direction. Let us assume these forces to be in their respective positive directions (if they are not in the positive directions likewe have assumed, then we will get a negative value for them).

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 anddirection of the reaction forces can be calculated.

Method of Joints[edit]

This 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 we have found the forces in each of the truss elements, it is a good practice to verify the results by completing the remaining force balances.

At B,

Method of Sections[edit]

This method can be used when the truss element forces of only a few members wants to be known. This method is used by introducing a single straight linecutting through the member whose force wants 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 us amaximum of 3 equations to find a maximum of 3 unknown truss element forces through which this cut is made. Let us try to find the forces FAB, FBD and FCDin the above example

Method 1: Ignore the right side[edit]

Method 2: Ignore the left side[edit]

Truss Structure Analysis, Method of Sections Right2.jpg

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[edit]

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. The equations of elasticity are a system of 15 partial differential equations. Due to the nature ofthe mathematics involved, analytical solutions may only be produced for relatively simple geometries. For complex geometries, a numerical solution method suchas the finite element method is necessary.

Methods using numerical approximation[edit]

It is common practice to use approximate solutions of differential equations as the basis for structural analysis. This is usually done using numerical approximationtechniques. 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. Thus, acontinuous 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. Thebehaviour of individual elements is characterised by the element's stiffness or flexibility relation, which altogether leads to the system's stiffness or flexibilityrelation. 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 bymeans of matrix algebra, solving partial differential equations.

Early applications of matrix methods were for articulated frameworks with truss, beam and column elements; later and more advanced matrix methods, referredto as "finite element analysis," model an entire structure with one-, two-, and three-dimensional elements and can be used for articulated systems together withcontinuous 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 the most popular by far thanks to its ease of implementation as well as of formulation for advanced applications. The finite-elementtechnology is now sophisticated enough to handle just about any system as long as sufficient computing power is available. Its applicability includes, but is notlimited 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 modeland the reliability of the data input.

Timeline[edit]

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: Isaac Newton published "Philosophiae Naturalis Principia Mathematica" which contains the Newton's laws of motion

Wikibooks has more onthe topic of: Structuralanalysis

1750: Euler–Bernoulli beam equation

1700–1782: 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 behaviors 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" 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[edit]

Structural integrity and failure

Stress–strain analysis

References[edit]

[1]

[2]

External links[edit]

Impact - Dynamic Finite Element Program Suite, for dynamic events like crashes, written in Java, GNU license

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