Structural engineering theory: Difference between revisions

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The criteria which govern the design of a structure are either serviceability (criteria which define whether the structure is able to adequately fulfill its function) or strength (criteria which define whether a structure is able to safely support and resist its design loads). A structural engineer designs a structure to have sufficient [[Strength of materials|strength]] and [[stiffness]] to meet these criteria.

Loads imposed on structures are supported yby means of forces transmitted through structural elements. These forces can manifest themselves as [[Tension (physics)|tension]] (axial force), [[compression (physical)|compression]] (axial force), [[Shear stress|shear]], and [[bending]], or [[flexure]] (a bending moment is a force multiplied by a distance, or lever arm, hence producing a turning effect or [[torque]]).

==Strength==

[[Strength of materials|Strength]] depends upon material properties. The strength of a material depends on its capacity to withstand axial [[stress (mechanics)|stress]], [[shear stress]], bending, and torsion. The strength of a material is measured in force per unit area (newtons per square millimetre or N/mm², or the equivalent megapascals or MPa in the [[SI system]] and often pounds per square inch psi in the United States Customary[[customary Unitsunits]] system).

A structure fails the strength criterion when the [[stress (mechanics)|stress]] (force divided by area of material) induced by the loading is greater than the capacity of the structural material to resist the load without breaking, or when the [[Strain (materials science)|strain]] (percentage extension) is so great that the element no longer fulfills its function ([[Yield (engineering)|yield]]).

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==Stiffness==

[[Stiffness]] depends upon material properties and [[geometry]]. The stiffness of a structural element of a given material is the product of the material's [[Young's modulus]] and the element's [[second moment of area]]. Stiffness is measured in force per unit length (newtons per millimetre or N/mm), and is equivalent to the 'force constant' in [[Hooke's Lawlaw]].

The [[Deflection (engineering)|deflection]] of a structure under loading is dependent on its stiffness. The [[dynamic response]] of a structure to dynamic loads (the [[natural frequency]] of a structure) is also dependent on its stiffness.

In a structure made up of multiple structural elements where the surface distributing the forces to the elements is rigid, the elements will carry loads in proportion to their relative stiffness - the stiffer an element, the more load it will attract. This means that load/stiffness ratio, which is deflection, remains same in two connected (jointed) elements. In a structure where the surface distributing the forces to the elements is flexible (like a wood -framed structure), the elements will carry loads in proportion to their relative tributary areas.

A structure is considered to fail the chosen serviceability criteria if it is insufficiently stiff to have acceptably small [[Deflection (engineering)|deflection]] or [[dynamics (mechanics)|dynamic]] response under loading.

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==Safety factors==

The safe design of structures requires a design approach which takes account of the [[statistics|statistical]] likelihood of the failure of the structure. Structural design codes are based upon the assumption that both the loads and the material strengths vary with a [[normal distribution]].{{Citation needed|date=March 2019}}

The job of the structural engineer is to ensure that the chance of overlap between the distribution of loads on a structure and the distribution of material strength of a structure is acceptably small (it is impossible to reduce that chance to zero).

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The safety factors for material strength vary depending on the material and the use it is being put to and on the design codes applicable in the country or region.

A more sophisticated approach of modeling structural safety is to rely on [[structural reliability]], in which both loads and resistances are modeled as probabilistic variables.<ref name=":0">{{Cite book|title=Melchers, R. E. (2002), "Structural Reliability Analysis and Prediction," 2nd Ed., John Wiley, Chichester, UK.}}</ref><ref name=":1">{{cite journal|last1=Piryonesi|first1=Sayed Madeh|last2=Tavakolan|first2=Mehdi|date=9 January 2017|title=A mathematical programming model for solving cost-safety optimization (CSO) problems in the maintenance of structures|journal=KSCE Journal of Civil Engineering|volume=21|issue=6|pages=2226–2234|doi=10.1007/s12205-017-0531-z|bibcode=2017KSJCE..21.2226P |doi-access=free}}</ref> However, using this approach requires detailed modeling of the distribution of loads and resistances. Furthermore, its calculations are more computation intensive.

===Load cases===

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{{Main|Newton's laws of motion}}

The most important natural laws for structural engineering are [[Newton's Lawslaws of Motionmotion]].

Newton's first law states that ''every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed.''

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Newton's third law states that ''all forces occur in pairs, and these two forces are equal in magnitude and opposite in direction.''

With these laws it is possible to understand the forces on a structure and how that structure will resist them. The Thirdthird Lawlaw requires that for a structure to be stable all the internal and external forces must be in [[Mechanical equilibrium|equilibrium]]. This means that the sum of all internal and external forces on a ''[[free-body diagram]]'' must be zero:

* <math> \sum \vec F = 0 </math>: the vectorial sum of the [[force]]s acting on the body equals zero. This translates to

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A structural engineer must understand the internal and external forces of a structural system consisting of structural elements and nodes at their intersections.

A statically determinate structure can be fully analysed using only consideration of equilibrium, from Newton's Lawslaws of Motionmotion.

A statically indeterminate structure has more unknowns than equilibrium considerations can supply equations for (see [[simultaneous equations]]). Such a system can be solved using consideration of equations of ''compatibility'' between geometry and deflections in addition to equilibrium equations, or by using [[virtual work]].

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<math>r + b = 2j</math>

It should be noted that evenEven if this relationship does hold, a structure can be arranged in such a way as to be statically indeterminate.<ref>{{cite book|title=Structural Modeling and Analysis|author=Dym, Clive L.|publisher=Cambridge University Press|date=1997|isbn=0-521-49536-9|pages=98}}</ref>

==Elasticity==

{{Main|Elasticity (physics)|l1=Elasticity}}

{{See also|Hooke's Lawlaw}}

Much engineering design is based on the assumption that materials behave elastically. For most materials this assumption is incorrect, but empirical evidence has shown that design using this assumption can be safe. Materials that are elastic obey Hooke's Lawlaw, and plasticity does not occur.

For systems that obey Hooke's Lawlaw, the extension produced is directly proportional to the load:

:<math> \vec{\mathbf{F}}=k\vec{\mathbf{x}} \ </math>

where

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{{Main|Plasticity (physics)|l1=Plasticity}}

Some design is based on the assumption that materials will behave [[Plasticity (physics)|plastically]].<ref name=Heyman1>{{cite book|title=Structural Analysis: A Historical Approach|author=Heyman, Jacques|publisher=Cambridge University Press|date=1998|isbn=0-521-62249-2}}</ref> A plastic material is one which does not obey Hooke's Lawlaw, and therefore deformation is not proportional to the applied load. Plastic materials are [[ductile]] materials. Plasticity theory can be used for some reinforced concrete structures assuming they are underreinforced, meaning that the steel reinforcement fails before the concrete does.

Plasticity theory states that the point at which a structure collapses (reaches yield) lies between an upper and a lower bound on the load, defined as follows:

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If the correct collapse load is found, the two methods will give the same result for the collapse load.<ref>{{cite book|title=Design of Concrete Structures|author1=Nilson, Arthur H. |author2=Darwin, David |author3=Dolan, Charles W. |publisher=McGraw-Hill Professional|date=2004|isbn=0-07-248305-9|pages=486}}</ref>

Plasticity theory depends upon a correct understanding of when yield will occur. A number of different models for stress distribution and approximations to the [[yield surface]] of plastic materials exist:<ref name=Heyman>{{cite book|title=The Science of Structural Engineering|author=Heyman, Jacques|url=https://books.google.com/books?id=Au34lwRovHIC&dqq=Leonhard+Euler+Daniel+Bernoulli+Beam+equation|date=1999|publisher=Imperial College Press|isbn=1-86094-189-3}}</ref>

* [[Mohr's circle]]

* [[Von Mises yield criterion]]

* [[Henri Tresca]]

==The Euler-BernoulliEuler–Bernoulli beam equation==

[[File:deflection.svg|thumb|right|Deflection of a cantilever under a point load (f) in engineering]]

{{Main|Euler-BernoulliEuler–Bernoulli beam equation}}

The Euler-BernoulliEuler–Bernoulli beam equation defines the behaviour of a beam element (see below). It is based on five assumptions:

(1)# [[continuumContinuum mechanics]] is valid for a bending beam<br />.

(2)# theThe [[Stress (physics)|stress]] at a [[Cross section (geometry)|cross section]] varies linearly in the direction of bending, and is zero at the [[centroid]] of every [[Cross section (geometry)|cross section]].<br />

(3)# theThe bending [[moment (physics)|moment]] at a particular cross section varies linearly with the second derivative of the deflected shape at that ___location.<br />

(4)# theThe beam is composed of an isotropic material<br.
# />(5) theThe applied load is orthogonal to the beam's neutral axis and acts in a unique plane.

A simplified version of Euler-BernoulliEuler–Bernoulli beam equation is:

:<math>EI \frac{d^4 u2}{dx^2}\left(EI\frac{d^2 xw}{dx^42}\right) = wq(x).\,</math>

Here <math>uw</math> is the deflection and <math>wq(x)</math> is a load per unit length. <math>E</math> is the [[elastic modulus]] and <math>I</math> is the [[second moment of area]], the product of these giving the [[stiffnessflexural rigidity]] of the beam.

This equation is very common in engineering practice: it describes the deflection of a uniform, static beam.

Successive derivatives of u<math>w</math> have important meaningmeanings:

:* <math>\textstyle{uw}\,</math> is the deflection.

:* <math>\textstyle{\frac{\partial udw}{\partial xdx}}\,</math> is the slope of the beam.

:* <math>\textstyle{-EI\frac{\partiald^2 uw}{\partial xdx^2}}\,</math> is the [[Bending|bending moment]] in the beam.

:* <math>\textstyle{-\frac{\partiald}{\partial xdx}\left(EI\frac{\partiald^2 uw}{\partial xdx^2}\right)}\,</math> is the [[Shearing (physics)|shear force]] in the beam.

A bending moment manifests itself as a tension force and a compression force, acting as a [[Couple (mechanics)|couple]] in a beam. The stresses caused by these forces can be represented by:

:<math>\sigma = \frac{My}{I} = -E y \frac{\partiald^2 uw}{\partial xdx^2}\,</math>

where <math>\sigma</math> is the stress, <math>M</math> is the bending moment, <math>y</math> is the distance from the [[neutral axis]] of the beam to the point under consideration and <math>I</math> is the [[second moment of area]]. Often the equation is simplified to the moment divided by the [[section modulus]] (<math>S)</math>, which is <math>I/y</math>. This equation allows a structural engineer to assess the stress in a structural element when subjected to a bending moment.

==Buckling==

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== See also ==

* [[Structural analysis]]

* [[Structural engineering software]]

== References ==

<References/>

* Castigliano, Carlo Alberto (translator: Andrews, Ewart S.) (1966). [https://books.google.com/books?id=wU1CAAAAIAAJ&q=The+Theory+of+Equilibrium+of+Elastic+Systems+and+Its+Applications&dq=The+Theory+of+Equilibrium+of+Elastic+Systems+and+Its+Applications&pgis=1 ''The Theory of Equilibrium of Elastic Systems and Its Applications'']. Dover Publications.

* Dym, Clive L. (1997). ''Structural Modeling and Analysis''. Cambridge University Press. {{ISBN|0-521-49536-9}}.

* Dugas, René (1988). ''A History of Mechanics''. Courier Dover Publications. {{ISBN|0-486-65632-2}}.

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* Mörsch, E. (Stuttgart, 1908). ''Der Eisenbetonbau, seine Theorie und Anwendung, (Reinforced Concrete Construction, its Theory and Application)''. Konrad Wittwer, 3rd edition.

* Nedwell, P.J.; Swamy, R.N.(ed) (1994). ''Ferrocement:Proceedings of the Fifth International Symposium''. Taylor & Francis. {{ISBN|0-419-19700-1}}.

* Newton, Isaac; Leseur, Thomas; Jacquier, François (1822). [https://books.google.com/books?id=TA-l3gysWaUC&printsec=frontcover&dqq=Philosophi%C3%A6+Naturalis+Principia+Mathematica ''Philosophiæ Naturalis Principia Mathematica'']. Oxford University.

* Nilson, Arthur H.; Darwin, David; Dolan, Charles W. (2004). ''Design of Concrete Structures''. McGraw-Hill Professional. {{ISBN|0-07-248305-9}}.

* Rozhanskaya, Mariam; Levinova, I. S. (1996). "Statics" in Morelon, Régis & Rashed, Roshdi (1996). ''Encyclopedia of the History of Arabic Science'', '''vol. 2-3''', Routledge. {{ISBN|0-415-02063-8}}

* Schlaich, J., K. Schäfer, M. Jennewein (1987). "[https://www.researchgate.net/profile/Michael_Kotsovos/publication/248122582_Toward_a_Consistent_Design_of_Structural_Concrete/links/59cc94c80f7e9bbfdc3f7515/Toward-a-Consistent-Design-of-Structural-Concrete.pdf Toward a Consistent Design of Structural Concrete]". ''PCI Journal'', Special Report, Vol. 32, No. 3.

* Scott, Richard (2001). ''In the Wake of Tacoma: Suspension Bridges and the Quest for Aerodynamic Stability''. ASCE Publications. {{ISBN|0-7844-0542-5}}.

* Turner, J.; Clough, R.W.; Martin, H.C.; Topp, L.J. (1956). "Stiffness and Deflection of Complex Structures". ''Journal of Aeronautical Science'' '''Issue 23'''.