Structural engineering theory: Difference between revisions

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The Euler-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-Bernoulli beam equation is:

:<math>EI \frac{d\partial^4 uw}{d\partial x^4} = 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 [[stiffness]] 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 meaning:

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

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

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

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

A bending moment manifests itself as a tension 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{\partial^2 uw}{\partial x^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==