Structural engineering theory: Difference between revisions

Revision as of 09:59, 11 May 2019

Revision as of 10:16, 5 June 2019

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A simplified version of Euler-Bernoulli beam equation is:

:<math>EI \frac{d^2}{dx^2}\partialleft(EI\frac{d^42 w}{\partial xdx^42}\right) = q(x).\,</math>

Here <math>w</math> is the deflection and <math>q(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 <math>w</math> have important meaningmeanings:

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

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

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

:* <math>\textstyle{-\frac{\partiald}{\partial xdx}\left(EI\frac{\partiald^2 w}{\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 w}{\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.