Deflection (engineering)

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Deflection (f) in engineering

In engineering, deflection is the degree to which a structural element is displaced under a load (due to its deformation). It may refer to an angle or a distance.

The deflection distance of a member under a load can be calculated by integrating the function that mathematically describes the slope of the deflected shape of the member under that load.

Standard formulas exist for the deflection of common beam configurations and load cases at discrete locations. Otherwise methods such as virtual work, direct integration, Castigliano’s method, Macaulay’s method or the direct stiffness method are used. The deflection of beam elements is usually calculated on the basis of the Eulerâ€“Bernoulli beam equation while that of a plate or shell element is calculated using plate or shell theory.

An example of the use of deflection in this context is in building construction. Architects and engineers select materials for various applications.

Contents

1 Beam deflection for various loads and supports

1.1 Cantilever beams

1.1.1 End-loaded cantilever beams

1.1.2 Uniformly-loaded cantilever beams

1.2 Simply-supported beams

1.2.1 Center-loaded simple beams

1.2.2 Off-center-loaded simple beams

1.2.3 Uniformly-loaded simple beams

1.3 Change in length

2 Units

2.1 International system (SI)

2.2 US customary units (US)

2.3 Others

3 Structural deflection

4 See also

5 References

6 External links

Beam deflection for various loads and supports

Beams can vary greatly in their geometry and composition. For instance, a beam may be straight or curved. It may be of constant cross section, or it may taper. It may be made entirely of the same material (homogeneous), or it may be composed of different materials (composite). Some of these things make analysis difficult, but many engineering applications involve cases that are not so complicated. Analysis is simplified if:

The beam is originally straight, and any taper is slight

The beam experiences only linear elastic deformation

The beam is slender (its length to height ratio is greater than 10)

Only small deflections are considered (max deflection less than 1/10 of the span).

In this case, the equation governing the beam’s deflection ({\displaystyle w}w) can be approximated as:

{\displaystyle {\cfrac {\mathrm {d} ^{2}w(x)}{\mathrm {d} x^{2}}}={\frac {M(x)}{E(x)I(x)}}}{\cfrac {{\mathrm {d}}^{2}w(x)}{{\mathrm {d}}x^{2}}}={\frac {M(x)}{E(x)I(x)}}

where the second derivative of its deflected shape with respect to {\displaystyle x}x ({\displaystyle x}x being the horizontal position along the length of the beam) is interpreted as its curvature, {\displaystyle E}E is the Young’s modulus, {\displaystyle I}I is the area moment of inertia of the cross-section, and {\displaystyle M}M is the internal bending moment in the beam.

If, in addition, the beam is not tapered and is homogeneous, and is acted upon by a distributed load {\displaystyle q}q, the above expression can be written as:

{\displaystyle EI~{\cfrac {\mathrm {d} ^{4}w(x)}{\mathrm {d} x^{4}}}=q(x)}

EI~\cfrac{\mathrm{d}^4 w(x)}{\mathrm{d} x^4} = q(x)

This equation can be solved for a variety of loading and boundary conditions. A number of simple examples are shown below. The formulas expressed are approximations developed for long, slender, homogeneous, prismatic beams with small deflections, and linear elastic properties. Under these restrictions, the approximations should give results within 5% of the actual deflection.