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 Euler–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.
Beam deflection for various loads and supports[edit]
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 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 () can be approximated as:
where the second derivative of its deflected shape with respect to ( being the horizontal position along the length of the beam) is interpreted as its curvature, is the Young's modulus, is the area moment of inertia of the cross-section, and is the internal bendingmoment in the beam.
If, in addition, the beam is not tapered and is homogeneous, and is acted upon by a distributed load , the above expression can be written as:
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.
The elastic deflection and angle of deflection (inradians) at the free end in the example image: A (weightless) cantilever beam, with an end load, can be calculated (at the free end B) using:[1]
Note that if the span doubles, the deflection increases eightfold. The deflection at any point, , along the span of an end loaded cantilevered beam can be calculated using:[1]
Note: At (the end of the beam), the and equations are identical to the and equations above.
The change in length of the beam is generally negligible in structures, but can be calculated by integrating the slope function, if the deflection function is known for all .
The formulas supplied above require the use of a consistent set of units. Most calculations will be made in the International System of Units (SI) or US customary units, although there are many other systems of units.
Other units may be used as well, as long as they are self-consistent. For example, sometimes the kilogram-force () unit is used to measure loads. In such a case, the modulus of elasticity must be converted to .
Building codes determine the maximum deflection, usually as a fraction of the span e.g. 1/400 or 1/600. Either the strength limit state (allowable stress) or the serviceability limit state (deflection considerations among others) may govern the minimum dimensions of the member required.
The deflection must be considered for the purpose of the structure. When designing a steel frame to hold a glazed panel, one allows only minimal deflection to prevent fracture of the glass.
The deflected shape of a beam can be represented by the moment diagram, integrated (twice, rotated and translated to enforce support conditions).
where the second derivative of its deflected shape with respect to 
(
is the Young's modulus, 
is the area moment of inertia of the cross-section, and 
is the internal bending moment in the beam.
![{\displaystyle {\begin{aligned}\delta _{B}&={\frac {FL^{3}}{3EI}}\\[1ex]\phi _{B}&={\frac {FL^{2}}{2EI}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ea57a95cc9eb248b177970939d5304e81900730)
where
= force acting on the tip of the beam
= length of the beam (span)![{\displaystyle {\begin{aligned}\delta _{x}&={\frac {Fx^{2}}{6EI}}(3L-x)\\[1ex]\phi _{x}&={\frac {Fx}{2EI}}(2L-x)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4a1595eb33e79fad8690e884f718423b401df58)
![{\displaystyle {\begin{aligned}\delta _{B}&={\frac {qL^{4}}{8EI}}\\[1ex]\phi _{B}&={\frac {qL^{3}}{6EI}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e78b299eec3a07598b4871a3e6d16392cfa35d6)
where
![{\displaystyle {\begin{aligned}\delta _{x}&={\frac {qx^{2}}{24EI}}\left(6L^{2}-4Lx+x^{2}\right)\\[1ex]\phi _{x}&={\frac {qx}{6EI}}\left(3L^{2}-3Lx+x^{2}\right)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d71623ff9b775b8a41a0d82f58f983d3f12fec)
for
where
where
where
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