# Deflection (engineering)

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

The deflection distance of a member under a load is directly related to the slope of the deflected shape of the member under that load and can be calculated by integrating the function that mathematically describes the slope of the member under that load. Deflection can be calculated by standard formula (will only give the deflection of common beam configurations and load cases at discrete locations), or by methods such as virtual work, direct integration, Castigliano's method, Macaulay's method or the direct stiffness method, amongst others. 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. The beams used for frame work are selected on the basis of deflection, amongst other factors.

## Beam deflection for various loads and supports

The elastic deflection $\delta$ and angle of deflection $\phi$ (in radians) 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]

$\delta_B = \frac {F L^3} {3 E I}$
$\phi_B = \frac {F L^2} {2 E I}$

where

$F$ = Force acting on the tip of the beam
$L$ = Length of the beam (span)
$E$ = Modulus of elasticity
$I$ = Area moment of inertia
Deflection of a cantilevered beam

Note that if the span doubles, the deflection increases eightfold. The deflection at any point, $x$, along the span of an end loaded cantilevered beam can be calculated using:[1]

$\delta_x = \frac {F x^2} {6 E I} (3L - x)$
$\phi_x = \frac {F x} {2 E I} (2L - x)$

Note that at $x = L$ (the end of the beam), the $\delta_x$ and $\phi_x$ equations are identical to the $\delta_B$ and $\phi_B$ equations above.

The deflection, at the free end B, of a cantilevered beam under a uniform load is given by:[1]

$\delta_B = \frac {q L^4} {8 E I}$
$\phi_B = \frac {q L^3} {6 E I}$

where

$q$ = Uniform load on the beam (force per unit length)
$L$ = Length of the beam
$E$ = Modulus of elasticity
$I$ = Area moment of inertia

The deflection at any point, $x$, along the span of a uniformly loaded cantilevered beam can be calculated using:[1]

$\delta_x = \frac {q x^2} {24 E I}(6L^2 - 4L x + x^2)$
$\phi_x = \frac {q x} {6 E I}(3L^2 - 3L x + x^2)$

The elastic deflection (at the midpoint C) of a beam, loaded at its center, supported by two simple supports is given by:[1]

$\delta_C = \frac {F L^3} {48 E I}$

where

$F$ = Force acting on the center of the beam
$L$ = Length of the beam between the supports
$E$ = Modulus of elasticity
$I$ = Area moment of inertia

The deflection at any point, $x$, along the span of a center loaded simply supported beam can be calculated using:[1]

$\delta_x = \frac {F x} {48 E I}(3L^2 - 4x^2)$

for

$0 \leq x \leq \frac{L}{2}$

The maximum elastic deflection on a beam supported by two simple supports, loaded at a distance $a$ from the closest support, is given by:[1]

$\delta_{max} = \frac {F a (L^2 - a^2)^{3/2}} {9\sqrt{3} L E I}$

where

$F$ = Force acting on the beam
$L$ = Length of the beam between the supports
$E$ = Modulus of elasticity
$I$ = Area moment of inertia
$a$ = Distance from the load to the closest support (i.e. $a \leq L/2$)

This maximum deflection occurs at a distance $x_1$ from the closest support and is given by:[1]

$x_1 = \sqrt{\frac{L^2 - a^2}{3}}$

Shows a statically determinate beam, deflecting under an evenly distributed load.

The elastic deflection (at the midpoint C) on a beam supported by two simple supports, under a uniform load (as pictured) is given by:[1]

$\delta_C = \frac{5 q L^4} {384 E I}$

where

$q$ = Uniform load on the beam (force per unit length)
$L$ = Length of the beam
$E$ = Modulus of elasticity
$I$ = Area moment of inertia

The deflection at any point, $x$, along the span of a uniformly loaded simply supported beam can be calculated using:[1]

$\delta_x = \frac{q x} {24 E I} (L^3 - 2L x^2 + x^3)$

## Structural deflection

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 amongst 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).