Deflection (engineering)

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In engineering mechanics, deflection is a term that is used to describe the degree to which a structural element is displaced under a load. The deflection of a member under a load is directly related to the slope of the deflected shape of the member under that load and can calculated by integrating the function that mathematically describes the slope of the member under that load. Deflection can be calculated by standard formulae (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 "matrix stiffness method" amongst others. (See structural analysis textbooks for procedure.)

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.

The elastic deflection f and angle of deflection φ (in radians) in the example image, a (weightless) cantilever beam, can be calculated (at the free end) using :

f_B = F·L³ / (3·E·I)

φ_B = F·L² / (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

The deflection at any point along the span can be calculated using the above-mentioned methods.

From this formula it follows that the span L is the most determinating factor; if the span doubles, the deflection increases 2³ = 8 fold.

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 deflective shape of a beam can be represented by the moment diagram, integrated.