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In
structural engineering Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and joints' that create the form and shape of human-made Structure#Load-bearing, structures. Structural engineers also ...
, deflection is the degree to which a part of a long
structural element In structural engineering, structural elements are used in structural analysis to split a complex structure into simple elements (each bearing a structural load). Within a structure, an element cannot be broken down (decomposed) into parts of dif ...
(such as beam) is deformed laterally (in the direction transverse to its longitudinal axis) under a load. It may be quantified in terms of an angle (
angular displacement The angular displacement (symbol θ, , or φ) – also called angle of rotation, rotational displacement, or rotary displacement – of a physical body is the angle (in units of radians, degrees, turns, etc.) through which the body rotates ( ...
) or a distance (linear
displacement Displacement may refer to: Physical sciences Mathematics and physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
). A longitudinal deformation (in the direction of the axis) is called '' elongation''. 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 In mechanics, virtual work arises in the application of the '' principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different fo ...
, 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

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 (w) can be approximated as: \frac = \frac where the second derivative of its deflected shape with respect to x (x being the horizontal position along the length of the beam) is interpreted as its curvature, E is the
Young's modulus Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the modulus of elasticity for tension or axial compression. Youn ...
, I is the
area moment of inertia The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. Th ...
of the cross-section, and M is the internal
bending In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external Structural load, load applied perpendicularly to a longitudinal axis of the element. The structural eleme ...
moment in the beam. If, in addition, the beam is not tapered and is
homogeneous Homogeneity and heterogeneity are concepts relating to the uniformity of a substance, process or image. A homogeneous feature is uniform in composition or character (i.e., color, shape, size, weight, height, distribution, texture, language, i ...
, and is acted upon by a distributed load q, the above expression can be written as: E I ~ \frac = 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.


Cantilever beams

Cantilever beams have one end fixed, so that the slope and deflection at that end must be zero.


End-loaded cantilever beams

The
elastic Elastic is a word often used to describe or identify certain types of elastomer, Elastic (notion), elastic used in garments or stretch fabric, stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rub ...
deflection \delta and
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
of deflection \phi (in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at ...
s) at the free end in the example image: A (weightless)
cantilever A cantilever is a rigid structural element that extends horizontally and is unsupported at one end. Typically it extends from a flat vertical surface such as a wall, to which it must be firmly attached. Like other structural elements, a cantilev ...
beam, with an end load, can be calculated (at the free end B) using: \begin \delta_B &= \frac \\ ex\phi_B &= \frac \end where 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: \begin \delta_x &= \frac (3L - x) \\ ex\phi_x &= \frac (2L - x) \end Note: 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.


Uniformly loaded cantilever beams

The deflection, at the free end B, of a cantilevered beam under a uniform load is given by: \begin \delta_B &= \frac \\ ex\phi_B &= \frac \end where The deflection at any point, x, along the span of a uniformly loaded cantilevered beam can be calculated using: \begin \delta_x &= \frac \left(6L^2 - 4L x + x^2\right) \\ ex\phi_x &= \frac \left(3L^2 - 3L x + x^2\right) \end


Simply supported beams

Simply supported beams have supports under their ends which allow rotation, but not deflection.


Center-loaded simple beams

The deflection at any point, x, along the span of a center loaded simply supported beam can be calculated using: \delta_x = \frac \left(3L^2 - 4x^2\right) for 0 \leq x \leq \frac The special case of elastic deflection at the midpoint C of a beam, loaded at its center, supported by two simple supports is then given by: \delta_C = \frac where


Off-center-loaded simple beams

The maximum elastic deflection on a beam supported by two simple supports, loaded at a distance a from the closest support, is given by: \delta_\text = \frac where This maximum deflection occurs at a distance x_1 from the closest support and is given by: x_1 = \sqrt


Uniformly loaded simple beams

The elastic deflection (at the midpoint C) on a beam supported by two simple supports, under a uniform load (as pictured) is given by: \delta_C = \frac where The deflection at any point, x, along the span of a uniformly loaded simply supported beam can be calculated using: \delta_x = \frac \left(L^3 - 2L x^2 + x^3\right)


Combined loads

The deflection of beams with a combination of simple loads can be calculated using the
superposition principle The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So th ...
.


Change in length

The change in length \Delta L of the beam, projected along the line of the unloaded beam, can be calculated by integrating the slope \theta_x function, if the deflection function \delta_x is known for all x. Where: If the beam is uniform and the deflection at any point is known, this can be calculated without knowing other properties of the beam.


Units

The formulas supplied above require the use of a consistent set of units. Most calculations will be made in the
International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...
(SI) or US customary units, although there are many other systems of units.


International system (SI)

*Force: newtons (\mathrm) *Length: metres (\mathrm) *Modulus of elasticity: \mathrm = \mathrm *Moment of inertia: \mathrm


US customary units (US)

*Force: pounds force (\mathrm) *Length: inches (\mathrm) *Modulus of elasticity: \mathrm *Moment of inertia: \mathrm


Others

Other units may be used as well, as long as they are self-consistent. For example, sometimes the kilogram-force (\mathrm) unit is used to measure loads. In such a case, the modulus of elasticity must be converted to \mathrm.


Structural deflection

Building code A building code (also building control or building regulations) is a set of rules that specify the standards for construction objects such as buildings and non-building structures. Buildings must conform to the code to obtain planning permis ...
s determine the maximum deflection, usually as a
fraction A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...
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 Steel frame is a building technique with a "skeleton frame" of vertical steel columns and horizontal I-beams, constructed in a rectangular grid to support the floors, roof and walls of a building which are all attached to the frame. The develop ...
to hold a glazed panel, one allows only minimal deflection to prevent
fracture Fracture is the appearance of a crack or complete separation of an object or material into two or more pieces under the action of stress (mechanics), stress. The fracture of a solid usually occurs due to the development of certain displacemen ...
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).


See also

* Slope deflection method


References


External links


Deflection of beams
{{Structural engineering topics Engineering mechanics Structural analysis