In
structural engineering
Structural engineering is a sub-discipline of civil engineering in which structural engineers are trained to design the 'bones and muscles' that create the form and shape of man-made structures. Structural engineers also must understand and cal ...
, deflection is the degree to which a part of a
structural element is displaced under a
load (because it
deforms). 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 As one of the methods of structural analysis, the direct stiffness method, also known as the matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type. It is a ...
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 (
) 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
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied ...
,
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. The ...
of the cross-section, and
is the internal
bending
In applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.
The structural element is assumed to ...
moment
Moment or Moments may refer to:
* Present time
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* ''Moment'' (Dark Tranquillity album), 2020
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* ''Moments'' (Christine Guldbrand ...
in the beam.
If, in addition, the beam is not tapered and is
homogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
, 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.
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 used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togethe ...
deflection
and
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
of deflection
(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. The unit was formerly an SI supplementary unit (before that ...
s) at the free end in the example image: A (weightless)
cantilever
A cantilever is a rigid structural element that extends horizontally and is supported at only 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 cant ...
beam, with an end load, can be calculated (at the free end B) using:
:
:
where
:
=
force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
acting on the tip of the beam
:
= length of the beam (span)
:
=
modulus of elasticity
:
=
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. The ...
of the beam's cross section
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:
:
:
Note: At
(the end of the beam), the
and
equations are identical to the
and
equations above.
Uniformly-loaded cantilever beams
The deflection, at the free end B, of a cantilevered beam under a uniform load is given by:
:
:
where
:
= uniform load on the beam (force per unit length)
:
= length of the beam
:
= modulus of elasticity
:
= area moment of inertia of cross section
The deflection at any point,
, along the span of a uniformly loaded cantilevered beam can be calculated using:
:
:
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,
, along the span of a center loaded simply supported beam can be calculated using:
:
for
:
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:
:
where
:
= force acting on the center of the beam
:
= length of the beam between the supports
:
= modulus of elasticity
:
= area moment of inertia of cross section
Off-center-loaded simple beams
The maximum elastic deflection on a beam supported by two simple supports, loaded at a distance
from the closest support, is given by:
:
where
:
= force acting on the beam
:
= length of the beam between the supports
:
= modulus of elasticity
:
= area moment of inertia of cross-section
:
= distance from the load to the closest support
This maximum deflection occurs at a distance
from the closest support and is given by:
:
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:
:
Where
:
= uniform load on the beam (force per unit length)
:
= length of the beam
:
= modulus of elasticity
:
= area moment of inertia of cross section
The deflection at any point,
, along the span of a uniformly loaded simply supported beam can be calculated using:
:
Change in length
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
.
Where:
:
= change in length (always negative)
:
= slope function (first derivative of
)
:
[Roark's Formulas for Stress and Strain, 8th Edition Eq 8.1-14]
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, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
(SI) or US customary units, although there are many other systems of units.
International system (SI)
:Force: newtons (
)
:Length: metres (
)
:Modulus of elasticity:
:Moment of inertia:
US customary units (US)
:Force: pounds force (
)
:Length: inches (
)
:Modulus of elasticity:
:Moment of inertia:
Others
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
.
Structural deflection
Building code
A building code (also building control or building regulations) is a set of rules that specify the standards for constructed objects such as buildings and non-building structures. Buildings must conform to the code to obtain planning permissi ...
s determine the maximum deflection, usually as a
fraction
A fraction (from la, fractus, "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 ...
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
Fracture is the separation of an object or material into two or more pieces under the action of stress. The fracture of a solid usually occurs due to the development of certain displacement discontinuity surfaces within the solid. If a displ ...
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