Deflection (engineering)
Encyclopedia
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 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 "direct stiffness method
" amongst others. The deflection of beam
elements is usually calculated on the basis of Euler-Bernoulli beam theory while that of a plate or shell element is calculated using plate theory
or shell theory
.
An example of the use of deflection in this context is in building construction. Architect
s and engineer
s 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 radian
s) in the example image, a (weightless) cantilever
beam, with an end load on it, can be calculated (at the free end B) using:
where
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 and height h are the most determining factors; if the span doubles, the deflection increases 2³ = 8 fold, and if the height doubles, the deflection decreases 2³ = 8 fold.
where
where:
where:
s 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.
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
, deflection is the degree to which a structural element is displaced under a load
Force
In physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
. 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 formulae (will only give the deflection of common beam
Beam (structure)
A beam is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment.- Overview...
configurations and load cases at discrete locations), or by methods such as "virtual work
Virtual work
Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the...
", "direct integration
Direct integration of a beam
Direct integration is a structural analysis method for measuring internal shear, internal moment, rotation, and deflection of a beam.For a beam with an applied weight w , taking downward to be positive, the internal shear force is given by taking the negative integral of the weight:The internal...
", "Castigliano's method
Castigliano's method
Castigliano's method, named for Carlo Alberto Castigliano, is a method for determining the displacements of a linear-elastic system based on the partial derivatives of the strain energy...
", "Macaulay's method
Macaulay's method
Macaulay’s method is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. Use of Macaulay’s technique is very convenient for cases of discontinuous and/or discrete loading...
" or the "direct stiffness method
Direct stiffness method
As one of the methods of structural analysis, the direct stiffness method , also known as the displacement method or matrix stiffness method, is particularly suited for computer-automated analysis of complex structures including the statically indeterminate type...
" amongst others. The deflection of beam
Beam
Beam may refer to:*Beam , a construction element*Beam , the most extreme width of a nautical vessel, or a point alongside the ship at the mid-point of its length*A narrow, propagating stream of particles or energy:...
elements is usually calculated on the basis of Euler-Bernoulli beam theory while that of a plate or shell element is calculated using plate theory
Plate theory
In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions . The typical thickness to width ratio of a plate...
or shell theory
Plate theory
In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat plates that draws on the theory of beams. Plates are defined as plane structural elements with a small thickness compared to the planar dimensions . The typical thickness to width ratio of a plate...
.
An example of the use of deflection in this context is in building construction. Architect
Architect
An architect is a person trained in the planning, design and oversight of the construction of buildings. To practice architecture means to offer or render services in connection with the design and construction of a building, or group of buildings and the space within the site surrounding the...
s and engineer
Engineer
An engineer is a professional practitioner of engineering, concerned with applying scientific knowledge, mathematics and ingenuity to develop solutions for technical problems. Engineers design materials, structures, machines and systems while considering the limitations imposed by practicality,...
s select materials for various applications. The beams
Beam (structure)
A beam is a horizontal structural element that is capable of withstanding load primarily by resisting bending. The bending force induced into the material of the beam as a result of the external loads, own weight, span and external reactions to these loads is called a bending moment.- Overview...
used for frame work are selected on the basis of deflection, amongst other factors.
End load cantilever beams
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....
deflection f and angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
of deflection φ (in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s) in the example image, a (weightless) cantilever
Cantilever
A cantilever is a beam anchored at only one end. The beam carries the load to the support where it is resisted by moment and shear stress. Cantilever construction allows for overhanging structures without external bracing. Cantilevers can also be constructed with trusses or slabs.This is in...
beam, with an end load on it, can be calculated (at the free end B) using:
where
- F = forceForceIn physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
acting on the tip of the beam - L = length of the beam (spanSpan (architecture)Span is the distance between two intermediate supports for a structure, e.g. a beam or a bridge.A span can be closed by a solid beam or of a rope...
) - E = modulus of elasticity
- I = area moment of inertia
From this formula it follows that the span L and height h are the most determining factors; if the span doubles, the deflection increases 2³ = 8 fold, and if the height doubles, the deflection decreases 2³ = 8 fold.
where
- b = width (x-dimension),
- h = height (y-dimension)
Centre loaded beam
The elastic deflection on a beam, loaded at its centre, supported by two simple supports is given by:where:
- δ = the deflection of the beam
- F = forceForceIn physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
acting on the centre of the beam - L = length of the beam between the supports
- E = modulus of elasticity
- I = area moment of inertia
Intermediately loaded beams
The elastic deflection on a beam supported by two simple supports, loaded at a distance a from one of the supports, is given by:where:
- δ = the deflection of the beam
- F = forceForceIn physics, a force is any influence that causes an object to undergo a change in speed, a change in direction, or a change in shape. In other words, a force is that which can cause an object with mass to change its velocity , i.e., to accelerate, or which can cause a flexible object to deform...
acting on the beam - L = length of the beam between the supports
- E = modulus of elasticity
- I = area moment of inertia
- a = the distance of the load (F) from one of the supports
Structural deflection
Building codeBuilding code
A building code, or building control, is a set of rules that specify the minimum acceptable level of safety for constructed objects such as buildings and nonbuilding structures. The main purpose of building codes are to protect public health, safety and general welfare as they relate to the...
s determine the maximum deflection, usually as a fraction
Fraction (mathematics)
A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
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
Steel frame
Steel frame usually refers to a building technique with a "skeleton frame" of vertical steel columns and horizontal -beams, constructed in a rectangular grid to support the floors, roof and walls of a building which are all attached to the frame...
to hold a glazed panel, one allows only minimal deflection to prevent fracture
Fracture
A fracture is the separation of an object or material into two, or more, pieces under the action of stress.The word fracture is often applied to bones of living creatures , or to crystals or crystalline materials, such as gemstones or metal...
of the glass.
The deflective shape of a beam can be represented by the moment
Bending Moment
A bending moment exists in a structural element when a moment is applied to the element so that the element bends. Moments and torques are measured as a force multiplied by a distance so they have as unit newton-metres , or pound-foot or foot-pound...
diagram, integrated.