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Information about Bending

Figure 1. A side view of a simply supported beam (top) bending under a distributed lateral load (bottom).
Figure 2. The internal forces and the axial stress distribution across the cross-section of a beam in bending.


This article is about structural behavior. For other meanings see Bending (disambiguation).

In engineering mechanics, bending (also known as flexure) characterizes the behavior of a structural subjected to an external load applied perpendicular to the axis of the element. A structural element subjected to bending is known as a beam. A closet rod sagging under the weight of clothes on clothes hangers is an example of a beam experiencing bending.

Bending produces reactive forces inside a beam as the beam attempts to accommodate the flexural load: in the case of the beam in Figure 1, the material at the top of the beam is being compressed while the material at the bottom is being stretched. There are three notable internal forces caused by lateral loads (shown in Figure 2): shear parallel to the lateral loading, compression along the top of the beam, and tension along the bottom of the beam. These last two forces form a couple or moment as they are equal in magnitude and opposite in direction. This bending moment produces the sagging deformation characteristic of compression members experiencing bending.

This stress distribution is dependent on a number of assumptions. First, that 'plane sections remain plane'. In otherwords, any deformation do to shear across the section is not accounted for (no shear deformation). Also, this linear distribution is only applicable if the maximum stress is less than the Yield (engineering) of the material. For stresses that exceed yield, refer to article Plastic Bending.

The compressive and tensile forces shown in Figure 2 induce stresses on the beam. The maximum compressive stress is found at the uppermost edge of the beam while the maximum tensile stress is located at the lower edge of the beam. Since the stresses between these two opposing vary linearly, there therefore exists a point on the linear path between them where there is no bending stress. The locus of these points is the neutral axis. Because of this area with no stress and the adjacent areas with low stress, using uniform cross section beams in bending is not a particularly efficient means of supporting a load as it does not use the full capacity of the beam until it is on the brink of collapse. Wide-flange beams (I-Beams) and truss girders effectively address this inefficiency as they minimize the amount of material in this under-stressed region.

Simple or Symmetrical Bending

Beam bending is analyzed with the Euler-Bernoulli beam equation. The classic formula for determining the bending stress in a member is:



simplified for a beam of rectangular cross-section to:

This equation is valid only when the stress at the extreme fiber (i.e. the portion of the beam furthest from the neutral axis) is below the yield stress of the material it is constructed from. At higher loadings the stress distribution becomes non-linear, and ductile materials will eventually enter a plastic hinge state where the magnitude of the stress is equal to the yield stress everywhere in the beam, with a discontinuity at the neutral axis where the stress changes from tensile to compressive. This plastic hinge state is typically used as a limit state in the design of steel structures.

Complex or Unsymmetrical Bending

The equation above is, also, only valid if the cross-section is symmetrical. For unsymmetrical sections, the full form of the equation must be used (presented below):



Complex Bending of Homogeneous Beams

The complex bending stress equation for elastic, homogeneous beams is given as where Mx and My are the bending moments about the x and y centroid axes, respectively. Ix and Iy are the second moments of area (also known as moments of inertia) about the x and y axes, respectively, and Ixy is the product of inertia. Using this equation it would be possible to calculate the bending stress at any point on the beam cross section regardless of moment orientation or cross-sectional shape. Note that Mx, My, Ix, Iy, and Ixy are all unique for a given section along the length of the beam. In other words, they will not change from one point to another on the cross section. However, the x and y variables shown in the equation correspond to the coordinates of a point on the cross section at which the stress is to be determined.

See also

Bending can refer to the following:
  • Bending - the behavior of a structural element subjected to a lateral load
  • Bending (metalworking) - a sheet metalworking process used in manufacture
  • Elemental Bending

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Applied mechanics is a branch of the physical sciences and the practical application of mechanics. Applied mechanics examines the response of bodies (solids and fluids) or systems of bodies to external forces.
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Structure is a fundamental and sometimes intangible notion covering the recognition, observation, nature, and stability of patterns and relationships of entities. From a child's verbal description of a snowflake, to the detailed scientific analysis of the properties of magnetic
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Load may refer to:
  • Structural load, forces which are applied to a structure
  • Cargo, Freight, or Lading
  • The load of a mutual fund (see Mutual fund fees and expenses)
  • The genetic load of a population
  • The parasite load of an organism

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beam is a structural element that carries load primarily in bending (flexure). Beams generally carry vertical gravitational forces but can also be used to carry horizontal loads (i.e. loads due to an earthquake or wind).
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closet (especially in North American usage) is a small and enclosed space, a cabinet, or a cupboard in a house or building used for general storage or hanging clothes. A closet for food storage is usually referred to as a pantry.
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Deflection or deflexion is a term used in physics, automotive, engineering, and weapon applications to describe four separate phenomena.

Structural engineering

In engineering mechanics, deflection
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clothes hanger, or coat hanger, is a device in the shape of:
  • Human shoulders designed to facilitate the hanging of a coat, jacket, sweater, shirt, blouse or dress in a manner that prevents wrinkles, with a lower bar for the hanging of trousers or skirts.

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In physics, force is an action or agency that causes a body of mass m to accelerate. It may be experienced as a lift, a push, or a pull. The acceleration of the body is proportional to the vector sum of all forces acting on it (known as net force or resultant force).
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Shear stress is a stress state where the stress is parallel or tangential to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. The variable used to denote shear stress is τ (tau).
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Physical compression is the result of the subjection of a material to compressive stress, resulting in reduction of volume. The opposite of compression is tension.

Explanation


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Tension is a reaction force applied by a stretched string (rope or a similar object) on the objects which stretch it. The direction of the force of tension is parallel to the string, towards the string.
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For other meanings, see Couple (disambiguation)

A Couple is a system of forces with a resultant moment but no resultant force. Another term for a couple is a pure moment. Its effect is to create rotation without translation.
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Principle of Moments redirects here. For the Robert Plant album, see The Principle of Moments. See also Moment (mathematics) for a more abstract concept of moments that evolved from this concept of physics.

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A bending moment exists in a structural element when a moment or torque 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 units such as newton.metres (N.m) and foot.pounds (ft.lb).
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A compression member is a general class of structural elements of which a column is the most common specific example.

Description

For a compression member, such as a column, the principal stress comes mainly from axial forces, that is forces that fall along one
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yield strength or yield point of a material is defined in engineering and materials science as the stress at which a material begins to plastically deform. Prior to the yield point the material will deform elastically and will return to its original shape when the applied
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Plastic bending is a nonlinear behavior peculiar to members made of ductile materials that frequently achieve much greater ultimate bending strength than indicated by a linear elastic bending analysis.
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Stress is a measure of force per unit area within a body. It is a body's internal distribution of force per area that reacts to external applied loads. Stress is often broken down into its shear and normal components as these have unique physical significance.
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prevew not available
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locus (Latin for "place", plural loci) is a collection of points which share a property. The term 'locus' is usually used of a condition which defines a continuous figure or figures, that is, a curve.
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I-beams (also known as double-T esp. in Polish and German) are beams with an I- or H-shaped cross-section. The horizontal elements are flanges, while the vertical element is the web.
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In architecture and structural engineering, a truss is a structure comprising one or more triangular units constructed with straight slender members whose ends are connected at joints.
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A girder is a large support beam used in construction, normally of iron or steel. Girders often have an I beam cross section for strength, but may also have a box shape, Z shape or other forms.
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Euler-Bernoulli beam theory or just beam theory is a simplification of the linear isotropic theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams.
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Stress is a measure of force per unit area within a body. It is a body's internal distribution of force per area that reacts to external applied loads. Stress is often broken down into its shear and normal components as these have unique physical significance.
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Principle of Moments redirects here. For the Robert Plant album, see The Principle of Moments. See also Moment (mathematics) for a more abstract concept of moments that evolved from this concept of physics.

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An axis in the cross section of a beam, shaft or the like along which there are no longitudinal stresses / strains. If the section is symmetric (in both geometry and materials) the neutral axis is at the geometric centroid.
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An axis in the cross section of a beam, shaft or the like along which there are no longitudinal stresses / strains. If the section is symmetric (in both geometry and materials) the neutral axis is at the geometric centroid.
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This article is about the moment of inertia as related to the bending of a plane. For the moment of inertia dealing with rotation of an object, see Moment of inertia.

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