Beam (structure)

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A statically determinate beam, bending (sagging) under a uniformly distributed load Bending.svg
A statically determinate beam, bending (sagging) under a uniformly distributed load

A beam is a structural element that primarily resists loads applied laterally across the beam's axis (an element designed to carry a load pushing parallel to its axis would be a strut or column). Its mode of deflection is primarily by bending, as loads produce reaction forces at the beam's support points and internal bending moments, shear, stresses, strains, and deflections. Beams are characterized by their manner of support, profile (shape of cross-section), equilibrium conditions, length, and material.

Contents

Beams are traditionally descriptions of building or civil engineering structural elements, where the beams are horizontal and carry vertical loads. However, any structure may contain beams, such as automobile frames, aircraft components, machine frames, and other mechanical or structural systems. Any structural element, in any orientation, that primarily resists loads applied laterally across the element's axis is a beam.

Overview

Historically a beam is a squared timber, but may also be made of metal, stone, or a combination of wood and metal [1] such as a flitch beam. Beams primarily carry vertical gravitational forces, but they are also used to carry horizontal loads such as those due to earthquake or wind, or in tension to resist rafter thrust (tie beam) or compression (collar beam). The loads carried by a beam are transferred to columns, walls, or girders, then to adjacent structural compression members, and eventually to the ground. In light frame construction, joists may rest on beams.

Classification based on supports

In engineering, beams are of several types: [2]

  1. Simply supported – a beam supported on the ends which are free to rotate and have no moment resistance.
  2. Fixed or encastré (encastrated) – a beam supported on both ends and restrained from rotation.
  3. Overhanging – a simple beam extending beyond its support on one end.
  4. Double overhanging – a simple beam with both ends extending beyond its supports on both ends.
  5. Continuous – a beam extending over more than two supports.
  6. Cantilever – a projecting beam fixed only at one end.
  7. Trussed – a beam strengthened by adding a cable or rod to form a truss. [3]
  8. Beam on spring supports
  9. Beam on elastic foundation

Second moment of area (area moment of inertia)

In the beam equation, the variable I represents the second moment of area or moment of inertia: it is the sum, along the axis, of dA·r2, where r is the distance from the neutral axis and dA is a small patch of area. It measures not only the total area of the beam section, but the square of each patch's distance from the axis. A larger value of I indicates a stiffer beam, more resistant to bending.

Stress

Diagram of stiffness of a simple square beam (A) and universal beam (B). The universal beam flange sections are three times further apart than the solid beam's upper and lower halves. The second moment of inertia of the universal beam is nine times that of the square beam of equal cross section (universal beam web ignored for simplification) BeamDiagram.svg
Diagram of stiffness of a simple square beam (A) and universal beam (B). The universal beam flange sections are three times further apart than the solid beam's upper and lower halves. The second moment of inertia of the universal beam is nine times that of the square beam of equal cross section (universal beam web ignored for simplification)

Loads on a beam induce internal compressive, tensile and shear stresses (assuming no torsion or axial loading). Typically, under gravity loads, the beam bends into a slightly circular arc, with its original length compressed at the top to form an arc of smaller radius, while correspondingly stretched at the bottom to enclose an arc of larger radius in tension. This is known as sagging; while a configuration with the top in tension, for example over a support, is known as hogging. The axis of the beam retaining its original length, generally halfway between the top and bottom, is under neither compression nor tension, and defines the neutral axis (dotted line in the beam figure).

Above the supports, the beam is exposed to shear stress. There are some reinforced concrete beams in which the concrete is entirely in compression with tensile forces taken by steel tendons. These beams are known as prestressed concrete beams, and are fabricated to produce a compression more than the expected tension under loading conditions. High strength steel tendons are stretched while the beam is cast over them. Then, when the concrete has cured, the tendons are slowly released and the beam is immediately under eccentric axial loads. This eccentric loading creates an internal moment, and, in turn, increases the moment-carrying capacity of the beam. Prestressed beams are commonly used on highway bridges.

A beam of PSL lumber installed to replace a load-bearing wall Parallam support beam.jpg
A beam of PSL lumber installed to replace a load-bearing wall

The primary tool for structural analysis of beams is the Euler–Bernoulli beam equation. This equation accurately describes the elastic behaviour of slender beams where the cross sectional dimensions are small compared to the length of the beam. For beams that are not slender a different theory needs to be adopted to account for the deformation due to shear forces and, in dynamic cases, the rotary inertia. The beam formulation adopted here is that of Timoshenko and comparative examples can be found in NAFEMS Benchmark Challenge Number 7. [4] Other mathematical methods for determining the deflection of beams include "method of virtual work" and the "slope deflection method". Engineers are interested in determining deflections because the beam may be in direct contact with a brittle material such as glass. Beam deflections are also minimized for aesthetic reasons. A visibly sagging beam, even if structurally safe, is unsightly and to be avoided. A stiffer beam (high modulus of elasticity and/or one of higher second moment of area) creates less deflection.

Mathematical methods for determining the beam forces (internal forces of the beam and the forces that are imposed on the beam support) include the "moment distribution method", the force or flexibility method and the direct stiffness method.

General shapes

Most beams in reinforced concrete buildings have rectangular cross sections, but a more efficient cross section for a beam is an I or H section which is typically seen in steel construction. Because of the parallel axis theorem and the fact that most of the material is away from the neutral axis, the second moment of area of the beam increases, which in turn increases the stiffness.

An I shaped beam of metal under a bridge Ahmaskoski road bridge.JPG
An I shaped beam of metal under a bridge

An I-beam is only the most efficient shape in one direction of bending: up and down looking at the profile as an I. If the beam is bent side to side, it functions as an H where it is less efficient. The most efficient shape for both directions in 2D is a box (a square shell); the most efficient shape for bending in any direction, however, is a cylindrical shell or tube. For unidirectional bending, the I or wide flange beam is superior.[ citation needed ]

Efficiency means that for the same cross sectional area (volume of beam per length) subjected to the same loading conditions, the beam deflects less.

Other shapes, like L (angles), C (channels), T-beam and double-T or tubes, are also used in construction when there are special requirements.

Walers and struts

This system provides horizontal bracing for small trenches, ensuring the secure installation of utilities. It's specifically designed to work in conjunction with steel trench sheets. [5]

Thin walled

A thin walled beam is a very useful type of beam (structure). The cross section of thin walled beams is made up from thin panels connected among themselves to create closed or open cross sections of a beam (structure). Typical closed sections include round, square, and rectangular tubes. Open sections include I-beams, T-beams, L-beams, and so on. Thin walled beams exist because their bending stiffness per unit cross sectional area is much higher than that for solid cross sections such a rod or bar. In this way, stiff beams can be achieved with minimum weight. Thin walled beams are particularly useful when the material is a composite laminate. Pioneer work on composite laminate thin walled beams was done by Librescu.

The torsional stiffness of a beam is greatly influenced by its cross sectional shape. For open sections, such as I sections, warping deflections occur which, if restrained, greatly increase the torsional stiffness. [6]

See also

Related Research Articles

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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 structures. Structural engineers also must understand and calculate the stability, strength, rigidity and earthquake-susceptibility of built structures for buildings and nonbuilding structures. The structural designs are integrated with those of other designers such as architects and building services engineer and often supervise the construction of projects by contractors on site. They can also be involved in the design of machinery, medical equipment, and vehicles where structural integrity affects functioning and safety. See glossary of structural engineering.

<span class="mw-page-title-main">Reinforced concrete</span> Concrete with rebar

Reinforced concrete, also called ferroconcrete, is a composite material in which concrete's relatively low tensile strength and ductility are compensated for by the inclusion of reinforcement having higher tensile strength or ductility. The reinforcement is usually, though not necessarily, steel bars (rebar) and is usually embedded passively in the concrete before the concrete sets. However, post-tensioning is also employed as a technique to reinforce the concrete. In terms of volume used annually, it is one of the most common engineering materials. In corrosion engineering terms, when designed correctly, the alkalinity of the concrete protects the steel rebar from corrosion.

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<span class="mw-page-title-main">Compressive strength</span> Capacity of a material or structure to withstand loads tending to reduce size

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<span class="mw-page-title-main">Truss</span> Rigid structure that consists of two-force members only

A truss is an assembly of members such as beams, connected by nodes, that creates a rigid structure.

<span class="mw-page-title-main">Stiffness</span> Resistance to deformation in response to force

Stiffness is the extent to which an object resists deformation in response to an applied force.

Stress–strain analysis is an engineering discipline that uses many methods to determine the stresses and strains in materials and structures subjected to forces. In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material.

<span class="mw-page-title-main">Buckling</span> Sudden change in shape of a structural component under load

In structural engineering, buckling is the sudden change in shape (deformation) of a structural component under load, such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have buckled. Euler's critical load and Johnson's parabolic formula are used to determine the buckling stress of a column.

<span class="mw-page-title-main">Bending</span> Strain caused by an external load

In applied mechanics, bending characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.

<span class="mw-page-title-main">Shear wall</span> A wall intended to withstand the lateral load

In structural engineering, a shear wall is a two-dimensional vertical element of a system that is designed to resist in-plane lateral forces, typically wind and seismic loads.

<span class="mw-page-title-main">Euler–Bernoulli beam theory</span> Method for load calculation in construction

Euler–Bernoulli beam theory is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral loads only. By ignoring the effects of shear deformation and rotatory inertia, it is thus a special case of Timoshenko–Ehrenfest beam theory. It was first enunciated circa 1750, but was not applied on a large scale until the development of the Eiffel Tower and the Ferris wheel in the late 19th century. Following these successful demonstrations, it quickly became a cornerstone of engineering and an enabler of the Second Industrial Revolution.

<span class="mw-page-title-main">I-beam</span> Construction element

An I-beam is any of various structural members with an I or H-shaped cross-section. Technical terms for similar items include H-beam, w-beam, universal beam (UB), rolled steel joist (RSJ), or double-T. I-beams are typically made of structural steel and serve a wide variety of construction uses.

This is an alphabetical list of articles pertaining specifically to structural engineering. For a broad overview of engineering, please see List of engineering topics. For biographies please see List of engineers.

<span class="mw-page-title-main">Bending moment</span> Force tending to bend a structural element

In solid mechanics, a bending moment is the reaction induced in a structural element when an external force or moment is applied to the element, causing the element to bend. The most common or simplest structural element subjected to bending moments is the beam. The diagram shows a beam which is simply supported at both ends; the ends can only react to the shear loads. Other beams can have both ends fixed ; therefore each end support has both bending moments and shear reaction loads. Beams can also have one end fixed and one end simply supported. The simplest type of beam is the cantilever, which is fixed at one end and is free at the other end. In reality, beam supports are usually neither absolutely fixed nor absolutely rotating freely.

<span class="mw-page-title-main">Shear and moment diagram</span> Structural design tool

Shear force and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear forces and bending moments at a given point of a structural element such as a beam. These diagrams can be used to easily determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure. Another application of shear and moment diagrams is that the deflection of a beam can be easily determined using either the moment area method or the conjugate beam method.

Section modulus is a geometric property for a given cross-section used in the design of beams or flexural members. Other geometric properties used in design include area for tension and shear, radius of gyration for compression, and second moment of area and polar second moment of area for stiffness. Any relationship between these properties is highly dependent on the shape in question. Equations for the section moduli of common shapes are given below. There are two types of section moduli, the elastic section modulus and the plastic section modulus. The section moduli of different profiles can also be found as numerical values for common profiles in tables listing properties of such.

<span class="mw-page-title-main">T-beam</span> T-shaped construction module

A T-beam, used in construction, is a load-bearing structure of reinforced concrete, wood or metal, with a T-shaped cross section. The top of the T-shaped cross section serves as a flange or compression member in resisting compressive stresses. The web of the beam below the compression flange serves to resist shear stress. When used for highway bridges the beam incorporates reinforcing bars in the bottom of the beam to resist the tensile stresses which occur during bending.

<span class="mw-page-title-main">Structural engineering theory</span>

Structural engineering depends upon a detailed knowledge of loads, physics and materials to understand and predict how structures support and resist self-weight and imposed loads. To apply the knowledge successfully structural engineers will need a detailed knowledge of mathematics and of relevant empirical and theoretical design codes. They will also need to know about the corrosion resistance of the materials and structures, especially when those structures are exposed to the external environment.

<span class="mw-page-title-main">Shell (structure)</span> Thin-walled geometric solid

A shell is a three-dimensional solid structural element whose thickness is very small compared to its other dimensions. It is characterized in structural terms by mid-plane stress which is both coplanar and normal to the surface. A shell can be derived from a plate in two steps: by initially forming the middle surface as a singly or doubly curved surface, then by applying loads which are coplanar to the plate's plane thus generating significant stresses. Materials range from concrete to fabric.

This glossary of structural engineering terms pertains specifically to structural engineering and its sub-disciplines. Please see glossary of engineering for a broad overview of the major concepts of engineering.

References

  1. "Beam" def. 1. Whitney, William Dwight, and Benjamin E. Smith. The Century dictionary and cyclopedia. vol, 1. New York: Century Co., 1901. 487. Print.
  2. Ching, Frank. A visual dictionary of architecture. New York: Van Nostrand Reinhold, 1995. 8–9. Print.
  3. The American Architect and Building News, Vol XXIII. Boston: James R. Osgood & Co. 1888. p. 159.
  4. Ramsay, Angus. "NAFEMS Benchmark Challenge Number 7" (PDF). ramsay-maunder.co.uk. Retrieved 7 May 2017.
  5. "Walers and Struts". MGF. 27 August 2020. Retrieved 24 October 2023.
  6. Ramsay, Angus. "The Influence and Modelling of Warping Restraint on Beams". ramsay-maunder.co.uk. Retrieved 7 May 2017.

Further reading