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Beam Structure & Theory

A statically determinate beam, bending under an evenly distributed load.

A 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). The loads carried by a beam are transferred to columns, walls, or girders, which then transfer the force to adjacent structural compression members. In Light frame construction the joists rest on the beam.

Beams are characterized by their profile (the shape of their cross-section), their length, and their material. In contemporary construction, beams are typically made of steel, reinforced concrete, or wood. One of the most common types of steel beam is the I-beam or wide-flange beam (also known as a "universal beam" or, for stouter sections, a "universal column"). This is commonly used in steel-frame buildings and bridges. Other common beam profiles are the C-channel, the hollow structural section beam, the pipe, and the angle.

Beam Structure

1 Structural Characteristics
2 General Shapes
3 See also
4 External links
5 References

Euler-Bernoulli Beam Theory

1 History
2 The beam equation
3 Stress
4 Boundary considerations
5 Loading considerations
6 Extensions
7 See also
8 Notes
9 References
10 External links

Timoshenko Beam Theory

Structural Characteristics

Internally, beams experience compressive, tensile and shear stresses as a result of the loads applied to them. Typically, under gravity loads, the original length of the beam is slightly reduced to enclose a smaller radius arc at the top of the beam, resulting in compression, while the same original beam length at the bottom of the beam is slightly stretched to enclose a larger radius arc, and so is under tension. The same original length of the middle of the beam, generally halfway between the top and bottom, is the same as the radial arc of bending, and so it 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 that are entirely in compression. 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 begun to cure, the tendons are 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. They are commonly used on highway bridges.

The primary tool for structural analysis of beams is the Euler-Bernoulli beam equation. 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 minimised for aesthetic reasons. A visibly sagging beam, though structurally safe, is unsightly and to be avoided. A stiffer beam (high modulus of elasticity and high second moment of area) produces 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 matrix stiffness method.

General Shapes

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

Mostly the beams have rectangular cross sections in reinforced concrete buildings, but the most efficient cross-section is an I-shaped beam. The fact that most of the material is placed away from the neutral axis (axis of symmetry in case of I beams) increases the second moment of area of the beam which in turn increases the stiffness.

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 and is less efficient. The most efficient shape for both directions in 2D is a box (a square shell) however the most efficient shape for bending in any direction is a cylindrical shell or tube. But, for unidirectional bending, the I beam is king.

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) or tubes, are also used in construction when there are special requirements.

External links

David Childs Ltd Consulting Civil Engineers: Tutorials
- Steel Beam Design
- Prestressed Concrete Beam Design Tutorial

American Wood Council: Free Download Library Wood Construction Data
- Wood Structural Design Data (pdf file)
- online Span Calculator

Introduction to Structural Design, U. Virginia Dept. Architecture
- Glossary
- Course Sampler Lectures, Projects, Tests
- Beams and Bending review points (follow using next buttons)
- Structural Behavior and Design Approaches lectures (follow using next buttons)

U. Maryland, J.A. Clark School of Engineering: HAMLET engineering simulations and models
- BeamAnalysis

U. Wisconsin-Stout, Strength of Materials online lectures, problems, tests/solutions, links, software
- Beams I - Shear Forces and Bending Moments
- Beams II - Bending Stress
Beam calculations in MS Excel from ExcelCalcs.com

References

Introduction to mechanics of solids, Egor P. Popov, Prentice-Hall, 1968

Euler-Bernoulli Beam Theory

This vibrating glass beam may be modeled as a cantilever beam with acceleration, variable linear density, variable section modulus, some kind of dissipation, springy end loading, and possibly a point mass at the free end.

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. 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.

Additional analysis tools have been developed such as plate theory and finite element analysis, but the simplicity of beam theory makes it an important tool in the sciences, especially structural and mechanical engineering.

History

The prevailing consensus is that Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue that Leonardo da Vinci was the first to make the crucial observations. Da Vinci lacked Hooke's law and calculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made.^[1]

The Bernoulli beam is named after Jacob Bernoulli, who made the significant discoveries. Leonhard Euler and Daniel Bernoulli were the first to put together a useful theory circa 1750.^[2] At the time, science and industrial art were generally seen as very distinct fields, and there was considerable doubt that a mathematical product of academia could be trusted for practical safety applications. Bridges and buildings continued to be designed by precedent until the late 19th century, when the Eiffel Tower and Ferris wheel demonstrated the validity of the theory on large scales.

The beam equation

The Euler-Bernoulli Beam Equation is based on 5 assumptions about a bending beam. Colloquially stated, they are that:

calculus is valid and is applicable to bending beams
the stresses in the beam are distributed in a particular, mathematically simple way
the force that resists the bending depends on the amount of bending in a particular, mathematically simple way
the material behaves the same way in every direction
the forces on the beam only cause the beam to bend, but not twist or stretch.

More rigorously stated, these assumptions are:

continuum mechanics is valid for a bending beam
the stress at a cross section varies linearly in the direction of bending, and is zero at the centroid of every cross section
the bending moment at a particular cross section varies linearly with the second derivative of the deflected shape at that location
the beam is composed of an isotropic material
the applied load is orthogonal to the beam's neutral axis and acts in a unique plane.

With these assumptions, we can derive the following equation governing the relationship between the beam's deflection and the applied load.

$\frac{\partial^2}{\partial x^2}\left(EI \frac{\partial^2 u}{\partial x^2}\right) = w\,$

This is the Euler-Bernoulli equation. The curve $u (x)$ describes the deflection $u$ of the beam at some position $x$ (recall that the beam is modeled as a one-dimensional object). $w$ is a distributed load, in other words a force per unit length (analogous to pressure being a force per area); it may be a function of $x$ , $u$ , or other variables.

Note that $E$ is the elastic modulus and that $I$ is the second moment of area. $I$ must be calculated with respect to the centroidal axis perpendicular to the applied loading. For an Euler-Bernoulli beam not under any axial loading this axis is called the neutral axis.

Often, $u = u (x)$ , $w = w (x)$ , and EI is a constant, so that:

$EI \frac{d^4 u}{d x^4} = w(x)\,$

This equation, describing the deflection of a uniform, static beam, is very common in engineering practice.

Successive derivatives of $u$ have important meanings:

$\textstyle{u}\,$ is the deflection.

$\textstyle{\frac{\partial u}{\partial x}}\,$ is the slope of the beam.

$\textstyle{EI\frac{\partial^2 u}{\partial x^2}}\,$ is the bending moment in the beam.

$\textstyle{-\frac{\partial}{\partial x}\left(EI\frac{\partial^2 u}{\partial x^2}\right)}\,$ is the shear force in the beam.

Stress

Besides deflection, the beam equation describes forces and moments and can thus be used to describe stresses. For this reason, the Euler-Bernoulli beam equation is widely used in engineering, especially civil and mechanical, to determine the strength (as well as deflection) of beams under bending.

Both the bending moment and the shear force cause stresses in the beam. The stress due to shear force is maximum along the neutral axis of the beam, and the maximum tensile stress is at either the top or bottom surfaces. Thus the maximum principal stress in the beam may be neither at the surface nor at the center but in some general area. However, shear force stresses are negligible in comparison to bending moment stresses in all but the stockiest of beams as well as the fact that stress concentrations commonly occur at surfaces, meaning that the maximum stress in a beam is likely to be at the surface.

It can be shown that the tensile stress experienced by the beam may be expressed as:

$\sigma = \frac{Mc}{I} = E c \frac{\partial^2 u}{\partial x^2}\,$

Here, $c$ , a position along $u$ , is the distance from the neutral axis to a point of interest; and $M$ is the bending moment. Note that this equation implies that "pure" bending (of positive sign) will cause zero stress at the neutral axis, positive (tensile) stress at the "top" of the beam, and negative (compressive) stress at the bottom of the beam; and also implies that the maximum stress will be at the top surface and the minimum at the bottom. This bending stress may be superimposed with axially applied stresses, which will cause a shift in the neutral (zero stress) axis.

Boundary considerations

The beam equation contains a fourth-order derivative in $x$ , hence it mandates at most four conditions, normally boundary conditions. The boundary conditions usually model supports, but they can also model point loads, moments, or other effects.

A cantilever beam.

An example is a cantilever beam: a beam that is completely fixed at one end and completely free at the other. "Completely fixed" means that at the left end both deflection and slope are zero; "completely free" implies (though it may or may not be obvious) that at the right end both shear force and bending moment are zero. Taking the $x$ coordinate of the left end as $0$ and the right end as $L$ (the length of the beam), these statements translate to the following set of boundary conditions (assume $E I$ is a constant):

$u|_{x = 0} = 0 \quad ; \quad \frac{\partial u}{\partial x}\bigg|_{x = 0} = 0 \qquad \mbox{(fixed end)}\,$

$\frac{\partial^2 u}{\partial x^2}\bigg|_{x = L} = 0 \quad ; \quad \frac{\partial^3 u}{\partial x^3}\bigg|_{x = L} = 0 \qquad \mbox{(free end)}\,$

Some commonly encountered boundary conditions include:

$\textstyle{u = \frac{\partial u}{\partial x} = 0}\,$ represents a fixed support.

$\textstyle{u = \frac{\partial^2 u}{\partial x^2} = 0}\,$ represents a pin connection (deflection and moment fixed to zero).

$\textstyle{\frac{\partial^2 u}{\partial x^2} = \frac{\partial^3 u}{\partial x^3} = 0}\,$ represents no connection (no restraint) and no load.

$\textstyle{-\frac{\partial}{\partial x}\left(EI\frac{\partial^2 u}{\partial x^2}\right)} = F\,$ represents the application of a point load F.

Loading considerations

Applied loading may be represented either through boundary conditions or through the distributed function $w$ . Using distributed loading is often favorable for simplicity. Boundary conditions are, however, often used to model loads depending on context; this practice being especially common in vibration analysis.

By nature, the distributed load is very often represented in a piecewise manner, since in practice a load isn't typically a "nice" continuous function. Point loads can be modeled with help of the Dirac delta function. For example, consider a static uniform cantilever beam of length $L$ with an upward point load $F$ applied at the free end. Using boundary conditions, this may be modeled through:

$EI \frac{d^4 u}{d x^4} = 0\,$

$u|_{x = 0} = 0 \quad ; \quad \frac{d u}{d x}\bigg|_{x = 0} = 0\,$

$\frac{d^2 u}{d x^2}\bigg|_{x = L} = 0 \quad ; \quad -EI \frac{d^3 u}{d x^3}\bigg|_{x = L} = F\,$

Using the Dirac function,

$EI \frac{d^4 u}{d x^4} = F \delta(x - L)\,$

$u|_{x = 0} = 0 \quad ; \quad \frac{d u}{d x}\bigg|_{x = 0} = 0\,$

$\frac{d^2 u}{d x^2}\bigg|_{x = L} = 0\,$

Note that shear force boundary condition (third derivative) is removed, otherwise there would be a contradiction. These are equivalent boundary value problems, and both yield the following solution:

$u = \frac{F}{6 EI}(3 L x^2 - x^3)\,$

The application of several point loads at different locations will lead to $u (x)$ being a piecewise function. Use of the Dirac function greatly simplifies such situations; otherwise the beam would have to be divided into sections, each with four boundary conditions solved separately. A well organized family of functions called Singularity functions are often used as a shorthand for the Dirac function, its derivative, and its antiderivatives.

Clever formulation of the load distribution allows for many interesting phenomena to be modeled. As an example, the vibration of a beam can be accounted for by using the load function:

$w(x, t) = -\mu \frac{\partial^2 u}{\partial t^2}\,$

where $μ$ is the linear mass density of the beam, not necessarily a constant. With this time-dependent loading, the beam equation will be a partial differential equation:

$\mu \frac{\partial^2 u}{\partial t^2} + \frac{\partial^2}{\partial x^2} \left( EI \frac{\partial^2 u}{\partial x^2} \right) = 0.$

Another interesting example describes the deflection of a beam rotating with a constant angular frequency of $ω$ :

$w(u) = \mu \omega^2 u\,$

This is a centripetal force distribution. Note that in this case, $w$ is a function of the displacement (the dependent variable), and the beam equation will be an autonomous ordinary differential equation.

Extensions

The kinematic assumptions upon which the Euler-Bernoulli beam theory is founded allow it to be extended to more advanced analysis. Simple superposition allows for three-dimensional transverse loading. Using alternative constitutive equations can allow for viscoelastic or plastic beam deformation. Euler-Bernoulli beam theory can also be extended to the analysis of curved beams, beam buckling, composite beams, and geometrically nonlinear beam deflection.

Euler-Bernoulli beam theory does not account for the effects of transverse shear strain. As a result it underpredicts deflections and overpredicts natural frequencies. For thin beams (beam length to thickness ratios of the order 20 or more) these effects are of minor importance. For thick beams, however, these effects can be significant. More advanced beam theories such as the Timoshenko beam theory (developed by the Russian-born scientist Stephen Timoshenko) have been developed to account for these effects.

Notes

^ Ballarini, Roberto (April 18, 2003). "The Da Vinci-Euler-Bernoulli Beam Theory?". Mechanical Engineering Magazine Online Retrieved on July 22, 2006.
^ Seon M. Han, Haym Benaroya and Timothy Wei (March 22, 1999). "Dynamics of Transversely Vibrating Beams using four Engineering Theories" (PDF). final version. Academic Press. Retrieved on April 15, 2007.

References

E.A. Witmer (1991-1992). "Elementary Bernoulli-Euler Beam Theory". MIT Unified Engineering Course Notes: pp. 5-114 to 5-164.

External links

Efunda's beam calculator

Timoshenko Beam Theory

Shear and bending deformation of a sandwich composite beam.

The Timoshenko beam theory was developed by Ukrainian/Russian-born scientist Stephen Timoshenko in the beginning of the 20th century. The model takes into account shear deformation and rotational inertia effects, making it suitable for describing the behaviour of short beams, sandwich composite beams or beams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of 4^th order, but unlike ordinary beam theory - i.e. Bernoulli-Euler theory - there is also a second order spatial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, why the result is a larger deflection under a static load and lower predicted eigenfrequencies for a given set of boundary conditions. The latter effect is more noticeable for higher frequencies as the wavelength becomes shorter, and thus the distance between opposing shear forces decreases.

If the shear modulus of the beam material approaches infinity - and thus the beam becomes rigid in shear - and if rotational inertia effects are neglected, Timoshenko beam theory converges towards ordinary beam theory.

This beam theory, allowing for vibrations, may be described with the coupled linear partial differential equations ^[1]:

$\rho A\frac{\partial^{2}u}{\partial t^{2}} = \frac{\partial}{\partial x}\left( A\kappa G \left(\frac{\partial u}{\partial x}-\theta\right)\right) + w$

$\rho I\frac{\partial^{2}\theta}{\partial t^{2}} = \frac{\partial}{\partial x}\left(EI\frac{\partial \theta}{\partial x}\right)+A\kappa G\left(\frac{\partial u}{\partial x}-\theta\right)$

where the dependent variables are $u$ , the translational displacement of the beam, and $θ$ , the angular displacement. Note that unlike the Euler-Bernoulli theory, the angular deflection is another variable and not approximated by the slope of the deflection. Also,

$ρ$ is the density of the beam material (but not the linear density).
$A$ is the cross section area.
$E$ is the elastic modulus.
$G$ is the shear modulus.
$I$ is the second moment of area.
$κ$ , called the Timoshenko shear coefficient, depends on the geometry. Normally, $κ = 5 / 6$ for a rectangular section.
$w$ is a distributed load (force per length).

These parameters are not necessarily constants.

Determining the shear coefficient is not straightforward (nor are the determined values widely accepted, ie there's more than one answer), generally it must satisfy:

$\int_A \tau dA = \kappa G A \theta\,$