Euler-Bernoulli beam equation

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The elementary Euler-Bernoulli beam theory is a simplification of the linear isotropic theory of elasticity which allows quick calculation of the load-carrying capacity and deflection of common structural elements called 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 structural analysis tools have been developed such as plate theory and finite element analysis, but the simplicity of E-B beam theory makes it a primary tool for preliminary sizing of all engineered structures. It remains known as the most fundamental lessons of civil engineering and mechanical engineering courses.

Contents 1 History 2 Assumptions 3 Predictions 3.1 Definitions 3.2 Final equations 4 Derivation 5 Practical simplifications 6 Extensions 7 See also 8 References 9 External links

[edit] 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]

Leonhard Euler and Daniel Bernoulli were the first to put together a useful theory circa 1750. 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 a large scale.

[edit] Assumptions

The full theory of elasticity is too complicated for routine design work. To simplify it, B-E beam theory makes six assumptions which are approximately true for most beams:

1. The beam is long and slender.
   • length >> width
   • length >> depth
   • therefore tensile/compressive stresses perpendicular to the beam are much smaller than tensile/compressive stresses parallel to the beam.
2. The beam cross-section is constant along its axis.
3. The beam is loaded in its plane of symmetry.
   • torsion = 0
4. Deformations remain small.
   • This simplifies the theory of elasticity to its linear form.
   • no buckling
   • no plasticity
   • no soft materials.
5. Material is isotropic
   • no wood
6. Plane sections of the beam remain plane.
   • approximately true for most solid beam forms
   • not necessarily true for a truss-beam
   • this was Bernoulli's critical contribution

Real-life structures never meet these assumptions exactly, but often approximate them well enough for the theory to make useful predictions. In practice, many engineers forget these assumptions after they graduate from school and often apply the theory inappropriately. The general conservative practices of applied engineering compensate for this, and structural failures due to design error are unusual.

[edit] Predictions

[edit] Definitions

• x = location along the beam axis
• y = location perpendicular to beam and to loading
• z = location perpendicular to beam, in load plane, with the axis origin at the centroid of the area of the cross-section
• u_x = deflection along beam axis
• u_z = deflection in load plane, perpendicular to beam
• F_x(x) = internal axial force as function of x
• M_y(x) = internal bending moment as function of x
• S_z(x) = internal shear force as function of x
• E = modulus of elasticity of the material
• A = cross sectional area perpendicular to beam axis
• I_y = area moment of inertia or second moment of inertia of the beam cross-section, taken about an axis perpendicular to the loading plane.

over the beam cross-section.

• b = beam width perpendicular to load plane
• Q_y(z) = first moment of inertia about the y axis of the cross-section area above height z.

[edit] Final equations

compression/extension along the length of the beam

bending deflection due to forces transverse to the beam

composite compressive/tensile stress due to compression/tension force + moment

shear stress in the beam

trivial outcome of linearity assumption, included for completeness

[edit] Derivation

[edit] Practical simplifications

The full E-B beam equations are still too complicated for routine application, but they can easily be simplified further with additional assumptions about the loading and geometry. For example, for a rectangular cantilevered beam with a transverse tip load P, the equations reduce to

occurring at the free tip (B), x = L

occurring at the fixed root (A), x = 0

with

Many books catalog simplified B-E equations for common structures. One of the best-known and most comprehensive is Roark's Formulas for Stress and Strain.

[edit] Extensions

Using different assumptions and derivations, the theory has been extended in a number of ways. A simple superposition allows 3D loading of the beam, although still without torsion. Other versions allow plastic bending, beam buckling, orthotropic materials (wood), and variable cross-sections.

[edit] See also

• Buckling
• Stress (physics), Strain.
• Applied mechanics

[edit] References

1. ^ Ballarini, Roberto (April 18 2003). "The Da Vinci-Euler-Bernoulli Beam Theory?". Mechanical Engineering Magazine Online. Retrieved on July 22, 2006.

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

[edit] External links

• Efunda's beam calculator