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Deformation of a Timoshenko beam (blue) compared with that of anEuler-Bernoulli beam (red).

Timoshenko beam theoryFrom Wikipedia, the free encyclopedia

The Timoshenko beam theory was developed by Ukrainian-born scientist and engineer Stephen Timoshenko early in the 20th century.[1][2] The model takesinto account shear deformation and rotational inertia effects, making it suitable for describing the behaviour of short beams, sandwich composite beams orbeams subject to high-frequency excitation when the wavelength approaches the thickness of the beam. The resulting equation is of 4th order, but unlikeordinary beam theory - i.e. Euler–Bernoulli beam theory - there is also a second order spatial derivative present. Physically, taking into account the addedmechanisms of deformation effectively lowers the stiffness of the beam, while the result is a larger deflection under a static load and lower predictedeigenfrequencies for a given set of boundary conditions. The latter effect is more noticeable for higher frequencies as the wavelength becomes shorter, and thusthe distance between opposing shear forces decreases.

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

Contents

1 Governing equations1.1 Quasistatic Timoshenko beam1.2 Dynamic Timoshenko beam

1.2.1 Axial effects1.2.2 Damping

2 Shear coefficient3 See also4 References

Governing equations

Quasistatic Timoshenko beam

In static Timoshenko beam theory without axial effects, the displacements of the beam are assumed to be given by

Timoshenko beam theory - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Timoshenko_beam_theory

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Deformation of a Timoshenkobeam. The normal rotates by anamount which isnot equal to .

where are the coordinates of a point in the beam, are the components of the displacement vector inthe three coordinate directions, is the angle of rotation of the normal to the mid-surface of the beam, and is thedisplacement of the mid-surface in the -direction.

The governing equations are the following uncoupled system of ordinary differential equations:

The Timoshenko beam theory for the static case is equivalent to the Euler-Bernoulli theory when the last term above isneglected, an approximation that is valid when

where is the length of the beam.

Combining the two equations gives, for a homogeneous beam of constant cross-section,

Derivation of quasistatic Timoshenko beam equationsFrom the kinematic assumptions for a Timoshenko beam, the displacements of the beam are given by

Then, from the strain-displacement relations for small strains, the non-zero strains based on the Timoshenko assumptions are


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Since the actual shear strain in the beam is not constant over the cross section we introduce a correction factor such that

The variation in the internal energy of the beam is

Define

Then

Integration by parts, and noting that because of the boundary conditions the variations are zero at the ends of the beam, leads to

The variation in the external work done on the beam by a transverse load per unit length is

Then, for a quasistatic beam, the principle of virtual work gives


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The governing equations for the beam are, from the fundamental theorem of variational calculus,

For a linear elastic beam

Therefore the governing equations for the beam may be expressed as

Combining the two equations together gives

Dynamic Timoshenko beam

In Timoshenko beam theory without axial effects, the displacements of the beam are assumed to be given by


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where are the coordinates of a point in the beam, are the components of the displacement vector in the three coordinate directions, isthe angle of rotation of the normal to the mid-surface of the beam, and is the displacement of the mid-surface in the -direction.

Starting from the above assumption, the Timoshenko beam theory, allowing for vibrations, may be described with the coupled linear partial differentialequations:[3]

where the dependent variables are , the translational displacement of the beam, and , 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,

is the density of the beam material (but not the linear density). is the cross section area. is the elastic modulus. is the shear modulus.

is the second moment of area., called the Timoshenko shear coefficient, depends on the geometry. Normally, for a rectangular section.

is a distributed load (force per length).

These parameters are not necessarily constants.

For a linear elastic, isotropic, homogeneous beam of constant cross-section these two equations can be combined to give[4][5]


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Derivation of combined Timoshenko beam equationThe equations governing the bending of a homogeneous Timoshenko beam of constant cross-section are

From equation (1), assuming appropriate smoothness, we have

From (3), assuming appropriate smoothness,

Differentiating equation (2) gives

From equations (4) and (6)

From equations (3) and (7)


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Plugging equation (5) into (8) gives

Rearrange to get

Axial effects

If the displacements of the beam are given by

where is an additional displacement in the -direction, then the governing equations of a Timoshenko beam take the form

where and is an externally applied axial force. Any external axial force is balanced by the stress resultant

where is the axial stress and the thickness of the beam has been assumed to be .


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The combined beam equation with axial force effects included is

Damping

If, in addition to axial forces, we assume a damping force that is proportional to the velocity with the form

the coupled governing equations for a Timoshenko beam take the form

and the combined equation becomes

Shear coefficient

Determining the shear coefficient is not straightforward (nor are the determined values widely accepted, i.e. there's more than one answer), generally it mustsatisfy:


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The shear coefficient is dependent to the Poisson's Ratio. The approaches of more precise expressions are made by many scientists, including StephenTimoshenko, Raymond D. Mindlin, G. R. Cowper, John W. Hutchinson, etc. In engineering practices, the expressions provided by Stephen Timoshenko[6] aregood enough for general cases.

For solid rectangular cross-section,

For solid circular cross-section,

See also

BendingEuler–Bernoulli beam theorySandwich theoryPlate theory

References

^ Timoshenko, S. P., 1921, On the correction factor for shear of the differentialequation for transverse vibrations of bars of uniform cross-section,Philosophical Magazine, p. 744.

1.

^ Timoshenko, S. P., 1922, On the transverse vibrations of bars of uniformcross-section, Philosophical Magazine, p. 125.

2.

^ Timoshenko's Beam Equations (http://ccrma.stanford.edu/~bilbao/master3.

/node163.html)^ Thomson, W. T., 1981, Theory of Vibration with Applications4.^ Rosinger, H. E. and Ritchie, I. G., 1977, On Timoshenko's correction for shearin vibrating isotropic beams, J. Phys. D: Appl. Phys., vol. 10, pp. 1461-1466.

5.

^ Stephen Timoshenko, James M. Gere. Mechanics of Materials. Van NostrandReinhold Co., 1972. Pages 207.

6.

Stephen P. Timoshenko (1932). Schwingungsprobleme der technik. Verlag von Julius Springer.


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