Timoshenko beam theory

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Orientations of the line perpendicular to the mid-plane of a thick book under bending.

The Timoshenko beam theory was developed by Stephen Timoshenko early in the 20th century.[1][2] 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 4th order but, unlike ordinary Euler–Bernoulli beam theory, there is also a second-order partial derivative present. Physically, taking into account the added mechanisms of deformation effectively lowers the stiffness of the beam, while 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.

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

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.

Quasistatic Timoshenko beam[edit]

Deformation of a Timoshenko beam. The normal rotates by an amount \theta_x = \varphi(x) which is not equal to dw/dx.

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


  u_x(x,y,z) = -z~\varphi(x) ~;~~ u_y(x,y,z) = 0 ~;~~ u_z(x,y) = w(x)

where (x,y,z) are the coordinates of a point in the beam, u_x, u_y, u_z are the components of the displacement vector in the three coordinate directions, \varphi is the angle of rotation of the normal to the mid-surface of the beam, and w is the displacement of the mid-surface in the z-direction.

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


 \begin{align}
    & \frac{\mathrm{d}^2}{\mathrm{d} x^2}\left(EI\frac{\mathrm{d} \varphi}{\mathrm{d} x}\right) = q(x,t) \\
    & \frac{\mathrm{d} w}{\mathrm{d} x} = \varphi - \frac{1}{\kappa AG} \frac{\mathrm{d}}{\mathrm{d} x}\left(EI\frac{\mathrm{d} \varphi}{\mathrm{d} x}\right).
  \end{align}

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


\frac{EI}{\kappa L^2 A G} \ll 1

where

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


   EI~\cfrac{\mathrm{d}^4 w}{\mathrm{d} x^4} = q(x) - \cfrac{EI}{\kappa A G}~\cfrac{\mathrm{d}^2 q}{\mathrm{d} x^2}

The bending moment M_{xx} and the shear force Q_x in the beam are related to the displacement w and the rotation \varphi. These relations, for a linear elastic Timoshenko beam, are:


    M_{xx} = -EI~\frac{\partial \varphi}{\partial x} \quad \text{and} \quad
    Q_{x}  = \kappa~AG~\left(-\varphi + \frac{\partial w}{\partial x}\right) \,.

Boundary conditions[edit]

The two equations that describe the deformation of a Timoshenko beam have to be augmented with boundary conditions if they are to be solved. Four boundary conditions are needed for the problem to be well-posed. Typical boundary conditions are:

  • Simply supported beams: The displacement w is zero at the locations of the two supports. The bending moment M_{xx} applied to the beam also has to be specified. The rotation \varphi and the transverse shear force Q_x are not specified.
  • Clamped beams: The displacement w and the rotation \varphi are specified to be zero at the clamped end. If one end is free, shear force Q_x and bending moment M_{xx} have to be specified at that end.

Example: Cantilever beam[edit]

A cantilever Timoshenko beam under a point load at the free end.

For a cantilever beam, one boundary is clamped while the other is free. Let us use a right handed coordinate system where the x direction is positive towards right and the z direction is positive upward. Following normal convention, we assume that positive forces act in the positive directions of the x and z axes and positive moments act in the clockwise direction. We also assume that the sign convention of the stress resultants (M_{xx} and Q_x) is such that positive bending moments compress the material at the bottom of the beam (lower z coordinates) and positive shear forces rotate the beam in a counterclockwise direction.

Let us assume that the clamped end is at x=L and the free end is at x=0. If a point load P is applied to the free end in the positive z direction, a free body diagram of the beam gives us


   -Px - M_{xx} = 0 \implies M_{xx} = -Px

and

 P + Q_x = 0 \implies Q_x = -P\,.

Therefore, from the expressions for the bending moment and shear force, we have


   Px = EI\,\frac{d\varphi}{dx} \qquad \text{and} \qquad -P = \kappa AG\left(-\varphi + \frac{dw}{dx}\right) \,.

Integration of the first equation, and application of the boundary condition \varphi = 0 at x = L, leads to


    \varphi(x) = -\frac{P}{2EI}\,(L^2-x^2) \,.

The second equation can then be written as


   \frac{dw}{dx} = -\frac{P}{\kappa AG} - \frac{P}{2EI}\,(L^2-x^2)\,.

Integration and application of the boundary condition w = 0 at x = L gives


   w(x) = \frac{P(L-x)}{\kappa AG} - \frac{Px}{2EI}\,\left(L^2-\frac{x^2}{3}\right) + \frac{PL^3}{3EI} \,.

The axial stress is given by


   \sigma_{xx}(x,z) = E\,\varepsilon_{xx} = -E\,z\,\frac{d\varphi}{dx} = -\frac{Pxz}{I} = \frac{M_{xx}z}{I} \,.

Dynamic Timoshenko beam[edit]

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


  u_x(x,y,z,t) = -z~\varphi(x,t) ~;~~ u_y(x,y,z,t) = 0 ~;~~ u_z(x,y,z,t) = w(x,t)

where (x,y,z) are the coordinates of a point in the beam, u_x, u_y, u_z are the components of the displacement vector in the three coordinate directions, \varphi is the angle of rotation of the normal to the mid-surface of the beam, and w is the displacement of the mid-surface in the z-direction.

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


\rho A\frac{\partial^{2}w}{\partial t^{2}} - q(x,t) = \frac{\partial}{\partial x}\left[ \kappa AG \left(\frac{\partial w}{\partial x}-\varphi\right)\right]

\rho I\frac{\partial^{2}\varphi}{\partial t^{2}} = \frac{\partial}{\partial x}\left(EI\frac{\partial \varphi}{\partial x}\right)+\kappa AG\left(\frac{\partial w}{\partial x}-\varphi\right)

where the dependent variables are w(x,t), the translational displacement of the beam, and \varphi(x,t), 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,

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]


   EI~\cfrac{\partial^4 w}{\partial x^4} + m~\cfrac{\partial^2 w}{\partial t^2} - \left(J + \cfrac{E I m}{k A G}\right)\cfrac{\partial^4 w}{\partial x^2~\partial t^2} + \cfrac{m J}{k A G}~\cfrac{\partial^4 w}{\partial t^4} = q(x,t) + \cfrac{J}{k A G}~\cfrac{\partial^2 q}{\partial t^2} - \cfrac{EI}{k A G}~\cfrac{\partial^2 q}{\partial x^2}

Axial effects[edit]

If the displacements of the beam are given by


  u_x(x,y,z,t) = u_0(x,t)-z~\varphi(x,t) ~;~~ u_y(x,y,z,t) = 0 ~;~~ u_z(x,y,z) = w(x,t)

where u_0 is an additional displacement in the x-direction, then the governing equations of a Timoshenko beam take the form


  \begin{align}
m \frac{\partial^{2}w}{\partial t^{2}} & = \frac{\partial}{\partial x}\left[ \kappa AG \left(\frac{\partial w}{\partial x}-\varphi\right)\right] + q(x,t) \\
J \frac{\partial^{2}\varphi}{\partial t^{2}} & = N(x,t)~\frac{\partial w}{\partial x} + \frac{\partial}{\partial x}\left(EI\frac{\partial \varphi}{\partial x}\right)+\kappa AG\left(\frac{\partial w}{\partial x}-\varphi\right)
  \end{align}

where J = \rho I and N(x,t) is an externally applied axial force. Any external axial force is balanced by the stress resultant


   N_{xx}(x,t) = \int_{-h}^{h} \sigma_{xx}~dz

where \sigma_{xx} is the axial stress and the thickness of the beam has been assumed to be 2h.

The combined beam equation with axial force effects included is


   EI~\cfrac{\partial^4 w}{\partial x^4} + N~\cfrac{\partial^2 w}{\partial x^2} + m~\frac{\partial^2 w}{\partial t^2} - \left(J+\cfrac{mEI}{\kappa AG}\right)~\cfrac{\partial^4 w}{\partial x^2 \partial t^2} + \cfrac{mJ}{\kappa AG}~\cfrac{\partial^4 w}{\partial t^4} = q + \cfrac{J}{\kappa AG}~\frac{\partial^2 q}{\partial t^2} - \cfrac{EI}{\kappa A G}~\frac{\partial^2 q}{\partial x^2}

Damping[edit]

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


   \eta(x)~\cfrac{\partial w}{\partial t}

the coupled governing equations for a Timoshenko beam take the form


m \frac{\partial^{2}w}{\partial t^{2}} + \eta(x)~\cfrac{\partial w}{\partial t} = \frac{\partial}{\partial x}\left[ \kappa AG \left(\frac{\partial w}{\partial x}-\varphi\right)\right] + q(x,t)

J \frac{\partial^{2}\varphi}{\partial t^{2}} = N\frac{\partial w}{\partial x} + \frac{\partial}{\partial x}\left(EI\frac{\partial \varphi}{\partial x}\right)+\kappa AG\left(\frac{\partial w}{\partial x}-\varphi\right)

and the combined equation becomes


  \begin{align}
   EI~\cfrac{\partial^4 w}{\partial x^4} & + N~\cfrac{\partial^2 w}{\partial x^2} + m~\frac{\partial^2 w}{\partial t^2} - \left(J+\cfrac{mEI}{\kappa AG}\right)~\cfrac{\partial^4 w}{\partial x^2 \partial t^2} + \cfrac{mJ}{\kappa AG}~\cfrac{\partial^4 w}{\partial t^4} + \cfrac{J \eta(x)}{\kappa AG}~\cfrac{\partial^3 w}{\partial t^3} \\
  & -\cfrac{EI}{\kappa AG}~\cfrac{\partial^2}{\partial x^2}\left(\eta(x)\cfrac{\partial w}{\partial t}\right) + \eta(x)\cfrac{\partial w}{\partial t} = q + \cfrac{J}{\kappa AG}~\frac{\partial^2 q}{\partial t^2} - \cfrac{EI}{\kappa A G}~\frac{\partial^2 q}{\partial x^2}
  \end{align}

A caveat to this Ansatz damping force (resembling viscosity) is that, whereas viscosity leads to a frequency-dependent and amplitude-independent damping rate of beam oscillations, the empirically measured damping rates are frequency-insensitive, but depend on the amplitude of beam deflection.

Shear coefficient[edit]

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

\int_A \tau dA = \kappa A G \varphi\, .

The shear coefficient depends on the Poisson's ratio. The attempts to provide precise expressions were made by many scientists, including Stephen Timoshenko,[6] Raymond D. Mindlin,[7] G. R. Cowper,[8] N. G. Stephen,[9] J. R. Hutchinson[10] etc. (see also the derivation of the Timoshenko beam theory as refined beam theory based on the variational-asymptotic method in the book by Khanh C. Le[11] leading to the different shear coefficients in the static and dynamic cases). In engineering practice, the expressions by Stephen Timoshenko[12] are sufficient in most cases. In 1975 Kaneko[13] published an excellent review of studies of the shear coefficient.

According to Cowper (1966) for solid rectangular cross-section,


\kappa = \cfrac{10(1+\nu)}{12+11\nu}

and for solid circular cross-section,


\kappa = \cfrac{6(1+\nu)}{7+6\nu}
.

See also[edit]

References[edit]

  1. ^ Timoshenko, S. P., 1921, On the correction factor for shear of the differential equation for transverse vibrations of bars of uniform cross-section, Philosophical Magazine, p. 744.
  2. ^ Timoshenko, S. P., 1922, On the transverse vibrations of bars of uniform cross-section, Philosophical Magazine, p. 125.
  3. ^ Timoshenko's Beam Equations
  4. ^ Thomson, W. T., 1981, Theory of Vibration with Applications, Second Edition, Prentice-Hall, New Jersey.
  5. ^ Rosinger, H. E. and Ritchie, I. G., 1977, On Timoshenko's correction for shear in vibrating isotropic beams, J. Phys. D: Appl. Phys., vol. 10, pp. 1461-1466.
  6. ^ Timoshenko, Stephen P., 1932, Schwingungsprobleme der Technik, Julius Springer.
  7. ^ Mindlin, R. D., Deresiewicz, H., 1953, Timoshenko's Shear Coefficient for Flexural Vibrations of Beams, Technical Report No. 10, ONR Project NR064-388, Department of Civil Engineering, Columbia University, New York, N. Y.
  8. ^ Cowper, G. R., 1966, The Shear Coefficient in Timoshenko’s Beam Theory, J. Appl. Mech., Vol. 33, No.2, pp. 335–340.
  9. ^ Stephen, N. G., 1980. Timoshenko’s shear coefficient from a beam subjected to gravity loading, Journal of Applied Mechanics, Vol. 47, No. 1, pp. 121–127.
  10. ^ Hutchinson, J. R., 1981, Transverse vibration of beams, exact versus approximate solutions, Journal of Applied Mechanics, Vol. 48, No. 12, pp. 923–928.
  11. ^ Le, Khanh C., 1999, Vibrations of shells and rods, Springer.
  12. ^ Stephen Timoshenko, James M. Gere. Mechanics of Materials. Van Nostrand Reinhold Co., 1972. Pages 207.
  13. ^ Kaneko, T., 1975, On Timoshenko’s correction for shear in vibrating beams, J. Phys. D: Appl. Phys., Vol. 8, pp. 1927–1936.