# Timoshenko beam theory

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 bending effects, making it suitable for describing the behaviour of thick 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 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.

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

Deformation of a Timoshenko beam (blue) compared with that of an Euler-Bernoulli beam (red).
Deformation of a Timoshenko beam. The normal rotates by an amount ${\displaystyle \theta _{x}=\varphi (x)}$ which is not equal to ${\displaystyle dw/dx}$.

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

${\displaystyle u_{x}(x,y,z)=-z~\varphi (x)~;~~u_{y}(x,y,z)=0~;~~u_{z}(x,y)=w(x)}$

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

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

{\displaystyle {\begin{aligned}&{\frac {\mathrm {d} ^{2}}{\mathrm {d} x^{2}}}\left(EI{\frac {\mathrm {d} \varphi }{\mathrm {d} x}}\right)=q(x)\\&{\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{aligned}}}

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

${\displaystyle {\frac {EI}{\kappa L^{2}AG}}\ll 1}$

where

• ${\displaystyle L}$ is the length of the beam.
• ${\displaystyle A}$ is the cross section area.
• ${\displaystyle E}$ is the elastic modulus.
• ${\displaystyle G}$ is the shear modulus.
• ${\displaystyle I}$ is the second moment of area.
• ${\displaystyle \kappa }$, called the Timoshenko shear coefficient, depends on the geometry. Normally, ${\displaystyle \kappa =5/6}$ for a rectangular section.
• ${\displaystyle q(x)}$ is a distributed load (force per length).

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

${\displaystyle EI~{\cfrac {\mathrm {d} ^{4}w}{\mathrm {d} x^{4}}}=q(x)-{\cfrac {EI}{\kappa AG}}~{\cfrac {\mathrm {d} ^{2}q}{\mathrm {d} x^{2}}}}$

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

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

### Boundary conditions

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

### Example: Cantilever beam

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 ${\displaystyle x}$ direction is positive towards right and the ${\displaystyle z}$ direction is positive upward. Following normal convention, we assume that positive forces act in the positive directions of the ${\displaystyle x}$ and ${\displaystyle z}$ axes and positive moments act in the clockwise direction. We also assume that the sign convention of the stress resultants (${\displaystyle M_{xx}}$ and ${\displaystyle Q_{x}}$) is such that positive bending moments compress the material at the bottom of the beam (lower ${\displaystyle z}$ coordinates) and positive shear forces rotate the beam in a counterclockwise direction.

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

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

and

${\displaystyle P+Q_{x}=0\implies Q_{x}=-P\,.}$

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

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

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

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

The second equation can then be written as

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

Integration and application of the boundary condition ${\displaystyle w=0}$ at ${\displaystyle x=L}$ gives

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

The axial stress is given by

${\displaystyle \sigma _{xx}(x,z)=E\,\varepsilon _{xx}=-E\,z\,{\frac {d\varphi }{dx}}=-{\frac {Pxz}{I}}={\frac {M_{xx}z}{I}}\,.}$

## Dynamic Timoshenko beam

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

${\displaystyle u_{x}(x,y,z,t)=-z~\varphi (x,t)~;~~u_{y}(x,y,z,t)=0~;~~u_{z}(x,y,z,t)=w(x,t)}$

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

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

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

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

• ${\displaystyle \rho }$ is the density of the beam material (but not the linear density).
• ${\displaystyle A}$ is the cross section area.
• ${\displaystyle E}$ is the elastic modulus.
• ${\displaystyle G}$ is the shear modulus.
• ${\displaystyle I}$ is the second moment of area.
• ${\displaystyle \kappa }$, called the Timoshenko shear coefficient, depends on the geometry. Normally, ${\displaystyle \kappa =5/6}$ for a rectangular section.
• ${\displaystyle q(x,t)}$ is a distributed load (force per length).
• ${\displaystyle m:=\rho A}$
• ${\displaystyle J:=\rho I}$

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]

${\displaystyle EI~{\cfrac {\partial ^{4}w}{\partial x^{4}}}+m~{\cfrac {\partial ^{2}w}{\partial t^{2}}}-\left(J+{\cfrac {EIm}{kAG}}\right){\cfrac {\partial ^{4}w}{\partial x^{2}~\partial t^{2}}}+{\cfrac {mJ}{kAG}}~{\cfrac {\partial ^{4}w}{\partial t^{4}}}=q(x,t)+{\cfrac {J}{kAG}}~{\cfrac {\partial ^{2}q}{\partial t^{2}}}-{\cfrac {EI}{kAG}}~{\cfrac {\partial ^{2}q}{\partial x^{2}}}}$

The Timoshenko equation predicts a critical frequency ${\displaystyle \omega _{C}=2\pi f_{c}={\sqrt {\frac {\kappa GA}{\rho I}}}.}$ For normal modes the Timoshenko equation can be solved. Being a fourth order equation, there are four independent solutions, two oscillatory and two evanescent for frequencies below ${\displaystyle f_{c}}$. For frequencies larger than ${\displaystyle f_{c}}$ all solutions are oscillatory and, as consequence, a second spectrum appears.[6]

### Axial effects

If the displacements of the beam are given by

${\displaystyle u_{x}(x,y,z,t)=u_{0}(x,t)-z~\varphi (x,t)~;~~u_{y}(x,y,z,t)=0~;~~u_{z}(x,y,z,t)=w(x,t)}$

where ${\displaystyle u_{0}}$ is an additional displacement in the ${\displaystyle x}$-direction, then the governing equations of a Timoshenko beam take the form

{\displaystyle {\begin{aligned}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{aligned}}}

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

${\displaystyle N_{xx}(x,t)=\int _{-h}^{h}\sigma _{xx}~dz}$

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

The combined beam equation with axial force effects included is

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

### Damping

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

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

the coupled governing equations for a Timoshenko beam take the form

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

and the combined equation becomes

{\displaystyle {\begin{aligned}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 AG}}~{\frac {\partial ^{2}q}{\partial x^{2}}}\end{aligned}}}

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

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:

${\displaystyle \int _{A}\tau dA=\kappa AG\varphi \,}$ .

The shear coefficient depends on the Poisson's ratio. The attempts to provide precise expressions were made by many scientists, including Stephen Timoshenko,[7] Raymond D. Mindlin,[8] G. R. Cowper,[9] N. G. Stephen,[10] J. R. Hutchinson[11] 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[12] leading to the different shear coefficients in the static and dynamic cases). In engineering practice, the expressions by Stephen Timoshenko[13] are sufficient in most cases. In 1975 Kaneko[14] published an excellent review of studies of the shear coefficient. More recently new experimental data show that the shear coefficient is underestimated.[15][16]

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

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

and for solid circular cross-section,

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

## References

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. ^ "Experimental study of the Timoshenko beam theory predictions", A. Díaz-de-Anda, J. Flores, L. Gutiérrez, R.A. Méndez-Sánchez, G. Monsivais, and A. Morales, Journal of Sound and Vibration, Volume 331, Issue 26, 17 December 2012, pp. 5732–5744.
7. ^ Timoshenko, Stephen P., 1932, Schwingungsprobleme der Technik, Julius Springer.
8. ^ 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.
9. ^ Cowper, G. R., 1966, "The Shear Coefficient in Timoshenko’s Beam Theory", J. Appl. Mech., Vol. 33, No.2, pp. 335–340.
10. ^ 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.
11. ^ Hutchinson, J. R., 1981, "Transverse vibration of beams, exact versus approximate solutions", Journal of Applied Mechanics, Vol. 48, No. 12, pp. 923–928.
12. ^ Le, Khanh C., 1999, Vibrations of shells and rods, Springer.
13. ^ Stephen Timoshenko, James M. Gere. Mechanics of Materials. Van Nostrand Reinhold Co., 1972. pages 207.
14. ^ Kaneko, T., 1975, "On Timoshenko's correction for shear in vibrating beams", J. Phys. D: Appl. Phys., Vol. 8, pp. 1927–1936.
15. ^ "Experimental check on the accuracy of Timoshenko’s beam theory", R. A. Méndez-Sáchez, A. Morales, J. Flores, Journal of Sound and Vibration 279 (2005) 508–512.
16. ^ "On the Accuracy of the Timoshenko Beam Theory Above the Critical Frequency: Best Shear Coefficient", J. A. Franco-Villafañe and R. A. Méndez-Sánchez, Journal of Mechanics, January 2016, pp. 1–4. DOI: 10.1017/jmech.2015.104.