Timoshenko–Ehrenfest beam theory

The Timoshenko beam theory was developed by Ukrainian-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, 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.

Governing equations

Quasistatic Timoshenko beam

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:

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

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

${\frac {EI}{\kappa LAG}}\ll 1$

where $L$ is the length of the beam.

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

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

Derivation of quasistatic Timoshenko beam equations

From the kinematic assumptions for a Timoshenko beam, the displacements of the beam are given by $u_{x}(x,y,z,t)=-z~\varphi (x,t)~;~~u_{y}(x,y,z,t)=0~;~~u_{z}(x,y,z)=w(x,t)$ Then, from the strain-displacement relations for small strains, the non-zero strains based on the Timoshenko assumptions are $\varepsilon _{xx}={\frac {\partial u_{x}}{\partial x}}=-z~{\frac {\partial \varphi }{\partial x}}~;~~\varepsilon _{xz}={\frac {1}{2}}\left({\frac {\partial u_{x}}{\partial z}}+{\frac {\partial u_{z}}{\partial x}}\right)={\frac {1}{2}}\left(-\varphi +{\frac {\partial w}{\partial x}}\right)$ Since the actual shear strain in the beam is not constant over the cross section we introduce a correction factor $\kappa$ such that $\varepsilon _{xz}={\frac {1}{2}}~\kappa ~\left(-\varphi +{\frac {\partial w}{\partial x}}\right)$ The variation in the internal energy of the beam is $\delta U=\int _{L}\int _{A}(\sigma _{xx}\delta \varepsilon _{xx}+2\sigma _{xz}\delta \varepsilon _{xz})~\mathrm {d} A~\mathrm {d} L=\int _{L}\int _{A}\left[-z~\sigma _{xx}{\frac {\partial (\delta \varphi )}{\partial x}}+\sigma _{xz}~\kappa \left(-\delta \varphi +{\frac {\partial (\delta w)}{\partial x}}\right)\right]~\mathrm {d} A~\mathrm {d} L$ Define $M_{xx}:=\int _{A}z~\sigma _{xx}~\mathrm {d} A~;~~Q_{x}:=\kappa ~\int _{A}\sigma _{xz}~\mathrm {d} A$ Then $\delta U=\int _{L}\left[-M_{xx}{\frac {\partial (\delta \varphi )}{\partial x}}+Q_{x}\left(-\delta \varphi +{\frac {\partial (\delta w)}{\partial x}}\right)\right]~\mathrm {d} L$ Integration by parts, and noting that because of the boundary conditions the variations are zero at the ends of the beam, leads to $\delta U=\int _{L}\left[\left({\frac {\partial M_{xx}}{\partial x}}-Q_{x}\right)~\delta \varphi -{\frac {\partial Q_{x}}{\partial x}}~\delta w\right]~\mathrm {d} L$ The variation in the external work done on the beam by a transverse load $q(x,t)$ per unit length is $\delta W=\int _{L}q~\delta w~\mathrm {d} L$ Then, for a quasistatic beam, the principle of virtual work gives $\delta U=\delta W\implies \int _{L}\left[\left({\frac {\partial M_{xx}}{\partial x}}-Q_{x}\right)~\delta \varphi -\left({\frac {\partial Q_{x}}{\partial x}}+q\right)~\delta w\right]~\mathrm {d} L=0$ The governing equations for the beam are, from the fundamental theorem of variational calculus, ${\frac {\partial M_{xx}}{\partial x}}-Q_{x}=0~;~~{\frac {\partial Q_{x}}{\partial x}}+q=0$ For a linear elastic beam ${\begin{aligned}M_{xx}&=\int _{A}z~\sigma _{xx}~\mathrm {d} A=\int _{A}z~E~\varepsilon _{xx}~\mathrm {d} A=-\int _{A}z^{2}~E~{\frac {\partial \varphi }{\partial x}}~\mathrm {d} A=-EI~{\frac {\partial \varphi }{\partial x}}\\Q_{x}&=\int _{A}\sigma _{xz}~\mathrm {d} A=\int _{A}2G~\varepsilon _{xz}~\mathrm {d} A=\int _{A}\kappa ~G~\left(-\varphi +{\frac {\partial w}{\partial x}}\right)~\mathrm {d} A=\kappa ~AG~\left(-\varphi +{\frac {\partial w}{\partial x}}\right)\end{aligned}}$ Therefore the governing equations for the beam may be expressed as ${\begin{aligned}{\frac {\partial }{\partial x}}\left(EI{\frac {\partial \varphi }{\partial x}}\right)+\kappa AG~\left({\frac {\partial w}{\partial x}}-\varphi \right)&=0\\{\frac {\partial }{\partial x}}\left[\kappa AG\left({\frac {\partial w}{\partial x}}-\varphi \right)\right]+q&=0\end{aligned}}$ Combining the two equations together gives ${\begin{aligned}&{\frac {\partial ^{2}}{\partial x^{2}}}\left(EI{\frac {\partial \varphi }{\partial x}}\right)=q\\&{\frac {\partial w}{\partial x}}=\varphi -{\cfrac {1}{\kappa AG}}~{\frac {\partial }{\partial x}}\left(EI{\frac {\partial \varphi }{\partial x}}\right)\end{aligned}}$

Dynamic Timoshenko beam

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

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

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 :

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

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

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,

• $\rho$ 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.
• $\kappa$, called the Timoshenko shear coefficient, depends on the geometry. Normally, $\kappa =5/6$ for a rectangular section.
• $q(x,t)$ 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

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

Derivation of combined Timoshenko beam equation

The equations governing the bending of a homogeneous Timoshenko beam of constant cross-section are ${\begin{aligned}(1)&&\quad m~{\frac {\partial ^{2}w}{\partial t^{2}}}&=\kappa AG~\left({\frac {\partial ^{2}w}{\partial x^{2}}}-{\frac {\partial \varphi }{\partial x}}\right)+q(x,t)~;~~m:=\rho A\\(2)&&\quad J~{\frac {\partial ^{2}\varphi }{\partial t^{2}}}&=EI~{\frac {\partial ^{2}\varphi }{\partial x^{2}}}+\kappa AG~\left({\frac {\partial w}{\partial x}}-\varphi \right)~;~~J:=\rho I\end{aligned}}$ From equation (1), assuming appropriate smoothness, we have ${\begin{aligned}(3)&&\quad {\frac {\partial \varphi }{\partial x}}&=-{\cfrac {m}{\kappa AG}}~{\frac {\partial ^{2}w}{\partial t^{2}}}+{\frac {\partial ^{2}w}{\partial x^{2}}}+{\cfrac {q}{\kappa AG}}\\(4)&&\quad {\frac {\partial ^{2}q}{\partial t^{2}}}&=m~{\cfrac {\partial ^{4}w}{\partial t^{4}}}-\kappa AG~\left({\cfrac {\partial ^{4}w}{\partial x^{2}\partial t^{2}}}-{\cfrac {\partial ^{3}\varphi }{\partial x\partial t^{2}}}\right)\end{aligned}}$ From (3), assuming appropriate smoothness, $(5)\qquad {\cfrac {\partial ^{3}\varphi }{\partial x^{3}}}=-{\cfrac {m}{\kappa AG}}~{\cfrac {\partial ^{4}w}{\partial x^{2}\partial t^{2}}}+{\cfrac {\partial ^{4}w}{\partial x^{4}}}+{\cfrac {1}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial x^{2}}}$ Differentiating equation (2) gives $(6)\qquad {\cfrac {\partial ^{3}\varphi }{\partial x\partial t^{2}}}={\cfrac {EI}{J}}~{\cfrac {\partial ^{3}\varphi }{\partial x^{3}}}+{\cfrac {\kappa AG}{J}}~\left({\frac {\partial ^{2}w}{\partial x^{2}}}-{\frac {\partial \varphi }{\partial x}}\right)$ From equations (4) and (6) $(7)\qquad {\cfrac {1}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial t^{2}}}-{\cfrac {m}{\kappa AG}}~{\cfrac {\partial ^{4}w}{\partial t^{4}}}+{\cfrac {\partial ^{4}w}{\partial x^{2}\partial t^{2}}}={\cfrac {EI}{J}}~{\cfrac {\partial ^{3}\varphi }{\partial x^{3}}}+{\cfrac {\kappa AG}{J}}~\left({\frac {\partial ^{2}w}{\partial x^{2}}}-{\frac {\partial \varphi }{\partial x}}\right)$ From equations (3) and (7) $(8)\qquad {\cfrac {1}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial t^{2}}}-{\cfrac {m}{\kappa AG}}~{\cfrac {\partial ^{4}w}{\partial t^{4}}}+{\cfrac {\partial ^{4}w}{\partial x^{2}\partial t^{2}}}={\cfrac {EI}{J}}~{\cfrac {\partial ^{3}\varphi }{\partial x^{3}}}+{\cfrac {m}{J}}~{\frac {\partial ^{2}w}{\partial t^{2}}}-{\cfrac {q}{J}}$ Plugging equation (5) into (8) gives $(9)\qquad {\cfrac {J}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial t^{2}}}-{\cfrac {mJ}{\kappa AG}}~{\cfrac {\partial ^{4}w}{\partial t^{4}}}+J~{\cfrac {\partial ^{4}w}{\partial x^{2}\partial t^{2}}}=-{\cfrac {mEI}{\kappa AG}}~{\cfrac {\partial ^{4}w}{\partial x^{2}\partial t^{2}}}+EI~{\cfrac {\partial ^{4}w}{\partial x^{4}}}+{\cfrac {EI}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial x^{2}}}+m~{\frac {\partial ^{2}w}{\partial t^{2}}}-q$ Rearrange to get $EI~{\cfrac {\partial ^{4}w}{\partial x^{4}}}+m~{\frac {\partial ^{2}w}{\partial t^{2}}}-\left(J+{\cfrac {mEI}{\kappa AG}}\right)~{\cfrac {\partial ^{4}w}{\partial x^{2}\partial t^{2}}}+{\cfrac {mJ}{\kappa AG}}~{\cfrac {\partial ^{4}w}{\partial t^{4}}}=q+{\cfrac {J}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial t^{2}}}-{\cfrac {EI}{\kappa AG}}~{\frac {\partial ^{2}q}{\partial x^{2}}}\quad \square$

Axial effects

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

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

where $J=\rho I$ 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

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

Damping

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

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

the coupled governing equations for a Timoshenko beam take the form

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

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

and the combined equation becomes

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

Shear coefficient

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:

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

See also

• Bending
• Euler-Bernoulli beam equation
• Sandwich theory
• Plate theory

References

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