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: : \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). • m := \rho A • J := \rho I • w is the displacement of the mid-surface in the z-direction. • \varphi is the angle of rotation of the normal to the mid-surface of the beam. 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(J + \cfrac{E I m}{\kappa A G}\right)\cfrac{\partial^4 w}{\partial x^2~\partial t^2} + \cfrac{m J}{\kappa A G}~\cfrac{\partial^4 w}{\partial t^4} = q(x,t) + \cfrac{J}{\kappa A G}~\cfrac{\partial^2 q}{\partial t^2} - \cfrac{EI}{\kappa A G}~\cfrac{\partial^2 q}{\partial x^2} : However, it can easily be shown that this equation is incorrect. Consider the case where q is constant and does not depend on x or t, combined with the presence of a small damping all time derivatives will go to zero when t goes to infinity. The shear terms are not present in this situation, resulting in the Euler-Bernoulli beam theory, where shear deformation is neglected. The Timoshenko equation predicts a critical frequency \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 f_c. For frequencies larger than f_c all solutions are oscillatory and, as consequence, a second spectrum appears.
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,t) = 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 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 ==