10.3. Timoshenko beam

10.3.1. Kinematic assumptions

Let \(z\) coordinate the axial coordinate of a beam, and \(x\), \(y\) a pair of cartesian coordinates to represent the points on a beam section.

Displacement. Displacement of the points of a beam can written as

\[\begin{split}\begin{aligned} \mathbf{s}(x,y,z,t) & = \mathbf{s}_P(z,t) + \boldsymbol{\theta}(z,t) \times \mathbf{r}_P(x,y) + \mathbf{s}^{\nu + w}(x,y,z,t) = \\ & = \mathbf{s}_P(z,t) + \hat{\mathbf{x}} \left( - y \theta_z \right) + \hat{\mathbf{y}} \left( x \theta_z \right) + \hat{\mathbf{z}} \left( + y \theta_x - x \theta_y \right) + \mathbf{s}^{\nu + w} \ , \end{aligned}\end{split}\]

where the first two contributions represent a rigid motion of the section identified by the value \(z\) of the axial coordinate, and \(\mathbf{s}^{\nu + w}\) the contribution of strain (due to non-zero Poisson ration) and warping of the section. Here the vector \(\mathbf{r}_P\)(x,y) lies in the same section of referencce point \(P\), i.e. \(\mathbf{r}_P = (x-x_P) \hat{\mathbf{x}} + (y-y_P) \hat{\mathbf{y}}\), so that the motion of points on section \(A(z)\) only depends on the displacement of \(P(z)\) and the rotation of the section \(A(z)\).

Strain.

\[\begin{split}\begin{aligned} \varepsilon_{zz} & = s'_{Pz} + y \theta'_x - x \theta'_y + s^{\nu+w}_{Pz/z} \\ \varepsilon_{xx} & = s^{\nu+w}_{Px/x} \\ \varepsilon_{yy} & = s^{\nu+w}_{Py/y} \\ 2 \varepsilon_{zx} & = s'_{Px} - \theta_y - y \theta'_z + s^{\nu+w}_{x/z} + s^{\nu+w}_{z/x} \\ 2 \varepsilon_{zy} & = s'_{Py} + \theta_x + x \theta'_z + s^{\nu+w}_{y/z} + s^{\nu+w}_{z/y} \\ 2 \varepsilon_{xy} & = s^{\nu+w}_{y/z} + s^{\nu+w}_{z/y} \ . \end{aligned}\end{split}\]

Stress. …todo… usually stiffness matrix is defined providing axial, bending, shear and torsion stiffness, and cross-coupling terms. Here, using a simplified (or modified, so that no contribution of \(\varepsilon_{xx}\), \(\varepsilon_{yy}\) exists) version of the constitutive law for elastic isotropic media,

\[\begin{split}\begin{aligned} \sigma_{zz} & = E \varepsilon_{zz} \\ \tau_{zx} & = 2 G \varepsilon_{zx} \\ \tau_{zy} & = 2 G \varepsilon_{zy} \\ \end{aligned}\end{split}\]

10.3.2. Internal actions

\[\begin{split}\begin{aligned} \mathbf{F} & = \int_{A} \hat{\mathbf{n}} \cdot \boldsymbol\sigma = \int_{A} \hat{\mathbf{x}} \tau_{zx} + \hat{\mathbf{y}} \tau_{zy} + \hat{\mathbf{z}} \sigma_{zz} \\ \mathbf{M} & = \int_{A} \mathbf{r} \times ( \hat{\mathbf{n}} \cdot \boldsymbol\sigma ) = \int_{A} \hat{\mathbf{x}} \left( y \sigma_{zz} \right) + \hat{\mathbf{y}} \left( - x \sigma_{zz} \right) + \hat{\mathbf{z}} \left( x \tau_{zy} - y \tau_{zx} \right) \\ \end{aligned}\end{split}\]

Internal actions as function of displacement - elastic isotropic media. Neglecting warping and strain due to non-zero Poisson ratio,

\[\begin{split}\begin{aligned} \mathbf{F} & = \int_{A} \hat{\mathbf{x}} \tau_{zx} + \hat{\mathbf{y}} \tau_{zy} + \hat{\mathbf{z}} \sigma_{zz} = \\ & = \int_{A} \hat{\mathbf{x}} G ( s'_{Px} - \theta_y - y \theta'_z ) + \hat{\mathbf{y}} G ( s'_{Py} + \theta_x + x \theta'_z) + \hat{\mathbf{z}} E ( s'_{Pz} + y \theta'_x - x \theta'_y) = \\ & = \hat{\mathbf{x}} \left( \chi_x GA ( s'_{Px} - \theta_y ) - G S_x \theta'_z \right) + \hat{\mathbf{y}} \left( \chi_y GA ( s'_{Py} + \theta_x ) + G S_y \theta'_z \right) + \hat{\mathbf{z}} \left( EA s'_{Pz} + E S_x \theta'_x - E S_y \theta'_y \right) \ , \\ \mathbf{M} & = \int_{A} \hat{\mathbf{x}} \left( y \sigma_{zz} \right) + \hat{\mathbf{y}} \left( - x \sigma_{zz} \right) + \hat{\mathbf{z}} \left( x \tau_{zy} - y \tau_{zx} \right) = \\ & = \int_{A} \hat{\mathbf{x}} y E \left( s'_{Pz} + y \theta'_x - x \theta'_y \right) - \hat{\mathbf{y}} x E \left( s'_{Pz} + y \theta'_x - x \theta'_y \right) + \hat{\mathbf{z}} G \left( x ( s'_{Py} + \theta_x + x \theta'_z ) - y ( s'_{Px} - \theta_y - y \theta'_z ) \right) = \\ & = \hat{\mathbf{x}} \left( E S_x s'_{Pz} + E J_{x} \theta'_x - E J_{xy} \theta'_y \right) + \hat{\mathbf{y}} \left(-E S_y s'_{Pz} - E J_{xy} \theta'_x + E J_{y} \theta'_y \right) + \hat{\mathbf{z}} \left( G S_y ( s'_{Py} + \theta_x ) - G S_x ( s'_{Px} - \theta_y ) + GJ_z \theta'_z ) \right) \end{aligned}\end{split}\]

or introducing matrix notation,

\[\begin{split}\begin{bmatrix} F_x \\ F_y \\ F_z \\ M_x \\ M_y \\ M_z \end{bmatrix} & = \begin{bmatrix} \chi_x^{-1} GA & & & & & -GS_x \\ & \chi_y^{-1} GA & & & & GS_y \\ & & EA & ES_x & -ES_y & \\ & & ES_x & EJ_x & -EJ_{xy} & \\ & & -ES_y & -EJ_{xy} & EJ_y & \\ -GS_x & GS_y & & & & GJ_z \end{bmatrix} \begin{bmatrix} s'_{Px} - \theta_y \\ s'_{Py} + \theta_x \\ s'_{Pz} \\ \theta'_{x} \\ \theta'_{y} \\ \theta'_{z} \end{bmatrix}\end{split}\]

Structural decoupling. \(S_i = 0\), \(J_{xy} = 0\)

\[\begin{split}\begin{aligned} \mathbf{F} & = \hat{\mathbf{x}} \chi_x GA ( s'_{Px} - \theta_y ) + \hat{\mathbf{y}} \chi_y GA ( s'_{Py} + \theta_x ) + \hat{\mathbf{z}} EA s'_{Pz} , \\ \mathbf{M} & = \hat{\mathbf{x}} E J_{x} \theta'_x + \hat{\mathbf{y}} E J_{y} \theta'_y + \hat{\mathbf{z}} G J_z \theta'_z \end{aligned}\end{split}\]

10.3.3. Balance equations

Balance equations for a beam can be obtained integrating indefinite balance equations for a 3-dimensional solid on the sections \(A(z)\) of the beam, with some further assumption on non-rigid contributions to displacement.

Momentum equation +

\[\mathbf{\rho}_0 \ddot{\mathbf{s}} = \nabla \cdot \boldsymbol\sigma + \mathbf{f}\]

gives

\[\begin{split}\begin{aligned} \mathbf{0} & = - \int_{\Delta V} \rho_0 \ddot{\mathbf{s}} + \int_{\Delta V} \nabla \cdot \boldsymbol\sigma + \int_{\Delta V} \rho_0 \mathbf{g} = \\ & = - \Delta z \, \int_{A} \rho_0 \left( \ddot{\mathbf{s}}_P - \mathbf{r}_P \times \ddot{\boldsymbol{\theta}} \right) + \int_{A(z)} \hat{\mathbf{n}} \cdot \boldsymbol\sigma + \int_{A(z+\Delta z)} \hat{\mathbf{n}} \cdot \boldsymbol\sigma + \int_{\Delta A_{lat}} \hat{\mathbf{n}} \cdot \boldsymbol\sigma + \Delta z \, \int_{A} \rho_0 \mathbf{g} = \\ & = \Delta z \left[ - m \ddot{\mathbf{s}}_P - \mathbf{S}_{P} \cdot \ddot{\boldsymbol\theta} + \mathbf{F}' + \mathbf{f} \right] \end{aligned}\end{split}\]

Angular momentum equation

\[\mathbf{r}_P \times \mathbf{\rho}_0 \ddot{\mathbf{s}} = \mathbf{r}_P \times \left( \nabla \cdot \boldsymbol\sigma + \mathbf{f} \right)\]

gives

\[\begin{split}\begin{aligned} \mathbf{0} & = - \int_{\Delta V} \rho_0 \mathbf{r}_P \times \ddot{\mathbf{s}} + \int_{\Delta V} \mathbf{r}_P \times \nabla \cdot \boldsymbol\sigma + \int_{\Delta V} \rho_0 \mathbf{r}_P \times \mathbf{g} = \\ & = - \Delta z \, \int_{A} \rho_0 \mathbf{r}_P \times \left( \ddot{\mathbf{s}}_P - \mathbf{r}_P \times \ddot{\boldsymbol{\theta}} \right) + \dots \\ & = \Delta z \left[ - \mathbf{S}_P^T \cdot \ddot{\mathbf{s}}_P - \mathbf{I}_{P} \cdot \ddot{\boldsymbol\theta} + \mathbf{M}' + \hat{\mathbf{z}} \times \mathbf{F} + \mathbf{m} \right] \ . \end{aligned}\end{split}\]

Contribution of stress to moment

\[\begin{split}\begin{aligned} \int_{V} \mathbf{r} \times \nabla \cdot \boldsymbol\sigma & = \hat{\mathbf{e}}_i \int_{V} \varepsilon_{ijk} r_j \sigma_{lk/l} = \\ & = \hat{\mathbf{e}}_i \int_{V} \varepsilon_{ijk} \left[ \left( r_j \sigma_{lk} \right)_{/l} - r_{j/l} \sigma_{lk} \right] = \\ & = \hat{\mathbf{e}}_i \int_{\partial V} \varepsilon_{ijk} r_j n_l \sigma_{lk} - \hat{\mathbf{e}}_i \int_{V} \varepsilon_{ijk} r_{j/l} \sigma_{lk} = \\ & = \int_{\partial V} \mathbf{r}_P \times ( \hat{\mathbf{n}} \cdot \boldsymbol\sigma ) - \hat{\mathbf{e}}_i \int_{V} \varepsilon_{ijk} \delta^{(2)}_{jl} \sigma_{lk} = \\ \end{aligned}\end{split}\]

For an elementary beam element \(\Delta z\), the first contribution contains internal moments on two sections at \(z\) and \(z+\Delta z\) and the contribution of the lateral surface, that can be summed with the volume contribution to get load from linear density loads,

\[\int_{\partial \Delta V} \mathbf{r}_P \times ( \hat{\mathbf{n}} \cdot \boldsymbol\sigma ) = \Delta z \, \mathbf{M}'(z) + \Delta z \, \int_{\partial A} \mathbf{r}_P \times ( \hat{\mathbf{n}} \cdot \boldsymbol\sigma ) \]

The second contribution becomes (as \(\delta^{(2)}_{xx} = \delta^{(2)}_{yy} = 1\), but \(\delta^{(2)}_{zz} = 0\)),

\[\begin{split}\begin{aligned} - \hat{\mathbf{e}}_i \int_{V} \varepsilon_{ijk} \delta^{(2)}_{jl} \sigma_{lk} & = - \int_{\Delta V} \left\{ \hat{\mathbf{z}} ( \sigma_{xy} - \sigma_{yx} ) + \hat{\mathbf{x}} ( \sigma_{yz} ) + \hat{\mathbf{y}} ( - \sigma_{xz} ) \right\} = \\ & = \Delta z \left[ - \hat{\mathbf{x}} \, T_y + \hat{\mathbf{y}} \, T_x \right] = \\ & = \Delta z \, \hat{\mathbf{z}} \times \mathbf{F} \ . \end{aligned}\end{split}\]

In components, for an inertially decoupled set of Cartesian coordinates,

\[\begin{split}\begin{aligned} 0 & = - m \ddot{s}_{Px} + F'_x + f_x \\ 0 & = - m \ddot{s}_{Py} + F'_y + f_y \\ 0 & = - m \ddot{s}_{Pz} + F'_z + f_z \\ 0 & = - I_x \ddot{\theta}_x + M'_x - T_y + m_x \\ 0 & = - I_y \ddot{\theta}_y + M'_y + T_x + m_y \\ 0 & = - I_z \ddot{\theta}_z + M'_z + m_z \\ \end{aligned}\end{split}\]

Using matrix formalism, momentum and angular momentum equations for an isotropic elastic beam read

\[\begin{split} \mathbf{0} = - \begin{bmatrix} m & & & & &-S_y \\ & m & & & & S_x \\ & & m & S_y & -S_x & \\ & & S_y & I_x & I_{xy} & I_{xz} \\ & &-S_x & I_{xy} & I_y & I_{yz} \\ -S_y & S_x & & I_{xz} & I_{yz} & I_z \\ \end{bmatrix} \begin{bmatrix} \ddot{s}_{Px} \\ \ddot{s}_{Py} \\ \ddot{s}_{Pz} \\ \ddot\theta_x \\ \ddot\theta_y \\ \ddot\theta_z \end{bmatrix} + \left( \begin{bmatrix} \chi_x^{-1} GA & & & & & -GS_x \\ & \chi_y^{-1} GA & & & & GS_y \\ & & EA & ES_x & -ES_y & \\ & & ES_x & EJ_x & -EJ_{xy} & \\ & & -ES_y & -EJ_{xy} & EJ_y & \\ -GS_x & GS_y & & & & GJ_z \end{bmatrix} \begin{bmatrix} s'_{Px} - \theta_y \\ s'_{Py} + \theta_x \\ s'_{Pz} \\ \theta'_{x} \\ \theta'_{y} \\ \theta'_{z} \end{bmatrix} \right)' + \begin{bmatrix} \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot &-\chi_y^{-1} GA & \cdot & \cdot & \cdot & -GS_y \\ \chi_x^{-1} GA & \cdot & \cdot & \cdot & \cdot & -GS_x \\ \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\ \end{bmatrix} \begin{bmatrix} s'_{Px} - \theta_y \\ s'_{Py} + \theta_x \\ s'_{Pz} \\ \theta'_{x} \\ \theta'_{y} \\ \theta'_{z} \end{bmatrix} + \begin{bmatrix} f_x \\ f_y \\ f_z \\ m_x \\ m_y \\ m_z \end{bmatrix} \end{split}\]

Structural and inertial simoultaneously decoupled isotropic elastic beam.

\[\begin{split}\begin{aligned} 0 & = - m \ddot{s}_{Px} + \left( \chi_x^{-1} GA ( s'_{Px} - \theta_y ) \right)' + f_x \\ 0 & = - m \ddot{s}_{Py} + \left( \chi_y^{-1} GA ( s'_{Py} + \theta_x ) \right)' + f_y \\ 0 & = - m \ddot{s}_{Pz} + \left( EA s'_{Pz} \right)' + f_z \\ 0 & = - I_x \ddot{\theta}_x + \left( EJ_x \theta'_x \right)' - \chi_y^{-1} GA ( s'_{Py} + \theta_x ) + m_x \\ 0 & = - I_y \ddot{\theta}_y + \left( EJ_y \theta'_y \right)' + \chi_x^{-1} GA ( s'_{Px} - \theta_y ) + m_y \\ 0 & = - I_z \ddot{\theta}_z + \left( GJ_z \theta'_z \right)' + m_z \\ \end{aligned}\end{split}\]