8.3 Kinematics of an Euler-Bernoulli beam

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8.3.1 Assumed Euler-Bernoulli beam bending kinematics

Beam kinematics

• An Euler-Bernoulli beam also assumes:
  • \[w(x,y,z) = w_0(z) + \alpha(z) x + \beta(z) y\]
  • Plane sections remain plane under load
  • Plane sections remain perpendicular to the reference line
• Therefore, the axial deformation is a linear function of the position in the plane of the cross section

Therefore: \[w(x,y,z) = w_0(z) - x {u {}_{,z}} - y {v {}_{,z}}\]

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Now, we can write strain:

\[\begin{align} {\varepsilon_{zz}}=& \, {\frac{\partial w}{\partial z}} \\ {\gamma_{zy}}=& \, {\frac{\partial v}{\partial z}} + {\frac{\partial w}{\partial y}} = {\frac{\partial v_0}{\partial z}} + \beta = 0\\ {\gamma_{zx}}=& \, {\frac{\partial u}{\partial z}} + {\frac{\partial w}{\partial x}} = {\frac{\partial u_0}{\partial z}} + \alpha = 0\\ {\varepsilon_{zz}}=& \, {w_0 {}_{,z}} - x {\alpha {}_{,z}} - y {\beta {}_{,z}} \\ \end{align}\]

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• When the assumptions of an Euler-Bernoulli beam hold, shear deformation is assumed negligible: \[\begin{align} {\gamma_{zy}}=& \, 0 = {\frac{\partial v_0}{\partial z}} + \beta \\ {\gamma_{zx}}=& \, 0 = {\frac{\partial u_0}{\partial z}} + \alpha \\ \end{align}\]

• thus: \[\begin{align} \alpha(z) =& \, - {u {}_{,z}} \\ \beta(z) =& \, - {v {}_{,z}} \\ \end{align}\] \[\begin{align} {\varepsilon_{zz}}=& \, {w_0 {}_{,z}} - x {\alpha {}_{,z}} - y {\beta {}_{,z}} \\ \end{align}\]

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8.3.2 Strain in a beam

\[\begin{align} {\varepsilon_{zz}}=& \, {w_0 {}_{,z}} - x {u {}_{,zz}} - y {v {}_{,zz}}\\ {\varepsilon_{zz}}=& \, {\varepsilon_{zz 0}}- x {u {}_{,zz}} - y {v {}_{,zz}} \end{align}\]

• The symbol \({\varepsilon_{zz 0}}\) is the axial strain of the reference line.
• The symbols \({u {}_{,zz}}\) and \({v {}_{,zz}}\) are the curvatures of the reference line in the \(x-z\) and \(y-z\) planes.

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8.3.3 Stress in a beam

The stress can now be written: \[\begin{align} {\sigma_{zz}}=& \, E {\varepsilon_{zz}}\\ {\sigma_{zz}}=& \, E \left({\varepsilon_{zz 0}}- x {u {}_{,zz}} - y {v {}_{,zz}}\right) \end{align}\]