33.1 Differential equations of static equilibrium


33.1.1 Axial static equilbrium

  • In the axial direction (\(z\)):
    • \(P = P(z)\) – axial force resultant
    • \(p_z = p_z(z)\) – distributed axial load [force/length]
      • e.g., \(p_z = \rho A g_z\)
      • Note \(\rho g_z\) is related to the stress equilibrium equations (body force per unit volume) as \(b_z = \rho g_z\)
    • \(\displaystyle\sum F_z = 0 = P(z + dz) - P(z) + p_z dz\)

Hence:

  • \(0 = \frac{P(z + dz) - P(z)}{dz} + p_z\)
  • \(\frac{d P}{d z} = - p_z(z)\)

  • For \(P = {\sigma_{zz}}\cdot A = E A \, {\varepsilon_{zz 0}}= E A \, {{w}_0 {}_{,z}}\)
  • \({\frac{d P(z)}{d z}} = {\frac{d }{d z}} \left[ E A {\frac{d w_0}{d z}} \right] = - p_z(z)\)
  • For constant \(EA\)
    • \(E A \, {w_0 {}_{,zz}} = - p_z(z)\)
  • Note, when \(p_z = 0\), the axial force does not vary (for small deflections)