Differential equations of static equilibrium
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)