Consider a 45 m long steel rod with density $\rho = 7280 \, \text{kg/m}^3$ and cross-sectional area $A = 0.1 \, \text{m}^2$ . The modulus of elasticity for steel is $E = 200 \, \text{GPa} = 200 \times 10^9 \, \text{N/m}^2$ .

Maximum Axial Force at $x = 0 \, \text{m}$ : Using $F_x = \rho g A (L - x)$ , $F_{x=0} = (7280 \, \text{kg/m}^3)(9.81 \, \text{m/s}^2)(0.1 \, \text{m}^2)(45 \, \text{m} - 0 \, \text{m}) = 321376 \, \text{N} \approx 321 \, \text{kN}$

Maximum Axial Stress at $x = 0 \, \text{m}$ : From $\sigma_x = \rho g (L - x)$ , $\sigma_{x=0} = (7280 \, \text{kg/m}^3)(9.81 \, \text{m/s}^2)(45 \, \text{m}) = 3213756 \, \text{Pa} \approx 3.2 \, \text{MPa}$

Axial Deformation at $x = 45 \, \text{m}$ : Using $d_x = \frac{\rho g x (2L - x)}{2E}$ , $d_{x=45} = \frac{(7280 \, \text{kg/m}^3)(9.81 \, \text{m/s}^2)(45 \, \text{m})^2}{2 \times 200 \times 10^9 \, \text{N/m}^2} = 0.00036 \, \text{m} = 0.4 \, \text{mm}$

Key Observations for Vertical Beams Under Self-Weight

Stress Independence from Cross-Section: The axial stress $\sigma_x = \rho g (L - x)$ is independent of the cross-sectional area $A$ . This simplifies analysis, as stress depends only on material density, gravity, and beam length.
Maximum Stress Location: Maximum axial stress occurs at the top of the beam ( $x = 0$ ), where $\sigma_{\text{max}} = \rho g L$ . This is critical for design, as it determines the highest load point.
Zero Stress at Free End: At the free end ( $x = L$ ), the axial stress is zero, reflecting no load from below.
Deformation Profile: The axial deformation $d_x = \frac{\rho g x (2L - x)}{2E}$ follows a quadratic distribution, with maximum deformation at the free end ( $x = L$ ), calculated as $d_{L} = \frac{\rho g L^2}{2E}$ .

References

Original Source

Source: engineeringtoolbox.com