Beam Analysis Calculator – Shear, Bending Moment & Deflection

Introduction

The EstiMate Civil Beam Analysis Calculator evaluates common idealized beam cases to provide maximum shear force, bending moment, and deflection parameters. It also generates shear-force (SFD) and bending-moment (BMD) visualizations to aid in structural behavior analysis.

Governing Formulas

Maximum shear ($V$), bending moment ($M$), and elastic deflection ($\Delta$) equations for determinate cases:

• Simply Supported Beam – Uniformly Distributed Load (UDL):
  \( V_{max} = \frac{w L}{2} \) (at supports)
  \( M_{max} = \frac{w L^2}{8} \) (at center)
  \( \Delta_{max} = \frac{5 w L^4}{384 E I} \) (at center)
• Simply Supported Beam – Point Load at Center ($P$):
  \( V_{max} = \frac{P}{2} \)
  \( M_{max} = \frac{P L}{4} \) (at center)
  \( \Delta_{max} = \frac{P L^3}{48 E I} \) (at center)
• Cantilever Beam – Uniformly Distributed Load (UDL):
  \( V_{max} = w L \) (at support)
  \( M_{max} = \frac{w L^2}{2} \) (at support)
  \( \Delta_{max} = \frac{w L^4}{8 E I} \) (at free end)
• Cantilever Beam – Point Load at Free End ($P$):
  \( V_{max} = P \) (at support)
  \( M_{max} = P L \) (at support)
  \( \Delta_{max} = \frac{P L^3}{3 E I} \) (at free end)

Structural Assumptions

This calculator utilizes classic engineering beam theory based on the following structural assumptions:

1. Euler-Bernoulli Beam Theory: Plane sections perpendicular to the longitudinal axis remain plane and perpendicular to the neutral axis after bending (neglecting shear deformation).
2. Linear Elastic Material: Material behavior complies strictly with Hooke's Law ($\sigma = E\epsilon$).
3. Small Deflections: Structural deflections are small relative to the span ($\Delta \ll L$), neglecting second-order axial force effects ($P-\Delta$).
4. Homogeneity & Isotropy: The beam material properties (Young's Modulus $E$) are uniform throughout the member.

Practical Example Calculation

Consider a simply supported steel beam with a span of \( L = 6\text{ m} \) carrying a uniform load of \( w = 15\text{ kN/m} \). The material is structural steel with \( E = 200\text{ GPa} \) (\( 2 \times 10^8\text{ kN/m}^2 \)) and a cross-section moment of inertia \( I = 1.2 \times 10^{-4}\text{ m}^4 \) (e.g., standard I-beam):

• Maximum Shear Force ($V_{max}$): \( \frac{15 \times 6}{2} = 45\text{ kN} \)
• Maximum Bending Moment ($M_{max}$): \( \frac{15 \times 6^2}{8} = 67.5\text{ kNm} \)
• Maximum Elastic Deflection ($\Delta_{max}$):
  \( \Delta = \frac{5 \times 15 \times 6^4}{384 \times (2 \times 10^8) \times (1.2 \times 10^{-4})} = 0.0105\text{ m} = 10.5\text{ mm} \)

Practical Limitations

This calculator does not account for:

• Timoshenko beam effects (shear deformation, which is significant for deep beams with span-to-depth ratios under 10).
• Support settlement, non-rigid boundaries, or dynamic impact loading.
• Lateral-torsional buckling or local plate buckling modes.

Standard References

Refer to the following standard texts for theoretical validation:

• Gere, J. M., & Timoshenko, S. P. Mechanics of Materials.
• AISC Steel Construction Manual (Design Formulas for determinate members).