Continuous Beam Moment Reaction Support Forces Distributed Point Loads

Overview

Engineering reference data for Continuous Beam Moment Reaction Support Forces Distributed Point Loads in material science and properties.

Key Formulas

Stress

$\sigma = \frac{F}{A}$

Force per unit area.

Strain

$\varepsilon = \frac{\Delta L}{L_0}$

Change in length per original length.

Hooke's Law

$\sigma = E \varepsilon$

Stress proportional to strain in elastic region.

Thermal Expansion

$\Delta L = \alpha L_0 \Delta T$

Length change due to temperature.

Variables

Coefficient Tables

The reaction force coefficient (cr) and moment coefficient (cm) depend on the number of supports and loading type. These coefficients are typically provided in structural engineering references.

Distributed Load Coefficients (3-Support Beam)

Point Load Coefficients (3-Support Beam)

Note: Coefficient values shown are for 3-support beams with uniform loading. For 4 or 5-support configurations, refer to specialized engineering tables or figures.

Worked Examples

Distributed Load Example

For a 3-support continuous beam with:

• Distributed load $q = 1000 \, \text{N/m}$
• Span length $L = 1 \, \text{m}$

End support reactions: $R_{end} = cr \cdot q \cdot L = 0.375 \times 1000 \, \text{N/m} = 375 \, \text{N} = 0.38 \, \text{kN}$

Center support reaction: $R_{center} = cr \cdot q \cdot L = 1.250 \times 1000 \, \text{N/m} = 1250 \, \text{N} = 1.25 \, \text{kN}$

Mid-span moment (end spans): $M_{end} = cm \cdot q \cdot L^2 = 0.070 \times 1000 \, \text{N/m} \times (1 \, \text{m})^2 = 70 \, \text{Nm}$

Moment at center support: $M_{center} = cm \cdot q \cdot L^2 = 0.125 \times 1000 \, \text{N/m} \times (1 \, \text{m})^2 = 125 \, \text{Nm}$

Point Load Example

For a 3-support continuous beam with:

• Point loads $F = 1000 \, \text{N}$ at mid-span
• Span length $L = 1 \, \text{m}$

End support reactions: $R_{end} = cr \cdot F = 0.313 \times 1000 \, \text{N} = 313 \, \text{N} = 0.31 \, \text{kN}$

Center support reaction: $R_{center} = cr \cdot F = 1.375 \times 1000 \, \text{N} = 1375 \, \text{N} = 1.4 \, \text{kN}$

Moment at load point (end spans): $M_{end} = cm \cdot F \cdot L = 0.156 \times 1000 \, \text{N} \times 1 \, \text{m} = 156 \, \text{Nm}$

Moment at center support: $M_{center} = cm \cdot F \cdot L = 0.188 \times 1000 \, \text{N} \times 1 \, \text{m} = 188 \, \text{Nm}$

Application Notes

• Coefficient Dependency: The coefficients cr and cm vary with:

  • Number of supports (3, 4, or 5)
  • Loading type (distributed vs. point)
  • Load positioning along spans

• Approximation vs. Exact Solutions: These coefficient methods provide quick approximations. For precise analysis, consider using slope-deflection equations or finite element methods.

• Conservation Check: For symmetric loading on symmetric beams, verify that sum of reactions equals total applied load:

  • Distributed: $\sum R = q \cdot L_{total}$
  • Point loads: $\sum R = \sum F$

References

• Original Source