Beam Deflection Formulas to Know for Structural Analysis

Understanding beam deflection is crucial in structural analysis. These formulas help predict how beams will bend under various loads, ensuring safety and stability in construction. Key scenarios include simply supported and cantilever beams, each with unique deflection characteristics.

1. Simply supported beam with point load at center

   • Maximum deflection occurs at the center of the beam.
   • Deflection formula: ( \delta = \frac{PL^3}{48EI} ), where ( P ) is the point load, ( L ) is the length, ( E ) is the modulus of elasticity, and ( I ) is the moment of inertia.
   • The beam experiences equal reactions at both supports.

2. Simply supported beam with uniformly distributed load

   • Maximum deflection occurs at the midpoint of the beam.
   • Deflection formula: ( \delta = \frac{5wL^4}{384EI} ), where ( w ) is the load per unit length.
   • The reactions at the supports are not equal due to the uniform load distribution.

3. Cantilever beam with point load at free end

   • Maximum deflection occurs at the free end of the beam.
   • Deflection formula: ( \delta = \frac{PL^3}{3EI} ).
   • The fixed support experiences both vertical and moment reactions.

4. Cantilever beam with uniformly distributed load

   • Maximum deflection occurs at the free end.
   • Deflection formula: ( \delta = \frac{wL^4}{8EI} ).
   • The fixed support must resist the total load and the moment caused by the distributed load.

5. Simply supported beam with point load at any location

   • Deflection varies based on the location of the load.
   • Use superposition to calculate deflection at any point.
   • The formula involves calculating contributions from both segments of the beam.

6. Fixed-end beam with point load at center

   • Maximum deflection occurs at the center, but the deflection is less than that of a simply supported beam.
   • Deflection formula: ( \delta = \frac{PL^3}{192EI} ).
   • Both ends of the beam are restrained, leading to higher stiffness.

7. Fixed-end beam with uniformly distributed load

   • Maximum deflection occurs at the center.
   • Deflection formula: ( \delta = \frac{5wL^4}{384EI} ), similar to simply supported but with reduced deflection.
   • The fixed supports provide additional resistance to deflection.

8. Propped cantilever with point load at free end

   • The deflection is influenced by both the cantilever and the support.
   • Deflection formula: ( \delta = \frac{PL^3}{3EI} - \frac{PL^2}{2EI} ) (considering the reaction at the prop).
   • The prop reduces the deflection compared to a free cantilever.

9. Moment-area method

   • A graphical method used to determine deflections and rotations in beams.
   • Involves calculating areas under the bending moment diagram.
   • Useful for complex loading and support conditions.

10. Conjugate beam method

    • A method that uses a fictitious beam to analyze deflections.
    • The slope and deflection of the real beam correspond to the bending moment in the conjugate beam.
    • Effective for beams with varying cross-sections and loading conditions.