Deflection of Beams

Why it matters

Understanding the deflection of beams is essential for civil engineers to ensure that structures can withstand loads without excessive bending, which could lead to structural failure or serviceability issues. Proper analysis helps in designing beams that are both safe and economical.

Key ideas

• Deflection: The displacement of a beam under load. It is crucial to keep deflection within permissible limits to ensure structural integrity and comfort.
• Types of Beams: Common types include simply supported beams, cantilever beams, and fixed beams, each with different deflection characteristics.
• Methods of Analysis: Various methods such as the double integration method, Macaulay's method, and moment-area method are used to calculate deflection.
• Material Properties: The modulus of elasticity (E) and moment of inertia (I) of the beam's cross-section significantly affect deflection.

Formulas

Assume a slender, linearly elastic Euler–Bernoulli beam with constant EI and small deflection. The first expression below is for a cantilever tip load; the second is the cantilever free-end deflection under full-span UDL.

• δ = (F·L³) / (3·E·I)

  • δ: Deflection at the cantilever free end (m)
  • F: Load applied at the cantilever free end (N)
  • L: Length of the beam (m)
  • E: Modulus of elasticity (Pa)
  • I: Moment of inertia (m⁴)

• δ = (w·L⁴) / (8·E·I)

  • δ: Maximum deflection (m)
  • w: Uniformly distributed load (N/m)
  • L: Length of the beam (m)
  • E: Modulus of elasticity (Pa)
  • I: Moment of inertia (m⁴)

Worked example

Given: A simply supported beam with a span of 6 m carries a central point load of 10 kN. The beam has a rectangular cross-section with a width of 150 mm and a depth of 300 mm. The modulus of elasticity is 200 GPa.

1. Calculate the moment of inertia (I):

   I = (b·d³) / 12

   I = (0.15 m · (0.3 m)³) / 12 = 3.375 × 10⁻⁴ m⁴

2. Calculate the deflection (δ) using the formula for a central point load:

   δ = (F·L³) / (48·E·I)

   δ = (10,000 N · (6 m)³) / (48 · 200 × 10⁹ Pa · 3.375 × 10⁻⁴ m⁴)

   δ = 0.0006667 m

   Final Answer: 0.667 mm

Common mistakes

• Unit Conversion Errors: Not converting units properly, especially when dealing with mm and m.
• Incorrect Formula Application: Using the wrong formula for the type of load or beam.
• Neglecting Material Properties: Ignoring the modulus of elasticity or moment of inertia.

For GATE CE

Questions often involve calculating deflection for different types of beams and loading conditions. Practice problems involving both point loads and distributed loads, and be familiar with different methods of analysis.

Quick check

1. What is the effect of increasing the moment of inertia on beam deflection?
2. How does a cantilever beam's deflection compare to a simply supported beam under the same load?
3. Why is it important to consider deflection in beam design?

Answers: 1. Decreases deflection. 2. For equal span, EI and point-load magnitude, a cantilever tip load gives 16 times the deflection of a simply supported beam with a central load; specify load positions before comparing. 3. To ensure safety and serviceability.