Deflection of Beams

Why it matters

Understanding the deflection of beams is crucial in civil engineering as it ensures the structural integrity and serviceability of buildings and bridges. Excessive deflection can lead to structural failures or serviceability issues, such as cracking of walls or misalignment of structural elements.

Key ideas

• Beam Deflection: The displacement of a beam under load. It is essential to predict and control deflection to ensure safety and functionality.
• Types of Beams: Common types include simply supported beams, cantilever beams, and fixed beams, each with different deflection characteristics.
• Load Types: Point loads, uniformly distributed loads (UDL), and varying distributed loads affect deflection differently.
• Boundary Conditions: The support conditions of a beam (e.g., fixed, simply supported) significantly influence its deflection.
• Superposition Principle: Used to calculate deflection for beams with multiple loads by summing the deflections due to individual loads.

Formulas

Assume a slender, linearly elastic Euler–Bernoulli beam, small deflection and constant EI. Superposition requires linear response. The first expression is for a cantilever with a tip load.

• δ = (F·L³) / (3·E·I)
  • δ: Deflection at the free end (m)
  • F: Force applied at the free end (N)
  • L: Length of the beam (m)
  • E: Modulus of elasticity (Pa)
  • I: Moment of inertia of the beam's cross-section (m⁴)
• δ = 5w·L⁴ / (384·E·I)
  • δ: Maximum deflection at the center (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 of length 6 m carries a uniformly distributed load of 2 kN/m. The beam has a rectangular cross-section with a width of 0.2 m and a height of 0.3 m. The modulus of elasticity is 200 GPa.

1. Calculate the moment of inertia (I):

   • Formula: I = (b·h³) / 12
   • Calculation: I = (0.2·0.3³) / 12 = 4.5 × 10⁻⁴ m⁴

2. Calculate the maximum deflection (δ):

   • Formula: δ = 5w·L⁴ / (384·E·I)
   • Calculation: δ = (5·2000·6⁴) / (384·200×10⁹·4.5×10⁻⁴)
   • Calculation: δ = 0.000375 m (0.375 mm)

Final Answer: 0.000375 m (0.375 mm)

Common mistakes

• Ignoring Units: Not converting units to SI units can lead to incorrect results.
• Incorrect Boundary Conditions: Misidentifying the type of beam or its supports can lead to incorrect deflection calculations.
• Neglecting Beam Self-weight: Sometimes the weight of the beam itself is not considered, which can affect the deflection.

For GATE CE

• Types of Questions: Expect questions on calculating deflection for different beam types and load conditions. Practice problems involving superposition and varying load types.
• What to Practice: Focus on understanding the derivation of deflection formulas and applying them to different scenarios.

Quick check

1. What is the effect of increasing the modulus of elasticity on beam deflection?
2. How does a point load at the center of a simply supported beam affect deflection compared to a UDL?
3. Why is it important to consider the moment of inertia in deflection calculations?

Answers: 1. Decreases deflection. 2. For the same total load P = wL, the central point load gives PL³/(48EI), which is 1.6 times the UDL deflection. 3. It determines the beam's resistance to bending.