Shear Stresses in Beams

Why it matters

Shear stresses in beams are critical for ensuring the structural integrity and safety of buildings, bridges, and other civil engineering structures. Understanding how these stresses distribute within a beam helps engineers design beams that can withstand transverse loads without failing.

Key ideas

• Shear Stress Distribution: In beams, shear stress is not uniform across the cross-section. For a rectangular section under transverse shear, the elementary beam result is parabolic, maximum at the neutral axis and zero at the top and bottom surfaces; other shapes need their own analysis.
• Shear Formula: The shear stress at any point in a beam's cross-section can be calculated using the shear formula.
• Assumptions: The analysis assumes that the beam is prismatic, the material is homogeneous and isotropic, and the beam carries transverse shear together with bending; this is not a pure-shear stress state.
• Neutral Axis: The line in the cross-section of a beam where the bending stress is zero.

Formulas

• τ = V·Q / (I·b)
  • τ: Shear stress at the point (N/m² or Pa)
  • V: Shear force at the section (N)
  • Q: First moment of area about the neutral axis (m³)
  • I: Moment of inertia of the entire cross-section about the neutral axis (m⁴)
  • b: Width of the section at the point where shear stress is calculated (m)

Worked example

Given: A simply supported beam with a rectangular cross-section of width 200 mm and height 400 mm is subjected to a shear force of 10 kN. Calculate the shear stress at a point 100 mm from the top fiber.

1. Convert units: Width b = 200 mm = 0.2 m, Height h = 400 mm = 0.4 m, Shear force V = 10 kN = 10000 N.
2. Calculate the moment of inertia I:
   • I = (b·h³) / 12 = (0.2·0.4³) / 12 = 0.001067 m⁴
3. Calculate the first moment of area Q:
   • Area above the point = b·y = 0.2·0.1 = 0.02 m²
   • Distance from neutral axis to centroid of area above = 0.2 - 0.05 = 0.15 m
   • Q = Area·distance = 0.02·0.15 = 0.003 m³
4. Calculate shear stress τ:
   • τ = V·Q / (I·b) = 10000·0.003 / ((0.2·0.4³/12)·0.2) = 140625 N/m²

Final Answer: 140.625 kPa

Common mistakes

• Forgetting to convert units, especially from mm to m.
• Incorrectly calculating the first moment of area Q.
• Assuming shear stress is uniform across the section.

For GATE CE

• Questions often involve calculating shear stress at a specific point in a beam's cross-section.
• Practice problems involving different cross-sectional shapes and loading conditions.

Quick check

1. What is the maximum shear stress location in a rectangular beam?
2. How does shear stress vary across a circular cross-section?
3. What is the unit of shear force V?

Answers: 1. At the neutral axis. 2. The elementary VQ/(Ib) result gives a width-averaged parabolic variation, maximum 4V/(3A) at the center; the exact local distribution is more complex. 3. Newton (N).