Structural Analysis Fundamentals / module 10
Shear in beams
Deriving τ = VQ/It, why shear stress is zero at the top and bottom faces, and why the web carries the shear.
What you will understand
- Explain why a vertical shear force also produces horizontal shear stress
- Derive τ = VQ/It from longitudinal equilibrium of a beam element
- Say what Q means and calculate it for common sections
- Explain the shear-stress distribution in a rectangular beam and the free-face boundary conditions
- Use shear flow to size welds, bolts and other connectors
Lessons
Why horizontal shear stress exists
The unbalanced push on a beam slice, and complementary shear.
Deriving τ = VQ/It
From longitudinal equilibrium of an element to the shear stress at any level.
Shear in unsymmetrical sections and the shear centre
Why a channel twists when you load it through its centroid.
Module summary
- Bending stress varies along the beam, so the horizontal force on a slice is unbalanced — shear stress makes up the difference
- τ = VQ/It, where Q is the first moment of the area beyond the level being considered
- In a rectangular beam, vertical shear stress is zero at the top and bottom and greatest at the neutral axis
- A rectangle peaks at 1.5 × average; a circle at 4/3 × average
- In an I-section the web carries almost all the shear, while the flanges carry the bending
- Shear flow q = VQ/I is a force per unit length, and is what connectors must resist
Teaching basis and review status
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesCheck your understanding
Practice
A rectangular beam 150 mm wide by 400 mm deep carries a shear force of 120 kN. What is the maximum shear stress?
Check yourself
What does Q represent in τ = VQ/It?
Check yourself
Why is shear flow used rather than shear stress when designing bolts joining a built-up beam?