Structural Analysis Fundamentals / Shear in beams
Why horizontal shear stress exists
The unbalanced push on a beam slice, and complementary shear.
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Read the lesson register and outstanding review notesWhy this matters
Slide a stack of loose planks and they shear past one another. Glue them and they act as one deep beam. The glue is carrying horizontal shear — and understanding that force is what lets you size the welds in a plate girder or the screws in a plywood box beam.
By the end of this lesson you should be able to
- Explain why a vertical shear force implies a horizontal one
- State the principle of complementary shear stress
- Describe the unbalanced force on a beam element
What you should already know
- Bending stress σ = My/I (Module 9)
- Shear force diagrams (Module 3)
- Second moment of area (Module 9)
Take two planks resting one on top of the other and bend them. The lower face of the top plank slides forwards over the upper face of the bottom plank. Each plank bends about its own neutral axis, and the pair is nowhere near as stiff as a single plank of the same total depth.
Now fix them together so they cannot slide. They behave as one deep beam, and the stiffness rises dramatically — the second moment of area of a solid section of depth 2d is eight times that of one of depth d, and two separate planks give only twice. Whatever stops the sliding must be carrying a horizontal shear force.
Predict first
Two identical planks of depth d are glued together to form a beam of depth 2d. Roughly how much stiffer is the glued beam than the two loose planks?
Here is the key observation. The bending moment is not the same at every section. If M is larger at one end of a short slice than the other, then the bending stresses on the two end faces are larger at one end too. Take the part of the slice above some level, and the horizontal push on its left face no longer balances the push on its right face.
Something has to make up the difference, and the only surface left is the horizontal plane at the bottom of that piece. The force acting there is the horizontal shear — and dividing it by the area it acts on gives the shear stress.
Practice
A rectangular beam 100 mm wide and 300 mm deep carries a shear force of 60 kN. What is the average shear stress, V/A, in N/mm²?
Practice
For the same beam, what maximum shear stress does the rectangular beam model predict, in N/mm²?
Practice
By what factor does the average shear stress underestimate the true peak in a rectangular section?
Summary
- Vertical shear force implies horizontal shear stress, through complementary shear
- The bending moment varies along the beam, so bending stresses on the two faces of a slice differ
- That imbalance is carried by horizontal shear on the plane below
- Preventing longitudinal sliding is what makes a built-up beam act as one member