Short Answer:
The maximum shear stress in a rectangular section occurs at the neutral axis (mid-depth) and is given by the simple formula
where is the transverse shear force and is the cross-sectional area (width , depth ). This value is 1.5 times the average shear , and the shear stress falls to zero at the top and bottom surfaces.
In plain words: when a rectangular beam carries a vertical shear force, the internal shear stress is not uniform — it is smallest at the outer surfaces and largest at the center. Designers must check against material shear strength to prevent shear failure or web buckling.
Maximum Shear Stress in Rectangular Section
Detailed Explanation :
What shear stress means
Shear stress in a beam is the internal force per unit area that resists sliding of one layer of the beam relative to the adjacent layer due to a transverse shear force . Unlike bending stress (which varies linearly with distance from the neutral axis), shear stress distribution depends on the cross-section shape and is generally non-uniform.
General formula used
The standard relation for shear stress at a horizontal level in any beam section is:
where
- = internal shear force at the section,
- = first moment of area of the portion of the cross-section on one side of the level about the neutral axis,
- = second moment of area (moment of inertia) of the whole section about the neutral axis,
- = width of the section at the level considered.
This relation comes from equilibrium of an infinitesimal slice and the assumption that plane sections remain plane (linear normal stress distribution).
Apply formula to a rectangle
Consider a rectangle of width and depth , neutral axis at mid-depth ( ). For a horizontal plane at distance from the neutral axis, take the area above that plane as the relevant area for . That area height is and its centroid distance from NA is . So:
The moment of inertia of the whole rectangle about the neutral axis is
Substitute and into :
This is the closed-form parabolic distribution of shear stress across the depth.
Maximum value and where it occurs
Set (neutral axis) to get the maximum shear stress:
since . At the extreme fibers , , as expected because shear on a free surface is zero. The shape between these points is a downward-opening parabola symmetric about the neutral axis.
Interpretation and practical meaning
- The average shear over the section is . For a rectangle, the highest shear is times that average value.
- The maximum shear occurs at the centroidal axis, so web or central material must resist the largest shear. In built-up sections such as I-beams, the web is sized to carry this shear while flanges mainly resist bending.
- Designers check against allowable shear stress (or convert to equivalent stresses when combined with bending) and provide reinforcements, thicker webs, or stiffeners if needed.
Assumptions and limitations
- Material is linear elastic and homogeneous; Hooke’s law applies.
- Plane sections remain plane (Euler–Bernoulli beam assumption).
- Section is straight and prismatic with constant over the depth.
- Shear deformation is small (valid for slender beams). For deep beams or sections where shear deformation is significant, Timoshenko beam theory gives better predictions.
- The formula ignores local effects like stress concentrations, holes, or discontinuities.
Examples and quick checks
- If mm, mm and kN, area mm², average shear N/mm², so N/mm².
- For comparison, a circular section has (≈1.333× average), showing section shape matters.
Design implications
- For rectangular beams under large shear, increasing width reduces directly. Increasing depth also reduces since area increases, but depth affects bending capacity more strongly (through in ).
- In practical steel or reinforced concrete beams, shear capacity and shear reinforcement are designed using these concepts; for RC beams, stirrups are added in regions of high shear (near supports) where and thus are largest.
Conclusion
The maximum shear stress in a rectangular section occurs at the neutral axis and equals . Its distribution is parabolic, zero at the surfaces and largest at mid-depth. This simple result, derived from , is widely used in beam design to check shear capacity and to size webs, flanges, or shear reinforcement. Remember the formula rests on elastic-beam assumptions and should be adjusted for deep beams or nonstandard geometries.