Short Answer:
Shear stress in a rectangular beam is the internal stress produced by a transverse shear force . Its distribution across the depth is parabolic, zero at the top and bottom surfaces and maximum at the neutral axis. The general beam formula is used and, for a rectangle (width , depth ), this gives a closed form:
where is the vertical distance from the neutral axis.
In plain words: shear stress is not constant through a rectangular section — it is smallest at the surfaces and largest at the mid-depth (neutral axis). The maximum shear stress equals (with ) and the shape of the variation is a parabola.
Shear Stress in Rectangular Beam
Detailed Explanation :
- Context and formula used
When a beam carries a transverse shear force , internal shear stresses develop over the cross-section so that equilibrium is maintained. The standard relation from elementary strength of materials is
where
- is the shear stress at the level where the cross-section width is ,
- is the internal shear force at the section,
- is the first moment of area of the part of the cross-section on one side of the level about the neutral axis,
- is the second moment of area (moment of inertia) of the whole section about the neutral axis, and
- is the width of the section at the level considered.
This formula follows from equilibrium of a thin slice and the linear variation of bending normal stresses across the depth (plane sections remain plane).
- Geometry and coordinates
Consider a rectangular cross-section of width and total depth . Put the origin at the neutral axis (mid-depth). Let measure vertical distance from the neutral axis (positive upward). The top surface is at and the bottom at . We want the shear stress at an arbitrary horizontal plane at coordinate . - Compute
To find at level we take the area of the portion of the cross-section above the level (you may also use the part below — result is same in magnitude). That area has height and width , so
The centroid of this area (measured from the neutral axis) lies at
Thus the first moment of area about the neutral axis is
Multiply out and simplify:
- Moment of inertia
For a rectangle about the centroidal horizontal axis,
- Substitute into
Now plug :
- Final expression and special values
Thus the shear stress distribution across the depth is
Key features:
- At the neutral axis :
- At the top or bottom surfaces :
as expected (no shear stress on free surface).
- The distribution is a parabola symmetric about the neutral axis.
- Interpretation and implications
- The maximum shear stress is times the average shear stress . Designers must check against material shear strength (or convert to equivalent stresses) when sizing beams.
- For wide thin sections, shear is small; for deep, narrow sections shear can be significant.
- In I-sections the flanges carry little shear; the web carries most shear. The rectangular derivation helps approximate web shear by taking the web as a narrow rectangle.
- Assumptions and limits
- The derivation assumes linear elastic material, small deformations, and that plane sections remain plane.
- It neglects shear deformation effects significant in deep beams (Timoshenko theory needed there).
- The formula applies for straight rectangular sections with constant width ; if varies with the general with appropriate and local must be used.
- Practical use
Engineers use this result to check shear capacity, to size webs and flanges, and to decide when to provide web stiffeners or increase thickness to avoid shear buckling or shear failure.
Conclusion
The shear stress distribution in a rectangular beam under a transverse shear force is parabolic and given by
with the maximum shear stress at the neutral axis and zero at the outer surfaces. This result follows directly from the general relation and is essential for safe beam design under shear loading.