Short Answer:

Shear stress in a beam is the internal tangential stress caused by transverse shear force. For any section the general relation is the shear formula , where  is the shear force at the section,  is the first moment of the area about the neutral axis of the portion above (or below) the point considered,  is the section moment of inertia about the neutral axis and  is the width at the point.

For a rectangular cross-section of width  and depth , this formula leads to a parabolic distribution of shear stress across the depth. The shear stress at distance  from the neutral axis is

and the maximum shear (at the neutral axis ) is .

Detailed Explanation :

Shear Stress Formula for Rectangular Beam

Objective: derive the shear stress distribution for a rectangular beam cross-section and obtain the maximum shear stress.

Assumptions (standard for shear derivation):

• Beam is straight, prismatic and made of homogeneous isotropic material.
• Small deformations, linear elastic behaviour.
• Cross sections remain plane (no warping considered for this derivation).
• Shear force  is known at the section under consideration.

1. Start from the general shear formula

The general expression used in beam theory for shear stress at a point in the cross section is:

where

•  = shear stress at the layer considered,
•  = internal shear force at the section,
•  = first moment of area of the portion of the cross section either above or below the layer (area × distance of its centroid from neutral axis),
•  = second moment of area (moment of inertia) of the whole cross section about the neutral axis,
•  = width of the cross section at the level where  is evaluated.

2. Geometry and co-ordinates for the rectangular section

Consider a rectangular cross section of width  (into the page) and overall depth . Place the origin at the neutral axis (centroidal axis), and let  be the vertical coordinate measured from the neutral axis (positive upward). The top fiber is at  and the bottom at .

We want shear stress at a horizontal layer located at coordinate .

3. Compute , the first moment of area

Take the area above the layer (from  to top at ). Its area is

The centroid of this area (distance from the neutral axis) is the midpoint of the interval from  to , so

Therefore the first moment  about the neutral axis is

Simplify the product inside:

so

4. Moment of inertia for rectangular section

The second moment of area about the neutral axis for rectangle is

5. Substitute into the shear formula

Now substitute :

Factor and simplify:

So

This is the standard parabolic distribution of shear stress for a rectangular section.

6. Key special values

• At the neutral axis :

This is the maximum shear stress in the section.

• At the extreme fibers :

so shear stress is zero at the top and bottom surfaces.

7. Interpretation and remarks

• The shear stress distribution is parabolic across depth, maximum at neutral axis and zero at surfaces.
• The average shear stress  is smaller than the maximum:  and .
• In I-sections and other non-rectangular shapes the shape of distribution is different and the web usually carries most shear.
• The derived formula assumes plane sections remain plane and neglects shear deformation effects important in deep or short beams (Timoshenko beam theory needed there).

Conclusion

Starting from the general shear formula  and applying it to a rectangular cross-section gives a parabolic shear distribution:

This result is widely used in beam design to check shear strength of rectangular members and to size web thickness in other section shapes.