The transverse shear stress in a beam of rectangular cross-section is variable and has a maximum value on the neutral axis. This is because the distribution of shear stress depends on the geometry of the beam and the applied load.

When a beam is subjected to a transverse shear load, the shear stress distribution is not uniform across the cross-section. It varies from zero at the top and bottom surfaces of the beam to a maximum value on the neutral axis.

Here is a detailed explanation of why the transverse shear stress is variable with a maximum on the neutral axis:

1. Shear stress distribution:
- Shear stress is a measure of the internal resistance of a material to shearing forces. In a beam, shear stress is caused by the shearing action of the applied load.
- The distribution of shear stress across the cross-section of a beam is not uniform. It varies from zero at the top and bottom surfaces to a maximum value on the neutral axis.
- This distribution is due to the shear force being transferred through the beam's cross-section. The shear force causes a deformation known as shear strain, which results in a distribution of shear stress.

2. Neutral axis:
- The neutral axis is an imaginary line that passes through the centroid of the cross-section and experiences no deformation when subjected to bending.
- The neutral axis is located at the center of the beam's depth for a rectangular cross-section.
- Since the neutral axis experiences no deformation, it is the location where the shear stress is maximum.
- The shear stress decreases linearly from the neutral axis towards the top and bottom surfaces of the beam.

3. Shear stress equation:
- The shear stress distribution can be calculated using the shear stress equation: τ = VQ / It, where τ is the shear stress, V is the shear force, Q is the first moment of area about the neutral axis, I is the moment of inertia, and t is the thickness of the beam.
- The first moment of area, Q, is zero at the top and bottom surfaces and maximum at the neutral axis. This contributes to the variable distribution of shear stress.

In conclusion, the transverse shear stress in a beam of rectangular cross-section is variable with a maximum value on the neutral axis. This is due to the distribution of shear forces and the deformation caused by the applied load. The shear stress decreases from the neutral axis towards the top and bottom surfaces of the beam.