**Basic Assumption of Plane Sections Normal to the Neutral Axis**

The basic assumption of plane sections normal to the neutral axis before and after bending is a fundamental concept used in the theory of bending of beams. This assumption states that the cross-sections of a beam that are normal to the neutral axis do not deform or rotate during bending. In other words, these cross-sections remain plane and perpendicular to the neutral axis before and after bending.

**Implications of the Assumption**

This assumption has important implications for the behavior of the beam during bending. It allows us to analyze the bending of a beam by considering the deformation of a single cross-section, rather than the entire beam. Based on this assumption, we can make the following conclusions:

1. **Uniform Strain over the Beam Cross-Section:** Since the plane sections normal to the neutral axis do not deform during bending, the strain experienced by these sections remains constant throughout the cross-section. This means that all points within the cross-section experience the same amount of deformation, resulting in a uniform strain distribution.

2. **Uniform Stress over the Beam Cross-Section:** According to Hooke's law, stress is directly proportional to strain. Therefore, if the strain is constant throughout the cross-section, the stress will also be constant. This implies that the stress distribution over the beam cross-section is uniform.

3. **Linearly Varying Strain over the Cross-Section:** The assumption of plane sections normal to the neutral axis does not imply a uniform strain distribution over the entire cross-section. Instead, it allows for a linear variation of strain across the cross-section. The strain is maximum at the extreme fibers (top and bottom) and decreases linearly towards the neutral axis.

4. **Stresses Proportional to Strains at All Cross-Sections:** Since the stress is constant and proportional to strain over the beam cross-section, the stresses at any given cross-section will be directly related to the strains at that cross-section. This means that the stress and strain at any point within the beam can be determined by knowing the stress-strain relationship and the strain distribution.

**Conclusion**

In conclusion, the basic assumption of plane sections normal to the neutral axis before and after bending leads to a linearly varying strain over the cross-section of a beam. This assumption allows for the analysis of beam bending by considering the behavior of a single cross-section, and it implies a uniform stress distribution over the beam cross-section.