Suppose a man of mass 50 kg climbs a stair of height 10 m.Clearly it gains 5000 J of potential energy.But since work done by normal reaction on the man is zero does'nt it violates the principle of conservation of mechanical energy because no non-conservative force is acting on the man and thus mechanical energy should be conserved.
By "work done by normal", you probably mean that since the foot does not move relative to the stairs, whatever the normal force on man exerted by the stairs, there cannot be work done by this force as displacement is zero. In the case of discrete stairs with horizontal top surface, that is correct.
However, when your muscles contract to climb each stair, energy is spent. There is no such thing as "conservation of mechanical energy". All that is required is that total energy is conserved.
Even with mechanical energy, suppose you have a compressed spring with two masses, one at each end: O///\O and release the spring. The balls will gain speed and have kinetic energy, even if there was no outside work performed on the total "spring plus balls" system. Energy stored in the spring was used to push the balls.
In your example, chemical energy is used to move your muscles in a very complex manner that results in a gain of gravitational potential energy. You could get the same result by having a complex mechanism that uses energy stored in a compressed spring to perform each step.
When the man applies force to the stairs, the stairs apply an equal force back.
The mans force down is acting on the stairs. If the stairs started moving downwards, you could say the man is doing work on the stairs. Instead, the stairs do not move; the force is static, and there is no work done on the stairs.
The stairs provide an equal and opposite reaction upwards; but the man is moving, so there is work done on the man by the stairs.
By pushing down with a force greater than his own weight, the man is able to use the stairs to generate an upwards push. The energy for the push doesn't come from the stairs though. It comes from the muscles, which get their energy through food (chemical energy).
the work done by normal force is not zero. since the displacement of man and normal force is in same direction and work done by a force is integral f.dx the work done by normal would be normal force*displacement in height. since man is applying external force to increase his mechanical energy so mechanical energy won't remain constant. mechanical energy of only those object which are isolated from other mass system and no external mechanical energy is provided remain constant.