Structural Analysis
The three moments equation is applicable only when

There is no discontinuity such as hinges within the span
The beam is prismatic
The spans are equal
There is no settlement of supports

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Structural Analysis
When a uniformly distributed load, shorter than the span of the girder, moves from left to right, then the conditions for maximum bending moment at a section is that

the head of the load reaches the section
the tail of the load reaches the section
the load position should be such that the section divides the load in the same ratio as it divides the span
the load position should be such that the section divides it equally on both sides

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