When a prismatic bar is hanging and subjected to self-weight then there is elongation in the beam due to the uniformly varying load i.e. weight of the beam.
Let the weight per unit volume of the bar is given by “=W/AL where W is the weight of the prismatic bar and A is the area of cross section and the initial length is L. The deflection of any element of the bar is given by the deflection due to the external surface force acting on that particular section and the load due to the uniformly varying weight above it.
CASE:1 DEFLECTION AT THE FREE END OF THE PRISMATIC BAR
The free end of the prismatic bar is not subjected to any surface force and only body force i.e. the uniform varying weight is acting on it, so we consider a thin strip of thickness dx at a distance x and integrate to get the result.
CASE:2 DEFLECTION AT MIDPOINT OF THE PRISMATIC BAR
If we consider the midsection of the prismatic bar than it can be made equivalent to a bar of length L/2 where a external surface load is acting W/2 then we take a section from the midpoint and integrate for the body force to get the result.
IMPORTANT CONCLUSIONS:
1. THE DEFLECTION DUE TO THE SELF WEIGHT OF BAR IS HALF OF THE DEFLECTION IF THE SAME AMOUNT OF LOAD IS ACTING EXTERNALLY
2. THE STRESS INDUCED IN THE BAR IS INDEPENDENT OF LENGTH
3. THE ELONGATION IS INDEPENDENT OF THE AREA OF THE CROSS SECTION IF THE WEIGHT DENSITY OF THE BAR IS KNOWN.
4. THE DEFLECTION DUE TO SELF WEIGHT IS IS PROPORTIONAL TO SQUARE OF LENGTH.
5. THE STRAIN ENERGY OF THE BAR IS PROPORTIONAL TO THE CUBE OF LENGTH
6. STRAIN ENERGY DEPENDS ON THE AREA OF CROSS SECTION.
7. IF THE BAR IS TAPERED THEN BY TAKING ELEMENT AT ANY SECTION WE CAN EASILY EVALUATE THE DEFLECTION AND THE STRAIN ENERGY OF THE BAR.
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