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Question
The sketch shows a column with a pin at the base and rollers at the top. It is subjected to an axial force P and a moment M at mid-height. The reactions at R is/are
 
 
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A.
A vertical force equal to P
B.
A vertical force equal to P/2
C.
A vertical force equal to P and a horzontal force equal to M/h
D.
A vertical force equal to P/2 and a horizontal force equal to M/h
Answer ( Option C)

A vertical force equal to P and a horzontal force equal to M/h

Solution

The three equilibrium conditions need to be satisfied are
∑H = 0,   H → Horizontal forces
∑V =0,     V → Vertical forces 
∑MZ = 0, MZ → Moment acting at a point
Calculation:
free body diagram is drawn,
Sign Convention
Vertical forces - ↑ - (+ve), ↓ - (–ve) 
Moments - clockwise moments are taken as positive, anticlockwise moments are taken as negative.
Since R is a hinged support it generates two reactions VR and HR 
Applying Equilibrium condition ∑V =0, we get
VR - P = 0
 →VR = P
Applying Equilibrium condition ∑MQ = 0,
we get, M - ( HR × h ) = 0 
HR = M / h

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