Simply Supported Beam Subjected to Couple at Mid-span:

Let us consider a simply supported beam AB of length l, subjected to a clockwise couple (μ) at mid-span. The tendency of the couple will be to uplift the beam from its support A and to depress the beam at its support B. Therefore, the reaction at A, will be downward and at B will be upward.

 Figure

Taking moment about A and equating to zero,

R_B  × l - μ = 0

R_B  =  + μ / l

R_A = Total downward load - R_B  = 0 - R_B  = 0 - μ/ l  = -    μ/ l

(Negative sign denote that the reaction at A is downward)

Shear Force (Starting from the Left End A)

SF at

A = F_A  = - μ/ l

SF just left of B,

 F _B   =- μ / l     (Reaction at the support B)

Bending Moment (beginning from Left End A)

 BM at A and B,         M_A = M_B = 0

BM just left of B,       M_B = (μ/l) × (l /2) =- μ/2

BM just right of B, M B = (μ/2) + μ = + (μ/2)

BM just right of B,  M_B = (μ/l) × (l /2) =- μ/2  (considering right side)