Support reactions in beams due to an applied couple or moment

 

couple of Forces

A couple is defined as two parallel forces with the same magnitude    but opposite in direction separated by a perpendicular distance d.

The net external effect of a couple is that the net force equals zero and the magnitude of the net moment equals = F.d 

The resultant of a couple at any point is always a pure constant moment  and equal to 

  

  M = F d

Problem : 2.7

A 6 m-long simply supported beam carries a anti clockwise couple 400 N-m and a point load as shown in Fig. Calculate support reactions

Solution:

Let the vertical reaction at support A is Ra and at support B is Rb.

Here the vertical forces are upwards reactive forces Ra ,Rb and downward point load of 1000 N. The horizontal pair of forces are formed as a couple and its resultant is a moment of 400N-m

Apply the  equilibrium equation is:

∑ Fy = 0

Ra + Rb - 1000 = 0      

Ra + Rb  = 1000  -----(1)

Taking moments of the forces about end A,

∑ MA = 0

-  400  + 1000 x 4 - 6.Rb  = 0

Rb  = 3600/6 = 600 N

substitute Rb value in equ.1

Ra   = 1000 -  Rb = 1000- 600 

Ra = 400N

Problem : 2.8

A simply supported beam carries a anti clockwise couple M as shown in Fig. Calculate support reactions

Let the vertical reaction at support A is Ra  and at support B is Rb

The equilibrium  equation is:

∑ Fy = 0

Ra + Rb  = 0      

Ra + Rb  = 0  -----(1)

Taking moments of the forces about end A,

∑ MA = 0

-  M - L.Rb  = 0

Rb  = - M / L (downward reaction)

substitute Rb value in equ.1

Ra   = + M/L (upward reaction)

Note: The application of a moment at any point in the beam, the support reactions doesn't change 

Problem : 2.9

A 5 m-long simply supported overhanging beam carries a clockwise moment of 6 N-m at point C and a concentrated load of 4KN at point B as shown in Fig. Calculate support reactions

Solution:

Let the vertical reaction at support A is Ra and at support D is Rd.

Apply the  equilibrium equation is:

∑ Fy = 0

Ra + Rd - 4 = 0      

Ra + Rd  = 4 KN  -----(1)

Taking moments of the forces about end A,

∑ MA = 0

 6  + 4x5 - 4.Rb  = 0

Rd  = 26/4 = 6.5 N (upward reaction)

substitute Rd value in equ.1

Ra   = 4 -  Rd = 4 - 6.5  

Ra = -2.5 KN (downward reaction)

Assignment :

Q 1. Determine the support reactions of a simply supported overhang beam loaded as shown in fig.

Q 2. Determine the support reactions of a simply supported overhang beam loaded as shown in fig.

Q 3. Determine the support reactions of a simply supported overhang beam loaded as shown in fig.