Determine reactions on compound beams:

Determine reactions at the support A, B, C, and D for arrangement of compound beams shown in the figure given below.

Sol.: This is the question of multiple beams (that is, beam on a beam). In this type of question, consider the top most beam, then 2^nd last beam as, In this problem on E and F, there are roller support, and this give reaction to both up and down beam. Consider free body digram of top most beams EB as shown in the given figure.

V = 0;

R_E  + R_B  - 10 - 4 - 6 - 8 = 0

R_E  + R_B = 28KN                                                                                                   ...(i)

Now taking moment about 'E'

10 × 1 + 4 × 2 + 6 × 3 + 8 × 4 - R_B  × 6  = 0

R_B = 11.33 KN                                   .......ANS

Putting the value of RB  in the equation (i)

R_E = 16.67 KN                                   .......ANS

Consider free body digram of second beam AF as shown in fig 8.56:

∑V = 0;

R_A  + R_F  - 6 - 8 - RE = 0

R_A  + R_F = 30.67 KN                                                                                          ...(ii)

Now taking moment about 'A'

6 × 1 + 8 × 2 + 16.67 × 3 - RF × 6  = 0

R_F = 12 KN                                        .......ANS

Putting the value of RF  in the equation (ii)

R_A = 18.67 KN                                   .......ANS

Consider FBD of third beam CD as shown in the given figure:

∑V = 0;

R_C  + R_D  - R_F = 0

R_C  + R_D = 12 KN                                                                                               ...(iii)

Now taking moment about 'C'

- R_D  × 5 + 12 × 3 = 0

R_D = 7.2 KN                                       .......ANS

Putting value of R_D  in the equation (iii)

R_C = 4.8 KN                                        .......ANS