Find the reactions at supports A, B, E, and F of the loaded compound beam, as shown in Figure

Find the reactions at supports A, B, E, and F of the loaded compound beam, as shown in Figure .

Fig.  Compound beam.

Solution

Free-body diagram. The free-body diagram of the entire beam is shown in Figure 3.15b.

Identification of primary and complimentary structure. The interaction diagram for the given structure is shown in Figure 3.15c. CD is the complimentary structure, while AC and DF are the primary structures.

Computation of reactions.

Analysis of complimentary structure CD.

Computation of support reaction. The isolated free-body diagram is shown in Figure . First, the distributed loading is replaced by a single resultant force (P), which is equal to the area of the rectangular loading, as shown in Figure . Applying the equations of equilibrium, and noting that due to symmetry in loading, the support reactions at point C and D are equal in magnitude, suggests the following:

Analysis of primary structure AC.

Computation of support reaction. Note that the reaction at C of the complimentary structure is applied as a downward force of the same magnitude at the same point on the primary structure. Applying the equation of equilibrium suggests the following:

Analysis of primary structure DF.

Computation of support reaction. The isolated free-body diagram is shown in Figure. First, the distributed loading is replaced by a single resultant force (P) equal to the area of the triangular loading, as shown in Figure . Applying the equations of equilibrium, and noting that the support reaction at point D of the complimentary structure is applied as a load on the primary structure, suggests the following: