Calculate the support reactions in the continuous beam ABC due to loading as shown in Fig.1.1 Assume EI to be constant throughout.

 Calculate the support reactions in the continuous beam ABC due to loading as shown in Fig.1.1 Assume EI to be constant throughout.

Select two reactions vise, at B(R1 ) and C(R2 ) as redundant, since the given beam is statically indeterminate to second degree. In this case the primary structure is a cantilever beam AC. The primary structure with a given loading is shown in Fig. 1.2

 

In the present case, the deflections (? L)1 and (? L) 2 of the released structure at B and C can be readily calculated by moment-area method. Thus

(? L) 1        = ? 819.16 / EI    

(? L) 2        = ? 2311.875/ EI  (1)

For the present problem the flexibility matrix is,

a11= 125/3EI ,a21= 625/6EI  

a12= 625/6EI , a22 = 1000/3EI        (2)

In    the   actual   problem   the   displacements   at B and Care  zero. Thus the

compatibility conditions for the problem may be written as, a11 R1+ a12 R2 + (? L) 1 = 0

a21 R1+ a22 R2+ (? L) 2  = 0(3)

 

Substituting the value of E and I in the above equation,

R1 = 10.609 KN and R2 = 3.620 KN

 

Using equations of static equilibrium, R3 = 0.771  KN m and R4  = ?0.755KN m