A special type of structure

Fact:

    You know for any beam, frame or arch if you add one loop for each of its constituting structural part you will end up with a structure which will have its static degree of indeterminacy equals to number of its support reactions. I mean for these type of structures you will get

                                                                      

Ds = r

Wow!!!

But

How???

Proof:

    Did you remember the equations I obtained by just adding the expressions for static and kinematic degree of indeterminacy, in case you forgot click here. 

Ds + Dk = f ( j + l - n)

    where; f is the number of degree of freedom;

                j is the number of joints;

                l is the number of loops; and

                n is number of structural parts 

Since we said that for these particular type of structures number of loops in them is equals to the number of its structural parts, we can say 

l = n

So the expressions for Ds + Dk will be reduced to,

Ds + Dk = f * j

And we know that Dk = f * j - r 

Now substituting its value we'll have,

Ds + ( f * j - r ) = f * j

So we'll end up with

Ds = r

Examples: