Indeterminate Beams and Plastic Hinges:

Indeterminate beams are those beams which cannot be analyzed using the equations of static equilibrium alone. These beams have one or more degrees of indeterminacy (unknowns) which must be solved using additional equations. One way to solve these additional equations is by using the concept of plastic hinges.

Plastic hinges are formed in a beam when the bending moment exceeds the yield moment of the material. At this point, the beam undergoes plastic deformation and a hinge is formed. The hinge allows the beam to rotate freely and redistribute the load to other parts of the beam.

Number of Plastic Hinges in an Indeterminate Beam:

To make an indeterminate beam determinate, we need to eliminate the degrees of indeterminacy. This can be done by introducing enough plastic hinges in the beam. The number of plastic hinges required to make an indeterminate beam determinate is given by:

Number of hinges = (Degree of Indeterminacy) - 1

where the degree of indeterminacy is the number of unknown reactions or internal forces in the beam.

For example, if an indeterminate beam with a degree of indeterminacy of 3 is to become determinate, we need to introduce 2 plastic hinges. This is because:

Number of hinges = (Degree of Indeterminacy) - 1
Number of hinges = 3 - 1
Number of hinges = 2

Answer to the Given Question:

In the given question, the degree of redundancy of the beam is R. The degree of redundancy is the number of extra unknowns in the beam. To make the beam determinate, we need to introduce enough plastic hinges. The number of plastic hinges required is given by:

Number of hinges = (Degree of Redundancy)

Substituting the value of the degree of redundancy, we get:

Number of hinges = R

Therefore, the correct answer is option B - R.