Solve Problem 13.15 by the method of least work. See Fig. P13.15.

Refer Figure (1).

Calculate the determinacy of the beam using the relation;

The reactions acting in the beam is 4.

The number of Equilibrium reaction is 3.

The degree of indeterminacy of the beam is 1.

Take the vertical reaction at B as the redundant.

Show the section located at a distance x from support C as shown in Figure (2).

Refer Figure (2).

Determine the bending moment at distance x for segment CB taken from the point C using the expression;

Differentiate with respect to .

Determine the bending moment at distance x for segment BA taken from the point C using the expression;

Differentiate with respect to .

Calculate the reaction at B using the method of least work using the expression;

…… (1)

Substitute for and 0 for inside the span CB and for and for inside the span BA.

Simplify it as below.

Calculate the bending moment at A using the expression;

Substitute for .

Take the sum of the forces in the vertical direction as zero.

Take the sum of the forces in the horizontal direction as zero.

Show the loading on the beam as shown in the Figure (3).

Refer Figure (3).

Calculate the shear force at support A using the expression;

Calculate the shear force at left of support B using the expression;

Calculate the shear force at right of support B using the expression;

The shear force at the free end C is zero.

Refer Figure (3).

The bending moment at the fixed support A is .

Calculate the bending moment at B using the expression;

The bending moment at the free end C is .

Show the shear force and the bending moment diagram as shown in Figure (4).

Therefore, the reactions acting on the beam are , , , and respectively.