A simply supported beam of span l is subjected to two point loads each of magnitude P at a distance of l/3 from either support. Find the maximum bending moment in the beam.



To find the maximum bending moment in the beam, we need to determine the location of the point of maximum bending moment.



The first step is to calculate the reactions at the supports. Since the beam is simply supported, the reactions will be equal and opposite.

Therefore, the reaction at each support is:

R = P/2



The bending moment at any point on the beam is equal to the algebraic sum of the moments due to all the loads on one side of the point.

At the left support, the bending moment due to the reaction is:

M1 = R x (l/2)

M1 = (P/2) x (l/2)

M1 = Pl/4

At the right support, the bending moment due to the reaction is:

M2 = R x (l/2)

M2 = (P/2) x (l/2)

M2 = Pl/4



The bending moment due to each point load can be calculated using the formula:

M = P x (l/3) x (l/2 - l/3)

M = P x l^2/18

The bending moment due to both point loads is:

M3 = 2 x P x l^2/18

M3 = P x l^2/9



The maximum bending moment occurs at the point where the bending moment due to the point loads is equal to the bending moment due to the reactions.

Therefore, we need to solve the equation:

M3 = M1 + M2

P x l^2/9 = Pl/4 + Pl/4

P = 27/4 x (M1 + M2)/l^2

P = 27/4 x (Pl/4 + Pl/4)/l^2

P = 27/32 x Pl

The maximum bending moment is then:

Mmax = P x l/2

Mmax = 27/64 x Pl^2



The maximum bending moment in the beam is 27/64 x Pl^2.

PL/3