Figure 10-1(a)

Solution:

The absolute maximum bending moment occurs under any of the wheal loads when that load and the resultant of the load system are equidistant from the centre of the beam.

First we will calculate the position of the resultant i.e. centroid (X) of the load system with reference to the leading load (in this case 4 kN load).

X = (moment of all the loads about 4 kN load)/(Total Load)

X = (3 x 3 + 2 x 5)/9 = 2.11 m from 4 kN load as shown in the following figure.

Now we will calculate the maximum bending moment (BM) under all the three loads and then the maximum of those values will be the absolute maximum bending moment.

First Trial: Calculate maximum BM under 2 kN load

Place the 2 kN load and the resultant at equal distance from the beam centre. The resultant is acting at the centroid of the loads i.e. 2.89 m from the 2 kN load. Therefore the 2 kN load should be placed at a distance of 2.89/2 = 1.445 m from the centre of the beam. The placement of loads is shown in the following figure.

For this load case the reaction at B, R_B = Resultant x distance of resultant from A /beam length

R_B = 9 x 7.445 / 12 = 5.584 kN and R_A = 9 - 5.584 = 3.416 kN

Therefore Bending moment under 2 kN = R_A x (6 - 1.445) = 3.416 x 4.555 = 15.56 kNm

Second Trial: Calculate maximum BM under 3 kN load

Place the 3 kN load and the resultant at equal distance from the beam centre. The 3 kN load is at a distance of 0.89 m from the resultant. Therefore the 3 kN load should be placed at a distance of 0.89/2 = 0.445 m from the centre of the beam. The placement of loads is shown in the following figure.

For this load case the reaction at B, R_B = Resultant x distance of resultant from A / beam length

R_B = 9 x 6.445 / 12 = 4.833 kN and R_A = 9 - 4.833 = 4.167 kN

Therefore Bending moment under 3 kN load = R_A x (6 - 0.445) - 2 x 2 = 4.167 x 5.555 - 4 = 19.148 kNm

Third Trial: Calculate Maximum BM under 4 kN load

Place the 4 kN load and the resultant at equal distance from the beam centre. The 4 kN load is at a distance of 2.11 m from the resultant. Therefore the 4 kN load should be placed at a distance of 2.11/2 = 1.05 m from the centre of the beam. The placement of loads is shown in the following figure.

For this load case the reaction at B, R_B = Resultant x distance of resultant from A /beam length

R_B = 9 x (6 - 1.055) / 12 = 9 x 4.945 / 12 = 3.708 kN and R_A = 9 - 3.708 = 5.292 kN

Therefore the bending moment under 4 kN load = R_A x 7.055 - 2 x 5 - 3 x 3 = 5.292 x 7.055 - 10 - 9 = 18.335 kNm

On the basis of above calculations it is found that the absolute maximum bending moment is 19.148 kNm which occurs under 3 kN load when it is placed at a distance of 0.445 m from the centre of the beam.

You can also use our online calculator for absolute maximum bending moment due to a series of upto 5 moving loads

Example 5-2 SFD and BMD for Simple beam

Example 5-3 SFD and BMD for Overhanging beam