A beam 6 m long is simply supported at the ends,and carries a uniform distributed load of 1500kg/m (including its own weight ) and three concentrated loads of 1000,2000 and 3000 kg acting respectively at the left quarter point,centre point and right quarter point.

Question

Draw the shear force ans bending moment diagram and determine the maximum bending moment.

Answer 1

get end forces up first

beam kg = 6*1500 = 9000 kg at center

forces F1 up at left and F2 up at right
moments about left to get F2
F2*6 = 1000*1.5 + 11000*3 + 3000*4.5
so
F2 = 48,000/6 = 8,000 up at right end
total mass = (1+11+3)10^3 = 15,000 kg
so
F1 = 15,000- 8000 = 7,000 up at left

shear in left 1/4 where 0<x<1.5
= 7000 - 1500x
integrate to get
BM in left 1/4 = 7000 x - 7,500 x^2

from 1.5<x<3
shear = 7,000 - 1500 x - 1000 = 6,000-1500x
BM, integrate again
7,000 x - 7,500 x^2 - 1000 (x-1.5)
= 6000 x - 7500 x^2 + 1500

check to see if BM is continuous at x = 1.5
BM at x = 1.5 from left
= 7000 (1.5) - 7,500 (2.25)
= -6375
BM at x = 1.5 from left of center 1.5<x<3
= 6000(1.5) -7500(2.25) + 1500
= 9000 - 16875 + 1500
= - 6375 whew, good
continue on to the right hald the same way
I am not sure I have designed the sign conventions the same as your class, if not you can remedy that

Answer 2

By the way I did all this as if a kg was a force
If you need Newtons and Newton meters, multiply final answers by 9.81