What are the bending moment and shear force functions for the different segments of the beam shown below? Where does the maximum bending moment occur and what is its value?
Bending Moment and Shear Force Functions
The given beam has three segments: AB, BC, and CD. To determine the bending moment and shear force functions for each segment, we need to calculate the reaction forces of the beam first.
Segment AB (0 ≤ x ≤ 6):
For this segment, the shear force function is V(x) = RA - wx, where w = 2 kN/m and RA = 16 kN. The bending moment function is M(x) = 16x - x².
Segment BC (6 ≤ x ≤ 9):
In this segment, the shear force function is V(x) = 10 kN. The bending moment function is M(x) = 16x - 2(x-6)²/2 - 10(x-6).
Segment CD (9 ≤ x ≤ 12):
For the last segment, the shear force function is V(x) = -2x + 40 kN. The bending moment function is M(x) = 16x - 2(x-6)²/2 - 10(x-6) - 12(x-9).
Maximum Bending Moment
The maximum bending moment occurs at x = 6 m. By substituting x = 6 into the bending moment function, we find:
M_max = 16 × 6 - 2(6-6)²/2 - 10(6-6) - 12(6-9) = 44 kN-m.
Therefore, the maximum bending moment value is 44 kN-m, which occurs at a distance of 6 m from the left end of the beam.