AFD, SFD, BMD of frame with triangular load

Reading of the figure: A is a hinge at (0, 0) and D is a roller at (10, 6). Column AB is 10 m high (5 + 5), with a 30 kN horizontal load to the right at mid-height (5 m). Beam B–C is 10 m long: a triangular load from 10 kN/m at B decreasing to zero at 8 m, a 20 kN point load at 8 m, and a 15 kN·m couple at C (taken counter-clockwise). Column CD is 4 m long, with the roller D at its foot. The roller gives a vertical reaction only.

Support reactions

Triangular load = ½ × 10 × 8 = 40 kN, acting at 8/3 = 2.667 m from B. Moments about A (anticlockwise positive): the 30 kN load at 5 m height pushes to the right, giving −30 × 5 = −150; the 40 kN triangular load gives −40 × 2.667 = −106.67; the 20 kN gives −20 × 8 = −160; the couple gives +15; D_y at 10 m gives +10 D_y.

ΣM_A = 0:  10 D_y − 150 − 106.67 − 160 + 15 = 0
           D_y = 401.67/10 = 40.17 kN ↑

ΣFx = 0: A_x + 30 = 0, so A_x = 30 kN ← ΣFy = 0: A_y + 40.17 − 40 − 20 = 0 → A_y = 19.83 kN ↑

Member forces

Column AB (0 to 10 m)

• N = −19.83 kN (compression), constant.
• V = 30 kN from A to mid-height, 0 from mid-height to B.
• M = 30y from A to mid-height: 0 → 150 kN·m at 5 m; constant 150 kN·m up to B (no shear).

Beam BC (x measured from B; w(x) = 10(1 − x/8))

• N = 0 (no horizontal force).
• V(x) = 19.83 − 10x + 0.625x², so V_B = 19.83 kN; V = 0 at x = 2.32 m; just left of the 20 kN load V(8) = −20.17 kN; just right of it −40.17 kN, constant to C.
• M(x) = 150 + 19.83x − 5x² + 0.2083x³
  • M_B = 150.00 kN·m
  • Maximum M = 171.70 kN·m at x = 2.32 m
  • M(4 m) = 162.67, M(8 m) = 95.33 kN·m
  • M_C (beam end) = 95.33 − 40.17 × 2 = 15.00 kN·m, balanced by the applied 15 kN·m couple, so the column has no moment.

Column CD (4 m)

• N = −40.17 kN (compression); V = 0; M = 0 (the roller reaction acts along the column).

Values at key points

BMD (kN·m)
        150 ----(171.7)---- 95.3 \ 15
         |                         C|
         |                          | (zero)
        150 (constant)              |
         |                          |
        0 A                       D (0)

Salient features: the maximum bending moment is 171.7 kN·m in the beam at 2.32 m from B, where the shear is zero; the moment in the column is constant at 150 kN·m between mid-height and B because the shear there is zero; the shear in the beam changes sign once; there is a sudden jump of 20 kN in the SFD at the point load, and the 15 kN·m couple at C produces a drop of 15 kN·m in the moment from the beam end to the column; the BMD in the beam is cubic (triangular load) and linear under no load.