AFD, SFD, BMD of frame with slanted column

Reading of the figure

• Origin at A: A(0,0) pin; B(3,4), so AB is slanted (3 m run, 4 m rise, length 5 m); beam BC is 4 m long at y = 4, ending at C(7,4); the column runs down from C with D(7,0) 4 m below C and the roller E(7,−2) 2 m below D.
• Loads: UDL 30 kN/m on BC (120 kN, 2 m from B); 50 kN ← at D.
• r = 3, determinate.

Reactions

      30 kN/m
    B=========C
   /          |
  /           |
 /            |
A (pin)       D←50kN
              |
              E (roller)

• ΣFx: Aₓ − 50 = 0, so Aₓ = 50.00 kN (→).
• ΣM_A: 7E_y − 120(5) + 50(0) = 0 (the 50 kN load acts along the level of A), so E_y = 85.71 kN (↑).
• ΣFy: A_y + E_y − 120 = 0, so A_y = 34.29 kN (↑).
• For the slanted member AB (cos θ = 0.6, sin θ = 0.8), resolve the reaction at A: axial force = Aₓcos θ + A_y sin θ = 50(0.6) + 34.29(0.8) = 57.43 kN (compression); shear V = Aₓ sin θ − A_y cos θ = 50(0.8) − 34.29(0.6) = 19.43 kN; moment at B = 19.43 × 5 = 97.14 kN·m.

Member forces and diagrams

Sign notes: N is + for tension and − for compression; V follows the member axis in the order of the letters (e.g. A→B); for M the side in brackets is the tension side, on which the BMD is drawn.

• AB (A to B, 5.00 m): N = 57.43 kN (compression); V = −19.43 to −19.43 kN; M = 0.00 (–) to 97.14 (top-left) kN·m, linear.
• BC (B to C, 4.00 m): N = 50.00 kN (compression); V = 34.29 to −85.71 kN; M = 97.14 (top) to 200.00 (top) kN·m, parabolic. V = 0 at 1.14 m from B, where M = −77.55 kN·m (hogging).
• CD (C to D, 4.00 m): N = 85.71 kN (compression); V = 50.00 to 50.00 kN; M = 200.00 (right) to 0.00 (–) kN·m, linear.
• DE (D to E, 2.00 m): N = 85.71 kN (compression); V = 0.00 to 0.00 kN; M = 0.00 (–) to 0.00 (–) kN·m, constant.

Salient features

• Maximum bending moment = 200.00 kN·m at the corner C (tension on the outside, i.e. the top of the beam and the right face of the column). The moment at B is 97.14 kN·m.
• In BC the shear is zero at 1.14 m from B, where the hogging moment has its smallest value, 77.55 kN·m. The beam is in hogging throughout (97.14 kN·m at B, 200.00 kN·m at C), so there is no sagging and no contraflexure.
• CD carries a constant shear of 50 kN and DE carries none; DE has no bending, only 85.71 kN compression (= E_y).
• AB is an inclined member with axial compression 57.43 kN and constant shear 19.43 kN; its bending moment is linear from 0 at A to 97.14 kN·m at B.