AFD, SFD, BMD of frame with hinge at C

Reading of the figure

• Origin at A: A(0,0) pin, B(0,3), C(0,6) hinge at the top of the left column, D(6,6), E(6,2) pin. Left column 6 m, beam CD 6 m, right column DE 4 m.
• Loads: 10 kN at B making 60° with the column, i.e. 30° below the horizontal pointing right: 8.66 kN → and 5.00 kN ↓; UDL 5 kN/m on CD (30 kN); 20 kN at D making 30° with the vertical, pointing down-left: 10.00 kN ← and 17.32 kN ↓; triangular load on DE, zero at D and 10 kN/m at E, acting to the left (total 20 kN, acting 2.67 m below D).
• r = 4 and the hinge at C gives one extra equation, so the frame is determinate.

Reactions

      5 kN/m          \ 20 kN
  C(hinge)==========D
  |                  |
  |                  | triangular load
  B --> 10 kN        | (0 at D, 10 kN/m at E)
  |    (30° below    |
  |     horizontal)  |
  A (pin)            E (pin)

• Hinge C (left part, M_C = 0): 6Aₓ + 8.66(3) = 0, so Aₓ = 4.33 kN (←).
• ΣFx: Aₓ + Eₓ + 8.66 − 10 − 20 = 0, so Eₓ = 25.67 kN (→).
• ΣM_A: 6E_y − 2Eₓ − 8.66(3) − 30(3) − 17.32(6) + 10(6) + 20(3.33) = 0, so E_y = 24.10 kN (↑).
• ΣFy: A_y + E_y − 5 − 30 − 17.32 = 0, so A_y = 28.22 kN (↑).

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, 3.00 m): N = 28.22 kN (compression); V = 4.33 to 4.33 kN; M = 0.00 (–) to 12.99 (right) kN·m, linear.
• BC (B to C, 3.00 m): N = 23.22 kN (compression); V = −4.33 to −4.33 kN; M = 12.99 (right) to 0.00 (–) kN·m, linear.
• CD (C to D, 6.00 m): N = 4.33 kN (compression); V = 23.22 to −6.78 kN; M = 0.00 (–) to 49.35 (bottom) kN·m, parabolic. V = 0 at 4.64 m from C, where M = 53.94 kN·m.
• DE (D to E, 4.00 m): N = 24.10 kN (compression); V = −5.67 to −25.67 kN; M = 49.35 (left) to 0.00 (–) kN·m, cubic.

Salient features

• The maximum moment in the beam is 53.94 kN·m at 4.64 m from the hinge C, where the shear is zero. At C the moment is zero (hinge) and the beam shear there is 23.22 kN.
• The moment at the knee D is 49.35 kN·m (tension on the inside); the right column has its largest moment at D and zero at the pin E.
• The left column has small moments (maximum 12.99 kN·m at B), because the hinge at C makes AC act like a member pinned at both ends (A and C) carrying the 10 kN load.
• The axial forces are large in DE (24.10 kN compression) and AB (28.22 kN compression).