AFD, SFD, BMD of frame with hinge

Reading of the figure

• Origin at A, x right, y up: A(0,0) pin, B(0,2), C(0,4), D(2,4) internal hinge, E(3,4), F(4,4), H(6,4); right column FG goes down 3 m to the pin G(4,1).
• Loads: 10 kN → at B; UDL 4 kN/m on CD (8 kN at 1 m from C); 15 kN ↓ at E; couple 20 kN·m at F (taken anticlockwise); 20 kN ↓ at the overhang end H.
• r = 4 reactions, 3 equilibrium equations + 1 hinge condition (M_D = 0): statically determinate.

Reactions

 4kN/m        15kN  20kNm    20kN
 C=====D(hinge)==E==F=====H ↓
 |                  |
 B→10kN             |
 |                  G (pin)
 A (pin)

• ΣFx: Aₓ + Gₓ + 10 = 0. ΣFy: A_y + G_y − 8 − 15 − 20 = 0.
• ΣM_A: 4G_y − 1·Gₓ − 10(2) − 8(1) − 15(3) − 20(6) + 20 = 0, i.e. 4G_y − Gₓ = 173.
• Hinge D (left part, M_D = 0): 4Aₓ − 2A_y + 10(2) + 8(1) = 0.
• Solving: Aₓ = 6.71 kN (←), A_y = 0.57 kN (↑), Gₓ = 3.29 kN (←), G_y = 42.43 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, 2.00 m): N = 0.57 kN (compression); V = 6.71 to 6.71 kN; M = 0.00 (–) to 13.43 (right) kN·m, linear.
• BC (B to C, 2.00 m): N = 0.57 kN (compression); V = −3.29 to −3.29 kN; M = 13.43 (right) to 6.86 (right) kN·m, linear.
• CD (C to D, 2.00 m): N = 3.29 kN (compression); V = 0.57 to −7.43 kN; M = 6.86 (bottom) to 0.00 (–) kN·m, parabolic. V = 0 at 0.14 m from C, where M = 6.90 kN·m.
• DE (D to E, 1.00 m): N = 3.29 kN (compression); V = −7.43 to −7.43 kN; M = 0.00 (–) to 7.43 (top) kN·m, linear.
• EF (E to F, 1.00 m): N = 3.29 kN (compression); V = −22.43 to −22.43 kN; M = 7.43 (top) to 29.86 (top) kN·m, linear.
• FH (F to H, 2.00 m): N = 0.00 kN (zero); V = 20.00 to 20.00 kN; M = 40.00 (top) to 0.00 (–) kN·m, linear.
• FG (F to G, 3.00 m): N = 42.43 kN (compression); V = 3.29 to 3.29 kN; M = 9.86 (right) to 0.00 (–) kN·m, linear.

Salient features

• Bending moment is zero at the hinge D and at the free end H. Maximum negative moment in the beam is 40.00 kN·m (tension on top) at F, due to the 20 kN overhang load.
• Maximum positive moment in CD is 6.90 kN·m at 0.14 m from C (V = 0 there); moment in the column is at most 13.43 kN·m at B.
• The beam from D to F has hogging moment (tension on top). Joint F check: 29.86 (EF) + 20 (couple) = 40.00 (FH) + 9.86 (FG).
• Axial force is largest in the right column FG (42.43 kN compression); the left column carries only 0.57 kN compression.