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AFD, SFD, BMD of frame with hinge at D

Short answer

Using equilibrium and hinge condition at D, reactions are Ay = 5.75 kN↓, Ax = 36.88 kN←, Gy = 30.75 kN↑, Gx = 13.13 kN←. Maximum bending moment is 73.75 kN·m at B in the left column (inside tension). The 20 kN·m couple causes a 20 kN·m jump in column moment, and moment is zero at hinge D.

A frame is loaded as shown in figure below. Draw the AFD, SFD and BMD and also show the salient features of each diagram. [Figure: frame with pin supports at A and G; left column A-B-C with 2 m + 2 m, 50 kN horizontal load and 20 kN·m moment at B; top beam C-D-E-F with 8 kN/m UDL on C-D, internal hinge at D, 9 kN downward load at E; horizontal dimensions 2 m, 1 m, 1 m.]

Reading of the figure

  • Origin at A: A(0,0) pin, B(0,2), C(0,4); beam C(0,4), D(2,4) internal hinge, E(3,4), F(4,4); right column FG down to the pin G(4,0).
  • Loads: 50 kN → and a couple of 20 kN·m (taken anticlockwise) at B; UDL 8 kN/m on CD (16 kN, 1 m from C); 9 kN ↓ at E.
  • r = 4 and a hinge at D (M_D = 0): determinate.

Reactions

 8 kN/m (hinge)   9kN
 C=====D===E=====F
 |                |
 B 50kN→, 20kNm   |
 |                |
 A (pin)          G (pin)
  • ΣFx: Aₓ + Gₓ + 50 = 0. ΣFy: A_y + G_y − 16 − 9 = 0.
  • ΣM_A: 4G_y + 20 − 50(2) − 16(1) − 9(3) = 0, so G_y = 30.75 kN (↑) and A_y = 5.75 kN (↓).
  • Hinge D (left part): 4Aₓ − 2A_y + 50(2) + 20 + 16(1) = 0, so Aₓ = 36.88 kN (←) and Gₓ = 13.13 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 = 5.75 kN (tension); V = 36.88 to 36.88 kN; M = 0.00 (–) to 73.75 (right) kN·m, linear.
  • BC (B to C, 2.00 m): N = 5.75 kN (tension); V = −13.13 to −13.13 kN; M = 53.75 (right) to 27.50 (right) kN·m, linear.
  • CD (C to D, 2.00 m): N = 13.13 kN (compression); V = −5.75 to −21.75 kN; M = 27.50 (bottom) to 0.00 (–) kN·m, parabolic.
  • DE (D to E, 1.00 m): N = 13.13 kN (compression); V = −21.75 to −21.75 kN; M = 0.00 (–) to 21.75 (top) kN·m, linear.
  • EF (E to F, 1.00 m): N = 13.13 kN (compression); V = −30.75 to −30.75 kN; M = 21.75 (top) to 52.50 (top) kN·m, linear.
  • FG (F to G, 4.00 m): N = 30.75 kN (compression); V = 13.13 to 13.13 kN; M = 52.50 (right) to 0.00 (–) kN·m, linear.

Salient features

  • The maximum moment is 73.75 kN·m at B in the left column (tension on the inside); at C it is 27.50 kN·m, and it is zero at the hinge D.
  • The moment in the right column is largest at F (52.50 kN·m, tension on the outside) and zero at G. The beam portion DF has a hogging moment (52.50 kN·m at F).
  • The 20 kN·m couple at B causes a step of 20 kN·m in the column BMD at B (73.75 below, 53.75 above).
  • The shear in AB (36.88 kN) is the largest in the frame; the vertical reaction at A is downward, so the left column is in tension.

✓ Answer checked by Claude Opus 5.5 (high effort)

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