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AFD, SFD, BMD of frame with 60 kN/m UDL

Short answer

Using equilibrium and hinge condition at C, reactions are Ax = 1.50 kN→, Ay = 244.60 kN↑, Ex = 6.50 kN←, Ey = 244.06 kN↑. The maximum bending moment is 463.89 kN·m at 3.93 m from C on beam CD (bottom tension). The 40 kN·m couple at B causes a 40 kN·m jump in column moment; moment is zero at hinge C.

Draw the Axial Force, Shear Force and Bending Moment diagram of the given frame. Also show the salient features if any. [Figure: frame with pin supports at A and E; column A-B-C with 5 m + 5 m, 10 kN at 30° at the upper part and 40 kN·m moment at B; top beam C-D of 8 m with 60 kN/m UDL and an internal hinge; column D-E of 5 m.]

Reading of the figure

  • Origin at A: A(0,0) pin, B(0,5), C(0,10) internal hinge, D(8,10), E(8,5) pin. Column AC 10 m, beam CD 8 m, column DE 5 m.
  • Loads: 10 kN at B at 30° to the vertical, pointing down-right: 5.00 kN → and 8.66 kN ↓; clockwise couple 40 kN·m at B; UDL 60 kN/m on CD (480 kN, 4 m from C).
  • r = 4 and a hinge at C: determinate.

Reactions

       60 kN/m
  C(hinge)==============D
  |                      |
  |\10kN 30°             |
  B (40kNm, clockwise)   E (pin)
  |
  A (pin)
  • Hinge C (left part, M_C = 0): 10Aₓ + 5(5) − 40 = 0, so Aₓ = 1.50 kN (→).
  • ΣFx: Aₓ + Eₓ + 5 = 0, so Eₓ = 6.50 kN (←).
  • ΣM_A: 8E_y − 5Eₓ − 5(5) − 40 − 480(4) = 0, so E_y = 244.06 kN (↑).
  • ΣFy: A_y + E_y − 480 − 8.66 = 0, so A_y = 244.60 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, 5.00 m): N = 244.60 kN (compression); V = −1.50 to −1.50 kN; M = 0.00 (–) to 7.50 (left) kN·m, linear.
  • BC (B to C, 5.00 m): N = 235.94 kN (compression); V = −6.50 to −6.50 kN; M = 32.50 (right) to 0.00 (–) kN·m, linear.
  • CD (C to D, 8.00 m): N = 6.50 kN (compression); V = 235.94 to −244.06 kN; M = 0.00 (–) to 32.50 (top) kN·m, parabolic. V = 0 at 3.93 m from C, where M = 463.89 kN·m. M changes sign (contraflexure) at 7.86 m from C.
  • DE (D to E, 5.00 m): N = 244.06 kN (compression); V = 6.50 to 6.50 kN; M = 32.50 (right) to 0.00 (–) kN·m, linear.

Salient features

  • The beam behaves almost as a simply supported beam: maximum moment = 463.89 kN·m (tension at the bottom) at 3.93 m from C, where V = 0. At D the moment is −32.50 kN·m (small hogging, tension on the outside of the corner).
  • The columns carry very large axial compression (244.60 kN in AB and 244.06 kN in DE) but small bending moments: at most 32.50 kN·m at B.
  • The 40 kN·m couple makes the column BMD jump by 40 kN·m at B (7.50 kN·m below B and 32.50 kN·m above, with opposite sides in tension).
  • Both horizontal reactions are small (1.50 kN and 6.50 kN).

✓ Answer checked by Claude Opus 5.5 (high effort)

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