AFD, SFD, BMD of frame with triangular load

Reading of the figure and assumptions

• Origin at A: A(0,0), left column AB 8 m up to B(0,8); beam BC 6 m to C(6,8) with an internal hinge at C; right column CDE down 8 m, D(6,4), E(6,0).
• Loads: UDL 5 kN/m acting to the right on AB (40 kN, at 4 m above A); clockwise couple 30 kN·m at B; triangular load on BC, 0 at B rising to 10 kN/m at C (30 kN, 4 m from B); 20 kN → at D.
• Supports are drawn as ground hatching at A and a small circle at E. A roller at E gives only a vertical reaction, and its line of action passes through the hinge C directly above it, so the column CDE could not resist the 20 kN horizontal load at D (it would rotate about C), even with a fixed base at A. For the frame to be stable and determinate, A and E are therefore taken as pin supports (r = 4, with one hinge condition).

Reactions

 triangular 10kN/m  (hinge)
 B================C
 |^30kNm          |
 | 5kN/m→         |
 |                D→20kN
 |                |
 A (pin)          E (pin)

• Hinge C (right part CDE, M_C = 0): 8Eₓ + 20(4) = 0, so Eₓ = 10.00 kN (←).
• ΣFx: Aₓ + Eₓ + 40 + 20 = 0, so Aₓ = 50.00 kN (←).
• ΣM_A: 6E_y − 40(4) − 30(4) − 30 − 20(4) = 0, so E_y = 65.00 kN (↑).
• ΣFy: A_y + E_y − 30 = 0, so A_y = 35.00 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, 8.00 m): N = 35.00 kN (tension); V = 50.00 to 10.00 kN; M = 0.00 (–) to 240.00 (right) kN·m, parabolic.
• BC (B to C, 6.00 m): N = 10.00 kN (tension); V = −35.00 to −65.00 kN; M = 270.00 (bottom) to 0.00 (–) kN·m, cubic.
• CD (C to D, 4.00 m): N = 65.00 kN (compression); V = −10.00 to −10.00 kN; M = 0.00 (–) to 40.00 (right) kN·m, linear.
• DE (D to E, 4.00 m): N = 65.00 kN (compression); V = 10.00 to 10.00 kN; M = 40.00 (right) to 0.00 (–) kN·m, linear.

Salient features

• The maximum bending moment is 270.00 kN·m, just to the right of B (tension at the bottom of the beam, i.e. inside). It equals the column moment at the top of AB (240.00 kN·m) plus the 30 kN·m couple. The beam moment falls to zero at the hinge C.
• Left column: the moment is 240.00 kN·m at B (tension on the right, i.e. inside); the 30 kN·m couple at B makes it jump to 270.00 kN·m in the beam at the joint.
• Right column CDE: the moment is zero at C and E, with a maximum of 40.00 kN·m at D (tension on the right, i.e. the outside face); the shear is a constant 10 kN, because the 20 kN load is shared equally by the hinge C and the pin E.
• The shear in AB is 50.00 kN at A and drops to 10.00 kN at B; the shear in the beam does not change sign, so there is no extra peak in BC.
• The vertical reaction at A is downward (35.00 kN), so AB is in tension (35.00 kN).