--- title: "AFD, SFD, BMD of frame with hinge at C" subject: "Applied Mechanics" chapter: 6 key: "semester_1/applied_mechanics:frame-afd-sfd-bmd-hinge-5knm-10knm-triangular" source: https://www.bctengineer.com/q/semester_1/applied_mechanics/frame-afd-sfd-bmd-hinge-5knm-10knm-triangular site: BCT Engineering --- # AFD, SFD, BMD of frame with hinge at C Applied Mechanics · Chapter 6: Analysis of Beams and Frames - Asked in: 2071 Shrawan, 14 marks - Video: [Elbaraa Structural analysis one hinge frame](https://www.youtube.com/watch?v=JN5WJx4eFcs) — Eng ElBaraa Louye - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/applied_mechanics/frame-afd-sfd-bmd-hinge-5knm-10knm-triangular - Whole chapter: https://www.bctengineer.com/md/semester_1/applied_mechanics/chapter-6 ## Question Draw axial force, shear force and bending moment diagram for the given frame. Also indicate salient features if any. [Figure: frame with pin at A and support at E; hinge at C; left column A-B-C, 3 m + 3 m, 10 kN at 60° at B; top beam C-D of 6 m with 5 kN/m UDL; 20 kN at 30° at D; right column D-E of 4 m with triangular load up to 10 kN/m.] ## Short answer Reactions: A_x = 4.33 kN ←, A_y = 28.22 kN ↑, E_x = 25.67 kN →, E_y = 24.10 kN ↑. Maximum beam moment is 53.94 kN·m at 4.64 m from hinge C. Moment at knee D is 49.35 kN·m (tension inside). ## Answer #### 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 ```text 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). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How does the hinge at C affect the left column AC? A. It makes it fixed B. It acts like a member pinned at both ends, so moments are small C. It doubles the moment D. It removes the column load Answer: B. With M_C = 0 and A pinned, AC carries only the moment from the 10 kN load at B. (Key point: Hinge at C) 2. What is the resultant of the triangular load on DE and where does it act? A. 20 kN, 2.67 m below D B. 40 kN, 2 m below D C. 20 kN, at mid-height D. 10 kN, at E Answer: A. Area = 1/2 x 10 x 4 = 20 kN, at 2/3 of the length from the zero end. (Key point: Triangular load resultant on DE) 3. Where is the beam moment maximum? A. At C B. At D, 49.35 kN·m C. 4.64 m from C, where shear is zero, 53.94 kN·m D. At 3 m from C Answer: C. V goes from +23.22 to -6.78 and is zero at 4.64 m, where M peaks. (Key point: Maximum beam moment) 4. What is the moment at knee D? A. 49.35 kN·m, the same in beam and column B. Zero C. 53.94 kN·m D. 12.99 kN·m Answer: A. The rigid knee transmits the same moment between members. (Key point: Knee D moment) ## Close variants set in other exams - [AFD, SFD, BMD of frame with triangular load](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aframe-afd-sfd-bmd-30knm-10knm-triangular-hinge) — Applied Mechanics, chapter 6, asked 2× ## Related in Applied Mechanics - [AFD, SFD, BMD of frame with 60 kN/m UDL](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aframe-afd-sfd-bmd-60knm-20knm-12kn) — Applied Mechanics, chapter 6 - [AFD, SFD, BMD of frame with 60 kN/m UDL](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aframe-afd-sfd-bmd-60knm-hinge) — Applied Mechanics, chapter 6 - [AFD, SFD, BMD of frame with hinge](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aframe-afd-sfd-bmd-hinge-4knm-20knm) — Applied Mechanics, chapter 6 - [AFD, SFD, BMD of frame with hinge at D](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aframe-afd-sfd-bmd-8knm-hinge-50kn) — Applied Mechanics, chapter 6 - [AFD, SFD, BMD for frame with internal hinge](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aframe-afd-sfd-internal-hinge-5kn-per-m) — Applied Mechanics, chapter 6 - [AFD, SFD, BMD for frame with UDL and couple](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Aframe-afd-sfd-bmd-30kn-4knm-8kn) — Applied Mechanics, chapter 6 ## Related in other subjects - [Frame AFD, SFD, BMD (10 kN/m, 60 kN, hinge)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aframe-afd-sfd-bmd-10knm-60kn) — Strength of Materials, chapter 1 - [Frame AFD, SFD, BMD (5 and 10 kN/m, hinge)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aframe-afd-sfd-bmd-5knm-10knm) — Strength of Materials, chapter 1, asked 2× - [Frame AFD, SFD, BMD (12 kN, 25 kNm, hinge)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aframe-afd-sfd-bmd-12kn-25knm) — Strength of Materials, chapter 1 - [Frame AFD, SFD, BMD (30 kN/m, 50 kNm, hinge)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aframe-afd-sfd-bmd-30knm-50knm) — Strength of Materials, chapter 1