Frame AFD, SFD, BMD (30 kN/m, 50 kNm, hinge)

Assumptions (taken from the figure):

• A(0,0) hinged; AB = 8 m; 50 kN·m couple at B (clockwise); beam B–C = 10 m with 30 kN/m downward, internal hinge at C (10, 8).
• Inclined member CD: horizontal projection 3 m and vertical height 5 m, so D = (13, 3) is the hinged support (length 5.83 m).
• The 15 kN force is taken perpendicular to CD at mid-length, pointing down and to the left.

Sign convention: N is positive in tension; V is positive when the forces on the part before the section (travelling A → end) point along the left-hand normal of the member axis (upward on a left-to-right beam); M is quoted as a magnitude with the tension face named. Support reactions are drawn on the free-body of the whole frame.

B-------------------------------Co
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A

Support reactions

Equilibrium of the whole frame gives $\Sigma F_x = 0$, $\Sigma F_y = 0$ and $\Sigma M = 0$; the internal hinge adds the condition $M_{hinge} = 0$ (moments of all forces on one side of the hinge), which makes the reactions determinate.

• $H_A$ = 68.75 kN →
• $V_A$ = 200.00 kN ↑
• $H_D$ = 55.89 kN ←
• $V_D$ = 107.72 kN ↑

Check: the three equilibrium equations of the complete frame and the hinge condition are satisfied to rounding.

Axial force, shear force and bending moment at the key sections

The moment drops from 549.98 to 499.98 kN·m across joint B because of the 50 kN·m couple applied there.

Shape of the diagrams

• AB: V constant, M linear
• BC: V linear, M parabolic
• CD: V constant on each half (it changes from +7.5 to -7.5 kN at the 15 kN load), M linear on each half (triangular, peak at mid-length)
• AFD: constant between loads on a member, changing only where a load component acts along the member.
• The BMD is drawn on the tension side of each member; it is continuous at rigid joints (apart from the jump caused by an applied couple) and zero at hinges and at free ends.

Salient features

• Maximum bending moment = 549.98 kN·m at B (on AB), tension on the left face
• V = 0 at 6.67 m from B along BC: M = 166.67 kN·m (bottom), the local maximum
• Point of contraflexure (M = 0) at 3.33 m from B along BC
• On CD the moment rises from 0 at C to $7.5\times2.92 = 21.87$ kN·m under the 15 kN load and falls back to 0 at D (tension on the lower-left, inner face)
• The bending moment is zero at the internal hinge C (from both sides).