Frame AFD, SFD, BMD (12 kN, 25 kNm, hinge)

Assumptions (taken from the figure):

• A(0,0) hinged; AB = BC = 3 m; beam C–D–E–F with CD = DE = 3 m, EF = 2 m; internal hinge at D; column from E (6,6) down to hinged support G, taken 5 m long, so G = (6,1).
• 15 kN at B acts at 30° to the horizontal, to the right and downward; triangular load on CD rises from 0 at C to 5 kN/m at D; 25 kN·m couple at E is clockwise; 12 kN downward at F (overhang EF).
• The overhang EF is replaced at E by the 12 kN force plus a moment of $12\times2 = 24$ kN·m (hogging); moment in EF is $-12x$.

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.

C------------D------------E
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B                         |
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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.

• $H_A$ = 12.54 kN ←
• $V_A$ = 2.09 kN ↓
• $H_G$ = 0.45 kN ←
• $V_G$ = 29.09 kN ↑

Check: the three equilibrium equations of the complete frame (and the hinge condition, where present) are satisfied to rounding (residuals below $10^{-9}$).

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

Shape of the diagrams

• AB: V constant (steps at point loads), M linear
• BC: V constant (steps at point loads), M linear
• CD: V parabolic, M cubic
• DE: V constant (steps at point loads), M linear
• EG: V constant (steps at point loads), M linear
• AFD: constant between loads on a member, changing only where a load component acts along the member.
• BMD is drawn on the tension side of each member; it is continuous at rigid joints (moment is transferred) and zero at hinges and at free ends.

Salient features

• Maximum bending moment = 51.26 kN·m at E (on DE), tension on the top
• Bending moment is zero at the internal hinge (value read from both sides: 0.00).
• Overhang EF: V = −12 kN (downward, constant), N = 0, M = 0 at F rising linearly to 24 kN·m (hogging, tension on top) at E. At joint E the 51.26 kN·m of DE is balanced by 24 (overhang) + 25 (couple) + 2.26 (column EG) kN·m.