Frame SFD, BMD (100 kN, 20 kN/m, hinge)

Assumptions (taken from the figure):

• A(0,0) hinged; column AB = 4 m; beam B–C–D = 2 m + 2 m; internal hinge at D (x = 4 m, y = 4 m).
• The inclined member DF slopes down at 60° to the horizontal to a hinged support F at the level of A, so F = (6.31 m, 0) and DF = 4.62 m.
• 20 kN/m acts horizontally (→) over the full 4 m of AB; 100 kN acts downward at C; 30 kN acts perpendicular to DF at its 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------------C------------Do
|                          \\
|                           \
|                            \\
|                             \\
|                               \
|                               \\\
|                                 \
|                                  \\
|                                   \
|                                    \\
|                                     \\
|                                       \
A                                       \F

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.

• $H_A$ = 36.08 kN ←
• $V_A$ = 53.92 kN ↑
• $H_F$ = 17.94 kN ←
• $V_F$ = 61.08 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

Shape of the diagrams

• AB: V linear, M parabolic
• BC: V constant, M linear
• CD: V constant, M linear
• DF: V constant on each half (it changes from +15 to -15 kN at the 30 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 (moment is transferred) and zero at hinges and at free ends.

Salient features

• Maximum bending moment = 92.15 kN·m at C (on BC), tension on the bottom
• V = 0 at 1.80 m from A along AB: M = 32.54 kN·m (right face), the local maximum
• On DF the moment rises from 0 at D to $15\times2.31 = 34.64$ kN·m under the 30 kN load and falls back to 0 at F (tension on the lower-left, inner face)
• Point of contraflexure (M = 0) at 3.61 m from A along AB
• Point of contraflexure (M = 0) at 0.29 m from B along BC
• The bending moment is zero at the internal hinge D (from both sides).