Frame AFD, SFD, BMD (15 kN/m, 75 kN)

Assumptions (taken from the figure):

• A(0,0) hinged; AB = 8 m, so B = (0, 8). B to C (7, 10) is inclined (rises 2 m in 7 m, length 7.28 m). The right column CD is 8 m high, so the hinged support D = (7, 2) is 2 m above the level of A.
• With two hinged supports and no internal hinge the frame is indeterminate, so a hinge is assumed at the middle of BC, (3.5, 9), giving a three-hinged frame.
• 15 kN/m acts downward per metre of horizontal projection of BC (total 105 kN); the 75 kN horizontal force acts to the left on CD, 3 m above D, i.e. at (7, 5).

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
          -----   |
   -------        |
B--               |
|                 |
|                 |
|                 |
|                 |
|                 |
|                 D  (2 m above A)
|
|
A

Support reactions

Moments about A: $7V_D - 2H_D - 105(3.5) + 75(5) = 0$. Hinge at the middle of BC (left part): $-3.5V_A + 9H_A + 52.5(1.75) = 0$. With $\Sigma F_x = 0$ ($H_A + H_D = 75$) and $\Sigma F_y = 0$ ($V_A + V_D = 105$):

• $H_A$ = 25.55 kN →
• $V_A$ = 91.94 kN ↑
• $H_D$ = 49.45 kN →
• $V_D$ = 13.06 kN ↑

Check: moments of the right part about the hinge: $13.06(3.5) + 49.45(7) - 75(4) - 52.5(1.75) \approx 0$ ✓

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

Shape of the diagrams

• AB: V constant, M linear
• BC: V linear, M parabolic
• CD: V constant on each side of the 75 kN load (it steps by 75 kN there), M linear on each side
• 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 = 204.38 kN·m at joint B (left face of AB, top of BC)
• V = 0 at 5.87 m from B along BC (5.64 m horizontally): M = 34.44 kN·m (bottom), the local maximum
• The moment on BC changes sign only at the internal hinge (3.64 m from B), where M = 0
• On CD the moment is 20.63 kN·m at C and 148.36 kN·m at the 75 kN load, falling to 0 at D (tension on the left, inner face)