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Frame AFD, SFD, BMD (50 kN, 15 kN/m)

Short answer

Reactions: HA=30 kN left, VA=27.5 kN up, VE=92.5 kN up. Maximum bending moment 205.21 kN·m occurs 1.83 m from B on BC (bottom tension). Point of contraflexure at 7.06 m from B on BC.

Draw axial force shear force and bending moment diagram for a given loaded frame. Also write the salient features. [Figure: frame with hinge support at A at the bottom of the left column (height 6 m to B); 50 kN horizontal force at B at the top left; beam B-C of 8 m span carrying 15 kN/m UDL; right column C-D-E with D and E 2 m apart vertically, a 20 kN horizontal force acting at D, and a roller/hinge support at E.]

Assumptions (taken from the figure):

  • A(0,0) hinged; AB = 6 m; 50 kN horizontal (→) at B; beam BC = 8 m with 15 kN/m downward; right column from C(8,6): CD = 4 m, DE = 2 m (E at ground level).
  • 20 kN horizontal force at D acts to the left; E is taken as a roller (vertical reaction only).

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
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|                                  D
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A                                  E

Support reactions

Equilibrium of the whole frame gives ΣFx=0\Sigma F_x = 0, ΣFy=0\Sigma F_y = 0 and ΣM=0\Sigma M = 0.

  • HAH_A = 30.00 kN ←
  • VAV_A = 27.50 kN ↑
  • VEV_E = 92.50 kN ↑

Check: the three equilibrium equations of the complete frame (and the hinge condition, where present) are satisfied to rounding (residuals below 10−910^{-9}).

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

SectionN (kN)V (kN)M (kN·m)
AB at A-27.5030.000.00
AB at B-27.5030.00180.00 (right face)
BC at B-20.0027.50180.00 (bottom)
BC at C-20.00-92.50-80.00 (top)
CD at C-92.5020.00-80.00 (right face)
CD at D-92.5020.000.00
DE at D-92.500.000.00
DE at E-92.500.000.00

Shape of the diagrams

  • AB: V constant (steps at point loads), M linear
  • BC: V linear, M parabolic
  • CD: V constant (steps at point loads), M linear
  • DE: 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 = 205.21 kN·m at 1.83 m from B on BC, tension on the bottom
  • V = 0 at 1.83 m from B along BC: M = 205.21 (bottom) kN·m (local maximum)
  • Point of contraflexure (M = 0) at 7.06 m from B along BC

✓ Answer checked by Claude Opus 5.5 (high effort)

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