Free-Body Diagram and Support Reactions — Worked Example

1. Problem

A simply supported beam of length L = 3 m carries a uniformly distributed load w = 5 kN/m over its entire span and a point load P = 20 kN at 1 m from the left support. Determine the support reactions and verify equilibrium.

2. Given

3. Required

• Vertical reaction at support A (R_A)
• Vertical reaction at support B (R_B)
• Horizontal reaction at support A (H_A)
• Verification of equilibrium

4. Assumptions

• Static loading (no dynamic effects)
• Rigid body (no deformation)
• All loads act in the vertical plane
• No horizontal loads — H_A = 0
• Supports are ideal (frictionless pin and roller)

5. Governing Equations

Equilibrium equations:
  ΣFx  =  0     (horizontal)
  ΣFy  =  0     (vertical)
  ΣM   =  0     (moment about any point)

Total UDL force:
  W_udl  =  w × L

UDL centroid:  L/2 from either support

6. Calculation

W_udl  =  5 × 3  =  15 kN   (acting at 1.5 m from A)

ΣM_A  =  0:
  R_B × 3  −  20 × 1  −  15 × 1.5  =  0
  3 R_B  =  20 + 22.5  =  42.5
  R_B  =  14.17 kN

ΣFy  =  0:
  R_A + R_B  =  20 + 15  =  35 kN
  R_A  =  35 − 14.17  =  20.83 kN

ΣFx  =  0:
  H_A  =  0   (no horizontal loads)

7. Result

R_A = 20.83 kN (upward). R_B = 14.17 kN (upward). H_A = 0. Total upward = 35 kN = total downward load (20 + 15). Equilibrium verified.

8. Check

• Vertical equilibrium: R_A + R_B = 35 kN = P + W_udl = 20 + 15 ✓
• Moment about B: R_A × 3 − 20 × 2 − 15 × 1.5 = 62.5 − 40 − 22.5 = 0 ✓
• R_A > R_B — expected because the point load is closer to A
• Dimensional check: kN/m × m = kN, kN × m = kN·m — consistent

9. Interpretation

The reactions are physically reasonable: the left support carries more load because the point load is closer to it. The UDL contributes equally to both supports (7.5 kN each), while the point load splits as R_A_point = 20 × 2/3 = 13.33 kN and R_B_point = 20 × 1/3 = 6.67 kN. The maximum bending moment occurs under the point load: M(x=1) = R_A × 1 − w × 1 × 0.5 = 20.83 − 2.5 = 18.33 kN·m.

10. Limitations

• Rigid body assumption — real beams deform, but for reaction calculation this is valid
• No support settlement or flexibility
• No dynamic effects
• The beam is assumed weightless (or self-weight is included in the UDL)

11. Related Resources

• Common Beam Deflection Formulae (Reference Library)
• How to Draw and Use a Free-Body Diagram
• How to Check Reactions and Equilibrium