Quick Reference: Beam Deflection & Stress
These are the formulas you’ll use most often in mechanical and structural design. Memorize the first two (center-loaded simply supported and end-loaded cantilever) — every other case is a variation.
Simply Supported Beam
Case 1: Center Point Load
| Parameter | Formula |
|---|---|
| Max Deflection (at center) | δ = PL³ / 48EI |
| Max Moment (at center) | M = PL / 4 |
| Reactions (each support) | R = P / 2 |
Case 2: Uniform Distributed Load
| Parameter | Formula |
|---|---|
| Max Deflection (at center) | δ = 5wL⁴ / 384EI |
| Max Moment (at center) | M = wL² / 8 |
| Reactions (each support) | R = wL / 2 |
Case 3: Off-Center Point Load (at distance a from left)
| Parameter | Formula |
|---|---|
| Max Deflection | δ = Pa²b² / 3EIL (at load point, approx) |
| Moment at load | M = Pab / L |
| Left reaction | R₁ = Pb / L |
| Right reaction | R₂ = Pa / L |
where b = L – a
Cantilever Beam (Fixed-Free)
Case 4: End Point Load
| Parameter | Formula |
|---|---|
| Max Deflection (at free end) | δ = PL³ / 3EI |
| Max Moment (at fixed end) | M = PL |
| Reaction (at fixed end) | R = P |
Case 5: Uniform Distributed Load
| Parameter | Formula |
|---|---|
| Max Deflection (at free end) | δ = wL⁴ / 8EI |
| Max Moment (at fixed end) | M = wL² / 2 |
| Reaction (at fixed end) | R = wL |
Fixed-Fixed Beam
Case 6: Center Point Load
| Parameter | Formula |
|---|---|
| Max Deflection (at center) | δ = PL³ / 192EI |
| Max Moment (at ends) | M = PL / 8 |
| Moment at center | M = PL / 8 |
Case 7: Uniform Distributed Load
| Parameter | Formula |
|---|---|
| Max Deflection (at center) | δ = wL⁴ / 384EI |
| Max Moment (at ends) | M = wL² / 12 |
| Moment at center | M = wL² / 24 |
Variable Definitions
| Variable | Description | Common Units |
|---|---|---|
| P | Point load (force) | lbs or N |
| w | Distributed load (force per unit length) | lbs/in or N/m |
| L | Beam span length | in or m |
| E | Young’s Modulus (see chart) | psi or Pa |
| I | Area moment of inertia | in⁴ or m⁴ |
| δ | Maximum deflection | in or m |
| M | Bending moment | in-lbs or N-m |
Comparison: How Support Conditions Affect Stiffness
For the same beam, load, and cross-section:
| Support | Center Point Load Deflection | Relative Stiffness |
|---|---|---|
| Cantilever (end load) | PL³/3EI | 1× (baseline) |
| Simply Supported | PL³/48EI | 16× stiffer |
| Fixed-Fixed | PL³/192EI | 64× stiffer |
Key insight: Fixing both ends of a beam makes it 4× stiffer than simply supported and 64× stiffer than a cantilever of the same length. Support conditions matter enormously.
Related: Young’s Modulus Chart | Tolerance Stack-Up Analysis