Beam Deflection Formulas — Simply Supported & Cantilever with Common Load Cases

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