Simply Supported Beam Deflection Calculator uses δ=PL³/48EI or δ=5wL⁴/384EI to calculate mid-span deflection, max moment, shear force, EI stiffness, and deflection ratio for point load or UDL.
ft
lbf
Mpsi
in4
Maximum Deflection (δ)
0.124 in
Calculated mid-span deflection for a simply supported beam.
Max Bending Moment (M)
2,500 lbf-ft
Location
Mid-span
Load Type
Point Load
The maximum internal bending moment causing stress in the beam section.
Max Shear Force (V)
500 lbf
Location
Supports
Load Type
Point Load
The maximum internal vertical shear force at the support reactions.
Beam Stiffness (EI)
2.90e+8 lbf-in2
Modulus (E)
29 Mpsi
Inertia (I)
10 in4
The flexural rigidity of the beam, representing its resistance to bending.
Deflection Ratio
L / 967
Total Length
120 in
Max Deflection
0.124 in
Standard engineering ratio used to verify if deflection meets code serviceability limits.
Engineering Note
Results assume linear elastic behavior of a simply supported beam with ideal boundary conditions. Self-weight of the beam is excluded unless incorporated into the distributed load.
What This Calculator Solves
This calculator finds the maximum deflection, bending moment, shear force, beam stiffness, and deflection ratio for a simply supported beam under two specific load cases: a center point load or a full-span uniformly distributed load (UDL).
Both supports are treated as simple supports — one pin, one roller — carrying equal reactions and providing no moment resistance at the ends.
Formulas Used
Point Load at Center
Load Case 1
δ = PL³ / 48EI
M = PL / 4
V = P / 2
M = PL / 4
V = P / 2
Uniformly Distributed Load (Full Span)
Load Case 2
δ = 5wL⁴ / 384EI
M = wL² / 8
V = wL / 2
M = wL² / 8
V = wL / 2
These are classical closed-form solutions from structural mechanics for a simply supported beam under symmetric loading. Source: Roark's Formulas for Stress and Strain, Table 8.
Input Reference
| Symbol | Input | Meaning |
|---|---|---|
L |
Beam Length | Distance between the two simple supports, measured center-to-center. |
P |
Point Load | A single concentrated force applied at the exact mid-span of the beam. |
w |
Distributed Load | A load spread uniformly along the full length of the beam, expressed per unit length. |
E |
Modulus of Elasticity | The material stiffness property. Steel is typically 29 Mpsi (200 GPa); concrete is around 4 Mpsi (27 GPa). |
I |
Moment of Inertia | The cross-section's geometric resistance to bending about its neutral axis. Larger sections have higher I. |
Output Reference
| Output | Meaning |
|---|---|
| Maximum Deflection (δ) | The peak vertical displacement at mid-span. For both load cases here, maximum deflection occurs at the center. |
| Max Bending Moment (M) | The highest internal bending demand in the beam, occurring at mid-span. Used to check bending stress in section design. |
| Max Shear Force (V) | The maximum vertical shear, which occurs at the support reactions for both load cases. |
| Beam Stiffness (EI) | The product of modulus and moment of inertia, representing overall flexural rigidity. Higher EI produces less deflection. |
| Deflection Ratio | Span length divided by maximum deflection, expressed as L/n. A larger denominator indicates stiffer behavior. |
Units and Conversions
The calculator converts all user inputs to a consistent base unit internally before computing results.
US Customary
| Quantity | Input → Internal |
|---|---|
| Length | ft → in |
| Point Load | lbf |
| Dist. Load | lbf/ft → lbf/in |
| Modulus | Mpsi → psi |
| Inertia | in⁴ |
| Deflection | in |
| Moment | lbf-ft |
Metric
| Quantity | Input → Internal |
|---|---|
| Length | m → mm |
| Point Load | N |
| Dist. Load | N/m → N/mm |
| Modulus | GPa → N/mm² |
| Inertia | cm⁴ → mm⁴ |
| Deflection | mm |
| Moment | N-m |
Worked Example
Inputs
| Unit System | Metric |
| Load Configuration | Point Load (Center) |
| Beam Length (L) | 10 m |
| Point Load (P) | 1,000 N |
| Modulus of Elasticity (E) | 29 GPa |
| Moment of Inertia (I) | 10 cm⁴ |
Internal Conversion
L = 10,000 mm | E = 29,000 N/mm² | I = 100,000 mm⁴
Calculation
δ = PL³ / 48EI
δ = (1,000 × 10,000³) / (48 × 29,000 × 100,000)
δ = 1.0 × 10¹² / 139,200,000 ≈ 7,183.908 mm
δ = (1,000 × 10,000³) / (48 × 29,000 × 100,000)
δ = 1.0 × 10¹² / 139,200,000 ≈ 7,183.908 mm
Results
| Maximum Deflection (δ) | 7,183.908 mm |
| Max Bending Moment (M) | 2,500.0 N-m |
| Max Shear Force (V) | 500.0 N |
| Beam Stiffness (EI) | 2.90e+9 N-mm² |
| Deflection Ratio | L / 1 |
Note: This very large deflection occurs because 10 cm⁴ is an extremely small moment of inertia for a 10 m span. The calculator is showing the mathematically correct result — it is not indicating the beam is usable or structurally adequate under these inputs.
Assumptions and Limits
- Simply supported beam only — one pin support, one roller support.
- Point load applied at the exact center, or UDL applied uniformly over the full span.
- Modulus of elasticity (E) and moment of inertia (I) are constant along the full beam length.
- Linear elastic material behavior is assumed throughout.
- Self-weight of the beam is not included unless added as part of the distributed load input.
- Results are for preliminary calculation only. They are not a substitute for full structural design or engineering review.
Cantilever beams, fixed-end beams, overhanging spans, partial loads, and non-symmetric loading are outside the scope of this tool.
References
- Young, W. C., & Budynas, R. G. — Roark's Formulas for Stress and Strain, 8th Edition, Table 8: Simply Supported Beams. McGraw-Hill. The source for the point load and UDL deflection, moment, and shear formulas used in this calculator.
- NIST — SI Units Reference (NIST). Used for metric unit definitions and conversions (N, mm, GPa, N/mm²).
- AISC — Steel Construction Manual. Referenced for standard steel section properties (E = 29 Mpsi / 200 GPa, and tabulated I values for W-shapes and other sections). Not a source for the deflection formulas themselves.