Calculator
Example Data Table
| Load Case | L | E | I | Load | x | Deflection at x | Maximum Deflection |
|---|---|---|---|---|---|---|---|
| Point Load | 2 m | 200 GPa | 8,500,000 mm^4 | 1 kN | 1 m | 0.490 mm | 1.569 mm |
| Uniform Load | 3 m | 70 GPa | 12,000,000 mm^4 | 0.5 kN/m | 1.5 m | 2.134 mm | 6.027 mm |
| End Moment | 1.5 m | 210 GPa | 6,000,000 mm^4 | 2 kN·m | 0.75 m | 0.446 mm | 1.786 mm |
Formula Used
Point load at free end: δ(x) = P x² (3L − x) / 6EI
Maximum point load deflection: δmax = P L³ / 3EI
Point load tip rotation: θ = P L² / 2EI
Uniform distributed load over full span: δ(x) = w x² (6L² − 4Lx + x²) / 24EI
Maximum distributed load deflection: δmax = w L⁴ / 8EI
Distributed load tip rotation: θ = w L³ / 6EI
End moment at free end: δ(x) = M x² / 2EI
Maximum end moment deflection: δmax = M L² / 2EI
End moment tip rotation: θ = M L / EI
Where: L is beam length, x is the selected position, E is Young’s modulus, I is second moment of area, P is point load, w is distributed load, and M is end moment.
How to Use This Calculator
1. Select the load case that matches your beam condition.
2. Enter beam length, material stiffness, and second moment of area.
3. Add the correct load value and unit.
4. Enter the distance x from the fixed end.
5. Choose the preferred output unit for displacement.
6. Click the calculate button to show results above the form.
7. Download the result set as CSV or PDF if needed.
Cantilever Beam Displacement Guide
Why beam displacement matters
Cantilever beam displacement is a key measure in mechanics, physics, and structural analysis. It shows how far a fixed beam moves when a force or moment acts on it. This helps users estimate serviceability, stiffness, and elastic response. Quick displacement checks reduce design errors and support better planning.
What controls cantilever deflection
Beam length has a strong effect on movement. Longer beams deflect more. Material stiffness also matters. A higher Young’s modulus reduces bending. The second moment of area is equally important. A larger section resists curvature better. Load type changes the deflection pattern too. A point load, distributed load, and end moment each bend the beam differently.
How this calculator helps
This cantilever beam displacement calculator solves common load cases in one place. It estimates displacement at a selected point and also reports maximum deflection at the free end. It adds tip rotation for better interpretation. Unit conversion is included. That makes it practical for classroom work, quick engineering checks, prototype frames, lab fixtures, and machine components.
Why point location is useful
Many beam tools only show maximum deflection. In real analysis, users often need displacement at a specific location. That value can affect sensor placement, clearance checks, optical alignment, or supported equipment limits. The x-position result gives better insight into beam behavior along the span.
Using results correctly
Use consistent units for length, modulus, inertia, and load. Small unit mistakes can create very large errors. This calculator assumes linear elastic behavior, small deflection theory, and standard cantilever boundary conditions. It is best for preliminary assessment. Final design should still consider stress, safety factors, material limits, and real support conditions.
Practical value
Because the page includes formulas, example data, and export options, it supports both learning and reporting. You can calculate, review, and save beam displacement outputs quickly. That makes the tool useful for physics students, analysts, and design teams who need fast and repeatable cantilever deflection estimates.
FAQs
1. What is cantilever beam displacement?
It is the vertical movement of a beam that is fixed at one end and free at the other. The value depends on load, beam length, material stiffness, and cross section geometry.
2. Why does beam length affect deflection so much?
Length has a strong mathematical effect in beam formulas. As span increases, deflection rises rapidly. Even a modest increase in length can cause a much larger movement.
3. What does Young’s modulus do in this calculation?
Young’s modulus measures material stiffness. A higher value means the material resists bending better. That usually reduces displacement for the same beam shape and load.
4. What is the second moment of area?
It describes how the cross section resists bending. A larger second moment of area makes the beam stiffer. It depends on section shape, size, and orientation.
5. Can I use this for steel, aluminum, or other materials?
Yes. Enter the correct Young’s modulus for the material and the proper section inertia. The formulas work for common elastic materials under standard cantilever assumptions.
6. Does this calculator support multiple load cases?
Yes. It handles a point load at the free end, a uniform distributed load across the full span, and an end moment at the free end.
7. Is the result valid for large deflection problems?
No. This tool uses small deflection beam theory. Very large movement, nonlinear material behavior, or unusual supports need a more advanced structural analysis approach.
8. Why export results as CSV or PDF?
Exports help with record keeping, sharing, and reporting. CSV is useful for spreadsheets. PDF is useful for quick documentation and static result snapshots.