Beam Deflection Calculator
Calculate beam deflection, bending stress, shear force, and bending moment for simply supported and cantilever beams. Uses the Euler-Bernoulli beam theory with linear-elastic, small deflection assumptions. Perfect for structural engineers, students, and designers.
📊 Common Material Properties
| Material | Elastic Modulus (E) | Typical Allowable Stress |
|---|---|---|
| Structural Steel (A36) | 200 GPa (29,000 ksi) | 250 MPa (36 ksi) |
| Aluminum (6061-T6) | 69 GPa (10,000 ksi) | 240 MPa (35 ksi) |
| Concrete | 25-35 GPa (3,600-5,000 ksi) | ~10 MPa (1.45 ksi) |
| Wood (Douglas Fir) | 13 GPa (1,900 ksi) | 8-12 MPa (1.2-1.8 ksi) |
| Stainless Steel (304) | 193 GPa (28,000 ksi) | 215 MPa (31 ksi) |
Understanding Beam Deflection
Beam deflection is the displacement of a beam under load. It is a critical consideration in structural design to ensure serviceability (preventing excessive sagging) and safety. This calculator uses the Euler-Bernoulli beam theory, which assumes linear-elastic material behavior, small deflections, and prismatic members.
The key formula for beam deflection depends on the support condition and load type:
Simply Supported, Point Load at Midspan:
Δmax = P × L³ / (48 × E × I)
Simply Supported, Uniform Load:
Δmax = 5 × w × L⁴ / (384 × E × I)
Cantilever, Point Load at Free End:
Δmax = P × L³ / (3 × E × I)
Cantilever, Uniform Load:
Δmax = w × L⁴ / (8 × E × I)
Where: P = point load, w = uniform load per unit length, L = span length,
E = elastic modulus, I = moment of inertia
The calculator also computes bending stress using the flexure formula: σ = M × c / I, where M is the bending moment, c is the distance from the neutral axis to the extreme fiber, and I is the moment of inertia. The section modulus S = I / c is also calculated.
Serviceability Checks
- Deflection Limit: Many design codes limit deflection to L/360 (span divided by 360) for floor beams and L/240 for roof beams.
- Stress Check: The calculated bending stress is compared to the allowable stress to ensure the beam is safe.
- Shear Check: Maximum shear force is also calculated for completeness.
❓ Beam Deflection FAQ
What is beam deflection?
Beam deflection is the displacement of a beam from its original position under the action of external loads. It is measured as the vertical distance the beam moves downward at a given point.
What is the Euler-Bernoulli beam theory?
The Euler-Bernoulli beam theory is a fundamental engineering theory that relates the deflection of a beam to the applied loads, beam geometry, and material properties. It assumes small deflections, linear-elastic behavior, and that plane sections remain plane.
What is the moment of inertia (I) for a rectangular cross-section?
For a rectangular cross-section with width b and height h, the moment of inertia about the neutral axis is I = b × h³ / 12. The distance to the extreme fiber is c = h / 2.
What is the moment of inertia (I) for a circular cross-section?
For a circular cross-section with diameter d, the moment of inertia is I = π × d⁴ / 64. The distance to the extreme fiber is c = d / 2.
What is the section modulus (S)?
The section modulus is a geometric property defined as S = I / c, where c is the distance from the neutral axis to the extreme fiber. It is used in the bending stress formula: σ = M / S.
What is the difference between a simply supported beam and a cantilever?
A simply supported beam rests on supports at both ends and is free to rotate. A cantilever beam is fixed at one end and free at the other, which produces higher deflections and bending moments for the same load.
What is a uniform load?
A uniform load (also called a uniformly distributed load or UDL) is a load that is spread evenly over the entire length of the beam. It is measured in force per unit length (e.g., N/m or lb/ft).
How does the point load position affect deflection?
For a simply supported beam, the maximum deflection occurs at the point of load application when the load is at midspan. As the load moves toward a support, the maximum deflection decreases. This calculator allows you to specify the load position.
What is the typical deflection limit for beams?
Common deflection limits: L/360 for floor beams (where L is the span), L/240 for roof beams, and L/180 for beams supporting plaster ceilings. This calculator shows the deflection ratio to help you check serviceability.
How does the elastic modulus (E) affect deflection?
Deflection is inversely proportional to the elastic modulus. A higher E means a stiffer material, resulting in less deflection. Steel (E = 200 GPa) deflects less than aluminum (E = 69 GPa) for the same geometry and load.
How does the moment of inertia (I) affect deflection?
Deflection is inversely proportional to the moment of inertia. A larger I means a stiffer cross-section, resulting in less deflection. This is why I-beams are used in construction — they have a high I for their weight.
What is bending stress and why is it important?
Bending stress is the internal stress developed in a beam due to bending moments. If the bending stress exceeds the material's allowable stress, the beam may fail. This calculator checks the stress against the allowable value.
What is the difference between stress and deflection?
Stress is the internal force per unit area within the material, measured in Pa or psi. Deflection is the physical displacement of the beam, measured in length units. Both are important in beam design — stress controls strength, deflection controls serviceability.
What assumptions does this calculator make?
This calculator assumes: linear-elastic material behavior, small deflections (less than about 5% of the span), prismatic beams (constant cross-section), and Euler-Bernoulli beam theory (shear deformation is neglected).
What is the superposition principle for beams?
The superposition principle states that for linear-elastic beams, the total deflection at any point is the sum of the deflections caused by each individual load. This calculator handles single load cases; for multiple loads, use superposition.
How do I choose between simply supported and cantilever?
Choose simply supported for beams with supports at both ends that can rotate (e.g., beams resting on walls). Choose cantilever for beams fixed at one end and free at the other (e.g., balconies, flagpoles, or overhanging beams).
What is the maximum shear force in a beam?
The maximum shear force depends on the support condition and load type. For a simply supported beam with a point load at midspan, Vmax = P/2. For a cantilever with a point load at the free end, Vmax = P.
Can I use this calculator for real-world beam design?
This calculator provides a good estimate for preliminary design and educational purposes. Always verify results with a structural engineer and follow applicable building codes (e.g., IBC, Eurocode, or AS/NZS) for final design.