Beam Deflection Calculator (Simply Supported)
Find the maximum deflection of a simply supported beam under a central point load or a uniformly distributed load.
🏗️ What is Beam Deflection?
Beam deflection is the amount a structural beam bends, or displaces vertically, under an applied load. For a simply supported beam, one end resting on a pin support and the other on a roller support, both allowing free rotation, deflection is greatest at the midpoint of the span for the two most common loading cases: a concentrated point load at the center, and a uniformly distributed load spread across the full length.
Structural and civil engineers check deflection constantly. A floor beam that deflects too much under furniture and foot traffic feels bouncy and can crack plaster ceilings below it. A roof purlin that sags excessively can pond rainwater instead of shedding it. Machine frames and crane girders need controlled deflection so precision equipment mounted on them stays properly aligned. In every case, the same underlying physics applies, just with different loads, spans, and material properties.
A common misconception is that beam strength (whether it will break) and beam stiffness (how much it bends) are the same design check. They are not. A beam can easily satisfy a strength check against yielding or fracture while still deflecting far more than is comfortable or serviceable, which is why deflection limits like L/360 exist as a separate, additional design requirement on top of strength calculations.
This calculator computes the maximum deflection of a simply supported beam under either a central point load or a uniformly distributed load, using the standard Euler-Bernoulli beam formulas, and shows the deflected shape along the full span as an interactive chart.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Steel Beam Under a Central Point Load
A 4 m steel beam (E = 200 GPa, I = 80,000,000 mm⁴) carrying a 20 kN point load at midspan
Example 2 — Same Steel Beam Under a Uniformly Distributed Load
The same 4 m steel beam carrying a 15 kN/m distributed load across the full span
Example 3 — Timber Beam Under a Central Point Load
A 3 m timber beam (E = 11 GPa, I = 60,000,000 mm⁴) carrying a 4 kN point load at midspan
❓ Frequently Asked Questions
🔗 Related Calculators
What is the formula for beam deflection under a central point load?
For a simply supported beam with a point load P at midspan, maximum deflection is delta_max = PL^3/(48EI), occurring at the center of the span (x = L/2). P is the load, L the span length, E the modulus of elasticity, and I the second moment of area.
What is the formula for beam deflection under a uniformly distributed load?
For a simply supported beam under a uniformly distributed load w (force per unit length), maximum deflection is delta_max = 5wL^4/(384EI), also occurring at midspan. This deflection is smaller than an equivalent total point load because the load is spread across the span.
What units should I use for E and I in this calculator?
Enter the modulus of elasticity E in gigapascals (GPa) and the second moment of area I in millimeters to the fourth power (mm^4), the standard units for structural steel and timber sections. The calculator converts internally to SI base units (Pa and m^4) before computing deflection in millimeters.
What is a typical deflection limit for beams?
A common serviceability limit is span/360 (L/360), meaning the maximum allowable deflection under live load is the span length divided by 360. For a 4 m span that is about 11.1 mm. Different codes and applications (roof beams, floor beams supporting plaster) use stricter limits such as L/240 or L/480.
Where does maximum deflection occur on a simply supported beam?
For both a central point load and a uniformly distributed load on a simply supported beam, maximum deflection occurs at the exact midspan (x = L/2), by symmetry of the loading and support conditions.
How does span length affect beam deflection?
Deflection is extremely sensitive to span length. Under a point load, deflection scales with L^3 (doubling the span increases deflection 8x). Under a uniformly distributed load, deflection scales with L^4 (doubling the span increases deflection 16x), assuming all other properties stay fixed.
What is the difference between a point load and a uniformly distributed load?
A point load (P) is a single concentrated force applied at one location, typically the midspan in this calculator. A uniformly distributed load (w) is spread evenly across the entire beam length in force-per-unit-length units (kN/m), such as the self-weight of a floor slab or a snow load on a roof.
Does increasing the modulus of elasticity always reduce deflection?
Yes, deflection is inversely proportional to E, so a stiffer material (higher E) under the same load and geometry deflects less. Steel (E is about 200 GPa) deflects roughly 18 times less than an equivalent timber beam (E is about 11 GPa) for identical span and section.
What assumptions does this beam deflection formula make?
This calculator uses classical Euler-Bernoulli beam theory, which assumes linear-elastic material behavior, small deflections relative to the span, a prismatic (constant cross-section) beam, and simple support conditions that allow free rotation at both ends with no moment restraint.
How is the second moment of area (I) used in this formula?
The second moment of area I measures how a cross-section's material is distributed relative to its bending axis. A larger I means greater resistance to bending, and since I appears in the denominator of the deflection formula, doubling I halves the deflection for the same load and span.
Can this calculator be used for cantilever or fixed-end beams?
No, this calculator specifically applies the simply supported beam formulas (pin support at one end, roller at the other, free rotation at both). Cantilever and fixed-end beams use different deflection formulas with different boundary conditions and would give incorrect results here.