What is beam deflection and beam bending?
In building construction, we usually use framing structures that are held in place by the foundations in the ground. These framing structures are like the skeletons of buildings, houses, and even bridges. In a frame, we call the vertical framing columns and the horizontal ones beams. Beams are the extended members of a structure that carry the loads brought by the horizontal slabs — solid concrete floors, wooden floor joist systems, and roofs.
When beams carry loads too heavy for them, they start to bend. We call the amount of beam bending beam deflection. Beam deflection is the vertical displacement of a point along the centroid of a beam. We can also consider the beam's surface as our reference point as long as there are no changes in the beam's height or depth during bending.
How to calculate the maximum beam deflection
Our beam deflection calculator uses standard engineering formulas to quickly determine the maximum deflection a specific beam will experience under the load it carries. These formulas apply to simple load configurations and their combinations.
Simply-supported beam deflection formulas
| Load Configuration | Maximum Deflection (δmax) |
|---|---|
| Point load P at center of span L | δmax = PL³ / (48EI) |
| Point load P at distance a from left support | δmax = Pa(L²−a²)^(3/2) / (9√3·L·E·I) |
| Uniform distributed load w over entire span | δmax = 5wL⁴ / (384EI) |
Cantilever beam deflection formulas
| Load Configuration | Maximum Deflection (δmax) |
|---|---|
| Point load P at the free end | δmax = PL³ / (3EI) |
| Uniform distributed load w over entire length | δmax = wL⁴ / (8EI) |
Method of superposition
To calculate the maximum deflection of a beam with a combination of loads, engineers use the method of superposition. This method states that the total deflection can be approximated by adding together all the deflections caused by each individual load configuration. For more complex loading scenarios, the double integration method or finite element analysis would be required.
Flexural rigidity of the beam (EI)
All deflection formulas include the term EI, known as the flexural rigidity of the beam:
- E — Modulus of Elasticity (Young's modulus) of the beam material, measured in psi, ksi, MPa or GPa
- I — Second Moment of Area (Moment of Inertia) of the cross-section, measured in in⁴ or mm⁴/cm⁴
A higher EI value means the beam is stiffer and will deflect less under the same load. Steel has a much higher E than wood, and a wide-flange I-beam has a higher I than a rectangular section of the same area.
Understanding the inputs
Beam length (L)
For a simply-supported beam, L is the distance between the two support points. For a cantilever, L is the length of the beam from the fixed wall to the free end. Supported in Imperial (ft, in) and Metric (m, mm).
Load (P or w)
A point load P is a concentrated force acting at a single location along the beam (lbf, kip, N, kN). A uniform distributed load w is a load that is spread evenly along the entire length of the beam (lbf/ft, lbf/in, kN/m, N/m).
Common modulus of elasticity values
| Material | E (Imperial) | E (Metric) |
|---|---|---|
| Steel | 29,000 ksi | 200 GPa |
| Aluminum | 10,000 ksi | 69 GPa |
| Wood (Douglas Fir) | 1,800 ksi | 12.4 GPa |
| Concrete | 3,600 ksi | 24.8 GPa |
Serviceability check (L/δ ratio)
Engineers use the span-to-deflection ratio L/δ to check serviceability. Common code limits include:
- L/360 — live load deflection limit for floors (most codes)
- L/240 — total load deflection limit for floors
- L/180 — roof deflection limit (non-sensitive)
Our calculator displays the computed L/δ ratio so you can quickly compare it against your applicable design code.
Sample beam deflection calculation
Problem: A simply-supported steel W8×31 beam spans 15 ft and carries a uniform distributed load of 2,000 lbf/ft. Find the maximum deflection.
Given:
- L = 15 ft = 180 in
- w = 2,000 lbf/ft = 166.67 lbf/in
- E = 29,000,000 psi (steel)
- I = 110 in⁴ (W8×31)
Formula: δmax = 5wL⁴ / (384EI)
Calculation:
δmax = 5 × 166.67 × (180)⁴ / (384 × 29,000,000 × 110)
δmax = 5 × 166.67 × 1,049,760,000 / 1,226,880,000,000
δmax ≈ 0.713 in
Serviceability: L/δ = 180 / 0.713 ≈ L/252, which satisfies the typical L/240 total load limit.
FAQs
What units should I use?
The calculator supports both Imperial (lbf, kip, ft, in, ksi, psi, in⁴) and Metric (N, kN, m, mm, GPa, MPa, cm⁴, mm⁴) systems. Select your preferred system and all unit options update automatically.
What is the moment of inertia of a beam?
The second moment of area (I) describes how a cross-section's area is distributed relative to its neutral axis. Standard W-shapes (wide-flange beams) have high I values for their weight, making them efficient structural members. Common lumber sections have much lower I values.
Can I use this for composite beams or non-prismatic beams?
No. These formulas assume a prismatic beam (uniform cross-section) made of a linear elastic material. For composite, tapered, or curved beams, specialized structural analysis software is required.
What is the difference between a simply-supported and a cantilever beam?
A simply-supported beam is supported at both ends (by pins or rollers) and free to rotate at supports. A cantilever beam is fixed at one end (built-in or welded) and completely free at the other. Cantilever beams experience much larger deflections for the same load and span because the fixed end resists rotation.