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.
Deflection Is the Check Most People Skip — Until the Floor Bounces
A beam can be perfectly strong — adequate bending capacity, no risk of yielding — and still fail in service because it deflects too much. Cracked drywall, sticking doors, a floor that feels springy underfoot: these are deflection problems, not strength problems. They’re governed by stiffness, not stress. This calculator solves for that deflection, along with the bending moment, shear force, and the L/n ratio that tells you at a glance whether your beam meets serviceability limits.
Two load configurations are supported: a single point load at mid-span, and a uniformly distributed load (UDL) spread across the full beam length. Both assume a simply supported beam — pinned at one end, roller at the other — with no intermediate supports.
The Formulas Behind the Output
Both deflection formulas come from classical Euler-Bernoulli beam theory for a simply supported span. The difference between them is significant.
For a center point load (P):
δ = PL³ / 48EI
For a uniformly distributed load (w):
δ = 5wL⁴ / 384EI
In both cases, deflection is proportional to the cube (point load) or fourth power (UDL) of the span length. That nonlinearity is why doubling a beam’s span doesn’t double its deflection — it multiplies it by 8 or 16 respectively. It’s also why span length is usually the most powerful lever for controlling sag, more so than section choice.
The denominator in both formulas is EI — flexural rigidity. E is the modulus of elasticity (material stiffness), and I is the second moment of area (geometric stiffness from cross-section shape and size). Increasing either one reduces deflection proportionally. A steel beam with E = 29 Mpsi is roughly 15 times stiffer in material terms than typical structural lumber, which is why steel sections can span much further for equivalent deflection.
The calculator performs all arithmetic in consistent base units internally regardless of what you enter. US inputs (feet, Mpsi, in⁴) are converted to inches before calculation. Metric inputs (meters, GPa, cm⁴) are converted to millimeters. Bending moment and shear force are derived from the same load and span used for deflection — they aren’t independent inputs.
For the point load case: maximum moment = PL/4, maximum shear = P/2, both at mid-span and supports respectively. For UDL: maximum moment = wL²/8, maximum shear = wL/2. These appear in the output cards alongside the deflection.
What the L/n Ratio Actually Tells You
The deflection ratio card — displayed as “L / 967” or similar — is the standard engineering shorthand for serviceability. Divide the span by the deflection and you get a dimensionless ratio that can be compared directly against code limits without worrying about units.
Common benchmark values: L/360 is the IBC limit for floor beams supporting brittle finishes (tile, plaster) under live load. L/240 applies to the same beams under total load. Roof members often use L/180. Headers over openings are sometimes checked to L/600 when supporting masonry. The higher the denominator, the stiffer the requirement.
A result of L/967 means the beam is deflecting less than 1/967th of its span — well within any typical limit. A result of L/180 on a floor joist under tile is a problem. The calculator shows you the ratio; the limit you check it against depends on the application and your jurisdiction’s adopted code.
Worked Example: Sizing a Steel Beam for a Residential Floor Opening
A 10-foot flush beam carries a tributary load of 1,000 lbf at mid-span from a post above (modeled as a point load). The beam is a steel section with E = 29 Mpsi (standard structural steel). The section being considered has I = 10 in⁴.
Entering those values — 10 ft span, 1,000 lbf point load, 29 Mpsi, 10 in⁴ — the calculator returns a maximum deflection of 0.124 inches. The deflection ratio works out to L/967. For a floor beam under live load with hardwood flooring above, the L/360 limit allows up to 10×12/360 = 0.333 inches. The beam passes comfortably.
The bending moment output of 2,500 lbf-ft is the number to carry into a separate stress check — divide by the section modulus S to get bending stress, then compare to the allowable. The shear of 500 lbf at the supports is relatively low for steel and won’t govern here. But for a wood beam with the same geometry, that moment value would be the first thing to check against Fb × S.
Switching to UDL mode with the same beam — 1,000 lbf/ft over 10 feet — increases deflection to 0.518 inches and the ratio drops to L/232. That’s fine for a roof but marginal for a tile floor. The difference illustrates why load distribution matters: the same total load (10,000 lbf spread vs. 1,000 lbf point) produces very different deflection profiles.
Where This Calculator Stops Being Accurate
The tool assumes linear elastic behavior throughout. Real beams start to behave nonlinearly as they approach yielding, and this formula gives no warning when that’s happening — the deflection output will simply be wrong on the high side if stresses exceed the elastic range. Always run a separate stress check.
More commonly overlooked: the simply supported boundary condition. A beam framing into a concrete wall or welded to a column isn’t simply supported — it has some degree of rotational fixity at the ends, which reduces actual deflection compared to the calculated value. A beam sitting on top of posts in a platform-frame floor, on the other hand, is close to simply supported and the formula applies well.
Self-weight is excluded. For short, heavily loaded beams this is usually negligible. For long, lightly loaded spans — a 20-foot glulam carrying only a modest roof load, for example — the beam’s own weight can contribute meaningfully to mid-span deflection. To include it, add the self-weight as a distributed load alongside any applied UDL, or switch to UDL mode and combine them before entering the value.
The calculator also models only the maximum deflection location. For a point load at center that’s always mid-span, and for UDL it’s also mid-span. Off-center point loads, cantilevers, continuous spans, and partial UDLs require different formulas entirely — this tool isn’t the right instrument for those configurations.
FAQs
When I switch between US and metric, do the input values convert automatically?
No. Unlike some unit-switching calculators, this one updates the unit labels and recalculates with whatever numbers are currently in the fields — it does not convert the numeric values themselves. If you entered 10 ft and switch to metric, the field still shows 10, but now interpreted as 10 meters. You need to manually re-enter the correct metric value (3.05 m in this case). The same applies to load, E, and I fields.
What E value should I use for common materials?
Structural steel is 29 Mpsi (200 GPa) — that’s the default and it’s essentially universal for carbon steel. Aluminum is roughly 10 Mpsi (69 GPa). Structural lumber varies considerably by species and grade, typically 1.2–1.9 Mpsi; Douglas Fir-Larch #2 is around 1.6 Mpsi. Glulam and LVL range from 1.5 to 2.0 Mpsi depending on grade. For concrete, E depends on compressive strength — a 4,000 psi mix gives roughly 3.6 Mpsi using the ACI formula. Always use the value for your specific material and grade, not a generic approximation.
The load field accepts zero — what does the calculator return?
A zero load is treated as valid (the validation only blocks negative loads). With P or w equal to zero, all outputs return zero: no deflection, no moment, no shear. The deflection ratio becomes mathematically undefined (division by zero) but the calculator handles this gracefully — the ratio card will show the span divided by a near-zero number, which produces an extremely large ratio effectively indicating no meaningful deflection. This is correct behavior for a no-load case.
For UDL mode, is the distributed load entered as load per unit length or total load?
Per unit length. In US mode the field expects lbf/ft; in metric mode it expects N/m. The calculator internally converts this to lbf/in or N/mm before applying the 5wL⁴/384EI formula. If you know only the total load, divide by the beam length before entering. Entering total load directly as if it were a UDL will produce a result that’s proportional to span length — a longer beam will appear to carry more load, which is the opposite of reality.
Why does the bending moment location always show “mid-span” for both load types?
Because both load configurations supported by this calculator — center point load and full-span UDL — produce their maximum bending moment at mid-span on a simply supported beam. This isn’t a simplification; it’s geometrically exact for these two specific cases. An off-center point load would shift the moment peak toward the load, but that scenario is outside the scope of this tool’s formulas.
References
The deflection formulas used (PL³/48EI for center point load; 5wL⁴/384EI for UDL) are standard results from structural mechanics, documented in references including the AISC Steel Construction Manual (Table 3-23) and Roark’s Formulas for Stress and Strain. The L/360 and L/240 serviceability limits referenced above derive from IBC Table 1604.3 (International Building Code), which tabulates allowable deflection ratios by structural member type and load combination. Local amendments may modify these values.