Beam Deflection (Point Load) Calculator
This calculator determines the maximum mid-span deflection of a simply supported beam subjected to a single concentrated (point) load at the centre of the span. The governing formula is delta = PL^3 / (48EI), derived from integration of the moment-curvature relationship for a prismatic elastic beam. Deflection control is a serviceability requirement in structural design: ASCE 7 and the International Building Code set limits such as L/360 for live loads on floors. Enter the load P, span length L, modulus of elasticity E, and second moment of area I in any consistent set of units to get the deflection.
Beam deflection formula (central point load)
delta = P × L³ / (48 × E × I)
Where: P = point load at mid-span, L = span length, E = modulus of elasticity, I = moment of inertia about the bending axis. The result is in the same length unit as L, provided P, E, and I are consistent.
Deflection limits and design practice
- AISC recommends checking live-load deflection against L/360 for floor beams and L/240 for roof beams (non-plastered).
- Total load deflection is often limited to L/240 for floors and L/180 for roofs per ASCE 7 commentary.
- The ratio L/delta computed above can be compared directly to these limits. If L/delta is less than 360, the beam does not meet the L/360 serviceability criterion.
- Deflection is highly sensitive to span: doubling L increases deflection by a factor of 8 (cubic relationship).
Frequently asked questions
What is the formula for mid-span deflection under a central point load?
For a simply supported beam with a point load P at mid-span, the maximum deflection at the centre is delta = PL^3 / (48 E I), where L is the span, E is Young's modulus, and I is the second moment of area.
What units should I use?
Use consistent units throughout. If P is in kips, L in inches, E in ksi, and I in in^4, the result is in inches. If P is in kN, L in mm, E in MPa (N/mm^2), and I in mm^4, the result is in mm.
Does this formula apply when the load is not at mid-span?
No. The formula delta = PL^3/(48EI) applies only when the point load is exactly at the centre of the span. For an off-centre load at position 'a' from one end, a different formula applies involving both 'a' and the span length.
What deflection limit should I check against?
AISC and IBC commonly limit live-load deflection to L/360 for floors and L/240 for roofs. Total load deflection limits are often L/240 or L/180 depending on member type and code requirements.
How does the moment of inertia affect deflection?
Deflection is inversely proportional to the moment of inertia I. Doubling I halves deflection. Selecting a deeper section with greater I is usually the most effective way to control deflection.
Official sources
- American Institute of Steel Construction (AISC): AISC Steel Construction Manual.
- American Society of Civil Engineers: ASCE 7 Minimum Design Loads.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.