Beam deflection calculator
Calculate support reactions, bending moment diagram, deflection curve and bending stress of straight beams per engineering beam theory (Euler-Bernoulli). Choose support and load case, enter the cross-section parametrically or specify I and W directly – the calculator performs the twin check against yield strength and deflection limit with traffic-light rating, live with every input.
Beam calculator (beam theory)
Simply supported – central point loadModel: straight prismatic beam per Euler-Bernoulli (first-order theory, linear-elastic, static load, bending about the major axis). No stability and no fatigue check; self-weight is not applied automatically. Sizing tool for machine and fixture design, not a structural code check (use Eurocode 3 for that). For rotating shafts use the DIN 743 shaft calculator.
Results
Calculating …
Formulas and fundamentals
The basis is the linearised differential equation of the deflection curve in Euler-Bernoulli theory: the curvature of the beam axis is proportional to the local bending moment, E·I·w″(x) = M(x). Integrating twice with the boundary conditions of the supports (pin: w = 0, fixed end: w = 0 and w′ = 0) yields a closed-form solution for every standard load case. The calculator evaluates these segment polynomials exactly – for the simply supported beam with a central point load M_max = F·L/4 and f_max = F·L³/(48·E·I), for the uniformly distributed load M_max = q·L²/8 and f_max = 5·q·L⁴/(384·E·I), for the cantilever with an end load f_max = F·L³/(3·E·I).
The section properties are computed from the dimensions as composite areas (outer rectangle minus cut-out): for the rectangle I_y = b·h³/12 and W_y = b·h²/6, for the round bar I = π·d⁴/64 and W = π·d³/32, and analogously for tube, rectangular hollow section and doubly symmetric I-section. The parametric I-section calculation without fillet radii is about 4 to 5 percent below the catalogue values of rolled IPE sections and therefore conservative. The role of the two properties matters: the deflection depends on the second moment of area I, the stress on the section modulus W – both are reported separately and can also be entered directly.
The stress check compares the maximum bending stress σ_b = |M|_max/W_y with the allowable stress R_e/S_req; the available safety factor is S = R_e/σ_b. For the beam fixed at both ends under distributed load the fixed-end moment q·L²/12 governs, not the midspan moment q·L²/24. The deflection check compares f_max with f_allow = L/k using a selectable criterion (typically L/300 for general beams, L/500 for sensitive equipment, L/150 for cantilevers). For an off-centre point load the maximum deflection is not under the load but in the longer segment at x = √((L²−b²)/3) – the calculator reports both values.
Worked example
A round bar of 50 mm diameter in S355 steel (R_e = 355 N/mm²) spans 2000 mm as a simply supported beam and carries a central point load of 5000 N. The section properties are A = 1963.5 mm², I = 306,796 mm⁴ and W = 12,272 mm³; both support reactions are 2500 N.
The maximum bending moment at midspan is M_max = F·L/4 = 2.5·10⁶ Nmm. This gives a bending stress of σ_b = M_max/W = 203.7 N/mm² and a safety factor of S = 355/203.7 = 1.74 – the stress check passes with S_req = 1.5. The maximum deflection at midspan is f_max = F·L³/(48·E·I) = 12.93 mm, the slope at the supports 19.4 mrad (1.11°).
Against the criterion L/300 = 6.67 mm, however, the deflection check fails (utilisation 194 percent). The example shows the typical situation in machine design: stiffness governs, not stress – a deeper cross-section (I grows with h³) helps far more than a stronger material.
Frequently asked questions
What is the difference between second moment of area I and section modulus W?
I describes the stiffness of the cross-section and governs deflection (f ~ 1/I); W describes the utilisation of the outer fibre and governs bending stress (σ = M/W). A common mistake is σ = M/I. For a doubly symmetric section, W = 2·I/h with the section depth h.
Why does deflection often govern instead of stress?
Because deflection grows with L³ or L⁴ while stress grows only with L. For longer beams the stiffness check is therefore almost always more critical. A stronger material does not help then, since the elastic modulus of all steels is practically identical – only a larger second moment of area (more depth) effectively reduces deflection.
Where is the maximum deflection for an off-centre point load?
Not under the load but in the longer span segment at x = √((L²−b²)/3) from the support remote from the load. The maximum, however, never lies farther than about 0.077·L from midspan, so the midspan deflection deviates by at most roughly 3 percent. The deflection directly under the load can be considerably smaller; the calculator reports both values.
Which beams does the calculation apply to?
Straight prismatic beams with constant cross-section under static load, linear-elastic with small deformations (first-order theory). Euler-Bernoulli theory neglects shear deformation; as a rule of thumb L/h should be at least 10, below that the calculator warns. If f_max exceeds L/50 the small-angle assumption is violated and a warning appears as well.
How realistic is the ideal fixed end?
Ideal fixed ends are rare: bolted end plates or short clamping lengths behave somewhere between fixed and pinned. The real deflection therefore lies between the two limiting cases – up to a factor of 5 for distributed load (q·L⁴/384 versus 5·q·L⁴/384). When in doubt, calculate both cases and use the less favourable one.
Does the calculator replace a structural code check?
No. It is a sizing tool for machine and fixture design with an elastic stress check and a global safety factor against yield. Civil structures must be verified to Eurocode 3 with partial safety factors, stability checks (lateral-torsional buckling, plate buckling) and regulated load assumptions.