Beam deflection calculator

Pick the supports, enter a point load and/or a uniform load, the section and the material, and get the maximum deflection and where it occurs, the end slope, bending moment, bending stress, reactions and a deflection and moment diagram. Inch (lbf) or metric (N, mm). Free, no sign-up, runs in your browser.

Maximum deflection

-At - from the left end; span/-

Maximum bending stress

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Maximum bending moment

-At - from the left end

Results

Left reaction force-
Right reaction force-
Left end moment (fixed end)-
Maximum shear force-
Maximum slope-
Second moment of area I-
Section modulus I/c-
Young's modulus E-

Checks

Deflected shape (exaggerated) and bending moment

How the calculator works

The beam is straight with a uniform cross-section, loaded in one plane, made of a linear elastic material, and deflects only a little compared with its length. Those are the assumptions of the classical Euler-Bernoulli beam equation E·I·y″ = −M(x). The calculator uses the closed-form solutions of that equation for the four support cases and adds the effect of the point load and the uniform load by superposition.

  1. Section. Rectangle: I = b·h3/12. Solid round: I = π·d4/64. Round tube: I = π(d4 − di4)/64 with di = d − 2t. Rectangular tube: I = (b·h3 − bi·hi3)/12. For any other shape (an I-beam or channel from a catalog) pick Custom and enter I and the distance c from the neutral axis to the extreme fibre.
  2. Reactions. Simply supported: RA = P·b/L + w·L/2 with b = L − a. Cantilever: R = P + w·L and a fixed-end moment P·a + w·L2/2. Fixed at both ends: RA = P·b2(3a + b)/L3 + w·L/2 and fixed-end moments P·a·b2/L2 + w·L2/12 at the left end. Propped cantilever: the support reaction at the pinned end is P·a2(3L − a)/(2L3) + 3w·L/8.
  3. Moment and shear. M(x) = RA·x − MA − w·x2/2 − P·(x − a) for x > a. The largest absolute value is searched along the beam, including the load point and the zero-shear point.
  4. Deflection and slope. Standard closed-form deflection curves (for example P·b·x(L2 − b2 − x2)/(6E·I·L) for a simply supported beam, x ≤ a) are evaluated along the beam; the peak is refined by a numerical search. The slope is the derivative of that curve.
  5. Stress. Bending stress σ = Mmax·c/I. The safety factor is the yield strength divided by σ. Stress concentrations at holes, steps and supports are not included, and shear stress is not checked.

Material values are typical: carbon steel E = 200 GPa (29.0 Mpsi), yield 250 MPa (36 ksi) as for a common structural grade; stainless 304 annealed 193 GPa, 215 MPa; aluminum 6061-T6 68.9 GPa, 276 MPa; titanium Ti-6Al-4V 113.8 GPa, 880 MPa; gray cast iron about 100 GPa, which has no usable yield (the safety factor is skipped). Use the certified values for your own material through Custom. Self-weight is not included: add it to the uniform load.

Worked example: steel bar, simply supported, load at mid-span

Span 1000 mm, section 25 mm × 25 mm, so I = 25·253/12 = 32,552 mm4; steel E = 200,000 MPa; a 1000 N load at mid-span. Reactions are 500 N each. The deflection is P·L3/(48·E·I) = 1000·10003/(48·200,000·32,552) = 3.20 mm, which is span/313. The bending moment is P·L/4 = 250,000 N·mm, so σ = 250,000·12.5/32,552 = 96.0 MPa and the safety factor against yield is 250/96.0 = 2.6. The end slope is P·L2/(16·E·I) = 0.0096 rad = 0.55°. Fixing both ends of the same bar cuts the deflection to 0.80 mm (a quarter).

Next step: size the welds that attach it with the fillet weld size calculator.

More free engineering calculators

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Questions

What is the formula for beam deflection?

It depends on the supports and the load. For a simply supported beam with a point load P at mid-span the maximum deflection is δ = P·L³ / (48·E·I). With a uniformly distributed load w (force per length) it is δ = 5·w·L⁴ / (384·E·I). For a cantilever with a load P at the free end it is δ = P·L³ / (3·E·I), and with a uniform load δ = w·L⁴ / (8·E·I). E is Young's modulus and I the second moment of area of the cross-section.

How do you calculate the deflection of a simply supported beam?

Example: a steel beam (E = 200 GPa) 1000 mm long with a 25 mm × 25 mm cross-section has I = 25·25³/12 = 32,552 mm⁴. A 1000 N load at mid-span gives δ = 1000·1000³ / (48·200,000·32,552) = 3.20 mm, which is span/313.

What is an acceptable beam deflection?

There is no single value; it comes from the function of the part. Building codes commonly limit floor beams to about span/360 under live load. Machine parts such as guides, shafts and frames carrying precision components often need span/1000 or less, and gears or bearings may need a slope below a fraction of a degree. Use the limit your application or standard requires.

How does the second moment of area affect deflection?

Deflection is inversely proportional to I. For a rectangular section I = b·h³/12, so doubling the height reduces the deflection to one eighth, while doubling the width halves it. For a round bar I = π·d⁴/64, so a 20 % larger diameter cuts the deflection by half.

Does this calculator include the weight of the beam?

No. Add the self-weight yourself as part of the uniform load w (cross-section area × density × g). It also assumes small deflections, a uniform section, a linear elastic material and bending about the axis you enter; shear deflection is ignored, which is accurate for slender beams with span over about ten times the depth.

Disclaimer

This calculator is an engineering aid based on classical beam theory. It does not replace a design check to the relevant standard (for example AISC, Eurocode 3 or a machine design handbook); you remain responsible for the design, and critical or safety-related designs must be verified independently. Nothing you enter leaves your browser except through a share link you copy yourself.