Beam Deflection Calculator: Shear & Moment

Beam deflection, shear and moment

UnitsSIUS customary

SupportSimply supportedCantilever

LoadUniform load (full span)Point load

Span L (m)Between supports, or fixed end to free end.

Modulus of elasticity E (GPa)Steel is about 200 GPa (29 000 ksi).

Moment of inertia I (mm⁴)About the bending axis. Scientific notation works, e.g. 8e6.

Uniform load w (kN/m)Acting downward over the whole span.

Maximum deflection

20.83 mm at x = 2 m

Maximum bending moment

20 kN·m at x = 2 m

Maximum shear

20 kN

Left reaction

20 kN

Right reaction

20 kN

x is measured from the left end. Sagging moment is positive; deflection is shown downward.

Loading and supports

Shear force V (kN)

Bending moment M (kN·m)

Deflected shape (mm)

Intermediate steps
Step	Formula	Value
Flexural rigidity	EI = 200 GPa × 8.000e+6 mm⁴	1600 kN·m²
Left reaction	R₁ = w·L / 2	20 kN
Right reaction	R₂ = w·L / 2	20 kN
Maximum shear	\|V\|max	20 kN
Maximum moment	M = w·L² / 8 (midspan)	20 kN·m at x = 2 m
Maximum deflection	δ = 5w·L⁴ / (384EI)	20.83 mm at x = 2 m

Inputs

Support — Simply supported (pin at the left, roller at the right) or cantilever (fixed at the left, free at the right).
Load — A single point load P at distance a from the left end, or a uniformly distributed load w over the whole span.
Span L (m or in) — Distance between supports, or from the fixed end to the free end.
Modulus of elasticity E (GPa or ksi) — From the material data sheet for your grade.
Moment of inertia I (mm⁴ or in⁴) — About the bending axis. Use the centroid and moment of inertia calculator for built-up sections.
P or w (kN, kN/m or lbf, lbf/in) — Load magnitude, acting downward.

Key formulas

Formulas used on this page, with the textbook each is cited from
Formula	Expression	Where	Cited from
Simple beam, UDL: maximum deflection	`δmax = 5·w·L⁴ / (384·E·I) at midspan`	w = load per unit length	Gere & Goodno, Mechanics of Materials
Simple beam, UDL: maximum moment	`Mmax = w·L² / 8`	at midspan; reactions R = w·L / 2	AISC Steel Construction Manual, Beam Diagrams and Formulas
Simple beam, point load: reactions and moment	`R₁ = P·b / L, R₂ = P·a / L, Mmax = P·a·b / L`	a = load position from the left support, b = L − a	Hibbeler, Mechanics of Materials
Simple beam, point load: maximum deflection	`δmax = P·c·(L² − c²)^(3/2) / (9·√3·E·I·L)`	c = the smaller of a and b; occurs in the longer segment	Roark’s Formulas for Stress and Strain
Cantilever, point load: tip deflection	`δtip = P·a²·(3L − a) / (6·E·I)`	a = load position from the fixed end; a = L gives P·L³ / (3·E·I)	Gere & Goodno, Mechanics of Materials
Cantilever, UDL: tip deflection and fixed-end moment	`δtip = w·L⁴ / (8·E·I), M₀ = −w·L² / 2`	moment is hogging (negative) at the support	Hibbeler, Mechanics of Materials

Assumptions

Euler–Bernoulli beam theory: plane sections stay plane, deflections and slopes are small, and shear deformation is ignored. For short, deep beams shear deformation adds noticeably to the deflection, so the result is low.
Linear-elastic, homogeneous material with constant E and a prismatic section with constant I along the span.
Loads act downward through the shear centre, so there is no twisting. The beam’s own weight is not included unless you add it to w.
Supports are ideal: pins and rollers allow rotation; a cantilever’s fixed end allows none.
Sign convention: sagging moment is positive, shear is the sum of forces left of the section (up positive), and deflection is reported downward.

Worked example

Example: A simply supported steel beam under a uniform load

Given

Simply supported span L = 4 m with a uniformly distributed load w = 10 kN/m over the full span.
Steel section with E = 200 GPa (a typical value for carbon steel) and I = 8.0 × 10⁶ mm⁴ about the bending axis.

Find

The support reactions.
The maximum bending moment.
The maximum deflection.

Solution

Convert to consistent units: I = 8.0 × 10⁶ mm⁴ = 8.0 × 10⁻⁶ m⁴, so the flexural rigidity isEI = 200 × 10⁹ Pa × 8.0 × 10⁻⁶ m⁴ = 1.6 × 10⁶ N·m²
The load is symmetric, so each support carries half of it.R₁ = R₂ = w·L / 2 = 10 kN/m × 4 m / 2 = 20 kN
Shear falls linearly from +20 kN to −20 kN and crosses zero at midspan, where the moment peaks.Mmax = w·L² / 8 = 10 × 4² / 8 = 20 kN·m
Maximum deflection is also at midspan.δmax = 5·w·L⁴ / (384·EI) = 5 × 10 000 × 4⁴ / (384 × 1.6 × 10⁶) = 0.02083 m
As a ratio of span, this is L/192.4000 mm / 20.83 mm = 192

Answer: R₁ = R₂ = 20 kN, Mmax = 20 kN·m at midspan and δmax = 20.83 mm at midspan (L/192).

Common mistakes

Mixing units in EI. Convert E and I to the same base, for example Pa and m⁴, before multiplying. I in mm⁴ is 10¹² times its value in m⁴.
Using the weak-axis I. Use I about the axis the beam bends around. For an I-beam loaded on its flange, that is the strong axis (Ix).
Assuming the point-load maximum is under the load. Maximum moment is under the load, but maximum deflection is in the longer segment, nearer midspan.
Treating a bolted or welded end as fully fixed. Real connections rotate a little. A cantilever result is a lower bound on deflection unless the support is truly rigid.
Leaving out self-weight. For long spans add the beam’s weight per length to w. The material weight calculator gives mass per length for common shapes.

Frequently asked questions

Which beam formulas does this calculator use?

The closed-form Euler–Bernoulli results for a prismatic beam: simply supported or cantilever, with a single point load or a full-span uniform load. They are the same formulas tabulated in mechanics of materials textbooks and the AISC beam diagrams.

How do I combine a point load and a UDL?

For linear-elastic beams, deflections and moments add. Run each load case separately and add the results at the same x. The maximum of the sum can fall between the two individual maxima, so check a few points.

What deflection is acceptable?

It depends on the application and the governing code or customer specification. Building codes and machine-design guides express limits as a fraction of span; check the one that applies to your job rather than using a general rule.

Why is my cantilever’s moment negative?

The calculator uses the sagging-positive convention. A cantilever bends with tension on the top fibre, which is hogging, so its moment is negative and largest at the fixed end.

Does it give bending stress?

Not directly. Divide the maximum moment by the section modulus S = I / c to get the peak bending stress. The centroid and moment of inertia calculator gives S for built-up sections.

More engineering calculators

Mechanics of materials lessons · All engineering calculators

Sources

J. M. Gere, B. J. Goodno, Mechanics of Materials, Cengage
R. C. Hibbeler, Mechanics of Materials, Pearson
W. C. Young, R. G. Budynas, A. M. Sadegh, Roark’s Formulas for Stress and Strain, McGraw Hill
Steel Construction Manual, Part 3: Beam Diagrams and Formulas, American Institute of Steel Construction (AISC)
NIST Special Publication 811, Guide for the Use of the International System of Units (conversion factors), NIST

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Beam Deflection Calculator

Beam deflection, shear and moment

Inputs

Key formulas

Assumptions

Worked example

Example: A simply supported steel beam under a uniform load

Given

Find

Solution

Common mistakes

Frequently asked questions

More engineering calculators

See also

Sources

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