Beam Deflection Calculator

Beam Deflection Calculator

Use this deflection calculator for beam analysis — calculate maximum deflection, bending stress, and support reactions for simply supported and cantilever beams under point loads and uniform distributed loads. Formulas from Machinery’s Handbook, 29th Edition.

Beam deflection is the displacement of a structural member under load, measured perpendicular to its original axis. It depends on load magnitude, span length, support conditions, material stiffness (modulus of elasticity E), and cross-section geometry (moment of inertia I). Engineers calculate deflection to verify that a beam meets serviceability limits — excessive deflection can cause vibration, misalignment, and damage to attached components even when stress is within allowable limits.

Beam Deflection Formulas

Simply Supported Beam — Center Point Load

δmax = WL³ / (48EI)

s = WL / (4Z)

where:
δmax = maximum deflection at midspan [length] — e.g. in, mm
s = bending stress at center [force / length²] — e.g. psi, MPa
(If cross-section is constant, this is the maximum stress.)
W = total concentrated load at center [force] — e.g. lbf, N
L = span length between supports [length] — e.g. in, mm
E = modulus of elasticity [force / length²] — e.g. psi, MPa
I = moment of inertia of cross-section [length⁴] — e.g. in⁴, mm⁴
Z = section modulus = I / c [length³] — e.g. in³, mm³

Machinery’s Handbook 29th Ed., p. 257, Table 1, Case 2

Simply Supported Beam — Uniform Distributed Load

δmax = 5WL³ / (384EI)

s = WL / (8Z)

where:
s = bending stress at center [force / length²] — e.g. psi, MPa
(If cross-section is constant, this is the maximum stress.)
W = total distributed load (= w × L) [force] — e.g. lbf, N

Machinery’s Handbook 29th Ed., p. 257, Table 1, Case 1

Cantilever Beam — End Point Load

δmax = WL³ / (3EI)

s = WL / Z

where:
s = stress at the fixed support [force / length²] — e.g. psi, MPa
(Maximum stress if cross-section is constant.)

Machinery’s Handbook 29th Ed., p. 261, Table 1, Case 11

Cantilever Beam — Uniform Distributed Load

δmax = WL³ / (8EI)

s = WL / (2Z)

where:
s = stress at the fixed support [force / length²] — e.g. psi, MPa
(Maximum stress if cross-section is constant.)

Machinery’s Handbook 29th Ed., p. 261, Table 1, Case 10

How to Use This Calculator

Select the load case that matches your beam setup, then enter the total load, span length, modulus of elasticity, moment of inertia, and section modulus. If you don’t know I and Z, use the cross-section helper to calculate them from basic dimensions (rectangle, circle, tube, or I-beam). The calculator returns the maximum deflection, maximum bending stress, bending moment, and support reactions.

Compare your calculated deflection against the allowable limit for your application. For general structural work, most codes limit deflection to L/360 of the span. For machine bases or precision equipment, tighter limits of L/600 to L/1000 are common. If you need to calculate the weight of a steel beam to determine the distributed self-weight load, use our metal weight calculator.

Worked Example

Problem: A simply supported A36 steel beam spans 48 inches with a 1,000 lbf load at the center. The beam is a 2″ × 4″ solid rectangular bar (2 in wide, 4 in deep). Find the maximum deflection and bending stress.

Given: W = 1,000 lbf, L = 48 in, E = 29,000,000 psi (A36 steel), b = 2 in, h = 4 in

I = bh³/12 = 2 × 4³ / 12 = 10.667 in⁴

Z = bh²/6 = 2 × 4² / 6 = 5.333 in³

δmax = WL³ / (48EI) = 1,000 × 48³ / (48 × 29,000,000 × 10.667) = 0.00747 in

smax = WL / (4Z) = 1,000 × 48 / (4 × 5.333) = 2,250 psi

The deflection ratio is L/δ = 48 / 0.00747 = 6,426, well within the L/360 limit. The bending stress of 2,250 psi is far below A36’s yield strength of 36,000 psi.

Frequently Asked Questions

How do you calculate beam deflection?

Beam deflection depends on the load type, support conditions, span length, modulus of elasticity (E), and moment of inertia (I). For a simply supported beam with a center point load W, the maximum deflection at midspan is δ = WL³ / (48EI). For a uniform distributed load, it is δ = 5WL³ / (384EI). These formulas are from Machinery’s Handbook 29th Ed., p. 257, Table 1 (Cases 1 and 2).

What is the formula for cantilever beam deflection?

For a cantilever beam with a concentrated load W at the free end, the maximum deflection at the tip is δ = WL³ / (3EI). For a cantilever with a uniform distributed load W over its full length, the maximum tip deflection is δ = WL³ / (8EI). Both formulas are from Machinery’s Handbook 29th Ed., p. 261, Table 1 (Cases 10 and 11).

What is the maximum allowable beam deflection?

Common deflection limits depend on the application. For general structural steel beams, AISC and IBC typically limit live-load deflection to L/360 of the span for floors and L/240 for roofs. Machine bases and precision equipment may require tighter limits such as L/600 or L/1000. Always check the applicable building code or machine design specification for your project.

What is the difference between moment of inertia and section modulus?

Moment of inertia (I), also called area moment of inertia or second moment of area, measures a cross-section’s resistance to bending deflection — it appears in deflection formulas. Section modulus (Z = I/c, where c is the distance from the neutral axis to the extreme fiber) measures resistance to bending stress — it appears in stress formulas. A beam with a large I deflects less; a beam with a large Z has lower bending stress. Both are properties of the cross-section geometry, not the material.

How do I find the moment of inertia for a rectangular beam?

For a solid rectangular cross-section, I = bh³/12, where b is the width and h is the height (depth) of the beam. The section modulus is Z = bh²/6. For example, a 2 in wide × 6 in deep bar has I = 2 × 6³ / 12 = 36 in⁴ and Z = 2 × 6² / 6 = 12 in³.

Related Calculators

References

  • Oberg, E. et al. Machinery’s Handbook, 29th Edition, Industrial Press, 2012, Table 1: “Stresses and Deflections in Beams,” pp. 256–267. Deflection and stress formulas for Cases 1, 2, 10, and 11.
  • Oberg, E. et al. Machinery’s Handbook, 29th Edition, pp. 237–240. Properties of cross-sections (moment of inertia, section modulus) for rectangle, circle, hollow circle, and I-beam shapes.
  • Oberg, E. et al. Machinery’s Handbook, 29th Edition, p. 422. Modulus of elasticity for common engineering metals (steel, aluminum, stainless steel, copper).

Data last verified: March 2026

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