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Category Engineering Calculators

Loading Condition

Load Type

Force F (N)

Span Length L (mm)

Bending Moment M (N·mm)

Cross-Section

Profile Shape

Width b (mm)

Height h (mm)

Diameter d (mm)

Outer Diameter d_o (mm)

Inner Diameter d_i (mm)

Outer Width B (mm)

Outer Height H (mm)

Inner Width b (mm)

Inner Height h (mm)

Total Depth H (mm)

Flange Width b_f (mm)

Flange Thickness t_f (mm)

Web Thickness t_w (mm)

Moment of Inertia I (mm⁴)

Distance to N.A. y (mm)

Material (Optional)

Yield Strength σ_y (MPa)

Results

Bending Moment M — N·mm

Moment of Inertia I — mm⁴

Distance to N.A. y — mm

Section Modulus S — mm³

Bending Stress σ — MPa

Safety Factor SF —

Cross-Section Diagram

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About

Miscalculating bending stress in a structural member leads to yielding, permanent deformation, or catastrophic failure. The Euler-Bernoulli beam equation σ = M ⋅ yI relates the internal bending moment M, the distance from the neutral axis y, and the second moment of area I to the normal stress at any fiber of the cross-section. This calculator computes I automatically for five standard cross-section profiles (solid rectangle, solid circle, I-beam, hollow rectangle, hollow circle) using exact closed-form formulas. It derives M from common loading conditions on simply-supported and cantilever beams, or accepts a direct user-supplied moment. Results assume linear-elastic, small-deflection behavior and homogeneous, isotropic material. The model breaks down for short, deep beams where shear deformation dominates (span-to-depth ratio below 4), and does not account for lateral-torsional buckling or stress concentrations at notches.

If you provide the material yield strength σ_y, the tool returns a safety factor. A factor below 1.0 means the section is overstressed. Most structural codes (AISC, Eurocode 3) require factors between 1.5 and 2.0 for static loads, and higher for fatigue or impact. Pro tip: always verify that the loading condition matches your actual support and load placement. A fixed-free (cantilever) beam under the same load produces a moment twice that of a simply-supported beam at midspan.

Formulas

The fundamental bending stress equation from Euler-Bernoulli beam theory, valid for slender beams under pure bending with linear-elastic, isotropic material:

σ = M ⋅ yI

Where σ = bending (flexural) stress at the fiber of interest Pa, M = internal bending moment at the section N⋅m, y = perpendicular distance from the neutral axis to the fiber m, and I = second moment of area (moment of inertia) of the cross-section about the neutral axis m⁴.

The section modulus S simplifies the expression for extreme-fiber stress:

S = Iy_max ⇒ σ_max = MS

The safety factor against yielding:

SF = σ_yσ_max

Where σ_y is the material yield strength. A value of SF < 1.0 indicates the section has yielded. Most design codes mandate SF ≥ 1.5 for static loading.

Moment of inertia for an I-beam (symmetric wide flange) about its strong axis:

I = b_f ⋅ H³ − (b_f − t_w) ⋅ h_w³12

Where b_f = flange width, H = total depth, t_w = web thickness, and h_w = clear web height (H − 2t_f).

Reference Data

Cross-Section	Moment of Inertia I	Neutral Axis Distance y_max	Section Modulus S	Typical Use
Solid Rectangle	bh³12	h2	bh²6	Timber joists, flat bars
Solid Circle	πd⁴64	d2	πd³32	Shafts, round bars, dowels
Hollow Circle (Tube)	π(d_o⁴ − d_i⁴)64	d_o2	π(d_o⁴ − d_i⁴)32d_o	Steel pipes, CHS columns
Hollow Rectangle (Box)	BH³ − bh³12	H2	BH³ − bh³6H	RHS/SHS structural tubing
I-Beam (Wide Flange)	Composite (flanges + web)	H2	Iy	Steel beams (W, S, HP shapes)
Common Loading - Maximum Bending Moment
Simply Supported - Center Point Load	M = F ⋅ L4			At midspan
Simply Supported - UDL	M = w ⋅ L²8			At midspan
Cantilever - End Point Load	M = F ⋅ L			At fixed end
Cantilever - UDL	M = w ⋅ L²2			At fixed end
Material Yield Strengths (Typical)
Mild Steel (A36)	250 MPa			Structural steel
High-Strength Steel (A992)	345 MPa			Wide-flange beams
Stainless Steel 304	205 MPa			Corrosion-resistant members
Aluminum 6061-T6	276 MPa			Lightweight structures
Aluminum 2024-T4	324 MPa			Aerospace
Titanium Ti-6Al-4V	880 MPa			Aerospace, medical
Douglas Fir (bending)	7.6 MPa			Timber construction (allowable)
Southern Pine (bending)	8.3 MPa			Timber framing (allowable)
Concrete (compressive)	20 - 40 MPa			Reinforced beams (compression fiber)
Cast Iron (grey)	130 MPa			Machine bases, frames

Frequently Asked Questions

The outermost fibers begin to yield plastically. Under continued loading the plastic zone migrates inward toward the neutral axis, forming a plastic hinge. The beam loses stiffness progressively. If the full cross-section yields, the beam can no longer resist additional moment and collapses. The safety factor SF = σ_yσ drops below 1.0, indicating failure. Design codes require SF ≥ 1.5 to 2.0 for static loads, and up to 3.0 for impact or fatigue.

The moment of inertia I scales with the cube of the distance from the neutral axis. An I-beam concentrates material in the flanges, far from the neutral axis, maximizing I while minimizing total cross-sectional area. A solid rectangle distributes material uniformly, including near the neutral axis where it contributes almost nothing to bending resistance. For equal mass per unit length, a W-shape can provide 3 - 5× the section modulus S of a solid bar.

The Euler-Bernoulli equation assumes plane sections remain plane and that shear deformation is negligible. This holds when the span-to-depth ratio L/h exceeds roughly 8 - 10 for isotropic materials. Below 4, shear deformation becomes significant and Timoshenko beam theory should be used instead. For composite or sandwich beams, the threshold may be higher due to low shear stiffness of the core.

This tool computes y as half the total depth, which is correct only for symmetric sections (rectangle, circle, I-beam with equal flanges). For asymmetric sections (T-beams, channels, unequal flanges), you must select the custom moment of inertia option, compute I about the true centroidal axis using the parallel-axis theorem externally, and enter both I and the correct y (distance from centroid to extreme fiber) manually.

All dimensional inputs (length, depth, width, thickness) are in millimeters. Force is in Newtons, distributed load in N/mm, and moment in N·mm. The calculator performs all arithmetic in consistent N-mm units internally. Stress output is in MPa (equivalent to N/mm²). Moment of inertia is displayed in mm⁴. No unit mixing occurs. If your input data is in meters or kN, convert before entry: 1 kN = 1000 N, 1 m = 1000 mm.

Temperature does not change the elastic bending stress for a given moment, since σ = My/I depends only on geometry and load. However, elevated temperature reduces the yield strength σ_y of most metals. For structural steel above 300 °C, yield strength drops significantly (to roughly 60% at 500 °C). This means the safety factor decreases at high temperature even if the applied stress stays constant. Fire-design codes (Eurocode 3 Part 1-2) provide reduction factors.