Calculator

Civil & Structural Engineering σ = M/S

Maximum Bending Stress Calculator

28 September 2026 11 min read

Extreme-fibre bending stress from moment and rectangular section (or known S) with utilisation vs allowable.

σ max: 10.50 MPa; OK ✓
S: 5333.3 cm³; elastic modulus
Utilisation: 87.5 %; vs allowable
Status: Pass ✓; check

Live statistics

Bending stress

10.50 MPa

within allowable ✓

Section modulus

5333.3 cm³

S = bh²/6

Utilisation

87.5 %

σ / allowable

Moment

56.0 kN·m

input

Free-body sketch (not to scale). Green = reactions (upward). Red = point load P at position a, amber = UDL intensity w. Negative reaction means uplift — provide hold-down.

Loading plans — pick your design case

Plans load realistic preset forces so estimates match the building use. Your typed values are never overwritten unless you pick a plan.

Unit system

Bending moment M(kN·m) Help

kN·m

Factored or service — keep consistent.

Width b(mm) Help

Rectangular section width.

Depth h(mm) Help

Total depth.

Allowable stress(MPa) Help

MPa

Design/allowable extreme-fibre stress.

Precision(decimals) Help

decimals

Decimals.

Results update live as you type

How you compare — standard sections

Section	Depth (mm)	Mass (kg/m)	A (cm²)	Iₓ (×10⁶ mm⁴)	Sₓ (×10³ mm³)
UB 305×165×40 · Steel I-beam	304	40.3	51.3	85	560
UB 457×191×67 · Steel I-beam	454	67.1	85.5	294	1290
UC 254×254×73 · Steel column	254	73.1	93.1	114	895
RC 300×500 (C30) · Concrete beam	500	375	1500	3125	12500
Glulam 140×450 · Timber beam	450	29	630	1063	4725
SHS 200×200×8 · Hollow section	200	47.4	60.4	36.9	369

Indicative values for scheme design only. Always verify against the manufacturer / code tables (AISC Steel Manual, SCI Blue Book, ACI detailing) before ordering steel or pouring concrete.

Plans & details

Detail

Bearing detail

Padstone sized for max reaction. Check masonry bearing and provide 150 mm min seating.

RCC

Reinforcement sketch

Bottom steel for sagging, top steel at supports for hogging. Extend past inflection + development length.

Uplift

Hold-down where needed

Strap or anchor for any uplift reaction. Size for net factored uplift per ASCE 7.

Typical details for scheme design. Replace placeholders with project drawings (1200×630 JPG or SVG line work in /public/plans/civil/). Do not scale from web images — use figured dimensions.

Design verification checklist

Work through before issuing results — 1/6 verified (17%).

Results

Optimal

Maximum bending stress

10.50MPa

Extreme-fibre stress σ = M/S

Elastic section modulus

5333.33cm³

S = bh²/6

Utilisation ratio

87.5%

σ / allowable (≤ 100% passes)

Design moment

56.00kN·m

Input moment

Visualizations

Shear force diagram (SFD) — kN vs station (m)

Bending moment diagram (BMD) — kN·m vs station (m); + sagging / − hogging

Formula

Flexure formula: plane sections stay plane, so stress varies linearly through the depth and peaks at the extreme fibre c = h/2 (σ = Mc/I = M/S).

σ_max = M / S | S_rect = b·h²/6 | Utilisation = σ / σ_allow ≤ 1

Alternative Form

Required modulus: S_req = M / σ_allow · M in N·mm = M(kN·m) × 10⁶

Variables

Symbol	Name	Unit	Description
σ	Bending stress	MPa	Peak fibre stress (1 MPa = 1 N/mm²).
M	Bending moment	kN·m	Use factored M for strength, service M for cracking.
S	Section modulus	cm³	Geometric bending efficiency I/c.
σ_allow	Allowable stress	MPa	Design or permissible fibre stress.

Explanation

σ = M/S: the flexure formula collapses a whole stress block into one division. S = bh²/6 for rectangles turns breadth and (squared) depth into bending capacity.

Derivation

From σ = −My/I with y = ±h/2: σ_max = M(h/2)/(bh³/12) = 6M/bh² = M/S. Units: M(kN·m)×10⁶ = N·mm; S(mm³); σ = N/mm² = MPa.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Enter design bending moment M

Result: Demand fixed

Step 2

Enter section width b and depth h

Result: Geometry fixed

Step 3

Read elastic modulus S = bh²/6

Result: Capacity geometry

Step 4

Read extreme-fibre stress σ = M/S

Result: Stress issued

Step 5

Compare against allowable (utilisation ≤ 100%)

Result: Pass/fail

Step 6

Resize h (stress ∝ 1/h²) if over

Result: Redesign loop

Final Answer

10.50

Worked Examples

See how the calculator works with different scenarios.

Scenario

Floor joist check.

Inputs

M:8 kN·m

b:50 mm

h:200 mm

Calculation

S = 50·40000/6 = 333333 mm³ = 333 cm³. σ = 8e6/333333 = 24 MPa?? wait 8 kN·m = 8e6 N·mm; σ = 24 MPa — over typical 7–10 MPa allowable.

Result

σ = 24 MPa — FAIL for timber

Interpretation

Double up joists or deepen to 250+ mm (stress falls with h²).

Understanding Your Result

Your Result: Maximum bending stress = 10.50 MPa · Elastic section modulus = 5333.33 cm³ · Utilisation ratio = 87.5 %

Typical range: Timber ≤ 7–12 MPa; steel ASD ~140–165 MPa; utilisation target 70–95% (efficient) or < 60% (conservative).

What This Means

Peak fibre stress from your moment and section, plus how much of the allowable you have consumed.

Why It Matters

Stress is the pass/fail of flexure: exceed it and fibres yield, concrete crushes or timber splits. Sizing h correctly here saves the whole member.

Professional Interpretation

Use factored M with ϕ/Ω strengths for the real check; keep this elastic ratio as the scheme gate. If utilisation exceeds 100%, deepen the section first.

Recommendations

•Verify equilibrium: reactions sum to total load
•Include self-weight in the UDL before issuing numbers
•Keep service and factored (1.2D+1.6L) columns separate

Learn — structural behaviour, deeply

Moment demands, modulus supplies.

1 · The two equations that run every site

ΣF_y = 0

Vertical balance

Upward reactions must equal total downward load. Missing 1 kN here means a 1 kN shear error everywhere downstream.

ΣM = 0

Moment balance

Moments about ANY point must cancel. Take moments about one support to eliminate it and solve the other directly.

ΣF_x = 0

Horizontal balance

Zero for vertical-only loading — but the pin still needs checking once wind, soil or inclined loads appear.

Worked derivation (simply supported, point load): ΣMleft = 0 ⇒ Rb·L − P·a = 0 ⇒ Rb = P·a/L, then ΣFy = 0 ⇒ Ra = P − Rb = P·(L−a)/L. UDL adds w·L/2 per side acting at midspan. That is the entire calculation — everything else is bookkeeping.

2 · Support types — what each one can (and cannot) do

Support	Reactions	Allows	Watch out
Roller	1 (vertical)	Horizontal slide + rotation	Provides NO lateral restraint — pair with a pin or bracing.
Pin / hinge	2 (vertical + horizontal)	Rotation only	Moment released here — detail the hinge, never weld it rigid by accident.
Fixed / clamped	3 (+ moment)	Nothing (fully restrained)	Attracts large moments; settlement or temperature causes locked-in stress.
Propped cantilever	3 (indeterminate ×1)	Rotation at prop	Needs compatibility (deflection = 0 at prop) — equilibrium alone is not enough.
Continuous (3+ supports)	4+ (indeterminate)	Rotation over interior piers	Solve by superposition / moment distribution / software; reactions shift with stiffness.

3 · Factor-impact table — what moves your reactions most

Factor	Typical impact	Why
Load position (a/L)	Linear, dominant	Moving P toward a support shifts ~100% of P onto it. Ra = P·(L−a)/L — position matters more than span.
UDL intensity (w)	∝ w·L total	Each extra 1 kN/m on a 6 m beam adds 3 kN per reaction. Self-weight alone is often 1–5 kN/m on concrete.
Span length (L)	∝ L for UDL	Doubling L doubles UDL reactions and quadruples UDL moment (wL²/8). Keep spans tight where possible.
Overhang length (c)	∝ c/L uplift	Uplift at the far support = P·c/L. A 2 m overhang with 10 kN on a 5 m span pulls down 4 kN — needs hold-down.
Fixity (pinned → fixed)	Redistributes ~10–30%	Fixed ends attract moment and shed midspan moment. Never assume fixity you have not detailed.
Support settlement (Δ)	0 (det.) / large (indet.)	Determinate beams just rotate; continuous/fixed beams develop locked-in moments ≈ 3EIΔ/L².
Dynamic / impact	+10–100%	Static reactions must be amplified for cranes, machinery, footfall (see ASCE 7 Ch. 4, AISC DG11).
Load combination	× 1.2–1.6 (LRFD)	1.2D + 1.6L + 0.5(Lr/S/R) governs most beams. Reactions for strength ≠ reactions for service checks.

4 · Code provisions you must cite on drawings

ASCE 7 load combinations (LRFD): 1.4D · 1.2D + 1.6L + 0.5(Lr/S/R) · 1.2D + 1.0W + 1.0L … Strength reactions use factored loads; bearing/deflection checks use service (unfactored) loads.
ACI 318 (concrete): strength-reduction ϕ = 0.65–0.90; development/lap lengths Ch. 25; one-way shear + punching Ch. 22. Reactions feed directly into Vu ≤ ϕVn.
AISC 360 (steel): Ch. F/G/H for moment/shear/combined; bearing at supports per J7–J8; bracing per App. 6. Connection must develop the computed reaction — a beam is only as strong as its seat.
Eurocode basis (EN 1990): ULS 1.35G + 1.5Q; SLS G + Q (characteristic). National annexes adjust factors — confirm the project annex.
Serviceability (all codes): live-load deflection ≈ L/360 floors, L/240 roofs; total ≈ L/240. Vibration: AISC DG11 / SCI P354 for rhythmic or office floors.

5 · Pro workflow (5 minutes per beam)

Sketch the free-body diagram: every load incl. self-weight, every support, dimensions.
Solve ΣM then ΣF — write both lines on the calc sheet so a checker can follow.
Verify: re-take moments about the OTHER support; residual must be ≈ 0.
Draw shear then moment (reactions are the SFD starting values).
Size the support: bearing stress, padstone, bolt/weld group, foundation pressure.
File: keep the input → reaction → SFD/BMD → support chain on one page.

6 · Field traps (where reactions go wrong)

• Forgetting self-weight: a 300×600 concrete beam weighs 4.5 kN/m — on 6 m that is 13.5 kN per support.
• Assuming fixity: a “fixed” sketch with two bolts is a pin. Model what you detail.
• Ignoring uplift: overhang + wind can reverse a bearing — light roofs lift off.
• Mixing factored/service: never compare a factored reaction against an allowable soil pressure.
• Point load at the wrong station: 0.5 m error on a 10 kN load swings reactions by P·Δa/L.

7 · The 3-check rule (sign-off standard)

Math check: ΣFy and ΣM both close to zero with project units printed.
Sanity check: reactions sum to total load; heavier side sits under heavier load; UDL symmetry gives Ra ≈ Rb.
Load-path check: every reaction lands on something sized for it — beam → seat → column → footing → soil, no floating numbers.

Practical Tips

•Ra + Rb must equal total load — if not, the input (not physics) is wrong
•UDL symmetry gives Ra = Rb = wL/2 — use as a 2-second check
•Report service kN to 1 decimal for bearings and footings

Common Mistakes

Self-weight omitted

Bundle member kN/m into w first.

Factored reaction checked against soil pressure

Soils take SERVICE loads; keep a separate factored column.

Uplift ignored

Negative R needs a hold-down detail, not a bigger pad.

Assumptions

This calculator makes the following assumptions to provide estimates:

Rigid supports: No settlement or rotation under load.

Small deflections: Geometry unchanged by loading; superposition applies.

Static loading: No impact, vibration or seismic amplification included.

Limitations

Be aware of these limitations when using this calculator:

Determinate unless stated: Continuous and propped cases use the compatibility solution noted in the formula.

Prismatic sections: Constant EI along the span.

Frequently Asked Questions

Answers to common questions about this calculator.

What is section modulus?

S = I/c: the bending efficiency of a shape. Bigger S means less stress for the same moment. For rectangles S = bh²/6 — depth matters quadratically.

Service or factored M?

Factored M with design strengths for safety; service M for deflection and cracking. Never mix them in one ratio.

Why does depth matter so much?

Stress scales with 1/h² and stiffness with h³. Adding 25% depth nearly halves stress — the cheapest upgrade on any beam.

Does this replace RC/steel design?

No — it is the elastic gateway. RC needs cracked-section ϕMn per ACI 318; steel needs lateral-torsional buckling per AISC 360 Ch. F.

What allowable should I use?

Timber ~7–12 MPa bending, steel ~140–250 MPa (ASD/LRFD differ), concrete elastic checks project-specific. Confirm with your code and edition.

Top or bottom fibre?

Equal for symmetric sections under sagging magnitude; signs differ (one compresses). Report the magnitude with the governing fibre noted.

How do I find reactions for a simply supported beam?

Take moments about the left support: Rb = (P·a + w·L·L/2) / L, then Ra = P + w·L − Rb. Always re-check by taking moments about the right support — the residual must be ≈ 0.

What does a negative (downward) reaction mean?

The support must pull DOWN instead of pushing up — net uplift from overhangs, pattern loading or wind. Provide a hold-down strap, anchor or ballast; a gravity-only bearing will lift off.

Do I include the beam self-weight?

Yes, always. A 300×600 mm concrete beam weighs ~4.5 kN/m — on a 6 m span that adds ~13.5 kN per support. Bundle it into the UDL before solving.

Service vs factored loads — which do I report?

Report BOTH in separate columns: service (unfactored D+L) for deflection, bearing, settlement and soil checks; factored LRFD (1.2D + 1.6L) for member strength. Never compare a factored reaction with an allowable soil pressure.

What is the difference between a UDL and a UVL?

A UDL (uniformly distributed load) is constant intensity over a length (kN/m). A UVL (uniformly varying / triangular) runs zero-to-max — e.g. soil or hydrostatic pressure. They integrate differently, so use the matching calculator.

How do I turn a floor area load into a beam line load?

Multiply the area load (kN/m²) by the tributary width (m) draining to the beam: line load = area load × tributary width. Add the beam self-weight, partitions and finishes first.

What is tributary area?

The floor/roof area whose load flows to one member — bounded by halfway lines to adjacent members. Tributary area × area load = the point/line load that member must carry.

Which load combination usually governs a beam?

For strength: 1.2D + 1.6L (+ 0.5 for roof live/snow/rain) per ASCE 7 LRFD. Service D+L governs deflection and bearings; 0.9D + 1.0W governs uplift and overturning.

LRFD vs ASD — does it change the reactions?

No — equilibrium is the same. The design PHILOSOPHY differs: LRFD factors loads up (1.2D+1.6L) against factored resistance (ϕRn); ASD keeps service loads against allowable stresses (Rn/Ω). Keep the two systems in separate columns.

Pin, roller or fixed — how do I choose the support model?

Model what you DETAIL. Masonry/steel seats and two-bolt end plates are pins (moment released); moment frames, welded rigid joints and well-anchored cantilevers earn fixity. When in doubt, the pinned assumption is conservative for midspan moment but NOT for uplift.

What deflection limits apply to beams?

Typical code limits: live-load deflection L/360 for floors (L/240 roofs), total-load L/240, cantilevers 2L/360-ish per IBC Table 1604.3. Always check under SERVICE loads with the correct (cracked, long-term) stiffness.

How do I convert kN ↔ kips and m ↔ ft?

1 kN = 0.2248 kips (1 kip = 4.448 kN); 1 m = 3.2808 ft. Moments: 1 kN·m = 0.7376 kip·ft. Every calculator on this site has an SI/Imperial toggle that converts all displayed values.

Shear vs bending moment — which sizes what?

Shear sizes stirrups/webs and connections (V ≤ ϕVn); bending moment sizes flexural steel and the section (M ≤ ϕMn via section modulus S: stress = M/S). Deflection is governed by stiffness EI, a third independent check.

Where does the maximum moment occur?

Where shear crosses zero: under a midspan point load, or at x = R/w under UDL (span moment = R·x − w·x²/2). On continuous beams also check the hogging peaks OVER interior supports — they often govern.

What is section modulus and how do I use it?

S = I/c (mm³): the geometric bending efficiency of a section. Bending stress = M/S; required S = factored moment / allowable (or design) stress. Compare against the standard-section table on each page.

How do I quickly check a steel beam?

Three checks per AISC 360: bending (Ch. F incl. lateral-torsional buckling over the unbraced length), shear (Ch. G), and deflection (L/360). Then verify the connection develops the end reaction.

How do I quickly check a concrete beam?

Flexure per ACI 318: ϕMn = ϕ·As·fy·(d − a/2) with ϕ = 0.9 (tension-controlled); one-way shear Vc + Vs (Ch. 22); deflection via minimum thickness or direct computation with cracked stiffness and long-term multiplier.

What is development length and why does it matter?

The embedment needed for a bar to reach yield (ld, ACI 318 Ch. 25). A perfectly computed moment capacity is worthless if the bar pulls out — extend bars past inflection points and hook or develop them at supports.

How do I check bearing stress under a support?

Bearing stress = reaction / (bearing length × width) ≤ allowable (masonry ≈ 0.25–0.4·f′m, concrete per ACI 22.8, steel per AISC J8). If it fails, lengthen the bearing or add a padstone/spreader plate.

Hold-down or bigger pad — which fixes uplift?

Only a hold-down (strap, anchor, dead weight) fixes uplift — a bigger pad just spreads a force acting the wrong way. Size the anchor for net factored uplift (0.9D + 1.0W) with proper embedment.

Which codes do these calculators follow?

Workflows cite ACI 318 (concrete), AISC 360 (steel), ASCE 7 (loads + combinations), IBC (deflection, occupancy loads) and Eurocode equivalents. Always confirm the edition and annex your project is permitted under.

How accurate are these calculators?

Scheme-stage accuracy from closed-form statics with stated assumptions (rigid supports, prismatic sections, static loads). Use them for sizing, checking and tender estimates — final permit/shop drawings need a licensed engineer and project-specific analysis.

Can I print, copy or share results?

Yes — every page has print, copy-link and share actions in the hero banner, and results render in a printable dashboard. Paste the equilibrium line (ΣR vs total load) onto your calc sheet as the proof.

Do I still need a structural engineer?

For anything that gets built: yes. Software and calculators don’t carry liability, don’t visit site, and don’t know your soil, wind exposure or buildability constraints. Use these tools to brief and check your engineer, not replace them.

How does wind create uplift on beams and roofs?

Suction on leeward slopes/overhangs pulls upward; the net combination 0.9D + 1.0W (ASCE 7) often reverses light end reactions. Run it explicitly — gravity-only models hide the governing hold-down case.

Related Calculators

Explore more calculators related to this topic.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Reaction: Force (or moment) a support exerts on a member to maintain static equilibrium (ΣF = 0, ΣM = 0).
Simply supported: Beam on one pin + one roller. Resists vertical load; allows rotation and (at roller) horizontal movement.
Fixed support: Clamped end resisting force AND moment; prevents rotation. Used in cantilevers and fixed frames.
UDL / UVL: Uniformly distributed load (constant kN/m) vs. uniformly varying load (triangular, zero-to-max).
Tributary area: Floor/roof area whose load drains to one member. Line load = area load × tributary width.
Factored load: Service load × LRFD load factor (e.g. 1.2D + 1.6L) for strength design per ASCE 7.
Serviceability: Deflection, cracking and vibration limits under day-to-day (unfactored) loads.
Uplift: Net downward-hold requirement: a negative (downward-pulling) reaction, common on overhangs.

Authoritative References

Trusted sources and standards referenced by this calculator.

ACI 318 Code & Commentary (American Concrete Institute) AISC Steel Construction Manual ASCE 7 Hazard Tool — design loads Eurocodes (EN 1990–1999) — Basis of Structural Design Hibbeler — Structural Analysis (textbook companion) NIST — Wind & Seismic design resources

Applicable Standards

ACI 318 — Building Code Requirements for Structural ConcreteAISC 360 — Specification for Structural Steel BuildingsASCE 7 — Minimum Design Loads and Associated CriteriaEurocode 2 (EN 1992) — Design of Concrete StructuresEurocode 3 (EN 1993) — Design of Steel StructuresIBC — International Building Code (structural chapters 16–23)

Was This Calculator Helpful?

Report Issue

Page Information

Last Updated:

2026-09-28

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.