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Civil & Structural Engineering Z · plastic

Plastic Section Modulus Calculator

28 September 2026 9 min read

Plastic modulus Z with elastic S and shape factor for rectangles and circles.

Z: 8000.0 cm³; plastic
S: 5333.3 cm³; elastic
k = Z/S: 1.500; reserve
Mp/My: 1.50×; extra

Live statistics

Plastic Z

8000.0 cm³

Mp = Z·fy

Elastic S

5333.3 cm³

My = S·fy

Shape factor

1.500

Z/S reserve

Area

800.0 cm²

gross

Schematic only — not to scale. Dimensions follow the inputs below.

Unit system

Shape

b / d(mm) Help

Width or diameter.

h(mm) Help

Depth (rect only).

Precision(decimals) Help

decimals

Decimals.

Precision

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 YOU	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

Geometry

Section sketch

Dimensioned cross-section with centroid and axes. Verify b/h/t against drawings before issuing.

Detail

Bar layout / rolling

For RC: cover, bar positions and crack basis. For steel: rolling direction and weld access.

Nett

Connection zones

Holes, copes and welds deduct from gross — use nett properties at connections.

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

Normal

Plastic section modulus

8000.0cm³

First moments above/below equal-area axis

Elastic section modulus

5333.3cm³

I/c for first yield

Shape factor

1.500

Plastic reserve over first yield

Area

800.00cm²

Gross area

Visualizations

Moduli (cm³)

8,000 cm³Z plastic

5,333 cm³S elastic

Formula

Plastic modulus assumes full yield above and below the equal-area axis — the true bending strength of compact sections.

Rect: Z = bh²/4 (k = 1.50) | Circle: Z = d³/6 (k = 1.698) | Mp = Z·fy, My = S·fy

Alternative Form

Equal-area axis splits A in half; Z sums first moments of both halves about it

Variables

Symbol	Name	Unit	Description
Z	Plastic modulus	cm³	Full-yield bending geometry.
S	Elastic modulus	cm³	First-yield geometry.
k	Shape factor	—	Z/S: hidden reserve past yield.
Mp / My	Capacities	kN·m	Times yield stress fy.

Explanation

Z = Σ first moments about the equal-area axis: full-plastic strength; k = Z/S prices the trip from first yield to collapse.

Derivation

Rectangle: halves bh/2 at h/4 arms give 2·(bh/2)(h/4) = bh²/4. Circle: semicircle centroid 4r/3π gives Z = d³/6. Shape factor normalises by elastic S.

Step-by-Step Calculation

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

Step 1

Pick rectangle or circle

Result: Family set

Step 2

Enter dims

Result: Geometry set

Step 3

Read elastic S (first yield)

Result: Yield gate

Step 4

Read plastic Z (full yield)

Result: Strength issued

Step 5

Note shape factor reserve

Result: Margin known

Step 6

Multiply by fy for Mp/My

Result: Capacity linked

Final Answer

8000.0

Worked Examples

See how the calculator works with different scenarios.

Scenario

Plastic reserve check.

Inputs

b:200 mm

h:400 mm

Calculation

Z = 200·160000/4 = 8.0e6 mm³ = 8000 cm³. S = 5333 cm³. k = 1.5.

Result

Z = 8000 cm³, k = 1.50

Interpretation

Compact rect carries 50% past first yield — if buckling and class allow.

Understanding Your Result

Your Result: Plastic section modulus = 8000.0 cm³ · Elastic section modulus = 5333.3 cm³ · Shape factor = 1.500

Typical range: k ≈ 1.5 rect, 1.7 solid circle, 1.1–1.2 I-beams; Z/S above 1.3 deserves a compactness check.

What This Means

Full-yield bending geometry with the elastic baseline and the reserve factor between them.

Why It Matters

Compact steel is bought and paid for at Mp, not My — Z unlocks 10–50% real capacity codes permit.

Professional Interpretation

Verify section class and LTB bracing before claiming Mp. Quote Z with the fy used for Mp so checkers close the loop.

Recommendations

•Verify dimensions against the drawing before issuing properties
•Use gross properties for analysis; cracked/nett for checks as codes require
•Compare required S/I against the standard-section table below

Learn — structural behaviour, deeply

Yield is a beginning, not an end.

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

•Depth beats width: I ∝ h³, S ∝ h², A ∝ h — deepen before widening
•State the basis: gross, nett, cracked or transformed
•Cross-check S_req = M/σ against the table below

Common Mistakes

Mixing mm and cm in one calc

Enter all dims in the selected unit; the solver converts once internally.

Using gross I for RC deflection

Use cracked Ie plus creep for service deflection.

Ignoring fillets then ordering tight

Fillets add 1–3%: fine for scheme, verify against mill tables for order.

Assumptions

This calculator makes the following assumptions to provide estimates:

Sharp-corner idealisation: Fillets, welds and chamfers ignored (1–3% on area, less on I for scheme).

Prismatic section: Properties constant along the member length.

Homogeneous elastic material: Single E unless a modular ratio n is entered.

Limitations

Be aware of these limitations when using this calculator:

Gross elastic properties: Nett (holes) and cracked/transformed variants need their own deductions.

Thin-wall torsion approx.: Open-section J ≈ Σbt³/3; warping Cw per standard approximations.

Frequently Asked Questions

Answers to common questions about this calculator.

Elastic vs plastic modulus?

S marks FIRST yield (My = S·fy); Z marks FULL yield (Mp = Z·fy). Compact sections reach Mp; slender ones buckle first.

What is the shape factor?

k = Z/S: the reserve past first yield — 1.5 rectangles, ~1.7 circles, ~1.1–1.2 wide-flange I-beams.

When can I use Mp?

Compact (Class 1) sections with bracing against lateral-torsional buckling (AISC 360 Ch. F / EC3). Otherwise My or buckling rules.

Equal-area axis?

The line splitting area in half (mid-depth for symmetric shapes). Z sums first moments of both halves about it.

I-beams?

Use the I-section page (Z ≈ flange area × lever + web term) or mill tables. This page covers solid rect/circle exactly.

Concrete?

RC uses Whitney-block ϕMn, not Z. Reserve Z thinking for steel and timber compact checks.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

How do I allow for impact and vibration?

Amplify static reactions per ASCE 7 Ch. 4 (cranes, machinery, forklifts: +10–100%) and check footfall vibration with AISC DG11/SCI P354 on office, gym and lab floors. These calculators report static values — apply the factors yourself.

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.

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.

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)

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Last Updated:

2026-09-28

Version:

2.0.0

References:

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Free to use for educational and professional purposes.