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Civil & Structural Engineering delta ns · ACI 318

Long Column Calculator

28 September 2026 11 min read

Magnified design moment for slender columns from cracked EI and Euler buckling.

Magnified Mc: 101.5 kN·m; delta 1.69

Live statistics

delta ns

1.69

101.5 kN·m

2609 kN

Unit system

Concrete fc(MPa) Help

MPa

Cylinder strength.

Width b(mm) Help

Column width.

Depth D(mm) Help

Depth in buckling plane.

Effective length kl(mm) Help

k times clear length.

Factored load Pu(kN) Help

Factored axial demand.

End moment M(kN·m) Help

kN·m

Larger factored end moment.

Cm factor(factor) Help

factor

Moment gradient factor, often 1.

Sustained load ratio(ratio) Help

ratio

Factored sustained over total axial.

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

Column section detail

Cover, ties and bar layout per ACI 318. Extend bars past the joint plus development length.

Path

Load path sketch

Slab to beam to column to footing. Verify tributary areas before issuing numbers.

Check

Site verification

Confirm concrete strength, cover and compaction against drawings and test records.

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

Caution

Magnifier delta ns

1.69ratio

Cm over 1 minus Pu over 0.75 Pc, at least 1.

Magnified moment Mc

101.5kN·m

Design the section for Mc with Pu.

Critical buckling Pc

2609kN

Euler load on cracked EI.

Visualizations

Results Overview

1.7ratioMagnifier delta ns

101.5ratioMagnified moment Mc

2,609ratioCritical buckling Pc

Formula

Cracked stiffness reduced for sustained load feeds Euler buckling; proximity of Pu to 0.75 Pc inflates the end moment into the magnified design moment.

Ec = 4700 sqrt(fc) | EI = 0.4 Ec Ig / (1 + beta d) | Pc = pi² EI / kl² | delta = Cm / (1 - Pu / 0.75 Pc) >= 1 | Mc = delta x M

Alternative Form

Ig = b D³ / 12 for the rectangle

Variables

Symbol	Name	Unit	Description
delta ns	Moment magnifier	ratio	Amplification of first-order moment, minimum 1.
Mc	Magnified moment	kN·m	Moment the slender section must resist.
Pc	Critical load	kN	Euler buckling on cracked EI.
EI	Flexural stiffness	N·mm2	Cracked and creep-reduced stiffness.

Explanation

Cracked EI sets the Euler load; the gap between Pu and 0.75 Pc sets the magnifier on the end moment.

Derivation

ACI 318 non-sway magnifier method: EI per 6.6.4.4, Euler Pc, then delta per 6.2.6 with the 0.75 stiffness reduction on Pc.

Step-by-Step Calculation

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

Step 1

Enter section, kl, Pu, M, Cm and sustained ratio

Result: Frame data fixed

Step 2

Read Pc and delta ns

Result: Amplification issued

Step 3

Design section for Pu with Mc

Result: Pair issued

Final Answer

1.69

Worked Examples

See how the calculator works with different scenarios.

Scenario

300 by 300, Pu 800 kN, end moment 60 kN m.

Inputs

fc:25

b:300

D:300

kl:4000

Pu:800

M:60

Cm:1

betad:0.5

Calculation

Ec is 23500 MPa and Ig is 6.75 x 10^8 mm4. EI is 4.23 x 10^12 N mm2 and Pc is 2609 kN. Delta is 1 / (1 - 800 / 1957) = 1.69, so Mc is 1.69 x 60 = 101.5 kN m.

Result

delta = 1.69, Mc = 101.5 kN·m

Interpretation

Design the section for 800 kN with 101.5 kN m, not the first-order 60.

Understanding Your Result

Your Result: Magnifier delta ns = 1.69 ratio · Magnified moment Mc = 101.5 kN·m · Critical buckling Pc = 2609 kN

Typical range: Braced office columns often show delta 1.1 to 1.7.

What This Means

Extra moment the slender column attracts through lateral deflection.

Why It Matters

Designing to first-order M alone under-reinforces tall columns and risks buckling failure.

Professional Interpretation

If delta tops 1.5, grow the section before detailing rather than fighting it with steel.

Recommendations

•Check slenderness before trusting short-column capacity
•Keep service and factored (1.2D + 1.6L) columns separate
•Verify one line by hand per schedule

Learn — structural behaviour, deeply

Tall columns bend further under load.

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

•Split dead vs live early — factoring depends on it
•Include self-weight; concrete often dominates
•Report service values to 1 decimal for bearings and footings

Common Mistakes

Service load checked against factored capacity

Keep a service column (settlement, soil, deflection) and a factored column (strength) side by side.

Self-weight omitted

Add member weight into the total first.

Wrong units in one term

Convert everything to the page unit system before combining terms.

Assumptions

This calculator makes the following assumptions to provide estimates:

Static gravity loading: No impact, vibration or seismic amplification unless the page applies one.

Nominal material properties: Code-grade materials per project data; confirm lightweight or high-strength cases.

Scheme-level accuracy: Closed-form code clauses for preliminary and check calculations.

Limitations

Be aware of these limitations when using this calculator:

Verify supports and pattern: Pattern loading, continuity and support settlement need their own runs.

Confirm local code values: Loads, factors and minimums vary by jurisdiction.

Frequently Asked Questions

Answers to common questions about this calculator.

What is beta d?

Ratio of factored sustained axial load to total factored axial load; creep lowers EI.

Why 0.75 on Pc?

ACI 318 applies a stiffness reduction factor of 0.75 in the magnifier denominator.

When is delta capped?

ACI 318 flags second-order analysis when delta exceeds about 1.4 to 1.5; enlarge the section early.

Is this a final design?

No, it is a preliminary magnifier per ACI 318. A licensed engineer must confirm final design.

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.

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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

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