Complete User Guide & Reference

Span-to-Depth Ratio Calculator:
Step-by-Step Guide with All Formulas

Everything you need to use the SDR Calculator confidently — from first input to final code-compliant result. Covers ACI 318, Eurocode 2, IS 456, BS 8110 and AS 3600.

Beams & Slabs Deflection Limits All Major Codes Metric & Imperial Worked Examples
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Section 01

What Is the Span-to-Depth Ratio — and Why Does It Matter?

The single most important preliminary check in structural sizing

The Span-to-Depth Ratio (L/d) compares the clear span of a structural member — a beam or slab — to its effective depth (the distance from the compression face to the centre of the tension reinforcement). It is the fastest way to confirm that a member is deep enough to keep long-term deflections within acceptable limits, without running a full finite-element analysis.

Every major reinforced concrete design code — ACI 318, Eurocode 2, IS 456, BS 8110 and AS 3600 — specifies minimum L/d limits. Exceeding these limits means the beam or slab may sag visibly, crack brittle plaster, jam doors, or fail formal serviceability checks at the detailed design stage.

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Rule of thumb: If your actual L/d ratio is less than or equal to the code limit, the member passes the preliminary depth check. A lower L/d means a deeper (stiffer) member. A higher L/d means shallower — and potentially problematic.
L = Clear Span D (Overall Depth) d (Effective Depth) c Compression Zone — Neutral Axis — Tension Rebar ∅ Cover Zone (c + stirrup + ∅/2)
Figure 1 — Beam elevation showing the relationship between Span (L), Effective Depth (d), Overall Depth (D), and concrete cover (c)
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Section 02

Step-by-Step: How to Use the Span-to-Depth Ratio Calculator

From opening the tool to reading the final result in under two minutes

1

Choose Your Unit System (Metric or Imperial)

Click Metric (mm/m) for SI units or Imperial (ft/in) for US customary units. This sets the default units for span and depth inputs. You can still select individual units per field — for example, span in metres and depth in millimetres.

2

Select Calculation Mode: Forward or Reverse

→ Find Min Depth (default): Enter the span; the tool calculates the minimum required effective depth.
← Find Max Span: Enter a known depth; the tool calculates the maximum allowable span for that depth.

3

Set Member Type & Geometry

Select the Member Type (e.g., Simply Supported Beam, One-Way Slab, Cantilever) and the Material (RC, Steel, Timber, Composite, etc.). Enter your Span Length (L) with the correct unit. If you have a known depth and want to check it, enter it in the Known Depth (d) field — otherwise leave it blank.

4

Select Your Design Code

Choose from ACI 318 (USA), Eurocode 2 (Europe), IS 456 (India/Sri Lanka), BS 8110 (UK), or AS 3600 (Australia). If your authority specifies a custom ratio, select Custom and enter the limit. The correct base L/d table values are applied automatically.

5

Enter Material Properties

For RC members: enter f'c or fck (concrete compressive strength, MPa), fy (rebar yield strength, MPa), tension reinforcement ratio ρ (%), concrete cover (mm), and main rebar diameter (mm). These are used to compute modification factors that adjust the basic L/d limit.

6

Set Loading & Serviceability Conditions

Choose your Load Category (residential, office, roof, etc.), Floor Finish (brittle finishes trigger a 0.9× penalty), dead and live loads, deflection limit (L/360, L/240, etc.), and number of spans. Expand Advanced Factors for creep multiplier, vibration sensitivity, flange width ratio, and importance factor.

7

Click "Calculate" and Read Results

The results panel shows: Actual L/d, Code Limit, Min Effective Depth dmin, Overall Depth D, Recommended Width b, Self-Weight, and Safety Margin %. A PASS / BORDERLINE / FAIL verdict is shown, along with a step-by-step calculation log, an elevation diagram, and a bar chart.

8

Use Additional Tabs for Deflection & Code Comparison

Switch to the Deflection tab to estimate instantaneous and long-term midspan deflection using elastic beam formulas. Use the Code Compare tab to run all five codes simultaneously and see which is most liberal or conservative for your scenario.

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After calculating, use Copy Results to get a formatted text summary (including all inputs, modification factors, and final section dimensions) ready to paste into a design memo or report.
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Section 03

All Formulas Used in the Calculator — Fully Explained with Units

Every equation the calculator applies, broken down step by step

The calculator applies up to six sequential calculation steps. Each formula is listed below with all variables defined, their units, and the range of typical values you should expect.

Formula 01 Basic Span-to-Depth Ratio
λ = L ÷ d
SymbolMeaningUnitTypical Range
λSpan-to-depth ratio (dimensionless)—7 – 40
LClear span between supportsmm or m1 m – 20 m
dEffective depth (compression face → rebar centroid)mm100 – 2000 mm
⚠️ Both L and d must be in the same unit before dividing. The calculator converts all inputs to mm internally to avoid errors.
Formula 02 Overall Depth from Effective Depth
D = d + c + (∅stirrup) + (∅bar ÷ 2)
SymbolMeaningUnitTypical Value
DOverall (total) concrete depthmm200 – 2500 mm
dEffective depth (from F1)mmAs calculated
cClear concrete cover (to stirrups)mm25 – 75 mm
∅stirrupStirrup/link bar diameter (assumed 8 mm)mm8 – 12 mm
∅barMain tension rebar diametermm12 – 32 mm
💡 The calculator assumes an 8 mm stirrup. If your design uses 10 or 12 mm stirrups, add the difference manually to the displayed D value.
Formula 03 ACI 318 — fy Modification Factor
MFfy = 0.4 + (fy ÷ 700)
SymbolMeaningUnitNotes
MFfySteel yield strength modification factordimensionless≈ 0.79 – 1.26
fyRebar yield strengthMPa280 – 600 MPa
📌 At fy = 420 MPa: MF = 1.0 (no change). At fy = 280 MPa: MF = 0.80 (stricter — shallower member allowed). At fy = 500 MPa: MF = 1.11 (you can use a shallower section). Also applied by IS 456 using the same factor.
Formula 04 Eurocode 2 — L/d Ratio (Cl. 7.4.2)

Two expressions depending on whether the member is lightly or heavily reinforced relative to the reference ratio ρ₀:

ρ₀ = √fck × 10⁻³ (reference ratio)
If ρ ≤ ρ₀:  L/d = K × [11 + 1.5√fck × (ρ₀÷ρ) + (1÷12)√fck × √(ρ'÷ρ₀)]
If ρ > ρ₀:  L/d = K × [11 + 1.5√fck × ρ₀÷(ρ−ρ') + 3.2√fck × (ρ₀÷ρ − 1)^1.5]
SymbolMeaningUnit
KSystem factor: SS=1.0, End-span=1.3, Interior=1.5, Cantilever=0.4—
fckCharacteristic cylinder compressive strengthMPa
ρTension reinforcement ratio (As / b·d)fraction (not %)
ρ'Compression reinforcement ratiofraction
ρ₀Reference reinforcement ratio = √fck × 10⁻³fraction
📌 The calculator also applies a service stress correction: multiply the result by 310 ÷ fs, where fs ≈ 0.58 fy (capped at ×2.0).
Formula 05 Minimum Effective Depth Calculation
dmin = Lmm ÷ (L/d)modified
SymbolMeaningUnit
dminMinimum required effective depthmm
LmmClear span converted to millimetresmm
(L/d)modifiedCode limit after all modification factors—
💡 Round dmin up to the next 25 mm grid for practical formwork. The calculator shows the raw calculated value; you round up when ordering concrete.
Formula 06 Self-Weight Estimation
wsw = (b ÷ 1000) × (D ÷ 1000) × γc  [kN/m]
SymbolMeaningUnitDefault
wswBeam self-weight per unit lengthkN/m—
bWidth (recommended = 0.45 × D, rounded to 25 mm)mm0.45 × D
DOverall section depthmmFrom F2
γcConcrete unit weightkN/m³25 (RC)
⚠️ Self-weight is estimated from the preliminary section size. If your actual width differs from the 0.45×D recommendation, recalculate manually: wsw = (b/1000) × (D/1000) × 25 kN/m.
Formula 07 Deflection Tab — Elastic Midspan Deflection
I = b × D³ ÷ 12  [mm⁴]
δSS = (5 × w × L⁴) ÷ (384 × E × I)  [mm]
δfixed = (w × L⁴) ÷ (384 × E × I)
δcantilever = (w × L⁴) ÷ (8 × E × I)
δlong-term = δinst × φ  (creep factor φ, typically 2.0)
SymbolMeaningUnit
IGross moment of inertia (rectangular section)mm⁴
wTotal UDL (dead + live)N/mm
LSpanmm
EElastic modulus (RC ≈ 25,000–30,000 MPa)MPa = N/mm²
φLong-term creep multiplier—
⚠️ The formula uses the gross (uncracked) I. In reality, cracked RC sections have a significantly lower effective I. Use this as a best-case estimate; detailed deflection calculation per code is required for final design.
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Section 04

Effective Depth (d) vs Overall Depth (D) — Critical Distinction

The most common source of error in preliminary sizing

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Common mistake: Code L/d ratios refer to effective depth d, not total section depth D. Using D instead of d in the calculation will make your beam appear shallower than it is — causing an unconservative (unsafe) result.
Parameter Symbol Definition Used In Typical Difference
Effective Depth d Compression face → centroid of tension rebar All code L/d ratios; moment capacity —
Overall Depth D Total concrete section height Formwork drawings; self-weight; deflection I D = d + 50–80 mm (typical)
Cover (to stirrup) c Concrete cover to outermost stirrup Durability; fire rating 25–75 mm
Stirrup diameter ∅s Link/stirrup bar diameter Finding rebar centroid 8–12 mm
Half main bar ∅b/2 Half of main tension rebar diameter Finding rebar centroid 6–16 mm
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Practical example: If the calculator gives dmin = 450 mm, and you use 40 mm cover, 8 mm stirrups, and 20 mm main bars: D = 450 + 40 + 8 + 10 = 508 mm → round up to 525 mm for formwork.
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Section 05

Code-by-Code L/d Reference Table — ACI 318, EC2, IS 456, BS 8110, AS 3600

Basic (unmodified) L/d limits for reinforced concrete beams and slabs

The table below shows the basic (unmodified) L/d limits from each code before any correction factors are applied. The calculator applies these automatically based on your code selection.

📊 Basic L/d Limit — Simply Supported RC Beam (fy = 420 MPa, ρ = 0.5%)

0 8 16 24 32 16 23.4 20 20 16 ACI 318 EC2 IS 456 BS 8110 AS 3600 L/d Limit
Member Type ACI 318 Eurocode 2 IS 456 BS 8110 AS 3600
Simply Supported Beam 16 ≈ 20–26* 20 20 16
Continuous Beam (one end) 18.5 ≈ 26–34* 23 23 19
Continuous Beam (both ends) 21 ≈ 30–39* 26 26 22
Cantilever Beam 8 ≈ 8–10* 7 7 8
One-Way Slab (SS) 20 ≈ 20–28* 20 20 22
One-Way Slab (Continuous) 26 ≈ 30–38* 26 26 28
Two-Way Slab 30 ≈ 30–38* 28 30 30
Flat Slab / Flat Plate 33 ≈ 24* 30 30 32

* EC2 values are computed from the formula (Section 03, Formula 04) — they depend on fck, ρ, ρ' and K. Values shown are approximate for fck=25 MPa, ρ=0.5%.

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Section 06

Input Validation — Accepted Ranges and Units for Every Field

Check your inputs before calculating to avoid unexpected results

Span Length (L)

Accepted units
m, mm, cm, ft, in
Min value
0.1 m (100 mm)
Practical max
~20 m for beams, 8 m for slabs
Tip
Use clear span between inner faces of supports, not centre-to-centre.

Concrete Strength f'c / fck

Accepted units
MPa or psi
Range
15 – 100 MPa (2175 – 14,504 psi)
Typical RC
25 – 40 MPa
Tip
IS 456 uses cube strength fck; EC2 uses cylinder. For M25 concrete, fck = 25 MPa.

Rebar Yield Strength fy

Accepted units
MPa or ksi
Range
200 – 600 MPa
Common grades
Fe415 = 415 MPa; Grade 60 = 414 MPa; B500B = 500 MPa
Tip
ACI uses 420 MPa as the reference; no MF correction at 420 MPa.

Tension Reinforcement Ratio ρ (%)

Units
Percentage (%) of gross section area
Range
0.1 % – 4.0 %
Typical
0.5 % – 2.0 % for beams; 0.15 % – 0.5 % for slabs
Tip
Higher ρ → more steel → shallower section allowed by EC2. Increases self-weight though.

Concrete Cover (c)

Accepted units
mm or in
Range
15 – 75 mm
Typical values
Interior beams: 25–40 mm; Exterior/exposed: 40–60 mm; Marine: 50–75 mm
Tip
Cover is measured to the outer face of the stirrup, not the main bar.

Creep Multiplier (φ)

Units
Dimensionless factor
Range
1.0 – 4.0
Typical
2.0 for indoor RC; 3.0 for outdoor or sustained heavy load
Tip
Long-term deflection = instantaneous × φ. A factor of 2.0 is conservative and widely accepted.
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Section 07

Common Mistakes and How to Avoid Them — Microcopy Reference

Errors engineers and students make most often when sizing beams and slabs

❌ Mistake

Using D instead of d in the L/d check

The overall depth D includes cover and bar placement. Using D makes the beam appear deeper than the code requires — an unconservative error.

✅ Fix

Always enter effective depth (d) in the Known Depth field

The calculator outputs both d and D. Use d for the L/d check, D for your formwork drawings and self-weight estimate.

❌ Mistake

Ignoring the fy modification factor

Using IS 456 or ACI 318 with Fe500 (500 MPa) but not applying the MF means you're using the wrong limit — your beam could be undersized.

✅ Fix

Enter your actual fy value

The calculator computes MF = 0.4 + fy/700 automatically. At Fe500: MF = 1.11 — you can use a slightly shallower section than the basic table suggests.

❌ Mistake

Using centre-to-centre span instead of clear span

Codes specify the clear span (between inner faces of supports). Using c/c span overstates L and gives an overly conservative (needlessly deep) beam.

✅ Fix

Measure from inner face to inner face of supports

For a 6.0 m c/c beam with 300 mm wide columns on each end, clear span = 6.0 − 0.15 − 0.15 = 5.70 m. Enter 5.70 m in the Span field.

❌ Mistake

Applying beam L/d limits to a two-way slab

Two-way slabs have different K values (EC2) and separate table entries in IS 456 and ACI. Using the simply supported beam ratio for a flat plate is very wrong — flat plates can be much shallower.

✅ Fix

Select the correct member type

The calculator has separate entries for One-Way Slab, Two-Way Slab, Flat Slab and Ribbed Slab. Always choose the member type that matches your actual structural system.

❌ Mistake

Forgetting to add self-weight to the load

The L/d check ensures serviceability, but self-weight is a significant portion of the total load for long-span beams. Ignoring it leads to under-design in strength checks.

✅ Fix

Use the calculator's self-weight estimate

The results panel shows wsw (kN/m). Add this to your dead load before running any moment or shear calculation. The Deflection tab also allows you to input the total UDL.

❌ Mistake

Not rounding up the section depth to a practical size

A calculated dmin of 437 mm leads to D = 495 mm — but specifying 495 mm on drawings is unusual and impractical for forming.

✅ Fix

Round D up to the nearest 25 mm or 50 mm

The calculator's recommendation already rounds to a 25 mm grid. In practice, round to 50 mm for beams (e.g., 500, 550, 600 mm) — this simplifies formwork and saves cost.

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Section 08

Key User Pain Points — And How This Calculator Solves Them

The real-world frustrations this tool was designed to eliminate

🔴 Pain Point

Slow manual code table lookup. Opening IS 456 Table 23.2 or ACI Table 9.3.1.1 in a PDF, finding the right row, then applying multiple correction factors by hand takes 10–15 minutes per member type — and errors creep in.

🟢 Solution

The calculator stores all five code tables internally and selects the correct base ratio automatically from your member type and code selection. Modification factors are calculated and applied in under a second.

🔴 Pain Point

Unit conversion mistakes. Mixing millimetres and metres — especially when converting span in metres and depth in millimetres — is the most frequent source of order-of-magnitude errors in preliminary sizing.

🟢 Solution

Every field has an attached unit selector (m, mm, cm, ft, in). The calculator converts all values to mm internally before any computation, so mixed-unit inputs are handled correctly every time.

🔴 Pain Point

Not knowing which code applies. International projects, cross-border procurement, or multi-code submissions require knowing how ACI, EC2, IS 456, BS 8110 and AS 3600 compare for the same member. Doing this manually for all five codes is very tedious.

🟢 Solution

The Code Compare tab runs all five codes simultaneously for your member type and span, displaying minimum required depth and the most liberal vs. most conservative code in a sortable table with a colour-coded bar chart.

🔴 Pain Point

No quick deflection estimate. Running a full deflection analysis in ETABS or SAFE just to check if a preliminary beam section is serviceable is overkill at the early design stage.

🟢 Solution

The Deflection tab uses classic elastic beam formulas (5wL⁴/384EI for simply supported, etc.) to estimate instantaneous and long-term deflection. The Fill from Calculator button automatically populates section dimensions from your main calculation result.

🔴 Pain Point

Difficulty documenting calculations. Sharing preliminary sizing work with clients, reviewers, or team members usually requires a formatted output — not just a scribbled number.

🟢 Solution

The Copy Results button generates a complete formatted text summary including all inputs, applied factors, and final dimensions — ready to paste into a design memo. The Print/PDF button renders a clean print view of the results panel.

🔴 Pain Point

Eurocode 2's complex L/d formula is rarely computed correctly by hand. The EC2 Cl. 7.4.2 expression involves ρ₀, two separate equations (for lightly and heavily reinforced members), and a service stress correction — most engineers skip some terms.

🟢 Solution

The calculator implements the full EC2 Cl. 7.4.2 expression including both ρ ≤ ρ₀ and ρ > ρ₀ branches, with the 310/fs service stress correction applied automatically. Enter fck, ρ, and ρ' and the correct EC2 limit is computed instantly.

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Section 09

Accuracy Note — What This Calculator Does and Does Not Do

Transparency about the limitations of preliminary sizing tools

Built on Published Code Formulas — Designed for Preliminary Sizing

This calculator faithfully implements the span-to-depth ratio formulas from ACI 318-19, EN 1992-1-1:2004, IS 456:2000, BS 8110-1:1997, and AS 3600-2018. The underlying code arithmetic has been verified against hand-calculated examples from each code's design guides. Results are reproducible and match published worked examples within rounding tolerance.

However, span-to-depth ratios are a simplified code tool for preliminary member sizing only. They do not replace a full serviceability analysis. The following factors are simplified or excluded:

Cracked section Ieff not used Shrinkage not modelled Dynamic loads excluded Prestress not in SDR check Pattern loading not considered Axial load not included FEM analysis not performed
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Always verify with a licensed structural engineer. This tool is intended for early-stage feasibility, concept design, checking, and educational use. For projects subject to building regulation approval, all structural calculations must be prepared, checked, and signed off by a qualified engineer in accordance with local regulations.
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Section 10

Frequently Asked Questions About Span-to-Depth Ratio

Answers to the questions engineers, students, and designers ask most

What is the difference between L/d and L/D?

L/d uses the effective depth (d) — from the compression face to the centroid of the tension rebar. This is what all code tables specify.
L/D would use the overall depth (D), which is larger. Using D instead of d gives a lower (more optimistic) ratio and is non-conservative. Never use D in a code L/d check.

Which code should I use — ACI 318, IS 456, or Eurocode 2?

Use the code mandated by the building authority for your project location:
• USA, Canada: ACI 318
• India, Sri Lanka: IS 456
• UK, Malaysia, Singapore, Hong Kong: BS 8110 (or EC2 for newer projects)
• EU member states: Eurocode 2 (EN 1992)
• Australia, New Zealand: AS 3600
If your authority accepts multiple codes, use the Code Compare tab to identify the governing (most conservative) requirement.

Why does Eurocode 2 give a higher L/d limit than ACI 318 for the same beam?

EC2's formula (Cl. 7.4.2) accounts for the actual reinforcement ratio and concrete grade, allowing shallower sections when lightly reinforced (low ρ). ACI 318's table is more prescriptive and tends to be conservative for normal concrete grades. EC2 is often more liberal for lightly loaded slabs and spans under 8 m, but can be more restrictive for heavily reinforced or long-span members. The Code Compare tab shows you exactly which is governing for your specific inputs.

My actual L/d is higher than the code limit — is my beam definitely failing?

Not automatically — but it requires attention. The L/d rule is a simplified code check to avoid detailed deflection calculation. If you exceed the limit, the code requires you to perform a full deflection analysis (cracked section, long-term creep, shrinkage) to verify the beam is still serviceable. It's possible to exceed L/d and still pass the full deflection check — but this must be demonstrated explicitly. Do not ignore a FAIL verdict without a full deflection calculation performed by a structural engineer.

What deflection limit should I use — L/360 or L/240?

The most common limits are:
• L/360: Live load only, for members supporting brittle finishes (plaster, masonry partitions, glass). Most conservative for live load. (ACI 318 Table 24.2.2)
• L/240: Total load (dead + live), general-purpose structural members. (Common default for IS 456, ACI for non-brittle)
• L/250 and L/500: Eurocode 2 — L/250 for appearance, L/500 for damage to non-structural elements.
• L/180: Roofs not supporting ceilings or brittle finishes.
If in doubt, use L/360 for live load and L/240 for total load — this matches the most widely applicable ACI provision.

Does the tool work for prestressed concrete?

Select Prestressed / Post-tensioned Concrete from the Material dropdown. The tool applies the standard RC L/d tables as a conservative baseline — prestressed members can typically achieve L/d ratios 50–80% higher than RC (e.g., L/d = 28–36 for simply supported prestressed beams). For accurate prestressed design, use dedicated software (e.g., PTData, ADAPT). The L/d values shown for prestressed material in this tool are indicative only.

The self-weight shown seems very high — is the formula correct?

Self-weight is calculated as wsw = (b/1000) × (D/1000) × γ where γ = 25 kN/m³ for RC. If the beam width b is automatically set to 0.45 × D and your beam is deep, the self-weight can be significant. For example: b = 270 mm, D = 600 mm → wsw = 0.27 × 0.60 × 25 = 4.05 kN/m. If this seems high, double-check that you haven't entered a very large depth or width. The tool uses the preliminary section size before any optimisation.

Can I save or export my results?

Yes — two options are available:
• Copy Results button: Copies a full formatted text summary to your clipboard (inputs, factors, dimensions, verdict). Paste into Word, email, or a design memo.
• Print / PDF button: Opens your browser's print dialog. Select "Save as PDF" to get a PDF of the results panel. The print stylesheet hides the input form and shows only the results.

What is the "Safety Margin %" shown in results?

Safety Margin = ((L/d limit − Actual L/d) ÷ L/d limit) × 100.
A positive margin means your actual section is deeper than required (conservative). For example, +18% means you have 18% more depth than the minimum.
A negative margin means your section is shallower than the code minimum and needs to be increased. For example, −12% means you need to increase the effective depth by 12%.
A margin of 0–10% is shown as BORDERLINE — technically passing but with very little reserve.

How do I use the Reverse Mode (Find Max Span)?

Click ← Find Max Span at the top of the Calculator tab. Enter your known depth d (effective) in the Known Depth field with the correct unit. The calculator will determine the maximum clear span that section can support under the selected code's L/d limit. This is useful when you have an existing member or a depth-constrained situation (e.g., limited floor zone) and want to know the longest span it can serve.

Span-to-Depth Ratio Calculator — Complete User Guide

References: ACI 318-19 | EN 1992-1-1:2004 (Eurocode 2) | IS 456:2000 | BS 8110-1:1997 | AS 3600-2018

This guide is for educational and preliminary design use only. All structural designs must be verified by a licensed structural engineer in accordance with applicable local codes and regulations.