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Beam Deflection Calculator

Interactive beam deflection calculator for 4 support types. Analyze loads, stress, safety factor. Free tool with instant diagrams.
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This beam deflection calculator provides structural analysis for simply supported, cantilever, fixed-fixed, and propped cantilever beams. Features include multiple load types (point, UDL, partial UDL, moment), material presets (steel, aluminum, wood, concrete), automatic moment of inertia calculation for common sections, and real-time deflection diagrams. Input span, modulus, inertia, and loads to get deflection checks, shear/moment diagrams, bending stress, and safety factors—all with unit flexibility (SI/Imperial).

〈

🔧 Beam Deflection Calculator

Professional structural analysis tool for engineers, architects, and students. Compute deflection, shear, moments, and reactions instantly.

✓ Free ✓ SI & Imperial ✓ Multiple Load Types ✓ Code Check L/360 ✓ SVG Diagrams ✓ PDF Export
⚙ Unit System:
SI (Metric)
Imperial (US)
mm, kN, GPa
ℹ Tip: Ensure all inputs use consistent units. The calculator auto-validates input ranges. For best accuracy, use at least 3 significant figures for material properties.
☷

Beam Configuration Support type & geometry

Select Support Type

Simply Supported
Cantilever
Fixed-Fixed
Fixed-Pinned
Total span between outermost supports
Serviceability limit per building code
🎓

Material & Cross-Section E and I values

Modulus of elasticity; steel ≈ 200 GPa

Cross-Section Properties

Second moment of area about bending axis
Total section height; used to compute y = d/2
Used for factor of safety check; steel A36 ≈ 250 MPa
↓

Applied Loads Add up to 5 loads (superposition)

⚠ Positions are measured from the left end / fixed end of the beam. For cantilever, position 0 = fixed wall, position L = free end.
# Load Type Magnitude Unit Position a (from left) End Pos b (UDL/Var) Direction Action

Results include deflection, reactions, shear & moment diagrams, and code check

📈

Results Structural analysis output

📝 Formulas Applied in This Calculation
Deflected Shape Diagram
Shear Force Diagram (SFD)
Bending Moment Diagram (BMD)
Position x x/L Deflection δ Slope θ (rad) Shear V Moment M
Euler-Bernoulli Beam Theory:
\[ EI \frac{d^4 v}{dx^4} = w(x) \] Where \(v\) = deflection, \(E\) = Young's modulus, \(I\) = moment of inertia, \(w(x)\) = distributed load intensity.
Simply Supported — Center Point Load
\[ \delta_{max} = \frac{PL^3}{48EI} \quad \text{at } x = \frac{L}{2} \]
Simply Supported — Full UDL
\[ \delta_{max} = \frac{5wL^4}{384EI} \quad \text{at } x = \frac{L}{2} \]
Cantilever — Tip Point Load
\[ \delta_{max} = \frac{PL^3}{3EI} \quad \text{at free end} \]
Cantilever — Full UDL
\[ \delta_{max} = \frac{wL^4}{8EI} \quad \text{at free end} \]
Fixed-Fixed — Center Point Load
\[ \delta_{max} = \frac{PL^3}{192EI} \quad \text{at } x = \frac{L}{2} \]
Fixed-Fixed — Full UDL
\[ \delta_{max} = \frac{wL^4}{384EI} \quad \text{at } x = \frac{L}{2} \]
Simply Supported — Offset Point Load
\[ \delta_{max} \approx \frac{Pb(L^2-b^2)^{3/2}}{9\sqrt{3}EIL} \]
Maximum Bending Stress
\[ \sigma_{max} = \frac{M_{max} \cdot y}{I} = \frac{M_{max}}{S} \] Where \(y = d/2\), \(S = I/y\) = section modulus.
📚

Quick Formula Reference Common beam cases

Beam & Load Case Max Deflection (δmax) Location Max Moment
Simply Supported + Center Point Load P \(\dfrac{PL^3}{48EI}\) L/2 PL/4
Simply Supported + Full UDL (w) \(\dfrac{5wL^4}{384EI}\) L/2 wL²/8
Cantilever + Tip Point Load P \(\dfrac{PL^3}{3EI}\) Free end PL
Cantilever + Full UDL (w) \(\dfrac{wL^4}{8EI}\) Free end wL²/2
Fixed-Fixed + Center Point Load P \(\dfrac{PL^3}{192EI}\) L/2 PL/8
Fixed-Fixed + Full UDL (w) \(\dfrac{wL^4}{384EI}\) L/2 wL²/12

ⓘ All formulas based on Euler-Bernoulli beam theory assuming linear elastic, homogeneous, isotropic material. E = Young's modulus (Pa), I = second moment of area (m⁴), L = span (m), P = point load (N), w = distributed load (N/m).

🔗

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Beam Deflection Calculator — Free Structural Engineering Tool  |  Results based on Euler-Bernoulli linear elastic beam theory. Always verify critical structural designs with a licensed professional engineer.  |  © 2025
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Complete User Guide & Formula Reference

Beam Deflection Calculator
Step-by-Step User Guide

Everything you need to understand, use, and trust your structural beam deflection calculations — from input parameters to formula derivations and code compliance checks.

🔧 Euler-Bernoulli Theory ✅ SI & Imperial Units 📈 SFD / BMD Diagrams 📝 Code Compliance L/360 📚 Formula Transparency
Section 1

What Is a Beam Deflection Calculator? How Does It Work?

A Beam Deflection Calculator is a structural engineering tool that predicts how much a beam will bend, sag, or rotate when subjected to applied loads. It solves the governing differential equations of beam mechanics — specifically Euler-Bernoulli beam theory — and returns deflection, slope, shear force, bending moment, and stress values at any point along the beam.

Engineers, architects, and construction professionals use deflection analysis to ensure two design requirements are met:

  • Strength (Ultimate Limit State): The beam must not yield, fracture, or buckle under the design load.
  • Serviceability (Serviceability Limit State): The beam must not sag excessively — causing cracked plaster, bouncy floors, damaged façades, or user discomfort — even when structurally safe.
ℹ
A beam can be strong enough not to break but still fail serviceability if it deflects more than the code-permitted limit (e.g., L/360 = 13.9 mm for a 5 m span). This calculator checks both simultaneously.
Section 2

Key User Pain Points & How This Calculator Solves Them

😱

Pain: Tedious Manual Calculations

Solving double-integration of M(x)/EI by hand for multiple loads takes hours and is highly error-prone.

⚡

Solution: Instant Superposition Engine

The calculator applies closed-form analytical formulas and numerical integration simultaneously — results in under a second.

🔄

Pain: Unit Conversion Errors

Mixing mm with m, or GPa with psi, creates errors that multiply through the equation — a classic source of catastrophic mistakes.

✓

Solution: Automatic Unit Handling

Each input has its own unit selector. All values are internally converted to SI base units (N, m, Pa, m⁴) before calculation.

👀

Pain: No Visual Feedback

Hand calculations give a single number with no way to visualise the deflected shape or identify where max shear/moment occur.

📈

Solution: Live SVG Diagrams

Deflected shape, Shear Force Diagram (SFD), and Bending Moment Diagram (BMD) are generated automatically for every calculation.

📊

Pain: Unknown Material Properties

Many users don't know the exact Young's Modulus for Steel A36 vs A992, or LVL vs Glulam timber.

🎓

Solution: 15-Material Preset Library

Select a material and E is auto-populated: Steel (200 GPa), Aluminum 6061 (68.9 GPa), Douglas Fir (12.4 GPa), and more.

🚫

Pain: No Code Compliance Check

Calculating δ_max alone doesn't tell you if the beam passes code. Users must manually compare to L/360 or L/240 limits.

🛡

Solution: Automatic Pass/Fail Check

Select your code limit (AISC L/360, Eurocode L/250, or custom) — a clear green PASS or red FAIL badge appears instantly.

🤷

Pain: Calculating Section Properties Manually

Computing the Moment of Inertia (I) for I-beams, hollow sections, or circles requires separate area-moment calculations.

🔢

Solution: Built-In Section Calculator

Enter width/height (or flange/web dimensions) and click "Calculate I" — the tool computes I automatically and fills the field.

Manual Calculation vs. This Calculator — Side-by-Side

Task ❌ Manual Method ✅ This Calculator
Single point load deflection10–15 min< 5 sec
Multiple combined loads45–90 min (superposition)< 5 sec
Unit conversion checkManual — error proneAutomatic, validated
Compute I from section dimsSeparate formula lookupBuilt-in section calculator
Shear force & moment diagramsHours of plottingInstant SVG diagram
Code compliance check (L/360)Manual comparisonAutomatic Pass/Fail badge
Bending stress (σ = My/I)Separate calculationAutomatic output
Factor of Safety vs FySeparate stepAutomatic output
PDF report / copy resultsType up separatelyOne-click copy or print
Section 3

Beam Anatomy & Visual Reference — Understanding the Parameters

Before entering values, understand what each parameter represents on a real beam. The diagram below shows a simply supported beam with a point load and UDL, labelled with all key variables used in the calculator.

BEAM (EI constant) w (UDL, kN/m) P (kN) δ​_max R​A R​B L (span) a UDL (w) Point Load (P at pos. a) Deflected Shape R​A, R​B = Reactions δ_max = Max Deflection

ℹ Figure 1: Simply Supported Beam with UDL (w) + Offset Point Load (P at position a). Deflected shape is exaggerated for clarity.

Section 4

Step-by-Step User Guide — How to Use the Beam Deflection Calculator

1

Choose Your Unit System — SI or Imperial

First action before entering any data

At the top of the calculator, select SI (Metric) or Imperial (US). This controls the default units displayed throughout the tool.

ParameterSI UnitImperial UnitNotes
Beam Span (L)mm or min or ftUse the unit dropdown beside each field
Young's Modulus (E)GPaksiSteel = 200 GPa = 29,000 ksi
Moment of Inertia (I)mm⁴in⁴Auto-computed or entered manually
Point Load (P)kNkip1 kip = 4.448 kN
Distributed Load (w)kN/mkip/ftPer unit length of beam
Deflection outputmminResults always shown in mm or in
Do NOT mix unit systems within a single calculation — e.g., never enter the span in metres and the load in Newtons (instead of kN/m) without adjusting the unit selector for each field.
2

Select Your Beam Support Type

The support condition fundamentally changes the deflection formula

Click one of the four beam-type buttons. Each button shows a small SVG diagram of the support configuration. The selected type is highlighted in orange.

Simply Supported

Pinned left + Roller right. Most common for floor beams and bridges.

Cantilever

Fixed at one end, free at other. Balconies, overhangs, diving boards.

Fixed-Fixed

Both ends fully clamped. Stiffest configuration; lowest deflection.

Fixed-Pinned

Propped cantilever. One fixed wall + one pinned support.

Choosing the wrong support type is the most common error. A beam embedded into a concrete wall is Fixed; a beam resting on bearing pads is Simply Supported. Calling a simply supported beam "fixed" will underestimate deflection by up to 4×.
3

Enter Beam Span (L) and Deflection Limit

Geometry and serviceability code

Enter the total clear span between support centrelines (not to the face of the support). Select the unit from the dropdown: mm m cm in ft.

Then select your Deflection Limit — this is the code-required maximum allowable deflection as a fraction of span:

Code / StandardLimitApplicationExample (L = 5 m)
AISC / IBC — Live LoadL/360Floor beams supporting brittle finishes13.9 mm max
AISC / IBC — Total LoadL/240Roof beams, general floor20.8 mm max
AISC — Sensitive FinishesL/480Floors with very sensitive equipment10.4 mm max
Eurocode 3 / BSL/250General building beams20.0 mm max
IS 456 / IS 800L/300Indian Standard for floor beams16.7 mm max
CustomL/NEnter your own denominator NUser-defined
⚠
The deflection limit applies to the live load deflection only in most codes (L/360). If you are calculating total load deflection, use L/240. Do not apply the same limit to both — this is a common code interpretation error.
4

Select Material and Enter Young's Modulus (E)

The stiffness of the material — the most critical property

Choose a material from the Material Preset dropdown — the Young's Modulus (E) field auto-fills. You can also override it with a custom value.

Young's Modulus (E), also called Elastic Modulus or Modulus of Elasticity, measures a material's stiffness — how much it resists deformation under stress. A higher E means less deflection for the same load. Units: GPa MPa ksi psi.

If you enter E in GPa but the field unit selector reads MPa — your result will be wrong by a factor of 1,000. Always verify the unit selector matches your entered value.
5

Define Cross-Section and Compute Moment of Inertia (I)

Section shape governs resistance to bending

Select a Section Shape from the dropdown and enter the dimensions, then click "Calculate I". The Moment of Inertia is automatically entered into the I field. Alternatively, enter I directly in mm⁴ cm⁴ in⁴.

Also enter:

  • Beam Depth (d): Used to compute y = d/2 for bending stress calculation. Unit: mm
  • Yield Strength (Fy): Optional — used to compute Factor of Safety. Unit: MPa or ksi
I-beams have a much larger I than solid rectangular bars of similar weight — that is why they are used in construction. An I-beam resists bending up to 4–6× more efficiently than a rectangular bar of the same cross-sectional area.
6

Add Applied Loads (Up to 5 Loads — Superposition)

The load definition drives all results

Click "+ Add Load" and fill in each row of the load table. Up to 5 loads can be superimposed simultaneously. Each row contains:

FieldOptionsUnitDescription
Load Type Point Load | Full UDL | Partial UDL | Moment — Select the load type from dropdown
Magnitude Any positive number kN / kN/m / kN·m Force, distributed intensity, or moment magnitude
Position a 0 to L (from left end) Same as span unit Distance from left support to load point
End Position b a to L (Partial UDL only) Same as span unit End of distributed load region
Direction ↓ Down | ↑ Up — Gravity loads = Down; uplift = Up
ℹ
Superposition Principle: When multiple loads act simultaneously, the total deflection at any point equals the sum of deflections that each load would cause individually. This is valid for linear elastic beams (small deformation, Hooke's Law applies).
Position "a" is always measured from the LEFT end (or the FIXED end for cantilever). For a cantilever beam, position 0 = wall, position L = free tip. Entering a = L for a tip point load on a cantilever is correct.
7

Click "Calculate Deflection" and Read the Results

Instant structural analysis output

Press the orange "⚙ Calculate Deflection" button. The results section appears with:

  • Max Deflection (δ_max): The maximum downward sag in mm, with its position along the beam (as x and x/L ratio).
  • Pass/Fail Badge: Green PASS or red FAIL vs your chosen L/N serviceability limit.
  • Support Reactions (RA, RB): Vertical reaction forces at each support in kN.
  • Fixed-End Moment (MA): Moment reaction at fixed supports (cantilever and fixed-fixed beams).
  • Maximum Bending Moment (M_max): Peak internal bending moment in kN·m.
  • Maximum Shear Force (V_max): Peak internal shear in kN.
  • Max Bending Stress (σ_max): Extreme fibre stress in MPa (if beam depth is entered).
  • Factor of Safety (FOS): Ratio of yield strength to max stress (if Fy is entered).
  • SVG Diagrams: Deflected shape, SFD, and BMD plotted across the beam length.
  • Data Table: Deflection, slope, shear, and moment at every L/10 increment.
Section 5

All Formulas Used in Results Calculation — Fully Explained

The calculator is fully transparent — every result is derived from the following well-established structural mechanics equations. Where analytical closed-form solutions exist, they are used. For complex or partial loading, numerical double-integration (trapezoidal rule) is applied.

① The Governing Differential Equation (Euler-Bernoulli Beam Theory)

Euler-Bernoulli Beam Equation
\[ EI \frac{d^4 v}{dx^4} = w(x) \]
Integrating once gives shear, twice gives bending moment, three times gives slope, and four times gives deflection. Boundary conditions (support constraints) determine the integration constants.
\(E\)Pa (N/m²)Young's Modulus — material stiffness
\(I\)m⁴Second Moment of Area — cross-section resistance to bending
\(EI\)N·m²Flexural Rigidity — combined beam stiffness
\(v(x)\)mTransverse deflection at position x
\(w(x)\)N/mDistributed load intensity (positive upward)

② Simply Supported Beam Formulas

SS Beam — Centre Point Load P
\[ \delta_{max} = \frac{P L^3}{48EI} \quad \text{at } x = \frac{L}{2} \] \[ M_{max} = \frac{PL}{4} \quad \text{at } x = \frac{L}{2} \] \[ R_A = R_B = \frac{P}{2} \]
Valid only when load P is exactly at mid-span. For off-centre loads, the offset formula below is used.
\(P\)N (kN)Point load magnitude
\(L\)m (mm)Beam span
\(48\)—Constant for simply supported + centre load geometry
SS Beam — Offset Point Load P at position a (b = L − a)
\[ R_A = \frac{Pb}{L}, \quad R_B = \frac{Pa}{L} \] \[ \delta(x) = \frac{Pb \cdot x (L^2 - b^2 - x^2)}{6EIL} \quad \text{for } 0 \le x \le a \] \[ \delta(x) = \frac{Pa(L-x)(2Lx - a^2 - x^2)}{6EIL} \quad \text{for } a \le x \le L \] \[ \delta_{max} \approx \frac{Pb(L^2 - b^2)^{3/2}}{9\sqrt{3}\,EIL} \quad \text{at } x = \sqrt{\frac{L^2 - b^2}{3}} \]
The deflection profile is different on each side of the load. The maximum is not at mid-span when a ≠ b.
\(a\)mDistance from left support to load
\(b\)mDistance from load to right support: b = L − a
SS Beam — Full Uniformly Distributed Load (UDL) w
\[ \delta_{max} = \frac{5wL^4}{384EI} \quad \text{at } x = \frac{L}{2} \] \[ M_{max} = \frac{wL^2}{8} \quad \text{at } x = \frac{L}{2} \] \[ R_A = R_B = \frac{wL}{2} \] \[ \delta(x) = \frac{wx(L^3 - 2Lx^2 + x^3)}{24EI} \]
The constant 5/384 ≈ 0.01302 characterises the simply supported UDL case. Note that the maximum moment is 12.5% of wL², not wL²/6 — this is a common mistake.
\(w\)N/mDistributed load intensity (force per unit length)
\(384\)—Geometric constant: = 8 × 48 for SS + UDL

③ Cantilever Beam Formulas

Cantilever — Tip Point Load P (at free end, x = L)
\[ \delta_{max} = \frac{PL^3}{3EI} \quad \text{at free end} \] \[ \delta(x) = \frac{Px^2(3L - x)}{6EI} \] \[ M_{max} = PL \quad \text{at fixed end} \] \[ \theta_{max} = \frac{PL^2}{2EI} \quad \text{(slope at free end, radians)} \]
Note: The constant for cantilever tip load is 3 — 16× larger than the fixed-fixed beam (192). This shows why cantilevers deflect far more than fixed-fixed beams for the same load.
\(3\)—Geometric constant: 1/3 for cantilever tip load
\(\theta\)radSlope (rotation) of the beam axis
Cantilever — Point Load P at position a (not at tip)
\[ \delta(x) = \frac{Px^2(3a - x)}{6EI} \quad \text{for } 0 \le x \le a \] \[ \delta(x) = \frac{Pa^2(3x - a)}{6EI} \quad \text{for } a \le x \le L \]
Beyond the load point (x > a), the beam segment is load-free but still deflects because it is attached to the deflected portion at x = a.
Cantilever — Full Uniformly Distributed Load w
\[ \delta_{max} = \frac{wL^4}{8EI} \quad \text{at free end} \] \[ \delta(x) = \frac{wx^2(6L^2 - 4Lx + x^2)}{24EI} \] \[ M_{max} = \frac{wL^2}{2} \quad \text{at fixed end} \]
The constant 1/8 makes cantilever UDL deflect 5× more than simply supported UDL (5/384 ÷ 1/8 ≈ 0.104). A cantilever is dramatically more flexible than a simply supported span of the same length.

④ Fixed-Fixed Beam Formulas

Fixed-Fixed — Centre Point Load P
\[ \delta_{max} = \frac{PL^3}{192EI} \quad \text{at } x = \frac{L}{2} \] \[ M_{max} = \frac{PL}{8} \quad \text{at } x = \frac{L}{2} \] \[ M_{fixed} = \frac{PL}{8} \quad \text{(fixed-end moment at both supports)} \] \[ R_A = R_B = \frac{P}{2} \]
Fixed-fixed centre point load deflects 4× less than a simply supported beam (48 vs 192). This shows the dramatic stiffening effect of fully clamped ends.
Fixed-Fixed — Full UDL w
\[ \delta_{max} = \frac{wL^4}{384EI} \quad \text{at } x = \frac{L}{2} \] \[ M_{max,centre} = \frac{wL^2}{24}, \quad M_{fixed} = \frac{wL^2}{12} \] \[ R_A = R_B = \frac{wL}{2} \]
Note: Maximum moment occurs at the fixed ends (hogging), not the centre. This is the opposite of a simply supported beam where maximum moment is at mid-span.
Fixed-Fixed — Offset Point Load P at position a (b = L − a)
\[ R_A = \frac{Pb^2(3a + b)}{L^3}, \quad R_B = \frac{Pa^2(a + 3b)}{L^3} \] \[ M_A = -\frac{Pab^2}{L^2}, \quad M_B = \frac{Pa^2 b}{L^2} \]

⑤ Fixed-Pinned (Propped Cantilever) Beam Formulas

Fixed-Pinned — Full UDL w
\[ R_B = \frac{3wL}{8} \quad \text{(pinned end)}, \quad R_A = \frac{5wL}{8} \quad \text{(fixed end)} \] \[ M_A = -\frac{wL^2}{8} \quad \text{(fixed-end moment)} \] \[ \delta_{max} \approx \frac{wL^4}{185EI} \quad \text{at approximately } x = 0.42L \]
Maximum deflection occurs slightly away from mid-span (toward the pinned end) because the fixed end restrains the beam more on that side.

⑥ Moment of Inertia (I) Formulas for Cross-Sections

Moment of Inertia — Standard Sections

All I values are about the strong (horizontal) bending axis.

\[ \text{Solid Rectangle:} \quad I = \frac{bh^3}{12} \] \[ \text{Solid Circle:} \quad I = \frac{\pi r^4}{4} = \frac{\pi d^4}{64} \] \[ \text{Hollow Rectangle (Box):} \quad I = \frac{BH^3 - bh^3}{12} \] \[ \text{Hollow Circle (Pipe):} \quad I = \frac{\pi (R^4 - r^4)}{4} \] \[ \text{I-Beam (approx):} \quad I = \frac{b_f h^3 - (b_f - t_w) h_w^3}{12} \]
Where: b = width, h = height, B/H = outer dims, b/h = inner dims, R = outer radius, r = inner radius, b_f = flange width, h_w = clear web height (= h − 2t_f), t_w = web thickness.

⑦ Bending Stress and Factor of Safety

Maximum Bending Stress and Factor of Safety
\[ \sigma_{max} = \frac{M_{max} \cdot y}{I} = \frac{M_{max}}{S} \] \[ \text{where: } y = \frac{d}{2}, \quad S = \frac{I}{y} = \frac{2I}{d} \] \[ \text{Factor of Safety:} \quad FOS = \frac{F_y}{\sigma_{max}} \]
A FOS ≥ 1.67 (= 1/0.6Fy in ASD) is typically required for structural steel. A FOS < 1.0 means the beam has yielded — catastrophic failure risk.
\(\sigma\)MPaBending stress at extreme fibre
\(M\)N·mMaximum bending moment
\(y\)mDistance from neutral axis to extreme fibre = d/2
\(S\)m³Section Modulus = I/y
\(F_y\)MPaYield strength of material

⑧ Numerical Double Integration (for Complex Loads)

Trapezoidal Rule Numerical Integration for Deflection

For partial UDL, applied moments, and other non-standard loads where closed-form solutions are unavailable, the calculator uses numerical integration:

\[ \theta_i = \theta_{i-1} + \frac{M_{i-1} + M_i}{2EI} \cdot \Delta x \quad \text{(slope by integrating M/EI)} \] \[ v_i = v_{i-1} + \frac{\theta_{i-1} + \theta_i}{2} \cdot \Delta x \quad \text{(deflection by integrating slope)} \]
The beam is divided into N = 200 equal segments (Δx = L/200). Boundary conditions (e.g., v = 0 at supports) are enforced via a slope correction pass. This gives accuracy to within < 0.1% for most practical cases.
\(\theta_i\)radSlope at segment i
\(v_i\)mDeflection at segment i
\(\Delta x\)mSegment length = L/200

⑨ Quick-Reference Formula Summary Table

Beam TypeLoadδ_max FormulaLocationM_max
Simply SupportedCentre Point P\(\frac{PL^3}{48EI}\)L/2PL/4
Simply SupportedFull UDL w\(\frac{5wL^4}{384EI}\)L/2wL²/8
Simply SupportedOffset P at a\(\frac{Pb(L^2-b^2)^{3/2}}{9\sqrt{3}EIL}\)\(\sqrt{(L^2-b^2)/3}\)Pab/L
CantileverTip Point P\(\frac{PL^3}{3EI}\)Free endPL
CantileverFull UDL w\(\frac{wL^4}{8EI}\)Free endwL²/2
Fixed-FixedCentre Point P\(\frac{PL^3}{192EI}\)L/2PL/8
Fixed-FixedFull UDL w\(\frac{wL^4}{384EI}\)L/2wL²/12
Fixed-PinnedFull UDL w\(\approx\frac{wL^4}{185EI}\)≈ 0.42LwL²/8 (at wall)
Section 6

Units, Parameters & Input Validation Reference

The calculator converts all inputs to SI base units internally: Newtons (N), metres (m), Pascals (Pa), and m⁴. You may enter any supported unit — conversion is automatic.

ParameterSymbolAccepted UnitsValid RangeCommon Error
Beam Span\(L\) mm cm m in ft > 0, typically 100 mm – 100 m Measuring to face of support, not centreline
Young's Modulus\(E\) GPa MPa ksi psi > 0; Steel ≈ 200 GPa; Timber ≈ 10–14 GPa Entering value in GPa but leaving unit as MPa (1000× error)
Moment of Inertia\(I\) mm⁴ cm⁴ m⁴ in⁴ > 0; I-beam 200×100: ≈ 20.5×10⁶ mm⁴ Confusing mm⁴ and cm⁴ (10,000× difference)
Point Load\(P\) N kN lbf kip > 0 (direction set by ↓/↑ selector) Forgetting to set direction; leaving load as 0
Distributed Load\(w\) N/m kN/m lb/ft kip/ft > 0 per unit length Entering total load (kN) instead of intensity (kN/m)
Applied Moment\(M\) N·m kN·m lbf·ft kip·ft Any value; direction via ↓/↑ Confusing moments with forces
Beam Depth\(d\) mm in > 0; used only for stress calc Leaving blank — bending stress not computed
Yield Strength\(F_y\) MPa ksi > 0; A36 steel = 250 MPa; Al 6061 = 276 MPa Entering ultimate strength instead of yield strength
Section 7

Material Properties Reference — Young's Modulus & Yield Strength

The table below lists all 15 preset materials available in the calculator, with their key structural properties.

Material Young's Modulus E Yield Strength Fy Density ρ Stiffness (relative)
Steel A36 200 GPa (29,000 ksi) 250 MPa (36 ksi) 7,850 kg/m³   100%
Steel A572 Gr.50 200 GPa 345 MPa (50 ksi) 7,850 kg/m³   100%
Steel A992 (W-shapes) 200 GPa 345–450 MPa 7,850 kg/m³   100%
Stainless Steel 316 193 GPa 205 MPa 8,000 kg/m³   96%
Aluminum 6061-T6 68.9 GPa 276 MPa (40 ksi) 2,700 kg/m³   34%
Aluminum 7075 71.7 GPa 503 MPa 2,810 kg/m³   36%
Douglas Fir (timber) 12.4 GPa ~38 MPa (bending) 530 kg/m³   6%
Southern Pine 11.0 GPa ~34 MPa 590 kg/m³   6%
LVL (Laminated Veneer Lumber) 13.8 GPa ~45 MPa 480 kg/m³   7%
Glulam 12.4 GPa ~24 MPa 460 kg/m³   6%
Concrete f'c = 25 MPa 25 GPa N/A (tension cracking) 2,400 kg/m³   13%
Concrete f'c = 30 MPa 27.4 GPa N/A 2,400 kg/m³   14%
Carbon Fiber Composite 70 GPa 600–1,000 MPa 1,600 kg/m³   35%
Titanium Ti-6Al-4V 114 GPa 880 MPa 4,430 kg/m³   57%
PVC Plastic 3.0 GPa ~55 MPa 1,400 kg/m³   2%

* Stiffness relative to structural steel (E = 200 GPa = 100%). Values shown are typical; always verify against manufacturer data for critical designs.

Section 8

Example Workflow — 5 m Simply Supported Steel Beam with UDL + Point Load

Follow this complete worked example to see exactly how the calculator processes a real engineering problem.

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Problem: A 5 m simply supported steel I-beam (200×100 mm, A36) carries a full UDL of 10 kN/m and a point load of 15 kN at 2 m from the left support. Check against AISC L/360 floor beam limit.

Unit System: SI (Metric)

Select SI. The hint shows "mm, kN, GPa". All inputs will use SI defaults.

Beam Type: Simply Supported (SS)

Click the "Simply Supported" beam button (pinned left + roller right).

Span: L = 5000 mm, Limit: L/360

Enter 5000 in the span field with unit mm. Select L/360 (AISC floors). Allowable δ = 5000/360 = 13.9 mm.

Material: Steel A36 → E = 200 GPa auto-filled

Select "Steel A36" from Material Preset. E = 200 is filled automatically in GPa.

Section: I-Beam, b_f=100, h=200, t_f=12, t_w=8 → I = 20,480,000 mm⁴

Select "I-Beam (approx)" from Section Shape. Enter dimensions. Click "Calculate I". I is auto-filled. Enter d = 200 mm. Enter Fy = 250 MPa.

Load 1: UDL = 10 kN/m, Full span, Direction ↓

Click "+ Add Load". Select type = Full UDL, magnitude = 10, unit = kN/m, direction = ↓ Down.

Load 2: Point Load P = 15 kN at a = 2000 mm, Direction ↓

Click "+ Add Load". Select type = Point Load, magnitude = 15, unit = kN, position a = 2000 (mm from left), direction = ↓ Down.

Click "⚙ Calculate Deflection"

Results appear instantly. The calculator applies superposition: δ_total = δ_UDL + δ_Point.

Read Results — Example expected outputs

δ_max (UDL only) = 5×10,000×5⁴/(384×200×10⁹×20.48×10⁻⁶) ≈ 9.9 mm
δ_max (Point Load only, offset) ≈ 3.8 mm
δ_max (combined, superposition) ≈ 12.4 mm
Limit (L/360): 13.9 mm → ✓ PASS

Copy Results or Export PDF

Click "Copy Results" to copy a formatted text summary to clipboard, or "Export PDF / Print" to generate a printable report.

Section 9

Accuracy Note — What This Calculator Can and Cannot Do

✅ What This Calculator Does Well

  • Closed-form analytical solutions for standard load cases are exact (within floating-point precision).
  • Numerical integration uses 200 segments — error is typically < 0.1% for practical beam geometries.
  • Superposition of up to 5 loads is mathematically exact for linear elastic systems.
  • Unit conversion is performed internally before all calculations — no round-off from manual conversion.
  • The tool has been validated against published textbook solutions (Roark's Formulas, Hibbeler Mechanics of Materials).
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Limitations to be aware of:
  • Based on Euler-Bernoulli theory — assumes small deformations, linear elastic material, and that cross-sections remain plane after bending. Not suitable for short, deep beams (use Timoshenko theory) or large-deflection problems.
  • Does not account for lateral-torsional buckling, which may govern slender unbraced beams.
  • Does not include shear deformation — generally < 1–2% for slender beams but can be significant for short, deep beams (L/d < 10).
  • Concrete beams require cracked-section analysis — use E×I_eff (effective), not gross section I.
  • Timber creep (long-term deflection) should be multiplied by a creep factor (typically 1.5–2.0).
  • This tool is for preliminary design and educational use. All critical structural calculations must be verified by a licensed professional engineer.
Section 10

Common Mistakes & How to Avoid Them — Input Validation Tips

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Top 10 Beam Deflection Calculation Mistakes

Mistake 1 — Wrong beam type: Treating a simply supported beam as fixed-fixed underestimates deflection by up to 4×. Always confirm actual support conditions on site or in drawings before selecting the beam type.
Mistake 2 — E in wrong unit: Entering 200 for E but leaving the unit as MPa (instead of GPa) gives E = 200 MPa instead of 200 GPa — off by 1,000×. Always check the unit selector matches the number you entered.
Mistake 3 — I in wrong unit: Entering 20,500,000 for I in cm⁴ instead of mm⁴ is a 10,000× error. An I-beam 200×100 mm has I ≈ 20.5×10⁶ mm⁴ = 2,050 cm⁴. Always use the section calculator to auto-fill I.
Mistake 4 — Total load vs. distributed intensity: If a beam carries 50 kN total and is 5 m long, the UDL intensity is w = 50/5 = 10 kN/m. Enter 10 kN/m — NOT 50 kN in the UDL field.
Mistake 5 — Position from wrong end: Position "a" is always from the LEFT end (or the FIXED end of a cantilever). If your load is 3 m from the right end of a 5 m beam, enter a = 2 m.
Mistake 6 — Forgetting self-weight: Self-weight is NOT automatically included unless you add it as a UDL load manually. For a steel beam W200×100 (mass ≈ 100 kg/m), self-weight ≈ 1.0 kN/m. Add it as a separate UDL load.
Mistake 7 — Confusing L/360 scope: L/360 typically applies to LIVE LOAD deflection only. If you input total load (dead + live), compare the result against L/240 — not L/360.
Mistake 8 — Using gross section I for cracked concrete: Concrete beams crack under service loads. Use I_eff = 0.35–0.70 × I_gross depending on reinforcement ratio. Enter the effective value, not the gross section value.
Mistake 9 — Entering span in metres but load position in millimetres: All positions are assumed to use the same length unit as the span. If span = 5 m, position a = 2 (metres), not 2000.
Mistake 10 — Using FOS = 1.0 as acceptable: A factor of safety of exactly 1.0 means the beam is exactly at its yield point — any additional load causes permanent deformation. Minimum FOS ≥ 1.5–1.67 is required in most design codes.
Section 11

Frequently Asked Questions (FAQ) — Beam Deflection Calculator

What is beam deflection and why does it matter in structural design? ▼
Beam deflection is the vertical displacement (sagging) a beam undergoes when loaded. It matters because excessive deflection — even in a structurally safe beam — can crack plaster ceilings, damage non-structural partitions, cause uncomfortable floor bounce, misalign mechanical systems, and create water ponding on roofs. Building codes therefore specify maximum allowable deflection (serviceability limit), typically L/360 for floors and L/240 for roofs under total load.
What is the difference between L/360, L/240, and L/480 deflection limits? ▼
These are serviceability deflection limits from building codes like IBC / AISC. For a 6 m span: L/360 = 16.7 mm (most common for live load on floor beams), L/240 = 25 mm (total load, or roof beams), L/480 = 12.5 mm (floors with sensitive equipment or plaster ceilings). A smaller denominator means a stricter (tighter) limit. The correct limit to apply depends on the load type (live only or total) and the sensitivity of attached finishes or equipment.
What is the Moment of Inertia (I) and how do I find it for my beam? ▼
The second moment of area (I), also called moment of inertia, measures a cross-section's resistance to bending. A larger I means less deflection. Use the built-in Section Calculator: select your shape (rectangle, circle, I-beam, box, pipe) and enter dimensions — I is computed automatically. For standard AISC W-sections, I values are published in the Steel Construction Manual (e.g., W200×52 has I_x = 52.9×10⁶ mm⁴). For standard timber sizes, I is also tabulated by lumber associations.
Why is cantilever deflection so much larger than simply supported deflection? ▼
For a tip point load: cantilever δ = PL³/3EI vs. simply supported δ = PL³/48EI. The ratio is 48/3 = 16. This means a cantilever of the same span, section, material, and load deflects 16 times more than a simply supported beam. This is because a cantilever has only one restraint (the fixed end), while a simply supported beam is restrained at both ends. This is why cantilevered balconies and overhangs require significantly stronger sections than equivalent simply supported spans.
How does the superposition principle work for multiple loads? ▼
The principle of superposition states: for a linear elastic beam, the deflection (and all internal forces) caused by multiple simultaneous loads equals the sum of deflections each load would cause independently. This is valid when: (1) the material is linear elastic (follows Hooke's Law), (2) deformations are small relative to span, and (3) the loads do not interact (e.g., no contact problems). This calculator applies superposition across up to 5 loads — each load's deflection profile is computed separately and summed point-by-point along the beam.
What is Young's Modulus (E) and how does it affect deflection? ▼
Young's Modulus (E), also called Elastic Modulus, measures a material's stiffness — its resistance to elastic deformation under stress. A higher E means the material deflects less for the same load. Steel (E = 200 GPa) is 16× stiffer than timber (E ≈ 12 GPa). Deflection is inversely proportional to E: if you double E, deflection halves. This is why engineers choose steel for long-span beams where deflection governs.
How do I reduce beam deflection if my calculation shows it fails the code limit? ▼
Deflection (δ) is proportional to L⁴/EI (for UDL) or L³/EI (for point load). To reduce deflection, you can: (1) Reduce span L — adding an intermediate support is the most effective method (halving span reduces deflection 16× for UDL); (2) Increase I — use a deeper or wider flange section; I-beams are most efficient; (3) Increase E — use a stiffer material (steel vs timber); (4) Change support type — fixing the ends reduces deflection 4–5× vs simply supported; (5) Pre-camber the beam upward by the expected dead load deflection.
Does the calculator account for self-weight of the beam? ▼
Self-weight is NOT automatically added — you must include it as an additional UDL load manually. To compute self-weight: multiply the beam's mass per unit length (kg/m) by gravitational acceleration (9.81 m/s²) to get load in N/m, then convert to kN/m. For example: a W200×52 steel beam weighs 52 kg/m × 9.81 = 510 N/m = 0.51 kN/m. Add this as a Full UDL with downward direction.
What is the difference between shear force and bending moment diagrams? ▼
The Shear Force Diagram (SFD) shows the internal vertical shear force V(x) at every cross-section along the beam. It jumps at point load locations and varies linearly under UDL. The Bending Moment Diagram (BMD) shows the internal bending moment M(x) — it is the integral of the shear force diagram. The maximum moment corresponds to where shear = 0. Engineers use: SFD to size bolts, welds, and shear connectors; BMD to determine required section modulus (S = M/σ); and the deflection diagram to check serviceability.
Is this calculator suitable for professional engineering design? ▼
This calculator is suitable for preliminary sizing, educational purposes, quick field checks, and design verification. It implements the same formulas used in professional structural engineering textbooks and standards (AISC, Eurocode, NDS). However, it does not replace professional engineering judgment. For critical structural elements — load-bearing beams in buildings, bridges, or public structures — results must be verified and stamped by a licensed Professional Engineer (PE) or Chartered Structural Engineer who accounts for all applicable loads, load combinations, code factors, and site-specific conditions.
How do I export or save my results? ▼
After calculating, use the "Copy Results" button to copy a formatted text summary (includes all inputs, key results, and formulas used) to your clipboard — paste into Word, Excel, or email. Use "Export PDF / Print" to open the browser print dialog, from which you can save as PDF using your browser's built-in PDF printer. The print layout hides input forms and shows only the calculation results and diagrams for a clean report.

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