AtlasCalcsUtility Calculation Suite
August 2026 11 min read

Beam Load Calculator

Free beam load calculator for mechanical and structural engineering. Compute support reactions, shear force, bending moment, deflection and bending stress for simply supported and cantilever beams under point and distributed loads.

Beam Load Calculator

Beam Loading Details

Beam Support Type
Simply Supported Cantilever (fixed left)
Unit System
Span Length (mm)
mm

Distance between support centres (wall-to-tip for cantilever)

Point Load P (N)
N

Concentrated force; use 0 for distributed-load-only beams

Position of P (from left) (mm)
mm
Distributed Load w (N/mm)
N/mm

Uniform over full span; include self-weight if significant

Section Width b (mm)
mm
Section Depth h (vertical) (mm)
mm
Material
Auto-calculate instantly

Maximum Bending Moment

37,462.45kN·m

Peak internal moment in the span — governs bending stress.

Design passes with comfortable margin
Reaction R₁

20,000N

Reaction R₂

20,000N

Max Deflection

9.703mm

Bending Stress

56.19MPa

Stress Utilization

34%of allowable

Total Distributed Load

30,000N

Section Modulus S

666.7cm³

Moment of Inertia I

6,667cm⁴

Reaction R₁

20,000N

Left support reaction

Reaction R₂

20,000N

Right support reaction

Max Deflection

9.703mm

≈ L/618 of span

Bending Stress

56.19MPa

Extreme-fibre σ = M/S

Stress Utilization

34%of allowable

Allowable 165 MPa

Total Distributed Load

30,000N

W = w × L

Section Modulus S

666.7cm³

Elastic section modulus

Moment of Inertia I

6,667cm⁴

Second moment of area

Result Summary

Beam TypeSimply Supported
Span L6,000 mm
Point Load P @ a10,000 N @ 3,000 mm
Distributed Load w5 N/mm
Cross-Section b × h100 × 200 mm
Deflection Limit Check (L/360)Pass
Strength CheckOK (34% used)
Estimated Result — Results use Euler–Bernoulli beam theory with linear-elastic material behaviour and idealised supports. Real beams include self-weight, connection flexibility, lateral-torsional buckling and load factors governed by building codes. Have final designs verified by a qualified engineer.

Interactive Diagram

Point load P Distributed load w Support reactions Deflected shape

Diagram updates live with your inputs — arrow positions, reaction sizes and the dashed deflected shape reflect the current load case.

Visualizations

Force Overview (N)

20,000NReaction R₁
20,000NReaction R₂
10,000NPoint Load P
30,000NUDL Total W
20,000NMax Shear V

Bending Moment Distribution Along Span

Strength Utilization

σ / σ_allow34%

Formula

Reactions follow from static equilibrium. Bending moment is integrated from the shear diagram, deflection comes from the Euler–Bernoulli elastic curve, and bending stress divides peak moment by elastic section modulus.

M_max = P·a·b/L + w·L²/8 | σ = M_max/S | δ_max = f(P, w, L, E, I)

Alternative Form

Simply supported: R₁ = P(L−a)/L + wL/2 , R₂ = P·a/L + wL/2 ; Cantilever: R₁ = P + wL , M_fix = P·a + wL²/2

Variables

SymbolNameUnitDescription
LSpan lengthmm / inDistance between supports (or wall to free end for cantilever)
PPoint loadN / lbfConcentrated force applied at distance a
aLoad positionmm / inDistance of P from the left support/fixed end
wDistributed loadN/mm / lbf/inUniformly distributed load over the full span
R₁, R₂Support reactionsN / lbfVertical forces supplied by the supports
M_maxMaximum bending momentN·mmPeak internal moment; governs bending stress
V_maxMaximum shear forceNPeak transverse internal force
ISecond moment of areamm⁴b·h³/12 for a rectangle
SSection modulusmm³I/(h/2); links moment to stress
EYoung's modulusMPaMaterial elastic stiffness
δ_maxMaximum deflectionmmLargest transverse displacement
σBending stressMPaM_max/S at extreme fibre

Explanation

Equilibrium gives the reactions; the shear diagram integrates the loads and the moment diagram integrates the shear. Where shear crosses zero the moment peaks — that station sets the required section modulus. Deflection follows the differential equation EI·v″ = M solved with boundary conditions for each support type, superposed for the two load components.

Derivation

Starting from EI·d²v/dx² = M(x): integrate once for slope θ(x) and again for deflection v(x), fixing constants from support conditions (v = 0 at both simple supports; v = θ = 0 at a fixed end). Closed-form results — PL³/48EI for a central point load and 5wL⁴/384EI for a full UDL — combine by superposition because the governing equation is linear.

Step-by-Step Calculation

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

Step 1

Convert all input loads and dimensions to a consistent unit basis.

L, P, a, w, b, h → SI base units
Span:6,000 mm
Point load:10,000 N
Position:3,000 mm
UDL:5 N/mm
Intermediate: Total distributed load W = w·L = 30,000 N
Result: Consistent input basis established
Step 2

Compute section properties of the rectangular cross-section.

I = b·h³/12 | S = I/(h/2)
b:100 mm
h:200 mm
Intermediate: I = 66,666,667 mm⁴
Result: S = 666,667 mm³
Step 3

Apply equilibrium to find both support reactions.

R₁ = P(L−a)/L + wL/2 ; R₂ = P·a/L + wL/2
Intermediate: ΣF_vertical = 0 and ΣM_support = 0
Result: R₁ = 20,000 N, R₂ = 20,000 N
Step 4

Evaluate shear and moment diagrams by superposition along the span.

V(x) = ΣR − ΣP − w·x ; M(x) = ∫V dx
Intermediate: Peak shear V_max = 20,000 N
Result: M_max = 37,462.45 kN·m
Step 5

Compute maximum deflection via exact elastic superposition.

δ(x) = δ_P(x) + δ_w(x), maximised numerically
Intermediate: E·I = 200,000 MPa × 66,666,667 mm⁴
Result: δ_max = 9.703 mm ≈ L/618
Step 6

Check extreme-fibre bending stress against the allowable value.

σ = M_max / S ≤ σ_allow
M_max:37,462.45 kN·m
S:666,667 mm³
Intermediate: σ_max = 56.19 MPa vs allowable 165 MPa
Result: Utilization = 34% → PASS
Final Answer
37,462.45 kN·m

Worked Examples

See how the calculator works with different scenarios.

Scenario

A library shelf beam carries a single central box load on a simple span.

Inputs

Support:Simply Supported
L:3000 mm
P:5000 N @ 1500 mm
w:0 N/mm
b×h:50 × 150 mm
Material:Steel

Calculation

R₁ = R₂ = 2500 N; M_max = P·L/4 = 5000×3000/4 = 3.75×10⁶ N·mm; I = 50×150³/12 = 1.406×10⁷ mm⁴; S = 187,500 mm³

Result

σ = 20 MPa (12% of allowable); δ_max ≈ 2.0 mm (L/1490)

Interpretation

A very comfortable design — the shelf could carry several times this load before approaching allowables.

Understanding Your Result

Your Result: 37,462.45 kN·m

Typical range: Healthy designs typically use 40–80% of allowable bending stress and keep total deflection below L/240–L/360. Yours: 34% utilization and L/618.

What This Means

Your beam experiences a peak bending moment of 37,462.45 kN·m, producing a maximum fibre stress of 56.19 MPa and deflection of 9.703 mm. That stress consumes 34% of the selected material's allowable value.

Why It Matters

Bending stress decides whether a beam breaks; deflection decides whether occupants feel it bounce and finishes crack. Checking both against allowables is the core of routine member sizing in mechanical and structural practice.

Professional Interpretation

The section is comfortably adequate — consider whether a lighter standard section would be more economical.

Reactions

The supports supply 20,000 N and 20,000 N. Their sum equals total applied load (40,000 N), confirming equilibrium.

Moment Diagram Shape

The moment diagram rises smoothly from zero at each support, peaking between the loads. The chart panel plots your exact curve.

Serviceability

Your deflection ratio is L/618. Floors commonly target L/360 (live) or L/240 (total); brittle finishes may demand L/480.

Recommendations

  • •If deflection governs, increasing depth helps far more than adding width because I grows with the cube of depth.
  • •Keep laterally unsupported compression flanges braced — long thin beams can buckle sideways well before bending stress limits are reached.
  • •Document your assumed loads — future modifications often exceed original design loads.
  • •Verify bearing at supports and connections; global analysis assumes these details can actually deliver the reactions.

Educational Content

Introduction

Beams are among the oldest and most universal structural elements — every shaft, chassis rail, floor joist and machine frame behaves as a beam under load. Understanding how loads map into reactions, internal shear, bending moment and deflection is foundational engineering knowledge, and a beam load calculator turns that classical theory into instant answers.

Concept Explanation

A loaded beam bends into an elastic curve. Internal forces develop to resist this curvature: transverse shear V balances vertical loads, and bending moment M balances the rotational effect of those loads. Equilibrium (ΣF = 0, ΣM = 0) fixes support reactions; integration of the load function yields shear and moment diagrams; and the Euler–Bernoulli relation EI·v″ = M converts moment into deflected shape. Stress follows from σ = M·y/I, maximal at the fibres farthest from the neutral axis.

Applications

▸Sizing floor joists, roof purlins and lintels in buildings
▸Checking machine frames, bases and gantry beams for rigidity
▸Verifying conveyor walkway and platform members
▸Preliminary bridge-stringer and trailer-chassis assessment
▸Selecting shelving, rack beams and workbench tops
▸Teaching aid for shear/moment diagram fundamentals

Benefits

  • •Instant equilibrium answers without manual diagram construction
  • •Combined point + distributed loading solved correctly by superposition
  • •Automatic conversion between SI and imperial units
  • •Immediate strength AND serviceability feedback (stress + deflection)
  • •Live diagrams that build intuition about load position effects

Limitations

  • •Linear-elastic, small-deflection theory only — no plastic redistribution
  • •One point load plus one UDL; multi-load cases need superposition software
  • •Idealised supports — no partial fixity or spring settlements
  • •No lateral-torsional buckling, shear crushing or local buckling checks
  • •Not a substitute for code-based design with factored load combinations

Best Practices

  • •Always include self-weight in the distributed load unless negligible (<5% of other loads)
  • •Check movable loads at several positions and design for the worst case
  • •Prefer deeper sections over wider ones when deflection governs
  • •Compare results against span/360 and span/240 serviceability targets
  • •Keep a safety factor of at least 1.5 against yield for static duties

Common Misconceptions

Myth: A bigger cross-sectional area always means a stiffer beam.
Truth: Stiffness depends on where the area sits, not just how much: I grows with depth cubed, so a tall thin section vastly outperforms a compact one of equal area.
Myth: If stress passes, the beam is fine.
Truth: Many beams pass strength yet fail serviceability — bouncing floors, cracked plaster — because deflection limits are stricter than strength limits.
Myth: Cantilevers are just half of a simply supported beam.
Truth: A cantilever tip load generates 4× the moment and 16× the deflection of the same load at the centre of a simple span of equal length.
Myth: Adding more load anywhere affects all reactions equally.
Truth: Reactions split according to load position: a load at quarter-span puts 75% onto the nearest support.

Practical Tips

  • •Current utilization is 34%. Aim for 50–80% in final designs to leave room for unforeseen loads.
  • •Keep live-load deflection within L/360 for occupied floors.
  • •Move the point load slider-style: try a = L/2 versus a = L/4 to feel how quickly reactions shift.
  • •For dynamic equipment multiply working loads by 1.3–2.0, or perform a proper vibration check.
  • •Record the governing load case in your calc package so reviewers can trace the numbers.

Common Mistakes

Mixing unit systems
Entering millimetres in a field labelled inches silently inflates results by 25.4³ in stiffness terms. Choose the unit system first, then enter matching values.
Forgetting self-weight
Steel weighs ~7850 kg/m³. Add a small UDL representing dead weight — for slender beams it can be 10–20% of the total load.
Reading M_max from the wrong location
Peak moment occurs where shear crosses zero, not necessarily under the point load when UDLs act simultaneously — use the plotted diagram.
Comparing deflection to the wrong limit
L/360 applies to LIVE load on floors; total load limits are looser (L/240) while brittle finishes need stricter ones (L/480).
Assuming fixed ends without fixed connections
Unless joints genuinely resist rotation, treat supports as pinned. Assuming fixity is unconservative and a classic cause of sagging real-world beams.

Assumptions

This calculator makes the following assumptions to provide estimates:

Euler–Bernoulli theory: Plane sections remain plane; shear deformation is neglected (valid for span ≥ ~10× depth).
Linear elasticity: Material stress-strain stays proportional; no yielding, creep or cracking is modelled.
Small deflections: Slopes are tiny enough that geometry changes do not alter the load path.
Full-length UDL: The distributed load covers the entire span uniformly — partial-span loads are not supported.
Ideal supports: Simple supports are frictionless pins/rollers; the cantilever wall is perfectly rigid.

Limitations

Be aware of these limitations when using this calculator:

Single point load: Multiple concentrated loads require repeated superposition or structural software.
Rectangular section only: I-shapes, channels and hollow tubes have different I and S — convert via their published properties.
No stability checks: Lateral-torsional buckling, web crippling and local buckling are excluded; deep slender beams may fail earlier than indicated.
Static loads only: Impact, fatigue and resonance require dynamic analysis beyond this tool.
No code verification: Output is a mechanics check, not a stamped design. Governing codes (AISC, Eurocode, NDS) take precedence.

Frequently Asked Questions

Answers to common questions about this calculator.

What is a beam load calculator?
A beam load calculator determines how applied loads — point forces and distributed loads — are carried by a beam. It computes support reactions, internal shear forces, bending moments, deflection and bending stress so you can verify that a beam is strong and stiff enough for its duty.
What is the difference between a point load and a distributed load?
A point load acts at a single location (for example a column bearing on a beam) and is measured in newtons or pounds. A distributed load spreads force along the span (for example self-weight or flooring) and is measured in N/mm or lbf/ft. This calculator superimposes both effects.
How are support reactions calculated?
For a simply supported beam, reactions follow from static equilibrium. Each reaction carries half of the total distributed load, plus a share of the point load inversely proportional to its distance from that support: R_left = P·(L−a)/L + wL/2 and R_right = P·a/L + wL/2.
What is bending moment?
Bending moment is the internal moment that develops in a beam as it resists loads trying to bend it. It peaks where shear crosses zero — at mid-span under a central point load, or at wL²/8 for a uniform load on a simply supported span.
Where does maximum deflection occur?
For a simply supported beam with a central point load or a full-span uniform load, maximum deflection occurs at mid-span. For an off-centre point load it shifts toward the load but stays within roughly L/2 of mid-span. For cantilevers it always occurs at the free end.
What is the relation between load and deflection?
Deflection scales linearly with load and with the cube of span length. Doubling the span increases deflection about eight-fold for distributed loading, which is why stiffness — not strength — often governs long beams.
What is section modulus and why does it matter?
Section modulus S = I/c measures how efficiently a cross-section resists bending. Bending stress equals M/S, so doubling S halves stress. Deep sections place material far from the neutral axis and are far more efficient than solid square sections of equal area.
How do I compute the second moment of area?
For a rectangle of width b and depth h: I = b·h³/12. For a circle of diameter d: I = π·d⁴/64. The calculator computes I automatically from the dimensions you enter.
Is a simply supported beam stronger than a cantilever?
For identical spans and loads, yes — dramatically. A cantilever point load produces four times the bending moment and sixteen times the deflection of the same load centred on a simply supported span. Support conditions dominate structural behaviour.
What safety factor should I use for beam design?
Typical allowable-stress practice applies factors of 1.5–2.0 against yield for static structural steelwork, higher for dynamic or fatigue service. Building codes may prescribe specific reduction factors; this calculator flags utilization against the allowable stress you select.
Does self-weight matter?
Often yes. Steel beams weigh 0.6–1.2 N/mm per metre-length equivalent and concrete much more. If unknown, add a conservative distributed load to represent self-weight rather than ignoring it.
Can I use this calculator for wood joists?
Yes, provided you enter appropriate values: wood E ≈ 9–12 GPa and allowable bending stress 6–15 MPa depending on grade and duration of load. Verify against the governing timber design standard such as the NDS.
Why is my deflection too large even though stress is low?
Long slender beams frequently fail a serviceability (deflection) limit before reaching any strength limit. Common practice limits live-load deflection to span/360 and total deflection to span/240. Increase depth — the most efficient fix — or add supports.
What is the maximum moment for a UDL?
For a uniformly distributed load w over a simply supported span L, maximum moment is wL²/8 at mid-span and each support carries wL/2. These two results come straight from equilibrium and symmetry.
How does moving a point load affect reactions?
The support nearest the load carries the larger share. A load at quarter span puts 75% on the near support and 25% on the far support; at mid-span the split is equal. Design both supports for the worst position of a movable load.
What units does this calculator use?
It accepts SI units (N, mm, MPa, GPa) or imperial units (lbf, inch, psi). All internal computation is metric; conversions use 1 in = 25.4 mm, 1 lbf = 4.448 N and 1 psi ≈ 0.006895 MPa.
What is shear force in a beam?
Shear force is the transverse internal force that slices one part of the beam relative to the other. Between supports it varies linearly under distributed load and jumps by the magnitude of any point load crossing. Peak shear sits at the supports.
How accurate is Euler-Bernoulli beam theory?
Very accurate for slender beams whose length exceeds roughly ten times their depth. For deep beams, short spans or concentrated loads near supports, shear deformation becomes significant and Timoshenko theory gives better predictions.
Can this calculator handle multiple point loads?
This version superposes one point load with one uniform load, which covers the vast majority of hand-check scenarios. For many complex load cases use superposition repeatedly or a frame-analysis program.
What is the deflection formula for a central point load?
δ_max = P·L³/(48·E·I). For a full-span uniform load δ_max = 5·w·L⁴/(384·E·I). The calculator uses exact superposition of these closed-form shapes evaluated along the whole span.
Why does orientation matter for rectangular sections?
I grows with the cube of depth, so a 50×200 board bent about its major axis is 16 times stiffer than the same board laid flat. Always enter b as the width and h as the vertical depth of bending.
How do I check deflection limits like L/360?
Divide your computed maximum deflection into the span length. A 6 m span with δ = 15 mm gives L/400, comfortably inside the common L/360 live-load limit. The summary panel reports this ratio automatically.
What happens if a point load sits directly over a support?
It transfers straight into that support, producing no span bending moment and no deflection contribution — only local bearing checks matter there. Try setting position a = 0 to see this behaviour.
Are impact and dynamic loads covered?
No. Static analysis assumes loads are applied slowly. Impact, vibration or moving loads need dynamic amplification factors or dedicated dynamic analysis; a common static shortcut multiplies live loads by 1.3–2.0.
What is the neutral axis?
It is the layer inside the bend where fibre strain is zero — above it material compresses, below it stretches (or vice versa). For symmetric sections it passes through the centroid, which is why S = I/(h/2) for rectangles.
How do holes and notches affect results?
Openings reduce net section and create stress concentrations not captured here. Reduce the section dimensions locally near the hole and apply a stress-concentration factor of roughly 2–3 for conservative checking.
What is the difference between total load and equivalent point load?
A UDL totalling W = wL placed at mid-span produces the same support reactions and mid-span moment as a point load W there — but different shear distribution and slightly smaller maximum deflection. The calculator reports W for quick comparisons.
When should a beam be considered fixed rather than simply supported?
Only when connections genuinely prevent end rotation — welded or heavy bolted moment connections. Most seated or pinned connections behave close to simply supported; assuming fixity without justification is unconservative.
Why do results change if I switch unit systems?
They should not — switching converts labels and default magnitudes but the underlying numbers you type are interpreted in the selected system. Enter values consistent with the chosen unit system to get correct output.
How is the bending stress computed?
σ = M_max/S using elastic section modulus. This is valid while stresses stay below yield. Once yielding starts, plastic redistribution raises the true capacity above the elastic prediction for ductile materials.
Can I analyse overhanging or continuous beams?
Not with this tool. Overhangs and continuity alter sign conventions and need three-moment or stiffness methods. Use specialised structural software for those configurations.
What is a typical allowable deflection for floors?
Common code guidance is span/360 for live load and span/240 for total load in residential and office floors. Brittle finishes such as plaster ceilings may require span/480. Always confirm with the governing building code.
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Glossary

Technical terms used in mortgage calculations explained in plain language.

Point Load (P)
A concentrated force applied at a single point on the beam, expressed in newtons (N) or pounds-force (lbf).
Distributed Load (w)
Load spread evenly along the span, expressed as force per unit length (N/mm or lbf/in).
Span (L)
Distance between the centres of the two supports of a simply supported beam, or wall-to-tip distance of a cantilever.
Reaction
The upward force a support exerts on the beam to maintain equilibrium, found from ΣF = 0 and ΣM = 0.
Shear Force (V)
Internal transverse force at a cut section; it equals the algebraic sum of vertical forces on either side of the cut.
Bending Moment (M)
Internal moment at a section causing the beam to curve; peak values drive bending-stress checks.
Second Moment of Area (I)
Geometric property measuring resistance to bending; units mm⁴. Larger I means less deflection and lower stress.
Section Modulus (S)
S = I divided by the extreme-fibre distance; bending stress equals moment divided by S.
Deflection (δ)
Transverse displacement of the beam under load, usually checked against span/360-type limits.
Young’s Modulus (E)
Material stiffness constant relating stress to elastic strain; steel ≈ 200 GPa, aluminium ≈ 69 GPa.
Utilization Ratio
Actual stress divided by allowable stress. Values below 1.0 indicate adequate strength; values above indicate overstress.
Superposition
Principle allowing combined-load effects to be obtained by summing the effects of each load acting separately.

Authoritative References

Trusted sources and standards referenced by this calculator.

Applicable Standards

ASTM A36 / A992EN 1993 (Eurocode 3)ANSI/AISC 360NDS 2018 Wood Design

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Page Information

Last Updated:

August 2026

Version:

2.1.0

License:

Free to use for educational and professional purposes.