Beam Calculator

Model a beam with multiple loads — point loads, uniform distributed loads, triangular loads, and applied moments. Choose material and section, set your span, and compute maximum bending moment, deflection, and bending stress, or solve directly for the required load to meet a deflection, moment, or stress limit.

Beam relationships
\[ \frac{dV(x)}{dx} = -w(x), \quad \frac{dM(x)}{dx} = V(x), \quad EI \frac{d^2 y(x)}{dx^2} = M(x) \]
Superposition
\[ M(x) = \sum_i M_i(x), \qquad \delta(x) = \sum_i \delta_i(x) \]
Bending stress & design
\[ \sigma_{\max} = \frac{M_{\max} \, c}{I} \]
Beam Visual

Material: Steel · Shape: Rectangular · Span: 6 m · Loads: 1 load defined

Overall Setup

Material

Section Shape

Span & Loads

Each row defines a load: Type → Magnitude → Units → Start position a → End position b (for distributed). Positions are measured from the left support / fixed end. In load-solve modes, only the first row is used and its magnitude is solved; leave additional rows empty.

Results

Structural Engineering Guide

Beam Calculator: From Loads to Safe Spans

A practical guide that mirrors how you actually use a beam calculator: define the span and supports, choose a material and section, add point and distributed loads, interpret shear, moment, and deflection, and sanity-check the results against code-style limits.

10–15 min read Updated 2025

Quick Start: Using the Beam Calculator Safely

The beam calculator above is designed for everyday structural checks: floor joists, roof rafters, simple lintels, and small cantilevers. It handles multiple loads (point loads, moments, uniform and non-uniform distributed loads) on common support conditions and shows the resulting reactions, shear, moment, and deflection.

  1. 1 Choose units, span, and supports.
    Set the beam length \(L\) (e.g., 4 m or 14 ft) and pick the support type: simply supported, cantilever, or fixed–fixed. Make sure your span in the calculator matches the clear span in the actual structure.
  2. 2 Select a material and section shape.
    For steel, you might pick a W-section (I-beam). For timber, select rectangular or flitched sections. The calculator uses the modulus of elasticity \(E\) and moment of inertia \(I\) behind the scenes.
  3. 3 Add your loads one by one.
    For each load, choose the type (point, uniform, triangular, moment), location (or start/end), and magnitude. Keep the sign convention consistent: downward loads negative, upward loads positive, moments positive in sagging.
  4. 4 Review the shear and moment diagrams.
    The calculator plots \(V(x)\) and \(M(x)\). Confirm that the diagram shapes match your intuition (e.g., triangular shear under uniform load, parabolic moment, kinks at point loads).
  5. 5 Check stresses and deflection.
    The tool compares bending stress \(\sigma = M_\text{max}c/I\) and deflection \(\delta_\text{max}\) against limits (for example, \(L/360\) for floors). If anything is over-stressed or too flexible, increase stiffness (bigger section, shorter span) or reduce loads.
  6. 6 Use “what-if” tweaks.
    Adjust material, section size, or load positions and rerun the calculator. Small changes in span or load location often have a big impact on moment and deflection.
  7. 7 Document your assumptions.
    Note which loads you included, which codes or deflection limits you used, and any simplifications (for example, “assumed uniform load over full span”).

Tip: Start with a simple model (one span, one or two loads) to get a feel for the calculator before modeling a fully loaded beam.

Warning: This calculator is for idealized, line-beam behavior. It does not replace a full code-compliant design or the judgment of a licensed professional.

Choosing Your Analysis Method

There are several ways to use a beam calculator depending on where you are in the design process. The tool above supports both “quick sizing” and more detailed checks.

Method A — Strength Check (Bending & Shear)

Use when you already know the span and loads and need to verify that the beam is strong enough.

  • Directly compares calculated bending and shear to allowable values.
  • Great for checking an existing beam or catalog section.
  • Works well with multiple point and distributed loads.
  • Requires reasonable estimates of load combinations (dead, live, snow, etc.).
  • Does not automatically account for lateral-torsional buckling or local checks.
Strength ratio: \( \phi M_n \ge M_u,\ \ \phi V_n \ge V_u \)

Method B — Serviceability (Deflection & Vibration)

Use when comfort or finishes control the design (floors, long roof members, fascia beams).

  • Ensures that deflections stay within code-style limits like \(L/240\), \(L/360\), or \(L/480\).
  • The calculator shows \(\delta_\text{max}\) and its location.
  • Helps reduce cracking of finishes and ponding on roofs.
  • Requires an appropriate deflection limit for your application.
  • Vibration and dynamic behavior are not explicitly modeled.
Typical limit: \( \delta_\text{max} \le \dfrac{L}{360} \) (live load on floor beams)

Method C — Rapid Sizing & Iteration

Use in early design to narrow down beam sizes before doing detailed code checks.

  • Quickly compares different materials and shapes for the same loading.
  • Ideal for “what-if” studies (longer span vs. deeper beam vs. heavier section).
  • Supports non-uniform loads to approximate realistic patterns.
  • Results must be followed by a full design per local codes.
  • Connection details and stability are outside the calculator’s scope.
Stiffness lever: \( EI \uparrow \Rightarrow \delta_\text{max} \downarrow \)

What Moves the Beam Calculator’s Numbers

The calculator responds very strongly to a few key variables. Understanding these lets you steer the design instead of just reacting to red and green checks.

Span length \(L\)

Bending moment and deflection both grow rapidly with span. For a simply supported beam with uniform load \(w\), \(M_\text{max} = wL^2/8\) and \(\delta_\text{max} \propto L^4\). A small increase in span can require a much stiffer section.

Load type & magnitude

Point loads create sharp peaks in the moment diagram. Uniform and triangular loads create smoother curves but usually larger total demand. Doubling the load roughly doubles reactions, shear, and moment.

Load position

A point load at midspan produces the largest bending moment in a simply supported beam. Moving the same load toward a support reduces midspan moment but increases reactions and local shear.

Material stiffness \(E\)

Steel has a much higher modulus of elasticity than timber or concrete. For the same section geometry, a steel beam will deflect far less than a timber member under the same loading.

Section geometry \(I\)

The second moment of area \(I\) drives both bending stress and deflection. Deep I-beams or rectangular sections have much larger \(I\) than shallow members, dramatically improving performance.

Support conditions

Cantilevers see higher deflections and moments than simply supported beams for the same span and load. Fixed–fixed beams are stiffer and share moments into the supports, reducing midspan demand.

Worked Examples with the Beam Calculator

Example 1 — Simply Supported Beam with Uniform Load and Point Load

  • Beam type: Simply supported
  • Span: \(L = 4.0\ \text{m}\)
  • Material: Steel, \(E = 200\ \text{GPa}\)
  • Section: I-beam with \(I = 8.0 \times 10^6\ \text{mm}^4\)
  • Loads: Uniform load \(w = 8\ \text{kN/m}\) over full span, plus a point load \(P = 12\ \text{kN}\) at midspan
  • Check: Bending stress and deflection vs. allowable limits
1
Set up the beam: In the calculator, choose simply supported, span \(L = 4\ \text{m}\), steel material, and the chosen I-beam section. Confirm the units (kN, m).
2
Add the uniform load: Select a uniform load, apply \(w = 8\ \text{kN/m}\) from \(x = 0\) to \(x = 4\ \text{m}\). The calculator uses
\[ M_{\max, w} = \frac{wL^2}{8} \]
for the base bending moment due to the distributed load.
3
Add the point load: Add a point load \(P = 12\ \text{kN}\) at \(x = 2\ \text{m}\) (midspan). For a symmetric point load at midspan,
\[ M_{\max, P} = \frac{PL}{4}. \]
The calculator superposes this with the uniform load.
4
Read reactions and diagrams: After solving, the reaction forces at each support should be equal due to symmetry. The shear diagram shows a step at the point load; the moment diagram shows a smooth parabolic curve with a kink at midspan.
5
Check bending stress: Let \(M_\text{max}\) be the peak positive moment from the calculator. Bending stress is
\[ \sigma_\text{max} = \frac{M_\text{max} c}{I}, \]
where \(c\) is the distance from the neutral axis to the extreme fiber. Compare \(\sigma_\text{max}\) with the allowable stress for the steel grade.
6
Check deflection: The calculator reports \(\delta_\text{max}\). Compare to a limit such as \(L/360\). For \(L = 4\ \text{m}\), \(L/360 \approx 11.1\ \text{mm}\). If \(\delta_\text{max} \le 11.1\ \text{mm}\), the deflection criterion is satisfied.

Example 2 — Cantilever Beam with Tip Load and Applied Moment

  • Beam type: Cantilever with fixed support at the left end
  • Cantilever length: \(L = 2.0\ \text{m}\)
  • Material: Glulam timber, \(E = 11\ \text{GPa}\)
  • Section: Rectangular, \(b = 90\ \text{mm}\), \(h = 270\ \text{mm}\)
  • Loads: Tip load \(P = 5\ \text{kN}\) at free end, plus an applied clockwise moment \(M_0 = 3\ \text{kN·m}\) at the same location
  • Checks: End rotation and tip deflection, stress at fixed support
1
Define the section: For a rectangular section, the calculator uses
\[ I = \frac{b h^3}{12}. \]
With \(b = 90\ \text{mm}\) and \(h = 270\ \text{mm}\), \(I\) is large enough for moderate spans.
2
Model the loads: Choose cantilever support. Add a point load \(P = 5\ \text{kN}\) at \(x = L = 2\ \text{m}\). Then add a concentrated moment \(M_0 = 3\ \text{kN·m}\) at the same position with the appropriate sign.
3
Check fixed-end moment and shear: For a tip load on a cantilever,
\[ M_\text{fixed} = PL,\quad V_\text{fixed} = P. \]
The applied moment \(M_0\) adds directly to \(M_\text{fixed}\) with the same sign convention. The calculator should report this combined moment at the support.
4
Check tip deflection: For a pure tip load,
\[ \delta_{P} = \frac{PL^3}{3EI}. \]
For the applied moment alone,
\[ \delta_{M_0} = \frac{M_0 L^2}{2EI}. \]
The calculator superposes these contributions and reports the total deflection at the free end.
5
Compare to timber limits: For timber beams, deflection limits are often \(L/240\) or \(L/180\) depending on load type. For \(L = 2\ \text{m}\), \(L/240 \approx 8.3\ \text{mm}\). Adjust the section size or span if the calculated deflection is larger.

Common Beam Layouts & Variations

The same beam calculator can model a wide range of real-world situations. The key is choosing a layout and load pattern that reasonably approximates the structure you are designing.

Layout / ScenarioHow to Model in the Beam CalculatorStrength & Serviceability Notes
Simple floor joist under uniform floor loadSimply supported beam, span equal to clear distance, uniform load from 0 to \(L\) representing dead + live load.Check bending, shear, and deflection vs. code limits (for example, \(L/360\) for live load).
Roof rafter with snow drift near ridgeSimply supported or fixed–pinned, several partial uniform loads over different regions to model drift and unbalanced snow.Snow patterns create asymmetric moment diagrams; pay attention to support reactions and uplift load cases.
Lintel over an opening with concentrated reactionsSimply supported span equal to clear opening plus bearing. Add two or more point loads at reaction locations from the wall above.Check bearing length, local flange/web buckling, and rotation limits at supports.
Cantilever balcony with railing loadsCantilever beam with tip point load for railing and uniform load for occupancy on the cantilevered portion only.Deflection is often critical; wind and crowd loading may require larger sections than strength alone suggests.
Crane runway or monorail beamSimply supported or continuous beam with moving point loads; approximate with a few critical positions using point loads at worst-case locations.Consider impact factors, fatigue, and lateral bracing in addition to the static analysis from the calculator.
Multi-span floor beam (approximate)Model each span separately as a simply supported beam with appropriate reactions, or use the fixed–fixed option as a first-order approximation.True continuous behavior needs frame analysis, but the calculator is useful for quick span-by-span checks and sizing.
  • Ensure that the modeled span matches the real clear span between effective supports.
  • Include self-weight in loads if it is not already built into the standard load values.
  • Be clear about factored vs. unfactored loads when comparing to allowable stresses.
  • Model critical load cases separately (maximum gravity, maximum snow, wind uplift, etc.).
  • Document which scenario each calculator run represents so you can revisit or show your work.
  • Use conservative assumptions where exact conditions are uncertain.

Specs, Logistics & Sanity Checks Before You Trust the Result

A beam calculator is only as reliable as the inputs and assumptions you give it. Before you treat any output as “good to build,” run through a quick checklist.

Key Design Inputs

  • Load sources: Dead, live, snow, wind, seismic, equipment loads, and any concentrated reactions from other members.
  • Load factors: Are you using service loads or factored loads? Be consistent with your design code.
  • Material properties: Confirm \(E\), yield stress, and allowable values match the grade specified in your project documents.
  • Section data: Use published section properties from reputable steel catalogs, timber tables, or manufacturer sheets.

Constructability & Detailing

  • Verify that connections (bolted, welded, or bearing) can safely transmit the reactions from the calculator.
  • Ensure the beam can actually be installed (depth, weight, crane access, handling limits).
  • Check for required camber, fire protection, corrosion protection, and vibration controls.
  • Confirm bearing lengths, bearing stresses, and compatibility with supporting members or walls.

Sanity Checks

  • Compare results to rule-of-thumb spans for similar beams in past projects.
  • Confirm the shape of the diagrams matches engineering intuition (no unexpected peaks or sign flips).
  • Run at least one hand-check using classic formulas for a simplified load case to validate the calculator setup.
  • When in doubt, have a licensed engineer review the model and outputs, especially for safety-critical structures.

The beam calculator is an excellent decision-support tool, but it is not a substitute for a complete design per AISC, ACI, NDS, Eurocode, or other applicable standards.

Frequently Asked Questions

Is this beam calculator enough for a code-compliant design?
No. The beam calculator is a powerful way to visualize reactions, shear, moment, and deflection, but it does not replace a full code check. You still need to verify load combinations, stability, buckling, connection design, and detailing per the governing building code.
What is the difference between a point load and a distributed load?
A point load acts at a single location along the span, such as a column reaction or heavy piece of equipment. A distributed load is spread over a length, like floor live load or roofing. The calculator lets you model both types (and non-uniform distributions) so that the resulting shear and moment diagrams reflect the actual loading pattern.
How should I choose the right deflection limit?
Deflection limits depend on the element and use. Floor beams often use limits like L/360 or L/480, roof members might use L/240, and special finishes may require tighter limits. Check the applicable design standard and use the calculator to make sure the computed maximum deflection is below the recommended limit.
Can this beam calculator handle multiple spans or continuous beams?
The calculator focuses on single-span beams with common support conditions. You can approximate a continuous system by analyzing each span separately or by using fixed-end conditions as a first pass, but a full multi-span frame should be analyzed with software that supports continuity and frame action.
What happens if I add several loads at the same location?
The calculator superposes all loads mathematically. Multiple loads at the same location are effectively combined into an equivalent resultant. This is useful when, for example, dead load and live load share the same distribution along the span but you want to track them separately during input.
How accurate are the shear and moment diagrams from the calculator?
For idealized prismatic beams with the specified loads and supports, the shear and moment diagrams are as accurate as the underlying formulas and numerical methods allow. Real structures may deviate due to connection flexibility, cracking, composite action, or non-prismatic members, so you should treat the diagrams as an idealized representation.
When should I involve a licensed structural engineer?
You should consult a licensed structural engineer for any safety-critical application, such as primary building frames, heavily loaded beams, unusual geometries, or when local codes require sealed calculations. The beam calculator is best used as a learning tool, a preliminary sizing helper, and a way to understand how loads interact on a beam.
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