Free Beam Calculator — SFD, BMD, Deflection & Reactions

Beam analysis is the backbone of structural engineering. Every building, bridge, and industrial structure relies on beams to carry loads across openings and transfer forces to columns and foundations.

This page covers the theory, formulas, and worked examples for the most common beam configurations. Use the links below to jump straight to a calculator:

• Simple Beam Calculator — reactions, SFD, BMD, deflection
• Beam Capacity Checker — flexure, shear, LTB per code
• Beam Deflection Tool — L/360, L/240 checks
• Continuous Beam Tool — multi-span analysis

Types of Beams by Support Condition

Simply Supported Beam

A beam with a pin support at one end and a roller at the other. The most common configuration in building construction.

• Reactions: Determinate. RA + RB = total load.
• Maximum deflection: At midspan (for symmetric loading)
• Zero moment: At both supports
• Use case: Floor joists, roof purlins, simple framing

Cantilever Beam

A beam fixed at one end and free at the other.

• Maximum moment: At the fixed support
• Maximum deflection: At the free end
• Sign convention: Negative moment (hogging)
• Use case: Balconies, canopies, overhanging beams, brackets

Fixed-Fixed (Built-In) Beam

A beam rigidly connected at both ends. Reduces midspan moment and deflection compared to simply supported.

• Maximum moment: At both supports (and possibly at midspan)
• Deflection: 5× less than simply supported for UDL
• Use case: Moment frames, rigid connections, continuous structures

Propped Cantilever

A cantilever with an additional simple support at the free end. One degree of indeterminacy.

• Moment: Negative at fixed end, positive near midspan
• Deflection: Less than pure cantilever
• Use case: Continuous beams with one cantilever end

Continuous Beam

A beam spanning over three or more supports. Statically indeterminate.

• Moment diagram: Negative over interior supports, positive between supports
• Redistribution: Moment redistribution may be permitted by code
• Use case: Multi-span floor systems, bridge girders

Load Types

Point Load (Concentrated Load)

A single force applied at a specific location along the beam.

Simply supported beam, point load P at midspan:

• Reactions: RA = RB = P/2
• Max moment: Mmax = PL/4 (at midspan)
• Max deflection: Δmax = PL³/(48EI) (at midspan)

Uniformly Distributed Load (UDL)

A constant load per unit length (w) applied across the full span.

Simply supported beam, UDL w:

• Reactions: RA = RB = wL/2
• Max moment: Mmax = wL²/8 (at midspan)
• Max deflection: Δmax = 5wL⁴/(384EI) (at midspan)

Partial UDL

A UDL applied over only a portion of the span. Requires integration or superposition.

Triangular Load

A linearly varying load, common in hydrostatic pressure and snow drift.

Simply supported, triangular load (0 to w₀):

• Total load: W = w₀L/2
• Max moment: Mmax = w₀L²/12√3 ≈ 0.0642w₀L² (at x = L/√3)

Moment Applied at Support

An external moment applied at a support point. Common in moment frames.

Simply Supported Beam Formulas Table

¹ Simplified for a ≤ b. The exact formula involves the position.

Cantilever Beam Formulas Table

Fixed-Fixed Beam Formulas Table

Note: Fixed-fixed deflection for UDL is exactly 1/5 of simply supported deflection. This is why rigid connections reduce floor vibration.

Worked Example: Simply Supported W18x55 Under UDL

Given:

• Beam: W18x55 (Ix = 890 in⁴, Sx = 98.3 in³)
• Span: L = 30 ft = 360 in
• Load: w = 1.5 kips/ft (factored)
• Steel: A992, Fy = 50 ksi

Step 1: Reactions RA = RB = wL/2 = 1.5 × 30/2 = 22.5 kips

Step 2: Maximum Moment Mmax = wL²/8 = 1.5 × 30²/8 = 168.75 kip-ft

Step 3: Bending Stress Check (LRFD) φMn = φFySx = 0.9 × 50 × 98.3 / 12 = 368.6 kip-ft Mu = 168.75 kip-ft < φMn = 368.6 kip-ft ✓

Step 4: Deflection (Service Load) Service w = 1.0 kip/ft (unfactored) Δmax = 5wL⁴/(384EI) = 5 × 1.0/12 × 360⁴ / (384 × 29000 × 890) = 5 × 0.0833 × 16,796,160,000 / 9,908,224,000 = 0.707 in

L/360 = 360/360 = 1.0 in Δmax = 0.707 in < 1.0 in ✓ (passes L/360 deflection limit)

Shear Force and Bending Moment Diagrams

The SFD and BMD are graphical representations of the internal forces in a beam:

• Shear Force Diagram (SFD): Shows the internal shear force V(x) along the beam length. The shear force at any section equals the algebraic sum of all transverse forces to one side of the section.
• Bending Moment Diagram (BMD): Shows the internal bending moment M(x) along the beam length. The moment at any section equals the sum of moments of all forces to one side.

Key relationships:

• dV/dx = -w(x) (shear slope equals negative load intensity)
• dM/dx = V(x) (moment slope equals shear force)
• d²M/dx² = -w(x) (moment curvature equals negative load)

Reading SFD/BMD:

• Where SFD crosses zero, BMD has a peak (maximum positive or negative moment)
• Point loads create discontinuities (jumps) in the SFD
• Concentrated moments create discontinuities in the BMD
• UDL creates a linear SFD and parabolic BMD

Beam Design Checklist

When designing a steel beam, verify all of the following:

1. Flexure (strong axis): Mu ≤ φMn where φMn = φFyZx (compact, LRFD)
2. Shear: Vu ≤ φVn where φVn = 0.6φFyAw (web shear)
3. Lateral-Torsional Buckling (LTB): Check if Lb ≤ Lp (compact), Lp < Lb ≤ Lr (inelastic), or Lb > Lr (elastic)
4. Deflection: Δmax ≤ L/360 (floors), L/240 (roofs), L/180 (wind)
5. Web crippling: Check concentrated loads per AISC Chapter J
6. Serviceability: Vibration, floor frequency, occupant comfort
7. Connection capacity: Verify end connections can deliver the assumed support conditions

Use the beam capacity calculator for automated checks per AISC 360, AS 4100, EN 1993, or CSA S16.

Beam Analysis Methods Comparison

For day-to-day beam design, the simple beam calculator handles equilibrium-based analysis instantly. For continuous beams, use the continuous beam tool.

Frequently Asked Questions

What is the difference between SFD and BMD? The Shear Force Diagram shows internal shear V(x) along the beam. The Bending Moment Diagram shows internal moment M(x). They are related: dM/dx = V.

How do I calculate beam reactions? For a simply supported beam, sum vertical forces (ΣFy = 0) and moments about one support (ΣM = 0). This gives two equations for two unknowns.

What is the maximum deflection of a simply supported beam with UDL? Δmax = 5wL⁴/(384EI). This occurs at midspan. For L/360 deflection limit, set this equal to L/360 and solve for the required I.

What is the strongest beam shape? For a given weight, the I-beam (W-shape) is the strongest in bending because material is concentrated at the flanges far from the neutral axis, maximizing Ix and Sx.

How do I account for lateral-torsional buckling? If the compression flange is not braced, the beam may buckle laterally before reaching its full plastic moment. Check LTB per AISC Chapter F using the beam capacity calculator.

What is the difference between elastic and plastic analysis? Elastic analysis keeps all stresses below Fy (uses Sx). Plastic analysis allows stress redistribution up to the plastic moment Mp = FyZx (uses Zx). LRFD uses plastic analysis for compact sections.

When should I use a continuous beam vs simply supported? Continuous beams reduce midspan moment and deflection. Use them when you have multiple spans and can detail rigid connections. Simply supported is simpler and more tolerant of foundation settlement.

Can I use these calculators for wood or concrete beams? The beam analysis (SFD, BMD, deflection) works for any material. But code-specific capacity checks are for steel per AISC 360, AS 4100, EN 1993, and CSA S16 only.

Related Pages

• Simple Beam Calculator — Reactions, SFD, BMD, deflection
• Beam Capacity Checker — AISC 360, AS 4100, EN 1993, CSA S16
• Beam Deflection Calculator — L/360, L/240 checks
• Steel Beam Sizes Chart — W, UB, IPE, HEA dimensions
• Cantilever Beam Formulas — Deflection and moment formulas
• Continuous Beam Tool — Multi-span analysis

Beam Selection Guide by Application

Residential Floor Beams

Typical spans: 10-25 ft. Typical loads: 40-50 psf live + 10-15 psf dead.

Roof Purlins

Typical spans: 15-30 ft. Typical loads: 20-30 psf snow + 10-15 psf dead. Often governed by deflection.

Crane Runway Beams

Governed by fatigue, deflection (L/800 or stricter), and lateral loads from crane. Must check web crippling, local flange bending, and lateral-torsional buckling simultaneously.

Deflection Limits Summary

The deflection calculator at /tools/beam-deflection/ checks all common limits automatically.

Connection Types and Their Effect on Beam Analysis

The assumed support condition directly affects the moment diagram and deflection:

Simple Shear Connection (Web Angle, Single Plate, End Plate)

• Transfers shear only
• No moment transfer to the support
• Model as pin/roller in analysis
• Most common connection type in building construction

Moment Connection (Fully Welded, Bolted End Plate, Extended End Plate)

• Transfers both shear and moment
• Creates continuity at the joint
• Reduces midspan moment and deflection
• Required for moment frames

Partially Restrained Connection

• Transfers some moment (typically 20-50% of full fixity)
• Intermediate stiffness between simple and rigid
• Permitted by AISC Chapter B with proper modeling

Always verify that your analysis assumptions match the actual connection detail. A beam modeled as simply supported but with stiff end plates will develop unintended negative moments at the supports.

Frequently Asked Questions (Extended)

How do I calculate beam reactions for multiple point loads? Use superposition. Calculate the reaction from each load separately, then sum. For a simply supported beam: RA = Σ(Pi × bi/L) where bi is the distance from load i to support B.

What is the difference between a beam and a girder? No technical difference. In practice, "girder" typically refers to a larger beam that supports other beams. Both are analyzed the same way.

How do I handle a beam with a notch or coped end? A cope reduces the section properties at the connection. Check the remaining web for shear, block shear, and local buckling. The coped section may govern the beam capacity. See coped beam reference.

What is beam camber? Camber is an upward curvature built into a beam at fabrication to offset anticipated dead load deflection. Typically 75-80% of calculated dead load deflection. Does not affect live load deflection checks.

How do I analyze a beam with a varying cross-section? Divide into segments with constant properties. Apply equilibrium on each segment with compatible boundary conditions. Alternatively, use the continuous beam tool with segment properties.

What about shear deformation in short beams? For span-to-depth ratio L/d < 10, shear deformation becomes significant. The Euler-Bernoulli beam theory (which neglects shear) overestimates stiffness. Timoshenko beam theory accounts for this.

How do concentrated loads near supports affect the beam? Concentrated loads near supports primarily stress the web in shear and crippling. Check web crippling per AISC Chapter J10. The moment diagram shows a steep gradient near the load point.

What is the effective length of a cantilever beam for LTB? For a cantilever with free end, Lb = the cantilever length. The LTB check uses the full cantilever length with appropriate Cb factors. Bracing the top flange at mid-span significantly improves LTB capacity.

How do I account for beam self-weight in calculations? Add the beam weight per foot (from AISC Table 1-1) to the dead load. For a W18x55, add 55 lb/ft = 0.055 kips/ft to the distributed load. This is typically small but should be included in deflection checks.

Disclaimer

This is a calculation tool, not a substitute for professional engineering certification. All results must be independently verified by a licensed Professional Engineer (PE) or Structural Engineer (SE) before use in construction, fabrication, or permit documents. The user is responsible for the accuracy of all inputs and the verification of all outputs.