Free Beam Calculator Guide — Bending Moment, Shear & Deflection

Quick access:

• What is a beam calculator?
• Beam types and support conditions
• Calculating bending moment and shear force
• Deflection calculations
• How the beam calculator works
• Worked example: continuous beam
• Frequently asked questions
• Try the beam calculator

What is a beam calculator?

A beam calculator is a structural analysis tool that determines the internal forces (bending moment, shear force) and deflections in a beam under specified loads and support conditions. It answers the essential structural question: For a given beam, loading, and support configuration, what are the maximum moment, shear, and deflection?

The three fundamental results from any beam analysis:

1. Bending moment diagram — Shows the variation of moment along the beam. The maximum moment determines the required section size (via Sx = M/Fb or Zx = M/phiFy).
2. Shear force diagram — Shows the variation of shear along the beam. The maximum shear determines the required web area or stiffener spacing.
3. Deflection curve — Shows the deformed shape of the beam under service loads. The maximum deflection must be within code limits (L/360 for floor live load, etc.).

The Steel Calculator beam tools perform all three analyses automatically for any support and loading configuration, covering AISC 360, AS 4100, EN 1993, and CSA S16 design codes.

Beam types and support conditions

Simply supported (pin-roller)

The most common structural beam. One end is pinned (restrained vertically and horizontally), and the other is on rollers (restrained vertically only). No moment is transferred at the supports.

• Maximum moment at midspan (uniform load): Mmax = wL²/8
• Maximum shear at supports: Vmax = wL/2
• Maximum deflection at midspan: Dmax = 5wL⁴/(384EI)
• Used for: Floor beams, roof beams, bridge girders, crane girders

Fixed-end beam

Both ends are fixed against rotation and translation. Fixed ends develop negative (hogging) moments that reduce the positive (sagging) midspan moment.

• Maximum negative moment at supports: Mmax = wL²/12
• Maximum positive moment at midspan: Mpos = wL²/24
• Maximum deflection at midspan: Dmax = wL⁴/(384EI) — 5x stiffer than simply supported
• Used for: Moment frame beams, continuous foundations, heavily restrained members

Cantilever beam

One end is fixed, the other is free. The cantilever carries load entirely in negative bending.

• Maximum moment at fixed end: Mmax = wL²/2 (uniform) or PL (point load at tip)
• Maximum shear at fixed end: Vmax = wL or P
• Maximum deflection at tip: Dmax = wL⁴/(8EI) or PL³/(3EI)
• Used for: Balconies, canopies, cantilevered roof overhangs, sign supports

Continuous beam

A beam spanning over three or more supports. Continuous beams are more efficient than simple spans because the negative moment over interior supports reduces the positive moment at midspan.

• Maximum positive moment: Approximately wL²/12 to wL²/16 (depends on number of spans)
• Maximum negative moment at interior supports: wL²/10 to wL²/12
• Maximum deflection: Typically 60-70% of a simple span of the same length
• Used for: Multi-span floor beams, bridge deck stringers, roof purlins

Propped cantilever

One end fixed, the other simply supported. Combines characteristics of fixed and simple supports.

• Maximum moment at fixed end: Mmax = wL²/8
• Maximum positive moment at approximately 0.375L from simple end: Mpos = 9wL²/128
• Used for: Eccentrically loaded beams, beams with mixed boundary conditions

Calculating bending moment and shear force

Sign conventions

• Positive moment: Causes tension on the bottom fiber (sagging)
• Negative moment: Causes tension on the top fiber (hogging)
• Positive shear: The left side of the section tends to move upward relative to the right side

Uniformly distributed load (UDL) on simple span

Shear force: V(x) = w(L/2 - x)

Varies linearly from +wL/2 at left support to -wL/2 at right support. Zero at midspan.

Bending moment: M(x) = w x (L - x) / 2

Varies parabolically, zero at supports, maximum at midspan: Mmax = wL²/8

Concentrated point load at midspan

Shear force: V(x) = +P/2 from left support to midspan, -P/2 from midspan to right support

Constant in each half, discontinuous at the load point.

Bending moment: M(x) = P x / 2 (left half), M(x) = P(L - x)/2 (right half)

Varies linearly, maximum at midspan: Mmax = PL/4

Partial UDL (load over portion of span)

For a UDL of intensity w over length a, starting at distance b from the left support:

• The reactions are found by taking moments about each support
• Shear and moment are calculated by integrating the load function
• The maximum moment occurs where shear = 0 (V(x) = 0)

Deflection calculations

Deflection is checked under service (unfactored) loads, typically the live load portion. The general formula for beam deflection is derived from the moment-curvature relationship:

d²y/dx² = M(x) / (E x I)

where M(x) is the bending moment as a function of position, E is the modulus of elasticity (29,000 ksi or 200 GPa for steel), and I is the moment of inertia.

Common deflection formulas

Allowable deflection limits

Per IBC Table 1604.3 and ASCE 7-22 Table CC.1:

How the beam calculator works

The Steel Calculator beam tools handle the full analysis pipeline:

Step 1: Define geometry

Select the beam type (simple, fixed, cantilever, or continuous with up to 5 spans). Enter the span lengths and support conditions.

Step 2: Define loads

Add load cases: uniform (distributed), point (concentrated), partial uniform, linearly varying, or moment loads. Each load is assigned to a category: dead, live, snow, wind, or seismic.

Step 3: Load combinations

The calculator applies the applicable load combinations based on the selected design code:

• AISC 360: ASCE 7-22 LRFD or ASD combinations
• AS 4100: AS/NZS 1170.0 combinations
• EN 1993: EN 1990 combinations
• CSA S16: National Building Code of Canada combinations

Step 4: Analysis

The calculator solves for reactions, shear, moment, and deflection at each point along the beam using direct integration (for simple/fixed/cantilever) or the stiffness method (for continuous beams). Results include:

• Shear and moment diagrams (with maximum values)
• Deflection profile (with maximum values)
• Support reactions for connection design
• Governing load combination for each limit state

Step 5: Section check

After selecting a steel section, the calculator performs:

• Flexural check (AISC 360 Chapter F)
• Shear check (AISC 360 Chapter G)
• Deflection check (serviceability)
• The utilization ratio for each check

Worked example: continuous beam

Problem: A three-span continuous beam carries a uniform dead load of 1.2 kip/ft (including self-weight) and a uniform live load of 1.8 kip/ft. Each span is 25 ft. Check a W24x55 (A992, Fy = 50 ksi).

Step 1: Factored loads

LRFD: wu = 1.2 x 1.2 + 1.6 x 1.8 = 1.44 + 2.88 = 4.32 kip/ft

Step 2: Approximate moments (3-span continuous, uniform load)

Using AISC Manual Table 3-22 (continuous beam coefficients):

• Positive moment at end spans: Coefficient = 0.080 Mpos_end = 0.080 x 4.32 x 25² = 216.0 kip-ft
• Positive moment at middle span: Coefficient = 0.025 Mpos_mid = 0.025 x 4.32 x 25² = 67.5 kip-ft
• Negative moment at first interior support: Coefficient = -0.100 Mneg = -0.100 x 4.32 x 25² = -270.0 kip-ft
• Negative moment at exterior support (if partially fixed): Coefficient = -0.045 Mneg_ext = -0.045 x 4.32 x 25² = -121.5 kip-ft

Step 3: Check W24x55 flexure

W24x55: Zx = 135 in³, Lp = 6.78 ft

Assume compression flange continuously braced by slab:

Mp = 50 x 135 / 12 = 562.5 kip-ft phiMp = 0.90 x 562.5 = 506.3 kip-ft

End span positive moment: 216.0/506.3 = 0.43 → OK Interior support negative moment: 270.0/506.3 = 0.53 → OK

Step 4: Deflection check (service live load)

w_LL = 1.8 kip/ft

Using the same coefficients for service load: Delta_LL_endspan = (Coefficient for deflection) x w_LL x L⁴ / (E x I)

For continuous beams, the approximate maximum deflection in the end span: Delta_max ≈ w_LL x L⁴ / (145 x E x I)

Ix = 1,350 in⁴ Delta_max = 1.8 x (25 x 12)⁴ / (145 x 29,000 x 1,350) = 1.8 x 8.1e9 / (145 x 29,000 x 1,350) = 0.60 in

Allowable = 25 x 12 / 360 = 0.83 in 0.60 < 0.83 → OK

The W24x55 passes all checks for this three-span condition. A simple-span beam of the same length would require a much heavier section (approximately W30x99).

Frequently asked questions

What is the difference between bending moment and shear force?

Bending moment is the internal couple that causes the beam to bend (tension on one face, compression on the other). Shear force is the internal vertical force that causes the beam to slide along a cross-section. The relationship between them is dM/dx = V — the slope of the moment diagram equals the shear at any point.

How do I calculate the maximum moment in a simply supported beam?

For a uniform load: Mmax = wL²/8 at midspan. For a single point load at midspan: Mmax = PL/4. For multiple loads, the moment diagram is the sum of individual load effects — the maximum occurs where the shear diagram crosses zero. For complex loading, use the beam calculator to compute and plot the moment diagram automatically.

What beam is most efficient for long spans?

For long spans (40 ft or more), composite steel beams (W-shape + shear studs + concrete slab) are the most efficient. For very long spans (60 ft+), plate girders or trusses may be more economical. The key is to maximize Ix per unit weight. Deep, relatively light sections (W24x55, W30x99) are more efficient than stocky sections (W12x65).

How does beam continuity reduce deflection?

Continuous beams have deflection approximately 40-60% of a simple span of the same length under the same loading. The continuity over supports creates negative moments that partially counteract the positive moment, reducing the net curvature and deflection. This efficiency makes continuous construction standard for bridges and multi-span buildings.

What is the relationship between load, span, and beam depth?

For steel beams under typical loads, a rough sizing rule is depth (in inches) = span (in feet). Example: a 30 ft span needs approximately a 30 in deep beam (W30x99). For roof beams with lighter loads, depth = 0.75 x span. For heavily loaded beams or strict deflection limits, depth = 1.25 x span.

Try the beam calculator

Use the free Beam Capacity Calculator for complete beam analysis including moment, shear, deflection, and section design. The calculator handles:

• Simple, fixed, cantilever, and continuous beams (up to 5 spans)
• Uniform, point, partial uniform, and moment loads
• All standard steel sections (W, S, HP, C, MC, HSS, pipe, UB, UC, IPE, HEA, HEB)
• AISC 360, AS 4100, EN 1993, and CSA S16 design codes
• LRFD and ASD load combinations
• Automatic shear and moment diagram computation

For additional beam analysis tools:

• Beam Deflection Calculator — detailed deflection analysis
• Continuous Beam Calculator — multi-span beam analysis
• Beam Sizes Reference — complete W-shape and metric section dimensions
• Steel Beam Calculator Guide — full design workflow
• Beam Bending Moment Formulas — formula reference

Disclaimer

This guide is for educational and reference use only. It does not constitute professional engineering advice. All design values must be verified against the governing building code, project specification, and applicable design standards. The Steel Calculator disclaims liability for any loss, damage, or injury arising from the use of this information. Always engage a licensed structural engineer for beam design on actual projects.

See Also

• Beam Capacity Calculator
• Beam Deflection Calculator
• Steel Beam Sizes Reference
• Beam Design Guide
• Beam Span Reference