2025-08-21 14 min

Bending Moment Calculator

Calculate bending moment at any section of a beam with point loads or UDL.

Calculator

Unit System

Beam Type

Point Load (P)(N)

Beam Span (L)(mm)

Load Position (a)(mm) Help

Distance from left support

UDL Intensity (w)(N/mm)

N/mm

Evaluate at Position (x)(mm)

Decimal Precision

Results update live as you type

Results

Normal

Moment at Position

7500000.00kNm

Maximum Moment

15000000.00kNm

Max Moment Location

2000.00mm

Visualizations

Results Overview

7,500,000kNmMoment at Position

15,000,000kNmMaximum Moment

2,000kNmMax Moment Location

Formula

Bending moment at any section is the algebraic sum of moments from all forces on one side of that section.

M(x) = Sum of moments to the left of section x

Alternative Form

Simply Supported: Mmax = P*a*(L-a)/L | UDL: Mmax = w*L^2/8

Variables

Symbol	Name	Unit	Description
M	Bending Moment	N*mm	Internal moment at a cross-section
P	Point Load	N	Concentrated force on the beam
w	UDL Intensity	N/mm	Load per unit length
L	Beam Span	mm	Length between supports
a	Load Position	mm	Distance from left support

Explanation

The moment diagram shows how internal bending varies along the beam. Maximum moment determines the required section size.

Derivation

From equilibrium of a beam segment, the sum of moments about the cut section must balance the external loads.

Step-by-Step Calculation

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

Step 1

Calculate support reactions using static equilibrium

Result: Reactions found

Step 2

Identify the section where moment is needed

Result: Section located

Step 3

Sum all moments about that section from one side

Result: Moment calculated

Step 4

Find zero-shear locations for maximum moment

Result: Max moment location found

Step 5

Compute maximum moment for beam design

Result: Design moment determined

Final Answer

7500000.00

Worked Examples

See how the calculator works with different scenarios.

Scenario

4 m beam with 15 kN at midspan.

Inputs

Load (P):15000 N

Span (L):4000 mm

Load Position (a):2000 mm

Calculation

Ra = Rb = 7500 N. Mmax = 15000*2000*2000/4000 = 15 kNm.

Result

Mmax = 15 kNm

Interpretation

Moment diagram is triangular with peak at midspan.

Understanding Your Result

Your Result: Moment at Position: 7500000.00 kNm

Typical range: Light beams: 5-50 kNm. Building beams: 50-500 kNm. Bridge girders: 500-5000+ kNm.

What This Means

Bending moment analysis determines internal forces that cause a beam to bend, essential for structural design.

Why It Matters

Bending moment governs beam sizing. Maximum moment determines required section modulus and material grade.

Professional Interpretation

Compare Mmax against section capacity. Check both positive and negative moments.

Recommendations

•Check moment at critical sections
•Account for self-weight
•Use consistent sign convention

Educational Content

Introduction

Bending moment analysis is the core of beam design in structural engineering.

Concept Explanation

Internal moments balance external loads through static equilibrium.

Applications

▸Building beams

▸Bridge girders

▸Machine beds

▸Aircraft spars

▸Ship structures

Benefits

•Identifies critical sections
•Determines required section modulus
•Foundation for deflection calculations

Limitations

•Assumes linear elastic behavior
•Does not capture stress concentrations
•Valid only for slender beams

Best Practices

•Draw complete moment diagram
•Check both positive and negative moments
•Include self-weight

Common Misconceptions

Myth: Moment is always maximum at midspan

Truth: For asymmetric loading, max moment may occur at the load location.

Practical Tips

•Zero shear marks maximum moment location
•UDL gives parabolic moment diagram
•Concentrated loads give linear segments
•Check moment at every support and load point

Common Mistakes

Wrong moment arm

Measure moment arms perpendicular from force to section.

Ignoring self-weight

Add self-weight as UDL (0.1-2 kN/m typical for steel).

Sign convention errors

Use consistent convention: sagging positive, hogging negative.

Assumptions

This calculator makes the following assumptions to provide estimates:

Linear Elastic: Material remains in elastic range.

Small Deflections: Geometry does not change significantly.

Plane Sections: Cross-sections remain plane after bending.

Limitations

Be aware of these limitations when using this calculator:

Plastic Behavior: Elastic formulas invalid beyond yield.

Lateral Torsional Buckling: Not captured by simple analysis.

Stress Concentrations: Local peaks near holes are not predicted.

Frequently Asked Questions

Answers to common questions about this calculator.

What is bending moment?

Bending moment at any section is the algebraic sum of all moments of forces on one side of that section about the section. It causes the beam to bend, creating tension on one face and compression on the other.

How is bending moment calculated?

Sum all moments about the section from forces on one side. M(x) = Sum(Fi * di) where di is the moment arm from the force to the section.

What is a bending moment diagram?

A BMD plots bending moment along the beam. Point loads create straight-line segments, UDLs create parabolas, and concentrated moments cause steps.

What is the maximum bending moment?

Maximum moment occurs where shear force is zero. For central point load: Mmax = PL/4. For full UDL: Mmax = wL^2/8 at midspan.

How does a point load affect the moment diagram?

A point load creates a kink in the moment diagram. The diagram consists of straight line segments between loads and supports.

How does a UDL affect the moment diagram?

A UDL creates a parabolic moment diagram. The maximum occurs where shear is zero, typically at midspan for symmetric loading.

What is the relationship between shear and moment?

dM/dx = V. The moment diagram slope equals the shear force value at any point. Zero shear means maximum moment.

What is the sign convention for bending moment?

Positive moment causes sagging (concave upward), with bottom fibers in tension. Negative moment causes hogging (concave downward).

How do you find the maximum moment?

Find where shear force equals zero. The moment at that location is the maximum. Check all zero-shear points for the absolute maximum.

What is the moment at a fixed support?

For a cantilever with end load P: M = PL at the fixed end. For UDL w: M = wL^2/2 at the fixed end.

How does load position affect maximum moment?

Maximum moment is greatest when load is at midspan. Moving the load toward a support reduces Mmax.

What is the moment for a simply supported beam?

Point load P at distance a: Mmax = Pa(L-a)/L. UDL w: Mmax = wL^2/8 at midspan.

What is the moment for a cantilever beam?

Point load P at free end: M = Px where x is from free end. UDL w: M = w(L-x)^2/2. Maximum at fixed end.

How do concentrated moments affect the diagram?

A concentrated moment causes a step change in the moment diagram at its location.

What is hogging vs sagging?

Sagging curves beam concave upward (positive). Hogging curves it concave downward (negative).

How does self-weight affect moment?

Self-weight adds wL^2/8 to midspan moment for simply supported beams. Often significant for long spans.

What is the moment area method?

Uses M/EI diagram area to find slopes and deflections. Change in slope equals area under M/EI.

How do you calculate moment from shear diagram?

Change in moment equals area under the shear diagram between two points.

What is the maximum moment for a continuous beam?

Continuous beams have both positive and negative moments. Use moment distribution or slope-deflection methods.

How does moment relate to bending stress?

sigma = My/I = M/S where M is moment, y is distance from neutral axis, I is moment of inertia, S is section modulus.

What is the plastic moment?

Mp = sigma_y * Zp where Zp is plastic section modulus. Maximum moment when entire section has yielded.

How do you handle multiple loads?

Use superposition: calculate moment for each load separately and sum them.

What is the relationship between moment and curvature?

M = EI*kappa where kappa is curvature (1/radius). This is the fundamental beam bending equation.

How does temperature affect moment?

In determinate beams, temperature causes deflection but no moment. In indeterminate beams, restrained expansion creates moments.

What is the maximum moment for fixed-fixed beam?

Central point load: M = PL/8 at supports and center. Fixed ends reduce moment vs simply supported.

What is moment distribution?

Hardy Cross method iteratively distributes unbalanced moments at joints based on member stiffness.

How do you verify a moment diagram?

Check M=0 at simple supports. Verify fixed-end moments match loads. Area under shear equals change in moment.

What is the typical moment range for steel beams?

Ranges from 10 kNm for light sections to 1000+ kNm for heavy plate girders.

What is the point of contraflexure?

Where moment changes sign (M=0). Occurs in fixed-end and continuous beams where hogging meets sagging.

How do overhanging beams affect moment?

Overhangs create negative moment at the support. The diagram extends beyond supports with sign change.

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Glossary

Technical terms used in mortgage calculations explained in plain language.

Bending Moment: Internal moment at a cross-section caused by external loads.
Sagging: Positive bending moment causing concave-up curvature.
Hogging: Negative bending moment causing concave-down curvature.
Point of Contraflexure: Location where bending moment changes sign.

Authoritative References

Trusted sources and standards referenced by this calculator.

Hibbeler, Mechanics of Materials AISC Steel Construction Manual Timoshenko, Strength of Materials

Applicable Standards

AISC 360 - Flexural DesignEurocode 3 - Bending ResistanceACI 318 - Flexural Strength

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Page Information

Last Updated:

2025-08-21

Version:

2.0.0

References:

License:

Free to use for educational and professional purposes.