Analyze a Beam: Moment, Shear and Deflection

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Beam Analysis

The Calcs.com Beam Analysis Calculator computes bending moment, shear force, and deflection along a single or multi-span beam using linear-elastic finite element analysis. Enter the span, section properties, supports, and loads, and it returns the maximum demands and the full diagrams. It handles cantilevers, continuous spans, and combined point, distributed, and applied-moment loads.

Worked example

Example 1: simply supported beam with a point load

A steel beam is 12 m long, pinned at the left end and on a roller at the right end. It carries one 8 kN point load, 9 m from the left support and 3 m from the right support. The loads and results follow a textbook example (R.C. Hibbeler, Structural Analysis, 8th ed., 2012) that Calcs.com uses as a test case for this calculator. The conditions are:

Simply supported beam, 12 m long, pinned at the left end and on a roller at the right end
One point load of 8 kN at 9 m from the left support
$E$ = 200 GPa and $I$ = 60 × 10⁶ mm⁴, so $EI$ = 12 000 kN·m²
Find the support reactions, the maximum shear and moment, and the maximum deflection

Add the Beam Analysis calculator

In your project, choose Add New Calculation, then choose Beam Analysis under Analysis.

Add a new calculation window with the US region selected and the Analysis group listing Beam Analysis, Portal Frame Analysis Wizard, Truss Analysis Wizard, Diaphragm Analysis (Load Linking), Steel Bolt Group Analysis (ASD) and Beam Analysis with Load Combinations.

Enter the section properties

In Section Properties, the Bending Stiffness

EI

starts at 1332. Replace it with the formula 200 GPa * 60e6 mm^4. The calculator converts it to 12 000 kN·m². The example gives no axial stiffness, so

EA

stays at its default of 356 000, and the Section Label stays as it is.

Key Properties with the Bending Stiffness field being edited, the formula 200 GPa * 60e6 mm^4 typed in and a tooltip showing 12 000, an Axial Stiffness of 356 000, a Length of Beam of 12 000 millimeters, Beam Orientation set to Horizontal, and supports of Pinned at 0 and Roller at 12 000 millimeters.

Set the length and the supports

Enter 12 000 mm for the Length of Beam and leave Beam Orientation as Horizontal. In Position of Supports from Left, keep the first support as Pinned at 0 mm. Change the second support to Roller at 12 000 mm. The Support Type list offers Fixed, Pinned, Roller and Hinge. The blank row at the bottom of the table stays empty.

Position of Supports from Left table with the Support Type list open on the second row, showing Fixed, Pinned, Roller and Hinge, with Pinned highlighted.

Key Properties with a Bending Stiffness of 12 000, an Axial Stiffness of 356 000, a Length of Beam of 12 000 millimeters, Beam Orientation set to Horizontal, and supports of Pinned at 0 and Roller at 12 000 millimeters.

Replace the default load with the point load

Simple Loads starts with a distributed load that runs from 0.1 to 1 kN/m over the full length. Clear that row. In Point Loads, add one load, Point Load 1, with a magnitude of 8 kN at a location of 9000 mm. The calculator applies the load as entered, with no load factor.

Simple Loads section with the default Distributed Load 1 of 0.1 kilonewtons per meter at the start and 1 kilonewton per meter at the end, from 0 to 12 000 millimeters, and an empty Point Loads table.

Simple Loads section with an empty Distributed Loads table and a Point Loads table with Point Load 1 of 8 kilonewtons at 9000 millimeters, and the beam diagram showing the load with a pinned support on the left and a roller on the right.

Read the summary

The Summary shows a Moment Demand

M^*

of 18 kN·m, a Shear Demand

V^*

of −6 kN, a Deflection

\delta

of −16.8 mm and a Maximum Lateral Reaction

R^*

of 6 kN. The deflection is negative because the load acts downward. The reactions diagram shows 2 kN at the left support and 6 kN at the right.

Summary with a Moment Demand of 18 kilonewton meters, a Shear Demand of minus 6 kilonewtons, a Deflection of minus 16.8 millimeters and a Maximum Lateral Reaction of 6 kilonewtons, a reactions diagram showing 2 kilonewtons at the left end and 6 kilonewtons at the right end, and the start of the shear graph.

Look at the graphs

Below the Summary, the shear force steps from 2 kN to −6 kN at the load, the bending moment peaks at 18 kN·m at 9 m, and the deflection curve is lowest between the supports at about −16.8 mm. Hover over a point to read its value. The last diagram shows the 8 kN load and the reactions of 2 kN and 6 kN.

Moment graph rising to a peak of 18 kilonewton meters at 9 meters, a deflection graph lowest at about minus 16.8 millimeters between the supports, and a diagram showing the 8 kilonewton load at 9 meters and reactions of 2 and 6 kilonewtons.

Shear graph with a tooltip at x of 9 meters reading a shear of 2 kilonewtons, just before the step down to minus 6 kilonewtons.

Moment graph with a tooltip at x of 9 meters reading a moment of 18 kilonewton meters.

Check the results per span and per support

Per-Span Analysis lists one span of 12 000 mm with a deflection of −16.8 mm, a positive moment of 18 kN·m and a negative moment of 0. Per-Support Analysis lists reactions of 2 kN at 0 mm and 6 kN at 12 000 mm, with shears of 2 kN and −6 kN.

Per-Span Analysis with a Deflection Per Span table showing a span length of 12 000 millimeters and a deflection of minus 16.8 millimeters, and a Moment Demands Per Span table with a positive moment of 18 kilonewton meters, values of 6.07, 12 and 18 at Q1, Q2 and Q3, and a negative moment of 0.

Per-Support Analysis with a Support Demands Table giving reactions of 2 kilonewtons at 0 millimeters and 6 kilonewtons at 12 000 millimeters, and shears of 2 and minus 6 kilonewtons.

Inputs, Calcs.com output and a hand check

Inputs

CalculatorBeam Analysis

Length of beam

L

12 000 mm

OrientationHorizontal

Left supportPinned at 0 mm

Right supportRoller at 12 000 mm

Bending stiffness

EI

12 000 kN·m²

E

200 GPa

I

60 × 10⁶ mm⁴

Axial stiffness

EA

356 000

Point load

P

8 kN

Load location9000 mm

Calcs.com output

Moment demand

M^*

18 kN·m

Shear demand

V^*

−6 kN

Deflection

\delta

−16.8 mm

Max lateral reaction

R^*

6 kN

Reaction at 0 mm2 kN

Reaction at 12 000 mm6 kN

Hand check

✓Taking moments about each support, the right reaction is

8 \times 9 / 12 =

6 kN and the left reaction is

8 \times 3 / 12 =

2 kN

✓Shear: 2 kN left of the load and

2 - 8 =

−6 kN right of it

✓Moment under the load:

M = 2 \times 9 =

18 kN·m

✓

EI = 12\,000

kN·m², from

200 \times 10^6

kN/m² and

60 \times 10^{-6}

m⁴. For a simply supported beam with one point load, the largest deflection is

\delta_{max} = \dfrac{P\,b\,(L^2 - b^2)^{3/2}}{9\sqrt{3}\,L\,EI}

,
where

b

is the distance from the load to the nearer support (3 m). This gives

\dfrac{8 \times 3 \times 135^{3/2}}{9\sqrt{3} \times 12 \times 12\,000}

= 0.0168

m, so 16.8 mm downward, at

\sqrt{(L^2 - b^2)/3} = 6.71

m from the left support

The reactions, shear, moment and deflection match the closed-form solution. The per-span and per-support tables are output from the calculator, with no hand check. This is a sanity check only. Beam Analysis version 63.

Method & scope

Property	Detail
Analysis type	Linear-elastic finite element analysis (direct stiffness method) using 1D beam elements.
Beam theory	Euler-Bernoulli (shear deformation not included).
What it returns	Bending moment, shear force, and deflection at every point along the member, plus maximum demands and reactions.
Supported models	Single span, multi-span continuous, and cantilever beams. Any number of supports. Combined point, uniform, linear (triangular or trapezoidal), and applied-moment loads.
Support types	Fixed, Pinned, Roller and Hinge in the Position of Supports table. A pinned support restrains translation with free rotation; a fixed support restrains translation and rotation. Linear and rotational spring supports and continuous rigid and continuous spring supports are also available.
Sign convention	In the Summary, a downward load gives a negative deflection, plotted below the axis. Full moment and shear sign convention shown in the diagrams and in Sign & Axis Conventions.
Key assumptions	Linear-elastic material, small displacements, prismatic member per span. Section properties ( $EI$ and $EA$ ) are taken as entered.
Out of scope	Material nonlinearity, large-displacement (P-delta) effects, torsion, dynamic and buckling analysis. This calculator returns analysis demands only. Code capacity checks are done in the design calculators.

Symbols

These are the symbols used on this page, with the label the calculator shows where it has one.

Symbol	Meaning
$L$	Length of Beam, the total length including any intermediate supports
$x$	Position of a support or a load, measured from the left of the beam
$w$	Distributed load magnitude, start and end
$P$	Point load magnitude
$b$	Distance from a point load to the nearer support, used in the hand check
$E$	Young’s modulus
$I$	Second moment of area
$EI$	Bending Stiffness
$EA$	Axial Stiffness
$M^*$	Moment Demand
$M$	Bending moment at a point, used in the hand check
$V^*$	Shear Demand
$\delta$	Deflection
$\delta_{max}$	Largest deflection of the beam, used in the hand check
$R^*$	Maximum Lateral Reaction

Calculation method

The Beam Analysis calculator uses the finite element method (FEM) matrix stiffness approach to solve for internal forces, reactions, and deflections in simple or continuous beams.

FEA solver

The solver assembles the global stiffness matrix from member bending stiffness

EI

and axial stiffness

EA

, applies boundary conditions at each support, and solves the system

\{F\} = [K]\{u\}

for nodal displacements. Internal forces (moment, shear, axial) and deflections are then recovered from the displacement solution at a dense mesh of points along the beam length. The loads are not factored: the support reactions in the Summary equal the applied loads. For factored load combination analysis, use the Beam Analysis with Load Combinations (ASD) or (LRFD) calculators instead.

Reported results

The governing demand outputs are envelope maxima across the full beam. The moment demand across all spans is:

\begin{aligned}M^* &= \max\bigl(\lvert\text{positive moment}\rvert,\\ &\qquad\ \lvert\text{negative moment}\rvert\bigr)\end{aligned}

The shear demand and the deflection are taken across the beam length:

$V^* = \max\lvert\text{shear}\rvert$
$\delta = \max\lvert\text{deflection}\rvert$

Per-span analysis reports positive moment, Q1, midpoint, Q3, and negative moment peaks for each bay between supports. Per-support analysis reports the vertical reaction, moment reaction (for fixed supports), and shear at each bearing point.

Supported configurations

Supports can be fixed, pinned, roller, or spring (linear or rotational). Continuous rigid supports are also available. Beams can be horizontal or inclined (Simple Slope or Hip/Corner Slope). Loads include distributed (uniform or linearly varying), point loads, moment loads, and axial loads, at any position along the span.

How to use it

The sheet has Key Properties, optional spring and continuous supports, Simple Loads, a second group for axial, moment, linked and oriented loads, and Summary, Per-Span Analysis and Per-Support Analysis results. Clicking any input label opens a reference explanation for that field.

Enter the key properties

Key Properties starts with the Section Properties table. The calculator analyzes the beam from the section’s Bending Stiffness

EI

and Axial Stiffness

EA

, so the table asks for those two values.

Key Properties section with a Section Properties table showing a section label, a Bending Stiffness EI of 1332 and an Axial Stiffness EA of 356 000, a Link icon, a Length of Beam of 2000 millimeters, Beam Orientation set to Horizontal, and a Position of Supports from Left table with two Pinned supports at 0 and 2000 millimeters.

Enter the values manually, or link a section from one of the cross-section calculators.Entering the values manually: type

EI

and

EA

for the section into the table. The values can be entered as formulas. For example, for a standard steel beam with a second moment of area of 6 660 000 mm⁴, you could enter 200000 MPa * 6660000 mm^4.Linking a cross-section: a section can be linked from any of the cross-section calculators under Cross-Sections in Add New Calculation, such as the Cross-Section Database. See Look Up Section Properties in the Cross-Section Database.

Add the cross-section calculator to the project and choose the section.
In Beam Analysis, select the chain-link icon at the end of the Section Properties row.
In the Link to custom cross-section window, choose the calculation, then the section row. The window lists the section with its Bending Stiffness, Axial Stiffness and Self-weight.

Once linked, the row shows the linked section’s label,

EI

and

EA

in grey, and the section drawing appears above the table.

Section Properties row after linking, with the section label XSect1-W12X26, a Bending Stiffness EI of 16 978 and an Axial Stiffness EA of 986 838 shown in grey fields, followed by Length of Beam, Beam Orientation and the supports table.

The calculator only analyzes the section for demands. It does not check the section against its capacity. Use a design calculator to design the member and check it against its capacity.Length of Beam is the total length from start to end, including any intermediate supports. Beam Orientation is Horizontal or Inclined.

Place the supports

In Position of Supports from Left, choose a Support Type for each support and enter its Position (x), the distance from the left of the beam to the center of the support. Add as many supports as you need to make a continuous beam with any number of spans.

Position of Supports from Left table with the Support Type list open, showing Fixed, Pinned, Roller and Hinge.

A pinned support is fixed in translation and free in rotation. A fixed support is fixed in both translation and rotation.If the first support is not at 0, a cantilever is assumed at the left side. In the same way, if the last support is not at the end of the beam length, a cantilever is assumed at the end.The reactions at each support update automatically as supports are added, changed or deleted, based on the specified loading.

Add spring and continuous supports (optional)

Open Spring Supports, Continuous Supports, and Inclined Roller Supports for four more tables.

Spring Supports, Continuous Supports, and Inclined Roller Supports section with four tables: Position of Linear Springs from Left, Rotational Spring Supports, Continuous Rigid Supports and Continuous Spring Supports.

Position of Linear Springs from Left: a linear spring is either perpendicular to the beam (resisting lateral forces) or parallel to the beam (resisting axial forces). It acts equally in compression and tension. Enter the Spring Type, Position (x) and Stiffness (k). A spring support may not restrict the same degree of freedom as a rigid support, so a lateral spring cannot be added at the location of a pinned support.
Rotational Spring Supports: a rotational spring resists moments at a location, equally in either direction of rotation. A rotational spring cannot be added at the location of a fixed support.
Continuous Rigid Supports: a length of the span that is entirely prevented from moving in one or more directions. Choose a Rigid Support Type (Pinned or Roller) and enter the Start Position and End Position. Calcs.com models the continuous support as a series of closely spaced individual supports.
Continuous Spring Supports: a length of the span restricted by a linear spring. Choose a lateral or axial Spring Type, enter the Start Position and End Position, and enter the Stiffness (k). These are also modeled as a series of closely spaced spring supports.

Apply the simple loads

The calculator supports different loading types, which can be applied in combination. Each load can be named. The sign convention used for loading is (positive values shown):

Loading sign convention diagram with an arrow pointing right, an arrow pointing down and a curved arrow in the corner between them.

Simple Loads section with a beam diagram showing a sloping distributed load, a Distributed Loads table with one row of start and end magnitudes and locations, and an empty Point Loads table.

Distributed Loads are in force per unit length, such as kN/m, along the beam. A distributed load starts and ends at any locations, and the start and end magnitudes may differ. Uniform loads have the same start and end magnitude. Linear loads have different start and end magnitudes, and can represent triangular or trapezoidal loads. All distributed loads entered in this table are applied perpendicular to the beam, whatever its orientation.Point Loads are in force units, such as kN, applied at one point along the beam. For example, a point load can represent a weight hung from a ceiling or a column supported by the beam. Enter the label, the magnitude and the location. All point loads entered in this table are applied perpendicular to the beam, whatever its orientation.To apply a load at an angle to the beam, use the tables in the next step.

Apply axial, moment, linked and oriented loads

The second group has two tables: Axial, Point & Moment Loads, and Distributed Loads.

Axial Loads, Moment Loads, Linked Loads, and Oriented Loads section with two tables: Axial, Point and Moment Loads with Location, Axial Eccentricity, Orientation Axis and Load Magnitudes columns, and Distributed Loads with Start Location, End Location, Orientation Axis and Load Magnitudes columns.

Axial, Point & Moment Loads are concentrated loads at a single point along the beam. Each row is one location. Axial eccentricity is the eccentricity at which the axial load is applied to the beam, so a non-zero eccentricity adds a moment equal to the axial load multiplied by the eccentricity.
Distributed Loads are applied along a continuous length of the beam. Each row is one pair of start and end locations.
Orientation Axis only matters if the beam is inclined. An Aligned orientation reads the inputs as exactly parallel and perpendicular to the beam. A Global orientation reads them along the global X and Y axes, which is useful because gravity always acts along the global Y axis however the beam is oriented.

Both tables have a Link from Calculator option.

Read the summary

Once the section, supports and loads are entered, the calculator uses the Calcs.com finite element analysis engine to find the moments, shear forces and deflections. The Summary gives the maximum values as Moment Demand

M^*

(the maximum of the positive and negative bending moments in the beam), Shear Demand

V^*

and Deflection

\delta

, with the reactions on the beam.

Summary section with a Moment Demand of 0.28 kilonewton meters, a Shear Demand of minus 0.7 kilonewtons, a Deflection of minus 0.00676 millimeters, and a beam diagram with a reaction labeled at each end.

Below them are the shear force, bending moment and deflection diagrams along the beam, and a diagram of the loads and reactions. Hovering over any point on a diagram gives the value at that location along the beam. The sign convention used in the shear force and bending moment diagrams is (positive values shown):

Sign convention diagram for shear force and bending moment, with a rectangle that has an upward arrow on its left side and a downward arrow on its right side, and a curved element with a curved arrow at each end.

Shear, moment and deflection graphs below the Summary, followed by a load diagram showing a sloping distributed load and a reaction at each end.

Check the results per span and per support

Per-Span Analysis and Per-Support Analysis list the results for each span and each support.

Per-Span Analysis with a Deflection Per Span table giving span length, span type and deflection, and a Moment Demands Per Span table giving the positive moment, the values at Q1, Q2 and Q3, and the negative moment.

Per-Support Analysis with a Support Demands Table giving the position, reaction, moment and shear at each support.

Beam Analysis with Load Combinations: Overview

This video covers the Beam Analysis with Load Combinations calculators (ASD and LRFD), and its examples use US units.

Video transcript

Timestamps are from the video captions and show roughly where each part begins.