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Beam Analysis Calculators

Beam Analysis using Macaulay's Theorem

Beam Analysis using Anastruct

2D Truss Analysis using OpenSeesPy

Column Buckling Calculator

Sample Project

Concrete Column P-M Diagram Calculator

Structural Engineering

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Structural Engineering Calculators

This calculator has two sub-pages that evaluate single-span and multi-span beams under user-defined loading conditions. It outputs the maximum moments and deflections along with shear, moment, slope and deflection diagrams. 

Beams can have three types of idealised support conditions: 

. Each support condition restrains the beam in different ways, thereby affecting the force development in the beam.

At a point of support, there are a maximum of 6 degrees of freedom, which describe the number of possible movements: 3 translations in x, y, z and 3 rotations about x, y, z.

If a certain degree of freedom is restrained at a support, then there is a corresponding reaction. For example, pinned supports allow rotation, so there will be no moment reaction. But since pinned supports restrict translation in both the x and y direction, there will be a reaction in both of those directions. 

When a beam is loaded, internal forces develop within it to maintain equilibrium. The internal forces in a beam have two components: 

We can represent the internal forces at any point in a beam with two resultant forces: a shear force, which is the resultant force of all internal vertical forces at the point of interest and a bending moment which is the resultant of the normal internal forces at the point of interest. Shear force and bending moment vary in magnitude along the length of the beam. 

If you apply a load on the topside of a beam, the beam will be curving downwards, also known as 

When a beam is sagging, the top of the beam will shorten and the forces at the top of the beam will be in compression, while the bottom side of the beam will stretch and the fibres at the bottom of the beam will be in tension. The inverse is known as 

when the load is applied from the underside of the beam or when a beam runs continuously over supports.

To design a beam, you need to know what shear forces and bending moments will exist in the beam under your desired loading. Beam analysis is the process to determine these forces. A common method is to plot the 

, which show you the shear force and bending moment at any point along the beam. 

There are several ways to determine the points needed to plot a SFD and BMD. You can use engineering principles, for example that fact that a uniformly distributed load creates a parabolic bending moment profile. You can also analyse the internal forces at any point in a beam directly, by making an imaginary ‘cut’ at a point of interest in the beam and then solve the three equations of equilibrium: ΣFx = 0, ΣFy = 0 and ΣM=0 with the applied loads and reactions/internal forces.

Sometimes you cannot solve the three equations of equilibrium because there is more than three unknown variables, this means you have a statically indeterminate beam. You must therefore use 

 to determine your internal forces by solving for your degrees of freedom in the stiffness equation F = k x. The maths quickly becomes too complex to do by hand, which is where computer FEA packages come into play!

Based on the spreadsheet "BEAMANAL.xls" by 

This calculator has two tools that evaluate single-span and multi-span beams under user-defined loading conditions.

Single and Multi-span Beam Analysis Tool

Beam Analysis Tool

 by the CalcTree engineering team on June 27, 2024

This calculator plots a Critical Buckling Force vs Length graph for a given steel section for the purpose of understanding the buckling phenomenon. Buckling occurs in compression members, such as columns.

Standards across the world address buckling in slightly different ways, but they are all based on the fundamental Euler buckling equation. For short columns, the Euler buckling equation does not represent real-life buckling accurately so modifiers are implemented, one example being the Johnson parabola. 

This calculator does not intend to replace Standards, but is to help understand the physics behind buckling.

OneSteel 300 Plus® Steel Column Sections

If your applied compressive load is below the line, then your column will 

Critical Force

Buckling occurs when slender elements suddenly collapse under compressive loads. Try holding a ruler at both ends and push your hands together. Notice how the ruler bends or snaps in the middle? That's what we call buckling. 

Buckling occurs due to a loss of stability which means a compression member will displace and continue to displace, that is, it is 

It is important to note that this failure mode is 

, which is why it is very dangerous and can be seemingly deceptive.

 means the member will buckle before it yields, while on the other hand an 

 (shorter) member will yield before it buckles. Buckling is an instantaneous failure mode, as oppose to yielding which is ductile, and so buckling is important to avoid.

To classify a member as slender, it's slenderness ratio is more than it's transition slenderness ratio.

The slenderness ratio is determined by: 


The transition slenderness ratio is determined by: 

The Euler buckling equation is the solution to a second-order differential equation of a compression member. 

The Euler critical stress approaches infinity as the column length approaches zero. Of course, this doesn't appropriately reflect the failure load that has been observed in practice. 

Therefore, the Euler buckling equation is modified by the Johnson parabola for shorter compression members. 

Thus, the Critical Buckling Force for a slender member is calculated from the Euler buckling equation and for an intermediate member is calculated from the Johnson parabola. The combined Euler-Johnson equation has demonstrated a good correlation with column buckling failures in practice. 

This calculator plots this Euler-Johnson equation for a given steel section. It defines the following output parameters:

This calculator analyses long and intermediate steel columns loaded centrally in compression with reference to the:  Euler formula (Long

Concrete Beam Designer to AS3600

Concrete Retaining Wall to AS4678

Concrete Shear Wall to AS3600

Concrete Wall to AS3600

Concrete Slabs to AS3600

Neutral Axis Calculator: RC Section to AS3600

Concrete Column Designer to AS3600

Concrete Slab-on-grade Designer to AS3600

Concrete Rectangular Footing Designer to AS3600

Concrete Punching Shear Calculator to AS3600

Reinforcement Development Length Calculator to AS3600

Steel Beam and Column Designer to AS4100

Fillet Weld Group Calculator to AS4100

Bolt Group Calculator to AS 4100

Steel Web Plate Connection to AS4100

Steel Moment End Plate Connection to AS4100

Steel Flexible End Plate Connection to AS4100

Timber Beam Calculator to AS1720.1

Timber Column Calculator to AS1720.1

Timber Joints to AS1720.1

Load Combinations to AS1170.0

Wind Load to AS/NZS 1170.2

Truss Analysis and Design

Check out CalcTree's library of structural engineering calculations to AS/NZ building and design codes here.

Structural engineering calculations to AS/NZ codes and standards

Concrete One-Way Slab Designer to EC2

Concrete Corbel Designer to EC2

Steel Base Plate Designer to EC3

Steel Section Designer to EC3

Elastic Critical Moment Calculator for Lateral Torsional Buckling to EC3

Timber Beam Calculator to EC5

Seismic Spectrum Calculator to EC8

Check out CalcTree's library of structural engineering calculations to Eurocode building and design codes here.

Structural engineering calculations to Eurocodes and standards

Concrete Rectangular Beam Calculator to ACI 318-19 (IMP)

Concrete Column P-M Diagram Calculator to ACI

Concrete Shear Wall to ACI 318

Concrete One-way Slab Designer to ACI318

Concrete Slab-on-grade Calculator to ACI 360R-10

Concrete Rectangular Spread Footing Designer to ACI

Purlin and Girts Size Calculator to AISI

Steel Beam and Column Designer to AISC 360

Composite Slab to ACI 318

Composite Beam to AISC 360

Wind Loading Calculator to ASCE 7-10

Design Response Spectrum to ASCE 7

Steel Baseplate Designer to AISC 360

CLT Properties Calculator to ANSI/APA PRG 320

Check out CalcTree's library of structural engineering calculations to North American building and design codes here.

Structural engineering calculations to North American codes and standards

Introducing our static analysis calculator for beams: a versatile tool tailored for engineers and students alike. With a capacity to analyze beams comprising up to 5 spans, it efficiently computes reactions, generating precise shear force and flexural moment diagrams.

matplotImage

Static analysis calculator for beams: a versatile tool tailored for engineers and students alike. With a capacity to analyse beams com

Multi-span beam analysis tool using Python

Beam Calculator [SI]

 by the CalcTree engineering team on July 23, 2024

On this page you will find a library of calculators that determine the shear force, bending moment and deflection of a single-span beam with different types of applied loading. The calculators are for a simply supported beam or a cantilever beam. 

Keep reading this page to discover the theory behind the calculators!

Beam Analysis Calculator for beam with multiple loads

Finding SFD and BMD Equations: Macaulay's Theorem

Macaulay's theorem enables us to describe the applied load, shear force diagram  (SFD) and bending moment diagram (BMD) for the full length of the beam with single equations. 

Macaulay's theorem is a singularity function, which means the function has discontinuities at singular points. Macaulay's theorem says:

The applied load, shear force and bending moments can be described by singularity functions as tabulated below. You will notice how the applied load equation is integrated once to get the shear force equation and then integrated again to get the bending moment equation. Note the integration rules are slightly different for singularity functions!

Now that we know the singularity functions for each type of force, we can form the 

For example, for a simply supported beam with a point load there will be two terms in the equations since there is 

. Use the singularity functions in the second row of the table to form the 

❗Sign convention: all terms in the example above are positive (+) even though a downwards load 

 is shown because when values are substituted in the equations, the negative (-) sign for 

Finding Deflection Equations: Double Integration Method

There are three main ways to calculate the deflection equations, but the most convenient is the Double Integration Method. 

Using the bending moment equation you created from the Macaulay's theorem, integrating it once gives you the slope equation and integrating a second time gives you the deflection equation. The formulas are given by:

A common mistake made when integrating is to forget the integration constants. Since there are two integrations there should be a the term 

To find these constants you need to apply boundary conditions (BCs) which differ to each support condition of your beam. A cantilever beam has a fixed support on one end and a simply supported beam has fixed/roller supports at both ends. The boundary condition can be stated as follows:

This library of calculators allow you to create the shear force, bending moment and deflection diagrams of either a cantilever or simply supported beam with various loading conditions. Macaulay's theorem, which is the theory behind how the loaded beams are analysed, and how the 

 equations are derived is also provided. 

A library of calculators that determine the shear force, bending moment and deflection of a single-span simply supported beam or cantilever beam

Beam Analysis Calculators for both individual loads and multiple

This calculator performs the analysis of a 2D planar truss using OpenSeesPy (a Python library), determining the nodal displacements and internal forces of each member.

Youngs modulus of elasticity of the truss elements

matplotImage (1)

👉Member force diagram: red (negative) = compression, blue (positive) = tension

displacement_list

internal_forces

This calculator performs the analysis of a 2D planar truss using python library OpenSeesPy & determines the nodal displacements and internal forces

Project Name

Design Stage

The structure is a one-storey reinforced concrete (RC) frame. Lateral stability is provided by moment-frame action, where seismic loads are resisted through ductile deformation of beams and columns. The foundation comprises of isolated pad footings.

👉Beam tag convention: uses the beam gridline with prefix "V". For example, the beam on gridline C is tagged "VC".

Column Analysis and Design [🚧coming soon]

Footing Analysis and Design  [🚧coming soon]

This page demonstrates a project example of using CalcTree and linking parameters

Linking Parameters: Simple Project Example

We are introducing our Reinforced Concrete Column P-M Diagram Calculator. This tool provides structural engineers and designers with a straightforward solution to assess the strength properties of reinforced concrete columns, streamlining the design process with efficiency and compliance in mind.

This calculator is in imperial [IMP] system. Currently columns can only be of a rectangular cross-section.

matplotImage (2)

The above calculation is according to the following theory:

The concrete is fully fissured when the ultimate axial tensile strength is reached, rendering its contribution null. Therefore, the ultimate strength is attained when the reinforcement reaches its yield stress.

The distribution of strains in a section (a-a) for an immediate failure state assumes that plain sections remain plain and vary linearly with the distance 'c' from the neutral axis.

By ensuring the compatibility of strains, the strains experienced by the reinforcement at any location are identical to those exhibited by the surrounding concrete.

The summation of forces adheres to the internal axial force equilibrium, where 

 represents the ultimate axial strength force.

 Stands for the concrete compression force given by the Whitney equivalent rectangular stress distribution and:

The summation of internal flexural moments and forces originating from the centre of the span must be equal and opposite to the internal force 

In reinforced concrete columns, the concept of "balanced failure in flexo-compression" refers to a condition where the concrete in compression and the tensile reinforcement reach their limit states simultaneously. In this balanced state, the column is designed to maximize the utilization of both materials, ensuring an efficient use of concrete and steel.

Given the above equation, the P-M interaction diagram is crafted by varying the "c" values within the range of (0, h).

The Diagram shows all the possible combinations of P-M valid for the concrete column cross-section. 

Reinforced Concrete Column P-M Diagram Calculator calculator according to geometry, materials and reinforcement. [Imperial Units]

Structural Engineering

Standards

General Analysis

Project Example