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Calculation templates

Welcome to the Physics library, here we have calculations and content for general physics concepts all the way through to topic areas like fluid mechanics and Astrophysics!

Statics

Dynamics

Thermodynamics and Fluid Dynamics

Acoustics and Optics

Relativity and Quantum Mechanics

Astrophysics

Physics Calculations

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Relationship between Young's modulus, bulk modulus and Poisson’s ratio

Moment of Inertia Calculators

Shear Modulus Calculator

Elastic Section Modulus Calculator

Poisson Ratio Calculator

Hooke's Law

Slenderness Ratio Calculator

Stress Calculator

Strain Calculator

Centre of Gravity Calculator

Centre of Mass Calculator

Radius of Gyration in Structural Engineering

Mass Moment of Inertia

Torsion of Shafts

Steel Wire Tension Calculator

Mass Timber Self-weight Calculator

Wood Density Calculator

Wood Moisture Content Calculator

Wood Specific Gravity Calculator

Wood Shrinkage Calculator

View and execute free statics calculations

Physics Calculators - Statics

This calculation tool is designed to help you quickly and accurately determine the slenderness ratio of a 1D element. 

The slenderness ratio or slenderness is the ratio of effective length to the least radius of gyration and it is calculated using the formula 

If you liked this, check out our other articles and resources!

Calculate slenderness ratio effortlessly on CalcTree. Free-to-Use Slenderness Ratio Calculator for Structural Analysis. Calculate now!

Slenderness Ratio Calculator 

The moment of inertia, also known as the second moment of area, also known as the moment of inertia, is a geometrical property that measures how "stiff" a cross-sectional shape is. In other words, a measure of a cross-sections resistance to bending and deflection. The second moment of area is typically denoted by 

, and its unit of dimension is a unit value of length to the fourth power, i.e. 

Explore our Moment of Inertia calculators below for different section types.

Moment of Inertia Calculator: Solid Rectangular Section

Moment of Inertia Calculator: Rectangular Hollow Section (SHS/RHS)

Moment of Inertia Calculator: Solid Circular Section

Moment of Inertia Calculator: Circular Hollow Section

Moment of Inertia Calculator: I Beam 

Moment of Inertia Calculator: T Section 

Moment of Inertia Calculator: Angle Section

Moment of Inertia Calculator: Channel Section

Moment of Inertia Calculator: Triangular Section

Moment of Inertia Calculator: Trapezoidal Area Section

Intuitively, you know that the arrangement on the left will bend less if you stood on it. The plank cross-sections are the same, the area, and its material properties are the same. Yet, by rotating the plank onto its side, its geometry in relation to the axis of bending, and therefore the moment of inertia, has changed significantly. 

 is not a fixed value that is unique to each cross-section; it changes depending on the location of the bending (or reference) axis. 

The parallel axis theorem involves relating the moment of inertia of a cross-section about an arbitrary reference axis to its moment of inertia about a parallel centroidal axis. This theorem is useful because knowing the centroidal moment of inertia of a shape means its moment of inertia can be calculated about any parallel axis by adding a correction factor. 

This theorem is particularly useful when the moment of inertia of a complex shape with multiple parts needs to be calculated. 

The moment of inertia of the entire cross-section about its centroid is equivalent to the sum of the moments of inertia of each segment about its own centroidal axis and the parallel axis terms.

If there are any holes or voids within the cross sections, their moment of inertia must also be calculated but subtracted from the total. 

Accurately calculate the moment of inertia on CalcTree. Free-To-Use online moment of inertia calculator. Try now!

Second Moment of Area Calculators

This template explores Hooke's Law both in general terms, and in the context of spring constants, it's most common usage.

Hooke's law can be rearranged to solve for each variable. Below is a calculation to solve for the force. Remember to account for the direction of your deflection and units.

Go ahead and duplicate this page to change it up and solve for k or s!

Hooke's law states that the strain/deformation of an elastic object or material is proportional to the stress applied to it. 

In this context, Hooke's Law is expressed as:

🏁 CalcTree's Free-to-use Hooke's Law Calculator to determine the force needed to extend or compress a spring. Input calculation values. Start now 💡

This calculation tool is designed to help you quickly and accurately determine the radius of gyration for a cross-section. Enter custom section values or select a cross-section for the AISC section size database.

A radius of gyration is defined as the distance from the centre of mass of a body at which the whole mass is concentrated (without changing the moment of rotational inertia) about an axis through the 

The radius of gyration is defined as the distance from the Neutral axis to the strip where the area is assumed and is calculated using the formula:

Calculate the radius of gyration for custom section sizes and AISC typical cross-section sizes.

Radius of Gyration for custom and AISC typical cross-sections

Radius of Gyration In Structural Engineering

This page provides insight into determining the centre of mass and some functions to help you calculate it.

The centre of mass is a position defined relative to an object or system of objects. It is the average position of all parts or objects of a system, weighted according to their masses.

This is demonstrated by the diagram below, depicting the coordinate system of an object:

The centre of mass is located at the centroid for simple rigid objects with uniform density and can be calculated with the following formula:

This calculator helps you determine the centre of gravity of a triangle and the centre of gravity between two masses. 

The centre of gravity refers to the point in a body, like a volume, area or line, that would be in balance if it were suspended. In a symmetrical body of a uniform material, the centre of gravity is the geometric centre (also known as the centroid).

This function calculates the centre of gravity of a body given the three side lengths of a triangle. 

Where all three medians meet at a single point (concurrent), is known as the centroid of a triangle. A centroid is also known as the centre of gravity.

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A simpler calculator to obtain the centre of gravity of a triangle is by using its height directly.

This function calculates the centre of gravity of two separate bodies, as a function of the masses and the distance between the centre of the bodies.

This calculation tool is designed to help you quickly and accurately determine the centre of gravity of objects.

Center of Gravity Calculator

This page provides some insight into determining stress, and provides some functions to help you calculate it!

Stress is defined as a force that changes the shape of an object. The units of stress are Pascals (Pa) or Newton per square metre (N/m^2), and is calculated according to the following formula

To use this as a shear stress calculator simply use the cross-sectional area parallel to the plane of the imposed force.  

Calculate stress with ease on CalcTree. Free-to-Use Online  Stress Calculator. Take control of your engineering analysis. Calculate now!

Engineering Stress Calculator Online 

Engineering Stress Calculator

Welcome to our strain and deformation calculator! This page will show you how to calculate strain as well as provide some insight into each of the relevant equations required to do so. 

Strain is the amount of deformation experienced by the body in the direction of force applied i.e. due to stress. It is defined as the ratio of change in shape to its original shape. The following equation determines the relation for deformation in terms of the length of a solid

Calculate strain in materials with ease on CalcTree. Free-to-use Engineering Strain Calculator. Get started now

Material Strain Calculator - 

This calculation tool is designed to help you quickly and accurately determine an object's general elastic section modulus.

It is a geometric property for a given cross-section for the design of beams or flexural members and is calculated according to the formula:

The Elastic Section Modulus Calculators For Other Cross Sections

To calculate the elastic section modulus for other shapes, check out these calculators:

Rectangle Elastic Section Modulus Calculator

I Beam Elastic Section Modulus Calculator

Circle Round Elastic Section Modulus Calculator

Hollow Round Elastic Section Modulus Calculator

Hollow Rectangle Square Elastic Section Modulus Calculator

Diamond Shape Elastic Section Modulus Calculator

C-Channel Shape Elastic Section Modulus Calculator

Calculate elastic section modulus for various section shapes. Look up elastic section modulus values for AISC cross-sections.

Elastic section modulus calculators - AISC Cross-section elastic modulus

This page provides insight and functions to determine the mass moment of inertia for a rigid body. Sub-pages with diagrams and calculators to calculate the mass moment of inertia for various shapes can also be found here!

The general equation for the mass moment of inertia is as follows:

Generial Mass Moment of Inertia, I

Analogous to the role of mass in dictating the necessary force for a given linear acceleration, the moment of inertia serves as a key parameter in determining the effort required to induce rotational motion around a specified axis. The selection of the rotational axis significantly influences the resultant moment of inertia value, underscoring the axis's integral role in these calculations.

The Mass Moment of Inertia For Other Shapes:

To calculate the mass moment of inertia for other shapes, check out these calculators:

This page provides explains mass moment of inertia for a rigid body and tools to calculate the value for various objects.

Mass moment of inertia Calculation tools for varying shapes

Mass moment of inertia

This page provides insight into determining the torsion. It will explore many things relating to torsion, such as:

This page will also provide some functions to help you calculate torsion.

Circular Shaft Maximum Torsion Calculation

Circular Shaft Polar Moment Of Inertia Calculation

Torsional Deflection of Shaft Calculation

Torsion is defined as the twisting of an object due to an applied torque, the shear stress or pressure on an object as a result of an applied twisting moment (torque)

Often used concerning shafts, torsion is also relevant to other areas.

The polar moment of inertia measures a beam's ability to resist torsion. This is necessary to determine the shear stress of a beam under torque.

Calculating the diameter of solid shafts is also possible using the maximum shear stress and maximum twisting moment.

The angular deflection of the shaft is also important to determine the warping of the cross-section of a beam.

This is the general formula for calculating the shear stress of a shaft using the imposed torsion (T), the distance from the centre of the surface (r) and the moment of inertia (J). 

Calculating the shear stress of shafts is important to ensure structural integrity and safety, as it helps design shafts that can withstand the applied forces without failing.

This is the formula for calculating the maximum torsion within a circular shaft. The torsion is calculated using the shear stress (τ), the radius (r) and the moment of inertia (J). 

Circular Shaft Polar Moment of Inertia Calculation

As the polar moment of inertia is needed to determine the torsion of a circular shaft, you may also like to use the formula and function below to calculate it!

The polar moment of inertia can be calculated by just knowing the diameter of the shaft.

From this, it is also possible to determine the required diameter of a circular shaft if you know the maximum torsion (T) and the maximum shear stress of the shaft (τ). The diameter can be calculated using the formula and function below.

Accurately calculating the torsional deflection of a shaft helps ensure the optimal performance and longevity of mechanical systems. 

The torsional deflection of a shaft is calculated using the length of the shaft (L), the torsion (T), the rigidity modulus (G) and the moment of inertia (J).

To calculate the torsion for other shaped shafts, check out these calculators:

Solid Cylinder Shaft Torsion

Hollow Cylinder Shaft Torsion

Ellipse Shaft Torsion

Rectangle Shaft Torsion

Square Shaft Torsion

Hexagon Shaft Torsion

C-Channel Section Shaft

I-Section Shaft Torsion

Rectangle Hollow Shaft Torsion

Triangle Shaft Maximum Allowable Torsion

This page provides insight into determining the torsion.

Calculate torsion of shafts

This calculator determines the shear modulus, a measure of the elastic shear stiffness of a material. For isotropic materials, the shear modulus is directly proportional to Young's modulus with Poisson's ratio as a multiplier. 

 is defined as the ratio of the shear stress, 

For isotropic materials (materials with the same properties in all directions), the shear modulus is directly proportional to Young's modulus, 

Take a segment of a material, as per image below, and apply an external force 

 (i.e. the segment is shearing) caused by an external force. How easily the segment shears is measured by its shear modulus. A higher shear modulus means a stiffer material and higher shear resistance.

This calculator determines the shear modulus, a measure of the elastic shear stiffness of a material. 

In isotropic materials, there is a relationship between Young's modulus, the bulk modulus and Poisson’s ratio. This calculator determines Young's modulus based on the relationship, and provides the derivation of the relationship.

Imagine a segment of rubber, in the usual shape of a unit cuboid. Let us consider, a tensile force 

 (such as you pulling on it) acting normally outward on each surface of the unit cube. 

 acting along the direction of x-axis produces an extension 

 along that direction. Since the cube is a unit cube, 

 indicate the tensile stress and tensile strain along that axis, respectively. By definition, Young’s modulus, 

Also by definition, the contraction of the cube in the other two directions (y-axis and z-axis) is given by 

 acting in the x-direction produces an extension 

 acting in the y-direction produces an extension 

 in the x-axis and z-axis. The tensile force 

 acting in the z-direction produces an extension 

 along each of the three axes is given by:

All the tensile forces acting together produce a volume stress of magnitude 

 is equal to three times the longitudinal strain, 

 along each direction. And therefore we get:

This calculator determines Young's modulus based on the relationship with the bulk modulus and Poisson's ratio.

This calculator computes the tension required in a steel wire to produce a certain frequency in the wire when it vibrates. 

The formula for tension incorporates the wire's physical dimensions, density, and vibrational frequency. This relationship is vital for assessing the wire's ability to endure different types of forces.

Example 1: calculating tension for a desired frequency

In musical instrument manufacturing, such as crafting a guitar or piano, achieving the correct pitch (frequency) for each string requires adjusting the tension. Here, desired frequency and the physical characteristics of the string is known, and is required to find the tension to apply.

 required in a string to produce the correct pitch (frequency) is given by the equation:

💡This calculator uses the equation above. 

Example 2: calculating frequency for a desired tension 

In civil engineering, particularly in the maintenance of bridge cables or suspension structures, knowing the current tension in a cable (through on-site measurements) can help calculate the fundamental frequency. This information is crucial for assessing the structure's resonance characteristics.

The equation in example 1 can be re-arranged such that the fundamental frequency, 

Check out our full library of CalcTree templates 

This calculator computes the tension required in a steel wire to produce a certain frequency in the wire when it vibrates.

This calculator presents three methods to determine the Poisson ratio of an object. Poisson ratio is the ratio of lateral strain and longitudinal (or axial) strain of a deformed object.

Three methods to calculate Poisson ratio:

Poisson ratio is a measure of how a material deforms when it is stretched or compressed. It is the ratio of the change in width to the change in length of a material. 

The ratio between the change in diameter, ΔD, and the original diameter, D is the lateral strain. The ratio between the change of the length, ΔL, and the original length, L, is the axial strain. The ratio between the lateral strain and the axial strain is the Possion ratio.


For isotropic materials (materials with the same properties in all directions), the Poisson ratio could also be calculated using the 

 and Stiffness of the object using the following formulae: 

Poisson ratio typically vary from 0 to 0.5.


Examples of Poisson ratio for common materials:

This calculator presents three methods to determine the Poisson ratio of an object, which is the ratio of lateral strain and longitudinal strain.

This calculator computes the self-weight of a mass timber panel, useful for when the value is not provided by the CLT or glulam manufacturer. Units are in the imperial system. 

 for values of G based on your wood species

The equation for self-weight of the CLT panel is:

This calculator computes the self-weight of a mass timber panel, useful for when the value is not provided by the CLT or glulam manufacturer.

This calculator computes the density of wood given its species and moisture content. Units are in the imperial system. 

Density is the material's mass-to-volume ratio. It fluctuates in materials like wood that absorb water as it's moisture content varies. The wood's total mass increases upon moisture absorption, which may also alter its volume. Contrary to specific gravity, density accounts for both mass and volume at the prevailing moisture content, inclusive of water-mass.

To compute wood density at any moisture content, employ its basic specific gravity Gb and its percent volumetric shrinkage from the green state to the desired moisture content SMC.