First Moment of Area

First Moment of Area

The reinforced frame of an aircraft body and sea vessel are some of the few critical CAD-related activities that employ the concept of the first moment of area.

Professions like civil, mechanical, nuclear, aerospace, and aeronautic engineers, all heavily depend on the concept of the first moment of area to carry out some of their very exclusive function.

However, a fair warning to readers with little background in science. The concept of the first moment of area is highly scientific and contains terms and methods that may be unfamiliar. But we’ll do our best to make them easier to understand.

And for our readers with an understanding of the sciences, most of our contents have been diluted to make them easier to follow.

Last but not least, the scope of our discussion has been limited to the peripherals and we would do our best to provide you with appropriate platforms to read up and discover more.

A Primer into Moments of Area

The concept of Moments of Area has its roots deep in Solid mechanics, which in turn is a branch of Continuum mechanics. Continuum mechanics look into the behavior of solid and physical materials and their resultant attitude when acted upon on forces, phase changes, and temperature changes, and other physics-related elements.  

In addition, the behavior of these physical materials includes motion, deformation, and other physical activities that the materials would undergo. For clarification, these materials include steel, wood, textiles, plastics, concrete, alloys, other geological, and biological materials, etc.

First moment of area is a necessary part of building construction
The First Moment of Area plays a Crucial Role in Building Construction.

The Build-Up to Moments of Area

To understand an object’s Moment of area, you need to first understand the terms that make them up, which are ‘Moments,’ and ‘Area’.

Firstly, Area is the total value of the dimensions of a plane surface, which includes triangles, squares, and circles, and also solid shapes like spheres, cubes, cylinders, etc. The Area of the shape is but one way of understanding the dimensions of the shape, the other is the Volume.

Also, the Moment of a material is the measure of the tendency of a material to rotate about a point or along an axis, as a result of the force acting on one part of the material.

Mathematically, the moment is the product of force and perpendicular distance. The idea of the moment is commonly used in door hinges, foldable ladders, etc.

The idea combines these two concepts, replacing the vague material, that isn’t defined in terms of area, with a specific one that has an exact area and the force responsible for the rotation of the material, is at the edges of the materials. 

Definition of The First Moment of Area

The Moment of Area of physical materials is sub-divided into two categories which are the first moment of area and the Second moment of area which, is usually regarded as the moment of inertia.

Furthermore, the first moment of area is the sum of the weighted distribution of the material involved. It probes into the object tested to derive the centroid, shear stress, and the first moment of area.

The weights are assigned about an arbitrarily defined point, which we derive by dividing the sum of the areas and the average of the dimensions, which we regard as the centroid.

The first moment of area assumes that if all the masses, forces, and other reacting physics quantities are acting on the centroid, their total value would be zero.

Again, the first moment of area is sometimes called the statical moment of area or static. 

The Second Moment of Area

On the other hand, the second moment of area is a geometrical property of an area that reveals how the points of a particular material are distributed about an arbitrary axis.

In better terms, it shows the effort required to bend or rotate a section of a given axis, and the value of this minimum effort is given in Mol.

The second moment of area is a physics-based phenomenon that finds practical application in structural and mechanical engineering, as well as other branches of engineering.

Interestingly, the second moment of area is also regarded as the area moment of inertia or inertia of plane area and is also a very well-known and important concept in the designs of aircraft and naval architecture, which in turn assess the co-relationship between the forces and the center of gravity acting on the vessel, which is provided by the first moment of area.

Furthermore, the second moment of the area provides the basis for the calculation of other concepts like the radius of gyration, polar moment of inertia, and the area moment of inertia for various shapes.

First moment of area is heavily involved in the building and analysis of airplanes.
Aeronautic Engineers Depend on the First Moment of Area in the Design of Airplanes.

Similarities and Differences between The First and Second Moments of Area

Calculations

The major difference between the first moment of area and the second moment of area or moment of inertia is that, for the first moment of area, the calculating process for a beam section is divided into four parts which relate to the axis.

To derive the centroid of a beam section or any other material, you would need to first calculate the first moment of area, which also important in deriving the shear stress of the beam section.

On the other hand, we will also need to derive the centroid of the beam section to derive the moment of inertia of the beam, but it only divides the beam materials into 3 sections.

Uses

The first moment of area is used in engineering applications to derive the centroid and shear stresses of the cross-sectional area of an object.

While the second moment of area is the basis to derive the radius of gyration and moment of inertia for regular and irregular shapes, as well as other quantities it can determine.

Moment

The moment of the object is the basis for both the first and second moments of area, correlates and integrates the power in the particular point, axis, or distance, concerning some quantities.

However, it’s different for both moments of area, as the first moment of area uses the first power of distance and the second moment of area uses the second power of distance.

Relationship between Centroid and the First Moment of Area

Before we go on with this article, let’s take the time to explain one of the terms that you’ll continue to see throughout this work and even in your physics, design, and engineering career, the Centroid.

The centroid is the geometric center of a plane figure or the arithmetic mean position of an object. The theory is that the centroid would provide the perfect balance for an object on the tip of a pin.

The centroid is regarded as the barycentre in astronomy and astrophysics, it’s regarded as the center of the mass between two or more bodies that orbit each other in space.

The centroid is typically used to substitute older terms like the center of gravity and the center of mass of an object.

Diagram Showing Centroid

The diagram below shows the cross-sectional area of the ith section and all parameters were done in terms of the ith section.

Diagram showing centroid as related to first moment of area
Diagram Depicting the Centroid of a Rectangle.

Formulas for the Centroid of a shape.

formula for centroid of a shape

The Centroid for composite cross-sections.

formula for the first moment of area in a composite cross section

Where

Xc,I and Yc,i­ – are the rectangular coordinates of the centroid’s location in the ith section.

Ai – this is the area of the section “I”.

The First Moment of Area in Formulas

As previously mentioned, the first moment of area is a very useful tool in the arithmetic calculations of many physics quantities and sees application in both plane shapes, solid shapes, and compound shapes, which is a combination of the two.

Additionally, the first moment of the area has an SI unit of a cubic foot (ft3) or inch3 in America and meters (m3) in other parts of the world that uses the imperial unit systems.

Theoretically, given an area, “A”, of any particular shape, which is in turn divided into several numbers of elemental areas regarded “dAi” denoted as “n”. We would use xi and yi to represent the distance or coordinate between the elemental area given from the x-y axis.

Diagram

The jth diagram below elaborates on the elements of the first moment of area concept for the object below.

Diagram Depicting the Elements of the First Moment of Area.
Diagram Depicting the Elements of the First Moment of Area.

The Concept Can Be Represented Mathematically in the Following Ways

Mathematical representation of the first moment of area

mathematical representation of the first moment of area

While for the y-axis it’s denoted as

mathematical representation of the first moment of area

Hence it’s calculated using the formula below

Use this formula to calculate the first moment of area.

The first moment of area for composite areas

First moment of area for composite areas

In this case, the letters represent the following;

Qj,x – It’s the statical moment of area “j” over the total body’s neutral x-axis.

dA – the elemental area from the complete area”j”.

y – the perpendicular distance from neutral axis x to the centroid of element dA.

Relationship Between the Shear Stress in a Semi-Monocoque Structure and the First Moment of Area.

Primarily, semi-monocoque is mostly utilized in the design and construction of modern, high-performance aircraft. The design is usually done with design software like SolidFace.

Furthermore, semi-monocoque is a half-single shell fuselage design in which the outer skin is prestressed to support the multitude of external and internal imposed stresses.

One more addition, semi-monocoque is the reinforced frames, formers, and longerons that are combined with the outer skin of the aircraft structure, which on its own can’t maintain these stresses and functions as a support mechanism.

Uses of the Semi-monocoque

The semi-monocoque fuselage is designed by CAD software and is popular in the design of large freight aircraft, commercial airliners, and military planes.

First Moment of Area in Semi-Monocoque Structure

Both the first and second moments of area are used in the design parameters of semi-monocoque structures and it’s based on the formula below. The equation provides for the shear flow which in turn is the basis for deriving the shear stress on the semi-monocoque.

The Formula for Semi-Monocoque Structure

formula semi-monocoque structure

q – this is the shear flow through a particular web section of the cross-section.

Vy – this is the shear force perpendicular to the neutral axis x, through the entire cross-section.

Sx – this is the first moment of area about the neutral axis x for a particular web section of the cross-section.

Ix – this is the second moment of area about the neutral axis x for the entire cross-section shear stress may now be calculated using the following equation.

Shear Stress

Shear stress is a unique type of stress that tends to cause a deformation of a material by slippage along a plane parallel to the imposed stress. The shearing stress may occur in solids, or also in liquids, where it’s related to fluid viscosity.  

Shear Stress Formula

Formula for shear stress. highly needed while calculating the first moment of area.

T – this the shear stress through a particular web section of the cross-section

q –  this the shear flow through a particular web section of the cross-section

t – this is the thickness of a particular web section of the cross-section at the point being measured.

Applications of the First Moment of Area

Mathematically

It’s used to calculate the center of gravity or centroid of a regular, irregular, plane, and even solid shapes.

What’s more, the first moment of area sees application in the derivation of shear stress in materials.

Real-Life

The mathematical elements derived from the first moment of area form the bases in calculating ‘I’, ‘T’, ‘L’ beams, channel, and hollow circular sections. Which is also used for designing various types of engineering works, ranging from structural buildings, shipbuilding, aircraft building, etc.

You can find more on this and more in this mechanical engineering blog.

Final Thoughts on the First Moment of Area

The first moment of area is one of the fundamental quantities in physics, covered in engineering courses like the strength of materials, which as we all know, has a far-reaching impact in every constructive aspect of engineering.

The idea of the first moment of area forms the basic concept behind the idea of the support mechanism used to support other elements in an engineering structure. You can get more info on the moments of area mechanicalc-blog and in Quora.

However, even though most of these calculations done by the moment of areas are now automated by various software, it’s still enough to know about their origins.