# Internal force, stress and strain

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What is internal force?

Internal force refers to the interaction force (additional internal force) between adjacent parts in an object caused by the action of external force. The external force exerted on the member is called external force.

As shown in the figure, any object is composed of an infinite number of particles. There is an interaction force between any two adjacent particles in the component, and the magnitude of the force is related to the relative position between the particles. When the object is subjected to external force, the object deforms, the relative position between its internal particles changes, and the interaction force between them changes. We call the change of the force generated by the action of external force as additional internal force, which is called internal force for short.

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Method for calculating internal force – Section Method

Obviously, the internal force is inside the component. If we want to solve the internal force, we can only expose the internal force. In this way, we use the section method to solve the section position of the internal force according to the needs. If the section is hypothetically cut off, the original member is balanced, and any part after cutting is also balanced, that is, any part on both sides of the section is in a balanced state under the action of external force and internal force on the section. Therefore, we can take either side of the section, study its equilibrium conditions, establish the equilibrium equation and solve the internal force on the section. The specific section solution steps are as follows.

- The imaginary cross section is generally used to divide the internal force of a rod into two parts.

- Substitution: take any part, and the effect of the abandoned part on the left part shall be replaced by the corresponding internal force (force or couple) acting on the section.

- Balance: establish a balance equation for the remaining part, and calculate the unknown internal force of the rod on the cut-off surface according to the known external force on it (at this time, the internal force on the cut-off surface is the external force for the retained part). According to the basic assumption of uniformity and continuity, an arbitrary force should be continuously distributed on the cut section, and there are internal forces at all points on the section, while there are only six equilibrium conditions of any force system in space, so we can’t solve the internal forces of all points. According to the simplification of the force system, we simplify the arbitrary force system of the internal force to a point of the section, usually to the centroid of the section, and obtain a principal vector and a principal moment, as shown in the figure below.

Taking the section centroid as the origin, establish a rectangular coordinate system. As shown in the figure, the X axis is perpendicular to the cross section, that is, along the axis direction of the rod, and the Y and Z axes are in the section plane. By decomposing the principal vector into three coordinate axes, three components can be obtained: axial force along the x-axis and shear force along the Y-axis and z-axis.

By decomposing the principal moment into three coordinate axes, three components can be obtained: torque along the x-axis and bending moment along the Y-axis and z-axis.

We also call these six components internal force, but we should note that these six components are the resultant force or combined moment of internal force. To solve the internal force of the member later is to calculate the axial force, shear force, torque and bending moment, because these internal forces correspond to the basic deformation of the member: tensile and compressive deformation, shear deformation, torsional deformation and bending deformation.

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What is stress?

Stress is the distribution concentration of internal force (stress is for a “point”. When we want to describe the stress of a point, we should point out the position of the point and the orientation of the plane passing through the point). In order to describe the stress of a point on the section, take a micro area Da around the point, as shown in the figure. On this micro area, the resultant force of the internal force system is DF. Because this area is small enough, we assume that its internal force is evenly distributed, then we can get its average stress, and then take the limit of the average stress to get the total stress or total stress of the point. The direction of the total stress changes with the position of the point taken. Obviously, the total stress is a vector, and the relationship between its direction and the section is arbitrary. Then we decompose the total stress into two components, one perpendicular to the section is called normal stress, and the other tangent to the section is called shear stress.

Average stress:

Total stress (total stress):

The total stress is decomposed into: the stress perpendicular to the section is called “normal stress”, and the stress located in the section is called “shear stress”.

Unit of stress: PA, usually MPa, GPA.

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What are displacement, deformation and strain?

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Displacement

The change of position of a point in the object before and after deformation, linear displacement and angular displacement in material mechanics, as shown in the figure below, the cantilever beam applies a concentrated force at the free end, and the beam is bent and deformed. If we investigate the displacement of a section, such as the displacement of the free end, it is obvious that the centroid of the section will produce a downward displacement, resulting in a linear displacement. At the same time, the normal direction of the section also changes, that is, the section rotates, resulting in angular displacement.

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Deformation

The size and shape of the object change under the action of external force.

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Strain

Measure the deformation degree of a component at a point, and the strain is also for a “point”.

(1) Linear strain (a measure of the change in the size of a point in an object).

As shown in the figure, we investigate any point a in the member, take any point B in the attachment of point a, and the length of AB is DX. Under the action of external force, the member deforms, and both points a and B are displaced to a new position, then the distance between the two points becomes DX + DS, and the deformation is uniform within the range of DX, and the average linear strain can be obtained.

Taking the limit of the above formula, we can get the linear strain at point A.

For plane problems, as shown in the figure, a small rectangle changes into a rectangle shown by the dotted line in the action line of external force (the size changes). If the deformation is uniform within the range of DX and Dy, there is an average linear strain along the X and Y directions.

Take the limit respectively to obtain the linear strain in x and y directions.

(2) Angular strain (a measure of the change in the shape of a point in an object) is also known as shear strain or shear strain.

Defined as the amount of change in a right angle.

As shown in Figure AB and BC, the edges are perpendicular to each other and are shown as dotted lines after deformation, then the angular strain is:

Its strict expression is:

(a) The angular strain shown in the figure is zero (there is rigid body displacement and no deformation); (b) The angular strain shown in the figure is a

Source: China **Flanges Manufacturer** – Yaang Pipe Industry (www.epowermetals.com)

(Yaang Pipe Industry is a leading manufacturer and supplier of nickel alloy and stainless steel products, including Super Duplex Stainless Steel Flanges, Stainless Steel Flanges, Stainless Steel Pipe Fittings, Stainless Steel Pipe. Yaang products are widely used in Shipbuilding, Nuclear power, Marine engineering, Petroleum, Chemical, Mining, Sewage treatment, Natural gas and Pressure vessels and other industries.)

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