1. Concepts about the science of "Strength of Materials" tasks, consistency, uniformity, priority, brief history

Static moments of plane sections

It is known that in order to define an arbitrary cross-sectional surface, it is divided into elementary dF surfaces. In it, the surface of the section is equal to the sum of the elementary surfaces that make up the plane section over the entire surface (Fig. 6.2).

F _i - simple cross-sectional surfaces.
m ² , cm ² (6.1)
23.Coefficient of transverse deformation in tension-compression. Relative change in volume (deformation, longitudinal, transverse, volume, initial, subsequent, relative change).

The coefficient of transverse deformation, also known as Poisson's ratio (ν), relates the relative deformation in the transverse direction to the longitudinal deformation in a material subjected to tension or compression. Additionally, the relative change in volume can be determined based on the longitudinal and transverse deformations. Here's an explanation of these concepts:

1. 
   Coefficient of transverse deformation/Poisson's ratio (ν):
   
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     Poisson's ratio (ν) is a material property that describes the ratio of transverse strain (ε_t) to longitudinal strain (ε_l) when a material is subjected to tension or compression.
     
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     It is denoted by the Greek letter ν.
     
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     Mathematically, ν = -ε_t/ε_l.
     
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     Poisson's ratio typically has a value between -1 and 0.5 for common materials. However, some materials, like auxetic materials, can have negative Poisson's ratios.
     
2. 
   Relative change in volume:
   
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     When a material is subjected to deformation, the volume of the material may change.
     
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     The relative change in volume (ΔV/V) is the ratio of the change in volume (ΔV) to the initial volume (V) of the material.
     
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     The relative change in volume can be calculated using the longitudinal and transverse deformations.
     
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     Mathematically, ΔV/V = ε_l + 2νε_t, where ε_l is the longitudinal strain and ε_t is the transverse strain.
     
3. 
   Longitudinal deformation:
   
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     Longitudinal deformation refers to the change in length or strain that occurs in the direction of the applied force.
     
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     It is denoted by ε_l.
     
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     Longitudinal deformation can be positive (tensile strain) or negative (compressive strain).
     
4. 
   Transverse deformation:
   
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     Transverse deformation refers to the change in width or strain that occurs perpendicular to the direction of the applied force.
     
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     It is denoted by ε_t.
     
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     Transverse deformation can be positive (lateral expansion) or negative (lateral contraction).
     
5. 
   Initial deformation and subsequent deformation:
   
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     Initial deformation refers to the deformation that occurs in a material due to the applied load.
     
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     Subsequent deformation refers to the additional deformation that occurs after the initial deformation has taken place.
     
6. 
   Relative change in volume (continued):
   
   • 
     The relative change in volume (ΔV/V) provides information about the overall volumetric response of a material under tension or compression.
     
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     A positive relative change in volume indicates an expansion, while a negative value indicates a contraction.
     

24-48 The relationship between moments of inertia with respect to parallel axes. Central moments of inertia of simple sections (central axis, arbitrary axis, rectangular, square, circle, round, triangular, rolled section).
Let’s guess the moments of inertia J _x , J _y, J _xy of the plane section relative to the central x and y axes are known. We determine the equatorial and centrifugal moments of inertia with respect to the new x ₁ and y ₁ axes parallel to the central axes and passing through arbitrary distances from them and .
coordinate axes x ₁ and y ₁ are equal to
x ₁ = x+
y ₁ =y+ will be.the coordinate axes x ₁ and y ₁

where: represents , since , is constant,
is equal to
because it is relative to the central axis
In that case, we will create the following formula by performing the above operations based on (6.6) to the 2nd

So, the moment of inertia with respect to the axis parallel to the central axis is equal to the sum of the moment of inertia with respect to the central axis and the cross-sectional area multiplied by the square of the distance between the axes.
We determine the moment of inertia (centrifugal) of the plane section parallel to the central axis with respect to new axes (6.9) based on the formula:

Here
we get the following

Therefore, the centrifugal moment of inertia with respect to the new axes parallel to the central axes is equal to the sum of the moments of inertia with respect to the central axes and the cross-sectional surface multiplied by the distances between the axes. Formulas (6.10, 6.11) are called Steiner's formula.
If it is necessary to determine the moment of inertia with respect to the x _{2 axis parallel to the central axis} x ₁ and passing through the distance "C" , first the moment of inertia with respect to the central x axis (6.13) formula determined based on Then, according to this formula, the moment of inertia with respect to the x ₂ axis is determined as follows:
It can be seen from these relations that the moment of inertia about any eccentric axis is smaller than the moment of inertia about the central axis. As the distance increases, the value of the moment of inertia also increases.