RU2553713C1 - Method of diagnostic representation and analysis of cantilever beam flexures - Google Patents

Abstract

FIELD: measurement equipment.

SUBSTANCE: invention relates to the field of material science and may be used to investigate impact of size factor at a modulus of elasticity of cantilever beam material. Substance: typical measurement and graphic representation of cantilever beam flexures is carried out with the point of origin in the cross section of closing up. The point of origin is transferred to the point of force application and moves with it, maximum camber with opposite sign is transferred to the cross section of closing up, the abscissa axis is directed from the new point of origin to the new point of maximum camber, and new values of flexures in intermediate control cross sections are counted from the new abscissa axis to the line of beam deflection. The reasons of flextures include maximum tangent stresses at inclined sites.

EFFECT: possibility of experimental research of dependence of flexure components on size characteristics of a beam and indices of elasticity of is material (for decoding of "size factor"), possibility to match value of flexure with stresses and deformations in a beam KS, possibility to decompose converted flexures in KS for the amount of components, proportional to voltages and deformations in KS.

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Description

The invention relates to the field of materials science and can be used to study the influence of the dimensional factor on the elastic modulus of the material of the cantilever beam.

A typical picture of the deflections y = f (x) (where x is the abscissa of the control cross section (CS)) of the cantilever beam l loaded on the cantilever by the concentrated force P (Fig. 1) is known: in the embedment section taken as the origin, i.e. at x = 0, the deflection is zero, and the maximum deflection y _max is observed at the point of application of force at x = l (Belyaev N.M. Resistance of materials, M .: 1953, p. 355, Fig. 282, f. (18.13, 18.14).

A typical picture of the change in bending moment and stress distribution (and hence strain) in the beam material is also known: in the embedment section, at x = 0, the bending moment M and stress (and strain!) Are maximum, and at the point of application of force, at x = l are equal to zero (Belyaev N.M. Resistance of materials, M .: 1953, p. 355, Fig. 282, f. (18.13, 18.14).

Thus, the typical picture of the observed deflections is diametrically opposite to the picture of stresses and strains in the beam material and (in principle) is not suitable for the technical analysis of the relationships between them.

In addition, shear stresses and deformations are considered only from the transverse force and they are usually neglected because of their smallness, which contradicts the well-known position on the presence of maximum tangential stresses even with simple uniaxial tension - compression on inclined platforms at an angle of 45 ° (A. Alexandrov Resistance of materials. M: Higher school. 2009, p. 56).

The consequences of these shortcomings are observed both in the structure and in the technical contradiction of the well-known formula (1), which for hundreds of years has been quite successfully used to describe the observed deflections:

where y is the amount of deflection in the control sections (CS) of the beam;

M = P · l - bending moment in the section of the seal;

x is the distance (abscissa) of the CS from the section of the embedment (reference);

EJ - section stiffness.

The contradiction is that the actual (y _f by formula 2) value of the maximum deflection on the console is twice that which the bending moment Mp = P · x of the force P (y _m by formula 3) can create:

.those. $$\raisebox{1ex}{${\text{y}}_{\text{F}}$}\!\left/ \!\raisebox{-1ex}{${\text{y}}_{\text{Mp}}$}\right.=\text{2}$$

.

Formally, this mismatch of the deflections is due to the bending moment of the embedment M, but it is exactly equal to the moment M _p of the force P and decreases to zero at the point of application of force in exact accordance with the change in M _p .

This means that the observed pattern of deflections, in addition to the component from normal stresses in bending, contains other components that are comprehensively taken into account in formula (1), but not disclosed by it.

These other components may include:

1. Free movement of the end of the beam when turning the COP during bending. For example: when you turn only one section of the embedment at a certain angle A, the end of the undeformed beam will move by the value Δy = A · l (figure 2).

2. Additional beam bending due to the maximum tangential stresses acting, as is known, simultaneously with normal, even with simple uniaxial tension-compression. Moreover, it is they that lead to the destruction of the test samples — plastic ductility and brittle cleavage at an angle of 45 ° to the tensile-compression line.

The diametrical opposite of the observed pattern of deflections in the standard coordinate system with the beginning in the section of the embedment with the picture of bending moment, stresses and strains in the material of the control sections of the beam, its basic unsuitability for technical analysis of the relationship of deflections with stresses and strains in the CS from the bending moment is noted; ignorance of the relation between normal stresses and maximum tangentials, acting even with simple uniaxial tension and compression of the samples, is noted.

Technical result

The possibility of an experimental study of the dependence of the deflection components on the dimensional characteristics of the beam and the elasticity parameters of its material (for deciphering the "dimensional factor"), the possibility of matching the deflection with stresses and strains in the CS of the beam, the possibility of decomposing the converted deflections in the CS into the sum of components proportional to stresses and deformations in the cop.

The technical result is achieved by the fact that the method for the diagnostic representation of the deflections of the cantilever beam includes a typical measurement and a graphical representation of the deflections of the cantilever beam with the origin in the embedment section, according to the invention, the origin is transferred to the point of application of force and moves with it; the maximum deflection arrow with the opposite sign is transferred to the embedment section, the abscissa axis is directed from the new origin to the new maximum deflection point, and the new deflection values in the intermediate control sections are counted from the new abscissa axis to the beam deflection line, and the maximum causes of deflection are included shear stresses on incline platforms.

Figure 1 shows a typical picture of the deflections y = f (x) of the cantilever beam l, loaded on the cantilever with a concentrated force P in the embedment section taken as the origin, i.e. at x = 0, the deflection is zero, and the maximum deflection y _max is observed at the point of application of force at x = l; figure 2 shows the free movement of the end of the beam by the value Δy = A · l when the section of the embedment is rotated through angle A; figure 3 shows the transformation of the observed pattern of deflections into the diagnostic one by transferring the origin 0 ₁ to the point of application of force, and the maximum deflection with the opposite sign into the section of the seal with the direction of the new abscissa axis from t. 0 ₁ to the new point of maximum deflection with the corresponding conversion deflections in the intermediate control sections (from the new abscissa axis to the deflection line of the beam).

The diagnostic analysis of the converted deflection values includes a mathematical description of the deflection line in the new coordinate system using a polynomial of the third degree, the elements of the polynomial correspond to deflections from the normal and tangential stresses in the CS of the beam when it is bent and the CS moves freely due to their rotation angle relative to the embedment section, which makes suitable for studying their dependence on the dimensional characteristics of the beam and the elasticity parameters of its material (i.e., to decipher the influence of the dimensional factor on the mode l elasticity).

определится как расстояние от новой оси абсцисс 0₁Z до линии прогибов балки:In intermediate CS, the magnitude of the deflections $${\text{y}}_{\text{i}}^{\text{''}}$$

defined as the distance from the new abscissa axis 0 ₁ Z to the deflection line of the beam:

rotation relative to the termination section.

Where Δy _1i is the correction for the slope of the new abscissa axis:

Δy _2i - correction for the position of the curved axis:

x _i , y _i - designation of abscissas and deflections in the old coordinate system.

2. To study the dependence of the converted values of the deflections y ′ = f (z) on the dimensional characteristics of the beam and the elasticity indices of its material using the regression equation in the form of a polynomial of the third degree (7), the elements of which are the decomposition of the total deflection in the CS of the beam into components corresponding to the stress state of the beam material in the COP:

- P · C · z ³ - corresponds to the deflection from normal stresses during bending according to f. (3);

- R · In · z ₂ - corresponds to the deflection from shear stresses in bending;

- P · A · z - corresponds to the free movement of the COP due to their angle.

1. The possibility of matching the magnitude of the deflections with stresses and deformations in the CS beam.

2. The possibility of decomposing the converted deflections in the CS into the sum of the components proportional to the stresses and strains in the CS.

3. The possibility of an experimental study of the dependence of the deflection components on the dimensional characteristics of the beam and the elasticity parameters of its material (for decoding the "dimensional factor").

Claims (1)

A method for the diagnostic representation of cantilever beam deflections, including a typical measurement and graphical representation of the cantilever beam deflections with the origin in the embedment section, characterized in that the coordinate origin is transferred to the point of force application and moves with it; the maximum deflection arrow with the opposite sign is transferred to the embedment section, the abscissa axis is directed from the new origin to the new maximum deflection point, and new values of deflections in the intermediate control sections are counted from the new abscissa axis to the beam deflection line, and the maximum tangents are included in the number of deflection reasons voltage on inclined platforms.

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