Tension (compression) the rod arises from the action of external forces directed along its axis. Tension (compression) is characterized by: - absolute elongation (shortening) Δ l;
relative longitudinal deformation ε= Δ l/l
relative transverse deformation ε`= Δ a/ a= Δ b/ b
With elastic deformations between σ and ε there is a dependence described by Hooke’s law, ε=σ/E, where E is the elastic modulus of the first kind (Young’s modulus), Pa. The physical meaning of Young’s modulus: The modulus of elasticity is numerically equal to the stress at which the absolute elongation of the rod is equal to its original length, i.e. E= σ with ε=1.
14. Mechanical properties of structural materials. Tension diagram.
The mechanical properties of materials include strength indicators tensile strength σ in, yield strength σ t, and endurance limit σ -1; stiffness characteristic elastic modulus E and shear modulus G; characteristics of contact stress resistance surface hardness NV, HRC; elasticity indicators relative elongation δ and relative transverse contraction φ; impact strength A.
Graphical representation of the relationship between the acting force F and elongation Δl called stretch diagram(compression) sample Δl= f(F).
X 
characteristic points and sections of the diagram: 0-1
section of the linear relationship between normal stress and relative elongation, which reflects Hooke’s law. Dot
1
corresponds to the proportionality limit σ pc =F pc /A 0, where F pc is the load corresponding to the proportionality limit. Dot
1`
corresponds to the elastic limit σ y, i.e. the highest stress at which there are still no residual deformations in the material. IN point 2 diagram, the material enters the plasticity region - the phenomenon of material fluidity occurs . Section 2-3 parallel to the x-axis (yield area). On section 3-4 strengthening of the material is observed. IN point 4 local narrowing of the sample occurs. The ratio σ in =F in /A 0 is called tensile strength. IN point 5 the sample ruptures under a destructive load F size.
15. Permissible stresses. Calculations based on permissible stresses.
The stresses at which a sample of a given material fails or at which significant plastic deformations develop are called extreme. These stresses depend on the properties of the material and the type of deformation. The voltage, the value of which is regulated by technical specifications, is called acceptable. Permissible stresses are established taking into account the material of the structure and the variability of its mechanical properties during operation, the degree of responsibility of the structure, the accuracy of the loads, the service life of the structure, the accuracy of calculations for static and dynamic strength.
For plastic materials, the permissible stresses [σ] are chosen so that, in the event of any calculation inaccuracies or unforeseen operating conditions, residual deformations do not occur in the material, i.e. [σ] = σ 0.2 /[n] t, where [n] t is the safety factor in relation to σ t.
For brittle materials, permissible stresses are assigned based on the condition that the material does not collapse. In this case, [σ] = σ in /[n] in. Thus, the safety factor [n] has a complex structure and is intended to guarantee the strength of the structure against any accidents and inaccuracies that arise during the design and operation of the structure.
The design of dynamometers - devices for determining forces - is based on the fact that elastic deformation is directly proportional to the force causing this deformation. An example of this is the well-known spring steelyard.
Relationship between elastic deformations and internal forces in the material was first established by the English scientist R. Hooke. Currently, Hooke's law is formulated as follows: mechanical stress in an elastically deformed body is directly proportional to the relative deformation of this body
The value characterizing the dependence of mechanical stress in a material on the type of the latter and on external conditions is called the elastic modulus. The elastic modulus is measured by the mechanical stress that must arise in the material when the relative elastic deformation is equal to unity.
Note that relative elastic deformation is usually expressed by a number much less than unity. With rare exceptions, it is almost impossible to get equal to one, since the material is destroyed long before that. However, the modulus of elasticity can be found from experience as a ratio and at a small value since in formula (11.5) it is a constant value.
The SI unit of elastic modulus is 1 Pa. (Prove it.)
Let us consider, as an example, the application of Hooke's law to deformation of unilateral tension or compression. Formula (11.5) for this case takes the form
where E denotes the modulus of elasticity for this type of deformation; it is called Young's modulus. Young's modulus is a measure of the normal stress that must occur in a material
at a relative deformation equal to unity, i.e., when the length of the sample is doubled, the numerical value of Young's modulus is determined from experiments carried out within the limits of elastic deformation, and in calculations is taken from tables.
Since from (11.6) we obtain
Thus, the absolute deformation during longitudinal tension or compression is directly proportional to the force and length of the body acting on the body, and inversely proportional to the area cross section body and depends on the type of substance.
The greatest stress in a material, after the disappearance of which the shape and volume of the body are restored, is called the elastic limit. Formulas (11.5) and (11.7) are valid until the elastic limit is passed. When the elastic limit is reached, plastic deformations occur in the body. In this case, a moment may come when, under the same load, the deformation begins to increase and the material collapses. The load at which the greatest possible mechanical stress occurs in the material is called destructive.
When building machines and structures, a safety margin is always created. The safety factor is a value that shows how many times the actual maximum load in the most stressed place of the structure is less than the breaking load.
Hooke's law usually called linear relationships between strain components and stress components.
Let's take an elementary rectangular parallelepiped with faces parallel to the coordinate axes, loaded with normal stress σ x, evenly distributed over two opposite faces (Fig. 1). Wherein σy = σ z = τ x y = τ x z = τ yz = 0.
Up to the limit of proportionality, the relative elongation is given by the formula
Where E— tensile modulus of elasticity. For steel E = 2*10 5 MPa, therefore, the deformations are very small and are measured as a percentage or 1 * 10 5 (in strain gauge devices that measure deformations).
Extending an element in the axis direction X accompanied by its narrowing in the transverse direction, determined by the deformation components
Where μ - a constant called coefficient lateral compression or Poisson's ratio. For steel μ usually taken to be 0.25-0.3.
If the element in question is loaded simultaneously with normal stresses σx, σy, σ z, evenly distributed along its faces, then deformations are added
By superimposing the deformation components caused by each of the three stresses, we obtain the relations
These relationships are confirmed by numerous experiments. Applied overlay method or superpositions to find the total strains and stresses caused by several forces is legitimate as long as the strains and stresses are small and linearly dependent on the applied forces. In such cases, we neglect small changes in the dimensions of the deformed body and small movements of the application points external forces and base our calculations on the initial sizes and initial form bodies.
It should be noted that the smallness of the displacements does not necessarily mean that the relationships between forces and deformations are linear. So, for example, in a compressed force Q rod loaded additionally shear force R, even with small deflection δ an additional point arises M = Qδ, which makes the problem nonlinear. In such cases, the total deflections are not linear functions of the forces and cannot be obtained by simple superposition.
It has been experimentally established that if shear stresses act along all faces of the element, then the distortion of the corresponding angle depends only on the corresponding components of the shear stress.
Constant G called the shear modulus of elasticity or shear modulus.
The general case of deformation of an element due to the action of three normal and three tangential stress components on it can be obtained using superposition: three shear deformations, determined by relations (5.2b), are superimposed on three linear deformations determined by expressions (5.2a). Equations (5.2a) and (5.2b) determine the relationship between the components of strains and stresses and are called generalized Hooke's law. Let us now show that the shear modulus G expressed in terms of tensile modulus of elasticity E and Poisson's ratio μ . To do this, consider special case, When σ x = σ , σy = -σ And σ z = 0.
Let's cut out the element abcd planes parallel to the axis z and inclined at an angle of 45° to the axes X And at(Fig. 3). As follows from the equilibrium conditions of element 0 bс, normal stress σ v on all faces of the element abcd are equal to zero, and the shear stresses are equal
This state of tension is called pure shear. From equations (5.2a) it follows that
that is, the extension of the horizontal element is 0 c equal to shortening vertical element 0b: εy = -ε x.
Angle between faces ab And bc changes, and the corresponding shear strain value γ can be found from triangle 0 bс:
It follows that
Force factors and deformations occurring in timber are closely related. This relationship between load and strain was first formulated by Robert Hooke in 1678. When a beam is stretched or compressed, Hooke's law expresses direct proportionality between stress and relative deformation , Where E the longitudinal modulus of elasticity of the material or Young’s modulus, which has the dimension [MPa]:
Proportionality factor E characterizes the resistance of the timber material to longitudinal deformations. The elastic modulus value is determined experimentally. Values E For various materials are given in table 7.1.
For homogeneous and isotropic materials E– const, then the voltage is also a constant value.
As shown earlier, during tension (compression), normal stresses are determined from the relation
and relative deformation - according to formula (7.1). Substituting the values of quantities from formulas (7.5) and (7.1) into the expression of Hooke’s law (7.4), we obtain
from here we find the elongation (shortening) obtained by the timber.
Magnitude EA , standing in the denominator, is called section rigidity in tension (compression). If a beam consists of several sections, then its total deformation will be determined as the algebraic sum of the deformations of individual i-x sections:
To determine the deformation of the beam in each of its sections, plots are drawn longitudinal deformations(diagram).
Table 7.2 – Values of elastic moduli for various materials
End of work -
This topic belongs to the section:
applied mechanics
Belorussian State University transport.. department of technical physics and theoretical mechanics..
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