Bend This type of deformation is called in which the initially straight axis of the rod is bent.

Rod with rectilinear the axis working in bending is called beam. Beams are one of the most important elements of all building structures, as well as many structures used in mechanical engineering, shipbuilding and other branches of technology.

The first question about the strength of beams was raised in 1638. Galileo in his book “Conversations and Mathematical Proofs Concerning Two New Branches of Science.” In 1826, that is, almost two centuries later, the French scientist Claude Louis Marie Henri Navier ( Navier, 1785 – 1836) practically completed the creation of the theory of beam bending. We essentially use this theory to this day.

The hypothesis of plane sections when bending a beam

Let us mentally draw a grid on the side surface of the undeformed beam, consisting of longitudinal and transverse (perpendicular to the axis of the beam) straight lines. As a result of bending the beam, we will see that the longitudinal lines will take on a curved outline, and the transverse lines practically will remain straight And perpendicular to the curved axis of the beam. Thus, cross sections that are flat and perpendicular to the axis of the beam before deformation remain flat and perpendicular to the curved axis after it is deformed.

This circumstance indicates that during bending (as during stretching and torsion) plane section hypothesis.

What displacements occur when a beam bends?

As a result of bending, an arbitrary point lying on the axis of the beam moves in the direction of the vertical axisy and longitudinal axisz . Vertical movement usually denoted by the letterv and call him deflection beams. Longitudinal movement dots are designated by letteru .

A tangent drawn to a point located on the curved axis of the beam will be rotated relative to the straight axis by a certain angle. This angle, as shown by numerous experimental data, turns out to be equal turning angle 𝜃 cross section of the beam passing through the point under consideration.

Thus, three sizes v , u Andθ are moving components arbitrary cross-section of a beam during bending.

In what follows we will show thatu << v , therefore, when calculating a beam for bending by longitudinal movementu neglected.

Which internal efforts occur in the cross section of a beam during straight bending?

Consider, for example, a beam (Fig. 1) loaded with a vertical concentrated forceP . For determining internal power factors, arising in a certain cross section located at a distancez from the place where the load is applied, we will use by section method. Let's demonstrate two options for using this method, which can be found in educational literature.

Fig.1. Internal force factors arising during straight bending

First option.

Let's cut it beam in the cross section we have outlined at a distancez from the left end (Fig. 1, A).

Let's discard mentally right part of the beam along with a rigid seal (or simply, for convenience, cover them with a piece of paper). Next we must replace the action of the discarded part on left by us leftpart of the beam by internal forces . (elastic forces) We see that the external load is trying to shift the part of the beam visible to us upward (in other words, to implement shiftP ) with a force equal to , and bendits convexity downwards, creating a moment equal to . Pz As a result, internal forces arise in the cross section of the beam, which resist the external load, that is, they counteract and shift , And. bending These forces obviously arise in everyone points beam cross section, And they are distributed across the cross section along unknown as long as we have the law. Unfortunately, immediately determine this endless system of forces impossible. So we'll bring all these forces together to the center of gravity cross section under consideration and let's replace their action statically equivalent internal forces: cutting force Q y And bending moment M.

xcutting force Q Andbending moment M As we have repeatedly noted above, the destruction of the rod in the section under consideration will not occur only if these internal forces will be able to balancecutting force Q= P external load.bending moment M = its convexity downwards, creating a moment equal to .Therefore we easily find that, Acutting force Q Andbending moment M Note that it is precisely thanks to these two

internal efforts option.

when unloading, the part of the beam we are considering will go down and straighten. Second Stilllet's cut it the beam in the place of interest to us into two parts. But left by us let's discardP . now not the right one, but part of the beam loaded with force We will replace. the action of the part we discarded on the left right part of the rod internal efforts We will find these efforts directly as the action of the discarded left side on the right side. P To do this we will parallel force transfer to the center of gravity) . According to the well-known lemma from the course of theoretical mechanics, a force applied at any point of a body is equivalent to the same force applied at any other point of this body, and a pair of forces whose moment is equal to the moment of this force relative to the new point of its application. Therefore, in the cross section of the rod we must apply a forceP and momentits convexity downwards, creating a moment equal to .cutting force Q= P Then the cutting force , Abending moment M = its convexity downwards, creating a moment equal to bending moment . That is, we get the same result, but without performing the procedure.

balancing , A And By what rules are they calculated? cutting force,emerging?

in the cross section of the beam during bending If we use first

1) option, then these rules are as follows: shear force numerically equal algebraic sum of all external forces (active and reactive) acting on the one under consideration;

2)us part of the beam bending moment numerically equal

the algebraic sum of the moments of the same forces relative to the main central axis passing through the center of gravity of the cross section under consideration. Note that bending, in which both a bending moment and a shear force occur in the cross section of the beam, is called. transverse If only a bending moment occurs in the cross section of the beam, then bending is called.

clean

What happens to the longitudinal fibers of a beam during bending? Many scientists have thought about this question. For example, Galileo believed that when a beam bends all its fibers stretch equally . Famous German mathematician (Gottfried Wilhelm Leibniz Leibnitz

, 1646 – 1716) believed that the outermost fibers located on the concave side of the beam do not change their length, and the elongations of all other fibers increase in proportion to the distance from these fibers. However, numerous experiments, for example, experiments (Arthur Jules Morin Morin, 1795 – 1880), carried out in the 40s.XIX c., showed that when bending a beam is deformed in such a way that some of its fibers experience tension, and some experience compression. The boundary between the areas of tension and compression is a layer of fibers that only bend without experiencing no stretching, no compression . These fibers form the so-called.

neutral layer The line of intersection of the neutral layer with the cross-sectional plane of the beam is called neutral axis or zero line.

When a beam bends, its cross sections rotate precisely relative to the neutral axis.

The strength of a beam is checked, as a rule, only according to the greatest normal stress. These stresses, as we already know, arise in the outermost fibers of the cross section of the beam in which the greatest force “acts.” absolute bending moment value. We determine its value from the diagram of bending moments.

During transverse bending in a beam, along with normal stresses, tangential stresses also arise, but in the overwhelming majority of cases they are small and, when calculating strength, are taken into account mainly only for I-beams, which we will discuss separately.

Condition for the strength of a beam when bending along normal voltages has the form:

where is the permissible stress [ σ ] is taken to be the same as when tensioning (compressing) a rod made of the same material.

Besides strength checks, according to formula (1) can be produced and selection of beam cross-section dimensions. At a given permissible voltage [ σ ] and known maximum absolute bending moment valuerequired moment of resistance beams in bending is determined from the following inequality:

It is necessary to keep in mind the following very important circumstance. When the position of the beam's cross-section changes relative to the acting load, its strength can change significantly, although the cross-sectional areaF and will remain the same.

Let, for example, a beam of rectangular cross-section with an aspect ratioh/ b=3 is located in relation to the force plane in such a way that its heighth perpendicular to the neutral axis x . In this case, the ratio of the moments of resistance of the beam during bending is equal to:

That is, such a beam is three times stronger than the same beam, but rotated by 90° .

Let us recall that in the expression for the moment of resistance of a beam of rectangular cross-section during bending squared its size is that which is perpendicular to the neutral axis.

Consequently, the beam section must be positioned in such a way that the force plane coincides with that of the main central axes about which the moment of inertia minimal. Or, what is the same, it is necessary to ensure that the neutral axis is the axis about which the main moment of inertia of the cross section maximum. In this case, the beam is said to be bending at planes of greatest rigidity.

The above once again emphasizes the importance of the topic “Determination of the position of the main central axes of inertia of the cross section of a rod,” which students usually treat superficially.

Having determined from the strength condition (1) the required moment of resistance during bending, we can move on to determining the dimensions and shape of the beam cross-section. At the same time, we need to strive to ensure that the weight of the beam is minimal.

For a given beam length, its weight is proportional to the cross-sectional areaF .

Let us show, for example, that a square cross-section is more economical than a round one.

In the case of a square cross-section, as we know, the moment of resistance during bending is determined by the formula

For a circular cross section it is equal to:

If we assume that the cross-sectional areas of a square and a circle are equal to each other, then the side of the squareacan be expressed in terms of the diameter of a circled : =0,125 Fd , we come to the conclusion that a square cross-section with the same area has a greater moment of resistance than a round one (almost 18%). Therefore, a square cross-section is more economical than a round cross-section.

Analyzing the distribution of normal stresses along the height of the cross section of the beam (), it is easy to come to the conclusion that that part of the material that is located near the neutral axis almost does not “work” (this, in particular, indicates the irrationality of a round cross-section compared to a square). To obtain the greatest savings in material, it should be placed as far as possible from the neutral axis. The most favorable case for a given cross-sectional area F and heighth obtained by placing each half of the area at a distanceh /2 from the neutral axis.

Then the moment of inertia and the moment of resistance will be respectively equal:

This is the limit that can be approached by using an I-beam cross-section with the largest amount of material in the flanges. However

, due to the need to allocate part of the material for the beam wall, the resulting limit value for the moment of resistance is unattainable. So, for rolled I-beams:

For such beams, the strength is checked as follows:At points farthest from the neutral axis

the strength of the I-beam is checked using formula (1);At the points where the shelf connects to the wall,

or one of the strength hypothesis formulas is used;

At points located on the neutral axis, – for the highest tangential stresses:

What is the potential strain energy during bending?

The potential deformation energy of a beam during transverse bending is determined by the following formula

where the first integral is the potential shear energy, and the second is the pure bending energy.

Dimensionless coefficient valuek , included in the first term of expression (2), depends on the cross-sectional shape of the beam and is calculated by the formula

For example, for a rectangular cross sectionk =1,2.

For most types of beams, the first term in formula (2) is significantly less than the second term. Therefore, when determining the potential strain energy during bending, the influence of shear (the first term) is often neglected.

Absolute deformation- the amount of change in the dimensions of bodies: length, volume, etc.

Anisotropy- the difference in the physical and mechanical properties of the material in different directions (wood, plywood, structural plastics, etc. - the variability of properties is due to the heterogeneity of the structure and the specifics of manufacturing).

Beam- This is a horizontal beam lying on supports and experiencing bending deformation.

Bolt— a rod with a head on one end and a thread on the other end for a nut (designed to connect parts of comparable thickness).

timber- this is an element in which one size (length) significantly exceeds the others. The main characteristics of the timber are its axis and cross-section. the shape can be straight or curved, the cross-section can be prismatic - constant cross-section and with a continuously changing cross-section (industrial pipes), as well as a stepped cross-section (bridge supports)

Shaft- this is a beam (usually shafts are straight bars with a circular or annular cross-section) that transmits torque to other parts of the mechanism. Most shafts experience a combination of bending and torsion deformations. When calculating shafts, tangential stresses from the action of transverse forces are not taken into account due to their insignificance.

Screw- a rod with a head on one (maybe without a head) and with a thread on the other end (usually along the entire length) for screwing into one of the parts to be fastened (intended mainly for connecting parts of incommensurate thickness, one of which is often a body) .

screw- a part with a threaded hole, screwed onto a bolt or stud and used to lock the parts being fastened.

Deformation (lat. Deformatio - distortion)- change in the shape and volume of the body under the influence of external forces. Deformation is associated with a change in the relative position of the particles of a body and is usually accompanied by a change in the magnitude of interatomic forces, the measure of which is elastic stress. There are four main types of deformation: tension/compression, shear, torsion and bending.

Solid body deformation— change in the size, shape and volume of a solid body. Deformation of a solid occurs when its temperature changes or under the influence of external forces.

Deformable body- a mechanical system that has, in addition to translational and rotational degrees of freedom, internal (oscillatory) degrees of freedom. Deformable bodies are divided into: absolutely elastic bodies without dissipative degrees of freedom; and on inelastic bodies with dissipation.

Section deplanation- during torsion - the phenomenon of violation of the flatness of cross sections. Section deplanation occurs when prismatic rods are torsioned.

Dynamics- a branch of mechanics that studies the influence of interactions between bodies on their mechanical motion.

Tension diagram- graph of the dependence of mechanical stress on the relative deformation of a solid body.

Rigidity- the ability of a body or structure to resist the formation of deformation. Stiffness is measured by the coefficient of proportionality between force and relative linear, angular, or curvature deformation.

Spring stiffness is the coefficient of proportionality between the deforming force and deformation in Hooke’s law. Spring stiffness: numerically equal to the force that must be applied to an elastically deformable sample to cause its unit deformation; depends on the material from which the sample is made and the dimensions of the sample.

Margin of safety- ratio: tensile strength of the material; to the maximum normal mechanical stress that the part will experience in operation.

(R. Hooke - English physicist; 1635-1703)- the relationship between the magnitude of elastic deformation and the force acting on the body. There are three formulations of Hooke's law: 1- the magnitude of the absolute deformation is proportional to the magnitude of the deforming force with a coefficient of proportionality equal to the rigidity of the deformed sample; 2 - the elastic force arising in the deformed body is proportional to the magnitude of the deformation with a proportionality coefficient equal to the rigidity of the deformed sample; 3 - elastic stress arising in the body is proportional to the relative deformation of this body with a proportionality coefficient equal to the elastic modulus.

Bend- in the strength of materials - a type of deformation of a beam, beam, slab, shell or other object, characterized by a change in the curvature of the axis or the middle surface of the deformed object under the influence of external forces or temperature.

Shear stress— force per unit cross-sectional area of ​​the sample, parallel to the direction of action of the external force.

Kinematics- a branch of mechanics that studies the geometric properties of the motion of bodies without taking into account their masses and the forces acting on them. Kinematics explores ways of describing movements and relationships between quantities that characterize these movements.

Classical mechanics- a physical theory that establishes the laws of motion of macroscopic bodies at speeds significantly lower than the speed of light in a vacuum.

Oblique bends to the center of gravity - in the resistance of materials - a type of deformation characterized by a change in the curvature of a beam under the influence of external forces passing through its axis and not coinciding with any of the main planes.

Torsion (torsion French)- in the strength of materials - a type of deformation characterized by mutual rotation of the cross sections of a rod (shaft, etc.) under the influence of pairs of forces acting in these sections. During torsion, the cross sections of round rods remain flat. Torsion- this is a type of deformation in which only a torque occurs in the cross sections of the beam.

Array- this is a body with dimensions of the same order (foundations, retaining walls, bridge abutments, etc.)

Mechanics— the main section of physics; the science of the mechanical movement of material bodies and the interactions that occur between them. As a result of interaction, the speeds of the bodies change or the bodies are deformed. Mechanics is divided into statics, kinematics and dynamics.

Continuum mechanics- a branch of mechanics that studies the movement and equilibrium of gases, liquids and deformable solids. In continuum mechanics, matter is considered as a continuous medium, neglecting its molecular-atomic structure; and consider the distribution of all its characteristics in a medium to be continuous: density, stress, particle velocities, etc. Continuum mechanics is divided into hydroaeromechanics, gas dynamics, elasticity theory, plasticity theory and other sections.

Mechanics of bodies of variable mass- a branch of mechanics that studies the movements of bodies whose mass changes over time due to the separation of material particles from the body (or attachment to it). Such problems arise during the movement of rockets, jet aircraft, celestial bodies, etc.

Mechanical stress- a measure of internal forces arising in a deformable body under the influence of external influences. Mechanical stress at a point on a body is measured by the ratio of: elastic force arising in the body during deformation; to the area of ​​a small cross-sectional element perpendicular to this force. In the SI system, mechanical stress is measured in pascals. There are two components of the mechanical stress vector: normal mechanical stress, directed normal to the section; and tangential mechanical stress in the section plane.

Moment of a couple of forces- the product of one of the forces that make up a pair of forces and the shoulder.

Modulus of elasticity (modulus of elasticity of the first kind, modulus of longitudinal elasticity of the material), Modulus(Coefficient of elasticity; Elastic modulus; Modulus of elasticity) - a coefficient of proportionality characterizing the tensile strength of a material. The modulus of elasticity characterizes the stiffness of the material. The greater the elastic modulus, the less the material deforms at the same stress.

Hardening— increase in the strength of crystals after plastic deformation. Hardening manifests itself in an increase in the limit of proportionality of the material and its fragility (ductility decreases).

Normal mechanical stress— force per unit cross-sectional area of ​​the sample, perpendicular to the direction of action of the external force.

Shell- a body bounded by two curved surfaces, whose thickness is significantly less than other dimensions (walls of tanks, gas tanks, etc.).

Homogeneous environment- a medium characterized by the equality of the considered physical properties at any point in space.

Relative deformation- the ratio of the amount of change in body size to its original size. Often the relative deformation is expressed as a percentage.

Plastic deformation

Couple of forces- two equal in numerical value and opposite in direction parallel forces applied to the same solid body. A couple of forces creates a moment of force.

Plate (plate)- this is a body bounded by two parallel surfaces, whose thickness is significantly less than other dimensions (the bottoms of vessels, for example). Thick plates are usually called slabs.

Plastic- the property of solids to change shape and size under load without the formation of ruptures and cracks; and maintain the changed shape and size after removing the load.

Plastic deformation- deformation that does not disappear after the cessation of external forces.

Shoulder couple- the shortest distance between the lines of action of the forces that make up a pair of forces.

Creep- the phenomenon of changes in the body under a constant load applied to the body. As temperature increases, the creep rate increases. Types of creep are relaxation and elastic aftereffect.

Potential energy of an elastically deformed body- a physical quantity equal to the work that elastic forces can do by the time elastic deformations are completely removed.

Transverse bend- bending that occurs in the presence of bending moments and shear forces.

Proportionality limit - mechanical stress, up to which is observed, the dependence of deformations on stresses is linear.

Elastic limit- the highest mechanical stress at which the material retains its elastic properties (the deformation disappears after the load is removed); when the limit is exceeded, the first signs of plastic deformation appear (in plastic materials).

Yield strength- stress at which strain increases without a noticeable increase in load.

Tensile strength (tensile strength)- the maximum mechanical stress that the material can withstand without collapsing.

Longitudinal-transverse bending- bending caused by the simultaneous action of forces directed along the axis of the rod and perpendicular to it.

Longitudinal bending- in the resistance of materials - the bending of an initially straight rod under the action of centrally applied longitudinal compressive forces due to its loss of stability.

span beams are the distance between supports; in frames, this is the distance between the axes of the posts.

Simple bending of a straight beam- bending of a straight beam, in which external forces lie in one of the planes passing through its axis and the main axes of inertia of the cross section (in one of the main planes of the beam). During plane bending, normal and shear stresses arise in the cross sections of the beam.

Work of force- a measure of the mechanical action of a force when moving the point of its application. The work of a force is a scalar physical quantity equal to the product of force and displacement.

Equilibrium of a mechanical system- the state of a mechanical system under the influence of forces, in which all its points are at rest relative to the reference system under consideration. Equilibrium of a mechanical system occurs when all forces and moments of force acting on the system are balanced. Under constant external influences, a mechanical system can remain in a state of equilibrium for as long as desired.

Frame is a system consisting of rods rigidly connected to each other.

Communication reaction- the force with which a mechanical connection acts on a body.

Tension-compression— in the strength of materials — a type of deformation of a rod under the action of forces, the resultant of which is normal to the cross section of the rod and passes through its center of gravity. Tension-compression is caused by: forces applied to the ends of the rod; or forces distributed throughout its volume: the own weight of the rod, inertial forces, etc.

Relaxation- in the resistance of materials - the process of spontaneous decrease in internal stress over time with constant deformation.

Rheology- the science of deformation and fluidity of matter. Rheology considers: - processes associated with irreversible residual deformations and the flow of various viscous and plastic materials: non-Newtonian fluids, dispersed systems, etc.; as well as the phenomena of stress relaxation, elastic aftereffect, etc.

Free torsion— torsion, in which the deplanation in all sections is the same. In this case, only shear stresses arise in the cross section.

Restricted torsion- torsion, in which, along with tangential stresses, normal stresses also arise in the cross sections of the rod.

Shift- in the resistance of materials - deformation of an elastic body, characterized by the mutual displacement of parallel layers (or fibers) of a material under the influence of applied forces at a constant distance between the layers.

Force- a measure of mechanical action: on a material point or body; provided by other bodies or fields; causing a change in the speed of points of the body or its deformation; occurring through direct contact or through fields created by bodies.

Force- physical vector quantity, which at each moment of time is characterized by: a numerical value; direction in space; and application point.

Elastic force- a force that arises in a deformable body and is directed in the direction opposite to the displacement of particles during deformation.

Complex resistance- in the resistance of materials - the deformation of a beam, rod or other elastic body that occurs as a result of several simple deformations occurring simultaneously: bending and stretching, bending and torsion, etc. Ultimately, any deformation can be reduced to tension-compression and shear.

Complex bending of a straight beam- bending of a straight beam caused by forces located in different planes. A special case of a complex bend is an oblique bend.

Strength of materials— the science of strength and deformability of elements (parts) of structures and machines. The main objects of studying the strength of materials are rods and plates, for which appropriate methods for calculating strength, rigidity and stability under the action of static and dynamic loads are established. The resistance of materials is based on the laws and conclusions of theoretical mechanics, and also takes into account the ability of materials to deform under the influence of external forces.

Statics- a branch of mechanics that studies the conditions of equilibrium of material points or their systems under the influence of forces.

Hardness- the ability of a material to resist mechanical penetration of foreign bodies into it.

Strain gauge— a testing device for determining the yield strength, tensile strength, elastic modulus and other physical and mechanical characteristics necessary for assessing the strength and deformability of materials.

Plasticity theory— branch of mechanics: studying the deformation of solids beyond the limits of elasticity; developing methods for determining the distribution of stresses and strains in plastically deformable bodies.

Elastic deformation- deformation that disappears after the cessation of external forces.

Elastic aftereffect- the process of spontaneous growth of deformation over time at constant stress.

Clean bend- bending that occurs in the presence of only bending moments.

General purpose washer- an annular plate designed to be placed under a nut or screw head in order to reduce the crushing of the part being fastened by the nut, if the part is made of a less durable material (plastic, aluminum, wood, etc.) to protect the clean surfaces of the part from scratches when screwing the nut ( screw) to close the hole when it is large.

Special purpose washer- these are lock or safety washers, the so-called nut locks (Grover spring washer, lock washer with teeth, etc.). These washers prevent the connection from unscrewing.

1. Beam - a beam loaded with external forces perpendicular to its axis, and working mainly in bending.

2. Shaft - a beam loaded with pairs of forces lying in the cross-sectional plane and working in torsion.

3. Eccentric tension or compression - tension or compression of a rod in which the resultant of internal forces is directed normal to the cross section, but does not pass through its center of gravity.

4. External forces - forces acting from any body or system on the body or system in question.

External forces include not only active forces (load), but also reactions of connections or supports.

5. Internal forces - forces of interaction between mentally dissected parts of the material body. In other words: elastic forces, resistance forces, efforts.

6. Endurance - the ability of materials to resist destruction under the action of repeated alternating stresses.

7. Hypothesis of plane sections - cross sections of a rod that are flat before deformation remain flat after it.

8. Deformation - in qualitative terms, is a change in the size and shape of a body under the influence of external forces or temperature.

9. Dynamic load - a load characterized by a rapid change in time in its value, direction or point of application and causing significant inertia forces in structural elements or machine parts.

10. Allowable voltage - the maximum value of voltage that can be allowed in a dangerous section to ensure the safety and reliability of operation required under operating conditions. F = ƒ(∆ℓ)

11. Rigidity - the ability of the material of structural elements to resist the formation of elastic deformations that occur under the influence of external forces.

12. Bending moment is a pair of internal forces perpendicular to the cross-sectional plane.

13. Distribution load intensity - distributed load acting per unit length or area.

14. Shear stress is a component of total stress located in the section plane.

15. Console - a beam with one pinched end and the other free end, or part of a beam that extends beyond the support.

16. Stress concentration is a local increase in stress that occurs with a sharp change in the cross-section of a body.

17. Critical force - the lowest value of force at which the rod loses stability.

18. Torque is a pair of internal forces lying in the cross-sectional plane. The torque in the cross section is equal to the sum of the moments of all external forces on one side of the section, taken relative to the central axis of the rod.

19. Torsion is a type of simple deformation in which only torques arise in the cross sections of the rod under the action of external pairs of forces located in planes perpendicular to the central axis of the rod.

20. Mechanical state of a material - the behavior of a material under mechanical load.

In relation to the central tension of a mild steel sample, the following mechanical states of the material are distinguished, for example: elasticity, general fluidity, hardening, local fluidity and fracture.

21. Load is a set of active external forces acting on the body in question.

23. Normal stress is a component of the total stress directed along the normal to the elementary section area on which this stress acts.

24. Dangerous section - the cross section of the rod where the greatest tensile and compressive stresses occur.

25. Zero-zero or pulsating voltage cycle - a change in time-varying voltage from zero to a maximum positive value (or from zero to a minimum negative value) during one period.

26. Plasticity is the property of a material under the influence of external forces to deform irreversibly without destruction.

27.Plane bending - bending under the action of external forces located in one plane - in the plane of symmetry of the rod or in the main plane passing through the line of bending centers.

28. Cross section - a section of a rod perpendicular (normal) to its central axis.

29. Fatigue limit (fatigue limit) - the highest value of the maximum cycle stress at which fatigue failure of a sample of a given material does not occur after an arbitrarily large number of cycles.

30. The limit of proportionality is the highest voltage up to which Hooke’s law is applicable.

31. Tensile strength is the ratio of the maximum force that a sample of a given material can withstand to the initial cross-sectional area of ​​the sample.

32. Yield strength is the stress at which a rapid increase in plastic deformation occurs without a noticeable increase in load.

33. The elastic limit is the highest stress at which only elastic deformations occur.

34. Limit state - a state in which a structure or structure ceases to meet specified operational requirements or requirements during construction.

35. The principle of independence of the action of forces (the principle of superposition, the principle of superposition, the principle of addition of the action of forces) - the principle according to which the total result obtained by the simultaneous action of several forces is the sum of the individual results obtained by the action of these forces separately.

36. Span - the entire beam or part of it located between two adjacent supports.

37. Strength is the ability of a material to resist destruction under the action of external forces. Strength is the ability of materials, within certain limits and conditions, to withstand external loads without collapsing. Strength is quantitatively characterized by stress (MPa).

38. Distributed load - a load applied continuously to a given surface or line.

39. Calculation model (diagram) - a simplified image of the structure, as well as its elements, taken to perform the calculation.

40. Symmetrical voltage cycle - a change in alternating voltage from a minimum to a maximum value during one period, with the maximum and minimum voltages being equal in magnitude and opposite in sign.

41. Crumple is a local plastic deformation that occurs on the contact surface under the action of compressive forces.

42. Concentrated load - a load applied to a very small area (point).

43. Shear - destruction resulting from shear in the plane of maximum tangential stresses.

44. Static load - a load whose value, direction and location of application changes so slightly that when calculating structural elements they are taken to be independent of time and therefore the influence of inertial forces caused by such a load is neglected.

45. Rod (bar) - a body whose shape is formed by the movement of a flat figure (constant or variable area), provided that the center of gravity of the figure moves along a certain line and the plane of the figure remains perpendicular to this line.

Another, simpler definition: a rod is a geometric object, two dimensions of which (transverse dimensions) are commensurate with each other and are much smaller than the third (length).

46. ​​Fluidity is a property of a material that manifests itself in the rapid increase in plastic deformations without a noticeable increase in load.

47. Strength theories are essentially hypotheses that seek to identify the mechanical state of a material under a complex stress state and thus determine the criteria for the strength of materials: the plasticity condition for elastoplastic materials, and the strength condition for brittle materials.

48. Angular strain is the angle of shear.

49. Impact strength is the ability of a material to resist impact, revealed on standard samples by impact with a falling load. Viscosity is the ability of a material to resist the formation of plastic deformations.

50. Elastic line - the curved axis of the beam within the limits of elastic deformations of the material.

51. Fatigue of materials is a change in the mechanical and physical properties of a material under the long-term action of stresses and strains that cyclically change over time.

52. Stability of a compressed rod - the ability of a compressed rod to resist the action of an axial force tending to remove it from its initial state of equilibrium.

53. Brittleness is the property of a material to collapse without previous significant plastic deformation.

54. Pure bending is a type of simple deformation in which only bending moments occur in the cross sections of the rod under the action of external forces.

1. Condition for tensile and compressive strength: N= ∑F i

a) σ max =N max /A ≤[G];

b) N max =σ max A;

c) N max = ∑N i .

2. Shear strength condition

a) Q ≤ [τ] ·А;

b) τ max = Q / A ≤ [τ];

c) τ max / [τ] ≤ 1.

3. Condition for shaft torsional strength:

a) τ max = M k · W ρ ≤ [τ] ;

b) τ max = | M k | max / W ρ ≤ [τ] ,

c) | M k | max ≤ [τ] · W ρ .

4. Strength condition for pure bending:

a) τ max + σ max ≤ [σ] ;

b) W ρ / σ max ≥ [σ] ;

c) σ max = | M max | / W z ≤ [σ] .

5. Euler’s formula for calculating the stability of a compressed rod:

a) F cr =π 2 E J min / (μℓ) 2 ;

b) F cr = π 2 E J max / μℓ 2 ;

c) F cr = π 2 E A / ί min.

6. Limits of applicability of Euler's formula

a) σ cr = σ t;

b) σ cr = a - bλ;

c) σ cr = π 2 E.

7. What characterizes W ρ:

a) cross-sectional area

b) torsional stress

c) maximum rotation angle

8. What characterizes J y and J z

a) moments of inertia during bending;

b) moments of inertia during torsion;

c) moments of inertia in dangerous sections, respectively, of the shaft and

9. What characterizes the limit of endurance

a) bending strength

b) maximum cycle stress for the base number of load cycles;

c) stress under a symmetrical load cycle.

10. Is Hooke's law valid beyond the limit of proportionality?

b) yes, with hardening

c) fair beyond the strength limit

11. Poisson’s ratio is the same for tension and compression

c) not the same up to the yield point.

12. The mechanical characteristics of brittle and ductile materials are numerically different

b) identical under compression,

c) are not the same when heated.

13. Does the rigidity of the part depend on the geometric characteristics of the section?

14. Diagrams of forces and moments are used to study strength and stiffness

b) when bending;

c) when identifying dangerous points and sections of timber.

15. For what types of deformations do the stresses in the section change according to a linear law?

a) in tension-compression, shear-shear;

b) during torsion and bending;

c) upon impact.

16. Polar moment of resistance is used to determine shear stresses in the shaft section

c) in the case of a circular section.

17. The polar moment of inertia of a shaft is used to determine its stiffness

c) to determine the relative angle of twist.

18. The safety factor is used to determine permissible stresses

c) to increase the weight of the structure.

19. Most often applicable 3 I and 4 I strength theory

b) 3 I strength theory;

20. Critical stresses during buckling are greater than the yield strength.

c) depend on the speed of application of the axial load.

21. The main parameters of cycles are:

a) σ max, σ min;

b) R= σ min /σ max , σ a ;

22. Which voltage cycle is the most dangerous:

a) asymmetrical,

b) pulsating,

c) symmetrical.

Answers to tests

Sections 1-2: 1 – b; 2 – a; 3 – a; 4 – b; 5 – a.

Section 3: 1 – b; 2 – a; 3 – in; 4 - a; 5 B.

Section 4: 1 – a; 2 – b; 3 – in; 4 – a; 5 B.

Section 5: 1 – a; 2 – a; 3 – b; 4 – a; 5 – a.

Section 6: 1 – a; 2 – b; 3 – b; 4 – b; 5 – a.

Section 7: 1 – a; 2 – b; 3 – in; 4 – b.

Section 8: 1 – b; 2 – in; 4 – in; 5 – a.

Sections 9-10: 1 – b; 2 – a; 3 – b; 4 – a; 5 B.

Section 11: 1 – b; 2 – a and b; 3 – in; 4 – a; 5 B.

Section 12: 1 – b; 2 – b; 3 – b; 4 – a; 5 – c.

Section 13: 1 – a; 2 – b; 3 – in; 4 – a.

Section 14: 1 – a; 2 – b and c; 3 – in; 4 – a; 5 – a.

Section 15: 1 – a and b; 2 – b; 3 – b; 4 – a; 5 – c.

Literature

Main

1. Volmir A.S., Grigoriev Yu.P., Stankevich A.I. Strength of materials: Publishing house: Bustard, 2007.

2. Mezhetsky G.D., Zagrebin G.G., Reshetnik N.N. and others. Strength of materials: Publishing house: Dashkov and Co., 2008.

3. Mikhailov A.M. Strength of materials: Academy Publishing House, 2009.

4. Podskrebko M.D. Strength of materials. Problem solving workshop. - M.: Higher School, 2009.

5. Kopnov V.A., Krivoshapko S.N. Strength of materials. A guide for solving problems and performing laboratory and computational and graphic work. - M.: Higher School, 2009.

6. Sapunov V.T. Classic course on strength of materials in problem solving. Publishing house: LKI, 2008.

Additional

1. Bulanov E.A. Solving problems on strength of materials. M.: Higher School, 1994, 206 p.

2. Darkov A.V., Shpiro G.S. Strength of materials. M.: Higher School, 1989, 624 p. (all years of publication)

3. Dolinsky F.V., Mikhailov N.M. Short course on strength of materials. M.: Higher School, 1988, 432 p.

4. Mirolyubov I.N. and others. A guide to solving problems on the strength of materials. M.: Higher School, 1969, 482 p.

5. Feodosiev V.I. Strength of Materials, M.: Nauka, 1986, 512 p. (all publication years)

6. Stepin P.A. Strength of materials. M.: Higher school. (all publication years)

7. Shevelev I.A. Reference tables for strength of materials. 1994, 40 p.

8. Shevelev I.A., Mozzhukhina G.L. Basics of strength calculations. 2003, 80 p.

For comments

Shevelev Ivan Andreevich

FEDERAL AGENCY FOR EDUCATION State educational institution of higher professional education

NORTHWESTERN STATE CORRESPONDENCE TECHNICAL UNIVERSITY

Department of Theoretical and Applied Mechanics

STRENGTH OF MATERIALS

TRAINING AND METODOLOGY COMPLEX

Mechanical Engineering and Technology Institute

Specialties:

151001.65 - mechanical engineering technology

150202.65 – equipment and technology for welding production

150501.65 – materials science in mechanical engineering Specializations:

151001.65-01; 151001.65-03; 151001.65-27;

150202.65-01; 150202.65-12; 150501.65-09

Institute of Transportation and Vehicle Organization

Specialties:

190205.65 – lifting and transport, construction, road machinery and equipment 190601.65 – cars and automotive industry

190701.65 – organization of transportation and transport management Specializations:

190205.65-03; 190601.65-01; 190701.65-01; 190701.65-02

Direction of bachelor's training 151000.62 - design and technological support of automated machine-building production

St. Petersburg Publishing house NWTU

Approved by the University's Editorial and Publishing Council

UDC 531.8.075.8

Strength of materials: educational and methodological complex / comp. L.G.Voronova, G.D. Korshunova, Yu.N. Sobolev, N.V. Svetlova. - St. Petersburg: Publishing house

SZTU, 2008. – 276 p.

The educational and methodological complex was developed in accordance with state educational standards of higher professional education.

The discipline is devoted to the study of the basic methods of calculating the strength, rigidity and stability of structural elements.

Considered at a meeting of the Department of Theoretical and Applied Mechanics on February 5, 2008, approved by the methodological commission of the Faculty of General Professional Training on February 7, 2008.

Reviewers: Department of Theoretical and Applied Mechanics of North-West Technical University (N.V. Yugov, Doctor of Technical Sciences, Prof.); Yu.A. Semenov, Ph.D. tech. Sciences, Associate Professor Department of TMM, St. Petersburg State Polytechnic University.

Compiled by: L.G. Voronova, associate professor; G.D. Korshunova, associate professor; Yu.N. Sobolev, associate professor; Art. teacher N.V. Svetlova

© Northwestern State Correspondence Technical University, 2008

© Voronova L.G., Korshunova G.D., Sobolev Yu.N., Svetlova N.V., 2008

1. Information about the discipline 1.1. Preface

The most important condition for the creation of new designs of machines, instruments and vehicles should be a comprehensive reduction in their cost per unit of power, a further increase in the efficiency of metal use when designing new types of machines, mechanisms and equipment through progressive solutions and calculations, as well as through the use of more economical profiles rolled products and advanced structural materials. All this requires specialists to have extensive knowledge in the field of strength calculations and sufficient training in experimental methods for studying stresses.

The purpose of studying the discipline is providing a base for engineering training.

The task of studying the discipline– mastering methods of calculations for strength, rigidity and stability.

As a result of studying the discipline, the student must master the fundamentals of knowledge in the discipline, formed at several levels:

Have an idea:

On the correct solution of problems related to the calculation of strength, rigidity and stability of structures used in difficult operating conditions under the influence of both static and dynamic loads, taking into account temperature influences and processes associated with the duration of operation, which is a necessary condition for reliability and durability machines and devices while simultaneously improving their weight characteristics.

Know: How to calculate the strength and stiffness of rods and rod systems under tension - compression, torsion, complex loading. For static and impact loads, calculations of rods for stability. Know the principles and methods of calculations.

Be able to: Determine deformations and stresses in rod systems under temperature influences using modern technology. Determine optimal system parameters.

Place of discipline in the educational process:

The theoretical and practical foundations of the discipline are courses

“Mathematics”, “Physics”, “Theoretical mechanics”. Purchased

mechanics”, “Strength reliability”, “Machine parts”, as well as in course and diploma design.

All majestic buildings of antiquity and the Middle Ages are characterized by monumentality, harmony, and proportions. These are monuments of human genius, but history has not preserved the memory of countless failures. Unique structures were built based on the experience and intuition of great architects.

As the years passed, the craftsmanship of builders—architects—improved, empirical and theoretical material gradually accumulated, and the prerequisites were created for the emergence of a science about the strength of materials and structures. Humanity has been forced to solve the problem of strength throughout the history of its existence.

For the first time, works that appeared during the Renaissance were devoted to the study of issues of strength and are associated with the name of Leonardo da Vinci (1452-1519). The first theoretical calculations of strength and experimental studies of the strength of beams were carried out by Galileo Galilei (1564-1642).

The fundamentals of the subject were developed in the 18th-18th centuries. works of Hooke R. (1635-1702), Newton I. (1642-1727), Bernoulli D. (1700-1782), Euler L. (1707-1783), Lomonosov M. V. (1711-1765), Young T. . (1773-1829).

The Strength of Materials course examines the basic methods of strength, stiffness, and stability calculations commonly used in machine parts courses and in many other specialized disciplines.

The main form of study for a part-time student is independent study of recommended literature. In-person classes held at the university and educational departments are also important in the learning process.

activities that significantly help the student in his independent work, making this work more effective and meaningful.

The study of theoretical material should begin with familiarization with the content of the curriculum.

When studying each topic of the course, it is necessary to comprehend the newly introduced concepts and assumptions, understand their physical essence, establish the connection between them and be able to derive the basic formulas of the topic.

After studying each topic, you should answer self-test questions. The student must be able to derive basic formulas and use their results when solving problems. Without studying theoretical issues, without mastering general research methods and without remembering the basic dependencies, it is impossible to count on successfully mastering the strength of materials course.

This educational complex is intended for students of specialties 151001.65, 150202.65, 190601.65, 190205.65 full-time and part-time forms of study in the amount of 170 hours and for students of specialties 150501.65, 261001.65, 190701.65 studying the course in the amount of 100 hours.

1.2. Contents of the discipline and types of academic work

Basic concepts. Section method. Central tension - compression. Shift. Geometric characteristics of sections. Straight transverse bend. Torsion. Oblique bending, eccentric tension-compression. Elements of rational design of simple systems. Calculation of statically determinate rod systems. Method of forces, calculation of statically indeterminate rod systems. Analysis of the stressed and deformed state at a point on the body. Complex resistance, calculation based on strength theories. Calculation of momentless shells of revolution. Stability of rods. Longitudinal-transverse bending. Calculation of structural elements moving with acceleration. Hit. Fatigue. Calculation based on bearing capacity.

Scope of discipline and types of academic work

For specialties 151001.65,150202.65,190601.65,190205.65

List of types of practical classes and control

- tests (general, by discipline, training, etc.);

- tests (number 3 if the course volume is 180 hours and 2 if

100 hours);

- practical lessons;

- laboratory works;

Exam (test).

2. Working training materials 2.1. Work program (180 hours)

Section 1. Introduction (14 hours). Basic concepts, p. 5.21

Course objectives. Assumptions and hypotheses in the strength of materials. Structural elements. External forces and their classification. Internal forces. Section method. The concept of stress. Deformations and their classification.

Section 2. Axial tension - compression of a straight rod (17 hours), s 48…71

Internal force factors in beam cross sections. Hooke's law. Stresses and strains. Diagram of tension and compression of materials in a ductile and brittle state. Condition of strength. Algorithm for solving problems.

Statically indeterminate rods. Stresses in inclined sections. Law of pairing of tangential stresses. Calculation based on bearing capacity.

pp. 63,341,377.

Stressed state at a point. Types of stress. Strength hypotheses. Deformed state at a point.

Section 4. Shift. Torsion (16 hours) p. 132…143

Pure shift. Torque. Construction of diagrams. Determination of stresses. Condition of strength. Determination of movements. Stiffness condition. Geometric characteristics of cross sections. Rational cross-sectional shapes.

Section 5. Flat straight bend. (38 hours), p.30…33, 108…128, 226…245.

Internal power factors. Sign rule. . Differential dependencies between q, Q and M. Constructing diagrams of shear force Q and

bending moment M. Determination of stresses in cross sections. Geometric characteristics of cross sections. Strength calculation. Analytical method for determining displacements. Graphic-analytical method for determining displacements.

Section 6. Statically indeterminate beams (20 hours), p.256…268.

Statically indeterminate beams. Degree of static indetermination. Method of forces. Three moment equation.

Section 7. Complex resistance (23 hours), p.168..197

Oblique bend. Determination of stresses and displacements. Neutral axis position. Eccentric loading. Bending with torsion. Calculation of momentless shells of revolution.

Section 8. Stability of compressed rods. (16 hours), p.403…422

Basic concepts. Euler's formula for critical force. Loss of stability beyond the limit of proportionality. Graph of the dependence of the critical stress on the flexibility of the rod. Rational cross-sectional shapes. Longitudinal - transverse bending.

Section 9. Dynamic load action (20 hours), p.470…482,499…506.

Accounting for inertial forces. Dynamic coefficient. Dynamic coefficient during oscillations. Dynamic coefficient at impact. The concept of metal fatigue. Fatigue failure. Types of voltage cycles and their parameters. Fatigue curves. Endurance limit. The influence of various factors on the endurance limit of a part. Testing strength under alternating stresses. Conclusion.

Technical mechanics

Glossary

for students of all forms of training in vocational training specialties: 150415 “Welding production”, 190631 “Maintenance and repair of motor vehicles”, 260203 “Technology of meat and meat products”, 260807 “Technology of public catering products”, 230401 “Information systems (by industry)

Light, 2013

Compiled by: Inkina G.V., teacher of special disciplines.

Methodist ___________ N.N. Pereboeva

Considered at a meeting of the Ministry of Defense

Protocol No.____ dated “___”___________20___

Chairman of the Moscow Region __________ M.S. Semko

Published by decision of the Methodological Council of the technical school, protocol No. __ dated “___” ___________ 20___.

©Inkina G.V., 2013

Terminological dictionary of technical mechanics

Statics

Kinematics

Dynamics

Strength of materials

Machine parts

Basic definitions and concepts of technical mechanics

STATICS

1. Theoretical mechanics is the science of the equilibrium of bodies in space, of systems of forces, and of the transition of one system to another.

2. Strength of materials - the science of calculating structures for strength, rigidity and stability.

3. Machine parts is a course that studies the purpose, classification and basic calculations of general types of parts.

Mechanical movements are changes in body position in space and time.

A material point is a body whose shapes and dimensions can be neglected, but which has mass.

An absolutely rigid body is a body in which the distance between any two points remains unchanged under any conditions.

Force is a measure of the interaction of bodies.

Force is a vector quantity that is characterized by:

1. application point;

2. size (modulus);

Axioms of statics.

1. An isolated point is a material point that, under the influence of forces, moves uniformly in a straight line, or is in a state of relative rest.

2. two forces are equal if they are applied to the same body, act along the same straight line and are directed in opposite directions, such forces are called balancing.

3. Without disturbing the state of the body, a balancing system of forces can be applied to it or rejected from it.

Corollary: any force can be transferred along the line of its action without changing the action of the force on a given body.

4. The resultant of two forces applied at one point is applied at the same point and is in magnitude and direction the diagonal of the parallelogram built on these forces.

5. Every action has a reaction equal in magnitude and direction.

Connections and their reactions.

A free body is a body whose movement in space does not change anything.

Those bodies that limit the movement of the selected body are called constraints.

The forces with which the connection holds bodies O, are called bond reactions.

When solving problems mentally, connections are discarded and replaced by reactions of connections.

1. Bond in the form of a smooth surface

2. Flexible communication.

3. Connection in the form of a rigid rod.

4. Support at a point or support at a corner.

5. Articulate movable support.

6. Articulated fixed support.

System of forces.

A system of forces is a totality.

Force system:

FlatSpatial

Converging Parallel Converging Parallel

KINEMATICS.

Kinematics studies types of movement.

Communication formulas:

DYNAMICS.

Dynamics studies the types of motion of a body depending on the applied forces.

Axioms of dynamics:

1. any isolated point is in a state of relative rest, or uniform linear motion, until applied forces bring it out of this state.

2. The acceleration of a body is directly proportional to the force acting on the body.

3. If a system of forces acts on a body, then its acceleration will be the sum of those accelerations that the body would receive from each force separately.

4. Every action has an equal and opposite reaction.

The center of gravity is the point of application of gravity; when the body turns, the center of gravity does not change its position.

The force of inertia.

The force of inertia is always directed in the opposite direction to acceleration and is applied to the connection.

With uniform motion, i.e. when a=0 the inertial force is zero.

During curvilinear motion, it is decomposed into two components: normal force and tangential force.

P u t =ma t =mεr

P u n =ma n =mω 2 r

Kinematics method: conventionally applying an inertial force to a body, we can assume that the external reaction forces of the connections and the inertial force form a balanced system of forces. F+R+P u =0

Friction force.

Friction is divided into two types: sliding friction and rolling friction.

Laws of sliding friction:

1. The friction force is directly proportional to the normal reaction of the support and is directed along the contacting surfaces in the opposite direction of movement.

2. The coefficient of static friction is always greater than the coefficient of motion friction.

3. The sliding friction coefficient depends on the material and physical and mechanical properties of the rubbing surfaces.

Self-braking condition.

Friction leads to a decrease in the service life of parts due to wear and heating. In order to avoid this, it is necessary to introduce lubricant. Improve the quality of surface treatment of parts. Use other materials in rubbing areas.

4. If possible, replace sliding friction with rolling friction.

Section method.

We mentally cut the load loaded with forces, in order to determine the internal force factors, for this we discard one part of the load. We replace the intermolecular force system with an equivalent system with a principal vector and a principal moment. When expanding the main vector and the main moment along the x, y, z axes. set the type of deformation.

Inside the section of the beam, force factors can arise within the beam; if force N (longitudinal force) occurs, then the beam is stretched or compressed.

If Mk (torque moment) occurs, then torsional deformation, force Q (lateral force) then shear or bending deformation. If M and x and M and z (bending moment) occur, then bending deformation occurs.

The section method allows you to determine the stress in the cross section of the load.

Stress is a quantity that shows how much load falls on a unit cross-sectional area.

A diagram is a graph of changes in longitudinal forces, stresses, elongations, torques, etc.

Tension (compression) is a type of deformation in which only longitudinal force occurs in the cross section of the beam.

Hooke's law.

Within the limits of elastic deformations, normal stress is directly proportional to longitudinal deformations.

to the center of gravity= Eε

E – Junck's modulus, a coefficient that characterizes the stiffness of a material under stress, depends on the material, the sample from the reference tables.

Normal voltage is measured in Pascals.

ε=Δ l/l

Δ l= l 1 - l

V=ε’/ε

Δ l=N l/AE

Strength calculation.

|b max |≤[b]

np – design safety factor.

[n] – permissible safety factor.

b max – calculation of the maximum voltage.

b max = N/A≤[b]

Torsion.

Torsion is a type of deformation in which only one internal force factor appears in the cross section of the beam - torque. Shafts and axles are subjected to torsion. And springs. When solving problems, torque diagrams are constructed.

Sign rule for torques: If the torque rotates the shaft from the cross-section side clockwise, then the torque will be equal to it with the “+” sign, and against it – with the “-” sign.

Condition of strength.

Τ cr =|M max |/W<=[ Τ кр ] – условие прочности

W=0.1d 3 - – moment of resistance of the section (for round)

Θ=|M to max |*e/G*Y x<= [Θ o ]

Y x – axial moment of inertia

G – shear modulus, MPa, characterizes the torsional rigidity of materials.

Bend.

Pure bending is a type of deformation in which only a bending moment occurs in the section of the beam.

Transverse bending is a bending in which a transverse force occurs in the cross section along with the bending moment.

Straight bending is a bending in which the force plane coincides with one of the main planes of the beam.

The main plane of a beam is a plane passing through one of the main axes of the cross section of the beam.

The main axis is the axis passing through the center of gravity of the beam.

Oblique bending is a bending in which the force plane does not pass through any of the main planes.

The neutral layer is the boundary passing between the compression and tension zones (the stress in it is 0).

The zero line is the line obtained by the intersection of the neutral layer with the cross-sectional plane.

Sign rule for bending moments and shear forces:

If the forces are directed from the beam, then F=+Q, and if towards the beam, then F=-Q.

If the edges of the beam are directed upwards and the middle downwards, then the moment is positive, and if vice versa, then the moment is negative.

MACHINE PARTS.

Detail– this is a product obtained from a material of a homogeneous brand without assembly operations.

Assembly unit- a product obtained using assembly operations.

Mechanism– a complex of parts and assembly units created for the purpose of performing a certain type of movement of the driven link with a predetermined movement of the leading link.

Car- this is a set of mechanisms created for the purpose of converting one type of energy into another, or to perform useful work, in order to facilitate human labor.

Mechanical gears.

Transfers- These are mechanisms designed to transmit movement.

1)According to the method of transmission of motion:

a) gearing (gear, worm, chain);

b) friction (friction);

2)By method of contact:

a) direct contact (tooth, worm, friction);

b) using a transmission link.

Serrated– consists of a gear and a cogwheel and is designed to transmit rotation.

Advantages: reliability and strength, compactness.

Flaws: noise, high requirements for manufacturing and installation precision, depressions are stress concentrators.

Classification.

1) cylindrical (11 axes), conical (crossed axes), screw (crossed axes).

2) According to the tooth profile:

a) involute;

b) cycloidal;

c) with a Novikov link.

3) According to the method of engagement:

a) internal;

b) external.

4) According to the location of the teeth:

a) straight toothed;

b) helical;

c) mevron.

5) By design:

a) open;

b) closed.

Used in machine tools, cars, watches.

Worm-gear consists of a worm and a worm wheel, the axes of which are crossed.

Serves to transmit rotation wheel.

Advantages: reliability and durability, the ability to create self-braking transmission, compactness, smooth and silent operation, the ability to create large gear ratios.

Flaws: low speed, high transmission heating, use of expensive anti-friction materials.

Classification.

1) Looks like a worm:

a) cylindrical;

b) globoidal.

2) According to the profile of a worm tooth:

a) involute;

b) covolutes;

c) Archimedes.

3) By number of visits:

a) single-pass;

b) Multi-pass.

4) Relation between worm and worm wheel:

a) with the bottom;

b) with the top;

c) with the side.

Used in machines and lifting devices.

Belting consists of pulleys and a belt. Serves to transmit rotation over a distance of up to 15 meters.

Advantages: smooth and silent operation, simplicity of design, possibility of smooth adjustment of the gear ratio.

Flaws: belt slippage, limited belt service life, need for tensioners, impossibility of use in explosive atmospheres.

It is used in convectors, machine drives, in the textile industry, and in sewing machines.

Instrumentation.

Belts– leather, rubber.

Pulleys– cast iron, aluminum, steel.

Chain transmission consists of a chain and gears. Serves to transmit torque over a distance of up to 8 meters.

Advantages: reliability and strength, no slipping, less pressure on shafts and bearings.

Flaws: noise, high wear, sagging, difficult lubrication supply.

Material– steel.

Classification.

1) By purpose:

a) freight,

b) tension,

c) traction.

2) By design:

a) roller,

b) bushings,

c) toothed.

They are used in bicycles, machine and car drives, and convectors.

Shafts and axles.

Shaft- This is a part designed to support other parts for the purpose of transmitting torque.

During operation, the shaft experiences bending and torsion.

Axis- this is a part intended only to support other parts mounted on it; during operation, the axis only experiences bending.

Shaft classification.

1) By purpose:

a) straight,

b) cranked,

c) flexible.

2) By form:

a) smooth,

b) stepped.

3) By section:

a) solid,

Shaft elements.

Shafts are often made of steel-20, steel 20x.

Shaft calculation:

tcr=|Mmax|\W<=

si=|Mmax|W<=

The axles are only for bending.

W – section moment of resistance [m3].

Couplings.

Couplings– these are devices designed to connect shafts for the purpose of transmitting torque and ensuring the unit stops without turning off the engine, as well as protecting the operation of the mechanism during overloads.

Classification.

1) Non-detachable:

a) hard

Advantages: simplicity of design, low cost, reliability.

Flaws: Can connect shafts of the same diameters.

Material: steel-45, gray cast iron.

2) Managed:

a) toothed

b) friction.

Advantages: simplicity of design, different shafts, the mechanism can be turned off when overloaded.

3) Self-acting:

a) safety,

b) overtaking,

c) centrifugal.

Advantages: reliability in operation, transmit rotation when a certain rotation speed is reached due to inertial forces.

Flaws: design complexity, high wear of cams.

In progress from gray cast iron.

4) Combined.

Couplings are selected according to the GOST table.

Permanent connections - these are connections of parts that cannot be disassembled without destroying the parts included in this connection.

These include: riveted, welded, soldered, adhesive joints.

Riveted connections.

1) By purpose:

a) durable,

b) dense.

2) According to the location of the rivets:

a) parallel,

b) in a checkerboard pattern.

3) By number of visits:

a) single row,

b) multi-row.

Advantages: they withstand shock loads well, are reliable and durable, provide visual contact for the quality of the seam.

Flaws: holes are stress concentrators and reduce the tensile strength, make the structure heavier, noisy production.

Welding connections.

Welding– this is the process of joining parts by heating them to the melting temperature, or by plastic deformation in order to create a permanent connection.

Welding:

a) gas,

b) electrode,

c) contact,

d) laser,

d) cold,

e) explosion welding.

Welded joints:
a) angular,

b) butt,

c) overlap,

d) T-bar,

d) point.

Advantages: provides a reliable sealed connection, the ability to connect any materials of any thickness, and a silent process.

Flaws: changes in physical and chemical properties in the weld area, warping of the part, difficulty in checking the quality of the seam, highly qualified specialists are required, poorly withstand repeated variable loads, the seam is a stress concentrator.

Adhesive joints.

Advantages: does not burden the structure, low cost, does not require specialists, the ability to connect any parts of any thickness, silent process.

Flaws: “aging” of the glue, low heat resistance, the need for preliminary cleaning of the surface.

All permanent connections are designed for shear.

tav=Q\A<=

Threads (classification).

1) By purpose:

a) fastenings,

b) running gear,

c) sealing.

2) By the angle at the apex:

a) metric(60°),

b) inch (55°).

3) By profile:

a) triangular,

b) trapezoidal,

c) stubborn

d) round,

d) rectangular.

4) By number of visits:

a) single-pass,

b) multi-pass.

5) In the direction of the helix:

b) right.

6) On the surface:

a) external,

b) internal,

c) cylindrical,

d) conical.

Threaded surfaces can be made:

a) manually,

b) on machines,

c) on automatic rolling machines.

Advantages: simplicity of design, reliability and strength, standardization and interchangeability, low cost, does not require specialists, the ability to connect any materials.

Flaws: thread is a stress concentrator, wear of contacting surfaces.

Material– steel, non-ferrous alloys, plastic.

Keyed connections.

There are dowels: prismatic, segmental, wedge.

Advantages: simplicity of design, reliability in operation, long keys - guides.

Flaws: keyway is a stress concentrator.

Spline connections.

There are: straight-sided, triangular, involute

Advantages: reliable operation, uniform distribution over the entire cross-section of the shaft.

Flaws: difficulty of manufacture.

R=sqr(x^2+y^2) for fixed supports

in x - cos of a given angle

by y - sin of this angle or cos (90-angle)

if the larger side of the triangle then take 2/3

if small then - 1/3

d'Alembert principle: F+R+Pu=0

P=F/A=sqrG^2+Tx^2+Tz^2 - total voltage

^L=(N*L)/(A*E) - second entry of Hooke’s law