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сопротивление материалов

сопромат

механика

деформация

напряжённо-деформированное состояние

напряжение

дистанционное обучение

кручение

касательные напряжения

депланация

момент сопротивения кручению

геометрический фактор жёсткости

напряжения в углах

кручение прямоугольного стержня

00:00:00

so now we move on to the torsion of non-

00:00:03

round rods, non-round rods, they

00:00:07

also occur, this is not and not only this, in

00:00:09

Volynsk, they have a non-round profile, although

00:00:12

you also have a non-round profile, they sometimes

00:00:14

have spline

00:00:18

grooves for moving gears, they sometimes have splined grooves for moving gears

00:00:21

and there are no grooves nor a

00:00:23

round profile, there are square ends,

00:00:26

but even when calculating the frames, a separate server,

00:00:32

some experience

00:00:33

both straight bending and torsion, we are faced with

00:00:36

the need to calculate the torsion of

00:00:38

rods of a non-round profile, a rod and a

00:00:41

round profile, here the solid stubbles are

00:00:43

still solid young, for example, a

00:00:45

rectangular triangular profile is drawn

00:00:48

profile the inclusion of such

00:00:51

solid so far solid rods,

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well, before we move on to

00:00:57

the methods of calculating them, well, we can

00:01:00

prove two provisions which

00:01:04

can be proven very easily based on the

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law of pairing of tangential stresses,

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the first is that the tangential stress at the

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protruding corners of the cross section

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will be equal to zero and it doesn’t matter which then the

00:01:19

protruding angle is equal to 90 degrees or

00:01:23

there is less than 90 or more than 90 120

00:01:26

anyway, the tangential stress in the

00:01:28

advancing corners will be equal to zero,

00:01:31

this is proven by contradiction,

00:01:34

that is, let’s assume that in the corner a

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tangential stress nevertheless arises

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in some area in the corner

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tangential stress there,

00:01:42

then we will decompose this tangential

00:01:45

stress into two components that

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will be perpendicular to the two adjacent ones,

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they are perpendicular, and then, according to the law of

00:01:56

pairing, regarding you the lead, then the

00:01:59

stroke itself should have the same

00:02:03

tangential stress on the side face,

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and the tangential stress melts two strokes

00:02:08

that arises in the transverse section,

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it is perpendicular to the rib,

00:02:12

the same tangential

00:02:15

stress should arise on the lateral 2 lateral faces of

00:02:19

the rods, they are free from stress on

00:02:22

them, nothing acts on them except

00:02:23

atmospheric air pressure,

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atmospheric air pressure on the

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tiny one is already more shearing stronger,

00:02:29

so these tangential

00:02:31

stresses on the lateral faces are equal to zero

00:02:34

and if they are equal to zero, it follows that the

00:02:36

tangential stresses in the

00:02:38

cross sections are equal to zero and thus

00:02:41

it turns out that the tangential stress at the

00:02:43

protruding corners will be equal to the glitch,

00:02:46

regardless of the size of the protruding angle,

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well, here an interesting question arises,

00:02:51

well, what if google is not protruding, but let’s

00:02:54

say internal, but how for example, on the

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keyways then what kind of

00:03:00

tension will there be, that is, if for example, here is the

00:03:03

keyway, here the shaft goes, then I wanted the

00:03:09

protruding corners, here we know that

00:03:11

regarding the work voltage is equal to zero,

00:03:13

and what about in the morning on the tangential tension in the

00:03:15

inner corners,

00:03:16

but here, according to the law, on parnassus

00:03:19

nothing can be proven, and if this is a

00:03:22

strictly right angle, this is such an acute

00:03:27

angle, then the tangential stress here

00:03:29

will tend to infinity and

00:03:32

therefore, of course, these internal angles are

00:03:34

always made rounded, there are always

00:03:37

fillets for rounding

00:03:40

and the larger the radius,

00:03:44

the less stress is very large this

00:03:46

cannot be done for operational

00:03:48

reasons,

00:03:49

but nevertheless, it forever provides for

00:03:52

some kind of rounding radius in the

00:03:54

internal corners they will be drilled,

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it turns out to be infinity, and the second

00:03:59

position that we can prove is

00:04:01

that the shear stress is directed

00:04:06

along the tangent contour, the flow of the

00:04:10

representative contour of the section is also

00:04:13

proven very easily, Pavel in games

00:04:16

based on the against the shaft, let’s assume that from

00:04:18

some area near the contour the

00:04:21

tangential stress is not directed tangentially to the

00:04:23

contour in some

00:04:26

unknown direction, we also

00:04:28

decompose it into two components 1

00:04:31

which is directed along the contour

00:04:34

exponentially 2 perpendicular to it, we

00:04:37

can find that this the

00:04:39

voltage of the metal is two lines that are

00:04:41

perpendicular to the contour should

00:04:43

correspond

00:04:44

should be in this area should

00:04:47

arise

00:04:48

so you tangential stress but

00:04:50

again the side surface is free

00:04:53

from stresses followed by this is u2 that e x

00:04:55

is equal to zero and the only thing left is the stroke

00:04:58

only that the rick means it acts

00:05:01

tangential stresses that are

00:05:03

directed only along the tangent contour,

00:05:07

knowing these basic

00:05:10

but some laws of distribution of

00:05:12

tangential stresses,

00:05:13

you can already approach the

00:05:16

question of torsion of a rod of a

00:05:19

rectangular cross section, and so the

00:05:24

torsion of a rod of a rectangular

00:05:27

cross section, how to approach the

00:05:30

solution of this problem if you accept the

00:05:33

same hypothesis of flat sections

00:05:36

that were used as the basis for solving the

00:05:39

problem, the

00:05:40

lesson of torsion of a round rod, there and

00:05:43

immediately we come to a contradiction, such

00:05:46

attempts have been made, and others, and were made by

00:05:48

such great scientists as Navi,

00:05:51

what happens if we accept the hypothesis of

00:05:54

plane sections, it

00:05:55

turns out that the maximum stress

00:05:58

will be at the greatest distance from the axis

00:06:00

of the rod, it is precisely the corner points most distant from the

00:06:03

axis of the rod,

00:06:04

but we have just proved

00:06:06

that at the corner points the tangential

00:06:08

stress is zero,

00:06:10

so this hypothesis cannot

00:06:13

be used as the basis for solving problems,

00:06:14

why, because in fact, when

00:06:18

torsion of rectangular profile rods, the

00:06:20

flow remains flat and

00:06:23

their diplo nation happens, what does diplo nation mean and

00:06:25

tell it the cross section, but it’s easy to

00:06:28

check if you take, say, an

00:06:30

eraser and untwist it, then you can see that

00:06:33

these are the ends of it, of course they will be

00:06:36

flat and distortion immediately occurs and

00:06:38

therefore this hypothesis cannot be accepted

00:06:40

but in solving the problem I will

00:06:42

then show how to solve this problem,

00:06:44

unfortunately, using the methods of resistance

00:06:46

of materials, as we solved the previous

00:06:48

problem

00:06:49

based on the Kalman truth of

00:06:51

plausible hypotheses of and using the

00:06:53

simplest relationships, and

00:06:56

this problem will be solved using the methods of the

00:06:59

theory of elasticity,

00:07:00

this is a science that

00:07:03

studies

00:07:05

stress deformed state, but

00:07:07

it was in a strict mathematical

00:07:09

formulation, and since

00:07:14

these equations of the theory of elasticity are

00:07:15

very far beyond the scope of the course on the

00:07:17

strength of materials, then

00:07:21

we will simply give the results according to the solution,

00:07:24

and the solution itself, of course, also

00:07:27

goes very far beyond the course of resistance

00:07:29

resistance of materials, so it

00:07:32

turns out that during torsion, the

00:07:35

restrained side of a rectangular profile,

00:07:37

side a, is considered larger than side b,

00:07:41

which is greater or equal for square b, the

00:07:48

maximum stress occurs

00:07:50

precisely at the points closest to the

00:07:53

earrings of the long sides, in the middle of the short

00:07:57

sides, the stress that is

00:07:59

indicated the fact that riho no smaller

00:08:01

they are smaller than the maximum in the corners the

00:08:04

stresses are equal to zero and a graph of the

00:08:07

stress distribution along the contour

00:08:10

can be drawn like this

00:08:21

the same for a

00:08:24

rectangular profile, the formula for

00:08:26

determining the tangential stresses is also

00:08:29

the same when a round beam with one, but

00:08:32

here it becomes not the polar moment of

00:08:35

torsional resistance, but is simply

00:08:37

called at the moment of torsional resistance, the moment of

00:08:40

torsional resistance, how is it

00:08:43

calculated, it is calculated as the

00:08:45

alpha coefficient which depends on the

00:08:47

aspect ratio the long side of the square is the short one,

00:08:51

it is also measured in cubic

00:08:53

form and so the formula is exactly the

00:08:57

same, but instead of the polar moment of

00:08:59

torsional resistance, say simply

00:09:00

the moment of torsional resistance, the

00:09:02

angle of rotation, again, the formula would

00:09:05

be exactly the same as for a round

00:09:08

rod, but instead of the polar moment of

00:09:11

inertia there is the so-called geometric

00:09:14

stiffness factor and

00:09:17

this river is called this river is called geometric

00:09:19

stiffness factor geometric

00:09:22

stiffness factor is defined as cassettes beta

00:09:25

which depends on the aspect ratio of the

00:09:27

long side and the short cube the

00:09:30

tension in the middle of

00:09:32

short litter is defined as

00:09:35

some pain from the maximum

00:09:37

stresses

00:09:38

that is, the maximum stress must be

00:09:40

multiplied by the coefficient this is this

00:09:42

the coefficient is less than or equal to unity

00:09:44

equal to unity for a square

00:09:46

it depends on the aspect ratio from in

00:09:50

this way you calculate the

00:09:53

stress for a rectangular profile from the

00:09:56

formula they remain the same when a

00:09:58

round rod but in place of the polar moment the

00:10:01

torsional resistance and the

00:10:03

polar moment of inertia are simply written down as the

00:10:05

moment of torsional resistance and the

00:10:08

geometric stiffness factor, what are

00:10:11

these coefficients equal to, well, they can be found

00:10:13

in reference books, here in the table the

00:10:16

most

00:10:17

frequently occurring values are given in

00:10:21

the relationship, we’ll take units, this is

00:10:24

the square of 24, and then we’ll count

00:10:27

zero, well, what can we say about the clamp,

00:10:31

the background varies from we can say about 0 21

00:10:34

to about one third a

00:10:37

recipe for troubles from a set of 41 to 1 also up to

00:10:42

1 3 games color it also varies not

00:10:45

very widely from one to 0

00:10:47

700 42 these are the most

00:10:50

common quantities

00:10:51

that will be needed when solving problems and

00:10:55

this means a rectangular profile

00:10:58

We didn’t get the results here, but here we

00:11:02

just wrote out the finished results

00:11:04

obtained by the method of elasticity theory

Description:

Обоснование нулевых касательных напряжений в выступающих углах сечения при кручении. Направление касательных напряжений по касательной к контуру сечения при кручении. Кручение стержня прямоугольного сечения. Невыполнение гипотезы плоских поперечных сечений, явление депланации. Результаты расчёта момента сопротивления кручению и геометрического фактора жёсткости на основе методов теории упругости. Лекцию читает Горбатовский Александр Александрович.

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