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shear stresses

Журавский

Сопротивление материалов

Изгиб

касательные напряжения

балка

МЭИ

Кирсанов М.Н.

00:00:09

when bending a beam, there is a beam on

00:00:13

two supports,

00:00:14

tanks, then the loads don’t matter besides

00:00:17

the moments, which here we know how to

00:00:19

calculate, they built a feast of moments, there are

00:00:22

also diagrams of the implementing forces, but everything

00:00:24

is clear that for us moments cause

00:00:27

normal

00:00:28

longitudinal stresses, we also

00:00:31

know the formula we now we will write down,

00:00:33

remember, but there is also a tangential

00:00:37

stress during someone if we

00:00:39

divide 1 and with something natural like this in

00:00:44

some place once and with something,

00:00:46

one piece of the beam

00:00:50

relative to the other will

00:00:51

resist such a shift, the question is

00:00:57

what kind of tangent praline there is

00:00:58

but the simplest 1 that comes to mind,

00:01:02

start if we know the cube, that is, the force

00:01:06

which is the total amount outside the force in this

00:01:09

section, the so-called experiencing force

00:01:11

of that, just divide the force by the area

00:01:14

and we get the concert voltage on in

00:01:18

principle, then of course it is possible, but this is a very

00:01:20

round approach First of all, what catches the eye

00:01:22

is that if we divide like this, it

00:01:25

turns out, well, roughly we divide you, it

00:01:27

seems like we are evenly and

00:01:29

tangentially, and the lesion is over the entire cross-section, therefore, and

00:01:30

tangentially, I will hide the stress

00:01:33

with the extreme as there cannot be some kind of

00:01:36

tangential stress, for example, here

00:01:39

such, for example, yes then all the dangers

00:01:41

here even to be such and here we have,

00:01:43

excuse me, the surface is free there

00:01:46

tangentially, the voltage is zero, which means it’s easy

00:01:48

to guess that in this place there will be

00:01:50

somehow zero tangentially, that is, they are

00:01:53

clearly uneven at the edges here there

00:01:56

will be 0 total idleness here is zero

00:01:58

here there will be some kind of maximum, this

00:02:00

formula for the extension of tangential

00:02:02

stresses was first derived by

00:02:05

Dmitry Ivanovich Zhuravsky Dmitry

00:02:07

Varshavsky born in 1821 and he lived for

00:02:15

70 years 1891 but he was one of the builders of the

00:02:20

first railway St. Petersburg

00:02:22

Moscow

00:02:23

there were problems with the construction of bridges

00:02:27

bridges and made wooden wooden

00:02:29

bridges, but wooden shelves that

00:02:32

naturally bend like that, they can

00:02:34

give such longitudinal cracks

00:02:36

because I said tangentially and lived

00:02:39

here, they arise,

00:02:40

but also longitudinal tangentially when living

00:02:42

now, I’ll show you with the example of just books,

00:02:45

not let’s say this ball, then of course it’s

00:02:47

not beam, but it’s comfortable and bendable, now we

00:02:51

’ll bend it, look

00:02:53

carefully and make it bigger, here’s

00:02:55

the cinnamon, well, what’s

00:03:00

happening is that from one

00:03:05

fiber to the other, God

00:03:07

shifts, of course, if we take

00:03:11

this book, we’ll carefully glue all the

00:03:14

pages spoil the thing, then this

00:03:17

shift will not happen, I will

00:03:20

bend everything very difficult, in principle, we will bend it,

00:03:22

of course, but due to the fact that this leads to

00:03:24

this bending, such things arise

00:03:27

about

00:03:28

tangent expressions, longitudinal failure,

00:03:32

this stress is longitudinal, naturally

00:03:34

they will coincide with the same

00:03:37

stresses

00:03:38

in which -at the place of the section, but that is,

00:03:41

naturally at the top there will be 0 at the top 0 and

00:03:43

here there will probably be a maximum this is

00:03:45

some kind of form the formula was

00:03:48

formal was first obtained by Dzhurovsky

00:03:53

cut out a

00:03:55

small section of the can like this

00:03:57

cut out larger make this one I am the

00:04:01

neutral axis of this remember when

00:04:02

you bend the beam thin then the load is Komi then there

00:04:05

will definitely be some place somewhere there will be

00:04:06

some

00:04:08

section of the People's Commissar growth at some

00:04:10

distance from the top or from the bottom there will be

00:04:13

a place where we will have a voltage

00:04:15

normally equal to 0 neutral axis and here

00:04:18

on some neutral

00:04:20

at some distance from the neutral

00:04:22

axis we have such a piece is cut out

00:04:24

here when we cut out we

00:04:26

replace all the acting external forces with

00:04:29

reactions below we will have these tangential

00:04:32

stresses that we are actually

00:04:34

looking for refusal of wounds so distributed well of

00:04:38

course we will approximately assume that they are

00:04:40

unevenly distributed that is why the

00:04:43

bit depth is very long and so and so,

00:04:45

that is, throughout the entire site they are

00:04:48

evenly distributed, but then it

00:04:52

will be directed downwards, here they will be

00:04:55

directed upwards, tangentially, and not

00:04:58

this, but we have written here tangentially,

00:05:01

now here a certain force we have cut to

00:05:04

protect in the amount of such normal efforts

00:05:08

will give us some kind of external reaction f

00:05:13

but here, as always, we have k we

00:05:14

take at a distance dx remember this

00:05:16

small piece of such a cool thing will be knocked down

00:05:19

about any small piece up to x dx

00:05:23

we will have along the x axis but the longitudinal axis

00:05:27

multics recoil from us dx here from here to here

00:05:33

all dx so but here it means we will

00:05:39

change by some forces of course from the

00:05:41

longitudinal then it f changes to in one

00:05:43

section 1 to run another clear to

00:05:45

will be let's say increases f + d f

00:05:49

were where the lion df here you start all the

00:05:54

quantities are balanced let's first let's

00:05:57

write the equilibrium equation projection on the

00:06:00

x axis

00:06:01

but if we write pro x asics then the

00:06:04

total by the tangential stress on

00:06:07

this lower lower surface will be

00:06:11

equal to the area of this surface

00:06:13

multiplied by there the total

00:06:16

force will begin it will mean tangentially this

00:06:18

stress si stress multiplied by the

00:06:20

area will be force means we have

00:06:22

it turns out that in this direction we have become

00:06:24

multiplied by the area and the area is

00:06:27

calculated this way dx this length and this is

00:06:30

the size this is a very important size this is the

00:06:33

very size that will enter the formula

00:06:34

now the size let's denote it b y was rick

00:06:39

b y but let's say y will be directed

00:06:43

here in this direction

00:06:44

perpendicular to y tan after all, but here I’m

00:06:48

really looking for a shout, this is

00:06:51

not a rectangle for us, this is some kind of

00:06:52

stick on the body, maybe

00:06:54

spatial figures, let’s

00:06:56

draw something like this,

00:06:57

but for example, you often draw,

00:06:59

see this video, that’s why I

00:07:02

was here I’ll also make the red one red

00:07:03

somewhere like this like this like this like this

00:07:08

a piece

00:07:12

like this this is the place where we have b

00:07:17

y calculated from the place where we cut

00:07:20

from below that is the width of this

00:07:24

rectangle that is right here

00:07:26

now and I’ll continue on here this is

00:07:30

shaded only we look from below,

00:07:32

of course the basis acts regarding the

00:07:35

advance of that means length dx width b

00:07:39

y means this is the area for the

00:07:43

voltage that turns out to be a force and here

00:07:46

this force can be balanced by something this is

00:07:49

plus this but this will be a plus with a

00:07:51

minus they destroyed it turns out df

00:07:54

about equals d.f.

00:07:57

df this is the first equation that you

00:08:02

received now we will decipher

00:08:05

df and tidy up with the leg let's clarify this a

00:08:09

little

00:08:10

misspoke the y axis we have at the top of the base we have

00:08:13

b with the y sign means that we have calculated

00:08:16

this size at the distance y from the

00:08:22

distance y from the neutral axis

00:08:24

that is from us this would be y means

00:08:27

that we have this distance y to the

00:08:31

neutral axis is very important then

00:08:33

when practically using this

00:08:35

formula and so y would be the distance to the

00:08:38

neutral axis and the transverse size

00:08:41

here in this in this direction the size of

00:08:43

this beam patrols or the beam of this

00:08:48

virus, so let's now let's

00:08:49

return to normal voltage, but the

00:08:51

well-known formula is this linear

00:08:53

distribution over sections, start if

00:08:59

we have these sections, the neutral axis here

00:09:02

will be with a plus, here there will be a zero here, a

00:09:04

minus, and here is such a cake of normal ones,

00:09:08

let's say like this, well depends on the sign of

00:09:11

the moment of course where to turn

00:09:12

you here sigma is normal but we have

00:09:15

them will be along the x axis then behind it sigma x

00:09:18

sigma x so the moment of inertia the moment of

00:09:21

inertia relative to this section

00:09:23

relative to this axis and y distance

00:09:26

night will cause y as we

00:09:30

agreed y you from the distance and so

00:09:32

we know this

00:09:33

now we wanted to decipher df in this formula,

00:09:36

which means f itself is calculated

00:09:39

as an integral of course the voltage over the

00:09:42

cross section is obtained as an integral over the cross section

00:09:46

but during the moment it does not depend on the

00:09:49

flow and from this they can take the

00:09:53

current moment too clarify the moment

00:09:57

which moment is around the z axis, that

00:10:00

is, around this one, which we have

00:10:01

directed I muse, let's clarify then z

00:10:04

m z and z

00:10:08

that is, the moment of inertia relative to I at

00:10:11

Tasi which passes through

00:10:13

neutrality and here we get an

00:10:15

interesting formula y him df no well

00:10:21

area or us f.f. strength, let's give

00:10:24

the area and we have hd

00:10:27

codes, and by the way, it is often used that the

00:10:29

area of the strength of material begins with the letter and the

00:10:31

big one will begin to fall under and this is the

00:10:35

static moment of the abstract part, the so-

00:10:37

called with at a distance y, that is, the

00:10:40

static moment of this

00:10:43

piece here, this one here relative to the

00:10:46

neutral axis y to the rescue thing

00:10:48

static moment is roughly speaking the

00:10:50

center of gravity only multiplied by the

00:10:52

area beginning if we, as you read, the center of

00:10:55

gravity started there here was y

00:10:57

started if we y c we calculate how the

00:11:00

static moment is, that is, the sum of the

00:11:03

areas there and what are they there on y from them to the

00:11:09

sum about cats, that is, this is our area,

00:11:12

but also,

00:11:14

but if we pass the sum to the integral of

00:11:16

and it turns out this one is called the

00:11:19

static moment, the static moment, the

00:11:21

static moments of the illuminated part, but

00:11:23

it is also calculated accordingly, the revolution is this, to

00:11:25

designate this

00:11:26

calculate it is necessary to multiply here too and

00:11:29

you get the coordinates of the center of gravity of this

00:11:32

figure on the area, that is, we find out

00:11:33

the area,

00:11:34

find the coordinate, multiply this is the coordinate of

00:11:38

this point and you can get the area, the

00:11:43

static moment

00:11:44

so well, let's rewrite m z and z to

00:11:52

c y begins in the section f a for and

00:11:57

of pride fc will change, which means we will

00:12:00

assume that we have such a

00:12:01

surface, this body is cylindrical, that

00:12:07

is, we won’t be used for this,

00:12:09

but in principle we can assume that

00:12:11

if the mobile is small dx, then even

00:12:13

if it doesn’t fit very well

00:12:14

here, this alone will already be so

00:12:17

naturally it will also be one thing that will already

00:12:19

change so much moment means

00:12:21

it turns out d e d fft it will be d m z

00:12:29

on and z on with y so

00:12:34

we can say that he parried this

00:12:37

d.f. but now that we know everything, we put

00:12:41

here further the elementary action, well, let's

00:12:44

continue until it equals this, I'll

00:12:46

continue here and we get the following

00:12:49

actual fat formula and we get dx we

00:12:54

used y about y we started needing to find that y

00:12:58

from here y equals about gangs / form already since

00:13:04

who will be ready now, so here I

00:13:10

am dividing the smoke under x,

00:13:12

let's remember that the smoke along dx x is the

00:13:18

longitudinal coordinates,

00:13:20

this is the transverse force and the so-called

00:13:23

differential

00:13:25

dependence for moments and transverse

00:13:28

forces, well, who is the transverse one, I

00:13:30

bandage it to do like and so this

00:13:32

dependence is known, let's we

00:13:35

will also put indices here m z z

00:13:38

therefore

00:13:39

we find from here then y divide by dx you

00:13:43

get

00:13:46

here moses

00:13:50

this will remain and

00:13:52

and b y

00:13:55

this is the famous Zhuravsky formula

00:13:58

which says that in order to

00:14:00

calculate the stress at some

00:14:04

place in the cross section of the beam it

00:14:07

is enough to find the transverse force well and the

00:14:10

most important thing is to find the static moment of the

00:14:13

illuminated part, that is, the one that is higher and

00:14:15

farther from the neutral window, divide the

00:14:19

moment of inertia for us and by this width, let’s

00:14:22

show with an example, for example, let’s take a

00:14:26

regular clip rectangle there are I-beams

00:14:29

by the way, I found interesting books where

00:14:32

there and triangles circle are also interesting

00:14:36

By the way, the formula turns out now, don’t

00:14:38

cry, it’s just the simplest case, this is

00:14:41

when we take a rectangle, so you

00:14:44

know the moment of inertia, cut off

00:14:48

static force, it’s easy to arrange and so the

00:14:51

rectangle is how it’s done in practice,

00:14:54

here it’s neutral for 8

00:14:56

rectangles, of course, the neutrality of

00:14:57

the accuracy in the middle it will be already

00:14:59

in half it’s already in half this is b then this is b

00:15:04

this would be y now and Krasnikov I

00:15:06

will highlight

00:15:07

that area the static element here

00:15:11

this is in this place I want to find the

00:15:14

voltage switch here this is what we have become

00:15:18

the most sacred part of this and and we must

00:15:20

find the static moment here this is the

00:15:24

distance y because remember this is the

00:15:28

neutral axis when you draw from the

00:15:31

picture some people don’t understand where

00:15:32

the owner of the beam is directed towards us

00:15:36

we will look at the cross section of the bane as much as

00:15:39

b scales and the balkan we have os x

00:15:42

x is directed from and from the heat and puppy c this is

00:15:45

the voltage here in this section of the night were

00:15:47

already agreed, most likely there

00:15:49

will be zero here there will be zero here there will be a

00:15:51

maximum now we will exceed it will be and sales of

00:15:54

three second then the values that we found

00:15:57

so to speak approximate we found as

00:15:59

q or for the area on a acacia Nikon and

00:16:04

there will be q by three second then and the old man,

00:16:07

now looking ahead,

00:16:09

needs to calculate using this formula so and so y

00:16:12

equals equal / granddaughter let’s just leave them the

00:16:18

static moment, this means the area

00:16:21

multiplied by the ordinate centers, well, the area

00:16:24

is that it would be very much by this

00:16:29

value and this is already in half minus y b like that

00:16:38

and hp multiplied by as much as half minus y

00:16:42

this is the area of this little red

00:16:46

part the area by the static ment how

00:16:49

the area is calculated so the coordinate the center

00:16:51

of gravity that is here from here further away

00:16:54

breathe how to find the coordinate of this

00:16:57

point but the coordinate is known this is known

00:17:00

and this is in the middle of the half-asleep itself

00:17:02

convenient so we write like this half the sum

00:17:06

here 1 2 immediately came here one cardio

00:17:11

then the other like this is cardio tash in half

00:17:13

already in half this is the y coordinate and so convenient

00:17:18

by the way it turns out because the world is here

00:17:20

plus then the proper name is just so

00:17:24

we figured out the moment of inertia it is the moment of

00:17:27

inertia let's take the entire cross-section of the Krishna

00:17:29

illuminated part, this is bh cube by 12 there,

00:17:31

give you complete as much as cube by 12, let me remind you that in the

00:17:36

cube we take this distance that

00:17:37

rotates and to the first power so, but

00:17:41

here we would have like this

00:17:42

by 6 we transform the right to take very

00:17:46

simple kukan that we will leave here b

00:17:50

perhaps will be reduced b will be reduced what else

00:17:54

can we do just here 1 2 1 6

00:18:00

get no 6 it turns out here 1 2 here

00:18:03

up to 6 here we get as much as a square

00:18:11

by 4 minus y squared Czechs we’ll do everything

00:18:15

right then it will even turn out to be three

00:18:17

second

00:18:18

so here, starting six, I wrote this

00:18:22

12 two to 36 went there so it has

00:18:26

n’t rolled off yet bh quad head trouble b&h in a

00:18:32

cube so by dimension let’s see by

00:18:36

dimension

00:18:37

we even leave this regarding and with the same not

00:18:39

and that is, this is the power to divide by area

00:18:43

here here is the second degree here 4 to

00:18:47

converges converges 2 4th degree converges

00:18:50

but probably everyone let's look at this

00:18:56

parabolic dependence naturally

00:18:58

where y equals plus minus h in half

00:19:00

then here and here it will be 0 great

00:19:03

to draw it turns out here zero

00:19:05

parabola

00:19:06

we know like this here there will be a

00:19:08

maximum and the maximum corresponds to y

00:19:10

equals zero, let's put 0 here, yes, perhaps it

00:19:12

turned out to be three second ones, you see there are 1

00:19:15

4 6 it turns out to be three second ones,

00:19:18

the square is already reduced and to

00:19:20

get it, it turns out q3 2 here

00:19:29

and your square rolls down dbh both shitao just

00:19:33

three second ones, and if it’s like this roughly- then

00:19:38

it turns out to be a unit, here is the formula for

00:19:45

tangential stresses during bending, but I’ll also

00:19:47

remind you of this ancient

00:19:51

aging mnemonic rules for the sword, my father

00:19:53

told me in the 20s 20 30 30s

00:19:58

when in Russia there was a

00:20:01

catastrophic shortage of tender

00:20:03

personnel but they were laid off, you mournful

00:20:06

course of engineers did not take collective farmers workers

00:20:10

very educated, so it was difficult for them to

00:20:14

give strength of evidence; not everyone came up with the formula; they came

00:20:18

up with a manic rule; it was in those years that the Turksib was built, if

00:20:19

you remember Ostap Bender, the golden calf

00:20:22

just built it and went to build the

00:20:24

Turksib, so they fill out the form for the

00:20:27

meaning of the fishes of the Turksib, so you have to go

00:20:32

remember the formula remember this

00:20:33

Turksib formula, since I started talking about the

00:20:36

attentional rule, that is, there is also a

00:20:40

formula for and you are metro on metro,

00:20:42

this is m divided by fuel and because

00:20:45

this is this time the moment of

00:20:46

resistance, that is, the normal

00:20:49

voltage is calculated using the metro

00:20:50

metro formula and the tangent according to the

00:20:53

Zhuravsky formula darkside

Description:

Выводим формулу Д. И. Журавского (1821-1891) для касательных напряжений при изгибе балки. В случае прямоугольного сечения получаем формулу tau=(3/2)Q/A. The formula for shear stresses in bending beams.

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