Bending is a type of deformation in which there is a curvature of the axes of straight beams or a change in the curvature of the axes of curved beams. Bending is associated with the appearance of bending moments in the cross-sections of the bar. A straight bend occurs when ... ... Wikipedia

Bend- - deformation of the part in the direction perpendicular to its axis. [Bloom E. E. Dictionary of basic metallographic terms. Yekaterinburg 2002] Bending is a deformation that occurs in beams, floor slabs, enclosing structures under ... ... Encyclopedia of terms, definitions and explanations of building materials

Bars, a deformed state that arises in a bar under the action of forces and moments perpendicular to its axis, and is accompanied by its curvature (about the I. plate and shell (see PLATE, SHELL)). Arising at I. in the cross-section of a bar ... Physical encyclopedia

BEND, bending, husband. 1. Arcuate curvature, rounded fracture, intricate turn. At the bend of the river. The beautiful curve of the swan's neck. Bends in the road. "Their (pine) roots lay in intricate bends like dead gray snakes." Maksim Gorky. 2. transfer ... Explanatory dictionary Ushakova

bend- KINK, twist, twist, twist, kink, twisted, serpentine, twisting, radiant, looped ... Dictionary-thesaurus of synonyms for Russian speech

In the resistance of materials, a type of deformation characterized by curvature (change in curvature) of the axis or the middle surface of an element (beam, plate, etc.) under the action of an external load. There are bends: clean, transverse, longitudinal, ... ... Big Encyclopedic Dictionary

BEND, ah, husband. Arcuate curvature. I. rivers. Bends of the Soul (trans.). Ozhegov's Explanatory Dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Ozhegov's Explanatory Dictionary

The stressed state of a rod or bar, accompanied by a curvature compared to its original shape. Distinguish between transverse I., occurring under the action of loads directed in most cases perpendicular to the axis of the rod, and ... ... Technical Railway Dictionary

bend- The type of body deformation, expressed in a change in its curvature in one or more directions [Terminological dictionary for construction in 12 languages ​​(VNIIIS Gosstroy USSR)] EN bendingflexure DE Biegung FR flexion ... Technical translator's guide

BEND- the type (see), in which the axis or the middle surface of the beam, rod, plate is bent under the action of external forces or temperature. The outer layers on the convex side of the deformed object experience the greatest stress. Deformation of the beam at ... Big Polytechnic Encyclopedia

Books

• Torsion and bending of thin-walled aircraft structures, A.A. Umansky Torsion and bending of thin-walled aircraft structures Reproduced in the original author's spelling of the 1939 edition (Oboronprom publishing house) ...
• Buckling. Torsion, A. N. Dinnik. Academician A. N. Dinnik's works "Buckling. Torsion" published in this volume, being a handbook for engineers, are now a bibliographic rarity. It…

bend, river curvature

Alternative descriptions

Saddle bend

Male name (lat.Luminiferous)

Bend in the river

The character of the play "At the Bottom"

Bend of the seashore

... "Krivulya" on the river

... Horse saddle toe

Saddle lug

Hero of the play "At the Bottom"

Hero, "At the Bottom"

Bent saddle lug

Arcuate bend of the river

Evangelist

J. bend, perished, curvature, bend; river turn, arc; low-lying and grassy or wooded cape; a drainage meadow, skirted by a river. Sometimes the onion is taken, back, in the meaning of a bay, a backwater, a backwater, or it is a novoros. grassy hollow, meadow. There are two saddle bows, a front and a back. Motri, heat the onion, do not freeze! tease the sissy. Church. guile, curvature of the soul. The bow of the river, like the knee, the elbow, is sometimes said instead of pleso, that is, not a bend, but a straight channel between the bends. Bow of the river (arc), bow, basket (bent, bent), bow (bent, that is, throwing arrows), bow, etc. of the common root. Bow m. Onion, strip bent into an arc; an elastic strip, wooden, horny, steel, tensioned with a bowstring, for shooting arrows. Bow with a butt, crossbow; a bow with a double bowstring, for throwing clay bullets, barracks. The bow is turning, boring, a kind of bow, with which the shells are turned back and forth. A saw bow, an iron tool for her. Fishing bow, karmak, oud for white fish. A woolen bow or onion, a three-arshin pole with a filly and a bowstring, which they push, beat the wool, hitting it with a katerinka, a mallet. Arch, bow, half-hoop, bend, for example. on the wagon. Onion on the spit, hook, rake, rake, rake for mowing bread. A net for catching songbirds (the cache is spread on two sticks, an onion on a half-hoop). A device in harness for horse breeding horses. Novg. a separate place for travelers, on a Tikhvinka boat, under a roof laid in arcs. equestrian soldier legs with bow. Who cares, and the arrow to the bow. A tight bow, then a heartfelt friend. Bow that king, arrows that messengers. The bow is good both for battle and in cabbage soup (play on words). The plow feeds, and the bow (weapon) spoils, old. about a man, a soldier. We didn’t use the bow, we didn’t squeak from the bow, but to drink, to dance, we cannot be found against us! The bow is small, but tight. Like an arrow from a bow. As if hiding from a bow. Both bows, both tight. Bow, old. measure of land; to the sowing. two fins,

River bend

Bend of the seashore

Bend in the river

River or saddle bend

Channel bend

Saddle bend

Bend in the river

Name of the Italian artist Signorelli

Which biblical character's name is translated from Latin as "light"

Gospel colleague Matthew

Colleague of Matthew, Mark and John

Steep meander of the river bed with ends close to the isthmus

Steep bend in the river

Which of the evangelists is depicted as a calf

Peacemaker in the play "At the Bottom"

Sea bend

Male name

Male name rhyming with beech

Male name: (latin) luminous

One of the evangelists

Character "At the bottom"

The turn of the river

Saddle swivel

River curvature

Russian sailor who opened the sea passage from the Northern Dvina to Northern Norway (XIV century)

Samara on the Volga

Companion of the apostle Paul

A good name for a Russian guy

Part of the saddle

The main peacemaker in Gorky's play "At the Bottom"

Character of Gorky's play "At the Bottom"

Wanderer in the play "At the Bottom"

Protruding bend of the front or rear edge of the saddle

Comforter

Mudischev

The character of the play "The Bear" by A.P. Chekhov

A character in the work of J. Moliere "The Reluctant Healer"

Greek of the Evangelists

River bend

Bend of the river channel

Coast bend

Saddle bend

Saddle squiggle

Curved saddle rim

Khlopov's name in "The Inspector General"

Curved seat lip

What is not a name for a Russian man

Russian male name

Male name rhyming with flour

Fit. Russian boy's name

Arched bend of the river

σ z = E ε z (\ displaystyle \ sigma _ (z) = E \ varepsilon _ (z))

that is, the voltages are also linearly distributed.

In the cross section of the beam (in the plane case), a bending moment occurs M x (\ displaystyle M_ (x)), lateral force Q y (\ displaystyle Q_ (y)) and longitudinal force N (\ displaystyle N)... An external distributed load acts on the section q (\ displaystyle q).

Consider two adjacent sections located at a distance d z (\ displaystyle dz) apart. In a deformed state, they are turned at an angle d θ (\ displaystyle d \ theta) relative to each other. Since the upper layers are stretched and the lower ones are compressed, it is obvious that there is neutral layer remaining unstretched. It is highlighted in red in the figure. The change in the curvature of the neutral layer is written as follows:

1 ρ = d θ d z (\ displaystyle (\ frac (1) (\ rho)) = (\ frac (d \ theta) (dz)))

The increment in the length of the segment AB located at a distance y (\ displaystyle y) from the neutral axis, is expressed as follows:

Δ l = (y + ρ) d θ - ρ d θ = y d θ (\ displaystyle \ Delta l = (y + \ rho) d \ theta - \ rho d \ theta = yd \ theta)

Thus, the deformation:

ε z = Δ ll = yd θ ρ d θ = y ρ (\ displaystyle \ varepsilon _ (z) = (\ frac (\ Delta l) (l)) = (\ frac (yd \ theta) (\ rho d \ theta)) = (\ frac (y) (\ rho)))

Force ratios

σ z = E ε z = E y ρ (\ displaystyle \ sigma _ (z) = E \ varepsilon _ (z) = E (\ frac (y) (\ rho)))

Let us associate the voltage with the force factors arising in the section. Axial force is expressed as follows:

N = ∫ A. σ z d A = ∫ A. E y ρ d A = E ρ ∫ A. yd A (\ displaystyle N = \ int \ limits _ (A) ^ (\ color (White).) (\ sigma _ (z)) \, dA = \ int \ limits _ (A) ^ (\ color (White ).) (E (\ frac (y) (\ rho))) \, dA = (\ frac (E) (\ rho)) \ int \ limits _ (A) ^ (\ color (White).) Y \, dA)

The integral in the last expression is the static moment of the section about the axis x (\ displaystyle x)... It is customary to take as an axis x (\ displaystyle x) the central axis of the section, such that

S x = ∫ A. y d A = 0 (\ displaystyle S_ (x) = \ int \ limits _ (A) ^ (\ color (White).) y \, dA = 0)

Thus, N = 0 (\ displaystyle N = 0)... The bending moment is expressed as follows:

M x = ∫ A. σ z y d A = E ρ ∫ A. y 2 d A = E ρ J x (\ displaystyle M_ (x) = \ int \ limits _ (A) ^ (\ color (White).) (\ sigma _ (z) y) \, dA = (\ frac (E) (\ rho)) \ int \ limits _ (A) ^ (\ color (White).) (Y ^ (2)) \, dA = (\ frac (E) (\ rho)) J_ (x ))

where J x = ∫ A. y 2 d A (\ displaystyle J_ (x) = \ int \ limits _ (A) ^ (\ color (White).) (y ^ (2)) \, dA)- moment of inertia of the section about the axis x (\ displaystyle x).

The stresses in the section can also be brought to the moment M y (\ displaystyle M_ (y))... To prevent this from happening, the following condition must be met:

M y = E ρ ∫ A. yxd A = E ρ J xy = 0 (\ displaystyle M_ (y) = (\ frac (E) (\ rho)) \ int \ limits _ (A) ^ (\ color (White).) (yx) \, dA = (\ frac (E) (\ rho)) J_ (xy) = 0)

that is, the centrifugal moment of inertia must be zero, and the axis y (\ displaystyle y) should be one of the main axes of the section.

Thus, the curvature of the bent axis of the beam is related to the bending moment by the expression:

1 ρ = M x E J x (\ displaystyle (\ frac (1) (\ rho)) = (\ frac (M_ (x)) (EJ_ (x))))

The stress distribution over the section height is expressed by the formula:

σ = M x J x y (\ displaystyle \ sigma = (\ frac (M_ (x)) (J_ (x))) y)

The maximum stress in the section is expressed by the formula:

σ max = M x J xh 2 = M x W x (\ displaystyle \ sigma _ (max) = (\ frac (M_ (x)) (J_ (x))) (\ frac (h) (2)) = (\ frac (M_ (x)) (W_ (x))))

where W x = J x h 2 (\ displaystyle W_ (x) = (\ frac (J_ (x)) (\ frac (h) (2))))- the moment of resistance of the section to bending, h (\ displaystyle h)- the height of the beam section.

The quantities J x (\ displaystyle J_ (x)) and W x (\ displaystyle W_ (x)) for simple sections (round, rectangular) are calculated analytically. For circular section with diameter d (\ displaystyle d):

J x = π d 4 64 (\ displaystyle J_ (x) = (\ frac (\ pi d ^ (4)) (64)))

W x = π d 3 32 (\ displaystyle W_ (x) = (\ frac (\ pi d ^ (3)) (32)))

For a rectangular section with a height h (\ displaystyle h) and width b (\ displaystyle b)

J x = b h 3 12 (\ displaystyle J_ (x) = (\ frac (bh ^ (3)) (12)))

W x = b h 2 6 (\ displaystyle W_ (x) = (\ frac (bh ^ (2)) (6)))

For more complex sections (for example, channel, I-beam) with standardized dimensions, these values ​​are given in the reference literature.

The bending moment in a section can be obtained by the section method (if the beam is statically definable) or by force / displacement methods.

Differential equilibrium equations. Definition of displacements

The main displacements that occur during bending are deflections v (\ displaystyle v) in the direction of the axis y (\ displaystyle y)... It is necessary to relate them to the bending moment in the section. Let's write down the exact relationship between the deflections and the curvature of the bent axis:

1 ρ = v ″ (1 + v ′ 2) 3 2 (\ displaystyle (\ frac (1) (\ rho)) = (\ frac (v "") ((1 + v "^ (2)) ^ ( \ frac (3) (2)))))

Since the deflections and angles of rotation are assumed to be small, the value

v ′ 2 = (tg (θ)) 2 ≈ θ 2 (\ displaystyle v "^ (2) = \ left (\ mathrm (tg) \, (\ theta) \ right) ^ (2) \ approx \ theta ^ (2))

is small. Hence,

1 ρ ≈ v ″ (\ displaystyle (\ frac (1) (\ rho)) \ approx v "")

We write the equilibrium equation of the section in the direction of the axis y (\ displaystyle y):

Q y + qdz - Q y - d Q y = 0 ⇛ d Q dz = q (\ displaystyle Q_ (y) + qdz-Q_ (y) -dQ_ (y) = 0 \ Rrightarrow (\ frac (dQ) (dz )) = q)

Let us write the equation of equilibrium of moments about the axis x (\ displaystyle x):

M x + Q dz + qdzdz 2 - M x - d M x = 0 (\ displaystyle M_ (x) + Q \, dz + q \, dz (\ frac (\, dz) (2)) - M_ (x ) - \, dM_ (x) = 0)

The quantity q d z d z 2 (\ displaystyle q \, dz (\ frac (\, dz) (2))) has the 2nd order of smallness and can be discarded. Hence,

d M x d z = Q y (\ displaystyle (\ frac (\, dM_ (x)) (\, dz)) = Q_ (y))

Thus, there are 3 differential equations. An equation for displacements is added to them:

d v d z = t g θ ≈ θ (\ displaystyle (\ frac (\, dv) (\, dz)) = \ mathrm (tg) \, \ theta \ approx \ theta)

In vector-matrix form, the system is written as follows:

d Z → dz + AZ → = b → (\ displaystyle (\ frac (\, d (\ overrightarrow (Z))) (\, dz)) + A (\ overrightarrow (Z)) = (\ overrightarrow (b) )) A = (0 0 0 0 - 1 0 0 0 0 - 1 EJ x (z) 0 0 0 0 - 1 0) (\ displaystyle A = (\ begin (Bmatrix) 0 & 0 & 0 & 0 \\ - 1 & 0 & 0 & 0 \\ 0 & - \ displaystyle (\ frac (1) (EJ_ (x) (z))) & 0 & 0 \\ 0 & 0 & -1 & 0 \ end (Bmatrix)))

System state vector:

Z → = (Q, M, θ, v) T (\ displaystyle (\ overrightarrow (Z)) = (Q, M, \ theta, v) ^ (T))

External load vector:

b → = (q, 0, 0, 0) T (\ displaystyle (\ overrightarrow (b)) = (q, 0,0,0) ^ (T))

This differential equation can be used to calculate multi-support beams with a variable length moment of inertia of the section and complexly distributed loads. Simplified methods are used to calculate simple beams. In the resistance of materials when calculating statically definable beams, the bending moment is found by the section method. The equation

v ″ = M x E J x (\ displaystyle v "" = (\ frac (M_ (x)) (EJ_ (x))))

integrates twice:

v ′ = θ (z) = ∫ M x (z) EJ xdz + C 1 (\ displaystyle v "= \ theta (z) = \ int (\ frac (M_ (x) (z)) (EJ_ (x) )) \, dz + C_ (1)) v (z) = ∫ (∫ M x (z) EJ xdz) dz + C 1 z + C 2 (\ displaystyle v (z) = \ int \ left (\ int (\ frac (M_ (x) (z) ) (EJ_ (x))) \, dz \ right) \, dz + C_ (1) z + C_ (2))

Constants C 1 (\ displaystyle C_ (1)), C 2 (\ displaystyle C_ (2)) are found from the boundary conditions imposed on the beam. So, for the cantilever beam shown in the figure:

M x (z) = - P (L - z) (\ displaystyle M_ (x) (z) = - P (L-z)) θ (z) = - PL z EJ x + P z 2 2 EJ x + C 1 (\ displaystyle \ theta (z) = - PL (\ frac (z) (EJ_ (x))) + P (\ frac ( z ^ (2)) (2EJ_ (x))) + C_ (1)) v (z) = - PL z 2 2 EJ x + P z 3 6 EJ x + C 1 z + C 2 (\ displaystyle v (z) = - PL (\ frac (z ^ (2)) (2EJ_ (x ))) + P (\ frac (z ^ (3)) (6EJ_ (x))) + C_ (1) z + C_ (2))

Border conditions:

θ (0) = 0 ⇛ C 1 = 0 (\ displaystyle \ theta (0) = 0 \ Rrightarrow C_ (1) = 0) v (0) = 0 ⇛ C 2 = 0 (\ displaystyle v (0) = 0 \ Rrightarrow C_ (2) = 0)

Thus,

θ (z) = - PL z EJ x + P z 2 2 EJ x (\ displaystyle \ theta (z) = - PL (\ frac (z) (EJ_ (x))) + P (\ frac (z ^ ( 2)) (2EJ_ (x)))) v (z) = - PL z 2 2 EJ x + P z 3 6 EJ x (\ displaystyle v (z) = - PL (\ frac (z ^ (2)) (2EJ_ (x))) + P (\ frac (z ^ (3)) (6EJ_ (x))))

Timoshenko's beam bending theory

This theory is based on the same hypotheses as the classical one, but Bernoulli's hypothesis is modified: it is assumed that sections that were flat before deformation and normal to the beam axis remain flat, but cease to be normal to the curved axis. Thus, this theory takes into account shear deformation and shear stresses. Consideration of shear stresses is very important for the design of composites and wood parts, since their destruction can occur due to the destruction of the binder during shear.

Main dependencies:

M = E J d θ d z (\ displaystyle M = EJ (\ frac (\, d \ theta) (\, dz))) Q = G F α (θ - d v d z) (\ displaystyle Q = (\ frac (GF) (\ alpha)) \ left (\ theta - (\ frac (\, dv) (\, dz)) \ right))

where G (\ displaystyle G)- shear modulus of the beam material, F (\ displaystyle F)- cross-sectional area, α (\ displaystyle \ alpha)- coefficient taking into account the uneven distribution of shear stresses over the section and depending on its shape. The quantity

γ = θ - d v d z (\ displaystyle \ gamma = \ theta - (\ frac (\, dv) (\, dz)))

represents the shear angle.

Bending of beams on an elastic foundation

This design model simulates railway rails, as well as ships (in the first approximation).

An elastic base is considered as a plurality of springs that are not connected to each other.

The simplest calculation method is based on Winkler's hypothesis: the reaction of an elastic foundation is proportional to the deflection at a point and is directed towards it:

P = - k ⋅ v (\ displaystyle p = -k \ cdot v)

where v (\ displaystyle v)- deflection;

P (\ displaystyle p)- reaction (per unit length of the beam);

K (\ displaystyle k)- coefficient of proportionality (called bed ratio).

In this case, the base is considered two-sided, that is, the reaction occurs both when the beam is pressed into the base, and when it is pulled off the base. Bernoulli's hypothesis persists.

The differential equation of bending of a beam on an elastic foundation has the form:

D 2 dz 2 (EJ x (z) d 2 vdz 2) + k (z) ⋅ v = q (z) (\ displaystyle (\ frac (d ^ (2)) (dz ^ (2))) \ left (EJ_ (x) (z) (\ frac (d ^ (2) v) (dz ^ (2))) \ right) + k (z) \ cdot v = q (z))

where v (z) (\ displaystyle v (z))- deflection;

E J x (z) (\ displaystyle EJ_ (x) (z))- flexural stiffness (which can be variable in length);

K (z) (\ displaystyle k (z))- the bed coefficient, variable along the length;

Q (z) (\ displaystyle q (z))- distributed load on the beam.

With constant stiffness and bedding ratio, the equation can be written as:

EJ xd 4 vdz 4 + k ⋅ v = q (z) (\ displaystyle EJ_ (x) (\ frac (d ^ (4) v) (dz ^ (4))) + k \ cdot v = q (z) )

D 4 vdz 4 + 4 m 4 ⋅ v = q (z) (\ displaystyle (\ frac (d ^ (4) v) (dz ^ (4))) + 4m ^ (4) \ cdot v = q (z ))

where indicated

4 m 4 = k E J x (\ displaystyle 4m ^ (4) = (\ frac (k) (EJ_ (x))))

Bending of a bar of large curvature

For beams, the radius of curvature of the axis of which ρ 0 (\ displaystyle \ rho _ (0)) commensurate with the section height h (\ displaystyle h), that is:

H ρ 0> 0.2 (\ displaystyle (\ frac (h) (\ rho _ (0)))> 0.2)

the distribution of stresses along the height deviates from the linear one, and the neutral line does not coincide with the axis of the section (which passes through the center of gravity of the section). Such a design model is used, for example, for the design of chain links and hooks for cranes.

File: Scheme of bending of a beam of large curvature.png

Transverse section

The formula for stress distribution is:

σ = M F ⋅ e ⋅ y R + y (\ displaystyle \ sigma = (\ frac (M) (F \ cdot e)) \ cdot (\ frac (y) (R + y)))

where M (\ displaystyle M)- bending moment in the section;

R (\ displaystyle R)- radius of the neutral section line;

F (\ displaystyle F)- cross-sectional area;

E = R 0 - R (\ displaystyle e = R_ (0) -R)- eccentricity;

Y (\ displaystyle y)- coordinate along the height of the section, measured from the neutral line.

The radius of the neutral line is determined by the formula:

R = ∫ d F u = ∫ R 1 R 2 b (u) duu (\ displaystyle R = \ int (\ frac (\, dF) (u)) = \ int \ limits _ (R_ (1)) ^ ( R_ (2)) (\ frac (b (u) \, du) (u)))

The integral is taken over the cross-sectional area, the coordinate u (\ displaystyle u) measured from the center of curvature. The approximate formulas are also valid:

E = J x R 0 ⋅ F (\ displaystyle e = (\ frac (J_ (x)) (R_ (0) \ cdot F)))

R 0 = R 0 - J x R 0 ⋅ F (\ displaystyle r_ (0) = R_ (0) - (\ frac (J_ (x)) (R_ (0) \ cdot F)))

Analytical formulas are available for frequently used cross sections. For a rectangular section with a height h (\ displaystyle h):

R = h ln R 0 + h 2 R 0 - h 2 = h ln R 2 R 1 (\ displaystyle R = (\ frac (h) (\ ln \ displaystyle (\ frac (R_ (0) + (\ frac ( h) (2))) (R_ (0) - (\ frac (h) (2)))))) = (\ frac (h) (\ ln \ displaystyle (\ frac (R_ (2)) (R_ (1))))))

where R 1, R 2 (\ displaystyle R_ (1), R_ (2)) are the radii of curvature of the inner and outer surface of the beam, respectively.

For a round section:

R = R 0 + R 0 2 - r 2 2 (\ displaystyle R = (\ frac (R_ (0) + (\ sqrt (R_ (0) ^ (2) -r ^ (2)))) (2) ))

where r (\ displaystyle r)- section radius.

Checking the strength of the beam

In most cases, the strength of a beam is determined by the maximum allowable stresses:

σ m a x< σ T n T {\displaystyle \sigma _{max}<{\frac {\sigma _{T}}{n_{T}}}}

where σ T (\ displaystyle \ sigma _ (T))- the yield strength of the beam material, n T (\ displaystyle n_ (T))- the safety factor for the fluidity. In the case of fragile materials:

σ m a x< σ b n b {\displaystyle \sigma _{max}<{\frac {\sigma _{b}}{n_{b}}}}

where σ b (\ displaystyle \ sigma _ (b))- ultimate strength of the beam material, n b (\ displaystyle n_ (b))- safety factor.

U = ∑ i = 1 4 C i K i (α z) (\ displaystyle u = \ sum _ (i = 1) ^ (4) C_ (i) K_ (i) (\ alpha z))

where the Krylov functions:

K 1 (α z) = 1 2 (ch ⁡ α z + cos ⁡ α z) (\ displaystyle K_ (1) (\ alpha z) = (\ frac (1) (2)) (\ operatorname (ch) \ alpha z + \ cos \ alpha z))

K 2 (α z) = 1 2 (sh ⁡ α z + sin ⁡ α z) (\ displaystyle K_ (2) (\ alpha z) = (\ frac (1) (2)) (\ operatorname (sh) \ alpha z + \ sin \ alpha z))

K 3 (α z) = 1 2 (ch ⁡ α z - cos ⁡ α z) (\ displaystyle K_ (3) (\ alpha z) = (\ frac (1) (2)) (\ operatorname (ch) \ alpha z- \ cos \ alpha z))

K 4 (α z) = 1 2 (sh ⁡ α z - sin ⁡ α z) (\ displaystyle K_ (4) (\ alpha z) = (\ frac (1) (2)) (\ operatorname (sh) \ alpha z- \ sin \ alpha z))

a C i (\ displaystyle C_ (i))- permanent.

Krylov's functions are linked by dependencies:

D dz K 1 (α z) = α K 4 (α z) (\ displaystyle (\ frac (d) (dz)) K_ (1) (\ alpha z) = \ alpha K_ (4) (\ alpha z) )

D dz K 2 (α z) = α K 1 (α z) (\ displaystyle (\ frac (d) (dz)) K_ (2) (\ alpha z) = \ alpha K_ (1) (\ alpha z) )

D dz K 3 (α z) = α K 2 (α z) (\ displaystyle (\ frac (d) (dz)) K_ (3) (\ alpha z) = \ alpha K_ (2) (\ alpha z) )

D dz K 4 (α z) = α K 3 (α z) (\ displaystyle (\ frac (d) (dz)) K_ (4) (\ alpha z) = \ alpha K_ (3) (\ alpha z) )

These constraints greatly simplify the writing of boundary conditions for beams:

C 1 = u z = 0; C 2 = 1 α (d u d z) z = 0; C 3 = 1 E J α 2 M z = 0; C 4 = 1 EJ α 3 Q z = 0 (\ displaystyle C_ (1) = u_ (z = 0); C_ (2) = (\ frac (1) (\ alpha)) \ left ((\ frac (du ) (dz)) \ right) _ (z = 0); C_ (3) = (\ frac (1) (EJ \ alpha ^ (2))) M_ (z = 0); C_ (4) = (\ frac (1) (EJ \ alpha ^ (3))) Q_ (z = 0))

At each end of the beam, two boundary conditions are specified.

The equation of natural vibrations has infinitely many solutions. In this case, of practical interest, as a rule, are only the first few of them, corresponding to the lowest natural frequencies.

The general formula for the natural frequency is:

P k = λ k 2 EJ m 0 l 4 (\ displaystyle p_ (k) = \ lambda _ (k) ^ (2) (\ sqrt (\ frac (EJ) (m_ (0) l ^ (4))) ))

For single-span beams:

Anchoring		λ k (\ displaystyle \ lambda _ (k))
Left end	Right end
Sealing	Sealing	λ 1 = 4.73; (\ displaystyle \ lambda _ (1) = 4.73;)λ 2 = 7, 853; (\ displaystyle \ lambda _ (2) = 7.853;)
Free	Free	λ 1 = 0; (\ displaystyle \ lambda _ (1) = 0;)λ 2 = 0; (\ displaystyle \ lambda _ (2) = 0;) λ k = 2 k + 1 2 π; (\ displaystyle \ lambda _ (k) = (\ frac (2k + 1) (2)) \ pi;)
Sealing	Articulated	λ 1 = 3.927; (\ displaystyle \ lambda _ (1) = 3.927;)λ 2 = 7, 069; (\ displaystyle \ lambda _ (2) = 7,069;) λ k = 4 k + 1 4 π; (\ displaystyle \ lambda _ (k) = (\ frac (4k + 1) (4)) \ pi;)
Articulated	Articulated	λ k = k 2 π 2 (\ displaystyle \ lambda _ (k) = k ^ (2) \ pi ^ (2))
Sealing	Free	λ 1 = 1, 875; (\ displaystyle \ lambda _ (1) = 1.875;)λ 2 = 4.694; (\ displaystyle \ lambda _ (2) = 4.694;) λ k = 2 k - 1 2 π; (\ displaystyle \ lambda _ (k) = (\ frac (2k-1) (2)) \ pi;)