Bending curvature. Clean bend. Cross bend. General concepts. Bending beams on an elastic foundation

bend, curvature of the river

Alternative descriptions

saddle edge bend

Male name (lat. Luminous)

bend in the river

The character of the play ""At the bottom""

The curve of the sea

. "Krivulya" on the river

. "Toe" of a horse saddle

seat protrusion

The hero of the play "At the bottom"

Hero, "Down Under"

Bent seat post

An arcuate turn of the river

Evangelist

Zh. bend, perished, curvature, bend; inversion of the river, arc; low-lying and grassy or wooded cape; floodplain meadow, skirted by the river. Sometimes the bow is taken, back, in the meaning of the bay, the backwater, the creek, or it is novoros. grassy hollow, meadow. There are two saddle bows, front and back. Motri, warm up to the bow, do not freeze! tease the sissy. Church. cunning, curvature of the soul. The bow of the river, like the knee, the elbow, is sometimes said instead of the 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. common root. Bow m. ray, a strip bent into an arc; elastic strip, wooden, horn, steel, strained by a bowstring, for shooting arrows. A bow with a butt, crossbow; bow with a double bowstring, for throwing clay bullets, barracks. A bow turning, drilling, a kind of bow, with which shells are turned back and forth. Saw bow, iron machine for it. Fishing onion, karmak, oud for white salmon. A woolly bow or bow, a three-yard pole with a filly and a bowstring, which they push, beat the wool, hitting it with a katerinka, mallet. Arc, shackle, half-hoop, bent, for example. on a tent. Onion on a scythe, hook, hornbeam, rakes, rakes for mowing bread. A net for catching songbirds (the hiding place is thrown on two sticks, a bow on a half-hoop). Device in the harness of horse-drawn horses. Novg. a separate place for travelers, on a Tikhvinka boat, under a roof laid in arcs. mounted soldier's legs with a beam. Who cares what, and the arrow to the bow. Tight bow, then a cordial friend. A bow that is a king, arrows that are messengers. The bow is good both for battle and in cabbage soup (play on words). The plow feeds, but the bow (weapon) spoils, old. about a man, a soldier. We are not from the bow, we are not from the squeak, but to drink, to dance, we cannot be found against us! A small bow, but tight. Like an arrow from a bow. As if hid from a bow. Both bows, both tight. Luke, old. measure of the earth; to the sowing two roasts,

Bend of the river

The curve of the sea

bend in the river

River bend or saddle

river bend

Saddle bend

river bend

The name of the Italian artist Signorelli

The name of which biblical character is translated from Latin as "light"

Matthew's colleague in the gospel

Colleague of Matthew, Mark and John

Steep meander of the riverbed with ends closely converging at the isthmus

A sharp 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 that rhymes with beech

Male name: (Latin) luminiferous

One of the evangelists

Character "Down"

river bend

Saddle rotation

river curvature

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

Samara on the Volga

Companion of the Apostle Paul

Nice name for a Russian guy

Saddle part

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

Character in Gorky's play "At the Bottom"

Wanderer in the play "At the Bottom"

Protruding curve of the front or rear edge of the saddle

Comforter

Mudishchev

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

The character of the work of J. Moliere "The Doctor involuntarily"

Greek of the Evangelists

river bend

River bend

coast bend

saddle bend

Squiggle saddle

Curved saddle collar

Khlopov's name in "Inspector"

Curved seat post

What is not a name for a Russian peasant

Russian male name

Male name that rhymes with flour

Suitable russian guy name

Curved bend of the river

Bending is a type of deformation in which there is a curvature of the axes of straight bars or a change in the curvature of the axes of curved bars. Bending is associated with the occurrence of bending moments in the cross sections of the beam. Direct bending occurs when ... ... Wikipedia

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

Bar, a deformed state that occurs in a bar under the action of forces and moments perpendicular to its axis, and is accompanied by its curvature (about I. plates and shells (see PLATES, SHELL)). Arising during I. in the cross section of the beam ... Physical Encyclopedia

BENDING, bending, husband. 1. Bow-shaped curvature, rounded kink, intricate twist. At the bend of the river. Beautiful curve of the swan neck. Road bends. "Their (pines) roots lay in intricate curves like gray dead snakes." Maksim Gorky. 2. transfer ... Dictionary Ushakov

bend- BENDING, twisting, meandering, twisting, twisting, twisting, serpentine, twisting, radiant, looping ... Dictionary-thesaurus of synonyms of Russian speech

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

BEND, eh, husband. Arcuate curvature. I. rivers. Curves of the soul (trans.). Explanatory dictionary of Ozhegov. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Explanatory dictionary of Ozhegov

The stressed state of a bar or bar, accompanied by a distortion from its original shape. There are transverse I., which occurs under the action of loads directed in most cases perpendicular to the axis of the rod, and ... ... Technical railway dictionary

bend- Type of deformation of the body, expressed in a change in its curvature in one or more directions [ Terminological dictionary on construction in 12 languages ​​(VNIIIS Gosstroy of the USSR)] EN bendingflexure DE Biegung FR flexion ... Technical Translator's Handbook

BENDING- view (see), in which the axis or middle surface of a beam, rod, plate is bent under the influence of external forces or temperature. The greatest stress is experienced by the outer layers on the convex side of the deformable object. Beam deformation at ... Great Polytechnic Encyclopedia

Books

• Torsion and bending of thin-walled aircraft structures, Umansky A.A. Torsion and bending of thin-walled aircraft structures Reproduced in the original author's spelling of the 1939 edition (Oboronprom publishing house) ...
• Longitudinal bend. Torsion, A. N. Dinnik. The works of academician A. N. Dinnik "Longitudinal bending. Torsion", published in this volume, being a reference book for engineers, are currently a bibliographic rarity. This…

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

that is, the stresses are also distributed linearly.

A bending moment occurs in the beam section (in the plane case) M x (\displaystyle M_(x)), transverse force Q y (\displaystyle Q_(y)) and longitudinal force N (\displaystyle N). The cross section is affected by an external distributed load q (\displaystyle q).

Consider two adjacent sections located at a distance dz (\displaystyle dz) from each other. In the deformed state, they are deployed 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, which remains 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 l l = y d θ ρ d θ = y ρ (\displaystyle \varepsilon _(z)=(\frac (\Delta l)(l))=(\frac (yd\theta )(\rho d\ theta ))=(\frac (y)(\rho )))

Power ratios

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

Let us relate the stress to 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 . y d 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 section about the axis x (\displaystyle x).

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

M y = E ρ ∫ A . y x d A = E ρ J x y = 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) must 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 distribution of stresses along the height of the section 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 m a x = M x J x h 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))))- moment of resistance of the section to bending, h (\displaystyle h)- beam section height.

Quantities J x (\displaystyle J_(x)) And W x (\displaystyle W_(x)) for simple sections (round, rectangular) are calculated analytically. For round 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) having 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 determinate) or by force/displacement methods.

Differential Equations of Equilibrium. Definition of displacements

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

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

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

v ′ 2 = (t g (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"")

Let us write the equilibrium equation for the section in the direction of the axis y (\displaystyle y):

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

We write the equation for the equilibrium of moments about the axis x (\displaystyle x):

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

Value q d z d z 2 (\displaystyle q\,dz(\frac (\,dz)(2))) has 2nd order smallness and may be discarded. Hence,

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

Thus, there are 3 differential equations. To them is added the equation for displacements:

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 → d z + A Z → = b → (\displaystyle (\frac (\,d(\overrightarrow (Z)))(\,dz))+A(\overrightarrow (Z))=(\overrightarrow (b) )) A = ( 0 0 0 0 − 1 0 0 0 0 − 1 E J 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 sectional moment of inertia variable along the length and loads distributed in a complex way. Simplified methods are used to calculate simple beams. IN resistance of materials when calculating statically determinate beams, the bending moment is found by the method of sections. The equation

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

integrated twice:

v ′ = θ (z) = ∫ M x (z) E J x d z + C 1 (\displaystyle v"=\theta (z)=\int (\frac (M_(x)(z))(EJ_(x) ))\,dz+C_(1)) v (z) = ∫ (∫ M x (z) E J x d z) d z + 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) = − P L z E J x + P z 2 2 E J x + C 1 (\displaystyle \theta (z)=-PL(\frac (z)(EJ_(x)))+P(\frac ( z^(2))(2EJ_(x)))+C_(1)) v (z) = − P L z 2 2 E J x + P z 3 6 E J 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) = − P L z E J x + P z 2 2 E J x (\displaystyle \theta (z)=-PL(\frac (z)(EJ_(x)))+P(\frac (z^( 2))(2EJ_(x)))) v (z) = − P L z 2 2 E J x + P z 3 6 E J x (\displaystyle v(z)=-PL(\frac (z^(2))(2EJ_(x)))+P(\ frac (z^(3))(6EJ_(x))))

Beam bending theory Tymoshenko

This theory is based on the same hypotheses as the classical one, but the Bernoulli hypothesis is modified: it is assumed that the sections that were flat and normal to the beam axis before deformation remain flat, but cease to be normal to the curved axis. Thus, this theory takes into account shear strain and shear stresses. Accounting for shear stresses is very important for the calculation composites and parts made of wood, 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)- beam material shear modulus, F (\displaystyle F)- cross-sectional area, α (\displaystyle \alpha )- coefficient that takes into account the uneven distribution of shear stresses over the section and depends on its shape. Value

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

is the angle of shear.

Bending beams on an elastic foundation

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

The elastic base is considered as a set of springs not connected with each other.

The simplest calculation method is based on the hypothesis Winkler: the reaction of the elastic base is proportional to the deflection at the 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 coefficient).

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 separated from the base. Bernoulli's conjecture holds.

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

D 2 d z 2 (E J x (z) d 2 v d z 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))- bending stiffness (which can be variable along the length);

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

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

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

E J x d 4 v d z 4 + k ⋅ v = q (z) (\displaystyle EJ_(x)(\frac (d^(4)v)(dz^(4)))+k\cdot v=q(z) )

D 4 v d z 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 beam of large curvature

For beams, the radius of curvature of the axis of which ρ 0 (\displaystyle \rho _(0)) commensurate with the height of the section 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 linear, and the neutral line does not coincide with the axis of the section (which passes through center of gravity sections). Such a calculation scheme is used, for example, to calculate chain links and hooks lifting cranes.

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

Cross 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, counted 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) d u u (\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. 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 commonly 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 surfaces of the beam, respectively.

For 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.

Beam strength check

In most cases, the strength of the 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)) - yield point beam material, n T (\displaystyle n_(T)) - safety factor by fluidity. For brittle materials:

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

Where σ b (\displaystyle \sigma _(b)) - tensile strength beam material, n b (\displaystyle n_(b)) - safety factor by strength.

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

where are 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 connected by dependencies:

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

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

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

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

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

C 1 = u z = 0; C 2 \u003d 1 α (d u d z) z \u003d 0; C 3 = 1 E J α 2 M z = 0 ; C 4 = 1 E J 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))

Two boundary conditions are specified at each end of the beam.

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

General formula for natural frequency looks like:

P k = λ k 2 E J 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
termination	termination	λ 1 \u003d 4, 73; (\displaystyle \lambda _(1)=4.73;)λ 2 \u003d 7, 853; (\displaystyle \lambda _(2)=7,853;)
Free	Free	λ1 = 0; (\displaystyle \lambda _(1)=0;)λ 2 \u003d 0; (\displaystyle \lambda _(2)=0;) λk = 2k + 1 2 π ; (\displaystyle \lambda _(k)=(\frac (2k+1)(2))\pi ;)
termination	Articulated	λ 1 \u003d 3, 927; (\displaystyle \lambda _(1)=3,927;)λ 2 \u003d 7, 069; (\displaystyle \lambda _(2)=7,069;) λk = 4k + 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))
termination	Free	λ 1 \u003d 1, 875; (\displaystyle \lambda _(1)=1,875;)λ 2 \u003d 4, 694; (\displaystyle \lambda _(2)=4,694;) λk = 2k − 1 2 π ; (\displaystyle \lambda _(k)=(\frac (2k-1)(2))\pi ;)