Cross bend rod. Transverse bending technical mechanics transverse bending solution

Forces acting perpendicular to the axis of the bar and located in the flat bone passing through this axis cause deformation called transverse bending. If the plane of the action of the mentioned forces – The main plane, then there is a straight (flat) transverse bending. Otherwise, the bending is called oblique transverse. Bar susceptible to bend is called beam 1 .

Essentially, the transverse bending is a combination of pure bending and shear. In connection with the revocation of cross-sections due to the unevenness of the distribution of shifts in height, the question arises on the possibility of using the normal voltage formula σ H.derived for pure bend on the basis of the hypothesis of flat sections.

1 single-break beam, having at the ends, respectively, one cylindrical fixed support and one cylindrical movable in the direction of the axis of the beam is called plain. The beam with one pinched and another free end is called console. A simple beam having one or two parts hanging behind the support is called console.

If, in addition, the cross sections are taken away from the location of the application of the load (at a distance not less than half the height of the cross section of the bar), then, as in the case of pure bend, it is possible that the fibers do not press each other. It means that each fiber is experiencing a uniaxial stretching or compression.

Under the action of a distributed load, transverse forces in two adjacent sections will differ by value equal to qDX . Therefore, the curvature of sections will also be somewhat different. In addition, the fibers will put pressure on each other. Careful question research shows that if the length of the bar l. great enough compared to his height h. (l./ h. \u003e 5), and during distributed load, these factors do not have a significant effect on normal stresses in cross section and therefore in practical calculations may not be taken into account.

a B C

Fig. 10.5 Fig. 10.6.

In sections under focused loads and near them distribution σ H. deviates from linear law. This deviation, which is local and not accompanied by an increase in the greatest stresses (in extreme fibers), is usually not taken into account in practice.

Thus, with transverse bending (in the plane hu.) Normal voltages are calculated by the formula

σ H.= – [M Z.(x.)/I Z.]y..

If we carry out two adjacent sections on the area of \u200b\u200bthe bar free from the load, the transverse force in both sections will be the same, which means the same and curvature of sections. In this case, any segment of the fiber aB (Fig.10.5) will move to a new position a "B", not undergoing additional elongation, and therefore, without changing the value of normal voltage.

We define the tangent stresses in cross section through the paired voltage, acting in the longitudinal section of the bar.

We highlight the element length from the bar dX (Fig. 10.7 a). Cut the horizon-lion cross section at a distance w. from neutral axis z.separated by the element into two parts (Fig. 10.7) and consider the equilibrium of the upper part having the base

width b.. In accordance with the law of partnership of tangent stresses, the voltage acting in the longitudinal section is equal to the stresses acting in cross section. Taking into account this suggesting that tangent stresses in the site b.it is uniformly used to use the condition σx \u003d 0, we obtain:

N * - (n * + dn *) +

where: n * is the resultant normal forces σ in the left transverse section of the DX element within the "cut-off" platform A * (Fig. 10.7 g):

where: s \u003d - the static moment of the "cut-off" part of the transverse section (shaded area in Fig. 10.7 V). Therefore, you can write:

Then you can write:

This formula was obtained in the XIX century Russian scientists and engineer D.I. Zhuravsky and carries his name. And although this formula is approximate, since there is averaging the voltage in the width of the section, but the obtained results of the calculation according to it are quite consistent with the experimental data.

In order to determine the tangent stresses in an arbitrary section of the cross section of a distance of Y from the Z axis:

Determine the magnitude of the transverse force Q acting in the section;

Calculate the moment of inertia I z of all sections;

Conduct a parallel plane through this point xz. and determine the width of the section b.;

Calculate the static moment of cut-off area of \u200b\u200bthyoughly main central axis z. And to substitute the found values \u200b\u200bin the formula of the Zhura-Bow.

We define the use of tangent stresses in a rectangular cross section (Fig. 10.6, c). Static moment relative to the axis z. Parts section above line 1-1, on which the voltage is determined to write in the form:

It changes under the law of a square parabola. The width of the section infor a rectangular bar is constant, it will also be a law of changing tangent stresses in the section (Fig.10.6, B). At y \u003d and y \u003d - casual voltages are zero, and on the neutral axis z. They achieve the greatest value.

For the beam of the circular cross section on the neutral axis we have.

Straight bend - This is a form of deformation, in which two internal power factor arise in the cross sections of the rod: bending moment and transverse force.

Pure bend - This is a particular case of direct bend, in which only bending moment appears in cross sections, and the transverse force is zero.

An example of pure bending - plot CD On the rod AB. Bending moment - This is a magnitude PA Couple of external forces causing bending. From equilibrium part of the rod to the left of the cross section mN. It follows that the inner efforts distributed through this section are statically equivalent to the moment M.equal and oppositely directed bending moment PA.

To find the distribution of these internal cross-sectional efforts, it is necessary to consider the deformation of the rod.

In the simplest case, the rod has a longitudinal plane of symmetry and is exposed to external bending pairs of forces in this plane. Then the bend will occur in the same plane.

Rod axis nN 1. - This is a line passing through the centers of gravity of its cross sections.

Let the cross section of the rod - a rectangle. I will apply two vertical lines on its face mm. and pp.. When bending, these lines remain straightforward and rotated so that they remain perpendicular to the longitudinal fibers of the rod.

Further bend theory is based on the assumption that not only lines Mm. and pp. , But the entire flat cross section of the rod remains flat and normal to longitudinal rod fibers. Consequently, with bending cross sections mm. and pp. Rotate relative to each other around the axes perpendicular to the bending plane (drawing plane). At the same time, longitudinal fibers on the convex side are tensile, and the fibers on the concave side are compression.

Neutral surface - This is a surface, not experiencing deformations in bending. (Now it is located perpendicular to the drawing, the deformed axis of the rod nN 1. belongs to this surface).

Neutral axis section - This is an intersection of a neutral surface with anyone with any cross section (now is also perpendicular to the drawing).

Let arbitrary fiber be at a distance y. from neutral surface. ρ - The radius of curvature of the curved axis. Point O. - Center for curvature. We carry out a line n 1 S 1parallel mm.. SS 1. - Absolute fiber elongation.

Relative extension ε X.fiber

It follows that deformations of longitudinal fibers Proportional to distance y. from neutral surface and inversely proportional to the radius of curvature ρ .

Longitudinal lengthening of the rod rod fibers is accompanied by side narray, and the longitudinal shortening of the concave side - side extensionAs in the case of simple stretching and compression. Because of this, the type of all transverse sections is changing, the vertical sides of the rectangle become inclined. Deformation in the lateral direction z.:

μ - Poisson's ratio.

Owing to such a distortion, all straight cross-section lines, parallel axes z., twist to remain normal to the sides of the section. The radius of curvature of this curve R. will be more than ρ In the same way, in which ε x in absolute value more than ε z and we get

These deformations of longitudinal fibers answer voltages

Voltage in any fiber is proportional to its distance from the neutral axis n 1 N 2. The position of the neutral axis and the radius of curvature ρ - Two unknown in the equation for σ x - can be determined from the condition that the efforts distributed according to any cross section form a couple of forces that balances the outer moment M..

All of the above is also fair if the rod does not have a longitudinal plane of symmetry, in which the bending moment acts, just only the bending moment acted in the axial plane, which concludes one of two main axes cross-section. These planes are called the main planes of bend.

When there is a plane of symmetry and the bending moment acts in this plane, the deflection occurs in it. Moments of domestic effort relative to the axis z. Balance the outer moment M.. Moments of effort relative to the axis y. mutually destroyed.

Classification of stem bends

Bend This type of deformation is called, in which bending moments appear in cross sections. Bend rod accepted bale. If the bending moments are the only internal power factors in cross-sections, then the rod is experiencing pure bending. If the bending moments arise in conjunction with the transverse forces, then such a bend is called transverse.

Beams, axles, shafts and other parts of structures work on bending.

We introduce some concepts. The plane passing through one of the main central axes of the section and the geometric axis of the rod is called the main plane. The plane in which external loads cause beam bending are called power plane. The crossing line of the power plane with the transverse cross section of the rod is called power line.Depending on the mutual position of the power and main planes, the beams distinguish between direct or oblique bending. If the power plane coincides with one of the main planes, then the rod is experiencing straight bend (Fig. 5.1, but) if it does not coincide - kosovo(Fig. 5.1, b).

Fig. 5.1. Rod bending: but - straight; b. - Kosovo

From a geometric point of view, the bending of the rod is accompanied by a change in the curvature of the axis of the rod. Initially, the straight axis of the rod becomes curvilinear with its bending. With a straight bending, the curved axis of the rod lies in the power plane, with a braid - in a plane other than the power.

Watching the bend of the rubber rod, it can be noted that part of its longitudinal fibers is stretched, and the other part is compressed. Obviously, between the stretched and compressed rod fibers, there is a layer of fibers that do not have a stretching, nor compression - the so-called neutral layer. The crossing line of the neutral layer of the rod with the plane of its cross section is called neutral cross section line.

As a rule, acting on the load beam can be attributed to one of three types: focused forces R, Concentrated moments M. Distributed loads intensity c. (Fig. 5.2). Part I beams located between the supports are called spanpart II beams located one way from the support - console.

As in § 17, suppose that the cross section of the rod has two axes of symmetry, one of which lies in the bend plane.

In the case of the transverse bending of the rod in cross section, there are tangent stresses, and during the deformation of the rod, it does not remain flat, as in the case of pure bend. However, for a bar of a continuous cross section, the effect of tangent stresses with transverse bending can be neglected and approximately adopted, which is the same as in the case of pure bending, the cross section of the rod during its deformation remains flat. Then, the formulas for stresses and curvature were derived in § 17, remain approximately valid. They are accurate for a particular case constant along the length of the transverse power rod 1102).

Unlike pure bend with a cross-bending, the bending moment and curvature remain constant along the length of the rod. The main task in the case of transverse bend is the definition of deflection. To determine the small deflection, you can use the known approximate dependence of the curvature of the curved rod from the deflection 11021. Based on this dependence, the curvature of the curved rod x C and the deflection V E. resulting from the creep material are associated with the ratio x C \u003d \u003d dV

Substituting into this ratio of curvature according to formula (4.16), we establish that

The integration of the last equation makes it possible to get a deflection resulting from the creep of the material beam.

Analyzing the above solution to the creep problem of the curved rod, it can be concluded that it is completely equivalent to solving the problem of bending a rod from the material in which the stretching diagrams of compression may be approximated by a power function. Therefore, the definition of the deflection arising from the creep in the case under consideration can be produced and using the Mora integral to determine the movement of the rods made from the material that does not obey the bike law

The plots of normal voltages operating by the venues 1-2 and 3-4 with a positive value of M, are shown in Fig. 39.7. For the same sites, tangent stresses also feature in Fig. 39.7. The magnitude of these stresses varies in the height of the section.

Denote the magnitude of the tangent stress at the lower points of 1-2 and 3-4 (at level). According to the law, the passage of tangent stresses it follows that the same by the magnitude of tangent stresses operate at the bottom site 1-4 of the dedicated element. Normal voltages on this platform are considered equal to zero, since in the theory of bending it is assumed that the longitudinal fibers of the beams do not have pressure on each other.

The platform 1-2 or 3-4 (Fig. 39.7 and 40.7), i.e., part of the cross section, located above the level (above the site 1-4), is called the cross-sectional part. Her area is denoted

The equilibrium equation for an element 1-2-3-4 in the form of the amount of the projections of all the forces attached to it on the axis beams:

Here is the resultant elementary forces arising from a platform of 1-2 elements; - the resultant elementary forces arising from the site 3-4 element; - the resultant elementary tangent forces arising from the site 1-4 element; - the width of the transverse section of the beam at the level of

Substitute expressions in the formulas (26.7) to equation (27.7):

But on the basis of the theorem of Zhuravsky [formula (6.7)]

The integral is a static moment of the area relative to the neutral axis of the beam cross section.

Hence,

Under the law of a partnership of tangent stresses of voltage at the cross-sectional points of the beam, disposable to the distance from the neutral axis are equal to (at an absolute value) that is.

Thus, the values \u200b\u200bof tangent stresses in the transverse sections of the beam and in cross-sectiones of its planes parallel to the neutral layer are determined by the formula

Here q is a transverse force in the transverse cross section of the beam; - static moment (relative to the neutral axis) of the cut-off part of the cross section, located one side from the level on which the tangent stresses are determined; J is the moment of inertia of the entire cross section relative to the neutral axis; - The width of the transverse section of the beam at the level on which the tangent stresses are determined.

The expression (28.7) is called the formula of Zhuravsky.

Determination of tangent stresses according to formula (28.7) is performed in the following order:

1) the cross section of the beam is carried out;

2) for this cross section, the values \u200b\u200bof the transverse force q are determined and the magnitude of the moment of inertia of the cross section relative to the main central axis coinciding with the neutral axis;

3) in cross section at the level for which the tangent stresses are determined, a straight line, cutting part of the section, is determined; The length of the segment of this direct concluded inside the cross-sectional circuit is a width included in the denominator of formula (28.7);

4) the static moment s is calculated with the cut-off (located one direction from the line specified in paragraph 3) of the section of the cross section relative to the neutral axis;

5) According to formula (28.7), the absolute value of the tangent voltage is determined. The sign of tangent stresses in the cross section of the beam coincides with the sign of the transverse force acting in this section. The sign of tangent stresses in sites parallel to the neutral layer is opposite to the sign of the transverse force.

We define the use of tangent stresses in the rectangular cross section of the beam shown in Fig. 41.7, a. The transverse force in this section acts parallel to the axis y and equal

The moment of inertia of the cross section relative to the axis

To determine the tangent stress at some point with spending 1-1, parallel axis through this point (Fig. 41.7, a).

We define the static moment s part of the cross section, cut-off direct 1-1, relative to the axis. Behind the cut-off can be taken as part of the section, located above the straight line 1-1 (shaded in Fig. 41.7, a) and the part below this direct.

For top

Substitute in formula (28.7) the values \u200b\u200bof Q, S, J and B:

From this expression it follows that tangent stresses vary in the height of the cross section under the law of the square parabola. At voltage the largest voltages are available at the points of the neutral axis, i.e.

where is the cross-sectional area.

Thus, in the case of a rectangular section, the greatest tangent voltage is 1.5 times larger than its average value equal to the incidence of tangent stresses, showing their change in the height of the cross section of the beam, is shown in Fig. 41.7, b.

To verify the obtained expression [see Formula (29.7)] will substitute it into equality (25.7):

The resulting identity indicates the correctness of the expression (29.7).

Parabolic escape of tangent stresses shown in Fig. 41.7, B, is a consequence of the fact that with a rectangular section, the static moment of the cut-off part of the section changes with a change in the position of the line 1-1 (see Fig. 41.7, a) according to the law of the square parabol.

During the sections of any other form, the nature of the change in tanning stresses in the height of section depends on how the situation changes the ratio of the width B is constant in some sections, the voltages in these areas are changed by the law of changing the static moment

At the cross-sectional points of the beams, the tangent stresses are zero, since when determining the stresses at these points in formula (28.7), the value of the static moment of the cut-off part of the cross section is substantiated.

The value 5 reaches a maximum for points located on a neutral axis, however, the tangent stresses in cross sections with a variable width B may not be maximal on the neutral axis. For example, the escape of tangent stresses for the section shown in Fig. 42.7, and has the appearance shown in Fig. 42.7, b.

Tangent stresses arising from transverse bending in planes parallel to the neutral layer characterize the interaction forces between the individual beam layers; These forces seek to move the neighboring layers of each other in the longitudinal direction.

If there are no sufficient bond between individual layers, then such a shift will occur. For example, boards, put on each other (Fig. 43.7, a), will resist the external load, as a whole bar (Fig. 43.7, b), until the efforts on the planes of the contamination of the boards do not exceed the friction forces between them. When the friction forces are surpassed, the boards will move one on the other, as shown in Fig. 43.7, in. At the same time, the brains of the boards will increase dramatically.

Tangent stresses acting in cross sections of beams and in sections parallel to the neutral layer cause shift deformations, as a result of which direct angles between these sections are distorted, i.e. they cease to be straight. The greatest distortions of the corners are available in those points of the cross section, in which the greatest tangent stresses operate; The upper and lower edges of the distortion beams are absent, as the tangent stresses are zero there.

As a result of shift deformations, the cross sections of the beam with transverse bending are curved. However, this does not significantly affect the deformations of longitudinal fibers, and therefore, on the distribution of normal stresses in the transverse sections of the beam.

Let us now consider the distribution of tangent stresses in thin-walled beams with cross-sections, symmetrical relative to the axis y, in the direction of which the transverse force Q, for example, in the beam of the 2-way section shown in Fig. 44.7, a.

For this, according to the Formula of Zhuravsky (28.7), we define tangent stresses at some characteristic points of the transverse section of the beam.

At the upper point 1 (Fig. 44.7, a) tangent stresses as the entire cross-sectional area is located below this point, and therefore static moment 5 relative to the axis (parts of the cross-section area located above point 1) is zero.

At point 2, located directly above the line passing through the bottom face of the upper shelf of the heap, tangent stresses, calculated by formula (28.7),

Between points 1 and 2 of the voltage [defined by formula (28.7)] varies in square parabole, as for a rectangular section. In the wall of the heateur at point 3, located directly under point 2, tangent stresses

Since the width B of the heap of the heap is much more than the thickness D of the vertical wall, then the escape of tangent stresses (Fig. 44.7, b) has a sharp jump in the level corresponding to the lower edge of the upper shelf. Below the point 3 tangent stresses in the wall of the heateur change according to the law of the square parabola, as for a rectangle. The greatest tangent stresses occur at the level of the neutral axis:

The escape of tangent stresses, constructed by the values \u200b\u200bobtained and, is shown in Fig. 44.7, b; It is symmetrical about the ordinary.

According to this scene, at points located in the inner edges of the shelves (for example, at points 4 in Fig. 44.7, a), there are tangent stresses perpendicular to the contour of the cross section. But, as already noted, such stresses near the cross section circuit cannot occur. Consequently, the assumption of the uniform distribution of tangent stresses in the width of the cross-sectional cross section, which is based on the output of formula (28.7), is not applicable to the heap regiments; It is not applicable to some elements of other thin-walled beams.

The tangent stresses of the TU in the hens of the hediar to determine the methods of resistance of materials cannot be. These voltages are very small in comparison with the voltages of the TU in the wall of the heap. Therefore, they are not taken into account and the essential of tangent stresses are built only for the heap of the heap, as shown in Fig. 44.7, in.

In some cases, for example, when calculating composite beams, the amount of tangent forces acting in the sections of the beam parallel to the neutral layer and per unit of its length are determined. This value will find, multiplying the value of the voltage on the width of the section B:

Substitute the value by formula (28.7):