Calculation of thin-walled vessels by Laplace's formula. Hydraulics problems with ready-made solutions. Calculation of thick-walled pipes

In engineering practice, structures such as tanks, water reservoirs, gas tanks, air and gas cylinders, building domes, chemical engineering apparatus, parts of turbine and jet engine housings, etc. are widely used. All these structures, from the point of view of their strength and rigidity calculations, can be classified as thin-walled vessels (shells) (Fig. 13.1, a).

A characteristic feature of most thin-walled vessels is that in shape they represent bodies of rotation, i.e. their surface can be formed by rotating some curve around the axis ABOUT-ABOUT. Section of a vessel by a plane containing an axis ABOUT-ABOUT, called meridional section, and sections perpendicular to meridional sections are called district. Circumferential sections, as a rule, have the shape of a cone. The lower part of the vessel shown in Fig. 13.1b is separated from the upper by a circumferential section. The surface dividing the thickness of the walls of the vessel in half is called middle surface. The shell is considered to be thin-walled if the ratio of the smallest principal radius of curvature at a given point on the surface to the thickness of the shell wall exceeds 10
.

Let us consider the general case of the action of some axisymmetric load on the shell, i.e. such a load that does not change in the circumferential direction and can only change along the meridian. Let us select an element from the shell body with two circumferential and two meridional sections (Fig. 13.1, a). The element experiences tension in mutually perpendicular directions and bends. Bilateral tension of an element corresponds to a uniform distribution of normal stresses across the wall thickness and the occurrence of normal forces in the shell wall. A change in the curvature of the element suggests the presence of bending moments in the shell wall. When bending, normal stresses arise in the beam wall, varying along the wall thickness.

Under the action of an axisymmetric load, the influence of bending moments can be neglected, since normal forces are predominant. This occurs when the shape of the shell walls and the load on it are such that a balance between external and internal forces is possible without the appearance of bending moments. The theory for calculating shells, based on the assumption that the normal stresses arising in the shell are constant throughout the thickness and, therefore, there is no bending of the shell, is called momentless theory of shells. The momentless theory works well if the shell does not have sharp transitions and hard pinches and, moreover, is not loaded with concentrated forces and moments. Moreover, this theory gives more accurate results, the smaller the shell wall thickness, i.e. the closer to the truth the assumption of a uniform distribution of stresses throughout the wall thickness.

In the presence of concentrated forces and moments, sharp transitions and pinching, solving the problem becomes much more complicated. In places where the shell is attached and in places of sudden changes in shape, increased stresses arise due to the influence of bending moments. In this case, the so-called moment theory of shell calculation. It should be noted that issues of the general theory of shells go far beyond the strength of materials and are studied in special sections of structural mechanics. In this manual, when calculating thin-walled vessels A momentless theory is considered for cases where the problem of determining the stresses acting in the meridional and circumferential sections turns out to be statically determinable.

13.2. Determination of stresses in symmetrical shells using the momentless theory. Derivation of Laplace's equation

Let us consider an axisymmetric thin-walled shell experiencing internal pressure from the weight of the liquid (Fig. 13.1, a). Using two meridional and two circumferential sections, we select an infinitesimal element from the shell wall and consider its equilibrium (Fig. 13.2).

In meridional and circumferential sections there are no tangential stresses due to the symmetry of the load and the absence of mutual displacements of the sections. Consequently, only the main normal stresses will act on the selected element: meridional stress
And hoop stress . Based on the momentless theory, we will assume that along the wall thickness the stress
And distributed evenly. In addition, we will refer all dimensions of the shell to the middle surface of its walls.

The middle surface of the shell is a surface of double curvature. Let us denote the radius of curvature of the meridian at the point under consideration
, the radius of curvature of the middle surface in the circumferential direction is denoted by . Forces act along the edges of the element
And
. Liquid pressure acts on the inner surface of the selected element , whose resultant is equal to
. Let us project the above forces onto the normal
to the surface:

Let us depict the projection of the element onto the meridional plane (Fig. 13.3) and, based on this figure, write the first term in expression (a). The second term is written by analogy.

Replacing the sine in (a) with its argument due to the smallness of the angle and dividing all terms of equation (a) by
, we get:

(b).

Considering that the curvatures of the meridional and circumferential sections of the element are equal, respectively
And
, and substituting these expressions into (b) we find:

. (13.1)

Expression (13.1) represents the equations of Laplace, named after the French scientist who obtained it at the beginning of the 19th century while studying surface tension in liquids.

Equation (13.1) includes two unknown voltages And
. Meridional stress
we will find by composing the equilibrium equation for the axis
forces acting on the cut-off part of the shell (Fig. 12.1, b). The circumferential area of ​​the shell walls is calculated using the formula
. Voltages
due to the symmetry of the shell itself and the load relative to the axis
distributed evenly over the area. Hence,

, (13.2)

Where - the weight of the part of the vessel and liquid lying below the section under consideration; fluid pressure, according to Pascal’s law, is equal in all directions and equal , Where depth of the section under consideration, and - weight per unit volume of liquid. If a liquid is stored in a vessel under some excess pressure compared to atmospheric , then in this case
.

Now knowing the tension
from the Laplace equation (13.1) one can find the voltage .

When solving practical problems, due to the fact that the shell is thin, instead of the radii of the middle surface
And substitute the radii of the outer and inner surfaces.

As already noted, circumferential and meridional stresses And
are the main stresses. As for the third principal stress, the direction of which is normal to the surface of the vessel, then on one of the surfaces of the shell (external or internal, depending on which side the pressure acts on the shell) it is equal to , and on the opposite – zero. In thin-walled shells, stress And
always much more . This means that the magnitude of the third principal stress can be neglected compared to And
, i.e. consider it equal to zero.

Thus, we will assume that the shell material is in a plane stressed state. In this case, to assess the strength depending on the state of the material, the appropriate strength theory should be used. For example, using the fourth (energy) theory, we write the strength condition in the form:

Let's consider several examples of calculations of momentless shells.

Example 13.1. A spherical vessel is under the influence of uniform internal gas pressure (Fig.13.4). Determine the stresses acting in the wall of the vessel and evaluate the strength of the vessel using the third theory of strength. We neglect the own weight of the walls of the vessel and the weight of the gas.

1. Due to the circular symmetry of the shell and the axisymmetric stress load And
are the same at all points of the shell. Assuming in (13.1)
,
, A
, we get:

. (13.4)

2. We carry out a test according to the third theory of strength:

.

Considering that
,
,
, the strength condition takes the form:

. (13.5)

Example 13.2. The cylindrical shell is under the influence of uniform internal gas pressure (Fig. 13.5). Determine the circumferential and meridional stresses acting in the wall of the vessel and evaluate its strength using the fourth theory of strength. Neglect the self-weight of the vessel walls and the weight of the gas.

1. Meridians in the cylindrical part of the shell are generatrices for which
. From Laplace’s equation (13.1) we find the circumferential stress:

. (13.6)

2. Using formula (13.2), we find the meridional stress, assuming
And
:

. (13.7)

3. To assess strength, we accept:
;
;
. The strength condition according to the fourth theory has the form (13.3). Substituting expressions for circumferential and meridional stresses (a) and (b) into this condition, we obtain

Example 12.3. A cylindrical tank with a conical bottom is under the influence of the weight of the liquid (Fig. 13.6, b). Establish the laws of changes in circumferential and meridional stresses within the conical and cylindrical part of the tank, find the maximum stresses And
and construct diagrams of stress distribution along the height of the tank. Neglect the weight of the tank walls.

1. Find the fluid pressure at depth
:

. (A)

2. We determine the circumferential stresses from the Laplace equation, taking into account that the radius of curvature of the meridians (generators)
:

. (b)

For the conical part of the shell

;
. (V)

Substituting (c) into (b) we obtain the law of change in circumferential stresses within the conical part of the tank:

. (13.9)

For the cylindrical part, where
the distribution law of circumferential stresses has the form:

. (13.10)

Diagram shown in Fig. 13.6, a. For the conical part, this diagram is parabolic. Its mathematical maximum occurs in the middle of the total height at
. At
it has a conditional meaning when
the maximum stress falls within the conical part and has a real value:

. (13.11)

3. Determine meridional stresses
. For a conical part, the weight of the liquid in the volume of a cone with a height is equal to:

. (G)

Substituting (a), (c) and (d) into the formula for meridional stresses (13.2), we obtain:

. (13.12)

Diagram
shown in Fig. 13.6, c. Plot maximum
, outlined for the conical part also along a parabola, occurs when
. It has real significance when
, when it falls within the conical part. The maximum meridional stresses are equal to:

. (13.13)

In the cylindrical part the voltage
does not change in height and is equal to the voltage at the upper edge at the place where the tank is suspended:

. (13.14)

In places where the surface of the tank has a sharp break, such as, for example, at the point of transition from a cylindrical part to a conical part (Fig. 13.7) (Fig. 13.5), the radial component of the meridional stresses
not balanced (Fig. 13.7).

This component along the perimeter of the ring creates a radial distributed load with an intensity
, tending to bend the edges of the cylindrical shell inward. To eliminate this bending, a stiffener (spacer ring) is installed in the form of an angle or channel that encircles the shell at the fracture site. This ring carries radial load (Fig. 13.8, a).

Let us cut out a part of it from the spacer ring using two infinitely closely spaced radial sections (Fig. 13.8b) and determine the internal forces that arise in it. Due to the symmetry of the spacer ring itself and the load distributed along its contour, shear force and bending moment in the ring do not occur. Only the longitudinal force remains
. Let's find her.

Let us compile the sum of the projections of all forces acting on the cut out element of the spacer ring onto the axis :

. (A)

Let's replace the sine of the angle angle due to its smallness
and substitute in (a). We get:

,

(13.15)

Thus, the spacer ring works in compression. The strength condition takes the form:

, (13.16)

Where radius of the midline of the ring; - cross-sectional area of ​​the ring.

Sometimes, instead of a spacer ring, a local thickening of the shell is created by bending the edges of the bottom of the tank into the shell.

If the shell experiences external pressure, then the meridional stresses will be compressive and the radial force will become negative, i.e. directed outward. Then the stiffening ring will work not in compression, but in tension. In this case, the strength condition (13.16) will remain the same.

It should be noted that installing a stiffening ring does not completely eliminate the bending of the shell walls, since the stiffening ring constrains the expansion of the shell rings adjacent to the rib. As a result, the forming shells near the stiffening ring are bent. This phenomenon is called the edge effect. It can lead to a significant local increase in stress in the shell wall. The general theory of taking into account the edge effect is discussed in special courses using the moment theory of calculating shells.

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St. Petersburg State Technological Institute (Technical University)

Hydraulics

Manual 578

The first training manual.
Issued at faculties 3 and 8.
Solving hydraulic problems 350 RUR. You can download the solution to problem 1 on hydraulics for free from this manual. Ready-made tasks from this manual are sold at a discount

Numbers of solved problems: 1 Download page 1 Download page 2, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 21, 22, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 35, 39, 43, 42, 44, 45, 46, 47, 50, 53, 54, 56, 57, 60, 61, 62, 65, 66, 68, 69, 74, 76, 80, 81, 83, 84, 85, 86, 89, 90, 93, 95, 97, 98, 99, 100, 101, 105, 109, 111, 112, 117, 120, 121, 129, 130, 133, 139, 140, 142, 152

Below are the conditions of solved hydraulic problems

Solved problems from 001 to 050

Conditions of problems 1-3: Three different devices for measuring pressure are attached to a tank filled with gasoline: a spring pressure gauge, a piezometric tube and a two-arm pressure gauge filled with gasoline, water and mercury. What advantage in operation does a two-arm pressure gauge provide compared to a piezometric tube at a given position of the levels?

Conditions of problems 4-7: Two reservoirs filled with alcohol and water are connected to each other by a three-arm pressure gauge, which contains alcohol, mercury, water and air. The position of liquid levels is measured relative to one common plane. The alcohol level in the left tank is h1=4m, the water level in the right tank is h6=3m. The pressure in the tanks is controlled using a pressure gauge and a vacuum gauge.

Conditions of problems 8-11: A mixture of oil and water in a volumetric ratio of 3:1 is poured into the settling tank under pressure controlled using a spring pressure gauge. Liquid levels and interfaces are determined using two measuring glasses; the first contains both liquids, the second only water. The interface between oil and water in the settling tank was established at a height of 0.2 m.

Conditions of problems 12-13: Pressure P at the surface of the water in the tank is measured with a mercury U-shaped manometer. Water density 1000 kg/m3; mercury 13600 kg/m3.

Conditions of problems 14-20: A cylindrical vessel with a diameter of 0.2 m, a height of 0.4 m is filled with water and rests on a plunger with a diameter of 0.1 m. The mass of the vessel lid is 50 kg, the cylindrical part is 100 kg, and the bottom is 40 kg. The pressure in the vessel is determined using a spring pressure gauge. The density of water is 1000kg/m^3.

Conditions of problems 21-22: A cylindrical vessel was initially installed on a fixed support and filled with water to the level with the top valve open. The valve was then closed and the support was removed. At the same time, the vessel dropped along the plunger to an equilibrium position, compressing the air cushion formed inside.

Conditions of problems 23-28: A tube is attached to a closed cylindrical vessel with a diameter of 2 m and a height of 3 m, the lower end of which is lowered below the liquid level in an open tank. The internal volume of the vessel can communicate with the atmosphere through valve 1. A valve 2 is also installed on the lower tube. The vessel is located at a height above the surface of the liquid in the tank and is initially filled with water through valve 1 to a level of 2 m with valve 2 closed (the pressure in the gas cushion is atmospheric) . Then the top tap is closed and the bottom tap is opened, and part of the liquid is drained into the reservoir. The gas expansion process is considered isothermal.

Conditions of problems 29-32: Two vessels whose cross-sectional areas are connected to each other horizontal pipe, inside which a piston of area can move freely without friction.

Conditions of problems 33-38: A cylindrical vessel with a diameter of 0.4 m is filled with water to a level of 0.3 m and hangs without friction on a plunger with a diameter of 0.2 m. The mass of the cover is 10 kg, the cylinder is 40 kg, and the bottom is 12 kg.

Conditions of problems 39-44: A thick-walled bell weighing 1.5 tons floats at atmospheric pressure on the surface of a liquid. The inner diameter of the bell is 1 m, the outer diameter is 1.4 m, its height is 1.4 m.

Conditions of problems 45-53: A vessel consisting of two cylinders, its lower end is lowered below the water level in tank A and rests on supports C located at height B above the level of the free surface of the liquid in the tank.

Goal: to form an understanding of the features of deformation and strength calculations of thin-walled shells and thick-walled cylinders.

Calculation of thin-walled shells

Shell - it is a structural element limited by surfaces located at close distances from each other. A shell is called thin-walled if the condition is satisfied for it p/h> 10, where h- shell thickness; p- the radius of curvature of the middle surface, which is the locus of points equidistant from both surfaces of the shell.

Parts whose shape model takes the shell include car tires, vessels, internal combustion engine liners, load-bearing car bodies, aircraft fuselages, ship hulls, floor domes, etc.

It should be noted that shell structures are optimal in many cases, since a minimum of materials are spent on their production.

A characteristic feature of most thin-walled shells is that in shape they are bodies of revolution, that is, each of their surfaces can be formed by rotating a certain curve (profile) around a fixed axis. Such bodies of revolution are called axisymmetric. In Fig. 73 shows a shell, the middle surface of which is obtained by rotating the profile Sun around the axis AC.

Let us select from the middle surface in the vicinity of the point TO., lying on this surface, an infinitesimal element 1122 two meridional planes AST And AST 2 s angle d(p between them and two sections normal to the meridians HO t And 220 2 .

Meridional called a section (or plane) passing through the axis of rotation AC. Normal called a section perpendicular to the meridian Sun.

Rice. 73.

Normal sections for the vessel in question are conical surfaces with vertices 0 And Oh g, lying on the axis AC.

Let us introduce the following notation:

r t- radius of curvature of the arc 12 in the meridional section;

p,- radius of curvature of the arc 11 in a normal section.

In general r t And p, are a function of the angle V- angle between axis AC and normal 0,1 (see Fig. 73).

A peculiarity of the operation of shell structures is that all its points, as a rule, are in a complex stress state and strength theories are used to calculate shells.

To determine the stresses arising in a thin-walled shell, the so-called momentless theory. According to this theory, it is believed that there are no bending moments among the internal forces. The walls of the shell work only in tension (compression), and the stresses are evenly distributed throughout the thickness of the wall.

This theory applies if:

• 1) the shell is a body of revolution;
• 2) shell wall thickness S very small compared to the radii of curvature of the shell;
• 3) the load, gas or hydraulic pressure is distributed polarly symmetrically relative to the axis of rotation of the shell.

The combination of these three conditions allows us to accept the hypothesis that the stress is constant across the wall thickness in a normal section. Based on this hypothesis, we conclude that the walls of the shell work only in tension or compression, since bending is associated with an uneven distribution of normal stresses across the wall thickness.

Let us establish the position of the main areas, i.e. those areas (planes) in which there are no tangential stresses (m = 0).

It is obvious that any meridional section divides the thin-walled shell into two parts, symmetrical both in geometric and force relationships. Since neighboring particles are deformed equally, there is no shear between the sections of the resulting two parts, which means that there are no tangential stresses in the meridional plane (m = 0). Therefore, it is one of the main platforms.

Due to the law of pairing, there will be no tangential stresses in sections perpendicular to the meridional section. Therefore, the normal section (platform) is also the main one.

The third main platform is perpendicular to the first two: at the outer point TO(see Fig. 73) it coincides with the side surface of the shell, in it r = o = 0, thus, in the third main area o 3 = 0. Therefore, the material at the point TO experiences a plane stress state.

To determine the principal stresses, we select a point in the vicinity TO infinitesimal element 1122 (see Fig. 73). Only normal stresses a„ and o, arise on the edges of the element. The first one a t called meridional, and the second A, - circumferential stress, which are the principal stresses at a given point.

Voltage vector A, directed tangent to the circle obtained from the intersection of the middle surface with a normal section. The voltage vector o„ is directed tangentially to the meridian.

Let us express the main stresses through the load (internal pressure) and the geometric parameters of the shell. To determine a t And A, two independent equations are needed. The meridional stress o„ can be determined from the equilibrium condition of the cut-off part of the shell (Fig. 74, A):

Substituting Mr. t sin 9, we get

The second equation is obtained from the equilibrium condition of the shell element (Fig. 74, b). If we project all the forces acting on the element onto the normal and equate the resulting expression to zero, we get

Due to small angles we accept

As a result of the mathematical transformations we get the following equation:

This equation is called Laplace equations and establishes the relationship between meridian and circumferential stresses at any point of a thin-walled shell and internal pressure.

Since the dangerous element of the thin-walled shell is in a plane stressed state, based on the results obtained with t And a h and also based on the dependence

Rice. 74. Fragment of a thin-walled axisymmetric shell: A) loading scheme; b) stresses acting along the edges of the selected shell element

So, according to the third theory of strength: a" 1 =&-st b

Thus, for cylindrical vessels of radius G and wall thickness AND we get

based on the equilibrium equation of the cut-off part, A"

therefore, a, a m, = 0.

When the maximum pressure is reached, the cylindrical vessel (including all pipelines) collapses along the generatrix.

For spherical vessels (r, = r t = g) application of Laplace's equation gives the following results:

_ R g rg _ rg

o, = o t =-, hence, = a 2 = u„= -,

2 h 2 h 2 h

From the results obtained, it becomes obvious that compared to a cylindrical vessel, a spherical one is more optimal design. The maximum pressure in a spherical vessel is twice as high.

Let's look at examples of calculating thin-walled shells.

Example 23. Determine required thickness receiver walls, if the internal pressure p- 4 atm = 0.4 MPa; R= 0.5 m; [a]= 100 MPa (Fig. 75).

Rice. 75.

• 1. In the wall of the cylindrical part, meridian and circumferential stresses arise, related by the Laplace equation: a t o, R
• -+-=-. It is necessary to find the wall thickness p.

RT P, h

2. Stressed state of the point IN - flat.

Strength condition: er" =cr 1 -et 3?[

• 3. It is necessary to express And o$ through sg„ And A, in letter form.
• 4. Size A", can be found from the equilibrium condition of the cut-off part of the receiver. Voltage value A, - from the Laplace condition, where r t = co.
• 5. Substitute the found values ​​into the strength condition and express the value through them AND.
• 6. For the spherical part, wall thickness h is determined similarly, taking into account p„= p,- R.

1. For a cylindrical wall:

Thus, in the cylindrical part of the receiver o, > o t and 2 times.

Thus, h= 2 mm - thickness of the cylindrical part of the receiver.

Thus, h 2 = 1 mm is the thickness of the spherical part of the receiver.

Calculation of thin-walled vessels using the momentless theory

Task 1.

The air pressure in the cylinder of the shock-absorbing strut of the aircraft landing gear in the parked position is equal to p = 20 MPa. Cylinder diameter d =….. mm, wall thickness t =4 mm. Determine the main stresses in the cylinder at rest and after takeoff, when the pressure in the shock absorber is ………………….

Answer: (in the parking lot); (after takeoff).

Task 2.

Water enters water turbine through a pipeline whose outer diameter at the machine building is equal to .... m, and the wall thickness t =25 mm. The machine building is located 200 m below the level of the lake from which water is drawn. Find the greatest voltage in ……………………….

Answer:

Task 3.

Check the strength of the wall …………………………… with a diameter of ….. m, under operating pressure p = 1 MPa, if the wall thickness t =12 mm, [σ]=100 MPa. Apply IV strength hypothesis.

Answer:

Task 4.

The boiler has a cylindrical diameter d =…. m and is under operating pressure p=….. MPa. Select the thickness of the boiler wall at the permissible stress [σ]=100 MPa, using III strength hypothesis. What would be the required thickness when using IV strength hypotheses?

Answer:

Task 5.

Steel spherical shell diameter d =1 m and thickness t =…. mm is loaded with internal pressure p = 4 MPa. Determine………………tension and………………..diameter.

Answer: mm.

Task 6.

Cylindrical vessel with diameter d =0.8 m has a wall thickness t =... mm. Determine the permissible pressure in the vessel based on IV strength hypothesis if [σ]=…… MPa.

Answer: [p ]=1.5 MPa.

Task 7.

Define ………………………….. material of a cylindrical shell, if, when loaded with internal pressure, the deformations in the direction of the sensors amounted to

Answer: ν=0.25.

Task 8.

Thick duralumin pipemm and inner diametermm reinforced with a thick steel jacket tightly placed on itmm. Find the limit ………………………..for a two-layer pipe according to the yield strength and ……………… stress between the layers at this moment, assuming E st = 200 GPa,E d =70 GPa,

Answer:

Task 9.

Conduit diameter d =…. mm during the launch period had a wall thickness t =8 mm. During operation, due to corrosion, the thickness in places……………………... What is the maximum column of water that a pipeline can withstand with a double safety margin, if the yield strength of the pipe material is

Problem 10.

Gas pipeline diameter d =……. mm and wall thickness t = 8 mm crosses the reservoir at a maximum ………………………….., reaching 60 m. During operation, gas is pumped under pressure p = 2.2 MPa, and during the construction of an underwater crossing there is no pressure in the pipe. What are they equal to? highest stress in the pipeline and when do they occur?

Problem 11.

A thin-walled cylindrical vessel has hemispherical bottoms. What should be the ratio between the thicknesses of the cylindrical and spherical parts so that in the transition zone there is no………………….?

Problem 12.

When manufacturing railway tanks, they are tested under pressure p = 0.6 MPa. Determine ………………………… in the cylindrical part and in the bottom of the tank, taking the test pressure as the calculated one. Calculate according to III strength hypotheses.

Problem 13.

Between two concentrically located bronze pipes a liquid flows under pressure p = 6 MPa. The thickness of the outer pipe isAt what thickness of the inner pipeis provided by …………………….. of both pipes? What are the highest voltages in this case?

Problem 14.

Determine ………………………… of the shell material if, when loaded with internal pressure, the deformation in the direction of the sensors was

Problem 15.

Thin-walled spherical vessel with diameter d =1 m and thickness t =1 cm is under internal pressure and external What is the ………………….. of the vessel P t if

Would the following solution be correct:

Problem 16.

A thin-walled pipe with plugged ends is under the influence of internal pressure p and bending moment M. Using III strength hypothesis, investigate …………………… stressesfrom the value of M for a given r.

Problem 17.

At what depth are the points with ………………….. meridional and circumferential stresses for the conical vessel shown on the right? Determine the magnitude of these stresses, assuming the specific gravity of the product is equal to γ=…. kN/m 3 .

Problem 18.

The vessel is subjected to gas pressure p = 10 MPa. Find………………………if [σ ]=250 MPa.

Answer: t =30 mm.

Problem 19.

A vertically standing cylindrical tank with a hemispherical bottom is filled to the top with water. Thickness of side walls and bottom t =2 mm. Define ………………………. stresses in the cylindrical and spherical parts of the structure.

Answer:

Problem 20.

A cylindrical reservoir is filled to a depth of H 1 = 6 m with liquid of specific gravityand on top - to a thickness of H 2 = 2 m - with water. Determine …………………….. of the tank at the bottom if [σ ]=60 MPa.

Answer: t =5 mm.

Problem 21.

A small gas holder for lighting gas has wall thickness t =5 mm. Find ……………… of the upper and lower vessels.

Answer:

Problem 22.

The valve float of the testing machine is a closed cylinder made of aluminum alloy with a diameter d =…..mm. The float is subjected to………………………pressure р =23 MPa. Determine the thickness of the float wall using the fourth strength hypothesis, if [σ]=200 MPa.

Answer: t =5 mm.

Problem 23.

Thin-walled spherical vessel with a diameter d =1 m and thickness t =1 cm is under the influence of internal ……………… and external What is the ……………….. of the vessel walls If

Answer: .

Problem 24.

Determine the maximum ………………… and circumferential stresses in a toroidal cylinder if p=…. MPa, t =3 mm, A=0.5 mm; d =0.4 m.

Answer:

Problem 25.

Steel hemispherical vessel of radius R =... m is filled with liquid with a specific gravity γ = 7.5 kN/m 3. Taking …………………………. 2 mm and using III using the strength hypothesis, determine the required thickness of the vessel wall if [σ]=80 MPa.

Answer: t =3 mm.

Problem 26.

Determine …………………… the points with the highest meridional and circumferential stresses and calculate these stresses if the wall thickness t =... mm, specific gravity of the liquid γ = 10 kN/m 3.

Answer: at a depth of 2 m; at a depth of 4 m.

Problem 27.

A cylindrical vessel with a conical bottom is filled with liquid with a specific gravity γ = 7 kN/m 3. The wall thickness is constant and equal t =...mm. Define …………………………….. and circumferential stresses.

Answer:

Problem 28.

A cylindrical vessel with a hemispherical bottom is filled with liquid with a specific gravity γ = 10 kN/m 3. The wall thickness is constant and equal t =... mm. Determine the maximum stress in the vessel wall. How many times will this voltage increase if the length………………………………, keeping all other dimensions constant?

Answer: will increase by 1.6 times.

Problem 29.

To store oil with a specific gravity γ = 9.5 kN/m 3, a vessel in the form of a truncated cone with a wall thickness is used t =10 mm. Determine the largest …………………………. stress in the vessel wall.

Answer:

Problem 30.

The thin-walled conical bell is located under a layer of water. Determine …………………………….. and hoop stresses if air pressure on the surface under the bell wall thickness t = 10 mm.

Answer:

Problem 31.

Shell thickness t =20 mm, shaped like an ellipsoid of rotation (Ox – axis of rotation), loaded with internal pressure р=…. MPa. Find ………………….. in longitudinal and transverse sections.

Answer:

Problem 32.

Using the third strength hypothesis, check the strength of a vessel shaped like a paraboloid of revolution with a wall thickness t =... mm, if the specific gravity of the liquid is γ = 10 kN/m 3, the permissible stress [σ] = 20 MPa, d = h =5 m. Check strength by height……………………………...

Answer: those. strength is guaranteed.

Problem 33.

A cylindrical vessel with spherical bottoms is designed to store gas under pressure p =... MPa. Under …………………, will it be possible to store gas in a spherical vessel of the same capacity with the same material and wall thickness? What kind of material savings does this achieve?

Answer: savings will be 36%.

Problem 34.

Cylindrical shell with wall thickness t =5 mm compressed by force F =….. kN. Due to manufacturing inaccuracies, the forming shells received little…………………………. Neglecting the influence of this curvature on the meridional stresses, calculatein the middle of the height of the shell, assuming that the generators are curved along one half-wave of the sinusoid, and f =0.01 l; l= r.

Answer:

Problem 35.

A vertical cylindrical vessel is designed to store liquid volume V And specific gravityγ. The total thickness of the upper and lower bases, assigned for design reasons, is equal toDetermine the most favorable height of the tank H opt, at which the mass of the structure will be minimal.Taking the height of the tank equal to H opt, find ………………………….. parts, assuming [σ]=180 MPa, Δ=9 mm, γ=10 kN/m 3, V =1000 m 3.

Answer: N opt =9 m, mm.

Problem 36.

Long thin tube thick t =…. mm is placed with a tightness Δ on an absolutely rigid rod of diameter d =…..mm . …………… must be applied to the tube to remove it from the rod if Δ=0.0213 mm; f =0.1; l=10 cm, E=100 GPa, ν=0.35.

Answer: F =10 kN.

Problem 37.

A thin-walled cylindrical vessel with spherical bottoms is subjected from the inside to gas pressure p = 7 MPa. By ……………………………….. diameter E 1 =E 2 =200 GPa.

Answer: N 02 =215 N.

Problem 38.

Among others structural elements Cylinders are used in aviation and rocketry high pressure. They usually have a cylindrical or spherical shape and for them, as for other structural units, it is extremely important to comply with the minimum weight requirement. The design of the shaped cylinder shown in the figure is proposed. The walls of the cylinder consist of several cylindrical sections connected by radial walls. Since the cylindrical walls have a small radius, the stress in them decreases, and it can be hoped that despite the increase in weight due to the radial walls, the total weight of the structure will be less than for an ordinary cylinder having the same volume……………………… …….?

Problem 39.

Determine ……………………… of a thin-walled shell of equal resistance containing liquid of specific gravity γ.

Calculation of thick-walled pipes

Task 1.

What is the pressure (internal or external)……………………. pipes? How many times are the greatest equivalent stresses according to III hypothesis of strength in one case more or less than in another if the pressure values ​​are the same? Will the largest radial displacements be equal in both cases?

Task 2.

The two pipes differ only in size cross section: 1st pipe – A=20 cm, b =30 cm; 2nd pipe – A=10 cm, b =15 cm. Which of the pipes has ……………………… ability?

Task 3.

Thick wall pipe with dimensions A=20 cm and b =40 cm cannot withstand the set pressure. In order to increase the load-bearing capacity, two options are proposed: 1) increase the outer radius by P times b ; 2) reduce the internal radius by P times A. Which option gives ……………………………. at same value P?

Task 4.

Pipe with dimensions A=10 cm and b =20 cm withstands pressure p=….. MPa. How much (in percent) ……………….. is the load-bearing capacity of the pipe if the outer radius is increased by … times?

Task 5.

At the end of the First World War (1918), Germany manufactured an ultra-long-range cannon for shelling Paris from a distance of 115 km. It was steel pipe 34 m long and 40 cm thick at the breech. The gun weighed 7.5 MN. Its 120-kilogram projectiles were a meter long with a diameter of 21 cm. The charge used 150 kg of gunpowder, which developed a pressure of 500 MPa, which ejected the projectile with an initial speed of 2 km/s. What should be the ……………………………. used to make a gun barrel, if not less than one and a half times the safety margin?

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Problem 1

Determine the difference in piezometer levels h.

The system is in equilibrium.

The piston area ratio is 3. H= 0.9 m.

Liquid water.

Problem 1.3

Determine the level difference h in piezometers when the multiplier pistons are in equilibrium, if D/d = 5, H= 3.3 m. Build a graph h = f(D/d), If D/d= 1.5 ÷ 5.

Problem 1. 5

A thin-walled vessel consisting of two cylinders with diameters d= 100 mm and D= 500 mm, the lower open end is lowered below the water level in tank A and rests on supports C located at a height b= 0.5 m above this level.

Determine the magnitude of the force perceived by the supports if a vacuum is created in the vessel, causing the water in it to rise to a height a + b= 0.7 m. Own weight of the vessel G= 300 N. How does a change in diameter affect the result? d?

Problem 1.7

Determine the absolute air pressure in the vessel if the reading of the mercury device h= 368 mm, height H= 1 m. Density of mercury ρ rt = 13600 kg/m 3. Atmospheric pressure p atm = 736 mm Hg. Art.

Problem 1.9

Determine the pressure above the piston p 01, if known: forces on the pistons P 1 = 210 N, P 2 = 50 N; instrument reading p 02 = 245.25 kPa; piston diameters d 1 = 100 mm, d 2 = 50 mm and height difference h= 0.3 m. ρ Hg /ρ = 13.6.

Problem 1.16

Determine pressure p in the hydraulic system and load weight G lying on the piston 2 , if to lift it to the piston 1 force applied F= 1 kN. Piston diameters: D= 300 mm, d= 80 mm, h= 1 m, ρ = 810 kg/m3. Build a graph p = f(D), If D varies from 300 to 100 mm.

Problem 1.17.

Determine the maximum height N max , to which gasoline can be sucked in by a piston pump if its saturated vapor pressure is h n.p. = 200 mmHg Art., a atmospheric pressure h a = 700 mm Hg. Art. What is the force along the rod if N 0 = 1 m, ρ b = 700 kg/m 3 ; D= 50 mm?

Build a graph F = ƒ( D) when changing D from 50 mm to 150 mm.

Problem 1.18

Determine diameter D 1 hydraulic cylinder required to lift the valve when there is excess fluid pressure p= 1 MPa, if the pipeline diameter D 2 = 1 m and mass of moving parts of the device m= 204 kg. When calculating the friction coefficient of the valve in the guide surfaces, take f= 0.3, the friction force in the cylinder is considered equal to 5% of the weight of the moving parts. The pressure behind the valve is equal to atmospheric pressure; neglect the influence of the stem area.

Build a dependency graph D 1 = f(p), If p varies from 0.8 to 5 MPa.

Problem 1.19

When the hydraulic accumulator is charged, the pump supplies water to cylinder A, lifting plunger B along with the load upward. When the battery is discharged, the plunger, sliding down, squeezes water out of the cylinder under the influence of gravity into hydraulic presses.

1. Determine the water pressure when charging p z (developed by the pump) and discharge p p (obtained by the presses) of the battery, if the mass of the plunger together with the load m= 104 t and plunger diameter D= 400 mm.

The plunger is sealed with a cuff, the height of which b= 40 mm and coefficient of friction on the plunger f = 0,1.

Build a graph p z = f(D) And p p = f(D), If D varies from 400 to 100 mm, the mass of the plunger with the load is considered unchanged.

Problem 1.21

In a sealed container A there is molten babbitt (ρ = 8000 kg/m3). When the vacuum gauge shows p vac = 0.07 MPa filling the ladle B stopped. At the same time H= 750 mm. Determine the height of the babbitt level h in the feeder vessel A.

Problem 1.23

Define strength F necessary to keep the piston at a height h 2 = 2 m above the surface of the water in the well. A column of water rises above the piston as high as h 1 = 3 m. Diameters: piston D= 100 mm, rod d= 30 mm. Ignore the weight of the piston and rod.

Problem 1.24

The vessel contains molten lead (ρ = 11 g/cm3). Determine the pressure force acting on the bottom of the vessel if the height of the lead level is h= 500 mm, vessel diameter D= 400 mm, pressure and vacuum gauge reading p vac = 30 kPa.

Construct a graph of the pressure force versus the diameter of the vessel if D varies from 400 to 1000 mm

Problem 1.25

Determine pressure p 1 fluid that must be supplied to the hydraulic cylinder to overcome the force directed along the rod F= 1 kN. Diameters: cylinder D= 50 mm, rod d= 25 mm. Tank pressure p 0 = 50 kPa, height H 0 = 5 m. Ignore friction force. Liquid density ρ = 10 3 kg/m 3.

Problem 1.28

The system is in equilibrium. D= 100 mm; d= 40 mm; h= 0.5 m.

What force must be applied to pistons A and B if a force acts on piston C P 1 = 0.5 kN? Ignore friction. Build a dependency graph P 2 from diameter d, which varies from 40 to 90 mm.

Problem 1.31

Define strength F on the spool rod if the vacuum gauge reading p vac = 60 kPa, overpressure p 1 = 1 MPa, height H= 3 m, piston diameters D= 20 mm and d= 15 mm, ρ = 1000 kg/m 3.

Build a graph F = f(D), If D varies from 20 to 160 mm.

Problem 1.32

A system of two pistons connected by a rod is in equilibrium. Define strength F, compressing the spring. The liquid located between the pistons and in the tank is oil with a density ρ = 870 kg/m 3. Diameters: D= 80 mm; d= 30 mm; height N= 1000 mm; overpressure r 0 = 10 kPa.

Problem 1.35

Define load P on the cover bolts A And B hydraulic cylinder diameter D= 160 mm, if to a plunger with a diameter d= 120 mm applied force F= 20 kN.

Build a dependency graph P = f(d), If d varies from 120 to 50 mm.

Task1.37

The figure shows the design diagram of a hydraulic lock, the flow section of which opens when fed into the cavity A control fluid flow with pressure p y. Determine at what minimum value p y piston pusher 1 will be able to open the ball valve if the spring preload is known 2 F= 50 H; D = 25 mm, d = 15 mm, p 1 = 0.5 MPa, p 2 = 0.2 MPa. Neglect frictional forces.

Problem 1.38

Determine gauge pressure p m, if the force on the piston P= 100 kgf; h 1 = 30 cm; h 2 = 60 cm; piston diameters d 1 = 100 mm; d 2 = 400 mm; d 3 = 200 mm; ρ m /ρ in = 0.9. Define p m.

Problem 1.41

Determine the minimum force value F, applied to the rod, under the influence of which a piston with a diameter of D= 80 mm, if the spring force pressing the valve to the seat is equal to F 0 = 100 H, and fluid pressure p 2 = 0.2 MPa. Valve inlet diameter (seat) d 1 = 10 mm. Rod diameter d 2 = 40 mm, fluid pressure in the rod cavity of the hydraulic cylinder p 1 = 1.0 MPa.

Problem 1.42

Determine the amount of preload of the differential spring safety valve(mm), ensuring that the valve begins to open at p n = 0.8 MPa. Valve diameters: D= 24 mm, d= 18 mm; spring stiffness With= 6 N/mm. The pressure to the right of the larger and to the left of the small pistons is atmospheric.

Problem 1.44

In a hydraulic jack with manual drive(Fig. 27) at the end of the lever 2 force applied N= 150 N. Pressure diameters 1 and lifting 4 plungers are respectively equal: d= 10 mm and D= 110 mm. Small lever arm With= 25 mm.

Taking into account the general efficiency of the hydraulic jack η = 0.82, determine the length l lever 2 sufficient to lift the load 3 weighing 225 kN.

Build a dependency graph l = f(d), If d varies from 10 to 50 mm.

Task 1.4 5

Determine height h column of water in a piezometric tube. A column of water balances a full piston with D= 0.6 m and d= 0.2 m, having a height H= 0.2 m. Neglect the self-weight of the piston and friction in the seal.

Build a graph h = f(D), if the diameter D varies from 0.6 to 1 m.

Problem 1.51

Determine the diameter of the piston = 80.0 kg; depth of water in cylinders H= 20 cm, h= 10 cm.

Build dependency P = f(D), If P= (20...80) kg.

Problem 1.81

Determine the reading of a two-fluid pressure gauge h 2, if the pressure on the free surface in the tank p 0 abs = 147.15 kPa, water depth in the tank H= 1.5 m, distance to mercury h 1 = 0.5 m, ρ rt / ρ in = 13.6.

Problem 2.33

Air is sucked in by the engine from the atmosphere, passes through an air cleaner and then through a pipe with a diameter of d 1 = 50 mm supplied to the carburetor. Air density ρ = 1.28 kg/m3. Determine the vacuum in the diffuser neck with diameter d 2 = 25 mm (section 2–2) at air flow Q= 0.05 m 3 /s. Accept the following resistance coefficients: air cleaner ζ 1 = 5; knees ζ 2 = 1; air damper ζ 3 = 0.5 (related to the speed in the pipe); nozzle ζ 4 = 0.05 (related to the velocity at the diffuser neck).

Problem 18

To weigh heavy loads 3 weighing from 20 to 60 tons, a hydrodynamometer is used (Fig. 7). Piston 1 diameter D= 300 mm, rod 2 diameter d= 50 mm.

Neglecting the weight of the piston and rod, construct a graph of pressure readings r pressure gauge 4 depending on weight m cargo 3.

Problem 23

In Fig. Figure 12 shows a diagram of a hydraulic valve with a spool diameter d= 20 mm.

Neglecting friction in the hydraulic valve and the weight of spool 1, determine the minimum force that the compressed spring 2 must develop to balance the oil pressure in the lower cavity A r= 10 MPa.

Draw a graph of spring force versus diameter d, If d varies from 20 to 40 mm.

Problem 25

In Fig. Figure 14 shows a diagram of a hydraulic distributor with a flat valve of 2 diameters d= 20 mm. In the pressure cavity IN hydraulic valve operates oil pressure p= 5 MPa.

Neglecting back pressure in the cavity A hydraulic distributor and the force of a weak spring 3, determine the length l lever arm 1, sufficient to open the flat valve 2 applied to the end of the lever by force F= 50 N if the length of the small arm a= 20 mm.

Build a dependency graph F = f(l).

Problem 1.210

In Fig. Figure 10 shows a diagram of a plunger pressure switch, in which, when plunger 3 moves to the left, pin 2 rises, switching electrical contacts 4. Spring stiffness coefficient 1 WITH= 50.26 kN/m. The pressure switch is activated, i.e. switches electrical contacts 4 with an axial deflection of spring 1 equal to 10 mm.

Neglecting friction in the pressure switch, determine the diameter d plunger, if the pressure switch should operate at oil pressure in cavity A (at exit) r= 10 MPa.

TaskI.27

A hydraulic intensifier (a device for increasing pressure) receives water from the pump overpressure p 1 = 0.5 MPa. In this case, the movable cylinder filled with water A with outer diameter D= 200 mm slides on a stationary rolling pin WITH, having a diameter d= 50 mm, creating pressure at the outlet of the multiplier p 2 .

Determine pressure p 2, taking the friction force in the seals equal to 10% of the force developed on the cylinder by pressure p 1, and neglecting the pressure in the return line.

Weight of moving parts of the multiplier m= 204 kg.

Build a dependency graph p 2 = f(D), If D varies from 200 to 500 mm, m, d, p 1 are considered constant.

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