The structural forms of modern machines and structures are extremely diverse. The choice of the shape of a part, assembly or structure is determined by many factors: their purpose, operating conditions, manufacturing technology, cost, as well as calculation methods. One of the most common types of modern and promising structures are thin-walled shells. Thin plates and shells find extremely wide application in the construction of a wide variety of engineering structures. For this reason, the creation of reliable, perfect structures directly depends on the level of development of the theory of thin plates and shells.
Thin shell can be defined as a body bounded by two curved surfaces, the distance between which is small compared to other dimensions. Thus, shell structures are characterized by thinness .
Shells include, in particular, thin-walled spatial systems outlined along curved surfaces. Shells are able to withstand various types of loads and provide insulation from the environment. They can be given a streamlined shape and based on them, relatively lightweight structures can be obtained, which is of great importance in the aerospace industry
Reducing the material consumption of a structure is an important factor for many machines and units. This is also beneficial in building structures. Shells make it possible to effectively solve the problem of mass minimization.
Nowadays shells can be seen everywhere. High-rise buildings and television towers, sports and concert complexes, indoor stadiums and markets, tanks and reservoirs, pipelines and cooling towers, airplanes and missiles, surface and underwater ships, and cars largely consist of shells. Transport structures are characterized not only by the ability to achieve high speeds, aerodynamic perfection of shape, and load capacity. They also embody the ideas of optimality, economy, and weight perfection.
Shells as structural elements have been known for a long time. This is both a steam boiler and water supply in ancient Rome. Since ancient times, containers for storing liquids and grains, and curved ceiling vaults in construction have been known. But shells have begun to play a decisive role in various areas of modern technology over the past few decades.
The term " shell" is one of the overloaded ones and can be given different meanings. In what follows, shells are understood as structures capable of performing power, operational, technological, architectural and aesthetic functions.
In mathematical modeling, the concept of a shell is primarily associated with the idea of geometric surface . In the mechanics of deformable solids and structural mechanics, the classification of objects (bodies) is based on the features of their shape and the ratio of characteristic sizes.
It is customary to distinguish and highlight structural elements, one size of which is much larger than the other two. These are rods, rings, arches. Bodies in which one size is much smaller than the others form the class of shells and plates.
The main problem of the theory of thin elastic shells is to reduce the three-dimensional problem of the theory of elasticity to a two-dimensional problem. Thus, the development of the general theory of thin elastic plates and shells follows the path of reducing the three-dimensional equations of the theory of elasticity to two-dimensional ones. To solve this problem, a large number of methods have been proposed, which, according to the classification of S.A. Ambartsumyan can be combined into three groups: the hypothesis method, the method of expanding the general equations of the theory of elasticity over the thickness of the shell, and the asymptotic method. All these methods are being intensively developed, complementing each other.
List of symbols
a 1 , a 2 - curvilinear orthogonal coordinates of the middle surface S o of the shell on the lines of principal curvatures; for a shell of revolution a 1 ─ longitudinal, a 2 - circumferential coordinates; z ─ normal coordinate
to S;A 1 , A 2 - Lame coefficients; k 1, k 2 - principal curvatures;
U, V, W - components of the displacement vector of an arbitrary point of the shell;
u, v, w are the components of the displacement vector of surface points S o ;
q 1, q 2 - angles of rotation of the normal
;e jk - strain tensor components;
E 11 , E 22 , E 12 - components of tangential deformation on S: tension-compression in the directions of coordinates a 1 and a 2 and shear;
K 11, K 22, K 12 - components of flexural deformation: changes in principal curvatures and torsion;
T 11, T 22, S - tangential internal forces reduced to S o: tension-compression and shear forces;
M 11, M 22, H - bending and torque moments;
Q 11, Q 22 - shearing forces;
q 1 , q 2 , q 3 - components of the external surface load, reduced to S;
E, n - Young's modulus and Poisson's ratios of the shell material;
y j - unified designations of the main independent variables in resolving systems of ordinary differential equations (ODE);
f j - operators of the right sides of canonical ODE systems;
Let us consider an element of an arbitrary thin shell, let in what follows
h is the thickness of the shell, which is assumed to be constant in the future.
Let us denote by R 1, R 2 the main radii of curvature of the middle surface of the shell S. R=min (R 1, R 2).
The main geometric parameter of the shell is the thin-wall parameter or relative thickness, determined by the ratio e=h/R.
A fairly conventional classification of shells according to their thickness into thin, medium-length and thick shells has been adopted.
We will consider the shell thin if its relative thickness is significantly less than unity. Shells are usually considered thin at e<1/20. Значения 1/20 < e < 1/10 соответствуют оболочке средней толщины, а e >1/10 - thick shell.
For open shells, you can set the characteristic size to size a. Then the thin-wall parameter can be defined as e = min (h/a, h/R).
The surface of the shell S, equidistant from the front surfaces S + and S - is called its middle surface.
Curvilinear, orthogonal coordinate systems
The rule for differentiating the basis vectors of a curvilinear orthogonal coordinate system is defined as follows:
e s,t = - (H t,s /H s) e t - d st ÑH t
Ñ = e m (…), m / H m
Here H m are the Lame parameters of the coordinate system, having the form
= (r, i) 2; Hi = ½ r, i ½ .
Here r, I - radius is the vector of an arbitrary point of the shell body. In particular:
e 1.1 = (H 1.1 /H 1) e 1 - (H 1.1 /H 1) e 1 - (H 1.2/H 2) e 2 - (H 1.3 /H 3) e 3
e 1.2 = (H 2.1 /H 1) e 2 ; e 3.2 = (H 2.3 /H 3) e 2 ; H i (a 1 , a 2 , a 3)
Let us write down the compatibility condition, which in the accepted notation has the form:
(e 1,1), 2 = (e 1,2), 1
(e 1.2), 1 = ((H 2.1 /H 1) e 2), 1 = (H 2.1/ H 1), 1 e 2 + (H 2.1 /H 1) (H 1.2 /H 2) e 1 ;
(e 1.1), 2 = - [ (H 1.2/ H 2) e 2 + (H 1.3/H 3) e 3 ], 2 =
= - (H 1.2 /H 2), 2 e 2 + (H 1.2 /H 2) ((H 2.1 /H 1) e 1 + (H 2.3 /H 3) e 3) -
(H 1.3 /H 3), 2 e 3 - (H 1.3 /H 3) (H 2.3 /H 3) e 2
Then, equating the coefficients of the basis vectors, we obtain.
Theory of calculation of thin shells of revolution
When designing steel shells, many general design and design issues arise that do not depend on the specific technological purpose of the shells. Let us therefore consider the theory of shell calculations, regardless of their technological purpose.
The surface of the shell of revolution has an axis of symmetry and two radii of curvature perpendicular to the surface: R 1 is the meridional radius forming the rotation curve, and R 2 is the annular radius of rotation, which originates on the axis of symmetry. Angles φ (latitude) and a (longitude) respectively characterize the location of the radii.
The spherical surface is characterized by the relation R 1 = R 2 ; cylinder - by the relations R 1 = ∞, R 2 = r and φ = n/2; cone relations = R 1 = ∞, R 2 sin φ = r and φ = const (constant angle).
Let us consider a cut out shell element (remote from the edges) of thickness δ with sides dS 1 and dS 2, the area of which is subject to a uniformly distributed load p. It turns out that in thin shells, which are characterized by a small ratio of the shell thickness to its radius (δ/R< 1/30) условия равновесия могут быть соблюдены при наличии только осевых сил — меридиональных Т 1 и кольцевых T 2 , направленных по касательной к срединной поверхности оболочки. Эти силы представляют собой равнодействующие нормальных напряжений, приложенных к сторонам элемента
Let us take the sum of the projections of all forces in the direction of the radius of curvature.
According to the equilibrium condition, this sum must be equal to zero:
Since at small angles
then, dividing both sides of the equation by dS 1 dS 2 we get:
Expressing T 2 in terms of stress, we obtain the basic equation for thin flexible shells
σ 2 is the hoop stress.
For a cylindrical shell with R 1 = ∞, we obtain hoop stresses
For a spherical shell, whose radius is the same in all directions (R 1 = R 2 = R), the operating conditions of each element are also the same in all directions and, therefore:
Thus, with the same radius, the spherical shell experiences 2 times less stress than the cylindrical shell.
The general equation (2.X) contains two unknowns σ 1 and σ 2, as a result of which it is necessary to have a second equation. This equation can be obtained by considering the section of the shell along a parallel circle and equating to zero the sum of the projections of all forces on the axis of symmetry:
Substituting equality (5.X) into equation (2.X), we establish the relationship between the hoop and meridional stresses
The resulting equations for thin shells, derived from equilibrium conditions in the presence of only axial forces (meridional and annular forces), assume that the shell is completely flexible, i.e., that its rigidity with respect to bending and torsion is zero.
The stresses in such a moment-free shell are uniformly distributed over the cross section; there is also freedom of axial deformation. Such prerequisites for the operation of the shell are valid for sections of it located far from supporting fastenings or places of kinks, i.e., from places where the center of the radius of curvature R 1 changes intermittently or the thickness of the shell changes, in a word, from all those places where the conditions for the axial deformation.
In these places, expansion forces and “edge” bending moments appear, causing the shell to bend due to constrained deformations under conditions of section continuity. Bending moments propagate over a relatively narrow zone of the shell, quickly fading due to the fact that the deformations of the shell have to overcome the elastic resistance of neighboring parts (similarly on an elastic base).
Determining these moments and shear forces from the condition of continuity of the cross-section of mating shells is a doubly statically indeterminate problem 1 .
The more severe the violation of the smooth surface of the shell, the greater the additional bending moments and shear forces. Therefore, when designing, sharp bends at the interfaces of the shells should be avoided. In cases where such connections are forced for design reasons, the connections should be checked and, if necessary, strengthened. Typically, reinforcement consists of thickening the sheet wall at the bend or installing a spacer ring.
1 S. P. Timoshenko, Plates and shells, Gostekhizdat, 1948; E. N. Lessig, A. F. Lileev, A. G. Sokolov, Steel sheet structures, Gosstroyizdat, 1956; K.K. Mukhanov, Applied methods for calculating the interfaces of shells of steel structures, collection of works No. 7 MISI, Gosstroyizdat, 1950.
A presentation of the general moment theory of shells can be found in the book by A. I. Lurie, Statics of thin-walled elastic shells, Gostekhizdat, 1947.
"Design of steel structures"
K.K. Mukhanov
Basic principles of shell theory
Most of the elements of engineering structures in the design scheme that are subject to strength calculations, as already noted, are associated with the calculation of beams, plates or shells.
The previous sections were devoted in some detail to the issues of calculating rods and rod systems. This section of the book is devoted to various issues of calculating plates and shells.
The shell is understood as a body, one of whose dimensions (thickness) is significantly smaller than the other two. The geometric locus of points equidistant from both surfaces of the shell is called median surface.
If the middle surface of the shell is a plane, then such a shell is called plate.
The geometric shape of objects that can be classified as shells or plates is extremely diverse: in mechanical engineering, these are the bodies of all kinds of machines; in civil and industrial construction - coverings and ceilings, awnings, cornices; in shipbuilding - ship hulls, dry and floating docks; in aircraft manufacturing - aircraft fuselages and wings; in rolling stock of railway transport, car bodies, tanks, load-bearing structures of locomotives; in nuclear energy - protective structure of nuclear plants, reactor vessels, etc.
If the middle surface of the shell forms a surface of revolution in the shape of a cylinder, then the shell is called cylindrical.
To the diagram axisymmetric A lot of engineering structures are reduced to a cylindrical shell, including: boilers, tanks, oil pipelines, gas pipelines, machine parts, etc.
The problem of calculating thin-walled shells of revolution is most easily solved in the case when it is possible to assume that the stresses arising in the shell are constant across the thickness and, therefore, there is no bending of the shell.
The theory of shells constructed under this assumption is called momentless shell theory.
If the shell has a sharp transition and hard pinching and, in addition, is loaded with concentrated force and moments, then in places where the shell is fastened, sharp changes in shape, and in places where concentrated forces and moments act, intense stresses arise due to bending effect. Accounting for bending effects can be obtained within moment theory of shells.
It should be noted that the smaller the thickness ratio h shell to its radius R, the more accurately the assumption of constant stress across the thickness is fulfilled and the more accurately the calculations are performed using the momentless theory.
Note that the shell is considered thin, if h /R ≤ 1/20.
Consequently, when calculating the strength of thin shells, depending on the nature of the distribution of external loads and supporting fastenings, either a moment-free or a moment theory is used. In this case, a uniform distribution of stresses is assumed over the longitudinal and transverse sections of the shells (the absence of bending, torsional moments and transverse forces in these sections).
With an axisymmetric load, there are also no shear forces. The determination of forces according to the momentless theory is carried out quite accurately at a distance exceeding the value (3÷ 5) from places of abrupt change in shape or cross-sectional area, rigid contour fastenings or from the place of application of external concentrated forces and moments. Near these places, additional stresses arise from the bending effect.
In the momentary and momentless theory of thin shells or the so-called technical theory of shells , consisting in a sharp difference in their thickness and overall dimensions, entails the possibility of simplifying the theory through some schematization of the actual operation of structures. This schematization is formed in the hypotheses used, similar to the hypotheses in the theory of rods, i.e. hypotheses of flat sections and hypotheses of “non-pressure” of shell layers on each other.
These hypotheses make it possible to reduce the three-dimensional problem of continuum mechanics to a two-dimensional one, just as in the theory of rods a three-dimensional problem is reduced to a one-dimensional one.
Shells to which the above hypotheses apply are called thin, and those to which these hypotheses do not apply are called thick.
The boundary between thin and thick shells is arbitrary and determined by the ratio h /R ≈1/ 20.
In cases where h /R ≥ 1/20, to obtain acceptable results in terms of accuracy, the apparatus of continuum mechanics is used, in particular the theory of elasticity or plasticity, depending on the formulation of the problem.
Thin-walled axisymmetric shell
Thin-walled axisymmetric is called a shell that has the shape of a body of rotation, the thickness of which is small compared to the radii of curvature of its surface (Fig. 8.1).
When calculating thin-walled shells, all loads acting on them are applied to median surface shells.
Thin shells can include such frequently occurring structural elements as reservoirs, cisterns, gas cylinders, casings of chemical units, etc.
When calculating such structural elements, it is used momentless shell theory, the main provisions of which are as follows:
1. loads acting on the surface of the shell can be considered perpendicular to them and symmetrical relative to the axis of rotation of the shell;
2. due to the small thickness of the shell, there is no bending resistance (no bending moment occurs);
From the shell shown in Fig. 8.1 we select two meridional planes nn 1 n 2 And nn 3 n 2, (i.e. planes passing through the axis of symmetry of the shell), with an angle dφ between them and two planes perpendicular to the axis of symmetry of the shell B.C. And AD, element ABCD.
Radii of curvature O2A And O2B element ABCD in the meridional plane we denote by R 2, and the radii of curvature O 1B And O 1C in a plane perpendicular to the meridian, denote by R 1. Normal stresses acting along the lateral faces AB And CD in contact with meridional planes are called circumferential stresses σ t. Normal stresses acting along the lateral faces B WITH And AD, are called meridional stresses σ s. In addition to stress σ s And σ t. the shell element is subject to load in the form of pressure q, perpendicular to the surface ABCD.
Fig.8.1
The basic equation of the momentless theory of shells is Laplace's equation, which has the following form
where δ is the thickness of the shell.
Before we move on to consider various options for determining stresses in shells, we will dwell on some differences caused by the presence of gas or liquid inside the shell.
In the case of gas pressure, the pressure value q constant at all points of the shell surface. For tanks filled with liquid, the value q variable according to their height.
For the case of filling a reservoir with liquid, it is necessary to take into account that if liquid pressure acts on any surface, then the vertical components of the pressure forces are equal to the weight of the liquid in the volume located above the surface. Therefore, the liquid pressure in different sections of the shell will be different, in contrast to the gas pressure.
Let us determine the stresses in spherical and cylindrical shells because they are most commonly used in industry.
Spherical shell
Let us cut off part of the spherical shell with a normal conical section with an angle 2φ at the apex and consider the equilibrium of this part of the shell together with the liquid contained in it with specific gravity γ. We separate the spherical part from the main shell by a plane perpendicular to the axis of symmetry.
Fig.8.2
Figure 8.2 shows the design diagram of a spherical shell with a radius R s . Height of the clipped surface. Pressure q on the cut-off part in this and subsequent cases is equal to the weight of the liquid in the volume located above the surface, which is equal to
where is the height of the liquid column above the cut-off part of the shell.
The equilibrium equation of the cut-off part can be written as the sum of the projections of all forces onto the vertical axis
In this equation the quantity G– the weight of the liquid filling the cut-off part of the spherical shell (see Fig. 8.2).
where is the volume of the lower cut-off part of the spherical shell.
By integration, the volume of a spherical segment can be determined by the formula
After substituting equation (8.5) into expression (8.4), and then, into (8.3), we obtain the final equilibrium equation for the spherical part of the segment
From this equation you can determine the value of the meridional stress, and, after substituting into the Laplace equation (16.1), find the value of the circumferential stress.
Cylindrical shell
Let us consider a cylindrical shell of radius , filled with liquid with specific gravity γ (see Fig. 8.3).
Fig.8.3
In this case, the cylindrical part is separated from the rest of the shell by a section perpendicular to the axis of symmetry.
The equilibrium equation of the cut-off part can be obtained as the sum of the projections of all forces onto the vertical axis.
where is the weight of the liquid filling the cut-off part of the cylindrical shell.
Cylinder volume with height x and the radius can be determined by the formula
Taking this into account, the equilibrium equation takes the form
In this equation, as in the previous case, there is one unknown
For the case of a cylindrical shell, when substituting into the Laplace equation, it is necessary to take into account that the quantity means
Conical shell
Let us cut off part of the conical shell with a normal conical section with an angle 2φ at the vertex and consider the equilibrium of the cut-off part.
Fig.8.4
As can be seen from Fig. 8.4 φ = π /2 - α.
The equilibrium equation for the cut-off part of the shell will have the form
where is the weight of the liquid filling the cut-off part of the cone.
Taking into account (8.11), expression (8.10) has the following form
It is possible to separate not the lower, but the upper part of the shell by a section, followed by writing the equilibrium equation. This is done so that when drawing up the equilibrium conditions for the cut-off element, the fastening of the shell does not fall into the diagram of the cut-off part. In such variants, in all cases considered, the sign of the force will change G, because in this case, its direction will coincide with the direction of the vertical component of stress.
In this case, when calculating the value G, the volume of the cut off upper part will be taken as the volume, and when calculating the value q in all cases, formula (8.2) will include the quantity - the height of the liquid column in the cut-off lower part of the shell. Otherwise, the calculation procedure will remain unchanged.
If the liquid is in a vessel under pressure P, then when calculating the value q pressure value is added P. Formula (8.2) will have the following form
In some problems, the cut part is not just one element, but two or more joined elements. In this case, the form of the equilibrium equations remains unchanged, and only the volume of the upper or lower part of the vessel changes, however, if the dependencies that determine the volumes of the elements are known, then finding the total volume is not difficult.
In Fig. 8.5, A shows a diagram of a shell of revolution, consisting of spherical, cylindrical and conical shells. The shell fastening is located at the level of the junction of the spherical and cylindrical shells. A container is filled with liquid under pressure R.
In Fig. 8.5, b An example of constructing voltage diagrams is shown. In the left half of the shell there is a diagram, and in the right half.
Fig.8.5
The resulting constructions are valid for areas located at some distance from the shell fastening line and the sphere-cylinder and cylinder-cone interface points. At the junction points, effects arise that cannot be taken into account by the theory of a momentless stress state. All this also applies to points immediately adjacent to the top of the cone.
Thick wall cylinder
A thick-walled cylinder is one for which the ratio of wall thickness to internal diameter is at least 1/20.
The problem of calculating a thick-walled cylinder is solved taking into account uniformly distributed external pressure and internal pressure. We assume that such a load cannot cause bending deformation of the cylinder.
Normal voltages. in sections by planes perpendicular to the axis of symmetry ABOUT cylinders cannot be considered uniformly distributed over the wall thickness, as is done when calculating thin-walled shells of rotation (Fig. 8.6).
Normal stresses acting on a cylindrical surface with a radius r can be of the same order and even exceed the voltage, which is impossible with thin-walled cylinders.
Fig.8.6
In the cross sections of the cylinder, tangential stresses are also assumed to be zero, however, it is possible that normal axial stresses exist, which arise as a result of loading the cylinder with forces acting along the axis. In what follows we will consider open cylinders, i.e. having no bottoms. The stresses in such cylinders are zero. The derivation of formulas for calculating stresses in thick-walled cylinders is based on the fact that for them plane section hypothesis, i.e. cross sections of a cylinder that are flat before loading will remain flat after loading.
The basic equations for calculating stresses in thick-walled cylinders are Lamé’s formulas:
When only external or internal pressure is applied to the cylinder, the signs of the diagrams are the same at all points of the cylinder. Diagrams of changes in radial and circumferential stress for the case of action of only external pressure are shown in Fig. 8.7. These stresses are negative at all points of the cylinder, which corresponds to compression.
Fig.8.7Fig.8.8
When loaded with internal pressure, diagrams of changes in radial hoop stress are shown in Fig. 16.8. Circumferential stress is expansive, and radial stress is compressive.
Analysis of Lame's formulas shows that increasing the thickness cannot in all cases provide the required strength of the cylinder. Therefore, for high-pressure vessels it is necessary to look for some other design solutions. One such solution is to create composite, tension-connected cylinders. This technique is used both in high pressure technology and in artillery practice to strengthen the barrels of powerful guns.
As a result of tension, normal stresses arise in the pipes, which partially compensate for the stresses in the pipe due to high pressure.
Composite cylinders. Autofretting. General provisions
From formulas (8.14) and (8.15) it follows that under the action of only internal pressure, the stresses at any points of the cylinder are positive and are greater in absolute value than stresses. The highest stress values are reached at points on the inner surface of the cylinder, where they are equal
At other points the voltage is less than this value.
The largest value can be reduced by using composite thick-walled cylinders consisting of thinner pipes placed on top of each other. In this case, the outer pipe is made with an inner diameter slightly smaller than the outer diameter of the inner pipe. The difference between these pre-assembly diameters is accepted before production and is called interference.
To connect the cylinders, the outer cylinder is usually heated, it expands and it becomes possible to put it on the inner cylinder. It is possible to cool the inner cylinder in liquid nitrogen or press the cylinders into each other. After assembly, the temperature is equalized, the outer cylinder tightly covers the inner one and a reliable connection is obtained.
As a result of the tension, initial stresses arise in the pipes, and the greater the value of the tension, the greater the initial stresses.
A method for reducing stress and, as a consequence, increasing the strength of thick-walled cylinders by replacing a solid cylinder with a composite one was proposed by academician A.V. Gadolin.
Let us denote by b And c radii of the outer cylinder, through a and b +∆/2 are the radii of the inner cylinder, and ∆ is the interference (see Fig. 8.9).
Fig.8.9
For the same length of connected cylinders, the contact pressure p k evenly distributed over the seating surface.
Substituting into formulas (8.14) and (8.15) the parameters characterizing the stresses in the outer cylinder, we obtain
Similarly, you can determine the stresses arising on the seating surface of the inner cylinder
If the inner and outer cylinders are made of the same material, then the contact pressure p k determined by dependency
Where E– elastic modulus of the material of the inner and outer cylinders.
Due to the tension, initial stresses arise in the composite cylinder, the nature of the change of which along the outer section is shown in Fig. 8.10.
Fig.8.10Fig.8.11
When internal operating pressure is applied, operating stresses are superimposed on the initial stresses (shown in dotted lines in Fig. 8.11). The total stresses are shown in Fig. 8.11.
At points located on the inner surface of a composite cylinder, the total circumferential stress is less than at the same points of the whole cylinder.
The optimal value of the tension can be determined from the condition of equal strength of the inner and outer cylinders, the optimal value of the radius of the contact surface - from the condition of the greatest reduction in the equivalent stress at the dangerous point.
In accordance with this, the optimal radius of the contact surface is:
Preload corresponding to this radius and internal pressure p V:
It should be noted that parts intended for tension connection must be manufactured with great precision, because even a slight deviation from the nominal interference value can lead to a decrease in the strength of the connection.
In high pressure technology, in addition to landing, the so-called autofrettage , which consists in preloading the cylinder with internal pressure greater than the working one, in such a way that plastic deformations occur in the internal layers of the cylinder. After the pressure is removed, elastic tensile stresses remain in the outer layers of the cylinder, and compressive strains occur in the inner layers (see Fig. 8.12).
Subsequently, when the cylinder is loaded with pressure, the residual stresses are added to the working stresses so that a net unloading takes place in the inner layers. The cylinder material does not undergo plastic deformation unless the operating pressure exceeds the pre-compression pressure.
Fig.8.12
Example of calculation of an element of a thin-walled shell of revolution
Fig.8.13
Solution:
Let's consider the cut-off part with force factors acting on it (see Fig. 8.4).
We pass through the point A first section.
; ; ; .
The second section is carried out at a distance x= 0.15 m.
v= 10 - 0.15 = 9.85 m.
Pressure .
In accordance with the equilibrium equation for the lower cut-off part of the shell (8.13), we have
According to Laplace's equation,
Radius of curvature R 2 for a cone is equal to ∞
Let us draw the third section through the point IN (x= 0.25 m).
Height of the liquid column above the section v= 10 - 0.25 = 9.75 m.
Pressure .
Solving the equilibrium equation (8.16) we have
In accordance with Laplace's equation we have,
Radius of curvature R 2 for a cone is equal to ∞
Example of calculation of a thick-walled steel pipe
For thick-walled steel pipe having an inner diameter d= 0.03 m and outer diameter D= 0.18 m, and made of plastic material with σ T= 250 MPa and with Poisson's ratio μ = 0.5, required:
1. Determine pressure p T, at which plastic deformation begins in the pipe material;
2. Determine the maximum internal pressure p ETC , in which all the material will be in a plastic state;
3. Construct stress distribution diagrams σ p, σ φ, σz by wall thickness for two states of the pipe, discussed in paragraphs 1 and 2;
4. Determine the permissible pressure value p a = p DOP at safety factor n = 1,5.
Solution.
1. According to the formula 
We determine the pressure at which plastic deformations will appear on the inner surface of the pipe:
2. Considering that p a = p T , from formulas
we determine the stresses corresponding to the beginning of plastic flow:
|
- 140,5 |
||
|
- 32 |
||
|
- 5,0 |
||
Stress diagrams σ p, σ φ, σz for the elastic state of the pipe material are shown in Fig. 1, A.
Let us now consider the limiting state of the pipe, when all the pipe material is in a plastic state. The maximum pressure in this case is determined by the formula
Fig.1
3. To determine voltages σ p, σ φ, σz let's use the formulas
We summarize the data for numerical calculations in a table
|
- 517,8 |
- 228,9 |
- 373,4 |
|
|
- 317,6 |
- 28,6 |
- 173,1 |
|
|
- 117,5 |
- 171,7 |
||
For a more accurate construction of diagrams, we will determine the points at which the indicated voltages are equal to zero:
for diagram
General concepts about shells. Classification of shells. Hypotheses in shell theory
Shell - a structural element limited by two curved surfaces, the distance between which h much smaller than the other two sizes b and I(Fig. 21.1, A). The surface equidistant from the outer and inner surfaces of the shell will be called the middle surface. We will consider shells of constant thickness h. Then the geometry of the shell will be completely determined if the shape of the middle surface, the thickness of the shell and the boundary contour are given (Fig. 21.1, a).
Normal section at some point M let's call the section a plane containing the normal to the surface at this point (Fig. 21.1, b). This section is some curved line on the surface of the shell. In the differential geometry of surfaces it has been proven that at any point M surface, you can specify two orthogonal (mutually perpendicular) directions, for which the normal to the surface drawn at an adjacent point intersects the normal at the point M. These directions are indicated 1-1 And 2-2, these are the main lines of curvature. If you draw lines along these directions on the surface, you can get two families of orthogonal lines called lines of curvature. Through a given point M runs along one line of each family. In Fig. 21.1, b marked: R And Ri- main radii of curvature, 0 And Oi- centers of curvature.
Quantities k - HR, kg= l/i?2 we call the principal curvatures, one of them has the maximum and the other has the minimum value. Product of principal curvatures TO = kfa Let's call it Gaussian curvature.
We classify shells according to Gaussian curvature.
Shells of zero Gaussian curvature (TO= 0) are shells of revolution (conical, Fig. 21.2, A) and transfer shells - translational (cylindrical, Fig. 21.2, b).
Shells of doubly curvature are shells of positive Gaussian curvature (K> 0) and negative Gaussian curvature (K 0). There are shells of positive Gaussian curvature: rotations (Fig. 21.2, V) and translational (Fig. 21.2, d), similarly for shells of negative Gaussian curvature (Fig. 21.2, d, f).
Note that shells of positive Gaussian curvature (Fig. 21.2, c, d) principal curvatures To And kj of the same sign (their centers of curvature are located on one side of the surface), and shells have negative Gaussian curvature (Fig. 21.2, d, e) principal curvatures To and ^2 different signs (their centers of curvature are located on different sides of the surface). Particular attention should be paid to folded surfaces (Fig. 21.3). Next, we will consider thin shells for which the shell thickness ratio is h to the minimum
main radius of curvature / 
The following hypotheses are introduced in shell theory.
- 1. Hypothesis about the absence of pressure between the layers of the shell. Normal stresses on areas parallel to the middle surface are negligible compared to other stresses.
- 2. Hypothesis of direct normals. A straight element perpendicular to the middle surface of the shell remains straight and perpendicular to the deformed middle surface and does not change its length.
Note that similar hypotheses are introduced in the theory of plates.
The shell is understood as a body, one of whose dimensions (thickness) is significantly smaller than the other two. The geometric locus of points equidistant from both surfaces of the shell is called median surface.
If the middle surface of the shell is a plane, then such a shell is called plate.
The geometric shape of objects that can be classified as shells or plates is extremely diverse: in mechanical engineering, these are the bodies of all kinds of machines; in civil and industrial construction - coverings and ceilings, awnings, cornices; in shipbuilding - ship hulls, dry and floating docks; in aircraft manufacturing - aircraft fuselages and wings; in rolling stock of railway transport, car bodies, tanks, load-bearing structures of locomotives; in nuclear energy - protective structure of nuclear plants, reactor vessels, etc.
If the middle surface of the shell forms a surface of revolution in the shape of a cylinder, then the shell is called cylindrical.
To the diagram axisymmetric A lot of engineering structures are reduced to a cylindrical shell, including: boilers, tanks, oil pipelines, gas pipelines, machine parts, etc.
The problem of calculating thin-walled shells of revolution is most easily solved in the case when it is possible to assume that the stresses arising in the shell are constant across the thickness and, therefore, there is no bending of the shell.
The theory of shells constructed under this assumption is called momentless theory of shells.
If the shell has a sharp transition and hard pinching and, in addition, is loaded with concentrated force and moments, then in places where the shell is fastened, sharp changes in shape, and in places where concentrated forces and moments act, intense stresses arise due to bending effect. Accounting for bending effects can be obtained within moment theory of shells.
It should be noted that the smaller the thickness ratio h shell to its radius R, the more accurately the assumption of constant stress across the thickness is fulfilled and the more accurately the calculations are performed using the momentless theory.
Note that the shell is considered thin, if h/R≤1/20.
Consequently, when calculating the strength of thin shells, depending on the nature of the distribution of external loads and supporting fastenings, either a moment-free or a moment theory is used. In this case, a uniform distribution of stresses is assumed over the longitudinal and transverse sections of the shells (the absence of bending, torsional moments and transverse forces in these sections).
With an axisymmetric load, there are also no shear forces. Determination of forces according to the momentless theory is carried out quite accurately at a distance exceeding a value of (3÷5) from places of abrupt change in shape or cross-sectional area, rigid contour fastenings or from the place of application of external concentrated forces and moments. Near these places, additional stresses arise from the bending effect.
In the momentary and momentless theory of thin shells or the so-called technical theory of shells , consisting in a sharp difference in their thickness and overall dimensions, entails the possibility of simplifying the theory through some schematization of the actual operation of structures. This schematization is formed in the hypotheses used, similar to the hypotheses in the theory of rods, i.e. hypotheses of flat sections and hypotheses of “non-pressure” of shell layers on each other.
These hypotheses make it possible to reduce the three-dimensional problem of continuum mechanics to a two-dimensional one, just as in the theory of rods a three-dimensional problem is reduced to a one-dimensional one.
Shells to which the above hypotheses apply are called thin, and those to which these hypotheses do not apply are called thick.
The boundaries between thin and thick shells are arbitrary and determined by the ratio h/R≈1/20.
In cases where h/R≥1/20, to obtain acceptable results in terms of accuracy, the apparatus of continuum mechanics is used, in particular the theory of elasticity or plasticity, depending on the formulation of the problem.
Thin-walled axisymmetric called a shell that has the shape of a body of rotation, the thickness of which is small compared to the radii of curvature of its surface (Fig. 1).
When calculating thin-walled shells, all loads acting on them are applied to median surface shells.
Thin shells can include such frequently occurring structural elements as reservoirs, cisterns, gas cylinders, casings of chemical units, etc.
When calculating such structural elements, it is used momentless shell theory, the main provisions of which are as follows:
1. loads acting on the surface of the shell can be considered perpendicular to them and symmetrical relative to the axis of rotation of the shell;
2. due to the small thickness of the shell, there is no bending resistance (no bending moment occurs);
From the shell shown in Fig. 1 we select two meridional planes nn 1 n 2 And nn 3 n 2, (i.e. planes passing through the axis of symmetry of the shell), with an angle dφ between them and two planes perpendicular to the axis of symmetry of the shell B.C. And AD, element ABCD.
Radii of curvature O 2 A And O2B element ABCD in the meridional plane we denote by R 2, and the radii of curvature O 1 B And O 1 C in a plane perpendicular to the meridian, denote by R 1. Normal stresses acting along the lateral faces AB And CD in contact with meridional planes are called circumferential stresses σt. Normal stresses acting along the lateral faces BC And AD, are called meridional stresses σs. In addition to stress σs And σt. the shell element is subject to load in the form of pressure q, perpendicular to the surface ABCD.
Fig.1 Thin-walled axisymmetric shell
The basic equation of the momentless theory of shells is Laplace's equation, which has the following form
where δ is the thickness of the shell,
σ t - circumferential stress
σs– meridional stress,
R 2 - radii of curvature O 2 A And O2B element ABCD,
R 1 - radii of curvature O 1 B And O 1 C in a plane perpendicular to the meridian.
Before we move on to consider various options for determining stresses in shells, we will dwell on some differences caused by the presence of gas or liquid inside the shell.
In the case of gas pressure, the pressure value q constant at all points of the shell surface. For tanks filled with liquid, the value q variable according to their height.
For the case of filling a reservoir with liquid, it is necessary to take into account that if liquid pressure acts on any surface, then the vertical components of the pressure forces are equal to the weight of the liquid in the volume located above the surface. Therefore, the liquid pressure in different sections of the shell will be different, in contrast to the gas pressure.
Let us determine the stresses in spherical and cylindrical shells, since they are most commonly used in industry.
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