DEVELOPMENT OF DIMENSIONALLY STABLE STRUCTURES OF MULTILAYER PIPELINES AND CYLINDRICAL PRESSURE VESSELS FROM CARBON FIBER REINFORCED PLASTIC

The main loads acting on pipelines and pressure vessels are temperature effects, associated with changes in temperature conditions, and internal pressure of a working medium [1]. Temperature effects and internal pressure cause deformations of a shell not only in the axial direction. The cylindrical shell is also deformed in the circumferential direction DEVELOPMENT OF DIMENSIONALLY STABLE STRUCTURES OF MULTILAYER PIPELINES AND CYLINDRICAL PRESSURE VESSELS FROM CARBON FIBER REINFORCED PLASTIC


Introduction
The main loads acting on pipelines and pressure vessels are temperature effects, associated with changes in tempera-ture conditions, and internal pressure of a working medium [1].Temperature effects and internal pressure cause deformations of a shell not only in the axial direction.The cylindrical shell is also deformed in the circumferential direction
Therefore, when designing pipelines and pressure vessels, the problem of compensation of elastic deformations takes an important place [1,2].For deformation compensation, special devices are traditionally used: floating or sliding supports, flexible inserts in the form of bellows, lens or packed expansion joints.With the same purpose, curvilinear sections in the form of L-, V-or U-shaped bends are "inserted" into the line.In this case, the greater the flexibility of the bends, the higher the ability of the structure to compensate elastic deformations, the lower the level of working forces and stresses.
The introduction of additional elements and units in the line increases dimensions and weight, the cost of the structure as a whole.Fundamentally new approaches to the design of pipelines and pressure vessels open up with the use of modern structural materials [3,4].The use of multilayer structures with dimensionally stable properties becomes an effective means of "compensation" of elastic deformations [5,6].Selection of materials [7], development of layer package structures and methods for designing such structures with weight and dimension limitations, without reducing the product strength, seems to be a rather urgent problem.

Literature review and problem statement
The most promising materials for dimensionally stable structures are fiber-reinforced composites (FRC) [8].Moreover, most of such structures are used in the rocket and space industry [9].
In [10], rod systems from FRC are considered.In [11], methods of optimization and design of dimensionally stable structures based on composite plates are presented.Matrix and binder selection questions are discussed in [6,10] (carbon fiber reinforced plastic), [12] (metal composite) and [13] (combined metal and FRC).In [14], the technique for measuring the deformation of composite specimens for designing dimensionally stable structures at low temperatures is proposed.
Systematization of the results of these studies [5] suggests that existing approaches to solving the problem of compensation of deformations of composite structures rely on uniaxial self-compensation.Package structures and reinforcement schemes that ensure dimensional stability have been developed only under temperature loads.There are practically no studies on the creation of composite pipelines and pressure vessels with dimensional stability under the combined effect of temperature and internal pressure [8].
All this suggests that it is expedient to conduct a study on the development of new layer package structures and design techniques of dimensionally stable structures from FRC under the combined effect of temperature and internal pressure.

The aim and objectives of the study
The aim of the study is to develop structures of multilayer cylindrical shells with axial dimensional stability.
To achieve this aim, it is necessary to accomplish the following objectives: -to choose package structures, reinforcement schemes and determine package parameters that provide geometric stability under two-component static loading; -to investigate two-way reinforced and combined multilayer pipes, made by winding of reinforced tape on a thinwalled metal mandrel; -to perform a design justification of dimensionally stable structures.

Derivation of relations for elastic deformations of the wall and analysis of the effect of reinforcement parameters
We consider the deformation of thin-walled pipelines and cylindrical pressure vessels under the action of temperature and internal pressure.
Fig. 1 shows the pipe section as a thin-walled cylindrical shell.The wall of the shell is formed by two-way spiral winding of two symmetric fiber (filaments or tapes) systems on the mandrel.Fibers make up angles ±q with the shell generator.
The number of the unidirectional layers is 2k+1.The inner layer (technological mandrel) is considered homogeneous and isotropic, the unidirectional layers from FRC (monolayers) -orthotropic and linearly elastic.The bonds of fibers and binder, as well as individual layers with each other, are assumed to be ideal.

Fig. 1. Thin-walled multilayer shell element
We use two right orthogonal systems of coordinate axes.Coordinates 1, 2, 3 (natural coordinates) are associated with the axes of the elastic symmetry of the monolayer: axis 1 is directed along the fibers, axis 2 -perpendicular to the fibers in the monolayer plane.Coordinates s, r, t are combined with the middle surface of the shell (Fig. 1).
We assume that the monolayer "works" under conditions of plane stress state (radial stresses are neglected).In this case, the relation between deformations and stresses has the following form [15]: or in the matrix form: Here {e 12 }, {s 12 }, {a 12 } are the vectors of deformations, stresses and coefficients of thermal expansion, respectively; [S 0 ] is the monolayer elastic compliance matrix; DТ is the temperature variation (measured from the initial level, corresponding to the stress-free state); Е 1 , Е 2 , G 12 , n 12 , n 21 , a 1 , a 2 are the technical thermoelasticity constants (effective thermoelastic constants: elastic moduli, shear modulus, Poisson's ratios and thermal expansion coefficients).In this case, n 12 E 2 =n 21 E 1 .A homogeneous field is assumed.The relations inverse to (1) are: or where [G 0 ] is the monolayer stiffness matrix, {b 12 } is the temperature stress vector.
The coefficients of the matrix [G 0 ] and the vector {b 12 } are related to the technical thermoelasticity constants by the following relations: Let us rewrite the dependences (1) and ( 2) from the natural coordinate system 1, 2, 3 to coordinates s, r, t.As a result, we get Transformations of matrices when the axes of coordinates s and t are rotated relative to the normal to the surface r have the known form [15]: Here [L 1 ] and [L 2 ] are rotation transformation matrices: In this case, m=cosq, n=sinq.The index "T" denotes the matrix transpose operation.
Note that, according to (5), with arbitrary reinforcement, the angular deformations g st of the unidirectional layer are related to normal stresses s s and s t , and the linear deformations e s and e t -to shear stresses t st .In the particular case of symmetric winding for the system composed of two monolayers with angles ±q, the coefficients As a result, these relations disappear.We define the linear deformations e s and e t of the section of the multilayer pipe reinforced with the symmetric filament system.To do this, using the method of sections, we write the following equilibrium equation [4]: Here i=1, 2,…, k+1 is the sequence number of the layer (composed of two monolayers), h i and r i are the thickness and radius of the middle surface of the layer, R is the pipe hole radius, p=p m +r f ×v m 2 is the intensity of internal pressure.In this case, p m is the static component, r f ×v m 2 is the dynamic component of pressure, equal to twice the velocity head (r f and v m are the density and velocity of the fluid flow).It is believed that the pipe end sections are closed with plugs.

r R g h pR g h r g h g h g h r g h g h pR g h g h g h
.
In the particular case of an isotropic body with k=1, the formulas for linear deformations (8) acquire the known form [1]: e = e =a D , where E, n, a are the elastic modulus, Poisson's ratio and thermal expansion coefficient, respectively.
In the case of an isotropic body, the equality e s = 0 holds: under the action of internal pressure, if n=0.5; under the action of temperature, if a = 0.
For the pipe made of two-way reinforced material containing two monolayers, we define the angles ±q * , ensuring e s =0.Consider internal pressure loading (temperature load is considered in [15]) on the pipe.In this case, based on (8), from the condition e s p =0 for k=1, it follows that: The coefficients g 12 and g 22 are determined by the formulas ( 5 where The solution of the quadratic equation ( 12) determines the angles ±q * , providing dimensional stability in the s direction.
Fig. 2 shows the graphs of the linear deformations e s and e t of the sample against the winding angle, constructed by the formulas (8).Graphs in Fig. 2, a correspond to the effect of temperature DТ=1 K.The points on the graph are the experimental results [5,15], averaged in the temperature range of 25¸150 о С. Graphs in Fig. 2 Table 1 shows the calculated values of the angles q * , providing the dimensional stability of the sample in the axial direction, under two-component static loading.The combined effect of temperature and internal pressure is taken into account.We state that the calculated angles for various combinations of loads are located in the interval of ±41 о 47 ' and ±52 о 16 ' .The boundaries of the interval are limited by the angles, calculated under the partial action of loads.We investigate the deformation of the combined multilayer pipe, made by two-way spiral winding of reinforced tape on the mandrel.The mandrel is made of technical titanium VT1-0 with the parameters [16]: E=1.121×10 5 MPa, a=8.0×10 -6 K -1 , n=0.32.The tape -from KMU-4l carbon fiber reinforced plastic.The inner diameter of the pipe is d=96 mm, mandrel thickness is h 1 =0.3 mm, and wall thickness is H=2.4 mm.Fig. 3 shows the graphs of the linear deformations e s and e t against the winding angle.Fig. 3, a corresponds to the effect of temperature DТ= 1 K, Fig. 3, b -to the action of internal pressure p=1 MPa.
Using ASCP software package [1] and the finite element [17], we analyze the stress-strain parameters of the pipeline with the flowing fluid.The design scheme of the pipeline is shown in Fig. 4 (dimensions in mm).We consider the mode of undisturbed internal flow.Fluid parameters: r f =1.02 g/cm 3 , v m =10 m/s, p m =0.898 MPa, DТ=100 K. Forces of friction of the liquid against the pipe walls are neglected.Let us compare three versions of structures differing in pipe design.Version I is a single-layer structure made of VT1-0 titanium (inner diameter d=96 mm, wall thickness H=2.4 mm).Version II is a multilayer combined structure made by two-way spiral winding of KMU-4l carbon fiber reinforced plastic on the VT1-0 titanium mandrel (d=96 mm, winding angles q=±60 о , mandrel thickness h 1 =0.3 mm, wall thickness H=2 4 mm).Version III is a multilayer combined dimensionally stable structure.Table 2 gives estimated displacements, longitudinal forces, maximum values of the bending moments and loads on supports for each version of the structure.
Each of the design versions of the pipeline gives noticeably different values of displacement and support reactions.

Discussion of the results of design justification of dimensionally stable structures
The results of numerical simulation (Fig. 2, 3, Tables 1, 2) indicate that the proposed design technique for multilayer composite pipelines allows creating dimensionally stable structures with two-component loading.
Indeed, from the analysis of the graphs in Fig. 2, it follows that: -At temperature load, dimensional stability of the sample (thermal stability) is ensured in both axial and circumferential directions (e s DТ =0 for q * =±41 о 47 ' , e t DТ =0 for q * =±48 о 13 ' ).In this case, in the region of angles q<41 о 47 ' with increasing temperature, the sample length decreases, and in the region q>48 о 13 ' its diameter decreases.
-Under the internal pressure, dimensional stability of the sample is ensured only in the axial direction.At the same time, in the region of angles q³±60 о , the circumferential deformations are rather small.From the graphs in Fig. 3 it can be seen that the winding angles corresponding to e s =0 are: under the action of temperature q * 1 =±12 о 46 ' and q * 2 =±35 о 58 ' , under the action of internal pressure -q * 1 =±12 о 53 ' and q * 2 =±50 о 34 ' .Note, in the region of q * 1 <q<q * 2 , the internal pressure causes a decrease in the pipe length.
Analysis of the results of the computational experiment on modeling the stress-strain state of the pipeline with the flowing fluid showed the following.For version III of the pipeline, the estimated displacements, bending moments and loads on the supports are practically zero.Obviously, the use of dimensionally stable multilayer pipes eliminates axial deformations.As a result, the forces and stresses in the pipes associated with the bending deformations disappear, the supporting structure is unloaded.
Single-layer metal and combined pipes from carbon fiber reinforced plastic are in tension.Moreover, the tensile force in the pipe with a dimensionally stable structure (version III) is equal to the compressive force in the channel, i. e., N»Р, where Р=рА о =7.238 kN.Here А о is the area of the pipe opening.In this case, the expansion loads from the pipe are not transferred to the fittings.
Of course, this method of compensation of deformations has some limitations: -According to the used hypotheses of the theory of thin shells, the normal stresses in the direction of the normal are considered to be zero.Only interlayer shear stresses are taken into account.In this, connections between the individual layers, as well as the connections between the fibers and the binder, are assumed to be ideal.In a real structure, this circumstance may require additional technological or design solutions to ensure joint deformation of the multilayer wall and exclude delamination.
-The problem considers the pipeline as an engineering object with low thermal conductivity.A homogeneous temperature field is assumed, that is, the same temperature is obtained in all layers.In real structures, when filling the pipeline, a temperature gradient and thermal shocks are obtained.But due to the small wall thickness of the pipeline and slow filling (quasi-static process), we can assume that the temperature field is aligned in thickness, and in the first approximation it can be considered as homogeneous.
-Internal pressure is considered as an action of a stationary internal fluid flow.In actual pipeline designs, hydraulic shocks and vibrations may occur during operation.Therefore, it is impossible to refuse all compensators, but their number can be reduced.
-The physicomechanical properties of materials vary with temperature.Therefore, the proposed pipeline designs will have dimensionally stable properties within the preservation of the properties of materials.For other temperature ranges, there will be other layer packages and reinforcement schemes.
Note, dimensionally stable multilayer pipes from FRC open up new approaches to the design of pipelines and pressure vessels.There are opportunities to create structures with preassigned (not necessarily zero) displacement fields [18].The coordination of these fields with the fields of the initial technological displacements, as well as with the displacements of the coupled elastic elements and equipment when the operation mode changes.The scope of such structures is not limited to "hot" pipes.The results can be used in cryogenic engineering.Thus, such structures can be used in products of aviation and rocket-space technology, oil and gas industry, that is, in structures where the operation of pipelines is characterized by an increased level of tension with stringent requirements to overall and mass characteristics.

Conclusions
1.The obtained calculated relations for linear deformations of multilayer cylindrical shells under separate and combined temperature and internal pressure loading allow estimating the influence of winding angles and package parameters on the values of elastic deformations.Analysis of dependencies and strain values revealed the package structures, reinforcement schemes and package parameters that ensure the geometric stability of the pipeline in the axial direction under two-component static loading.
The presented structures were obtained on the basis of technical VT1-0 titanium and KMU-4l carbon fiber reinforced plastic.However, this is not the only combination of materials.Structures composed of stainless steels, niobium, organoplastic, and other materials have similar properties (dimensional stability).
2. The study of two-way reinforced and combined multilayer pipes, made by winding of reinforced tape on a thinwalled metal mandrel, allowed obtaining the dependences of the elastic strains on the reinforcement parameters.The possibility of self-compensation of axial deformations both under separate action of temperature and internal pressure, as well as with their combined action is demonstrated.
3. The calculated substantiation of the obtained dimensionally stable structures of multilayer cylindrical shells was performed.Using the example of the pipeline with the flowing fluid, the advantages of dimensionally stable structures are demonstrated.Thermal power loading of several versions of reinforcement schemes showed: the use of dimensionally stable multilayer pipes eliminates axial deformations.As a result, the forces and stresses in the pipes associated with the bending deformations disappear, the supporting structure is unloaded.
11) into equality(10) and introducing a new variable l= n 2 , we obtain the algebraic equation of the following form l + l + =

Fig. 2 .
Fig. 2. Change of linear deformations from the winding angle for the carbon fiber reinforced plastic sample: a -temperature loading; b -pressure loading

Fig. 3 .Fig. 4 .
Fig. 3. Change of linear deformations from the winding angle for the combined sample: a -temperature loading; b -pressure loading

Table 1
Angles q * ensuring axial dimensional stability

Table 2
Pipeline stress-strain parameters