**Article citation information:**

Lolov, D.,
Lilkova-Markova, S. Dynamic stability of a cracked pipe conveying
fluid and resting on a Pasternak elastic foundation. *Scientific Journal of Silesian University of Technology. Series
Transport*. 2022, **117**, 103-113. ISSN:
0209-3324. DOI: https://doi.org/10.20858/sjsutst.2022.117.7.

Dimitar LOLOV**[1]**,
Svetlana LILKOVA-MARKOVA**[2]**

**DYNAMIC STABILITY OF A CRACKED PIPE CONVEYING FLUID AND RESTING ON A
PASTERNAK ELASTIC FOUNDATION**

**Summary****.** Pipeline
transport is used worldwide in many sectors of the economy. Its main advantages
are continuity of transport, large transportation volumes, small energy
consumption, safety, reliability and high environmental benefits. However, the
safety problems of pipes attract much interest in science and industry. This
paper deals with a cracked pipe with a static scheme of a simply supported
beam. It rests along its entire length on a Pasternak elastic foundation. The
flowing fluid is considered non-compressible and heavy. The Galerkin method is
employed to approach the problem numerically. Conclusions are drawn based on
the influence of the crack and the parameters of the Pasternak elastic
foundation on the critical flow velocity of the fluid.

**Keywords:** pipe,
fluid, dynamic stability, crack, Pasternak elastic foundation, critical
velocity

**1. INTRODUCTION**

Pipelines conveying fluid have many engineering
applications. They are widely used in the petroleum industry for the
transportation of oil and gas. Another established use of them is in the
transport of water. They also find applications in
nuclear engineering, aviation and aerospace.

Nanoscale tubes find applications in
nanophysics, nanobiology and nanomechanics as nanofluidic devices in
nanocontainers for gas storage and nanopipes conveying fluid. The experiments
at the nanoscale are difficult and expensive. That is why the continuum elastic
models have been used to study the fluid-structure interaction. The carbon
nanotubes are considered with Euler- and Timoshenko-beam models.

Tubes conveying fluid may also be found in
pulmonary and urinary systems and haemodynamics.

The interaction of a tube and the
fluid flowing in it has been the subject of much research. The flow of the fluid in the tube
causes oscillations in it. The dynamic characteristics of the pipe’s
oscillations depend on the velocity and the mass of the conveyed fluid. The
system is stable for flow velocities that are less than a certain value called
critical flow velocity. The research on the dynamic
stability of pipes conveying fluid is branched into two directions: a) dynamic
stability of pipes with a rectilinear axis, and b) dynamic stability of curved
pipes.

The most common methods used for the dynamic
analysis of pipes conveying fluid are the transfer matrix method (TMM) and the
generalized differential quadrature method (GDQM). Both methods
have a significant advantage over the finite element method (FEM). The conventional FEM can be very
time-consuming when it comes to the investigation of a pipeline with a high
number of spans. The order of the overall property matrices for the whole
multi-span pipeline increases with the number of spans. This
is unlike the TMM, in which the order of the overall transfer matrix is
independent of the number of spans and is kept unchanged.

The GDQM approximates a derivative
of a function in the partial differential equation of the lateral vibration of
the pipe at any discrete point as a weighted sum of the function values at all
discrete values at the domain. The main advantage of this method is its high convergence with a few
grid points.

Pipelines often rest with their entire length
or with part of it on an elastic medium. The first suggested model of that
medium is the Winkler elastic foundation. Although it
has some shortcomings, it is still being widely used in civil engineering since
its introduction in 1867. The Winkler model of the
elastic medium consists of mutually independent vertical linear springs.

In 1954, a refined model of the elastic medium
was introduced by Pasternak. He introduced shear
interaction between adjacent linear springs in the Winkler model. The Pasternak foundation is a two-parameter model. The values of the parameters for practical application are
the subject of much research in the field of geotechnical mechanics.

Cracks are the most encountered damage in the
structures. They reduce the stiffness of the structural
element, causing a decrease in its natural frequencies and a change in the mode
shapes. In pipes conveying fluid, cracks lead to a decrease in the critical
velocity of the fluid. The cracks could be hazardous for the system. They might lead to loss of stability if the reduced
critical velocity of the transported fluid, due to the crack, is exceeded. This is why crack detection is a topic of great interest
in scientific research. Some of the studies for
crack detection deal with the changes in the natural frequencies and
eigenforms, and others with dynamic responses to harmonic loads.

The book [16] deals with the
dynamics of slender cylindrical bodies in contact with axial flow. It not only covers the fundamentals of the problem but
also solves some examples that have direct applications in engineering and
physiological systems.

M. Paidoussis and N. Issid in [15]
investigated the dynamic stability of pipes with internal flow. They considered
clamped-clamped and pinned-pinned pipes.

M. Siba et al. [19]
reviewed studies of the oscillations of a tube conveying fluid. The
need for more experiments in this area is justified.

L. Shiwen et al. [18] studied the flow-induced vibration characteristics of a pipeline system.
Fluid-structure
interaction numerical simulation is conducted for a typical fluid-conveying
pipe network with the help of software.

Bing Chen et al. [2] used Galerkin’s
method and the complex mode method to find the natural frequencies of a
pinned-pinned pipe conveying fluid and lying on a Pasternak foundation.

In this paper, a fluid-conducting tube resting
on a Pasternak elastic foundation is investigated. The results obtained reflect
the dependence of the critical fluid velocity on the parameters of the
Pasternak elastic foundation. They also show the effect
of an open crack on the critical velocity of the fluid.

This paper is structured as follows. First, the model of the pipe and the governing
differential equation of its transverse vibration is presented. The Galerkin method is employed to approach the solution
to the problem. It is shown how to obtain the characteristic equation of
the problem. Based on its roots, conclusions could be
drawn about the stability of the system. Second, it
is shown how to model the crack with the help of Castigliano’s theorem.
Finally, the obtained results from the numerical
solution are presented, and several important conclusions are summarized.

**2. PROBLEM FORMULATION**

This
paper uses the Euler-Bernoulli beam theory to investigate the dynamic stability
of a pipe of length

The pipe is divided into two
segments. The first segment is the left-hand side of the crack, and the second
– the right-hand side of the crack.

The transverse vibration of a
straight pipe conveying inviscid fluid and lying on a Pasternak elastic
foundation is governed by the following differential equation:

where

Fig. 1. Static scheme and
cross-section of the investigated pipe

For simplicity, the following
dimensionless parameters are introduced:

The dimensionless equations for
transverse vibration in the two segments of the pipe are:

In (3) and in the sequel, primes
denote derivatives with respect to

The
spectral Galerkin method is applied to approximate the solution of the
differential equation (3). The solution for each segment of the pipe
is sought in the following form:

where:

The boundary conditions of the
cracked simply supported beam, shown in Figure 1 are:

For the left end of the beam:

For the right end of the beam:

For the cracked section of the pipe
[21]:

Inserting equation (4) in equation
(3), yields:

where the elements of the matrices in equation
(8) are:

The general solution of the system
(8) is expressed through the roots of the equation:

The elements of the matrix

Based on the
obtained roots could be concluded the stability of the system. The
system is stable if the real part of all the roots of the characteristic
equation (13) is negative.

**3. CRACK MODELLING**

It is considered that the bending
vibrations of the Euler-Bernoulli beam is in the plane

where

where

The equivalent rotational spring
stiffness:

**4. RESULTS AND DISCUSSION**

Numerical studies have been carried
out for the system in Figure 1.

The geometric and material characteristics
of the pipe are: the inner and the outer radii of the cross-section of the
pipes -

The finite element method was used
to obtain the basic functions

At first, the position of the crack
is fixed with the coordinate

Based on
the obtained roots of the characteristic equation (12) could be drawn conclusions
about the stability of the system. The system is stable if the real part of all the roots is negative. If
one or more roots have positive real parts, then the system is unstable. When one or more roots of the characteristic equation have
real parts equal to zero, the system is at the edge of loss of stability, the
corresponding fluid velocity is the critical fluid velocity. The roots
depend on all the parameters of the system. If all of them are fixed, except
the velocity of the conveyed fluid

The obtained results for a cracked
pipe are compared with the results of an undamaged pipe.

For the pipe in Figure 1 are
obtained the critical velocities for different values of the parameters of the
Pasternak elastic foundation. The results shown in Figure 2 are calculated for
a crack fixed with a coordinate

The obtained results show that the Pasternak
foundation has a stabilizing effect on the pipe - by increasing the parameters of the
foundation, the critical velocity increases. The crack has a
destabilizing effect on the system, leading to decreasing in the critical
velocity.

For the Winkler
elastic foundation (

The same dependence between the rigidity of the
rotational foundation and the critical velocity of the fluid is observed.

The second part
of the survey investigated the influence of the position of the crack on the
stability of the system. It considered not only the position
of the crack along the length of the beam but also if the crack is on the top
or bottom edge of the pipe. The results are shown in Figure 3.

When the coordinate of the crack

There is a slight difference in the
critical velocities when the position of the crack is on the top and bottom
edge of the pipe. For all investigated coordinates of
the crack

Fig. 2. Critical velocity versus the
rigidity of the Pasternak elastic foundation

Fig. 3. Critical velocity versus the
position of the crack (

**5. CONCLUSION**

The classical Winkler foundation is often used
as a model in geotechnical analyses. It states that the deflection at any point
at the surface of an elastic medium is proportional to the load applied at the
point and does not depend on the applied loads at other points of the surface, and that
is its major shortcoming. To
overcome this, the Pasternak model is introduced. It is
an improved two-parameter model of the elastic medium.

Cracks are the most
encountered damage in the structures. When a structure is cracked, its stiffness is reduced,
with a consequent reduction in the natural frequencies and a change in the eigenforms.

This paper investigated the influence of the
parameters of the Pasternak elastic foundation on the stability of a cracked
pipe conveying fluid.

The pipe is modelled as two segments connected by a
rotational elastic spring at the cracked cross-section. Castigliano’s
theorem is employed to calculate the stiffness of the spring. The spring
stiffness depends on the geometry of the cross-section of the pipe and the
severity of the crack.

The results obtained in this study could be summarized
as follows:

1.
The Pasternak foundation has a stabilizing effect on
the system. This means that the fluid could flow
through the pipe at a higher velocity without causing a loss of stability in
the system.

2.
When the parameter of the Pasternak elastic foundation

5.
The position of the crack affects the stability of the
system. If the crack severity remains unchanged, the
critical velocity of the fluid is higher when the crack is located at the
bottom edge of the pipe than when the crack is at the upper edge of the pipe.
Also, the position of the crack along the length of the pipe affects the
stability of the system. The closer the crack is to the
middle of the span, the more unstable the system becomes.

It is
worth mentioning that the damping of the Pasternak elastic foundation also
affects the stability of the system; however, this effect was not considered in
this paper.

The results obtained contribute to the safety of pipes
conveying fluid. To avoid damages, the operator of the
pipe should not allow higher transportation velocities than the critical
velocity of the system. As the critical velocity depends on many
parameters of the system, among which is the severity and position of the
crack, the operator of the pipe should perform strict crack detection tests,
then based on the results, correct the velocity of the fluid to the damaged
system, not to lose stability.

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Received 29.06.2022; accepted in
revised form 02.10.2022

Scientific Journal of Silesian University of Technology. Series
Transport is licensed under a Creative Commons Attribution 4.0
International License

[1] Faculty of Hydraulic Engineering,
University of Architecture, Civil engineering and Geodesy, Hristo Smirnenski 1 Street,
1046 Sofia, Bulgaria. Email: dlolov@yahoo.com. ORCID:
https://orcid.org/0000-0002-8138-0265

[2] Faculty of Hydraulic Engineering,
University of Architecture, Civil engineering and Geodesy, Hristo Smirnenski 1
Street, 1046 Sofia, Bulgaria. Email: lilkovasvetlana@gmail.com. ORCID:
https://orcid.org/0000-0003-0582-8176