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An Efficient and Comprehensive Methodology to Evaluate the Misalignment
of Pump Shafts using FEM
Conference Paper · December 2014
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An efficient and comprehensive methodology to evaluate the misalignment of pump
shafts using FEM
Barbara Röhrnbauer, Maurizio Sciancalepore, Michael Singer, Torsten Johne, Philippe Dupont
Sulzer
1. Introduction
Quality and efficiency are two major aspects in the evaluation of the mechani cal integrity of pump
casings. In the present context mechanical integrity comprises structural intactness and
functionality in all relevant loading conditions, including temperature loads, internal pressures and
nozzle loads. Quality is achieved by applying modern analysis tools such as FEM and by
developing robust templates and standardized methodologies. Efficiency requires smart use of
such tools based on engineering knowledge. In this contribution, an efficient methodology based
on FEM is presented to evaluate the misalignment of the pump shaft with respect to an installed
motor-coupling system due to nozzle loads.
1.1. Mechanical integrity based on failure modes
Mechanical integrity of a pump, i.e. its intactness and functionality represents a wide and abstract
concept. It is to be interpreted with respect to the relevant loading conditions, the releva nt effects
and consequences of specific modes of failure and also with respect to the tools available for
evaluating those. With the increasing use of finite element analyses (FEA), design by analysis
(DbA) methods have gained importance over design by rules methods (DbR) for the analysis of
pressure retaining parts and are also comprised in the pressure vessel codes such as ASME VIII,
Div. 2 [1] and EN 13445 [2]. An efficient use of FEA requires a precise definition of the results
aimed at and therefore a clear idea of what is meant by mechanical integrity. A systematic and
structured way of summarizing the causes, effects, and consequences of relevant and possible
modes of failure is the so-called failure mode table or cause-effect-consequence (CEC) table,
which has been described for pumps by Singer and Johne [3] (Table 1). The CEC table represents
a basis for the selection of evaluation procedures and criteria adjusted to the specific pump type
and customer needs.
Table 1 Failure mode table [3]
Cause
Effect
Consequence(s)
Fatigue stress
Cracked case wall
Leakage of working fluid
1.2. Shaft Misalignment due to nozzle loads
1.2.1. Cause-Effect-Consequence(s)
The present work focuses on one specific failure mode, the failure of the coupling caused by the
displacement of the pump shaft end due to nozzle loads.
Figure 1 schematically shows the system investigated, a pump equipped with two nozzles and a
coupling connecting the pump shaft to the motor shaft.
In general, causes for shaft misalignment can be grouped into two categories: on the one hand
factors characterized by defined directions, such as internal pressure, thermal growth, gravity,
shaft bending or shaft lift-off, both rotordynamic phenomena; and on the other hand factors with
undefined directions, such as seismic accelerations, mounting tolerances to be considered for the
installation of the pump on the baseplate and for the bearing housings on the casing, shaft orbits
and associated vibrations, which again arise from the rotordynamics of the pump. The second
category additionally contains nozzle loads which are caused by thermal expansion of the
connected piping, their imprecise installation or by other undesired effects, such as pipe loads
2
during seismic events. As nozzle loads are major contributors to the overall shaft misalignment,
they are treated within one separate failure mode.
Figure 1 System investigated: A pump with two nozzles, the pump shaft is connected to the motor
shaft by a coupling for torque transmission.
A misalignment of the pump shaft with respect to the motor shaft might lead to increased
temperatures at the coupling and premature failure of the coupling. Moreover, malfunction or
failure of the mechanical seal, impaired tightness of the casings, rotordynamic problems and even
rupture of the shaft close to the coupling are possible consequences of shaft misalignment. In
general, it reduces the overall lifetime of the pump significantly. For a detailed description of
causes and consequences of this failure mode the reader is referred to [4].
1.2.2. Nozzle load specification
Nozzle loads are in general specified by the pump purchaser or by a third party in charge of the
piping design. However, reference values for respective force and moment components (six per
nozzle) are provided by ISO 13709 for different pump types and nozzle configurations [5]. These
components refer to defined coordinate systems at the nozzles but are specified as unsigned
values.
1.2.3. Allowable values
There are three types of shaft misalignment: axial, radial and angular misalignment, which are
illustrated in Figure 2.
Figure 2 Types of shaft misalignment: axial, radial, angular.
3
The tolerances for the misalignment in operating conditions are provided by the coupling
manufacturer. Amongst others, they depend on the type of coupling, the length of the coupling and
on the rotational speed of the motor and the pump. Allowable values for axial misalignment are in
the range of millimeters, whereas allowable values for radial misalignment are typically around
20% of this value. The allowable angular misalignment depends on the axial and radial
misalignment and is typically in the range of a few tenths of a degree. These values are often
further reduced by a safety factor adjusted to the type of pump and its application.
1.3. State of the art
According to ISO 13709 “the pressure casing shall be designed to operate … while subject
simultaneously to the MAWP (maximum allowable working pressure) (…) and the worst-case
combination of twice the allowable nozzle loads…” [5]. Finding this worst-case combination from a
set of twelve (in case of two nozzles, suction and discharge) unsigned nozzle load components
possible combinations. The decision on the loading conditions for
requires to consider
the analysis of shaft misalignment is commonly based on longtime experience of the analysts and
agreement with the customers. It is therefore subjective and difficulties might arise for new
developments or with less experienced parties involved. Moreover, during the development
process requirements in terms of nozzle load values might change demanding for a re-definition of
worst–case loading conditions and for additional analyses.
1.4. Motivation
The described methodology for the evaluation of shaft misalignment based on an estimated worst case combination of nozzle loads is subjective and might therefore lack reproducibility. Moreover,
the worst-case combination might even be missed producing non-conservative results and thus
compromising the reliability of the analysis. On the other hand calculating a number of
analyses would require a non-affordable effort in terms of calculation time.
The present contribution proposes an efficient approach based on linear superposition of FE
analyses to extract the worst-case combination of nozzle loads and to calculate the resulting shaft
misalignment. This new approach therefore addresses two major requirements for mechanical
analyses, effectivity and quality.
2. Calculation methodology
The effectivity of the present approach is based on reducing the number of costly FE analyses
from to (where
) combined with a less costly and time-efficient postprocessing procedure. This means, for two nozzles, suction and discharge,
( ) costly FE
analyses are replaced by
. Quality is assured by checking all possible combinations of nozzle
loads and identifying the most critical one during post-processing.
2.1. Assumptions
Direct reference measurements of shaft misalignments caused by nozzle moments have shown
good agreement with the findings of purely linear analyses. Therefore, only linear effects (linear
material properties, small deflections and linear contact behavior) are taken into account for the
underlying analyses. Consequently, results can be scaled and superposed.
2.2. System equations
The introduced structural system (i.e. the pump and a motor positioned on a baseplate) can be
described by a structural stiffness matrix . It is applied to the vector of the degrees of freedom of
the system , which results in the right hand side , the load vector of applied forces and moments,
here the nozzle loads.
4
!! = !
(1)
Solving this linear system of equations for the vector of degrees of freedom
following system of equations with the corresponding structural compliance matrix
! = !!! !
results in the
-1.
(2)
Thus, applied to any vector of nozzle loads, this structural compliance matrix provides the solution
vector in terms of degrees of freedom. The structural stiffness matrix is typically setup within the
finite element software solving the system of equations (2).
2.3. System compliance matrix
Choosing a specific set of nozzle loads ! = {!! , !! , … , !! } where !! = (1,0,0, … ,0)! , !! =
(0,1,0, … ,0)! , … , !! = (0,0,0, … ,1)! are unit loads, allows for directly determining all components of
the structural compliance matrix by
!! = !!! !!
! = 1…!
(3)
and
! !!!! = !! !
(4)
According to equation (4), the unit load solutions in terms of degrees of freedom directly represent
the columns of the structural compliance matrix. Therefore, the structural compliance matrix of the
present system can be determined by finite element analyses applying unit loads.
2.4. Structural degrees of freedom
In order to increase the efficiency, the order of the system, i.e. the number of degrees of freedom
(number of rows of the structural compliance matrix) is reduced to those needed for calculating the
shaft misalignment (Figure 3). Considering two shafts, the pump shaft and the motor shaft, the
axial misalignment is calculated by the difference between the corresponding axial shaft end
displacements. The components refer to the coordinate systems introduced in Figure 3.
!!!!!
!!!!!
!!!!!! = !!!
− !!!
(5)
The radial misalignment is calculated as the magnitude of the differential vector of in-plane shaft
end displacements.
!
!!!!!
!!!!! )!
!!!!!
!!!!!
!
!!!!!!! = !(!!!
− !!!
+ !!!!
− !!!
(6)
For the angular misalignment
the angle between the vectors normal to the two shaft end cross
sectional areas is calculated. The normal vectors are calculated as the vector connecting the
5
vertex at the shaft end
shaft centerline.
and a vertex
adjacent to it (Figure 3). Both vertices are located on the
!!!!!
!!!!!
− !!!!
!!!
!!!!!
!!!!!
− !!!!
!!!
!! = !
!
!!!!!
!!!!!
!!!!!!!!!!⃗
+ !!′! !! !
−!
!
!!!
!!
⎧
!! =
(7)
!!!!!!!!!!
!!!!!
!!!!!
!!!
− !!!
!
!!!!!
!!!!!
!!!
− !!!
!
⎫
(8)
⎨ !!!!!
⎬
!!!!!
!!!!!!!!!!⃗
!!′
!
+
!
−
!
!
!
!
!!
!!!
⎩
⎭!!!!!!!!!!
cos ∆! = !! ∙ !!
(9)
The indices
refer to the directions of the local coordinate systems at the shaft ends introduced
in Figure 3, the indices
refer to shaft and respectively and the prime indicates the vertex
adjacent to the shaft end vertex (without prime). The distance between these two vertices, is
!!!!!!!!!!!⃗
expressed by !!′
.
! !! !, where
The vector of reduced structural degrees of freedom can thus be written as
!!!!!
! = !!!!
!!!!!
!!!
!!!!!
!!!
!!!!!
!!!!
!!!!!
!!!!
!!!!!
!!!!
!!!!!
!!!
!!!!!
!!!
!!!!!
!!!
!!!!!
!!!!
!!!!!
!!!!
!
!!!!!
!!!!
!
(10)
The resulting structural compliance matrix is a
of nozzles.
matrix, where is determined by the number
6
Figure 3 Calculation of shaft misalignment, a) Two local coordinates are defined at the end of each
shaft end, the pump shaft end, referred to by (1) and the motor shaft end, referred to by (2), to
determine the components of the shaft end displacement vectors. b) The axial and the radial shaft
end displacements are calculated using the differential vectors of shaft end displacement. c) For the
calculation of the angular shaft end displacement the normal vectors to the shaft end cross sectional
areas are used. They are defined as the connecting vectors between the shaft end vertex and a
vertex adjacent to it, both vertices located at the shaft centerline.
2.5. Linear superposition
Applied to any vector of nozzle load, the structural compliance matrix determines the
corresponding solution in terms of structural degrees of freedom and finally shaft end
displacements. As equations (2) and (4) hold, the degrees of freedom vector !! for any nozzle
load vector !! = (!!! , !!! , … !!! )! can be written as
!! = !!! !!
!! = !!! !! + !!! !! + !!! !! + ⋯ + !!! !!
!! = !!! !!
(11)
(12)
(13)
7
Consequently, the solution in terms of degrees of freedom is a linear combination or a
superposition of the unit load displacements multiplied by the nozzle load components. These
matrix multiplications or linear combinations represent numerically cheap calculations which are
efficiently applied to different combinations of nozzle load components!!! .
3. Implementation
Turning the previous theory into praxis requires finite element analyses applying unit loads and a
post-processing procedure evaluating different combinations of signed nozzle load components
!!! . The implementation is shown using a Sulzer axial split pump.
3.1. Example case: axial split pump
The Sulzer Type HSB pump is a single stage, double suction, axial split, double volute, between
bearings pump (Figure 4a). It is commonly used in pipelines at speeds between 1800-3600 rpm.
The size of the pump is small relative to its nozzle size. Additionally, connected piping emerging
from underground can cause high moments applied to the nozzles. Therefore, excessive shaft
misalignment resulting from nozzle loads represents a major mode of failure for this type of pump.
Figure 4 Sulzer Type HSB pump: a single stage, double suction, axial split, double volute, between
bearings pump. a) Photograph, b) Geometry model of the pump on the baseplate, used for the finite
element analysis. The coordinate system defined by ISO 13709 for the specification of nozzle load
components is shown.
3.2. Numerical calculation of unit load cases
The geometry of the pump mounted on a baseplate is derived from 3D CAD data (NX, Siemens)
and prepared for FEA using DesignModeler (Figure 4b). For this example no motor is considered.
The displacements are calculated with respect to the undeformed reference configuration instead.
The commercial finite element software ANSYS Workbench (ANSYS Inc.) is used for this analysis.
The full pump is modeled using SOLID186 and SOLID187 elements. Moreover, the shaft is
modeled by BEAM188 elements, assigning the local cross-sections. It is connected to the pump
casing at the locations of the bearings by using joint elements constraining the respective degrees
of freedom for radial and axial bearings. For this analysis the solution in terms of displacements is
evaluated, which allows for a relatively coarse mesh (here approximately 500’000 nodes). Two
) unit load cases applying unit forces and unit moments in
nozzles require setting up twelve (
the three coordinate directions of each nozzle coordinate system (Figure 4b and Figure 5). It is to
be noted that at this point the values for the nozzle loads are not required yet. Thus, the FE
8
analyses set up a generic framework which is to be “filled” during the post -processing procedure
with the specified loads. Following the solution of the unit load cases, the vectors of structural
degrees of freedom are extracted by inserting user defined post -processing commands (command
snippet) which allow for accessing the nodal results of the FE analysis. The notation “vertex” of
paragraph 2.4 is therefore replaced by “node”. These structural solution vectors of the unit load
cases are exported as .txt-files, readable by any application for further data processing.
Figure 5 Definition of twelve unit load cases, six unit loads are applied at the suction nozzle (TOP)
and six unit loads are applied at the discharge nozzle. The components of the loads refer to the local
nozzle coordinate systems defined according to ISO 13709 [5].
3.3. Implementation of the post-processing procedure
In the present work, Microsoft Excel (Microsoft Corporation) is used to perform the post-processing
procedure. The structural vectors of degrees of freedom are imported and combined as the
columns of the structural compliance matrix (equation 4), representing the overall compliance of
the system investigated. It is explicitly given in equation 14.
9
!!!!!!
⎧ !!
⎫
!!!!!
⎪!!! ⎪
!!!!! ⎪
⎪!!!
⎪ !!!!! ⎪
⎪!!!! ⎪
!!!!!
!!!!! ⎪
⎪!!!!
⎡ !!!!!!!!
⎪ !!!!! ⎪
!!!!!
⎢ !!!!
!!!!
!!!
!
=
⎢
!!!!!
⎨!!! ⎬ ⎢ ⋮
!!!!! ⎪
!!!!!
⎪!!!
!!!!!
!!!
!
⎪ !!!!! ⎪ ⎣
⎪!!! ⎪
!!!!! ⎪
⎪!!!!
⎪ !!!!! ⎪
⎪!!!! ⎪
!!!!!
⎩!!!!
⎭
!!!!!
!!!!
!!!
!
!!!!!
!!!!
!!!
⋮
⋮
⋮
…
…
!!!!!
!!!!
!!!
!
…
…
!
…
!!!!!
⎧ !!! ⎫
!
⎪ !!!! ⎪
⎪ !! ⎪
⎪!!!!! ⎪
!!!!!
!!!
⎪
!!!!
!!! ⎪!!
! ⎤
⎪!!!!! ⎪
⋮ ⎥
⎥ ! !!!
⋮ ⎥ ⎨ !!!! ⎬
!!!!!
⎪ !! ⎪
!!!!!
!!!
! ⎦
⎪ !!!!! ⎪
⎪ !!!! ⎪
⎪ !!!! ⎪
⎪ !! ⎪
⎩ !!!!! ⎭
(14)
Nozzle loads are specified as unsigned components as exemplarily shown in Table 2. Allowable
values are determined based on the type of coupling selected and on internal safety factors ( Table
3).
Table 2 Nozzle loads specified by unsigned components referring to the local nozzle coordinate
system shown in Figure 4b. Note: these values are exemplary values based on twice the values
specified in ISO 13709 [5].
Nozzle
Size
Fx [N]
Fy [N]
Fz [N]
Mx [Nm]
My [Nm]
Mz [Nm]
Suction
580
26'689
30'249
19'890
21'632
11'294
15'771
Discharge
490
21'634
25'194
16'508
18'022
9'242
13'225
Table 3 Allowable values for shaft misalignment. Note: these values are exemplary values.
Type of misalignment
Allowable limit
Axial [mm]
3
Radial [mm]
0.7
Angular [°]
0.165
Using the matrix multiplication tool in Microsoft Excel (Microsoft Corporation), the structural
compliance matrix is applied to all possible combinations of signed nozzle load vectors ( ), a
time-efficient procedure even if performed
times. The corresponding axial, radial and angular
shaft misalignments are calculated according to equations 5-9. Finally the worst-case is chosen
according to the most critical type of shaft misalignment, in this case the radial misalignment.
3.4. Results
The different types of shaft misalignment for all combinations of signed nozzle loads are shown in
Figure 6. The worst case in terms of radial misalignment is marked by the blue circle surrounding
all cases (Figure 6a). The corresponding displacement vector is
[mm]. It
can be seen in Figure 6b that the original, unsigned case represents any arbitrary case, which
might be far from the worst case. For the same case, the diagrams of angular and axial
misalignment are shown in Figure 6c and d.
10
Figure 6 a) All cases of radial misalignment due to all combinations of signed nozzle load
components are drawn as blue circles. The worst case is marked by the blue circle surrounding all
cases. The red circle shows the maximum allowable radial misalignment. b) Zoomed-in view, The
green dot shows the radial misalignment due to the original, unsigned nozzle load combination. c)
Angular misalignment: The values represent the angular misalignment corresponding to the worst
case radial misalignment. d) Axial misalignment: The values represent the axial misalignment
corresponding to the worst case radial misalignment.
3.5. Validation
For validation of the resulting shaft misalignment and in order to evaluate the full results of the
worst case scenario the solution superposition functionality in ANSYS Workbench (ANSYS Inc.) is
used. Following the idea of equation 12, the signed nozzle load components are the multipliers for
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the unit load cases (Figure 7). The calculated shaft end displacement vector is equal to
. [mm], resulting in a total deformation of 0.405 mm.
Figure 7 The superposition functionality in ANSYS Workbench (ANSYS Inc.) is used to calculate the
full model solution applying the worst case combination of nozzle loads. Furthermore, the results are
used for validating the results found by the post-processing procedure using Microsoft Excel
(Microsoft Corporation).
In general, the worst-case nozzle loads identified through this linear post-processing methodology
can be taken as inputs for any finite-element analysis, e.g. non-linear analyses.
4. Discussion and conclusions
A new methodology to comprehensively and efficiently extract the worst case nozzle load
combination with respect to shaft misalignment is presented. Efficiency is derived by reducing the
number of necessary and costly FEA significantly and replacing them by a linear superposition
approach. The proposed method ensures quality by comprehensively covering all possible load
combinations. Moreover, the procedure is standardized by implementation in a spreadsheet. It is
expected to be well accepted by customers in particular as specifications of worst -case nozzle load
combinations are not needed any more. As FE calculations are only needed to set up the
framework, i.e. the structural compliance matrix, this calculation scheme allows for quick and
flexible reactions to changes in nozzle load specifications.
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References
[1] Boiler and Pressure Vessel Code, American Society of Mechanical Engineers, 2011.
[2] EN13445 Unfired Pressure Vessels, European Committee for Standardization, 2012.
[3] M. Singer und T. Johne, «Design of Pump Casings: Guidelines for a Systematic Evaluation of
Centrifugal Pump Pressure Boundary Failure Modes and their Mechanisms,» Proceedings of
the Twenty-Ninth International Pump Users Symposium, 2013.
[4] J. Piotrowski, Shaft Alignment Handbook, CRC Press, Third Edition, 2006.
[5] ISO 13709: Centrifugal Pumps for Petroleum, Petrochemcial and Natural Gas Industries,
Geneva Switzerland: International Organization for Standardization, Second Edition, 2009.
Acknowledgements
Disclaimer
All data provided in this document is non-binding.
This data serves informational purposes only and is especially not guaranteed in any way.
Depending on the subsequent specific individual projects, the relevant data may be subject to
changes and will be assessed and determined individually for each project. This will depend on the
particular characteristics of each individual project, especially specific site and operational
conditions.
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