Stress Analysis of a Hydraulic Actuator Based on
Accumulator Response
by
John C. Connor
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December, 2015
(For Graduation May 2016)
i
© Copyright 2015
by
John C. Connor
All Rights Reserved
ii
CONTENTS
List of Tables ..................................................................................................................... v
List of Figures ................................................................................................................... vi
List of Symbols ................................................................................................................ vii
Acknowledgements........................................................................................................... ix
Abstract .............................................................................................................................. x
1. Introduction.................................................................................................................. 1
2. Methodology ................................................................................................................ 2
2.1
Hydraulic System Assumptions ......................................................................... 3
2.2
Analysis Input Values ........................................................................................ 3
2.3
Material Properties and Allowable Stresses ....................................................... 3
2.4
Traditional Stress Calculation Methodology ..................................................... 4
2.5
2.4.1
Stress at Sections Far from an End Condition ....................................... 4
2.4.2
Principal Stress Combination ................................................................. 6
2.4.3
Stresses at the End Condition................................................................. 6
Finite Element Analysis ..................................................................................... 6
3. Results and Discussion ................................................................................................ 9
3.1
3.2
3.3
Actuator System without an Accumulator ......................................................... 9
3.1.1
Traditional Stress Analysis .................................................................... 9
3.1.2
Abaqus Stress Analysis and Optimization ............................................. 9
3.1.3
Weight Evaluation and Comparison .................................................... 12
Actuator System with an Accumulator ............................................................ 13
3.2.1
Traditional Stress Analysis .................................................................. 13
3.2.2
Abaqus Stress Analysis and Optimization ........................................... 13
3.2.3
Weight Evaluation and Comparison .................................................... 15
Determination of the Appropriate Hydraulic System ...................................... 16
4. Conclusion ................................................................................................................. 17
iii
4.1
Suggestions of Future Research ....................................................................... 18
5. References.................................................................................................................. 19
Appendices ...................................................................................................................... 20
A – Determination of Inputs ...................................................................................... 20
B – Traditional Strength of Material Calculation ...................................................... 23
C – Abaqus CAE Finite Element Analysis ................................................................ 28
iv
List of Tables
Table 1 - Material Properties ............................................................................................. 3
Table 2 - Allowable Stresses Based on the Design Criteria .............................................. 4
Table 3 - Traditional Stress Analysis of the Actuator System without an Accumulator ... 9
Table 4 - Abaqus Analysis based on Traditional Approach ............................................ 10
Table 5 - Optimized Actuator without an Accumulator .................................................. 12
Table 6 - Actuator without an Accumulator Weight Comparison ................................... 12
Table 7 - Traditional Stress Analysis of the Actuator System without an Accumulator . 13
Table 8 - Abaqus Analysis based on Traditional Approach ............................................ 14
Table 9 - Optimized Actuator with an Accumulator ....................................................... 15
Table 10 - Actuator without an Accumulator Weight Comparison ................................. 15
Table 11 - Summary of Design Considerations ............................................................... 16
Table 12 - Input Variables from Previous Analysis ........................................................ 20
Table 13 - Maximum Pressure from Previous Analysis .................................................. 22
Table 14 - Abaqus Variables for the Analysis ................................................................. 29
Table 15 - Abaqus Material Properties ............................................................................ 29
Table 16 - Actuator Geometry Based on Abaqus Initial Results..................................... 33
v
List of Figures
Figure 1 - System Diagram ................................................................................................ 2
Figure 2 – Eight Node Linear Element [7] ........................................................................ 7
Figure 3 - Twenty Node Quadratic Element [7] ................................................................ 8
Figure 4 - Stress Plot of the Actuator without an Accumulator using Linear Elements . 10
Figure 5 - Stress Plot of the Actuator without an Accumulator using Quadratic Elements
......................................................................................................................................... 10
Figure 6 - Stress Plot of the Mesh Excitation Study ........................................................ 11
Figure 7 – Stress Plot of the Optimized Actuator without an Accumulator for Abaqus
Results.............................................................................................................................. 12
Figure 8 - Stress Plot of the Actuator with an Accumulator............................................ 14
Figure 9 - Stress Plot of the Optimized Actuator with an Accumulator using Abaqus
Results.............................................................................................................................. 15
Figure 10 - Actuator Mockup .......................................................................................... 17
Figure 11 - Complete Schematic of the Hydraulic System Previously Analyzed ........... 20
Figure 12 - Screen Shot of Euler Method Analysis ......................................................... 22
Figure 13 - Free Body Diagram Stress in a Cylinder ...................................................... 24
Figure 14 - Free Body Diagram Pressure on a Flat Plate ................................................ 26
Figure 15 - Actuator Modeled in Abaqus Based on Maple Calculated Dimensions ....... 28
Figure 16 - Mesh of the Modeled Actuator ..................................................................... 30
Figure 17 - Boundary Conditions Placed on the Actuator ............................................... 31
Figure 18 - Loads Placed on the Actuator ....................................................................... 32
vi
List of Symbols
σrr
Radial stress in the actuator (Pa)
σθθ
Tangential stress in the actuator (Pa)
σzz
Axial stress in the actuator (Pa)
σvon
Von Mises stress (Pa)
σB
Bending stress (Pa)
p1
Internal pressure (Pa)
p2
External pressure (Pa)
a
Inner radius (m)
b
Outside radius (m)
r
Radius where the stress is being evaluated (m)
M
Bending moment per unit length (N*m/m)
KM
Special case plate constant from Roark’s
q
Pressure differential across the flat plate (Pa)
t
Thickness of the end cap (m)
Acyl
Area of the hydraulic actuator piston (m2)
Ap
Area of the accumulator piston (m2)
Arod
Area of the hydraulic actuator rod (m2)
h
Time step interval (s)
L
Length/ height of the accumulator (m)
mcyl
Mass of actuator piston and attached component (kg)
mp
Mass of the accumulator piston (kg)
Patm
Atmospheric Pressure (Pa)
Pl
Fluid pressure in the actuator with an accumulator (Pa)
Pg
Gas pressure in the accumulator (Pa)
Pl_NA
Fluid pressure in the actuator without an accumulator (Pa)
s
Length/stroke of the linear actuator (m)
x
Position of the actuator piston (m)
x’
Velocity of the actuator piston (m/s)
x’’
Acceleration of the actuator piston (m/s2)
vii
y
Position of the accumulator piston (m)
y’
Velocity of the accumulator piston (m/s)
y’’
Acceleration of the accumulator piston (m/s2)
rho
Unit weight of the material (N/m3)
viii
Acknowledgements
I would like to thank all of my professors at Rensselaer Polytechnic Institute for their
support over the last three years, with special thanks to Professor Ernesto GutierrezMiravete who advised me over the last several months. I would also like to thank my
girlfriend Stefanie who always pushed me to focus and work on my project when I was
struggling.
ix
Abstract
Hydraulic shock is a phenomenon which can occur when an external force is applied to
an actuator which in turn causes a sudden pressure transient. In Leonid Simkin’s
Analysis of Accumulator Response to an External Force acting on a Hydraulic Actuator,
the hydraulically operated suspension system for a dump truck was analyzed to
determine the response of an accumulator using several methods. This project will build
upon Mr. Simkin’s analysis and analyze the stresses in the hydraulic actuator due to the
external force. Inputs based on the previous analysis will be used to evaluate the
mechanical stresses in the actuator using traditional strength of material calculations and
a Finite Element Analysis (FEA). Microsoft Excel will be used to compute inputs for
both types of analyses and calculate the results based on the traditional approach. The
FEA will utilize Abaqus Standard to determine the stresses using the peak pressures in
the hydraulic systems with and without an accumulator. The results of the analysis will
conclude if the accumulator is an appropriate addition into the hydraulic system based on
the actuator stresses.
x
1. Introduction
A hydraulic actuator system is uses hydraulic fluid to achieve a motion. These systems
can be a very complex system where several fields of engineering can intersect. In this
project the system outlined in Analysis of Accumulator Response to an External Force
acting on a Hydraulic Actuator [1] will be further evaluated to determine the mechanical
stresses in the actuator due to an external load. The stresses will then be evaluated
against established criteria in the American Society of Mechanical Engineers (ASME)
Boiler Pressure Vessel Code (BPVC) [2].
Pressure transients are the primary loads in hydraulic systems and can be due to several
types of loading conditions which are not typically seen in normal operation. The
pressure rise the in hydraulic system that was evaluated in the previous paper was due to
an external force being applied to the actuator. This pressure spike can occur when a
sudden impact is imparted into the system. An example of this is when a hydraulic shock
in a car absorbs the force of hitting a pot hole. Pressure transients must be evaluated and
the appropriate factor of safeties must be used to ensure that no part of the system
ruptures or bursts during operation which could cause failure and/or bodily harm.
This paper will evaluate the stresses in the hydraulic actuator for a system with and
without an accumulator. By determining the minimum required wall thickness to ensure
that all of the safety factors outlined in the ASME BPVC are met. The analysis will
ensure that the actuator is designed with the appropriate strength and robustness, while
also minimizing the weight and space envelope the actuator encompasses.
1
2. Methodology
Utilizing the previous, this paper will evaluate the hydraulic actuator with and without
an accumulator, by using the traditional strength of materials calculations for the peak
pressures seen in the actuator. Once a baseline wall thickness is calculated the actuator
will be modeled in Abaqus CAE, using the peak static pressures. By using these two
analysis methods the hydraulic actuator will be optimized to provide the appropriate
factors of safety. Figure 1 shows a diagram of the two actuator systems that will be
evaluated.
Fin
Fin
Actuator
Closed System
Actuator with
Accumulator
ṁ
Figure 1 - System Diagram
2
2.1 Hydraulic System Assumptions
The following assumptions are for the system which affects both types of stress analyses
preformed:
1. The actuator has zero gauge pressure at the beginning of the analysis.
2. The liquid in the hydraulic system is incompressible.
3. The actuator has an external pressure equal to one atmosphere.
4. The actuator body is made of aluminum.
2.2 Analysis Input Values
The input and calculated output values are derived from the previous paper. Table 12 in
Appendix A shows the inputs used in Analysis of Accumulator Response to an External
Force acting on a Hydraulic Actuator to determine the pressure in the cylinder. Using
these inputs the previously calculated pressures were recreated and are shown in
Appendix A. The Euler method was chosen for its simplicity and the step time to
determine the peak pressure was set to 0.005 seconds. The step time was chosen based
on the previous research which, had shown the Euler and Runge-Kutta method as stable
and produce similar peak pressures for the step size.
2.3 Material Properties and Allowable Stresses
The actuator system will be evaluated for an aluminum body; this will guarantee that the
actuator has the adequate strength required and does not weigh an excessive amount.
Table 1 shows the material properties that will be used in this analysis.
Table 1 - Material Properties
Material
2011 T6
Aluminum Alloy
Yield Strength,
Ultimate
Unit Weight,
MPa
Strength, MPa
kN/m3
169
324
26.6
3
Source
[3]
The ASME BPVC [2] states that the working pressures of actuators shall be designed to
the following factors of safety:

1.5 factor of safety on material yield strength

3.5 factor of safety on material ultimate strength
Using these design criteria Table 2, was developed to determine the maximum allowable
stresses that the actuators can experience under the pressure rise for each material. Table
2 shows the ultimate stress criteria govern for the aluminum alloy chosen.
Table 2 - Allowable Stresses Based on the Design Criteria
Material
2011 T6
Aluminum Alloy
Yield Allowable
Ultimate Allowable
Stress, MPa
Stress, MPa
112.7
92.6
The design criterion assumes the actuator body is forged; this removes the requirement
in the ASME BPVC to apply a quality factor to the design criteria. This quality factor
(up to 80% of the material strength [2]) applies only to cast pressure vessels. It should be
be noted that the quality factor for a casting can be reduced if significant quality testing
is done as specified in the ASME BPVC.
2.4 Traditional Stress Calculation Methodology
The following sections will outline the methodology used to determine the stress in a
thick walled cylinder, due to internal pressure loading, as well as the flat plate equations
used to determine the stress at the actuator end cap. Since, no information could be
found to evaluate the interface between the end cap and the cylinder wall it is assumed,
using engineering judgement, that this region is stiffened by the end cap and will have
less stress then the cylinder wall. This assumption will be validated or disproven with the
finite element analysis.
2.4.1
Stress at Sections Far from an End Condition
To evaluate the stress at the midsection rotational symmetry must be assumed and the
effects of the end of the cylinder must be neglected [4]. Due to the rotational symmetry
4
the stress will be evaluated using a rotational coordinate system. The cylindrical stresses
acting in the actuator are:

Circumferential (Tangential) Stress

Axial Stress

Radial Stress
For thin wall cylinders, which have a ratio of inner radius over wall thickness less than
20 [4], a reduced equation which neglects radial stresses can be used. Since, the final
thickness is not known at this time the simplified equations will be ignored and the
“thick-walled” equations will be used. The general equations for the cylinder stresses are
[3]:
𝜎𝑟𝑟 =
𝑝1 𝑎2 − 𝑝2 𝑏 2 𝑎2 𝑏 2 (𝑝1 − 𝑝2 )
− 2 2
𝑏 2 − 𝑎2
𝑟 (𝑏 − 𝑎2 )
𝜎𝜃𝜃 =
𝑝1 𝑎2 − 𝑝2 𝑏 2 𝑎2 𝑏 2 (𝑝1 − 𝑝2 )
+ 2 2
𝑏 2 − 𝑎2
𝑟 (𝑏 − 𝑎2 )
𝜎𝑧𝑧 =
𝑝1 𝑎2 − 𝑝2 𝑏 2
𝑃
−
2
2
2
𝑏 −𝑎
𝜋(𝑏 − 𝑎2 )
Where: σrr is the radial stress in the actuator
σθθ is the tangential stress in the actuator
σzz is the axial stress in the actuator
p1 is the internal pressure
p2 is the external pressure
a is the inner radius
b is the outside radius
r is the radius where the stress is being evaluated
P is the load applied to the end cap
Since the maximum stress will be at the inner surface for a pressurized cylinder “r” is set
equal to “a”. It is then assumed that there is no force applied to the end cap; the thickwalled equations can be simplified further to:
𝜎𝑟𝑟
𝑝1 𝑎2 − 𝑝2 𝑏 2 𝑎2 𝑏 2 (𝑝1 − 𝑝2 )
=
− 2 2
𝑏 2 − 𝑎2
𝑎 (𝑏 − 𝑎2 )
𝜎𝜃𝜃 =
𝜎𝑧𝑧
𝑝1 𝑎2 − 𝑝2 𝑏 2 𝑎2 𝑏 2 (𝑝1 − 𝑝2 )
+ 2 2
𝑏 2 − 𝑎2
𝑎 (𝑏 − 𝑎2 )
𝑝1 𝑎2 − 𝑝2 𝑏 2
𝑃
=
−
𝑏 2 − 𝑎2
𝜋(𝑏 2 − 𝑎2 )
5
2.4.2
Principal Stress Combination
It is important to note that the calculated stresses are principal stresses and need to be
transformed into a Von Mises stress, σvon, so that it can be compared with the finite
element model. The Von Mises stress equation is shown below:
1
𝜎𝑣𝑜𝑛 = √ [(𝜎𝜃𝜃 − 𝜎𝑟𝑟 )2 + (𝜎𝑟𝑟 − 𝜎𝑧𝑧 )2 + (𝜎𝑧𝑧 − 𝜎𝜃𝜃 )2 ]
2
2.4.3
Stresses at the End Condition
The stress analysis the end condition will assume that the end cap is a flat plate with a
fixed inner diameter and a simply supported outer diameter. This assumption is
appropriate because the outlet of the actuator is thought to increase the relative stiffness
of the end cap, while the intersection with the end cap and cylindrical body will cause
the actuator to flex more. Roark’s Formulas for Stress and Strain [5] gives the formulas
for a flat plate under pressure loads. Case 2d in Table 11.2 of Reference [5], defines the
formulas for a simply supported plate with a fixed inner diameter. Since the pressure is
applied to the entire area of the plate, there are special cases can such as, the bending
moment equation for the flat plate is:
𝑀 = 𝐾𝑀 𝑞𝑎2
Where: M is the bending moment per unit length
KM is the special case plate constant from Roark’s
q is the pressure differential across the flat plate
Roark’s specifies that when the flat plate equations are used the following bending stress
equation must be used to determine the maximum stress:
𝜎𝐵 =
6𝑀
𝑡2
Where: σB is the bending stress
t is the thickness of the end cap
2.5 Finite Element Analysis
Abaqus CAE will be used for the finite element analysis to prove that the system is
sufficiently designed. The Abaqus manual [6] was used to determine if the actuator
should be modeled using Abaqus/Standard (a static model) or Abaqus/Explicit (a
6
dynamic model). After reviewing the software manual it was determined that both types
of modeling methods could have be used. However, Abaqus/Standard however reduces
the computational time required to run the analysis.
Abaqus/Standard is a solution based finite element analysis that is ideal for static and
low speed dynamics events, which due to the solver type yields highly accurate results.
Due to the nature of Abaqus/Standard the problem to be analyzed must be sufficiently
constrained to allow the solver to reach a steady state. The actuator impact load is a
dynamic event; however, since the maximum pressure over time will govern the stress in
the actuator, and external loads that are not present, the actuator can be evaluated using
peak pressures in a static analysis.
Abaqus can use several types of elements to compute the results of the analysis: beams
(1 dimensional) elements, shell (2 dimensional) elements, and solid (3 dimensional)
elements. Solid elements will be used to show the stress through the cylinder wall and
will allow a discrete pressure to be applied to the inside and outside surface of the
modeled actuator, as well as, symmetry boundary conditions to reduce the total model
size.
Two types of solid elements will be used and compared in this analysis: linear and
quadratic. The linear elements used are eight node cubic elements with incompatible
modes referred to as C3D8I elements in Abaqus (see Figure 2).
Figure 2 – Eight Node Linear Element [7]
7
C3D8I utilize bubble functions that have zero value at all nodes and nonzero values in
between, which prevent shear locking and reduce volumetric locking [7]. These elements
are ideal to reduce computational resources by reducing the need for multiple elements
through the membrane. C3D8I elements tend to produce accurate results in shear and
bending. Since the pressure load causes bending and shear in the actuator depending on
the location being evaluated this element is ideal to get a basic understanding of the
stresses in the cylinder. However, due to the formulation of the linear elements, without
increasing the mesh density, the peak stresses may not be shown. To better understand
the peak stresses in the cylinder a quadratic formulation will provide the most accurate
prediction.
The quadratic elements that will be used consist of twenty node cubic elements, known
as C3D20 (see Figure 3). These elements use full integration and are excellent for linear
elastic analyses due to the location of the integration points. These locations allow for
accuracy of stress concentrations at the surface of structures. Which is ideal for the stress
analysis on the actuator, since the peak stress in the cylinder due to pressure will be
located on the surface.
Figure 3 - Twenty Node Quadratic Element [7]
8
3. Results and Discussion
This section will determine if the accumulator will reduce the space envelope and weight
of the actuator. The accumulator must reduce the actuator weight and or envelope
enough to justify the addition of the accumulator into the system.
3.1 Actuator System without an Accumulator
This section will provide the dimensions of the actuator based on the results from
Appendix B and C for the system without an accumulator. The traditional stress analysis
was run and the dimensions that would satisfy the stress criteria were determined. These
dimensions were then used to develop the initial Abaqus model to evaluate the actuator
without the accumulator.
3.1.1
Traditional Stress Analysis
The stress analysis on the actuator using the hand calculated method is outlined in
Appendix B for the system without an accumulator. The results of the traditional
analysis are shown in Table 3.
Table 3 - Traditional Stress Analysis of the Actuator System without an Accumulator
Actuator Area
Thickness (mm)
Stress (MPa)
Cylinder Body
9.95
92.3
End Cap
7.20
92.4
The cylinder body and end cap dimensions are similar with the end cap slightly thinner.
This is to be expected since the end cap was assumed to be stiffer than the cylinder wall.
3.1.2
Abaqus Stress Analysis and Optimization
To evaluate the results of the traditional analysis the actuator was modeled with the
dimensions specified in Section 3.1. The model was run with linear and quadratic
elements to help validate the model without the direct correlation to the traditional
approach. The results of the stress analysis using Abaqus are shown in Figure 4, Figure
5, and Table 4.
9
Figure 4 - Stress Plot of the Actuator without an Accumulator using Linear Elements
Figure 5 - Stress Plot of the Actuator without an Accumulator using Quadratic Elements
Table 4 - Abaqus Analysis based on Traditional Approach
Actuator Area
Thickness (mm)
Linear Elements
Quadratic Elements
Peak Stress (MPa)
Peak Stress (MPa)
Cylinder Body
9.95
93.4
99.3
End Cap
7.20
101.0
146.6
10
The stress results of the Abaqus model show that the linear and quadratic models differ
to a point where the models should be put into question. To resolve this issue another
model with a higher mesh density of 0.0025 instead of 0.01 with quadratic elements. The
results of this analysis are shown below in Figure 6. The results of this analysis show the
quadratic elements were the appropriate choice for the analyses, even though the stresses
are higher. This is because of the stress concentration where the actuator body meets the
end cap grows exponentially. However, the main stresses in the cylinder wall and end
cap do not change with a higher mesh density. As such the quadratic elements with a
mesh density of 0.01 will be used in the subsequent analyses.
Figure 6 - Stress Plot of the Mesh Excitation Study
Figure 5 identified several areas of overstresses in the cylinder body and the end cap
location; primarily where the endcap and the cylinder body intersect. To correct for the
overstresses found using Abaqus the part was modified with the thicknesses in Table 5.
These modifications allowed the actuator to meet the design criteria when evaluated in
Abaqus. The final stress plot is shown in Figure 7.
11
Figure 7 – Stress Plot of the Optimized Actuator without an Accumulator for Abaqus Results
Table 5 - Optimized Actuator without an Accumulator
Actuator Area
3.1.3
Thickness (mm)
Quadratic Elements
Peak Stress (MPa)
Cylinder Body
10.85
92.5
End Cap
11.75
91.4
Weight Evaluation and Comparison
The weight of the actuators designed using the two methods were determined in
Appendix B and C. Table 6 shows that the minor changes to the actuator body had a
minimal effect on the weight of the actuator. The 10% increase in the weight is
significant; however, the overall weight of the actuator is inconsequential even though it
is higher.
Table 6 - Actuator without an Accumulator Weight Comparison
Analysis Method
Weight (N)
Traditional
389
Abaqus
428
12
3.2 Actuator System with an Accumulator
The stress analyses were reanalyzed assuming an accumulator was added to the system,
reducing the peak pressures in the actuator. In theory, the addition of the accumulator
will allow the actuator space envelope to decrease and justify the addition of the
accumulator.
3.2.1
Traditional Stress Analysis
The stress analysis on the actuator using the traditional method is outlined in Appendix
B for the system with an accumulator. The results of the analysis are shown in Table 7.
Table 7 - Traditional Stress Analysis of the Actuator System without an Accumulator
Actuator Area
Thickness (mm)
Stress (MPa)
Cylinder Body
7.05
92.3
End Cap
6.25
91.6
The results are similar to the previous analysis and give a good foundation for the
Abaqus model. Based on the results in Section 3.2.1 the actuator dimensions are
expected to increase.
3.2.2
Abaqus Stress Analysis and Optimization
To evaluate the results of the traditional analysis the actuator was modeled again, this
time with the dimensions specified in Section 3.2.1. Since the basic modeling technique
was validated in Section 3.1.2, the actuator was only run with quadratic elements. The
results of the stress analysis using Abaqus are shown in Figure 8 and Table 8.
13
Figure 8 - Stress Plot of the Actuator with an Accumulator
Table 8 - Abaqus Analysis based on Traditional Approach
Actuator Area
Thickness (mm)
Quadratic Elements
Peak Stress (MPa)
Cylinder Body
7.05
107.7
End Cap
6.25
107.7
The results of the Abaqus model show overstresses in the areas previously seen in
Section 3.1.2. To correct for the overstresses found using Abaqus the actuator was
modified with the thicknesses in Table 9. These modifications allowed the actuator to
meet the design criteria when evaluated in Abaqus. The final stress plot is shown in
Figure 9.
14
Figure 9 - Stress Plot of the Optimized Actuator with an Accumulator using Abaqus Results
Table 9 - Optimized Actuator with an Accumulator
Actuator Area
3.2.3
Thickness (mm)
Quadratic Elements
Peak Stress (MPa)
Cylinder Body
7.65
88.9
End Cap
7.65
92.6
Weight Evaluation and Comparison
The weights of the actuators with an accumulator in the system were determined in
Appendix B and C for the two analysis methods. Table 10 shows that the minor changes
to the actuator body had an approximately 9% increase in the actuator weight. However,
the actuator is still relatively light and the change in weight would not affect the system.
Table 10 - Actuator without an Accumulator Weight Comparison
Analysis Method
Weight (N)
Traditional
271
Abaqus
295
15
3.3 Determination of the Appropriate Hydraulic System
The main points of comparison between the two actuators that were designed with and
without an accumulator are the outer diameter and the actuator weight. Table 11 shows
these parameters and the percent difference between them. This indicates that the outer
diameter will not be a major factor in the decision for whether an accumulator is needed
in the system or that it is justified. However, the weight of the actuator is a significant
consideration with a 31% difference between the system types. Even though the actuator
without an accumulator weighs more than one with an accumulator the weight difference
does not justify the need for an accumulator since the weight of the piping and the
accumulator will total higher than the weight that is saved. As such the actuator without
an accumulator is the appropriate choice without evaluating the other components of the
system.
Table 11 - Summary of Design Considerations
Design
Percent
No Accumulator
With Accumulator
Outer Diameter [m]
0.1667
0.1603
3.8%
Actuator Weight [N]
428
295
31%
Considerations
16
Difference
4. Conclusion
Based on the results in Section 3 it can be shown that the Abaqus model predicted high
stresses in the actuators then the traditional stress calculations. The traditional
calculations may not have predicted the precise peak stresses due to the boundary
conditions at the end cap. It can be concluded based on the required factors of safety in
the ASME BPVC [2]; the actuator would not have failed even if the peak stresses in the
actuator were slightly higher than predicted using the traditional analysis. Since the
ASME BPVC is standard for all analyses, the overstresses identified using Abaqus
should be removed to produce a product that meets the criteria explicitly.
Section 3 also identified that the ideal system when looking at only the actuator stresses,
is the system without an accumulator. This is due to the relatively minute change in
outer diameter between the actuators that were designed. Additionally the weight of the
actuator without an accumulator was not excessive; which means that the weight of the
accumulator cannot be justified.
Figure 10 shows a mockup of the final designed actuator.
Figure 10 - Actuator Mockup
17
4.1 Suggestions of Future Research
Future research could go through a complete stress analysis of a hydraulic system with
an actuator and an accumulator. This analysis could couple the stress analysis done in
this paper to the previous analysis [1] where the pressures in the cylinder and
accumulator were determined. This would provide a complete view of the hydraulic
system to see how the system reacts in a dynamic environment to a given force input.
Additional research could also focus on verifying and reevaluating the assumptions
made for the end cap in the traditional calculations. This would be beneficial since no
common end cap assumptions were found while researching for this project.
18
5. References
[1] L. Simkin, "Analysis of Accumulator Reponse to an External Force acting on a
Hydraulic Actuator," Rensselaer Polytechnic Institute, Hartford, 2012.
[2] American Society of Mechanical Engineers, Boiler Pressure Vessel Code, Rules for
Construction of Pressure Vessels, Section II Part D, New York: American Society
of Mechanical Engineers, 2011.
[3] R. G. Budynas, J. K. Nisbett and J. E. Shigley, Shigley's Mechanical Engineering
Design, New York: McGraw-Hill, 2011.
[4] A. P. Boresi and R. J. Schmidt, Advanced Mechanics of Materials, Hoboken: John
Wiley & Sons, Inc., 2003.
[5] W. C. Young and R. G. Budynas, Roark's Formulas for Stress and Strain, New
York: McGraw Hill, 2002.
[6] Simulia, Abaqus 6.12 User Manual, Providence: Dassault Systèmes, 2012.
[7] G. Dhondt, "Element Types," Massachusetts Institute of Technology, 02 03 2013.
[Online].
Available:
http://web.mit.edu/calculix_v2.7/CalculiX/ccx_2.7/doc/ccx/node25.html. [Accessed
25 12 2015].
[8] "Hydraulic Cylinders: WW: Specification," WEEMAC, [Online]. Available:
http://www.weemac.fi/pages/products/hydraulic-cylinders/ww/specification.php.
[Accessed 25 March 2012].
[9] "Piston Tyle Accumulators: Type-AP," EPE Process Filters and Accumulators,
[Online].
Available:
http://www.accumulatorsandfilters.com
/accumulators/Piston.pdf. [Accessed 10 March 2012].
[10] Maplesoft, Maple User Manual, Waterloo: Waterloo Maple Inc., 2008.
[11] Abaqus,
Inc.,
"imechanica,"
Dassault
Systèmes,
http://imechanica.org/files/0-overview%20Explicit.pdf.
2010].
19
[Online].
[Accessed
Available:
25
October
Appendices
A – Determination of Inputs
The results from the previous analysis were recreated to provide the necessary inputs for
the stress calculation. Table 12 provides the input variables from the previous analysis
and Figure 11 shows a schematic of the analysis.
Table 12 - Input Variables from Previous Analysis
Variable
Value [Unit]
Source
Acyl
0.0165 [m2] Based Cylinder Dimensions 125/70, Model WW [8]
Ap
0.0491 [m2]
Arod
h
Const
L
mcyl
mp
Patm
s
Vg
γ
Based Accumulator bore size 250 mm, Model AP100 [9]
2
0.0038 [m ] Based Cylinder Dimensions 125/70, Model WW [8]
0.005 [s] Chosen based on the results of the previous analysis
2440509568 [Pa*L1.4] Const = PgVgγ ,where Pg = 3,867,947 Pa
2.037 [m] L = (Vg/(1000 𝑚^3/𝑙))/𝐴𝑝
Mass acting on the center axle of a Caterpillar 740
Articulated Dump Truck
Based on Accumulator bore size 250mm Model
AP-100, aluminum density of 2810 kg/m3 and
57.9 [kg]
scaled dimensions in [1].
101353 [Pa] Standard Atmospheric Pressure
6380 [kg]
Stroke chosen for the actuator to make sure the
actuator piston does not reach the end of the
3 [m]
actuator during the impulse force.
Based on Accumulator bore size of 250mm, Model
100 [L]
AP-100 [9]
1.4 Ratio of Specific Heat for Air
[1]
Figure 11 - Complete Schematic of the Hydraulic System Previously Analyzed
20
Once the input variables are set the following initial conditions are used in the previous
analysis:
𝑥0 = 3 𝑚
𝑥0′ = 0 𝑚/𝑠
𝑦0 = 0 𝑚
𝑦′0 = 0 𝑚/𝑠
𝑦′′0 = 0 𝑚/𝑠 2
Where: x0 is the initial position of the actuator piston
x0’ is the initial velocity of the actuator piston
y0 is the initial position of the accumulator piston
y0’ is the initial velocity of the accumulator piston
y0’’ is the initial acceleration of the accumulator piston
Using the above initial conditions and the inputs provided in Table 12 the following
equations detailed in the previous analysis were solved for k=1 to N; to determine the
pressure rise in the hydraulic system with an accumulator.
𝑉𝑔,𝑘−1 = 1000𝐴𝑝 (𝐿 − 𝑦𝑘−1 )
−𝛾
𝑃𝑔,𝑘−1 = (𝐶𝑜𝑠𝑛𝑡)𝑉𝑔,𝑘−1
𝑚𝑝 ′′
𝑃𝑙,𝑘−1 = ( ) 𝑦𝑘−1
+ 𝑃𝑔,𝑘−1
𝐴𝑝
𝐴𝑐𝑦𝑙
𝐴𝑐𝑦𝑙 − 𝐴𝑎𝑡𝑚
𝐹𝑘−1
′′
𝑥𝑘−1
=(
) 𝑃𝑙,𝑘−1 − (
) 𝑃𝑎𝑡𝑚 −
𝑚𝑐𝑦𝑙
𝑚𝑐𝑦𝑙
𝑚𝑐𝑦𝑙
𝑦𝑘′ − 1 = − (
𝑦𝑘′′ =
𝐴𝑐𝑦𝑙 ′
) 𝑥𝑘−1
𝐴𝑝
′
𝑦𝑘′ − 𝑦𝑘−1
ℎ
′
(ℎ)
𝑦𝑘 = 𝑦𝑘−1 + 𝑦𝑘−1
′
′′
𝑥𝑘′ = 𝑥𝑘−1
+ 𝑥𝑘−1
(ℎ)
′
(ℎ)
𝑥𝑘 = 𝑥𝑘−1 + 𝑥𝑘−1
Where: x is the position of the actuator piston
x’ is the velocity of the actuator piston
x’’ is the acceleration of the actuator piston
21
y is the position of the accumulator piston
y’ is the velocity of the accumulator piston
y’’ is the acceleration of the accumulator piston
Pl is the fluid pressure in the actuator with an accumulator
Pg is the gas pressure in the accumulator
For the case of no accumulator the pressure of the liquid will equal the sum of the
impulse force as shown below:
𝐴𝑐𝑦𝑙 −𝐴𝑟𝑜𝑑
𝑃𝑙_𝑁𝐴 ,𝑘−1 = (
𝐴𝑐𝑦𝑙
) 𝑃𝑎𝑡𝑚 +
𝐹𝑘−1
𝐴𝑐𝑦𝑙
Where: Pl_NA is the fluid pressure in the actuator without an accumulator
Figure 12 shows a snapshot of the excel analysis used to compute the pressures given the
force and time step used in the previous analysis.
Figure 12 - Screen Shot of Euler Method Analysis
Table 13 shows the maximum pressures in the actuators for the system with and without
the accumulator.
Table 13 - Maximum Pressure from Previous Analysis
No Accumulator [Pa] With Accumulator [Pa]
12189404
9129089
22
B – Traditional Strength of Material Calculation
The following stress analysis was done using Maplesoft Math Solver [10] as a way to
compute the stresses in the wall of the cylinder and at the end caps. The analysis also
includes a weight assessment of the actuator designed using Maplesoft.
Actuator Analysis
Inputs for the analysis
[Pa]
>
[Pa]
>
[Pa]
>
[m]
>
Inner Radius of the Actuator Outlet [m]
>
Ratio of Inner to Outer Radius of the Flat Plate
>
Roark's Flat Plate Special Case Constant - 0.7 Case 2d
>
Length of the Actuator [m]
>
Unit Weight of Aluminum [N/m^3]
>
Stress Analysis of the Pressurized Cylinder Wall
Figure 13 is a free body diagram of the stress analysis for the actuator wall. This analysis
accounts for an internal and external pressure in the cylinder. The analysis optimizes the
actuator wall thickness with and without an accumulator such that the actuator meets the
design criteria in the ASME BPVC.
23
a
b
p1
p2
Figure 13 - Free Body Diagram Stress in a Cylinder
No Accumulator
[m]
>
[m]
>
[Pa]
>
[Pa]
>
[Pa]
>
[Pa]
>
24
With Accumulator
[m]
>
[m]
>
[Pa]
>
[Pa]
>
[Pa]
>
[Pa]
>
Stress Analysis of the Cylinder End Cap
To evaluate the end cap it will be assumed that the end cap is a flat plate with pinned
boundary conditions at the outside radius of the actuator body, see Figure 14. The plate
will also have a hole in the center with a radius equal to the size of the outlet stated in
the previous analysis (0.05m). This analysis will be run for the actuator with and without
an accumulator, and optimize the end cap thickness to meet the ASME BPVC design
criteria.
25
ro
p1
b
b
p2
Figure 14 - Free Body Diagram Pressure on a Flat Plate
No Accumulator
Thickness of the End Cap [m]
>
Pressure Differential Across the Flat Plate [Pa]
>
Moment per Unit Circumference [N*in/in]
>
Bending Stress in the Flat Plate [Pa]
>
With Accumulator
Thickness of the End Cap [m]
>
Pressure Differential across the Flat Plate [Pa]
>
Moment per Unit Circumference [N*in/in]
>
Bending Stress in the Flat Plate [Pa]
>
26
ro
tEC
Weight Evaluation of the Actuator Designed using Maple
The weights of the actuators which were designed using the pressures with and without
the accumulator are calculated below.
No Accumulator
[m]
>
[m]
>
Thickness of the End Cap [m]
>
Volume of the Hollow Cylinder [m^3]
>
Volume of the Spherical End Cap [m^3]
>
Weight of the Actuator Body [N]
>
With Accumulator
[m]
>
[m]
>
Thickness of the End Cap [m]
>
Volume of the Hollow Cylinder [m^3]
>
Volume of the Spherical End Cap [m^3]
>
Weight of the Actuator Body [N]
>
27
C – Abaqus CAE Finite Element Analysis
Abaqus Actuator Part Geometry and Material Properties
Utilizing the symmetry of the actuator cylinder only half of the cylinder will be modeled
as a 3-dimensional solid. Furthermore because the stress in the cylinder wall will not
change once the end conditions are sufficiently far away, the length of the cylinder was
reduced from 3m to 0.5m. These edits to the part geometry are based off the traditional
stress analysis formulations and allow the actuator model to be optimized for Abaqus, to
reduce the overall run time of the finite element model without effecting the results.
Figure 15 shows the actuator modeled in Abaqus and defines the geometric variables;
the values of which are shown in Table 14.
t
a
L
tEC
Lo
ro
t
Figure 15 - Actuator Modeled in Abaqus Based on Maple Calculated Dimensions
28
Table 14 - Abaqus Variables for the Analysis
Variable
Value [Unit]
a
t
L
0.0725 [m] Inner radius of the actuator
0.00995 [m] Thickness of the actuator wall calculated using Maple –
No Accumulator
0.00705 [m] Thickness of the actuator wall calculated using Maple –
Accumulator
0.5 [m] Length of the modeled actuator
0.05 [m] Inner radius of the piping outlet
ro
tEC
Lo
Description
0.00720 [m] Thickness of the actuator end cap using Maple – No
Accumulator
0.00625 [m] Thickness of the actuator end cap using Maple Accumulator
0.1 [m] Length of the outlet
The part was then partitioned to simplify the geometric shapes, allowing Abaqus to mesh
the part using a structured formulation. The material properties applied to the actuator in
the finite element model are shown in Table 15.
Table 15 - Abaqus Material Properties
Material
Material
Elastic Modulus,
Type
GPa
2011 T6
Solid,
Aluminum Alloy
Homogeneous
71.7
Poisson’s Ratio
Source
0.333
[3]
The values in Table 14 will be reevaluated after the model is run to ensure that all that
the safety factors have been appropriately met.
Mesh Attributes
The partitioned actuator then needed to have a mesh generated which would produce
accurate and reliable results without costing considerable computational resources. It
was determined based on the geometry and the element type that was being considered,
that nodes every 0.01m (global seed = 0.01) would be acceptable. The seed choice
creates a part with 1 element thick mesh which is approximately cubic. The cubic shape
29
is ideal for a hexahedral element, but due to the element thickness of the part the default
linear C3D8R (an 8-node linear brick, reduced integration, hourglass control) cannot be
used. Instead two other element types were considered for the initial evaluation; a linear
C3D8I (an 8-node linear brick, incompatible modes) and a quadratic C3D20 (A 20-node
quadratic brick). The mesh was evaluated for both methods and no mesh warnings were
generated. An example of the meshed actuator part is shown in Figure 16.
Mesh Density: 0.01
Number of Elements: 1197
Mesh Warnings: 0
Figure 16 - Mesh of the Modeled Actuator
Assembly Characteristics
The analysis consists of the actuator body which has loads and boundary conditions
applied to it to simulate the pressurized actuator. Since the actuator is the only part in the
30
analysis the Abaqus model contains no interactions or constraints. The analysis does
consist of the following static load steps which are used to apply the loads and boundary
conditions:
1. Initial
2. Pressure Load
Boundary Conditions
The initial step in the analysis creates the two boundary conditions used to define and
constrain the model and do not change in the subsequent steps. The first boundary
condition is the “Fixed” boundary condition (black surface, Figure 17) which prevents
the top and bottom surface of the modeled actuator from moving in the vertical direction
(U2=0). The second boundary condition establishes the symmetry (blue surface, Figure
17) about the z-axis used to simplify the model (U3=UR1=UR2=0).
Fixed
Boundary
Condition
Symmetry
Boundary
Condition
Figure 17 - Boundary Conditions Placed on the Actuator
31
Applied Loads
Two static pressure loads were then created in order to apply the simulated the loads into
the actuator. The internal pressure load (red surface, Figure 18) and an external load
(green surface, Figure 18) were created in the “Pressure Load” step. The internal
pressure load was set equal to the maximum pressure calculated in Appendix A with and
without an accumulator. The external pressure load applied to the actuator was set equal
to atmospheric pressure shown in Table 12.
Internal
Pressure
Surface
External
Pressure
Surface
Figure 18 - Loads Placed on the Actuator
32
Actuator Geometry based on Abaqus Initial Results
Based on the initial results in Sections 3.1.2 and 3.2.2 the actuator models were modified
such that the Abaqus results meet the design criteria. As such the dimensions highlighted
in red in Table 16, were found to meet the design criteria given their respective loads.
The weight analysis for the actuator with the updated dimensions is shown below.
Table 16 - Actuator Geometry Based on Abaqus Initial Results
Variable
a
t
L
Value [Unit]
0.0725 [m] Inner radius of the actuator
0.01085 [m] Thickness of the actuator wall calculated using
Abaqus – No Accumulator
0.00765 [m] Thickness of the actuator wall calculated using
Abaqus – Accumulator
0.5 [m] Length of the modeled actuator
0.05 [m] Inner radius of the piping outlet
ro
tEC
Lo
Description
0.01175 [m] Thickness of the actuator end cap using Abaqus –
No Accumulator
0.00765[m] Thickness of the actuator end cap using Abaqus –
Accumulator
0.1 [m] Length of the outlet
No Accumulator
[m]
>
Inner Radius of the Actuator Outlet [m]
>
Thickness of the End Cap [m]
>
Volume of the Hollow Cylinder [m^3]
>
Volume of the Spherical End Cap [m^3]
>
Weight of the Actuator Body [N]
>
33
With Accumulator
[m]
>
Inner Radius of the Actuator Outlet [m]
>
Thickness of the End Cap [m]
>
Volume of the Hollow Cylinder [m^3]
>
Volume of the Spherical End Cap [m^3]
>
Weight of the Actuator Body [N]
>
34