Stress Analysis of an Elliptical Pressure Vessel Under Internal Pressure

By

Jonathan C. Wang

A Seminar submitted to the Faculty of Rensselaer at Hartford in partial fulfillment of the requirements for the Degree of MASTER of Science.

Major Subject: Mechanical Engineering

The original of the seminar is on file at the Rensselaer at Hartford Library.

Approved by Seminar Advisor, Prof. Ernesto Gutierrez

Rensselaer at Hartford, Hartford, CT December 8, 2005

2

Table of Contents List of Figures 3 Abstract 5 1 Introduction 6 2 Theory 7

2.1 Axi-Symmetric Pressure Vessels 7 2.2 Non-Circular Pressure Vessels 9

2.2.1 Obround Vessels 10 3 Analysis 12

3.1 Overview of Finite Element Theory 12

3.1.1 The Basis 12 3.1.2 The Ritz Method and FEM 12 3.1.3 The Computer Algorithm 13

3.2 ANSYS Model Setup & Optimization 14

3.2.1 Hoop Stress Uniformity in the Z-direction 14 3.2.2 Symmetry Conditions 14 3.2.3 Mesh Selection 17

4 Results 20 4.1 Obround Pressure Vessel 20 4.2 Elliptical Pressure Vessel 22

4.2.1 Vessels with Wall thickness = .1” 22 4.2.2 Vessels with Wall thickness = .3” 24 4.2.3 Vessels with Wall thickness = .5” 26

4.3 Interpretation of ANSYS Data 28

5 Discussion & Conclusions 29 References 31 Appendices

Appendix A: ANSYS Model Parameters 32 Appendix B: Mesh Sensitivity Data 33 Appendix C: Excel Data for Obround Results 34 Appendix D: Excel Data for Ellipse Results 35

3

List of Figures Figure 1.1 Engine installation inside oblong cavity Figure 2.1.1 Representative element of an axi-symmetric pressure vessel Figure 2.1.2 Circumferential cut-away section of element abcd Figure 2.1.3. Longitudinal cut-away section of element abcd Figure 2.2.4. Equilibrium force balance on element abcd Figure 2.2.1.1 Obround vessel with applied internal pressure, P Figure 3.2.1.1 3D mesh using Solid 95 brick elements Figure 3.2.1.2 1st principal stress contour plot. Figure 3.2.2.1 2D quarter model mesh using Plane 82 elements Figure 3.2.2.2 1st principal stress contour plot Figure 3.2.2.3 2D full hoop mesh using Plane 82 elements Figure 3.2.2.4 1st principal stress contour plot Figure 3.2.3.1 Hoop stress sensitivity to circumferential mesh density at locations A and D Figure 3.2.3.2 Hoop stress sensitivity to circumferential mesh density at locations B and C Figure 3.2.3.3 Hoop stress sensitivity to mesh density through thickness at locations A and D Figure 3.2.3.4 Hoop stress sensitivity to mesh density through thickness at locations B and C Figure 3.2.3.5 Mesh applied to ellipse having major axis=20, minor axis=20 and wall

thickness=.3” Figure 4.1.1 Hoop stress at location A for an obround pressure vessel Figure 4.1.2 Hoop stress at location B for an obround pressure vessel Figure 4.1.3 Hoop stress at location C for an obround pressure vessel Figure 4.1.4 Hoop stress at location D for an obround pressure vessel Figure 4.2.1.1 Hoop stress vs Eccentricity at location A for a .1” thick elliptical pressure vessel Figure 4.2.1.2 Hoop stress vs Eccentricity at location B for a .1” thick elliptical pressure vessel Figure 4.2.1.3 Hoop stress vs Eccentricity at location C for a .1” thick elliptical pressure vessel Figure 4.2.1.4 Hoop stress vs Eccentricity at location D for a .1” thick elliptical pressure vessel Figure 4.2.1.5 Hoop stress vs Eccentricity at location A for a .3” thick elliptical pressure vessel

4

Figure 4.2.1.6 Hoop stress vs Eccentricity at location B for a .3” thick elliptical pressure vessel Figure 4.2.1.7 Hoop stress vs Eccentricity at location C for a .3” thick elliptical pressure vessel Figure 4.2.1.8 Hoop stress vs Eccentricity at location D for a .3” thick elliptical pressure vessel Figure 4.2.1.9 Hoop stress vs Eccentricity at location A for a .5” thick elliptical pressure vessel Figure 4.2.1.10 Hoop stress vs Eccentricity at location B for a .5” thick elliptical pressure vessel Figure 4.2.1.11 Hoop stress vs Eccentricity at location C for a .5” thick elliptical pressure vessel Figure 4.2.1.12 Hoop stress vs Eccentricity at location D for a .5” thick elliptical pressure vessel Figure 4.3.1 1st principal stress plot for an internally pressurized obround vessel Figure 4.3.2 3rd principal stress plot for an internally pressurized obround vessel Figure 5.1 Location of max compressive stress for ellipse of eccentricity=1.33 Figure 5.2 Location of max compressive stress for ellipse of eccentricity=4 Figure 5.3 Location of max compressive stress for ellipse of eccentricity=6.66 Figure 5.4 Stress concentration near point C for a high-eccentricity ellipse

5

Abstract

This paper presents the results of a study performed using finite element analysis to determine the hoop stress in a pressure vessel of elliptical cross-section. The motivation for the subject matter is derived from certain applications of gas turbine engines, where the outer exhaust duct assumes a non-circular shape. To form a basis for comparison to the ellipse, theoretical formulations for hoop stress in similar cross-sectional shapes are first introduced, followed by a review of the finite element modeling parameters and finally a discussion of the finite element results performed for the ellipse.

6

1 Introduction Internally pressurized vessels in use today generally assume two basic shapes: spheres and cylinders. The reason for this is simply that a sphere will impart the lowest value of membrane stress on its walls compared to any other shape. (The derivation of the formula for a sphere can be found in a strength of materials textbook). Hence, for a given volume enclosed, a sphere represents the most weight-efficient design for a pressure vessel. In many situations, a straight cylinder is the preferred shape, as it is much more easily produced than a sphere while still providing a reasonably weight-efficient design. However, there arise situations where neither a cylindrical or spherical vessel may be best suited to the particular application due to additional design constraints that must be satisfied. One example of this occurs in the design of aircraft propulsion systems where the engine is to be packaged within the airframe structure. This type of arrangement is most commonly seen in the design of military aircraft (as opposed to commercial aircraft, where the engine is located external to the airframe). Shown below in Figure 1.1 is a sketch of what an engine cavity might look like for a military aircraft. To maintain a low overall profile for the aircraft, the cavity is significantly oblong-shaped rather than circular. For military propulsion systems, maximizing thrust is often the foremost performance requirement, and in general, a greater amount of air mass passing through the engine will result in a larger amount of thrust produced. Therefore, with respect to the engine cavity shown in Fig. 1.1, it would be desirable to design the engine cross-section to be shaped as an ellipse, rather than a circle, in order to enclose a greater area. The above example establishes the motivation for this paper, which is to investigate the stress imparted on an internally pressurized vessel of elliptical cross-section. As the Analysis and Results sections of this paper will show, a pressure vessel designed with a non-circular cross-section will produce significantly higher wall stresses than that of an axi-symmetric (circular) pressure vessel. To fully appreciate this difference, it is helpful to first understand the principles of axi-symmetric pressure vessel theory, which will be reviewed at the beginning of this paper. Following this will be an overview of non-circular pressure vessel theory, as a precursor to the discussion of the finite element methodology and analysis of various pressure vessel models. Finally, the results of the finite element analyses for elliptical pressure vessels will be presented in graphical format for discussion and interpretation.

Figure 1.1 Engine installation inside oblong cavity

Airframe structureEngine Engine cavity

7

2 Theory 2.1 Axi-symmetric Pressure Vessels This section provides the derivation of the governing equations for membrane stress in pressure vessels having circular cross-section, which includes cylinders and any other shape having a revolved axis of symmetry. Consider an element of size ds1 by ds2 by thickness t, extracted from the internally pressurized thin-shelled enclosure shown in Figure 2.1.1. Note that for computational simplicity, the chosen element is oriented along the principal (longitudinal and circumferential) directions of the part, so that only normal forces act on its sectioned faces. By examining the view of the element along side a-b as shown in Figure 2.1.2, the resultant circumferential force can be evaluated as simply the product of the circumferential stress and the elemental facial area: Moreover, the vector component of this force that directly opposes the applied internal pressure is:

1ds1θd

21θd

t

b

a

21θd

21θd

21θd

circF1F

1F

circF

p

2ds2θd

t

a

d

22θd

longF

22θd

22θd

22θd

2F

2FlongF

p

1σ1σ

2σ

2σ

a b

cd

p

( )21 dstFcirc ×= σ )1(

×=

2sin 1

1θdFF circ

)2(

Figure 2.1.1 Representative element of an axi-symmetric pressure vessel

Figure 2.1.2 Circumferential cut-away section of element abcd

Figure 2.1.3. Longitudinal cut-away section of element abcd

8

where θ1 is the angle subtended by the arc produced by ds1. Figure 2.1.3 shows a cut-away view of the longitudinal face of the element, and by similar observation to that of Figure 2.1.2, it is seen that the resultant longitudinal force is simply and that the vector component acting opposite the pressure is Considering now the inner surface of the element, as bounded by points a,b,c and d, the resultant forcing acting outward on the element is simply product of the internal pressure and the surface of the element. From Figures 2.1.2 and 2.1.3, it can be seen that arc lengths, ds1 and ds2, can be computed as where r1 and r2 are the radii of curvature in the circumferential and longitudinal directions, respectively, and therefore the force F3 due to internal pressure is For static equilibrium to prevail on the element, the resultant forces from the circumferential and longitudinal membrane stresses must balance the outward force due to the pressure, and therefore: or, in expanded form

2F 3F1F 1F2F

ab

c d

( )12 dstFlong ×= σ )3(

)4(

=

2sin 2

2θdFF long

=

2sin2 1

11θdrds )5(

=

2sin2 2

22θdrds )6(

( )( )213 dsdspF = )7(

213 22 FFF += )8(

+

=

2sin2

2sin2

2sin2

2sin2 2

121

212

21

1θσθσθθ dtdsdtdsdrdrp )9(

Figure 2.2.4. Equilibrium force balance on element abcd

9

Equation 9 can be simplified by observing that

which results in the final form of the pressure vessel equation. In the case of an exhaust duct for a gas turbine engine, the pressure vessel is an air conduit and is therefore free of end restraints. Hence σ1=0, since no longitudinal stress would be induced. With this simplification, only the circumferential stress remains, and Eq. 12 reduces to: or, expressed in terms of the stress: 2.2 Non-circular pressure vessels At first glance, deriving the governing equation for hoop stress in a non-circular vessel may appear to be a straightforward application of Eq. 14, whereby the constant value, r2, would be replaced by the variable value for the local radius of curvature for the non-circular shape. Furthermore, in the case of an ellipse, since both its algebraic curve equation and the equation for the radius of curvature can be expressed in Cartesian coordinates, it would even be possible to modify Eq. 14 to directly calculate the value of hoop stress, σ2, as a function of a given x-y position on the curve, thus bypassing the intermediate calculation for the radius of curvature. The above reasoning, however, is an improper extrapolation of Eq. 14. The error would become apparent upon performing a finite element analysis (FEA) on the geometry. However, without resorting to FEA, the logical flaw can be found by noting that for an ellipse (or any non-circular geometry), the internal pressure induces a bending moment on the wall, which is only absent from the circle due to its uniformity of curvature. In other words, rather than regarding the ellipse as an extension of the theory for a circle, it is the circle that is regarded as a contraction of the theory for an ellipse, whereby the component of hoop stress due to bending is zero as result of having constant curvature. Physically, the presence of bending can be accepted by considering the extreme case of a high eccentricity ellipse under internal pressurize and recognizing that the deformed shape would not be uniformly expanded, since the “flatter” regions would undergo much larger deflection than the “corner” regions. From beam theory, it is clear a bending moment would produce an additional circumferential stress on the pressure vessel, and therefore the total hoop stress for a non-circular pressure vessel would the sum of the stress due to bending and the surface membrane stress due to internal pressure.

1

11

22sin

rdsd

=

θ

2

22

22sin

rdsd

=

θ

tp

rr=+

2

2

1

1 σσ

tp

r=

2

2σ

)11,10(

)12(

)13(

tpr2

2 =σ )14(

10

Having identified the behavioral differences between circular and non-circular pressure vessels, it should be noted that research into analytical methods for calculating hoop stress in non-circular pressure vessels has been very limited to date. In the latest (2004) edition of the ASME Boiler and Pressure Vessel Code [1], only two non-circular geometries, rectangles and obrounds, are cited. However, due to the geometric similarity to an ellipse, it is useful to examine the obround geometry in further detail, as one would expect the magnitude of hoop stress to be comparable to that of an ellipse of equivalent eccentricity. 2.2.1 Obround Vessels As mentioned in the preceding general discussion of non-circular vessels, the hoop stress in an obround is comprised of a shape-dependent bending component superimposed on the pressure-induced membrane component. A representative obround shape is shown in Figure 2.2.1.1, and the corresponding stress components at the noted critical points are as follows: Membrane Stress (σm)

Locations A,B:

Locations C,D:

Bending Stress (σb) Locations A,B:

Locations C,D:

*one side will be in compression, the other in tension

( )t

PRBAm =,σ

( ) ( )t

LRPDCm

2,

+=σ

( )AI

cCPLBAb 6

12*, ±=σ

( ) ( )

−+±=

ACRL

IcPL

DCb1

22*

, 236

σ

12

3tI = ( ) 2221 1232 RLC ++= π R

RLA π+

=

222

surfaceoutertoaxisneutralfromdisttc .2

≡=

)15(

)17(

)18(

)16(

Figure 2.2.1.1. Obround vessel with applied internal pressure, P

where

A

B

C D

t

2L

R

2L

t

p

11

Total Hoop Stress (σT):

Locations A,B: Locations C,D:

( ) ( ) ( ) BAbBAmBAT ,,, σσσ +=

( ) ( ) ( ) DCbDCmDCT ,,, σσσ +=

)19(

)20(

12

3 Analysis

3.1 Overview of Finite Element Theory

The Finite Element Method (FEM) is a technique that is currently used to solve engineering problems in a variety of fields such as solid mechanics, fluid mechanics and heat transfer. The acceptance and growth of FEM has occurred almost concurrently with advancements in computer technology and processing power in the recent decades, which has enabled the solutions to increasingly complex problems to be analyzed and solved within a reasonable timeframe. However, FEM was first developed as a tool for evaluating linear elastic solids, which is the type of the problem being discussed in this paper. Therefore, the following subsections will provide an explanation of FEM as it pertains to this fundamental application.

3.1.1 The Basis

To solve a problem involving the deformation of linear elastic solids, generally the most common approach is that which is first introduced in an undergraduate Strength of Materials course. This method involves defining the three basic types of field equations—equilibrium, constitutive (stress-strain), and compatibility (strain-displacement)—and solving for the unknowns after appropriate substitutions have been made. The second method involves the application of energy principles, which equates the state of equilibrium to the condition of having minimum potential energy. A simple example of this is that of a ball being at rest at the lowest point (or “valley”) of an uneven terrain. It is this fundamental concept that forms the basis of the solution criteria of FEM. In mathematical terms, the energy principle is often stated as follows, where the potential energy, Π, is defined as the internal energy (U) minus the external work (W) performed on the system. In FEM, the problem is expressed as the variational form of the above energy equation. The variational form is analogous to expressing a function in differential form. Thus, in Eq. (21), given that U and W have been properly expressed as functions of the unknown displacement, u, a state of minimum potential energy exists only when an infinitesimal change in u will yield no corresponding change in Π. In other words, FEM seeks the condition at which δΠ=0.

3.1.2 The Ritz Method and FEM To obtain a solution for δΠ=0, classical methods for solving differential equations can be used, or alternatively, approximate methods can be invoked. In the case of FEM and its reliance on computers in practice, approximate methods tend to be the preferred approach. One such approximate method, called the Ritz method, involves estimating the form of the solution for u (assuming an appropriate expression of Eq. 21 as a function of u has first been developed). The general form of this estimate is expressed as

WU −=∏ )21(

∑=k

kk xaxu )()( ψ )22(

13

where ψ is a function that must satisfy the principal boundary conditions of the problem and ak are the unknown coefficients. Next, the partial derivatives of Π are taken with respect to each of the coefficients This sets up a system of simultaneous algebraic equations that can be solved for ak to determine the value of the displacement, and subsequently the value of stress if desired. FEM essentially follows the same logic as the Ritz method, but introduces the notion of discretization, where the solid body is considered as an assemblage of distinct sub-geometries (or elements), that are user-specified with respect to size and shape. The elements are interconnected by nodes, and similar to the Ritz method, the individual nodal displacements—and hence the displacement of the overall structure—can be obtained by solving the systems of algebraic equations resulting from the partial derivatives as follows, where the subscripts represent the individual nodes.

3.1.3 The Computer Algorithm

A common approach to linear algebraic systems of equations is to directly solve for the unknown variables through substitution and elimination. There are a number of methods that fall under this category such as the Gaussian Elimination, Choleski and Givens Factorization methods. However, within the context of FEM, one must consider that a body can easily consist of thousands of nodes, depending on the desired level of accuracy (with a greater number of nodes resulting in a more accurate solution) and the geometric complexity of the body. Likewise, the number of operations required to obtain the overall solution would also be of the same order. Because computers can only store a finite number of decimal places for each variable, a computer solution obtained using the direct approach could yield significantly erroneous results simply due to round-off error magnified over the course of thousands of sequential calculations. Therefore, to avoid this potential pitfall, software designed for FEM utilizes an indirect, iterative approach to converge to a solution of the systems of equations. The procedure begins with an initial “guess” for a solution. With each subsequent iteration, a better approximation results—assuming that each successive approximation satisfies the established criteria for convergence—and convergence to a solution is attained after a finite number of iterations. This algorithm for solving simultaneous algebraic equations represents the final step in the basic sequence of logic that comprises the Finite Element Method. The following sections will now discuss the preparation of ANSYS finite element models, which were generated for both obround and elliptical pressure vessel geometries, and whose stress results will be the topic of Section 4.

.0.....0,021

=∂

∏∂=

∂∏∂

=∂

∏∂

kaaa

0.....0,021

=∂

∏∂=

∂∏∂

=∂

∏∂

iuuu

14

3.2 ANSYS Model Setup & Optimization The selected parameters used in the construction of all ANSYS finite element models for this project are provided in Appendix A. However, prior to utilizing the stress data obtained from these models, it was first necessary to first verify that the chosen modeling and meshing procedure was sound. Below is a summary of a number of checks that were performed, to ensure that the stress values obtained from ANSYS were legitimate and not the result of improper setup of the finite element model. 3.2.1 Hoop stress uniformity in the Z-direction Due to the sheer number of finite element models that were required for data collection, it was desirable to make exclusive use of 2D models for the analyses. In order to justify this simplification, a number of 3D elliptical models were generated, meshed, and loaded with internal pressure. Figure 3.2.1.2 below shows a plot of the 1st principal stress (hoop tensile stress) for one of these test models, demonstrating that the hoop stress does not vary in the lengthwise direction.

3.2.2 Symmetry conditions For additional modeling efficiency, symmetry conditions were applied along the x and y axes, to reduce the complexity of the model from a full hoop to a single quadrant. As shown in Figure 3.2.2.1, appropriate boundary conditions were placed along the lines of symmetry to correctly constrain the model. A comparison of the stress values between the full hoop model and quarter model shows very good correlation.

Figure 3.2.1.1. 3D mesh using Solid 95 brick elements

Figure 3.2.1.2. 1st principal stress contour plot.

15

Ux=0

Uy=0

Figure 3.2.2.1. 2D quarter model mesh using Plane 82 elements

Figure 3.2.2.2. 1st principal stress contour plot

16

Figure 3.2.2.3. 2D full hoop mesh using Plane 82 elements

Figure 3.2.2.4. 1st principal stress contour plot

17

3.2.3 Mesh Selection To determine the mesh size required to obtain an accurate stress plot, several models were run with varying mesh density. The first series of trials involved varying the number of elements along the circumference, with the number of elements through the wall thickness held constant, and the results of this are shown in Figures 3.2.3.1 and 3.2.3.2, where the degrees of freedom (DOF) plotted on the x-axis represents the number of nodal points present on the curve surface. The second series involved varying the number of elements through the wall thickness, with the number of circumferential elements held constant, and those results are shown in Figures 3.2.3.3 and 3.2.3.4. The model used to generate these plots was an ellipse having major axis=20, minor axis=19.9, and wall thickness=.1”, with an applied internal pressure of 20 psi.

Hoop stress vs. Circumferential DOF

4920

4940

4960

4980

5000

5020

5040

0 200 400 600 800 1000 1200

Degrees of Freedom

Stre

ss (p

si)

Stress at B Stress at C

Hoop Stress vs Circumferential DOF

-1050

-1040

-1030

-1020

-1010

-1000

-9900 200 400 600 800 1000 1200

Degress of Freedom

Stre

ss(p

si)

Stress at A Stress at D

AC

Figure 3.2.3.1. Hoop stress sensitivity to circumferential mesh density at locations A and D

AB

C D

Figure 3.2.3.2. Hoop stress sensitivity to circumferential mesh density at locations B and C

18

In the circumferential direction, the data from Figures 3.2.3.1 and 3.2.3.2 suggests that stability is attained at approximately 150-200 DOF. Through the thickness, the data from Figures 3.2.3.3 and 3.2.3.4 suggests that about 9 DOF is sufficient. Based on these trials, the number of circumferential and radial DOF selected all finite element models used to generate data for the Results section was 801 (400 element divisions) and 13 (6 element divisions), respectively. They were purposely chosen to be somewhat higher than the perceived stability threshold as an added measure of safety, to account for possible variation when applied to ellipse models having different eccentricity. Figure 3.2.3.5 below shows how the chosen mesh size would appear on an ellipse of eccentricity=2 and wall thickness=0.3.”

Hoop stress vs DOF Through Thickness

49204930494049504960497049804990

0 5 10 15 20 25

Degrees of Freedom

Stre

ss (p

si)

Stress at B Stress at C

Hoop Stress vs DOF Through Thickness

-1015

-1010

-1005

-1000

-995

-9900 5 10 15 20 25

Degrees of Freedom

Stre

ss (p

si)

Stress at A Stress at D

AB

C D

Figure 3.2.3.3. Hoop stress sensitivity to mesh density through thickness at locations A and D

Figure 3.2.3.4. Hoop stress sensitivity to mesh density through thickness at locations B and C

19

Figure 3.2.3.5. Mesh applied to ellipse having major axis=20, minor axis=20 and wall thickness=.3”

20

4 Results 4.1 Obround Pressure Vessel To further test the fidelity of the ANSYS modeling methodology discussed in the preceding section, a series of finite element models was run using the obround pressure vessel geometry, for the purpose of comparing the resulting magnitudes of hoop stress to their predicted values, as stated earlier in Eqn’s (19) and (20). A range of aspect ratios was examined, and results are shown below in Figures 4.1.1-4.1.4 at the 4 critical points labeled in Figure 6. The raw data corresponding to these plots can be found in Appendix C.

Obround: Hoop Stress at A

-300000

-250000

-200000

-150000

-100000

-50000

0

50000

0 1 2 3 4 5 6 7 8 9

Aspect Ratio

Stre

ss (p

si)

Calculated Stress ANSYS Stress

Obround: Hoop Stress at B

0

50000

100000

150000

200000

250000

300000

0 2 4 6 8 10

Aspect Ratio

Stre

ss (p

si)

Calculated Stress ANSYS Stress

tRtRL

heightTotalwidthTotalRatioAspect

22222 2

+++

=≡ (see Fig. 2.2.1.1)

Figure 4.1.1. Hoop stress at location A for an obround pressure vessel

Figure 4.1.2. Hoop stress at location B for an obround pressure vessel

21

Overall, the plots show good correlation between the predicted values for hoop stress and those generated by the ANSYS model, which further substantiates the ANSYS modeling rationale outlined in the preceding section and suggests that an elliptical model generated in the same manner would yield fairly accurate results.

Obround: Hoop Stress at C

050000

100000150000200000250000300000350000400000

0 1 2 3 4 5 6 7 8 9

Aspect Ratio

Stre

ss (p

si)

Calculated Stress ANSYS Stress

Obround: Hoop Stress at D

-400000-350000-300000-250000-200000-150000-100000

-500000

50000

0 1 2 3 4 5 6 7 8 9

Aspect Ratio

stre

ss

Calculated Stress ANSYS Stress

Figure 4.1.3. Hoop stress at location C for an obround pressure vessel

Figure 4.1.4. Hoop stress at location D for an obround pressure vessel

22

4.2 Elliptical Pressure Vessel For the case of an ellipse, the hoop stress at four critical surface points was examined relative to eccentricity and varying wall thickness. The resulting data obtained from ANSYS is shown below in Figures 4.2.1.1-4.2.1.12. (See Appendix D for raw data) 4.2.1 Vessels with Wall Thickness = .1”

AB

C D

Hoop stress at B

0

50000

100000

150000

200000

250000

0 1 2 3 4 5 6 7 8 9

Eccentricity

Stre

ss (p

si)

Hoop Stress at A

-250000

-200000

-150000

-100000

-50000

0

50000

0 2 4 6 8 10

Eccentricity

Stre

ss (p

si)

AB

C D

Figure 4.2.1.1. Hoop stress vs Eccentricity at location A for a .1” thick elliptical pressure vessel

Figure 4.2.1.2. Hoop stress vs Eccentricity at location B for a .1” thick elliptical pressure vessel

23

Hoop Stress at D

-400000-350000

-300000-250000-200000-150000-100000-50000

050000

0 1 2 3 4 5 6 7 8 9

Eccentricity

Stre

ss (p

si)

Hoop Stress at C

0

100000

200000

300000

400000

500000

600000

0 1 2 3 4 5 6 7 8 9

Eccentricity

Stre

ss (p

si)

AB

C D

Figure 4.2.1.3. Hoop stress vs Eccentricity at location C for a .1” thick elliptical pressure vessel

Figure 4.2.1.4. Hoop stress vs Eccentricity at location D for a .1” thick elliptical pressure vessel

24

4.2.2 Vessels with Wall Thickness = .3”

Hoop stress at B

0

5000

10000

15000

20000

25000

0 1 2 3 4 5 6 7 8 9

Eccentricity

Stre

ss (p

si)

Hoop Stress at A

-25000

-20000

-15000

-10000

-5000

0

5000

0 1 2 3 4 5 6 7 8 9

Eccentricity

Stre

ss (p

si)

AB

C D

Figure 4.2.1.5. Hoop stress vs Eccentricity at location A for a .3” thick elliptical pressure vessel

Figure 4.2.1.6. Hoop stress vs Eccentricity at location B for a .3” thick elliptical pressure vessel

25

Hoop Stress at D

-35000

-30000

-25000

-20000

-15000

-10000

-5000

0

5000

0 1 2 3 4 5 6 7 8 9

Eccentricity

Stre

ss (p

si)

Hoop Stress at C

0

20000

40000

60000

80000

100000

120000

140000

0 1 2 3 4 5 6 7 8 9

Eccentricity

Stre

ss (p

si)

AB

C D

Figure 4.2.1.7. Hoop stress vs Eccentricity at location C for a .3” thick elliptical pressure vessel

Figure 4.2.1.8. Hoop stress vs Eccentricity at location D for a .3” thick elliptical pressure vessel

26

Vessels with Wall Thickness = .5”

Hoop Stress at A

-9000

-8000

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

0 2 4 6 8 10

Eccentricity

Stre

ss (p

si)

Hoop stress at B

0100020003000400050006000700080009000

0 2 4 6 8 10

Eccentricity

Stre

ss (p

si)

AB

C D

Figure 4.2.1.9. Hoop stress vs Eccentricity at location A for a .5” thick elliptical pressure vessel

Figure 4.2.1.10. Hoop stress vs Eccentricity at location B for a .5” thick elliptical pressure vessel

27

Hoop Stress at C

05000

100001500020000250003000035000400004500050000

0 2 4 6 8 10

Eccentricity

Stre

ss (p

si)

Hoop Stress at D

-12000

-10000

-8000

-6000

-4000

-2000

0

2000

0 2 4 6 8 10

Eccentricity

Stre

ss (p

si)

AB

C D

Figure 4.2.1.11. Hoop stress vs Eccentricity at location C for a .5” thick elliptical pressure vessel

Figure 4.2.1.12. Hoop stress vs Eccentricity at location D for a .5” thick elliptical pressure vessel

28

4.3 Interpretation of ANSYS data To determine the maximum value of hoop stress at each location (A,B,C,D) from the ANSYS model, both the 1st principal stress and 3rd principal stress contour plots had to be examined, due the fact that a large compressive hoop stress would be regarded as the numerically lowest stress (and hence assigned to the 3rd principal stress) because of the negative sign. An example of this is illustrated in Figures 4.3.1 and 4.3.2 below, which show plots of the 1st and 3rd principal stresses, respectively, for an obround vessel. The hoop stress at C corresponds to the 1st principal stress value, as shown by Figure 4,3,1. However, the hoop stress at D is found under the 3rd principal stress because it is compressive. It is clearly not 0, as indicated by Figure 4.3.1. Therefore, the data in Figures 18-33 represents…. Therefore, the data points appearing in Figures 4.1.1-4.2.1.12 represent a combination of 1st and 3rd principal stress values due to the manner in which principal stresses are ordered. From this, it follows that whenever both tensile and compressive stresses are present, an accurate portrayal of hoop stress can only be obtained by superposition of the 1st and 3rd principal stress plots.

C D

C D

Figure 4.3.2. 3rd principal stress plot for an internally pressurized obround vessel.

Figure 4.3.1. 1st principal stress plot for an internally pressurized obround vessel

29

5 Discussion & Conclusions From the data plots, several useful observations can be made. In comparing the plots for a .1” thick obround (Figs. 4.1.1-4.1.4) to that of an “equivalent” ellipse (Figs. 4.2.1.1-4.2.1.4), the curves are very similar in shape and the stress values are comparable in magnitude. This strongly suggests that the same underlying physical mechanisms that govern the obround hoop stress equations (Eq. 19-20) are present in the ellipse as well. Further evidence of this can be seen by examining the sensitivity of hoop stress to wall thickness, such as by comparing the plots of Figures 4.2.1.1, 4.2.1.5 and 4.2.1.9. By noting the relative stress magnitudes, it is apparent that the relationship is highly non-linear. Following the rationale of an obround vessel, the non-linearity could be explained by noting that for smaller wall thicknesses, the hoop stress is dominated by bending, which is exponentially sensitive to the wall thickness due to the moment of inertia term (I) found in Equations 17 and 18. In comparing the stress curves for the ellipse over the range of thicknesses evaluated, there is some shape dissimilarity in the curves for locations C and D. At D, the hoop stress seems to attenuate more rapidly for higher eccentricities with increasing wall thickness. Rather than being an indication that the magnitude of maximum compressive stress on the outer surface of the ellipse actually decreases above a certain eccentricity, this is more likely the result of the location of the peak stress shifting away from D for higher eccentricities. This is illustrated below in Figures 5.1-5.3. Note that the magnitude of maximum compressive stress increases with eccentricity.

Figure 5.1. Location of max compressive stress for ellipse of eccentricity=1.33

Max compressive stress location Max compressive

stress location

Max compressive stress location

Figure 5.2. Location of max compressive stress for ellipse of eccentricity=4

Figure 5.3. Location of max compressive stress for ellipse of eccentricity=6.66

30

At location C, Figures 4.2.1.7 and 4.2.1.11 show some graphical instability at higher eccentricities. From the ANSYS stress corresponding to these apparent outlying points, it is seen that the stress distribution is very narrow, except near point C, where the bands converge and the stress concentrates (see Figure 5.4). This much resembles a crack initiation site, and the true value of stress may be indiscernible due to its highly localized nature. Therefore, although the points plotted represent actual data obtained from ANSYS, this is an area that may benefit from additional scrutiny. In summary, the results of this study have shown that the an internally pressurized ellipse will exhibit significantly higher hoop stresses than that of an equivalently sized circle due its the non-uniformity in curvature, which induces additional bending stresses. From the results obtained from ANSYS, the net effect appears similar to that of an obround pressure vessel, for which some prior analytical work has been performed. From an engineering standpoint, the consequence of having to accommodate higher stresses is often a heavier and costlier design due to the need for material reinforcement in the high stress areas. However, as stated in the opening of this paper, there are situations where these drawbacks can be tolerated, because the perceived benefits of utilizing a non-circular design more than adequately compensate for them.

Figure 5.4. Stress concentration near point C for a high-eccentricity ellipse

31

References [1] American Society of Mechanical Engineers (ASME), Mandatory Appendix

13: Vessels of Noncircular Cross Section, from the ASME Boiler & Pressure Vessel Code, ASME International, 2004.

[2] Barkanov, E., Introduction to the Finite Element Method, Riga Technical

University, 2001. [3] Harvey, J., Theory and Design of Modern Pressure Vessels, 2nd Ed., Van

Nostrand Reinhold Co., New York, 1974. [4] Harvey, J., Pressure Component Construction: Design and Materials

Application, Van Nostrand Reinhold Co., New York, 1980 [5] Mubeezi, J., Finite Element Analysis (FEA) method, and its Application in

Evaluating Stress-Strain Characteristics, Rensselaer at Hartford Engineering Seminar, 2004.

32

Appendix A: ANSYS Model Parameters The parameters apply to all ANSYS models used to generate data in Section 4 Type of analysis: Linear Elastic Element type: Plane 82 Material Properties: E=1.5x107 psi, ν=.33 Number of circumferential element divisions – N=400 Number of element divisions through wall thickness – N=6 Internal pressure applied: 20 psi Model geometry: Obround – Ellipse –

"1.

20”

20”

From 2.5” to 20”

From 2.5” to 20”

From .1” to .5”

33

Appendix B: Mesh Sensitivity Data The following is the raw data for the plots found in Section 3.2.3: For Figure 3.2.3.1:

Major Axis

Minor Axis

Stress at D

Stress at B

Stress at C

Stress at A

Circ DOF

20 19.9 -1042.8 4982.1 5020.8 -1024.5 17 20 19.9 -1028.4 4961 4999.1 -1010.8 23 20 19.9 -1020.6 4948.6 4986.7 -1002.8 33 20 19.9 -1017.2 4943.2 4981 -999.58 43 20 19.9 -1013.8 4938.2 4975.4 -996.6 65 20 19.9 -1012.8 4936.7 4973.8 -995.66 81 20 19.9 -1012.1 4935.5 4972.6 -994.93 107 20 19.9 -1011.5 4934.6 4971.7 -994.38 159 20 19.9 -1011.2 4934.1 4971.2 -994.04 315 20 19.9 -1011.1 4934 4971.2 -993.98 449 20 19.9 -1011.1 4934 4971.1 -993.95 629 20 19.9 -1011.1 4933.9 4971.1 -993.94 1047

For Figure 3.2.3.2:

Major Axis

Minor Axis

Stress at D

Stress atB

Stress at C

Stress at A

Dof thru thk

20 19.9 -1011.1 4934 4971 -993.94 3 20 19.9 -1003.5 4926 4978.7 -1001.4 5 20 19.9 -1002.1 4925.2 4980 -1002.8 7 20 19.9 -1001.6 4924.5 4981 -1003.2 9 20 19.9 -1001.4 4924.4 4981 -1003.5 11 20 19.9 -1001.3 4924.3 4981.1 -1003.6 13 20 19.9 -1001.2 4924.3 4981.1 -1003.7 15 20 19.9 -1001.1 4924.3 4981.1 -1003.8 17 20 19.9 -1001.1 4924.2 4981.2 -1003.8 19 20 19.9 -1001.1 4924.2 4981.2 -1003.8 21

34

Appendix C: Data for Obround Pressure Vessel Results The following is the raw data for the plots found in Section 4.1:

Major Minor a/b Predicted

Stress at APredicted

Stress at C Predicted

Stress at BPredicted

Stress at D 20 20 1 1980 1980 1980 1980 20 19 1.053 -34316 23683.4 38076 -19723.4 20 18 1.111 -67405.21 45594.8 70965.21 -41634.8 20 15 1.333 -147908.4 112091.6 150868.4 -108131.6 20 10 2 -223101 221899 225061 -217939 20 6 3.333 -235822.2 303177.8 236982.2 -299217.8 20 5 4 -233283.6 321716.4 234243.6 -317756.4 20 4 5 -228725.6 339274.4 229485.6 -335314.4 20 3 6.667 -222307.2 355693.8 222867.2 -351733.8 20 2.5 8 -218455 363420 218915 -359460

Major Minor a/b ANSYS

Stress at AANSYS

Stress at C ANSYS

Stress at BANSYS

Stress at D 20 20 1 1990 1990 1970 1970 20 19 1.053 -34501 23877 38261 -19769 20 18 1.111 -67781 45981 71341 -41707 20 15 1.333 -148850 113130 151810 -108220 20 10 2 -224950 224490 226910 -217670 20 6 3.333 -238320 308220 239948 -297680 20 5 4 -235920 327850 236880 -315490 20 4 5 -231500 346990 232260 -331890 20 3 6.667 -225200 365970 225760 -346290 20 2.5 8 -221400 375730 221860 -352350

35

Appendix D: Data for Elliptical Pressure Vessel Results The following is the raw data for the plots found in Section 4.2: For wall thickness = .1”

Major Axis

Minor Axis a/b

Stress at D

Stress at B

Stress at C

Stress at A

20 20 1 1970 1970 1990 1990 20 19.99 1.0005 1672.7 2266.6 2289.3 1689.5 20 19.95 1.002506 483.5 3450.1 3486.3 490.31 20 19.93 1.003512 -110.65 4040.5 4084 -107.94 20 19.9 1.005025 -1001.3 4924.4 4981 -1003.6 20 19.5 1.025641 -12811 16518 16873 -12750 20 18 1.111111 -56000 56808 60429 -53571 20 15 1.333333 -137050 122350 142640 -119980 20 10 2 -254240 188090 265070 -186740 20 6 3.333333 -322269 205210 352410 -204450 20 5 4 -332212 205500 376120 -204880 20 4 5 -333920 204600 407040 -204100 20 3 6.666667 -317520 202820 471100 -202450 20 2.5 8 -291540 201700 565420 -201410

For wall thickness = .3”

Major Axis

Minor Axis a/b

Stress at D

Stress at B

Stress at C

Stress at A

20 20 1 636.81 636.81 656.81 656.81 20 19.99 1.0005 604.32 669.01 689.95 623.32 20 19.95 1.002506 474.36 797.44 822.51 489.63 20 19.93 1.003512 409.44 861.52 888.75 422.95 20 19.9 1.005025 312.11 957.44 988.04 323.1 20 19.5 1.025641 -977.91 2215.3 2305.4 -986 20 18 1.111111 -5687.8 6583.1 7138 -5530.4 20 15 1.333333 -14473 13666 16314 -12901 20 10 2 -26770 20675 30434 -20248 20 6 3.333333 -32128 22366 43269 -22136 20 5 4 -31486 22350 49562 -22165 20 4 5 -28291 22210 66289 -22069 20 3 6.666667 -17764 21937 123060 -21840 20 2.5 8 -7171.3 21589 127600 -21515

36

For wall thickness = .5”

Major Axis

Minor Axis a/b

Stress at D

Stress at B

Stress at C

Stress at A

20 20 1 370.25 370.25 390.25 390.25 20 19.99 1.0005 358.74 381.57 402.13 378.15 20 19.95 1.002506 312.74 426.71 449.71 329.86 20 19.93 1.003512 289.75 449.23 473.48 305.76 20 19.9 1.005025 255.3 482.94 509.12 269.7 20 19.5 1.025641 -201.21 924.99 982.08 -203.09 20 18 1.111111 -1865 2458.5 2720 -1842.5 20 15 1.333333 -4949 4936.9 6040.9 -4493.5 20 10 2 -9113.4 7355.1 11353 -7113.5 20 6 3.333333 -10160 7890.3 18019 -7766.8 20 5 4 -9250 7870.7 24293 -7774 20 4 5 -6650 7794.7 42875 -7725.2 20 3 6.666667 -1258.9 7479 45045 -7435 20 2.5 8 0 7109.5 39462 -7078