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
Table of Contents List of Figures
3
Abstract
5
1
Introduction
6
2
Theory
7
2.1 2.2
3
4
7 9 10
Analysis
12
3.1
Overview of Finite Element Theory 3.1.1 The Basis 3.1.2 The Ritz Method and FEM 3.1.3 The Computer Algorithm
12 12 12 13
3.2
ANSYS Model Setup & Optimization 3.2.1 Hoop Stress Uniformity in the Z-direction 3.2.2 Symmetry Conditions 3.2.3 Mesh Selection
14 14 14 17
Results
20
4.1 4.2
20 22 22 24 26 28
4.3
5
Axi-Symmetric Pressure Vessels Non-Circular Pressure Vessels 2.2.1 Obround Vessels
Obround Pressure Vessel Elliptical Pressure Vessel 4.2.1 Vessels with Wall thickness = .1” 4.2.2 Vessels with Wall thickness = .3” 4.2.3 Vessels with Wall thickness = .5” Interpretation of ANSYS Data
Discussion & Conclusions
29
References
31
Appendices Appendix A: ANSYS Model Parameters Appendix B: Mesh Sensitivity Data Appendix C: Excel Data for Obround Results Appendix D: Excel Data for Ellipse Results
2
32 33 34 35
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
3
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
4
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.
5
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.
Engine Engine cavity
Airframe structure
Figure 1.1 Engine installation inside oblong cavity
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 crosssection 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.
6
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 crosssection, which includes cylinders and any other shape having a revolved axis of symmetry.
p
σ2
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.
b
a
σ1
d
σ1
c
σ2
Figure 2.1.1 Representative element of an axi-symmetric pressure vessel
F1
Fcirc
F2
dθ1 2
Flong
dθ 2
ds1
dθ1 t
p
dθ1 2
F1
ds2 t
p
dθ 2 2
a Fcirc
a
dθ 2 2
b
d θ1 2
dθ 2 2
d Flong
d θ1 2
F2
Figure 2.1.2 Circumferential cut-away section of element abcd
dθ 2 2
Figure 2.1.3. Longitudinal cut-away section of element abcd
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:
Fcirc = σ 1 × t (ds 2 )
(1)
Moreover, the vector component of this force that directly opposes the applied internal pressure is:
dθ F1 = Fcirc × sin 1 2
7
(2)
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
Flong = σ 2 × t (ds1 )
(3)
and that the vector component acting opposite the pressure is
dθ F2 = Flong sin 2 2
(4)
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
dθ ds1 = 2 r1 sin 1 2
(5)
dθ ds 2 = 2 r2 sin 2 2
(6)
c
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
F3 = p (ds1 )(ds 2 )
d
a
b
(7 )
F2
F1
F2 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:
F1
F3
Figure 2.2.4. Equilibrium force balance on element abcd
(8)
F3 = 2 F1 + 2 F2 or, in expanded form
dθ dθ dθ dθ p 2r1 sin 1 2r2 sin 2 = 2σ 1tds2 sin 1 + 2σ 2tds1 sin 2 2 2 2 2
8
(9)
Equation 9 can be simplified by observing that
dθ ds sin 1 = 1 2 2r1
dθ ds sin 2 = 2 2 2r2
(10,11)
which results in the final form of the pressure vessel equation.
σ1 r1
+
σ2 r2
=
p t
(12)
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:
σ2 r2
p t
(13)
pr2 t
(14)
=
or, expressed in terms of the stress:
σ2 =
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 noncircular pressure vessel would the sum of the stress due to bending and the surface membrane stress due to internal pressure.
9
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
t B
t
As mentioned in the preceding general discussion of non-circular vessels, the hoop stress in an obround is comprised of a shapedependent bending component superimposed on the pressureinduced 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:
A p
R
L2
L2
Figure 2.2.1.1. Obround vessel with applied internal pressure, P
Membrane Stress (σm) Locations A,B:
(σ m )A, B = PR
Locations C,D:
(σ m )C ,D = P(R + L2 )
(16)
Locations A,B:
(σ b )A,B = ± * PL2C1c
(17)
Locations C,D:
(σ b )C ,D = ± * PL2 c 3(L2 + 2 R ) − C1
(15)
t
t
Bending Stress (σb)
6 AI
6I
t3 I= 12
C1 = L2 (2 + 3π ) + 12 R 2
where
c=
A
2
(18)
L2 2 + πR A = 2 R
t ≡ dist . from neutral axis to outer surface 2
*one side will be in compression, the other in tension
10
C
D
Total Hoop Stress (σT): Locations A,B:
(σ T )A,B = (σ m )A,B + (σ b )A,B
(19)
Locations C,D:
(σ T )C ,D = (σ m )C ,D + (σ b )C ,D
(20)
11
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,
∏ = U −W
( 21)
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
u ( x ) = ∑ a kψ k ( x ) k
12
( 22 )
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
∂∏ ∂∏ ∂∏ = 0, = 0..... = 0. ∂a1 ∂a 2 ∂a k 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 subgeometries (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,
∂∏ ∂∏ ∂∏ = 0, = 0..... =0 ∂u1 ∂u 2 ∂u i 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.
13
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.
Figure 3.2.1.1. 3D mesh using Solid 95 brick elements Figure 3.2.1.2. 1st principal stress contour plot.
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.
14
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
15
Figure 3.2.2.3. 2D full hoop mesh using Plane 82 elements
Figure 3.2.2.4. 1st principal stress contour plot
16
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.
A
Hoop Stress vs Circumferential DOF
C
-990
Stress(psi)
-1000 0
200
400
600
800
1000
1200
-1010 -1020 -1030 -1040 -1050 Degress of Freedom Stress at A
Stress at D
Figure 3.2.3.1. Hoop stress sensitivity to circumferential mesh density at locations A and D
B A C D
Hoop stress vs. Circumferential DOF 5040
Stress (psi)
5020 5000 4980 4960 4940 4920 0
200
400
600
800
1000
1200
Degrees of Freedom Stress at B
Stress at C
Figure 3.2.3.2. Hoop stress sensitivity to circumferential mesh density at locations B and C
17
Hoop Stress vs DOF Through Thickness -990
Stress (psi)
-995
0
5
10
15
20
25
-1000 -1005 -1010 -1015 Degrees of Freedom Stress at A
Stress at D
Figure 3.2.3.3. Hoop stress sensitivity to mesh density through thickness at locations A and D
B A C D
Hoop stress vs DOF Through Thickness 4990 Stress (psi)
4980 4970 4960 4950 4940 4930 4920 0
5
10
15
20
25
Degrees of Freedom
Stress at B
Stress at C
Figure 3.2.3.4. Hoop stress sensitivity to mesh density through thickness at locations B and C
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.”
18
Figure 3.2.3.5. Mesh applied to ellipse having major axis=20, minor axis=20 and wall thickness=.3”
19
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 50000
Stress (psi)
0 -50000 0
1
2
3
4
5
6
7
8
9
-100000 -150000 -200000 -250000 -300000 Aspect Ratio Calculated Stress
ANSYS Stress
Figure 4.1.1. Hoop stress at location A for an obround pressure vessel
Aspect Ratio ≡
Total width 2 L2 + 2 R + 2t = Total height 2 R + 2t
(see Fig. 2.2.1.1)
Obround: Hoop Stress at B 300000
Stress (psi)
250000 200000 150000 100000 50000 0 0
2
4
6 Aspect Ratio
Calculated Stress
ANSYS Stress
Figure 4.1.2. Hoop stress at location B for an obround pressure vessel
20
8
10
Stress (psi)
Obround: Hoop Stress at C 400000 350000 300000 250000 200000 150000 100000 50000 0 0
1
2
3
4
5
6
7
8
9
8
9
Aspect Ratio
Calculated Stress
ANSYS Stress
Figure 4.1.3. Hoop stress at location C for an obround pressure vessel
stress
Obround: Hoop Stress at D 50000 0 -50000 0 -100000 -150000 -200000 -250000 -300000 -350000 -400000
1
2
3
4
5
6
7
Aspect Ratio Calculated Stress
ANSYS Stress
Figure 4.1.4. Hoop stress at location D for an obround pressure vessel
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.
21
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” Hoop Stress at A 50000
Stress (psi)
0 -50000
0
2
4
6
8
10
-100000
B A
-150000 -200000
C D
-250000 Eccentricity Figure 4.2.1.1. Hoop stress vs Eccentricity at location A for a .1” thick elliptical pressure vessel
B A C D
Hoop stress at B 250000
Stress (psi)
200000 150000 100000 50000 0 0
1
2
3
4
5
6
7
8
Eccentricity Figure 4.2.1.2. Hoop stress vs Eccentricity at location B for a .1” thick elliptical pressure vessel
22
9
Hoop Stress at C 600000
Stress (psi)
500000 400000 300000 200000 100000 0 0
1
2
3
4
5
6
7
8
9
Eccentricity Figure 4.2.1.3. Hoop stress vs Eccentricity at location C for a .1” thick elliptical pressure vessel
B A C D
Hoop Stress at D 50000 0
Stress (psi)
-50000 0
1
2
3
4
5
6
7
8
-100000 -150000 -200000 -250000 -300000 -350000 -400000 Eccentricity Figure 4.2.1.4. Hoop stress vs Eccentricity at location D for a .1” thick elliptical pressure vessel
23
9
4.2.2 Vessels with Wall Thickness = .3” Hoop Stress at A 5000 0 Stress (psi)
0
1
2
3
4
5
6
7
8
9
-5000 -10000 -15000 -20000 -25000 Eccentricity Figure 4.2.1.5. Hoop stress vs Eccentricity at location A for a .3” thick elliptical pressure vessel
B A C D
Hoop stress at B 25000
Stress (psi)
20000 15000 10000 5000 0 0
1
2
3
4
5
6
7
8
Eccentricity Figure 4.2.1.6. Hoop stress vs Eccentricity at location B for a .3” thick elliptical pressure vessel
24
9
Hoop Stress at C 140000
Stress (psi)
120000 100000 80000 60000 40000 20000 0 0
1
2
3
4
5
6
7
8
9
Eccentricity Figure 4.2.1.7. Hoop stress vs Eccentricity at location C for a .3” thick elliptical pressure vessel
B A C D
Hoop Stress at D 5000 0 Stress (psi)
-5000 0
1
2
3
4
5
6
7
8
-10000 -15000 -20000 -25000 -30000 -35000 Eccentricity Figure 4.2.1.8. Hoop stress vs Eccentricity at location D for a .3” thick elliptical pressure vessel
25
9
Vessels with Wall Thickness = .5” Hoop Stress at A 1000 0 -1000 0
2
4
6
8
10
Stress (psi)
-2000 -3000 -4000 -5000 -6000 -7000 -8000 -9000 Eccentricity Figure 4.2.1.9. Hoop stress vs Eccentricity at location A for a .5” thick elliptical pressure vessel
B A C D
Hoop stress at B 9000
Stress (psi)
8000 7000 6000 5000 4000 3000 2000 1000 0 0
2
4
6
8
Eccentricity Figure 4.2.1.10. Hoop stress vs Eccentricity at location B for a .5” thick elliptical pressure vessel
26
10
Hoop Stress at C 50000 45000
Stress (psi)
40000 35000 30000 25000 20000 15000 10000 5000 0 0
2
4
6
8
10
Eccentricity Figure 4.2.1.11. Hoop stress vs Eccentricity at location C for a .5” thick elliptical pressure vessel
B A C D
Hoop Stress at D 2000 0 Stress (psi)
-2000
0
2
4
6
8
-4000 -6000 -8000 -10000 -12000 Eccentricity Figure 4.2.1.12. Hoop stress vs Eccentricity at location D for a .5” thick elliptical pressure vessel
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10
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.
C
D
Figure 4.3.1. 1st principal stress plot for an internally pressurized obround vessel
C
D
Therefore, the data in Figures 18-33 represents…. Figure 4.3.2. 3rd principal stress plot for an internally pressurized obround vessel.
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.
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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.
Max compressive stress location
Max compressive stress location
Figure 5.2. Location of max compressive stress for ellipse of eccentricity=4
Figure 5.1. Location of max compressive stress for ellipse of eccentricity=1.33
Max compressive stress location
Figure 5.3. Location of max compressive stress for ellipse of eccentricity=6.66
29
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.
Figure 5.4. Stress concentration near point C for a high-eccentricity ellipse
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 nonuniformity 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.
30
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.
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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 – . 1"
From 2.5” to 20”
20”
Ellipse – From .1” to .5”
From 2.5” to 20”
20”
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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 20 20 20 20 20 20 20 20 20 20 20 20
Minor Stress at Stress at Stress at Stress at Axis D B C A 19.9 -1042.8 4982.1 5020.8 -1024.5 19.9 -1028.4 4961 4999.1 -1010.8 19.9 -1020.6 4948.6 4986.7 -1002.8 19.9 -1017.2 4943.2 4981 -999.58 19.9 -1013.8 4938.2 4975.4 -996.6 19.9 -1012.8 4936.7 4973.8 -995.66 19.9 -1012.1 4935.5 4972.6 -994.93 19.9 -1011.5 4934.6 4971.7 -994.38 19.9 -1011.2 4934.1 4971.2 -994.04 19.9 -1011.1 4934 4971.2 -993.98 19.9 -1011.1 4934 4971.1 -993.95 19.9 -1011.1 4933.9 4971.1 -993.94
Circ DOF 17 23 33 43 65 81 107 159 315 449 629 1047
For Figure 3.2.3.2: Major Axis 20 20 20 20 20 20 20 20 20 20
Minor Axis 19.9 19.9 19.9 19.9 19.9 19.9 19.9 19.9 19.9 19.9
Stress at Stress at Stress at D B C -1011.1 4934 4971 -1003.5 4926 4978.7 -1002.1 4925.2 4980 -1001.6 4924.5 4981 -1001.4 4924.4 4981 -1001.3 4924.3 4981.1 -1001.2 4924.3 4981.1 -1001.1 4924.3 4981.1 -1001.1 4924.2 4981.2 -1001.1 4924.2 4981.2
33
Stress at Dof thru A thk -993.94 3 -1001.4 5 -1002.8 7 -1003.2 9 -1003.5 11 -1003.6 13 -1003.7 15 -1003.8 17 -1003.8 19 -1003.8 21
Appendix C: Data for Obround Pressure Vessel Results The following is the raw data for the plots found in Section 4.1:
Major 20 20 20 20 20 20 20 20 20 20
Minor 20 19 18 15 10 6 5 4 3 2.5
Predicted Predicted Predicted Predicted a/b Stress at A Stress at C Stress at B Stress at D 1 1980 1980 1980 1980 1.053 -34316 23683.4 38076 -19723.4 1.111 -67405.21 45594.8 70965.21 -41634.8 1.333 -147908.4 112091.6 150868.4 -108131.6 2 -223101 221899 225061 -217939 3.333 -235822.2 303177.8 236982.2 -299217.8 4 -233283.6 321716.4 234243.6 -317756.4 5 -228725.6 339274.4 229485.6 -335314.4 6.667 -222307.2 355693.8 222867.2 -351733.8 8 -218455 363420 218915 -359460
Major 20 20 20 20 20 20 20 20 20 20
Minor 20 19 18 15 10 6 5 4 3 2.5
ANSYS ANSYS ANSYS ANSYS a/b Stress at A Stress at C Stress at B Stress at D 1 1990 1990 1970 1970 1.053 -34501 23877 38261 -19769 1.111 -67781 45981 71341 -41707 1.333 -148850 113130 151810 -108220 2 -224950 224490 226910 -217670 3.333 -238320 308220 239948 -297680 4 -235920 327850 236880 -315490 5 -231500 346990 232260 -331890 6.667 -225200 365970 225760 -346290 8 -221400 375730 221860 -352350
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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 20 20 20 20 20 20 20 20 20 20 20 20 20 20
Minor Axis 20 19.99 19.95 19.93 19.9 19.5 18 15 10 6 5 4 3 2.5
a/b 1 1.0005 1.002506 1.003512 1.005025 1.025641 1.111111 1.333333 2 3.333333 4 5 6.666667 8
Stress at D 1970 1672.7 483.5 -110.65 -1001.3 -12811 -56000 -137050 -254240 -322269 -332212 -333920 -317520 -291540
Stress at Stress at B C 1970 1990 2266.6 2289.3 3450.1 3486.3 4040.5 4084 4924.4 4981 16518 16873 56808 60429 122350 142640 188090 265070 205210 352410 205500 376120 204600 407040 202820 471100 201700 565420
Stress at A 1990 1689.5 490.31 -107.94 -1003.6 -12750 -53571 -119980 -186740 -204450 -204880 -204100 -202450 -201410
For wall thickness = .3” Major Axis 20 20 20 20 20 20 20 20 20 20 20 20 20 20
Minor Axis 20 19.99 19.95 19.93 19.9 19.5 18 15 10 6 5 4 3 2.5
a/b 1 1.0005 1.002506 1.003512 1.005025 1.025641 1.111111 1.333333 2 3.333333 4 5 6.666667 8
Stress at Stress at Stress at Stress at D B C A 636.81 636.81 656.81 656.81 604.32 669.01 689.95 623.32 474.36 797.44 822.51 489.63 409.44 861.52 888.75 422.95 312.11 957.44 988.04 323.1 -977.91 2215.3 2305.4 -986 -5687.8 6583.1 7138 -5530.4 -14473 13666 16314 -12901 -26770 20675 30434 -20248 -32128 22366 43269 -22136 -31486 22350 49562 -22165 -28291 22210 66289 -22069 -17764 21937 123060 -21840 -7171.3 21589 127600 -21515
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For wall thickness = .5”
Major Axis 20 20 20 20 20 20 20 20 20 20 20 20 20 20
Minor Axis 20 19.99 19.95 19.93 19.9 19.5 18 15 10 6 5 4 3 2.5
a/b 1 1.0005 1.002506 1.003512 1.005025 1.025641 1.111111 1.333333 2 3.333333 4 5 6.666667 8
Stress at Stress at Stress at D B C 370.25 370.25 390.25 358.74 381.57 402.13 312.74 426.71 449.71 289.75 449.23 473.48 255.3 482.94 509.12 -201.21 924.99 982.08 -1865 2458.5 2720 -4949 4936.9 6040.9 -9113.4 7355.1 11353 -10160 7890.3 18019 -9250 7870.7 24293 -6650 7794.7 42875 -1258.9 7479 45045 0 7109.5 39462
36
Stress at A 390.25 378.15 329.86 305.76 269.7 -203.09 -1842.5 -4493.5 -7113.5 -7766.8 -7774 -7725.2 -7435 -7078
