WO2014100553A1

WO2014100553A1 - Turbine disc with reduced neck stress concentration

Info

Publication number: WO2014100553A1
Application number: PCT/US2013/076845
Authority: WO
Inventors: Loc Quang Duong; Xiaolan Hu
Original assignee: United Technologies Corporation
Priority date: 2012-12-20
Filing date: 2013-12-20
Publication date: 2014-06-26
Also published as: EP2935786A1; CA2887943A1; US20140174098A1; EP2935786A4

Abstract

A disc with two sides includes a hub having a bore and a bore radius, a neck, and a rim. The neck is connected to and radially outward of the hub and has an inner wedge with a curved section on one side of the disc, an outer wedge with a curved section on that same side of the disc, and a center section between the wedges with a flat side on that same side of the disc. The rim is connected to and radially outward of the neck, the rim having a radius that is no more than seven times greater than the bore radius.

Description

TURBINE DISC WITH REDUCED NECK STRESS CONCENTRATION

BACKGROUND

The present invention relates to discs, and, more particularly, to a disc for a gas turbine engine.

Discs are included in many types of rotary machines, and in many applications, discs must rotate at high speeds during operation. This rotation requires the disc to have enough structural integrity to generate the necessary centripetal force to keep the disc intact. Otherwise the reaction to the centripetal force, known as centrifugal force (an imaginary force created by the inertia of the disc itself) will cause stress within the disc to exceed the material strength of the disc, breaking the disc apart. In addition to the mass of the disc itself, other components may be attached to the outer periphery of the disc. This increases the amount of rotating mass, requiring that the disc have greater strength. This can be solved by adding material to the disc, but doing so also adds cost and weight. Adding cost to a design is always undesirable, and, in the case of a vehicle application such as a gas turbine engine, adding weight may not be an option.

SUMMARY

According to one embodiment of the present invention, a disc with two sides includes a hub having a bore and a bore radius, a neck, and a rim. The neck is connected to and radially outward of the hub and has an inner wedge with a curved section on one side of the disc, an outer wedge with a curved section on that same side of the disc, and a center section between the wedges with a flat side on that same side of the disc. The rim is connected to and radially outward of the neck, the rim having a radius that is no more than seven times greater than the bore radius.

In another embodiment, a gas turbine engine includes a compressor section, combustor section downstream of the compressor section with an inner radius, and a turbine section downstream of the combustor section with a rotor with an outer radius. The outer radius is no more than 0.83 times as large as the inner radius of the combustor. The turbine section also includes a disc with a neck that has an inner wedge with a curved section on one side of the disc, an outer wedge with a curved section on that same side of the disc, and a center section between the wedges with a flat side on that same side of the disc.

In another embodiment, a gas turbine engine includes a compressor section, combustor section downstream of the compressor section, and a turbine section downstream of and substantially surrounded by the combustor section. The turbine section includes a disc with a neck that has an inner wedge with a curved section on one side of the disc, an outer wedge with a curved section on that same side of the disc, and a center section between the wedges with a flat side on that same side of the disc.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a partial cross-section view of a gas turbine engine of an auxiliary power unit.

FIG. 2 is a perspective view of a section of a rotor of the gas turbine engine section, including a disc and a blade.

FIG. 3 is a cross-section view of a neck of the disc.

DETAILED DESCRIPTION

In FIG. 1, a partial cross-section view of gas turbine engine 10 of an auxiliary power unit (APU) is shown. In the illustrated embodiment, the APU is designed for use on an aircraft. The APU includes gas turbine engine 10 that provides rotational force that can drive auxiliary equipment (not shown), such as an electrical generator or a pump.

Gas turbine engine 10 extends along engine axis 12 and includes compressor section 14, combustor section 16 downstream of compressor section 14, and turbine section 18 downstream of combustor section 16. Compressor section 14 includes impeller 19, and turbine section 18 includes first rotor 20 and second rotor 22. Impeller 19, first rotor 20, and second rotor 22 are all connected to shaft 23, which is rotatably positioned in gas turbine engine 10. More specifically, impeller 19, first rotor 20, second rotor 22 are connected to shaft 23 with a plurality of joints 25A-25D, respectively. In addition, first rotor 20 and second rotor 22 are connected to each other at joint 25C. Each joint 25 is a mechanical joint that prevents relative rotation between the connecting components, such as a spigot fit, a spline, a curvic coupling, or an axially toothed Hirth joint.

In one embodiment, gas turbine engine 10 is a compact gas turbine engine. In general, a compact gas turbine engine has a proportionally shorter axial length when compared to a more traditional gas turbine engine. In a traditional gas turbine engine, the whole combustor section is axially aft of the compressor section and the whole turbine section is axially aft of the combustor section, with the three sections having similar outer diameters. As shown in FIG. 1, compact gas turbine engine 10 has combustor section 16 inside of turbine section 18. More specifically, turbine section 18 is substantially surrounded by combustor section 16. This reduces the axial length of gas turbine engine 10 because turbine section 18 is not wholly axially aft of combustor section 16. This is possible because turbine section 18 is radially smaller than combustor section 16. Second rotor 22 has outer radius 24 that is no more than 0.83 times as large as inner radius 26 of combustor section 16. In equation form, (rotor outer radius 24) < 0.83 * (combustor inner radius 26).

During operation of gas turbine engine 10, gas G enters compressor section 14 and is compressed. Then gas G enters combustor section 16 and is mixed with fuel (not shown) and ignited, turning gas G into high pressure exhaust. Gas G is then expanded through turbine section 18 where energy is extracted and utilized to drive compressor section 24 and the auxiliary equipment (not shown). More specifically, as gas G expands through turbine section 18, first rotor 20 and second rotor 22 are rotated at high speed.

The components and configuration of gas turbine engine 10 allow for gas G and fuel to drive the auxiliary equipment by rotating first rotor 20 and second rotor 22. In addition, gas turbine engine 10 can have a compact size by positioning turbine section 18 at least partially inside of combustor section 16.

Depicted in FIG. 1 is one embodiment of the present invention, to which there are alternative embodiments. For example, gas turbine engine 10 can be used for propulsion. In such an embodiment, shaft 23 can be connected to a fan.

In FIG. 2, a perspective view of a section of second rotor 22 of gas turbine engine 10 is shown, including disc 28 and blade 30. While first rotor 20 and second rotor 22 are not identical, for the purposes of this discussion it will be understood that the below embodiments are applicable to both first rotor 20 and second rotor 22.

Second rotor 22 includes disc 28, which is a body of revolution about engine axis 12. Disc 28 has hub 32 with bore 33 and bore radius 34, rim 38 with rim radius 40, and neck 36 extending between hub 32 and rim 38. More specifically, neck 36 is connected to and radially outward of hub 32, and rim 38 is connected to and radially outward of neck 36.

Hub 32 includes front ring 44 to interface with first rotor 20 (shown in

FIG. 1) and rear ring 46 to interface with shaft 23 (shown in FIG. 1). First rotor 20 may have a similar front and rear rings, although the front ring on first rotor 20 interfaces with shaft 23 and the rear ring interfaces with second rotor 22 (at front ring 44). Extending radially outward from hub 32 is neck 36. The portion of neck 36 that is adjacent to hub 32 is inner wedge 48. Inner wedge 48 has a concave side and serves as a transition from hub 32 to center section 50, which is radially outward from inner wedge 48. Center section 50 is a thin, flat ring that extends between inner wedge 48 and outer wedge 52, which is radially outward from center section 50. Outer wedge 52 has a concave side and serves as a transition between center section 50 and rim 38.

Rim 38 includes root cut 42 into which blade 30 is positioned and serves to attach blade 30 to disc 28. (While only one blade 30 and one root cut 42 is shown in FIG. 2 for clarity, it will be understood that there are a plurality of blades 30 around the circumference of rim 38 and a plurality of circumferentially spaced root cuts 42, with each blade 30 being connected at a root cut 42.) The assembled second rotor 22 has outer radius 24, with outer radius 24 being the distance from engine axis 12 to the radially outer tip of blade 30.

As stated previously, in one embodiment, gas turbine engine 10 (shown in FIG. 1) is a compact engine. Thereby, in that embodiment, outer radius 24 is no more than ten times greater than bore radius 34. In equation form, (outer radius 24) < 10 * (bore radius 34). In the illustrated embodiment, the ratio of outer radius 24 to bore radius 34 is 7.9. In addition, in this embodiment, rim radius 40 is no more than seven times greater than bore radius 34. In equation form, (outer radius 24) < 7 * (bore radius 34). In the illustrated embodiment, the ratio of rim radius 40 to bore radius 34 is 4.9.

When gas turbine engine 10 (shown in FIG. 1) is operating, second rotor 22 rotates about engine axis 12. During rotation, there is stress on disc 28 caused by items attached to disc 28 (for example, blade 30) as well the body force of disc 28 itself. In other words, all of the rotating mass in second rotor 22 desires to move away from engine axis 12, but the structural integrity of disc 28 must prevent this from happening to avoid a catastrophic event. While this stress is spread throughout disc 28, different regions of disc 28 have different magnitudes of stress. For example, the stress is at a maximum magnitude at neck 36 (more specifically, at center section 50) because this region has the smallest cross-sectional area of any portion of disc 28.

The components and configuration of second rotor 22 allow for second rotor 22 to spin without fracturing. In addition, because center section 50 is narrower than rim 38 and hub 32, the thickness of neck 36 is minimized which reduces the volume of material and the weight of disc 28. In FIG. 3, a cross-section view of neck 36 of disc 28 is shown. In this view, both front side 54 and rear side 56 of disc 28 are visible. Between front side 54 and rear side 56 is radial line 58, which is perpendicular to engine axis 12 (shown in FIG. 1).

With respect to front side 54, inner wedge 48 has a concave curved section 60 with radius Ri, center section 50 has flat side 61, and outer wedge 52 has a concave curved section 62 radius R₂. Extending between curved sections 60, 62 is flat side 61, and, more specifically, flat side 61 abuts curved section 60 at P] at one end and flat side 61 abuts curved section 62 at P₂ at the opposite end. Flat side 61 is straight (which is an infinite radius of curvature) and has length Li. In the illustrated embodiment, length Li is at least 0.1 times greater than the smaller of either radii of curvature Ri, R₂. In equation form, Li > 0.1 * R(_smaii_{er of l and ¾}· In addition, flat side 61 is substantially radial (as it is parallel to radial line 58), and flat side 61 is continuous with and tangent to both curved sections 60, 62.

With respect to rear side 56, in the illustrated embodiment, rear side 56 is substantially the same as front side 54, although rear side 56 has the opposite orientation from front side 56. More specifically, inner wedge 48 has a concave curved section 64 with radius R3, center section 50 has flat side 65, and outer wedge 52 has a concave curved section 66 radius Ri. Extending between curved sections 64, 66 is flat side 65, and, more specifically, flat side 65 abuts curved section 64 at P3 at one end and flat side 65 abuts curved section 66 at P₄ at the opposite end. Flat side 65 is straight (which is an infinite radius of curvature) and has length L₂. In the illustrated embodiment, length L₂ is at least 0.1 times greater than the smaller of either radii of curvature R₃, R₄. In equation form, L₂ > 0.1 * R(_smaiier of 3 and )- In addition, flat side 65 is substantially radial (as it is parallel to radial line 58), and flat side 65 is continuous with and tangent to both curved sections 64, 66.

During operation (i.e. rotation) of second rotor 22 (shown in FIG. 2), the inertia of rim 38 would separate rim 38 from hub 32 (both shown in FIG. 2) but for neck 36 providing a reactionary force that prevents disc 28 (shown in FIG. 2) from structurally failing. This reactionary force generates stress that, according to the fluid flow theory of stress, can be said to flow through neck 36. Also according to the fluid flow theory of stress, a radiused feature such as one of curved sections 60, 62, 64, 66 redirects the flow of stress, creating a stress concentration. In addition, if neck 36 were configured such that curved sections 60, 64 were adjacent to curved sections 62, 66, respectively, (i.e. with no center section 50 or flat sides 61, 65) then there would be a single radial location at which neck 36 would have the smallest cross-sectional area. Unfortunately, the result would be a sharply increasing flow of stress intensity through neck 36 due to the meeting of two curved sections 60, 62 and 64, 66 and the small cross-sectional area thereat. The stress flow through this location would be choked, resulting in one large compound stress concentration at the junction of curved sections 60, 62 and another large compound stress concentration at the junction of curved sections 64, 66, wherein the former is on front side 54 and the latter is on rear side 56. Although, the remainder of neck 36 would have a much lower magnitude of stress than at these compound stress concentrations.

In accordance with the present invention, during rotation, disc 28 (shown in FIG. 2) lacks the aforementioned large compound stress concentrations. This is because flat sides 61, 65 have infinite radii of curvature and geometrically separate curved sections 60, 62 and 64, 66, respectively. Therefore, flat sides 61, 65 also separate the stress concentrations resulting from curved sections 60, 62, 64, 66, preventing the large compound stress concentration that would be present if curved sections 60, 62 and 64, 66 were adjacent, respectively. Instead, the embodiment illustrated in FIG. 3 smoothes the flow of stress through neck 36. More specifically, there is a stress concentration proximate to each of Pi, P₂, P₃, and P₄. The exact magnitudes of these stress concentrations depends on the specific geometry and loading of neck 36, including radii of curvature 60, 62, 64, and 66. But because curved sections 60, 62 are separated by flat side 61, the magnitudes of the stress concentrations proximate to Pi and P₂ are less than the compound stress that would result if flat side 61 did not separate Pi and P₂. Similarly, because curved sections 64, 66 are separated by flat side 65, the magnitudes of the stress concentrations proximate to P₃ and P₄ are less than the compound stress that would result if flat side 61 did not separate P₃ and P₄.

In the illustrated embodiment, having curved sections 60, 62 separated by flat side 61 creates an region proximate to the center of flat side 61 that has a magnitude of stress that is less than the magnitude of stress in the stress concentrations that are proximate to Pi and P₂. Similarly, having curved sections 64, 66 separated by flat side 65 creates a region proximate to the center of flat side 65 that has a magnitude of stress that is less than the magnitude of stress in the stress concentrations that are proximate to P₃ and P₄.

The configuration of disc 28 allows for a more homogenous stress distribution in neck 36. This reduces the maximum magnitudes of stress concentrations, which lowers the amount of material necessary to withstand the forces within neck 36 during operation of gas turbine engine 10 (shown in FIG. 1). In the illustrated embodiment, the reduction in the maximum magnitude of stress in neck 36 is at least fifteen percent when compared to a conventional disc neck that lacks any flat sides separating the curved sections. Although, in other embodiments, other stress reduction levels are possible.

Depicted in FIG. 3 is one embodiment of the present invention, to which there are alternative embodiments. For example, radius R] of curved section 60 can be substantially the same as radius R₂ of curved section 62, and radius R₃ of curved section 64 can be substantially the same as radius R₄ of curved section 66. For another example, radius Ri of curved section 60 can be substantially different from radius R3 of curved section 64, and radius R₂ of curved section 62 can be substantially different from radius R4 of curved section 66. For a further example, length Li can be substantially different from length L₂. For yet another example, flat sides 61, side 65 can be oriented such that one or both of them extend in a generally radial direction that is not substantially radial.

It will be recognized that the present invention provides numerous benefits and advantages. For example, the maximum stress concentration in neck 36 is reduced, which allows neck 36 to be thinner. This reduces the weight and cost of disc 28.

While the invention has been described with reference to an exemplary embodiment(s), it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the invention. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment(s) disclosed, but that the invention will include all embodiments falling within the scope of the appended claims.

Discussion of Various Embodiments

A disc according to an exemplary embodiment of this disclosure, among other possible things includes: a first side and a second side, the disc comprising: a hub including a bore with a bore radius; a neck connected to and radially outward of the hub, the neck comprising: an inner wedge with a first concave curved section on the first side of the disc including a first radius of curvature; an outer wedge with a second concave curved section on the first side of the disc including a second radius of curvature; and a center section extending between the inner wedge and the outer wedge, the center section including a first flat side on the first side of the disc; and a rim connected to and radially outward of the neck, the rim including a rim radius that is no more than seven times greater than the bore radius.

A further embodiment of the foregoing disc, wherein the first flat side can include a length that is at least 0.1 times greater than the smaller of the first radius of curvature and the second radius of curvature.

A further embodiment of any of the foregoing discs, wherein the first flat side can be tangent to the first concave curved section and tangent to the second concave curved section.

A further embodiment of any of the foregoing discs, wherein the first flat side can extend substantially radially.

A further embodiment of any of the foregoing discs, wherein the disc can further comprise: a third concave curved section on the inner wedge on the second side of the disc that includes a third radius of curvature that is substantially the same as the first radius of curvature; a fourth concave curved section on the outer wedge on the second side of the disc that includes a fourth radius of curvature that is substantially the same as the second radius of curvature; and a generally radial second flat side on the center section on the second side of the disc extending between the inner wedge and the outer wedge.

A further embodiment of any of the foregoing discs, wherein the first radius of curvature can be substantially the same as the second radius of curvature.

A further embodiment of any of the foregoing discs, wherein can be configured to react to a stress in the neck by distributing the stress to a first stress concentration of a first magnitude in the neck and a second stress concentration of a second magnitude in the neck, the first stress concentration and the second stress concentration separated by a region in the center section, the region having stress of a third magnitude that is lower than both the first magnitude and the second magnitude.

A gas turbine engine according to an exemplary embodiment of this disclosure, among other possible things includes: a compressor section; a combustor section downstream of the compressor section, the combustor section including an inner radius; and a turbine section downstream of the combustor section, the turbine section including a rotor with an outer radius that is no more than 0.83 times as large as the inner radius of the combustor, the rotor including a disc with a first side, a second side and a neck comprising: an inner wedge with a first concave curved section on the first side of the disc; an outer wedge with a second concave curved section on the first side of the disc; and a center section extending between the inner wedge and the outer wedge, the center section including a first flat side on the first side of the disc.

A further embodiment of the foregoing gas turbine engine, wherein the first flat side can include a length that is at least 0.1 times greater than the smaller of the first radius of curvature and the second radius of curvature.

A further embodiment of any of the foregoing gas turbine engines, wherein the disc can be configured to react to a stress in the neck by distributing the stress to a first stress concentration of a first magnitude in the neck and a second stress concentration of a second magnitude in the neck, the first stress concentration and the second stress concentration separated by a region in the center section, the region having stress of a third magnitude that is lower than both the first magnitude and the second magnitude.

A further embodiment of any of the foregoing gas turbine engines, wherein the first flat side can be tangent to the first concave curved section and tangent to the second concave curved section.

A further embodiment of any of the foregoing gas turbine engines, wherein the first flat side can extend substantially radially.

A further embodiment of any of the foregoing gas turbine engines, wherein the gas turbine engine can further comprise: a third concave curved section on the inner wedge on the second side of the disc that includes a third radius of curvature that is substantially the same as the first radius of curvature; a fourth concave curved section on the outer wedge on the second side of the disc that includes a fourth radius of curvature that is substantially the same as the second radius of curvature; and a generally radial second flat side on the center section on the second side of the disc extending between the inner wedge and the outer wedge.

A further embodiment of any of the foregoing gas turbine engines, wherein the first radius of curvature can be substantially the same as the second radius of curvature.

A further embodiment of any of the foregoing gas turbine engines, wherein the gas turbine engine can further comprise: a hub with a bore radius connected to and radially inward of the neck; and a rim connected to and radially outward of the neck, the rim including a rim radius that is no more than seven times greater than the bore radius.

A further embodiment of any of the foregoing gas turbine engines, wherein the turbine section can be substantially surrounded by the combustor section. A gas turbine engine according to an exemplary embodiment of this disclosure, among other possible things includes: a compressor section; a combustor section downstream of the compressor section; and a turbine section downstream of and substantially surrounded by the combustor section, the turbine section including a disc with a first side, a second side and a neck comprising: an inner wedge with a first concave curved section on the first side of the disc; an outer wedge with a second concave curved section on the first side of the disc; and a center section extending between the inner wedge and the outer wedge, the center section including a first flat side on the first side of the disc.

A further embodiment of the foregoing gas turbine engines, wherein the first flat side can include a length that is at least 0.1 times greater than the smaller of the first radius of curvature and the second radius of curvature.

A further embodiment of any of the foregoing gas turbine engines, wherein the combustor section includes an inner radius, and the turbine section includes a rotor that includes the disc, the rotor including an outer radius that can be no more than 0.83 times as large as the inner radius of the combustor.

Claims

CLAIMS:

1. A disc with a first side and a second side, the disc comprising:

a hub including a bore with a bore radius;

a neck connected to and radially outward of the hub, the neck comprising: an inner wedge with a first concave curved section on the first side of the disc including a first radius of curvature;

an outer wedge with a second concave curved section on the first side of the disc including a second radius of curvature; and a center section extending between the inner wedge and the outer wedge, the center section including a first flat side on the first side of the disc; and

a rim connected to and radially outward of the neck, the rim including a rim radius that is no more than seven times greater than the bore radius.

2. The disc of claim 1, wherein the first flat side includes a length that is at least 0.1 times greater than the smaller of the first radius of curvature and the second radius of curvature.

3. The disc of claim 1, wherein the first flat side is tangent to the first concave curved section and tangent to the second concave curved section.

4. The disc of claim 1 , wherein the first flat side extends substantially radially.

5. The disc of claim 1, further comprising:

a third concave curved section on the inner wedge on the second side of the disc that includes a third radius of curvature that is substantially the same as the first radius of curvature;

a fourth concave curved section on the outer wedge on the second side of the disc that includes a fourth radius of curvature that is substantially the same as the second radius of curvature; and a generally radial second flat side on the center section on the second side of the disc extending between the inner wedge and the outer wedge.

6. The disc of claim 1 , wherein the first radius of curvature is substantially the same as the second radius of curvature.

7. The disc of claim 1, wherein the disc is configured to react to a stress in the neck by distributing the stress to a first stress concentration of a first magnitude in the neck and a second stress concentration of a second magnitude in the neck, the first stress concentration and the second stress concentration separated by a region in the center section, the region having stress of a third magnitude that is lower than both the first magnitude and the second magnitude.

8. A gas turbine engine comprising:

a compressor section;

a combustor section downstream of the compressor section, the combustor section including an inner radius; and

a turbine section downstream of the combustor section, the turbine section including a rotor with an outer radius that is no more than 0.83 times as large as the inner radius of the combustor, the rotor including a disc with a first side, a second side and a neck comprising:

an inner wedge with a first concave curved section on the first side of the disc;

an outer wedge with a second concave curved section on the first side of the disc; and

a center section extending between the inner wedge and the outer wedge, the center section including a first flat side on the first side of the disc.

9. The gas turbine engine of claim 8, wherein the first flat side includes a length that is at least 0.1 times greater than the smaller of the first radius of curvature and the second radius of curvature.

10. The gas turbine engine of claim 8, wherein the disc is configured to react to a stress in the neck by distributing the stress to a first stress concentration of a first magnitude in the neck and a second stress concentration of a second magnitude in the neck, the first stress concentration and the second stress concentration separated by a region in the center section, the region having stress of a third magnitude that is lower than both the first magnitude and the second magnitude.

11. The gas turbine engine of claim 8, wherein the first flat side is tangent to the first concave curved section and tangent to the second concave curved section.

12. The gas turbine engine of claim 8, wherein the first flat side extends substantially radially.

13. The gas turbine engine of claim 8, further comprising:

14. The gas turbine engine of claim 8, wherein the first radius of curvature is substantially the same as the second radius of curvature.

15. The gas turbine engine of claim 8, further comprising:

a hub with a bore radius connected to and radially inward of the neck; and a rim connected to and radially outward of the neck, the rim including a rim radius that is no more than seven times greater than the bore radius.

16. The gas turbine engine of claim 8, wherein the turbine section is substantially surrounded by the combustor section.

17. A gas turbine engine comprising:

a compressor section;

a combustor section downstream of the compressor section; and a turbine section downstream of and substantially surrounded by the

combustor section, the turbine section including a disc with a first side, a second side and a neck comprising:

18. The gas turbine engine of claim 17, wherein the first flat side includes a length that is at least 0.1 times greater than the smaller of the first radius of curvature and the second radius of curvature.

19. The gas turbine engine of claim 17, wherein the combustor section includes an inner radius, and the turbine section includes a rotor that includes the disc, the rotor including an outer radius that is no more than 0.83 times as large as the inner radius of the combustor.

20. The gas turbine engine of claim 17, wherein the disc is configured to react to a stress in the neck by distributing the stress to a first stress concentration of a first magnitude in the neck and a second stress concentration of a second magnitude in the neck, the first stress concentration and the second stress concentration separated by a region in the center section, the region having stress of a third magnitude that is lower than both the first magnitude and the second magnitude.