# Patent

## Title
Z-shaped sheet piling

## Abstract

An improved Z-shaped sheet piling (15) comprising a first flange (16), a second flange (18), a web (19), a junction (20) between the first flange and the web, the junction having an inner surface (44), the inner surface defined by at least a first radius (23a) and a second radius (23b). The inner surface may be further defined by a third radius (23c). The sheet piling may comprise a first flange (16) having a substantially planar flange surface (37), a second flange (18), a web (19) having a substantially planar web surface (39), a junction (20) between the first flange and the web, the junction having a fillet portion (43) defined by the planes of the web surface and flange surface extended to an intersection (59) and an inner substantially arcuate surface (44) intersecting the web surface and flange surface, the inner arcuate surface defined by at least a first radius (23a) and a second radius (23b). The inner arcuate surface may be further defined by a third radius (23c).

## Background
FIELD OF THE INVENTION
The present invention relates generally to the field of sheet pilings, and more particularly to an improved sheet piling having a substantially Z-shaped transverse cross section.
BACKGROUND ART
A variety of Z-shaped steel sheet pilings are known in the prior art. Z-shaped sheet pilings are typically produced in different sizes characterized by their approximate weight in pounds per square foot (“psf”). Typical sizes include the PZ22, PLZ23, PLZ25, PZ27, PZ35, and the PZ40. Such sheet pilings have been produced by Bethlehem Steel Corporation and United States Steel Corporation.
However, sheet pilings known in the prior art do not provide much versatility with respect to the placement of steel near the junction. This has been found to limit the ability to strengthen the piling with respect to transverse stresses (i.e., those stresses oriented perpendicular to the longitudinal axis of the sheet piling).
Hence, it would be useful to provide sheet pilings which can be manufactured efficiently and with greater selectivity for strength.
DISCLOSURE OF THE INVENTION
With parenthetical reference to the corresponding parts, portions or surfaces of the disclosed embodiment, merely for the purposes of illustration and not by way of limitation, the present invention provides an improved Z-shaped sheet piling (15) comprising a first flange (16), a second flange (18), a web (19), a junction (20) between the first flange and the web, the junction having an inner surface (44), the inner surface defined by at least a first radius (23a) and a second radius (23b). The inner surface may be further defined by a third radius (23c).
The present invention also provides a Z-shaped sheet piling comprising a first flange (16) having a substantially planar flange surface (37), a second flange (18), a web (19) having a substantially planar web surface (39), a junction (20) between the first flange and the web, the junction having a fillet portion (43) defined by the planes of the web surface and flange surface extended (37a, 39a) to an intersection (59) and an inner substantially arcuate surface (44) intersecting the web surface (60b) and flange surface (60a), the inner arcuate surface defined by at least a first radius (23a) and a second radius (23b). The inner arcuate surface may be further defined by a third radius (23c).
Accordingly, the general object of the present invention is to provide an improved Z-shaped sheet piling in which the thickness of the web and flange at the junction can be increased more selectively to provide greater strength.
Another object is to provide Z-shaped sheet pilings which are strengthened more selectively.
Another object is to provide improved Z-shape sheet pilings in which steel is extended at the junction along the web or flange in a more case specific manner.
These and other objects and advantages will become apparent from the foregoing and ongoing written specification, the drawings, and the appended claims.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a perspective view of the improved sheet piling under loading conditions.
FIG. 2 is a left side elevation of the sheet piling shown in FIG. 1.
FIG. 3 is a plan view of the sheet piling shown in FIG. 1.
FIG. 4 is a plan view of the sheet piling with load applied for finite testing and analysis.
FIG. 5 is a transverse horizontal sectional view of a sheet piling shown in FIG. 1.
FIG. 6 is a detailed view of the first junction shown in FIG. 5.
FIG. 6A is a second detailed view of the first junction shown in FIG. 5.
FIG. 7 is a detailed view of the second junction shown in FIG. 5.
FIG. 7A is a second detailed view of the second junction shown in FIG. 5.
FIG. 8 is a schematic of the three radii defining the inner surface of the junction shown in FIG. 6.
FIG. 9 is a schematic of the three radii defining the inner surface of the junction shown in FIG. 7.
FIG. 10 is a transverse horizontal sectional view of an alternate embodiment of a sheet piling shown in FIG. 1.
FIG. 11 is a detailed view of the first junction shown in FIG. 10.
FIG. 11A is a second detailed view of the first junction shown in FIG. 10.
FIG. 12 is a second detailed view of the second junction shown in FIG. 10.
FIG. 12A is a second view of the second junction shown in FIG. 10.
FIG. 13 is a schematic of the two radii defining the inner surface of the junction shown in FIG. 11.
FIG. 14 is a schematic of the two radii defining the inner surface of the junction shown in FIG. 12.
FIG. 15 plots the allowable moment of a first embodiment of the improved piling versus applied pressure at the wale or support location.
FIG. 16 plots the allowable moment of the first embodiment of the improved piling versus applied pressure at the span location.
FIG. 17 plots the allowable moment of a second embodiment of the improved piling versus applied pressure at the wale or support location.
FIG. 18 plots the allowable moment of the second embodiment of the improved piling versus applied pressure at the span location.
FIG. 19 plots the allowable moment of a third embodiment of the improved piling versus applied pressure at the wale or support location.
FIG. 20 plots the allowable moment of the third embodiment of the improved piling versus applied pressure at the span location.
FIG. 21 plots the allowable moment of a fourth embodiment of the improved piling versus applied pressure at the wale or support location.
FIG. 22 plots the allowable moment of the fourth embodiment of the improved piling versus applied pressure at the span location.
FIG. 23 plots the allowable moment of a fifth embodiment of the improved piling versus applied pressure at the wale or support location.
FIG. 24 plots the allowable moment of the fifth embodiment of the improved piling versus applied pressure at the span location.
FIG. 25 plots the allowable moment of a sixth embodiment of the improved piling versus applied pressure at the wale or support location.
FIG. 26 plots the allowable moment of the sixth embodiment of the improved piling versus applied pressure at the span location.



DESCRIPTION OF THE PREFERRED EMBODIMENTS
At the outset, it should be clearly understood that like reference numerals are intended to identify the same structural elements, portions or surfaces consistently throughout the several drawing figures, as such elements, portions or surfaces may be further described or explained by the entire written specification, of which this detailed description is an integral part. Unless otherwise indicated, the drawings are intended to be read (e.g., cross-hatching, arrangement of parts, proportion, debris, etc.) together with the specification, and are to be considered a portion of the entire written description of this invention. As used in the following description, the terms “horizontal”, “vertical”, “left”, “right”, “up” and “down”, as well as adjectival and adverbial derivatives thereof, (e.g., “horizontally”, “rightwardly”, “upwardly”, etc.), simply refer to the orientation of the illustrated structure as the particular drawing figure faces the reader. Similarly, the terms “inwardly” and “outwardly” generally refer to the orientation of a surface relative to its axis of elongation, or axis of rotation, as appropriate.
Referring now to the drawings, and, more particularly, to FIG. 1 thereof, this invention provides an improved Z-shaped sheet piling, of which the presently preferred embodiment is generally indicated at 15. As shown in FIG. 5, the improved sheet piling 15 broadly includes a first flange 16, a web 19, and a second flange 18. The left marginal end of flange 16 is provided with a socket connection 32. The right marginal end of second flange 18 is provided with a ball connection 31. As shown in FIGS. 3–4, ball 31 and socket 32 connections, or other similar interlocks known in the art, allow the joining of individual sections of sheet piling to form a continuous steel wall, which may be employed in the construction of bridge piers, cofferdams, bridge abutments, bulkheads or the like.
As shown in FIGS. 5 and 10, in each of the two general embodiments flange 16 and web 19 are connected at arcuate junction 20. Similarly, web 19 and flange 18 are connected at arcuate junction 21. Flange 16 is a substantially-planar steel member having a thickness dimension 28. Similarly, web 19 and flange 18 are substantially-planar members with thickness dimensions 29 and 30, respectively. Flange 16 and flange 18 are generally parallel to each other. Web 19 transversely connects flanges 16 and 18. However, rather than a perpendicular connection between flanges 16 and 18, web 19 intersects flange 16 at a web angle 22.
As shown in FIGS. 6 and 11, in each of the two general embodiments junction 20 is generally defined by web angle 22, inner surface 44 and outer arcuate surface 46. In each embodiment, junction 20 is the substantially arcuate portion connecting web 19 and flange 16. As shown in FIGS. 7 and 12, in each of the two general embodiments junction 21 is generally defined by web angle 22, inner surface 49 and outer arcuate surface 50. Junction 20 has an inner surface 44 and an outer arcuate surface 46. In each embodiment, junction 21 is the substantially arcuate portion connecting web 19 and flange 18. Junction 21 has an inner surface 49 and an outer arcuate surface 50. Flange 18 has substantially parallel inner and outer surfaces 41 and 42. Flange 16 has substantially parallel inner and outer surfaces 37 and 38. Similarly, web 19 has substantially parallel inner and outer surfaces 39 and 40.
FIGS. 5–9 show a first general embodiment characterized by junctions 20 and 21 having three inner radii. In this first general embodiment, as shown in FIGS. 6 and 8, surface 44 is generated about three center points, indicated at 55a, 55b and 55c. Surface 44 is thus defined by three inner radii, indicated at 23a, 23b and 23c. Surface 44 thus comprises a first arc distance from 60a to 70 of a cylinder having a radius 23a, a second arc distance from 70 to 71 of a cylinder having a radius 23b, and a third arc distance from 71 to 60b of a cylinder having a radius 23c. 
Point 55a is located at the intersection of imaginary line 51a and radius 23a. Line 51a extends perpendicular to flange surface 37 at tangent point 60a. Point 55c is located at the intersection of imaginary line 51c and radius 23c. Line 51c extends perpendicular to inner web surface 39 at tangent point 60b. Tangent point 60a is located at the intersection of surface 37 and arcuate surface 44, which is the point at which the inner surface 37 of flange 16 begins to bend towards inner web surface 39. Similarly, tangent point 60b is located at the intersection of surface 44 and surface 39. Point 55b is located at the intersection of imaginary line 51b and radius 23b. Line 51b extends perpendicularly from tangent point 70, the point at which the curvature of surface 44 changes from being defined by radius 23a to being defined by radius 23b. It also can extend from tangent point 71, the point at which the curvature of surface 44 changes from being defined by radius 23b to being defined by radius 23c. 
As shown in FIG. 6, the plane of inner flange surface 37 and of inner web surface 39, respectively, may be extended into junction 20 to imaginary intersection point 59. Arcuate surface 44, which is in turn defined by radii 23a–c, the extension 39a of inner web surface 39, and the extension 37a of inner flange surface 37, define fillet 43.
As shown in FIG. 6A, outer junction surface 46 is generated about center point 56 and has a radius 24. Surface 46 is defined by a single arc distance of a cylinder having a radius 24. Flange 16 has an outer surface 38 and web 19 has an outer surface 40. Outer surface 38 and outer surface 40 are joined by arcuate outer surface 46. Center point 56 is located at the intersection of imaginary lines 52a and 52b. Line 52a extends perpendicular to flange outer surface 38 at tangent point 61a and line 52b extends perpendicular to outer web surface 40 at tangent point 61b. Tangent points 61a and 61b are located at the intersections of surface 46 and surfaces 38 and 40, respectively.
In this first general embodiment, as shown in FIGS. 7 and 9, surface 49 is generated about three center points, indicated at 57a, 57b and 57c. Surface 49 is thus defined by three inner radii, indicated at 25a, 25b and 25c. Surface 49 thus comprises a first arc distance from 62b to 74 of a cylinder having a radius 25a, a second arc distance from 74 to 73 of a cylinder having a radius 25b, and a third arc distance from 73 to 62a of a cylinder having a radius 25c. 
Point 57a is located at the intersection of imaginary line 53a and radius 25a. Line 53a extends perpendicular to web surface 40 at tangent point 62b. Point 57c is located at the intersection of imaginary line 53c and radius 25c. Line 53c extends perpendicular to flange surface 41 at tangent point 62a. Tangent point 62a is located at the intersection of surface 41 and arcuate surface 49, which is the point at which the inner surface 41 of flange 18 begins to bend towards web surface 40. Similarly, tangent point 62b is located at the intersection of surface 40 and surface 49. Point 57b is located at the intersection of imaginary line 53b and radius 25b. Line 53b extends perpendicularly from tangent point 74, the point at which the curvature of surface 40 changes from being defined by radius 25a to being defined by radius 25b. It also can extend from tangent point 73, the point at which the curvature of surface 49 changes from being defined by radius 25b to being defined by radius 25c. 
As shown in FIG. 7, the plane of inner flange surface 41 and of web surface 40, respectively, may be extended into junction 21 to imaginary intersection point 64. Arcuate surface 49, which is in turn defined by radii 25a–c, the extension 40a of web surface 40, and the extension 41a of inner flange surface 41, define fillet 48.
As shown in FIG. 7A, outer junction surface 50 is generated about center point 58 and has a radius 26. Surface 50 is defined by a single arc distance of a cylinder having a radius 26. As mentioned above, flange 18 has an outer surface 42 and web 19 has a surface 39. Surface 42 and surface 39 are joined by arcuate outer surface 50. Center point 58 is located at the intersection of imaginary lines 54a and 54b. Line 54a extends perpendicular to flange outer surface 42 at tangent point 63a and line 54b extends perpendicular to web surface 39 at tangent point 63b. Tangent points 63a and 63b are located at the intersections of surface 50 and surfaces 42 and 39, respectively.
Sheet pilings may be analyzed to calculate transverse (perpendicular to the interlock) stresses and the calculation for the allowable longitudinal moment (“ML”) of the pilings has been expanded to include the effect of transverse stresses: 



  
    M
    L
  
  =
  
    
      I
      y
    
    ⁢
    
      (
      
        
          Fy
          FS
        
        -
        
          
            (
            Ts
            )
          
          ⁢
          
            (
            p
            )
          
        
      
      )
    
  




where “Ts” is the transverse stress contribution, “I” is the moment of inertia of the cross section, “y” is the distance from the centroidal axis to the point of calculating the stresses, “Fy” is the yield stress. “FS” is the factor of safety, and “p” is the normal pressure. The “transverse stress contribution” is a value calculated mathematically. The formulation of allowable longitudinal bending moment in the piling is based on use of the Maximum Shear Stress Failure Criterion.

FIG. 4 shows the improved sheet piling for both general embodiments under loading conditions of one psi oriented normal to the longitudinal surface of the piling. This is the applied pressure load. FIGS. 1–2 show and generally differentiate between wale positions 12 and span positions 13. Wale positions 12 are at those longitudinal points on the piling at which the piling is constrained by a wale 14, and span positions 13 are at those longitudinal points at which the piling is not constrained by a wale 14. The wale location is meant to be that location in the piling which controls the allowable moment of the piling at wale position 12. The span location is meant to be that location in the piling which controls the allowable moment of the piling at span positions 13.
Six different specific embodiments of the invention are provided, which are delineated by weight per square foot of wall. The embodiments are hereafter identified as PZ35, PZ40, PZC21.7, PZC24.2, PZC31.8 and PZC39.7. Using linear finite element analysis, Applicant tested each of these embodiments, the results of which are provided in FIGS. 15–26, with the allowable longitudinal moment on the y axis and pressure on the x axis. The design curve plots the allowable longitudinal moment as a function of pressure for a standard steel yield strength of 50 ksi (kilopounds per square inch) and a factor of safety FS of 1.538 (allowable stress being 65 percent (65%) of the steel yield stress). The pressure is applied normal to the surfaces of the piling. Accordingly, Applicant has discovered that it is highly beneficial to form the pilings with at least two radii, rather than just one. Such a new design allows for more selective reduction of the transverse stresses flowing through the cross-section of the piling. The sets of two graphical depictions shown in FIGS. 15–26 illustrate the allowable moment as a function of pressure for the improved piling at the span and wale locations, respectively, for the six specific sheet pilings.
Of the six specific embodiments, the PZ35, PZ40, PZC21.7 and PZC24.2 employ junctions defined by an inner surface having three inner radii as generally shown in FIGS. 5–9. As shown, the arcs from 60a to 70 and from 71 to 60b in this general embodiment have substantially less curvature than the arc from 70 to 71. As a result, steel is extended further along surface 37 of flange 16 and surface 39 of web 19, respectively, and steel is concentrated less at the central portion of junction 20. The greater the radius, the flatter the arc and the thinner and further along the subject surface the steel extends. It is contemplated that surface 44 could consist of one or more linear rather than arcuate sections, with the subject radius thereby approaching infinite.
The structural dimensions and the data from the finite analysis, as more fully shown in FIGS. 15–22, of these four specific embodiments are summarized in following charts A1–A4 and B1–B4, respectively.







CHART A1





PZ35- Physical Characteristics



















Weight
35.0
psf



Moment of Inertia
369.5
in4/ft



Section Modulus
48.9
in3/ft



Web Angle
62.7
degrees



First Junction Inner Radii



a. radius 1 (web)
10
in



b. radius 2 (central)
1.5
in



c. radius 3 (flange)
10
in



First Junction Outer Radius
1.957
in



Second Junction Inner Radii



a. radius 1 (web)
10
in



b. radius 2 (central)
1.5
in



c. radius 3 (flange)
10
in



Second Junction Outer Radius
1.957
in



First Flange Thickness
0.605
in



Second Flange Thickness
0.605
in



Web Thickness
0.5
in



Cross-Sectional Area
19.4
in2



Distance from Centroid to the First
7.55
in



Flange Outer Surface



Distance Between the First Flange Outer
15.1
in



Surface and the Second Flange Outer



Surface

















CHART B1





PZ35- Strength
















Transverse Stress at Wale Location
131.8 psi per psi applied pressure


between 0 and 6.59 psi


Transverse Stress at Span Location
343.7 psi per psi applied pressure


between 0 and 36.92 psi


Total Area Under Curve at Wale
41,650


Location


Area Under Curve between 0 psi and
29,050


20 psi at Wale Location


Total Area Under Curve at Span
69,650


Location


Area Under Curve between 0 psi and
28,450


20 psi at Span Location
















CHART A2





PZ40- Physical Characteristics



















Weight
40.1
psf



Moment of Inertia
504.2
in4/ft



Section Modulus
61.5
in3/ft



Web Angle
74.1
degrees



First Junction Inner Radii



a. radius 1 (web)
10
in



b. radius 2 (central)
0.875
in



c. radius 3 (flange)
10
in



First Junction Outer Radius
1.5
in



Second Junction Inner Radii



a. radius 1 (web)
10
in



b. radius 2 (central)
0.875
in



c. radius 3 (flange)
10
in



Second Junction Outer Radius
1.5
in



First Flange Thickness
0.605
in



Second Flange Thickness
0.605
in



Web Thickness
0.5
in



Cross-Sectional Area
19.3
in2



Distance from Centroid to the First
8.2
in



Flange Outer Surface



Distance Between the First Flange Outer
16.4
in



Surface and the Second Flange Outer



Surface

















CHART B2





PZ40- Strength
















Transverse Stress at Wale Location
423.1 psi per psi applied pressure


between 6.37 and 39.21 psi


Transverse Stress at Span Location
335.2 psi per psi applied pressure


between 0 psi and 39.8 psi


Total Area Under Curve at Wale
67,850


Location


Area Under Curve between 0 psi and
37,100


20 psi at Wale Location


Total Area Under Curve at Span
84,800


Location


Area Under Curve between 0 psi and
35,850


20 psi at Span Location
















CHART A3





PZC21.7- Physical Characteristics



















Weight
21.7
psf



Moment of Inertia
15.2
in4/ft



Section Modulus
24.2
in3/ft



Web Angle
41.9
degrees



First Junction Inner Radii



a. radius 1 (web)
18
in



b. radius 2 (central)
1
in



c. radius 3 (flange)
6
in



First Junction Outer Radius
2.2
in



Second Junction Inner Radii



a. radius 1 (web)
18
in



b. radius 2 (central)
1
in



c. radius 3 (flange)
6
in



Second Junction Outer Radius
1.4
in



First Flange Thickness
0.375
in



Second Flange Thickness
0.375
in



Web Thickness
0.375
in



Cross-Sectional Area
14.8
in2



Distance from Centroid to the First
6.285
in



Flange Outer Surface



Distance Between the First Flange Outer
12.557
in



Surface and the Second Flange Outer



Surface

















CHART B3





PZC21.7- Strength
















Transverse Stress at Wale Location
1561 psi per psi applied pressure


between 1.23 and 20.20 psi


Transverse Stress at Span Location
1017 psi per psi applied pressure


between 1.68 and 22.94 psi


Total Area Under Curve at Wale
8,650


Location


Area Under Curve between 0 psi and
8,650


20 psi at Wale Location


Total Area Under Curve at Span
12,500


Location


Area Under Curve between 0 psi and
10,800


20 psi at Span Location
















CHART A4





PZC24.2- Physical Characteristics



















Weight
24.2
psf



Moment of Inertia
255.5
in4/ft



Section Modulus
33.5
in3/ft



Web Angle
54.4
degrees



First Junction Inner Radii



a. radius 1 (web)
18
in



b. radius 2 (central)
1
in



c. radius 3 (flange)
6
in



First Junction Outer Radius
1.75
in



Second Junction Inner Radii



a. radius 1 (web)
18
in



b. radius 2 (central)
1
in



c. radius 3 (flange)
6
in



Second Junction Outer Radius
1.135
in



First Flange Thickness
0.375
in



Second Flange Thickness
0.375
in



Web Thickness
0.375
in



Cross-Sectional Area
14.8
in2



Distance from Centroid to the First
7.625
in



Flange Outer Surface



Distance Between the First Flange Outer
15.25
in



Surface and the Second Flange Outer



Surface

















CHART B4





PZC24.2- Strength
















Transverse Stress at Wale Location
1400 psi per psi applied pressure


between 1.15 and 23.22 psi


Transverse Stress at Span Location
1006 psi per psi applied pressure


between 0 and 8.94 psi


Total Area Under Curve at Wale
13,250


Location


Area Under Curve between 0 psi and
13,000


20 psi at Wale Location


Total Area Under Curve at Span
16,950


Location


Area Under Curve between 0 psi and
14,900


20 psi at Span Location









Of the six specific embodiments, PZC31.8 and PZC39.7 employ junctions defined by an arcuate surface having only two radii, as generally shown in FIGS. 10–14. As shown in FIGS. 10–14, the general structure of the Z-shaped piling is similar as the structure of the first general embodiment shown in FIGS. 5–9. However, in this second general embodiment, junctions 20 and 21 and inner surfaces 44 and 49 are defined by two radii rather than three. In particular, in this second general embodiment, as shown in FIGS. 11 and 13, surface 44 is generated about two center points, indicated at 55a and 55b. Surface 44 is thus defined by two inner radii, indicated at 23a and 23b. Surface 44 thus comprises a first arc distance from 60a to 75 of a cylinder having a radius 23a and a second arc distance from 75 to 60b of a cylinder having a radius 23b. 
As shown, the arc from 75 to 60b in this embodiment has substantially less curvature than the arc from 60a to 75. As a result, steel is extended further along surface 39 of web 19, and steel is extended less along the inner surface 37 of flange 16. The greater the radius, the flatter the arc and the thinner and further along the subject surface the steel extends. It is contemplated that surface 44 could consist of one or more linear rather than arcuate sections, with the subject radius approaching infinite.
In this second general embodiment, point 55a is located at the intersection of imaginary line 51a and radius 23a. Line 51a extends perpendicular to flange surface 37 at tangent point 60a. It also can extend from tangent point 75, the point at which the curvature of surface 44 changes from being defined by radius 23a to being defined by radius 23b. Point 55b is located at the intersection of imaginary line 51b and radius 23b. Line 51b extends perpendicular to inner web surface 39 at tangent point 60b. It too can extend from tangent point 75. Tangent point 60a is located at the intersection of surface 37 and arcuate surface 44, which is the point at which the inner surface 37 of flange 16 begins to bend towards inner web surface 39. Similarly, tangent point 60b is located at the intersection of surface 44 and surface 39.
As shown in FIG. 11, again the plane of inner flange surface 37 and of inner web surface 39, respectively, may be extended into junction 20 to imaginary intersection point 59. In this general embodiment, arcuate surface 44, which is in turn defined by radii 23a and 23b, the extension 39a of inner web surface 39, and the extension 37a of inner flange surface 37, define fillet 43.
In this second general embodiment, as shown in FIGS. 12 and 14, surface 49 is generated about two center points, indicated at 57a and 57b. Surface 49 is thus defined by two inner radii, indicated at 25a and 25b. Surface 49 thus comprises a first arc distance from 62b to 76 of a cylinder having a radius 25a and a second arc distance from 76 to 62a of a cylinder having a radius 25b. 
Point 57a is located at the intersection of imaginary line 53a and radius 25a. Line 53a extends perpendicular to web surface 40 at tangent point 62b. It also can extend from tangent point 76, the point at which the curvature of surface 49 changes from being defined by radius 25a to being defined by radius 25b. Point 57b is located at the intersection of imaginary line 53b and radius 25b. Line 53b extends perpendicular to flange surface 41 at tangent point 62a. Tangent point 62a is located at the intersection of surface 41 and arcuate surface 49, which is the point at which the inner surface 41 of flange 18 begins to bend towards web surface 40. Similarly, tangent point 62b is located at the intersection of surface 40 and surface 49.
As shown in FIG. 12, the plane of inner flange surface 41 and of web surface 40, respectively, may be extended into junction 21 to imaginary intersection point 64. Arcuate surface 49, which is in turn defined by radii 25a and 25b, the extension 40a of web surface 40, and the extension 41a of inner flange surface 41, define fillet 48.
The structural dimensions and the data from the finite analysis, as more fully shown in FIGS. 23–26, of these two specific embodiments are summarized in following charts A5–A6 and B5–B6, respectively.







CHART A5





PZC31.8- Physical Characteristics



















Weight
31.8
psf



Moment of Inertia
397.9
in4/ft



Section Modulus
48.5
in3/ft



Web Angle
58.6
degrees



First Junction Inner Radii



a. radius 1 (web)
10
in



b. radius 2 (flange)
0.833
in



First Junction Outer Radius
1.125
in



Second Junction Inner Radii



a. radius 1 (web)
10
in



b. radius 2 (flange)
0.833
in



Second Junction Outer Radius
1.125
in



First Flange Thickness
0.56
in



Second Flange Thickness
0.56
in



Web Thickness
0.46
in



Cross-Sectional Area
19.5
in2



Distance from Centroid to the First
8.21
in



Flange Outer Surface



Distance Between the First Flange Outer
16.42
in



Surface and the Second Flange Outer



Surface

















CHART B5





PZC31.8- Strength
















Transverse Stress at Wale Location
691.1 psi per psi applied pressure


between 0 and 47.04 psi


Transverse Stress at Span Location
546.7 psi per psi applied pressure


between 0 and 31.68 psi


Total Area Under Curve at Wale
39,600


Location


Area Under Curve between 0 psi and
26,450


20 psi at Wale Location


Total Area Under Curve at Span
14,350


Location


Area Under Curve between 0 psi and
26,200


20 psi at Span Location
















CHART A6





PZC39.7- Physical Characteristics



















Weight
39.7
psf



Moment of Inertia
614.1
in4/ft



Section Modulus
67
in3/ft



Web Angle
71.5
degrees



First Junction Inner Radii



a. radius 1 (web)
10
in



b. radius 2 (flange)
0.833
in



First Junction Outer Radius
1.2
in



Second Junction Inner Radii



a. radius 1 (web)
10
in



b. radius 2 (flange)
0.833
in



Second Junction Outer Radius
1.2
in



First Flange Thickness
0.6
in



Second Flange Thickness
0.6
in



Web Thickness
0.505
in



Cross-Sectional Area
20.8
in2



Distance from Centroid to the First
9.170
in



Flange Outer Surface



Distance Between the First Flange Outer
18.3
in



Surface and the Second Flange Outer



Surface

















CHART B6





PZC39.7- Strength
















Transverse Stress at Wale Location
539.6 psi per psi applied pressure


between 3.96 and 60.23 psi


Transverse Stress at Span Location
461.9 psi per psi applied pressure


between 0 and 70.35 psi


Total Area Under Curve at Wale
69,850


Location


Area Under Curve between 0 psi and
38,550


20 psi at Wale Location


Total Area Under Curve at Span
76,550


Location


Area Under Curve between 0 psi and
37,350


20 psi at Span Location









By selectively modifying the amount and distribution of steel at the junction with a two or three radii design, it has been found that a substantial increase in the allowable moment as a function of pressure is obtained, especially with respect to the reduction of transverse stresses.
The present invention contemplates that many changes and modifications maybe made. Therefore, while the presently-preferred forms of the Z-shaped piling has been shown and described, those skilled in this art will readily appreciate that various additional changes and modifications may be made without departing from the spirit of the invention, as defined and differentiated by the following claims.

## Claims

1. A Z-shaped sheet piling comprising:
a first flange;
a second flange;
a web;
a junction between said first flange and said web;
said junction having substantially concave inner surface;
said inner surface comprising at least a first concave surface of a first radius and a second concave surface of a second radius that is different from said first radius;
thereby allowing for selective strengthening of the piling with respect to transverse stresses.

2. The Z-shaped sheet piling of claim 1, wherein said inner surface further comprises a third concave surface of a third radius that is different from said second radius.

3. A Z-shaped sheet piling comprising:
a first flange having a substantially planar flange surface;
a second flange;
a web having a substantially planar web surface;
a junction between said first flange and said web;
said junction having a fillet portion defined by said planes of said web surface and said flange surface extended to an imaginary intersection of said planes and substantially concave inner surface extending between said web surface and said flange surface;
said inner surface comprising at least a first concave surface of a first radius and a second concave surface of a second radius that is different from said first radius;
thereby allowing for selective strengthening of the piling with respect to transverse stresses.

4. The Z-shaped sheet piling of claim 3, wherein said inner surface further comprises a third concave surface of a third radius that is different from said second radius.