Hat-shaped steel sheet pile
The hat-shaped steel sheet pile design enhances buckling strength against axial forces by optimizing the geometric relationships of its web, flanges, and arms, addressing manufacturing complexity and maintaining structural integrity.
Patent Information
- Application Number
- JP2022093793
- Authority / Receiving Office
- JP · JP
- Patent Type
- Patents
- Current Assignee / Owner
- Priority Date
- 2018-08-31
- Filing Date
- 2022-06-09
- Publication Date
- 2025-11-25
- Estimated Expiration
- 2039-08-16
AI Technical Summary
Existing hat-shaped steel sheet piles face challenges in improving buckling strength against axial forces while maintaining their basic cross-sectional shape, which can lead to reduced bending performance and complex manufacturing processes.
The design incorporates a web, flanges, and arms with specific geometric relationships, including a flange angle of θ≦70°, an equivalent width-thickness ratio of 120 or more, and a relationship between effective width, height, and angle that satisfies certain formulae, enhancing buckling strength without altering the basic shape.
This configuration significantly improves the buckling strength against axial forces while maintaining the hat-shaped steel sheet pile's structural integrity and manufacturing efficiency.
Smart Images

Figure 0007775149000014 
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Abstract
Description
[Technical Field]
[0001] The present invention relates to a hat-type steel sheet pile. [Background technology]
[0002] Hat-shaped steel sheet piles are widely used in civil engineering and construction projects to construct walls for earth retaining and watertight construction by driving them underground. Therefore, the bending performance of the wall has traditionally been emphasized. In addition, to improve economic efficiency, i.e., to reduce the weight of the wall, the cross section of hat-shaped steel sheet piles tends to be wider and thinner.
[0003] Meanwhile, as the performance and economy of hat-type steel sheet piles have improved, peripheral technological developments have been made for the hat-type steel sheet piles, and their range of application and uses have expanded. One example of peripheral technological development is the increase in the size of construction heavy machinery. The increase in size of construction heavy machinery aims to enable construction in harder or deeper strata, for example, and to improve construction efficiency. As construction heavy machinery increases in size, not only are bending stresses applied to the hat-type steel sheet piles after driving, but large axial forces also act on them during construction.
[0004] Furthermore, efforts are being made to expand the scope of application of hat-shaped steel sheet piles. Specifically, for example, in addition to conventional wall structures, hat-shaped steel sheet piles are increasingly being used as foundations. For example, Patent Document 1 describes a technique for using steel sheet piles as foundation structures. Patent Document 2 describes a technique for improving the bearing capacity of hat-shaped steel sheet piles by providing a blocking portion only at the tip of the hat-shaped steel sheet pile in order to improve the performance when used as a foundation. In this way, when hat-shaped steel sheet piles are used as foundations to bear bearing capacity, it is necessary to improve the performance of the hat-shaped steel sheet piles as axial force members.
[0005] Generally, consideration must be given to buckling in axial force members. In particular, in cross sections composed of thin-walled plate materials such as hat-shaped steel sheet piles, sufficient attention must be paid to local buckling, among other buckling problems. From this perspective, Patent Document 3 proposes a technology for improving the buckling strength of hat-shaped steel sheet piles by forming bent portions at the corners of the cross section. Considering the trend toward wider and thinner cross sections of hat-shaped steel sheet piles as mentioned above, it is thought that improving the buckling strength of hat-shaped steel sheet piles against axial force will become increasingly necessary in the future. [Prior art documents] [Patent documents]
[0006] [Patent Document 1] Patent No. 3832845 [Patent Document 2] Patent No. 4916932 [Patent Document 3] Patent No. 4088041 Summary of the Invention [Problem to be solved by the invention]
[0007] However, while the technology described in Patent Document 3 is efficient when considering hat-type steel sheet piles as axial force members, it reduces the amount of steel material in the web, which may result in a decrease in the bending performance of the wall, which is the performance originally expected of hat-type steel sheet piles. Furthermore, because the cross-sectional shape becomes complex, highly advanced technology is required in the rolling manufacturing process, and it is difficult to improve manufacturing efficiency. Considering these points, it is desirable to improve the buckling strength against axial force while maintaining the basic cross-sectional shape of the hat-type steel sheet pile, which consists of the web, flange, and arm.
[0008] Therefore, an object of the present invention is to provide a new and improved hat-shaped steel sheet pile that can improve the buckling strength against axial force while maintaining the basic cross-sectional shape of the hat-shaped steel sheet pile. [Means for solving the problem]
[0009] According to an aspect of the present invention, a hat-shaped steel sheet pile has, in a cross section perpendicular to the longitudinal direction, a web extending along the width direction on a first side in a cross-section height direction, and a flange extending from both ends of the web in the width direction to both sides in the width direction and toward a second side in the cross-section height direction, and forming a flange angle θ (θ≦70°) a pair of arms extending from each end of the pair of flanges along the width direction and toward both sides in the width direction on a second side in the cross-sectional height direction, and a pair of fitting joints formed at each end of the pair of arms on the opposite side from the pair of flanges. The ratio of the total length of the web, the pair of flanges, and the pair of arms in the cross section to the average plate thickness of the web, the pair of flanges, and the pair of arms is 120 or more, the effective width B and effective height h in the cross section, and the flange angle θ satisfy the condition of the following formula (i) using a constant C (1.01≦C≦1.13) when the plate thickness ratio of the flange to the web is in the range of 0.6 or more and 1.0 or less, and the effective width B, the web length Bw in the cross section, the cross-sectional height H, and the flange angle θ satisfy the relationship B-Bw-2H / tan θ>0. TIFF0007775149000001.tif22165
[0010] According to another aspect of the present invention, a hat-shaped steel sheet pile has, in a cross section perpendicular to the longitudinal direction, a web extending along the width direction on a first side in a cross-sectional height direction, and a flange extending from both ends of the web in the width direction to both sides in the width direction and toward a second side in the cross-sectional height direction, and forming a flange angle θ (θ≦70°)the web and the flanges; a pair of arms extending from the ends of the pair of flanges along the width direction and toward both sides in the width direction on a second side in the cross-sectional height direction; and a pair of fitting joints formed at the ends of the arms opposite the pair of flanges. The ratio of the total length of the web, the pair of flanges, and the pair of arms in the cross section to the average plate thickness of the web, the pair of flanges, and the pair of arms is 120 or more. The effective width B and effective height h in the cross section, and the flange angle θ satisfy the condition of the following formula (i) using a constant C (1.03≦C≦1.13) when the plate thickness ratio of the flange to the web is in the range of 0.7 or more and 1.0 or less, and the effective width B, the web length Bw in the cross section, the cross-sectional height H, and the flange angle θ satisfy the relationship B-Bw-2H / tan θ>0. TIFF0007775149000002.tif22165
[0011] According to the above configuration, it is possible to improve the buckling strength against axial force while maintaining the basic cross-sectional shape of the hat-shaped steel sheet pile. [Brief explanation of the drawings]
[0012] [Figure 1] FIG. 1 is a cross-sectional view of a hat-shaped steel sheet pile according to an embodiment of the present invention. [Figure 2] 1 is a graph showing the relationship between the converted width-thickness ratio and the web width-thickness ratio in a conventional steel sheet pile product. [Figure 3] FIG. 2 is a diagram schematically showing a cross-sectional shape of a hat-shaped steel sheet pile in the first studied example. [Figure 4] 10 is a graph showing the results of a buckling strength analysis in the first study example. [Figure 5] 5 is a graph showing the local buckling strength based on the results shown in FIG. 4. [Figure 6] 10 is a graph showing the local buckling strength based on the results of the buckling strength analysis in the second study example. [Figure 7] 10 is a graph showing the relationship between the flange angle and the aspect ratio in the third study example. [Figure 8]10 is a graph showing the relationship between the flange angle and the aspect ratio in the fourth study example, together with a part of the third study example. [Figure 9] 10 is a graph showing the relationship between the flange angle and the aspect ratio in the third and fourth study examples, together with an approximation curve of the correlation for each plate thickness ratio. [Figure 10] 10 is a graph showing the relationship between the flange angle and the aspect ratio in the third and fourth study examples, together with an approximation curve of the correlation for each plate thickness ratio. DETAILED DESCRIPTION OF THE INVENTION
[0013] Hereinafter, preferred embodiments of the present invention will be described in detail with reference to the accompanying drawings. In this specification and drawings, components having substantially the same functional configurations are designated by the same reference numerals, and redundant description will be omitted.
[0014] Fig. 1 is a cross-sectional view of a hat-shaped steel sheet pile according to one embodiment of the present invention. As shown in Fig. 1, the hat-shaped steel sheet pile 1 includes, in a cross section perpendicular to the longitudinal direction (z direction in the figure), a web 2 extending along the width direction (x direction in the figure) on a first side in the cross-sectional height direction (the rear side in the y direction in the figure), flanges 3A and 3B extending from both ends of the web 2 in the width direction to both sides in the width direction and toward a second side in the cross-sectional height direction (the front side in the y direction in the figure) and forming a flange angle θ (acute angle side) with the width direction, arms 4A and 4B extending along the width direction from the ends of the flanges 3A and 3B on the second sides in the cross-sectional height direction and toward both sides in the width direction, and fitting joints 5A and 5B formed at the ends of the arms 4A and 4B opposite to the flanges 3A and 3B.
[0015] 1 shows the dimensions of each part of the hat-shaped steel sheet pile 1, specifically, the length Bw and thickness tw of the web 2, the length Bf and thickness tf of the flanges 3A and 3B, and the length Ba and thickness ta of the arms 4A and 4B. Here, the length Bw is the distance between two intersections formed between the thickness center line of the web 2 and each of the flanges 3A and 3B. Similarly, the length Bf is the distance between two intersections formed between the thickness center line of the flange 3A and each of the web 2 and the arm 4A. Furthermore, the length Ba is the distance between the intersections formed between the thickness center line of the arm 4A and the thickness center line of the flange 3A and each of the fitting center E of the fitting joint 5A. A Since the cross-sectional shape of the hat-shaped steel sheet pile 1 is symmetrical about the neutral axis in the width direction (y-axis in the figure), the flange 3B also has a length Bf like the flange 3A, and the arm 4B also has a length Ba like the arm 4A.
[0016] 1 also shows the effective width B, cross-sectional height H, and effective height h of the hat-shaped steel sheet pile 1. Here, the effective width B is the distance between the fitting centers E of the fitting joints 5A and 5B. A ,E B The cross-sectional height H is the cross-sectional height of the hat-shaped steel sheet pile 1, including the thickness of the web 2 and arms 4A, 4B but not including the overhang of the fitting joints 5A, 5B. The effective height h is the cross-sectional height H minus half the thickness of the web 2 and arms 4A, 4B, i.e., h = H - (tw / 2 + ta / 2). When the shape of the hat-shaped steel sheet pile 1 shown in Figure 1 is geometrically valid, the overall width B, web length Bw, cross-sectional height H, and flange angle θ satisfy the relationship B - Bw - 2H / tanθ > 0.
[0017] Here, for the hat-shaped steel sheet pile 1 according to this embodiment, the total length B TTL Average plate thickness t AVE Ratio to B TTL / t AVE is defined as the "equivalent width-thickness ratio." As will be described later, in the hat-type steel sheet pile 1, the equivalent width-thickness ratio B TTL / t AVEis 120 or more, and the effective width B and effective height h in the cross section, as well as the flange angle θ, satisfy a predetermined relationship. TTL is the total length of the web 2, flanges 3A and 3B, and arms 4A and 4B, and is calculated by the following formula (1). AVE is the average of the plate thicknesses of the web 2, the flanges 3A and 3B, and the arms 4A and 4B, and is calculated by the following formula (2).
[0018]
number
[0019] Figure 2 is a graph showing the relationship between the converted width-thickness ratio and the web width-thickness ratio (Bw / tw) of conventional steel sheet pile products. The inventors have studied methods for improving the buckling strength against axial force while maintaining the basic cross-sectional shape of hat-shaped steel sheet piles. As already mentioned, this issue becomes more apparent as the cross-section of hat-shaped steel sheet piles tends to become wider and thinner. Therefore, it is reasonable to consider hat-shaped steel sheet piles that are wider or thinner than currently known hat-shaped steel sheet piles. Here, as shown in Figure 2, steel sheet piles with an converted width-thickness ratio of 120 or more have not yet been industrially realized, not only for hat-shaped steel sheet piles but also for U-shaped steel sheet piles. Therefore, the following study focuses on hat-shaped steel sheet piles with an converted width-thickness ratio of 120 or more, which are wider or thinner than conventional steel sheet pile products.
[0020] In this study, buckling strength analysis (eigenvalue analysis based on elastic theory) was carried out for hat-shaped steel sheet piles with an equivalent width-thickness ratio of 120 or more, for various cross-sectional shapes with different effective widths, cross-sectional heights, and plate thicknesses, and after identifying the local buckling mode, the relationship between the buckling strength of the hat-shaped steel sheet piles and the representative elements of the cross-sectional shape was analyzed. As a result, the following findings were made.
[0021] (First example) First, the effective width B of the hat-shaped steel sheet pile 1 was set to 1350 mm, and the thickness of the web plate thickness tw, flange plate thickness tf, and arm plate thickness ta were all set to 9.0 mm. Under these conditions, the moment of inertia per meter of wall width of the steel sheet pile wall, in which multiple hat-shaped steel sheet piles 1 are fitted together with fitting joints 5A and 5B and connected in the width direction, was set to approximately 10,000 cm 4 The effective height h (mm) and flange angle θ (degrees) of the cross-sectional shapes are shown in Table 1, and each cross-sectional shape is shown schematically in Figure 3.
[0022] [Table 1]
[0023] As shown in Table 1 and Figure 3, if the magnitude of the moment of inertia per meter of wall width is maintained, the effective height h increases as the flange angle θ decreases. This is because the decrease in the moment of inertia caused by the smaller flange angle θ must be compensated for by increasing the effective height h.
[0024] Figure 4 is a graph showing the results of the buckling strength analysis for the first study example. In Figure 4, the horizontal axis shows the length (longitudinal dimension) of the hat-shaped steel sheet pile, and the vertical axis shows the buckling strength of the hat-shaped steel sheet pile (results of elastic analysis). In the graph in Figure 4, it can be seen that a local buckling mode occurs at the location indicated by arrow (1), resulting in a decrease in strength. In this study, the strength in this local buckling mode (local buckling strength) was treated as the result, and the local buckling strength for each cross-sectional shape was compared. Note that arrow (2) in the graph in Figure 4 indicates that the results are shown for Example 1 (θ = 70°), Example 2 (θ = 60°), ..., and Example 9 (θ = 29°) at the point where the arrows intersect, in that order.
[0025] Figure 5 is a graph showing the local buckling strength based on the results shown in Figure 4. In Figure 5, the horizontal axis represents the flange angle θ for each cross-sectional shape, and the vertical axis represents the local buckling strength. The graph in Figure 5 shows that the local buckling strength increases rapidly as the flange angle θ increases from a small value, but then reaches a maximum at a certain angle and gradually decreases as the flange angle θ increases. In the example shown in Figure 5, the optimal flange angle θ at which the local buckling strength is considered to reach its maximum is 33.5°. These results suggest that the flange angle θ significantly contributes to suppressing the out-of-plane deformation associated with local buckling at each component of the hat-shaped steel sheet pile 1 (web 2, flanges 3A and 3B, and arms 4A and 4B). In other words, based on the above results, the optimal solution for maximizing the local buckling strength can be obtained in the cross-sectional design of the hat-shaped steel sheet pile 1 by appropriately determining the flange angle θ and then determining the effective height h based on the relationship between the flange angle θ and the moment of inertia.
[0026] (Second example) Next, the effective width B of the hat-shaped steel sheet pile 1 remains 1350 mm, and the thickness of the web plate tw and arm plate ta is set to 10.0 mm, and the flange plate thickness is set to tf8.0 mm. Under these conditions, the moment of inertia of the steel sheet pile wall per 1 m of wall width is approximately 10,000 cm 4 The effective height h (mm) and flange angle θ (deg) of the cross-sectional shapes that were set are shown in Table 2.
[0027] [Table 2]
[0028] Figure 6 is a graph showing the local buckling strength based on the results of the buckling strength analysis for the second study example. As with Figure 5, the graph in Figure 6 also shows that the local buckling strength varies, reaching a maximum value with respect to the flange angle θ. In the example shown in Figure 6, the optimal flange angle θ at which the local buckling strength is thought to reach a maximum value is 35.7°. According to the results shown in Figures 5 and 6, even when the plate thickness, which has a significant effect on local buckling, is changed, an optimal solution that maximizes the local buckling strength can be obtained using the flange angle θ and the effective height h, which is determined by the relationship between the flange angle θ and the moment of inertia.
[0029] (Third example) Next, when the effective width B of the hat-shaped steel sheet pile 1 is 1100mm, 1300mm, and 1500mm, the moment of inertia per 1m of wall width of the steel sheet pile wall is approximately 9,000cm 4 / m (9,000 class) ~ approx. 55,000 cm 4 / m (55,000 class) were set. The effective width B (mm), effective height h (mm), web thickness tw (mm), aspect ratio B / h, flange angle θ (degrees), and second moment of area class of the set cross-sectional shapes are shown in Tables 3 to 5.
[0030] [Table 3]
[0031] [Table 4]
[0032] [Table 5]
[0033] In the example shown in Table 3, the web thickness tw and arm thickness ta are all 9.0 mm, but the flange thickness tf is thinned to 6.3 mm (the flange to web thickness ratio is 0.7). In the example shown in Table 4, the web thickness tw, flange thickness tf, and arm thickness ta are all 9.0 mm (the flange to web thickness ratio is 1.0). In the example shown in Table 5, the web thickness tw and arm thickness ta are all 12.5 mm, but the flange thickness tf is 7.5 mm (the flange to web thickness ratio is 0.6). For each example in Tables 3 to 5, a buckling strength analysis similar to the examples shown in Figures 4 to 6 was performed on the cross-sectional shape, and the optimal flange angle θ and corresponding effective height h were determined, which were considered to maximize the local buckling strength.
[0034] Figure 7 is a graph showing the relationship between the flange angle and the aspect ratio in the third study example. In the graph in Figure 7, it is observed that there is a correlation between the flange angle θ and the aspect ratio B / h of the hat-shaped steel sheet pile 1, which is independent of the effective width B or the moment of inertia of area, when the plate thickness ratio is 1.0, 0.7, and 0.6.
[0035] (Fourth example) Next, the plate thickness was increased within the range where the converted width-thickness ratio was 120 or more, with the same effective width B as in the third study example above. Table 6 shows the effective width B (mm), effective height h (mm), web plate thickness tw (mm), aspect ratio B / h, flange angle θ (degrees), and second moment of area class of the cross-sectional shape set in this example.
[0036] [Table 6]
[0037] In all of the examples shown in Table 6, the plate thickness ratio is 0.7. That is, in each example, the web plate thickness tw and the arm plate thickness ta are equal, and the flange plate thickness tf is 0.7 times the web plate thickness tw. In each example in Table 6, a buckling strength analysis was performed on the cross-sectional shape in the same way as the examples shown in Figures 4 to 6, and the results set the optimal flange angle θ and the corresponding effective height h that are thought to maximize the local buckling strength.
[0038] Figure 8 is a graph showing the relationship between the flange angle and the aspect ratio in the fourth study example, along with a part of the third study example (an example with a plate thickness ratio of 0.7 shown in Table 3). The graph in Figure 8 also shows that there is a correlation between the flange angle θ and the aspect ratio B / h of the hat-shaped steel sheet pile 1 that is independent of the effective width B or the moment of inertia of area. Furthermore, the graph in Figure 8 shows that, even though the web plate thickness tw is different, in each example with a common plate thickness ratio (0.7), the flange angle θ and the aspect ratio B / h show a correlation on an almost common curve.
[0039] (Summary of study examples) Figures 9 and 10 are graphs showing the relationship between the flange angle and the aspect ratio in the third and fourth study examples, along with the approximation curves of the correlation for each thickness ratio. As mentioned above, considering economic considerations and constraints such as manufacturing facilities, the thickness ratio of the hat-type steel sheet pile 1 can realistically range from approximately 0.6 to 1.0. However, the third and fourth study examples include the upper and lower limits of this thickness ratio range. Therefore, the conditions available for realistic design of the hat-type steel sheet pile 1 can be expressed by the following formula (3) using a constant C based on the two approximation curves shown in the graphs of Figures 9 and 10. Figure 9 shows the case where the thickness ratio of the hat-type steel sheet pile 1 is in the range of 0.6 to 1.0, and the range of constant C is 1.01 ≤ C ≤ 1.13. On the other hand, Figure 10 shows the case where the thickness ratio of the hat-type steel sheet pile 1 is in the range of 0.7 to 1.0, and the range of constant C is 1.03 ≤ C ≤ 1.13.
[0040]
number
[0041] The cross-sectional shape of an actual hat-shaped steel sheet pile is not determined solely by considering its buckling strength against axial force; for example, its effective width and effective height are greatly affected by manufacturing constraints. Furthermore, the effective height of a hat-shaped steel sheet pile must be set so that the moment of inertia required for the wall design is obtained. From an economical standpoint, it is desirable to make the web and arm thicknesses thicker than the flange thickness, i.e., to make the thickness ratio smaller than 1.0. However, the range of thickness ratios that can be achieved is limited by the constraints of manufacturing facilities and the technological level.
[0042] In contrast, the correlation between the optimal flange angle and aspect ratio that maximizes the local buckling strength found in the above study examples is independent of the effective width or moment of inertia of the hat-shaped steel sheet pile, as shown in the third study example. Furthermore, as shown in the fourth study example, although the correlation is affected by the thickness ratio, the absolute value of the thickness has little effect. In other words, according to the results of the above study, the cross-sectional shape conditions that maximize the local buckling strength are determined by the flange angle θ and two dimensionless quantities (the aspect ratio B / h and the thickness ratio tf / tw). By utilizing these conditions, it is possible to design a cross-sectional shape that maximizes the local buckling strength, taking into account the constraints of the manufacturing equipment and the required moment of inertia.
[0043] Although the preferred embodiments of the present invention have been described in detail above with reference to the accompanying drawings, the present invention is not limited to these examples. It is clear that a person skilled in the art to which the present invention pertains can conceive of various modifications and alterations within the scope of the technical ideas set forth in the claims, and it is understood that these also naturally fall within the technical scope of the present invention. [Explanation of symbols]
[0044] 1...Hat-shaped steel sheet pile, 2...Web, 3A, 3B...Flange, 4A, 4B...Arm, 5A, 5B...Fitting joint, E A ,E B…chimerism center.
Claims
1. A hat-shaped steel sheet pile, In a cross section perpendicular to the longitudinal direction, the cross section comprises: a web extending along the width direction on a first side in a cross-sectional height direction; a pair of flanges extending from both ends of the web in the width direction to both sides in the width direction and toward a second side in the cross-sectional height direction, the flanges forming a flange angle θ (θ≦70°) with the width direction; a pair of arms extending from each end of the pair of flanges on the second side in the cross-sectional height direction along the width direction and toward both sides in the width direction; and a pair of fitting joints formed at each end of the pair of arms opposite the pair of flanges, a ratio of a total length of the web, the pair of flanges, and the pair of arms in the cross section to an average plate thickness of the web, the pair of flanges, and the pair of arms is 120 or more; The effective width B and effective height h in the cross section, and the flange angle θ satisfy the condition of the following formula (i) using a constant C (1.01≦C≦1.13) in a range where the plate thickness ratio of the flange to the web is 0.6 or more and 1.0 or less, and the effective width B, the web length Bw in the cross section, the cross-sectional height H, and the flange angle θ satisfy the relationship of B−Bw−2H / tan θ>0, A hat-shaped steel sheet pile that does not include a joint in the cross section.
2. A hat-shaped steel sheet pile, In a cross section perpendicular to the longitudinal direction, the cross section comprises: a web extending along the width direction on a first side in a cross-sectional height direction; a pair of flanges extending from both ends of the web in the width direction to both sides in the width direction and toward a second side in the cross-sectional height direction, the flanges forming a flange angle θ (θ≦70°) with the width direction; a pair of arms extending from each end of the pair of flanges on the second side in the cross-sectional height direction along the width direction and toward both sides in the width direction; and a pair of fitting joints formed at each end of the pair of arms opposite the pair of flanges, a ratio of a total length of the web, the pair of flanges, and the pair of arms in the cross section to an average plate thickness of the web, the pair of flanges, and the pair of arms is 120 or more; The effective width B and effective height h in the cross section, and the flange angle θ satisfy the condition of the following formula (i) using a constant C (1.03≦C≦1.13) in a range where the plate thickness ratio of the flange to the web is 0.7 or more and 1.0 or less, and the effective width B, the web length Bw in the cross section, the cross-sectional height H, and the flange angle θ satisfy the relationship of B−Bw−2H / tan θ>0, A hat-shaped steel sheet pile that does not include a joint in the cross section.
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