Combined pile for collapsible loess

Abstract

The utility model provides a kind of for collapsible loess combined pile, including upper section pile and lower section pile, the both ends of the upper section pile are equipped with first link section, the both ends of the lower section pile are equipped with second link section, the first link section of the upper section pile is butt joint with the second link section of lower section pile, butt joint forms the pile joint recess of combined pile;Second non-link section in the pile joint recess is located in non-collapsible soil layer, first non-link section in the pile joint recess is located in collapsible loess layer.The utility model pile joint recess effectively reduces the side friction loss at collapsible loess boundary line, and inhibits the transmission of negative friction force downward, while it is beneficial to relieve end stress concentration when hammering construction.

Description

Technical Field

[0001] This utility model relates to the field of building engineering technology, and in particular to a composite pile for collapsible loess. Background Technology

[0002] Collapsible loess strata possess high strength in their natural state, but their structure rapidly deteriorates and significant settlement occurs upon contact with water, posing a serious threat to superstructures. Therefore, pile foundations have become a widely used and crucial technical measure in engineering construction in such areas. Among these, precast concrete piles (including pipe piles, hollow square piles, and solid square piles) are extensively used in pile foundation engineering for industrial and civil buildings, railways, highways, and municipal projects due to their advantages such as stable quality, fast construction speed, and high bearing capacity.

[0003] However, with the continuous expansion of the application scale of precast piles in collapsible loess areas and the deepening of engineering practice, a series of inherent technical problems have become increasingly prominent, seriously restricting their economy and reliability, mainly in the following three aspects:

[0004] First, the significant negative skin friction effect leads to poor design economy and high construction risks for pile foundations. The settlement deformation of collapsible loess layers exceeds that of the pile itself, generating downward negative skin friction on the pile. This results in a severe "negative at the top, positive at the bottom" stress distribution on the pile; that is, the upper part of the pile bears downward tensile force, while the lower part provides upward positive skin friction and end resistance. To offset the weakening of bearing capacity due to the upper negative skin friction, the pile length is often significantly increased during design to ensure sufficient positive skin friction and end resistance at the pile bottom to meet load requirements. This increase in pile length brings a chain of problems: First, the project cost rises significantly due to the increased use of concrete and steel, making it less economical; second, the longer piles often need to penetrate hard interlayers such as calcareous nodules and gravel layers to reach the bearing layer. When using hammering or static pressure methods, it is very easy to continue construction after reaching a reasonable hammering standard or pile driving force because the pile length does not meet the design value, resulting in "pile bursting" accidents such as crushed pile heads or cracked pile bodies. This not only causes direct economic losses, but also brings great uncertainty to the project quality and construction progress.

[0005] Second, existing composite pile structures have defects, severely sacrificing the horizontal bearing capacity and seismic performance of the pile top. To address these issues, some existing technologies attempt to adopt a "larger at the bottom, smaller at the top" composite pile configuration, where the lower pile section uses a larger diameter (or side length) to provide higher bearing capacity, while the upper pile section uses a smaller diameter (or side length) to reduce the impact of negative skin friction. However, this structure introduces new technical shortcomings: the horizontal forces on the building (such as wind loads and seismic forces) are mainly transmitted to the foundation through the pile top area. Reducing the pile diameter at the top directly leads to a decrease in the section modulus of the pile body, thus severely weakening the horizontal bearing capacity of the pile foundation. In seismic fortification areas, this weakening is fatal, making it difficult for the pile foundation to effectively resist the enormous horizontal shear forces generated by earthquakes, posing a serious threat to the seismic safety of the overall structure.

[0006] Third, the poor compatibility between product structure and construction technology, as well as production molds, restricts production efficiency and construction quality. The aforementioned "larger at the bottom, smaller at the top" combined pile or irregular pile structure typically requires extensive and complex modifications to existing standardized precast pile production line molds. This modification not only incurs high initial investment costs but also severely disrupts the existing efficient production rhythm, leading to a decline in production efficiency. Furthermore, during construction, the variable cross-section of such irregular piles is prone to mismatch with the clamping stroke of the static pressure pile driver. If the clamp happens to be clamped in the transition area of ​​the cross-section change, stress concentration can easily cause the pile concrete to collapse. Simultaneously, if an excessively long groove is formed at the joint of the upper and lower pile sections, it will create a significant weak point at the critical stress-bearing area where collapsible loess and non-collapsible soil layers meet, further exacerbating the adverse effects of negative skin friction and the loss of side skin friction.

[0007] Existing precast pile technologies for collapsible loess areas, whether in the application of single pile types or in the exploration of combined piles, have failed to systematically resolve the contradiction between offsetting negative skin friction, ensuring horizontal bearing capacity, and facilitating production and construction.

[0008] Therefore, there is an urgent need for an innovative precast pile combination scheme that can simultaneously achieve a high efficiency increase in bearing capacity and a comprehensive optimization of structural performance without significantly increasing costs and production complexity. Summary of the Invention

[0009] To address the shortcomings of existing technologies, this utility model provides a composite pile for collapsible loess. The composite pile is formed by connecting the piles in a joint groove. This joint groove effectively reduces the loss of lateral friction at the boundary of collapsible loess and inhibits the downward transmission of negative friction. It also helps to alleviate the stress concentration at the end during hammering construction.

[0010] This utility model achieves the above-mentioned technical objectives through the following technical means.

[0011] A composite pile for collapsible loess includes an upper pile and a lower pile. The upper pile has a first connecting section at both ends, and the lower pile has a second connecting section at both ends. The first connecting section of the upper pile and the second connecting section of the lower pile are connected to form a splicing groove for the composite pile. The second non-connecting section in the splicing groove is located in a non-collapsible soil layer, and the first non-connecting section in the splicing groove is located in a collapsible loess layer.

[0012] Furthermore, the ratio of the sum of the axial lengths of the second connecting sections at both ends of the lower pile to the total pile length of the lower pile is less than 5%.

[0013] Furthermore, the middle part of the upper pile is the first non-connecting section, which is a continuous, equal-diameter structure; the length of the first non-connecting section accounts for more than 90% of the total length of the upper pile.

[0014] Furthermore, the axial length of the splicing groove of the combined pile is less than 500mm.

[0015] Furthermore, the middle part of the lower pile is the second non-connecting section, which is an irregularly shaped pile with axially variable diameter. The second non-connecting section includes a convex section and a concave section, which are arranged alternately along the axial direction.

[0016] Furthermore, the outer contour of the second connecting segment is smaller than the outer contour of the protrusion. Here, the outer contour generally refers to the outer diameter or side length.

[0017] Furthermore, the outer contour of the second connecting segment is at least 40 mm smaller than the outer contour of the protrusion.

[0018] Furthermore, the sum of the axial length of the convex joint and the axial length of the adjacent concave joint is 1.8m to 2.2m.

[0019] Furthermore, the ratio of the net length of the convex joint to the net length of the concave joint is greater than 1.2.

[0020] Furthermore, the axial dimension of the protrusion adjacent to the second connecting section in the lower pile section is not less than 800mm; the axial dimension of the remaining protrusions in the lower pile section is not less than 1m.

[0021] Furthermore, the ratio of the axial length of the connecting groove to the axial length of its adjacent protrusion is less than 0.5.

[0022] The beneficial effects of this utility model are as follows:

[0023] 1. The composite pile for collapsible loess described in this utility model effectively reduces the loss of side friction at the boundary of collapsible loess by forming a joint groove in the joint, and inhibits the downward transmission of negative friction, while also helping to alleviate the stress concentration at the end during hammering construction.

[0024] 2. The composite pile for collapsible loess described in this utility model significantly improves the side friction resistance of the pile in stable soil layers by designing the non-connecting section of the lower pile as a longitudinally variable diameter concave-convex joint structure. According to industry standards, the side friction resistance of this concave-convex joint structure can be calculated based on the circumference of its larger diameter (or side length) (outer diameter or side length of the convex joint), and can be multiplied by an improvement factor of 1.1 to 1.3, thereby fundamentally enhancing the vertical bearing capacity of the pile foundation structurally.

[0025] 3. The composite pile for collapsible loess described in this utility model achieves the effect of completely maintaining the original horizontal bearing capacity of the pile top by designing the non-connecting section of the upper pile as a continuous, uniform diameter structure, with its outer diameter being the same as or similar to the outer diameter (or side length) of the protruding section of the lower pile. This design ensures that the pile top has the maximum section modulus, thereby effectively resisting horizontal loads and seismic forces, and solving the inherent defect of weakened seismic performance in "larger at the bottom and smaller at the top" composite piles.

[0026] 4. The composite pile for collapsible loess described in this utility model achieves a controllable joint groove at the joint by designing the outer diameter (or side length) of the connecting section of the upper and lower piles to be more than 40mm smaller than the outer diameter (or side length) of the non-connecting section and limiting the length of the connecting section. This structure effectively reduces the loss of side friction at the boundary of collapsible loess and inhibits the downward transmission of negative friction, while also helping to alleviate the stress concentration at the end during hammering construction.

[0027] 5. The combined pile for collapsible loess described in this utility model achieves a perfect match with the stroke of existing static pressure pile driver clamps by setting the periodic length (1.8m~2.2m) of the concave and convex sections of the lower pile and the ratio of the net length of the convex section to the concave section (greater than 1.2). This ensures that the clamp can stably hold the concrete on the convex section during construction, avoiding damage to the pile concrete caused by clamping on a variable cross-section slope, and greatly improving the reliability and success rate of construction.

[0028] 6. The combined pile for collapsible loess described in this utility model achieves the effect of ensuring that the pile end has sufficient length to stably enter the bearing layer by limiting the size of the protruding section at the tail end of the lower pile to not less than 800mm. This makes the theoretical calculation model of end resistance highly consistent with the actual stress state, giving full play to the bearing capacity of the pile end and ensuring the safety and accuracy of the design.

[0029] 7. The composite pile for collapsible loess described in this utility model, through the combination design of the upper section with a uniform diameter (or side length) and the lower section with an irregular shape, achieves a technical effect of increasing the overall bearing capacity by more than 25% compared with traditional precast pile combinations under the same engineering geological conditions. At the same time, this structure maximizes the use of existing production lines, requires minimal mold modification, and comprehensively achieves multiple economic benefits such as low production cost, high production efficiency, convenient construction, and low pile burst rate. Attached Figure Description

[0030] To more clearly illustrate the technical solutions in the embodiments of this utility model or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. The drawings described below are some embodiments of this utility model. For those skilled in the art, it is obvious that other drawings can be obtained from these drawings without creative effort.

[0031] Figure 1 This is a schematic diagram of the combined pile for collapsible loess described in this utility model.

[0032] Figure 2 This is a schematic diagram of the upper section pile described in this utility model.

[0033] Figure 3 This is a schematic diagram of the lower section pile described in this utility model.

[0034] In the picture:

[0035] 1-Upper pile; 11-First connecting section; 12-First non-connecting section; 2-Lower pile; 21-Second connecting section; 22-Second non-connecting section; 221-Convex joint; 222-Concave joint; 3-Pile joint groove. Detailed Implementation

[0036] The embodiments of this utility model are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain this utility model, and should not be construed as limiting this utility model.

[0037] In the description of this utility model, it should be understood that the terms "center," "longitudinal," "transverse," "length," "width," "thickness," "upper," "lower," "axial," "radial," "vertical," "horizontal," "inner," and "outer," etc., indicating orientation or positional relationships, are based on the orientation or positional relationships shown in the accompanying drawings and are only for the convenience of describing this utility model and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of this utility model. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of indicated technical features. Thus, a feature defined with "first" or "second" may explicitly or implicitly include one or more of that feature. In the description of this utility model, "a plurality of" means two or more, unless otherwise explicitly specified.

[0038] In this utility model, unless otherwise explicitly specified and limited, the terms "installation," "connection," "linking," and "fixing," etc., should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this utility model according to the specific circumstances.

[0039] like Figure 1 As shown, the composite pile for collapsible loess of this invention includes an upper pile 1 and a lower pile 2. The upper pile 1 has a first connecting section 11 at both ends, and the lower pile 2 has a second connecting section 21 at both ends. The first connecting section 11 of the upper pile 1 and the second connecting section 21 of the lower pile 2 are joined to form a splicing groove 3 for the composite pile. The second non-connecting section 22 in the splicing groove 3 is located in a non-collapsible soil layer, and the first non-connecting section 12 in the splicing groove 3 is located in a collapsible loess layer. The splicing groove 3 of this invention effectively reduces the lateral friction loss at the boundary of collapsible loess and inhibits the downward transmission of negative friction, while also helping to alleviate end stress concentration during hammering construction. Generally, the axial length of the splicing groove 3 of the composite pile is less than 500 mm.

[0040] Example

[0041] like Figure 1As shown, in this embodiment, the total length of the combined pile is 24 meters, consisting of an upper pile 1 and a lower pile 2 connected by a splice. According to the project's geological survey report, the boundary line of collapsible loess is approximately located 12 meters below the ground surface. Therefore, the length of both the upper and lower pile sections is designed to be 12 meters to ensure that the structural design matches the stress characteristics of the strata.

[0042] like Figure 2 The upper pile 1 has a structure with first connecting sections 11 at both ends and a first non-connecting section 12 in the middle. The outer diameter of the first connecting section 11 is smaller than the outer diameter of the first non-connecting section 12, and the difference between the outer diameters of the first connecting section 11 and the first non-connecting section 12 is not less than 40 mm (i.e., radial thinning is not less than 20 mm). This design creates a stress buffer zone during pile splicing or driving, which helps reduce stress concentration at the ends and prevents concrete cracking. When the upper pile 1 is a square pile, the side length of the first connecting section 11 is smaller than the side length of the first non-connecting section 12, and the difference between the side lengths of the first connecting section 11 and the first non-connecting section 12 is not less than 40 mm.

[0043] The axial length of the first connecting section 11 is less than 250 mm. This compact length design maximizes the proportion of the uniform cross-section pile body while ensuring the connection function. The first non-connecting section 12 is a continuous section with a uniform diameter (or equal side length), accounting for more than 90% of the total length of the upper pile section 1. This ensures that the upper part of the pile body has a consistent and maximum section modulus, thereby guaranteeing no loss of horizontal bearing capacity in the pile top area, which is crucial for the seismic performance of the structure. At the same time, the uniform cross-section structure also facilitates the clamping and construction of static pressure equipment, improving construction efficiency and safety.

[0044] like Figure 3 Before describing the structure of the lower pile 2, the term "net length" needs to be defined: "Net length" specifically refers to the axial length of the central cylindrical section (or equilateral rectangular column section) of the convex or concave section, excluding the length of the inclined or arc transition sections connecting with other sections. Unless otherwise specified, the lengths and proportions of the convex section 221 and concave section 222 mentioned below refer to their net lengths.

[0045] The lower pile 2 has a second connecting section 21 at both ends and a second non-connecting section 22 in the middle. The second non-connecting section 22 adopts a longitudinally variable diameter (or variable side length) concave-convex joint structure, formed by alternating convex joints 221 and concave joints 222. This bamboo-like structure can significantly increase the contact area and mechanical interlocking force between the pile body and the surrounding soil, thereby greatly improving the positive and lateral friction resistance of the underlying stable soil layer on the pile body. The outer diameter (or side length) of the second connecting section 21 is smaller than the outer diameter (or side length) of the convex joint 221 in the second non-connecting section 22, and the diameter difference between the outer diameter (or side length) of the second connecting section 21 and the outer diameter (or side length) of the convex joint 221 is not less than 40 mm.

[0046] The sum of the axial lengths of a single cycle (i.e., one convex section and an adjacent concave section) of the convex section 221 and concave section 222 is 1.8 meters to 2.2 meters. This design matches the stroke of existing static pressure pile driver clamps, enabling the clamp to stably hold the convex section during construction, avoiding crushing of the pile concrete caused by clamping on the slope of the concave-convex transition, and improving the tolerance and reliability of construction. The ratio of the net length of the convex section 221 to the net length of the concave section 222 is greater than 1.2. In the concave-convex section cycle, the convex section dominates. A longer convex section means a larger effective bearing diameter (or side length) and a more reliable clamping surface.

[0047] The convex section 221 at the tail end (i.e., pile end) of the lower pile 2 has a size of not less than 800 mm, which ensures that the pile end has a sufficiently long stable section to enter the bearing layer, so that the theoretical calculation value of the end resistance matches the actual bearing capacity, and fully utilizes the bearing capacity of the pile end. The ratio of the total length of the second connecting section 21 (including the connecting sections at both ends) to the total pile length of the lower pile 2 is less than 5%, which ensures that the concave-convex section structure (i.e., the second non-connecting section 22) which bears the main function of improving side friction resistance occupies an absolutely dominant position in the lower pile.

[0048] The upper pile 1 and the lower pile 2 are connected by a first connecting section 11 and a second connecting section 21 at their ends. After splicing, a splicing groove 3, consisting of the connecting sections at both ends, is formed at the splice. The axial length of the splicing groove 3 is controlled within 500 mm. The relatively short groove length (compared to the concave section 222) helps reduce the loss of side friction at the critical stress interface between the collapsible loess and non-collapsible soil layers, avoiding the formation of obvious weak links due to unreasonable splicing structure. The ratio of the length of the splicing groove 3 to the length of the adjacent convex section is less than 0.5. This ratio further optimizes the stress at the boundary, effectively suppressing the downward transmission of negative skin friction that may be generated by the upper collapsible loess, thereby ensuring the effective utilization of the overall side friction resistance of the pile body.

[0049] In this embodiment of the composite pile, the first non-connecting section 11 of the upper pile 1, with its continuous length and equal diameter, effectively ensures the horizontal bearing capacity and seismic performance of the pile top. The second non-connecting section 21 of the lower pile 2, with its concave-convex joint, significantly improves the positive side friction resistance in stable soil layers. Reasonably controlled dimensions of the connecting sections and splice grooves minimize the adverse effects of negative skin friction and optimize construction adaptability. When the pile length exceeds 30 meters, the lower pile 2 is a lower composite pile, which consists of two or more lower pile sections.

[0050] This utility model also provides a construction method for the above-mentioned combined piles for collapsible loess, which works in conjunction with the pile structure design and includes the following steps: Step 1: Based on the soil performance parameters provided in the geological survey report, especially the depth and thickness of the collapsible loess layer, determine the boundary line of the collapsible loess. Step 2: Determine the length of the upper pile 1 based on the boundary line of the collapsible loess (the pile length is from the location of the boundary line of the collapsible loess to the design elevation of the top of the upper pile), and calculate the negative bearing capacity (i.e. negative skin friction) generated by the upper pile 1. Step 3: After adding the total load required by the single composite pile to the negative bearing capacity generated by the upper pile 1, calculate the bearing capacity that the lower pile 2 should bear based on the soil parameters, determine the pile length of the lower pile 2, and thus determine the total pile length required for the composite pile and the lengths of the convex section 221 and concave section 222 of the lower pile. Step 4: Transport the prefabricated upper pile 1 and lower pile 2 to the designated locations on the construction site. To avoid confusion between the upper and lower sections during construction, clearly mark the lower pile 2 (e.g., by spraying different colors or lettering). Step 5: Using static pressure pile driver or hammer pile driver, lift the lower pile section 2 and drive it vertically to the pre-marked pile position. Construction should continue until the top of the lower pile section 2 reaches a depth of 0.5 to 1.0 m below the ground. This depth provides the necessary working space for the splicing operation of the upper pile section. During this process, because the period length of the concave-convex joint of the lower pile section 2 (1.8 m to 2.2 m) and the design of the convex joint are matched with the stroke of the static pressure pile driver's clamp, the clamp can stably hold the convex joint 221, effectively avoiding damage to the pile body concrete. After construction, the second non-connecting section 22 of the lower pile section 2 completely passes through the boundary of the collapsible loess layer and anchors into the stable bearing layer below it. Step 6: Lift the upper pile 1, align its lower end second connecting section 21 with the upper end second connecting section 21 of the already positioned lower pile 2, and connect the two into a whole composite pile using reliable methods such as welding or mechanical connection. After splicing, the length of the splicing groove 3 formed at the splice point is controlled within 500mm, effectively reducing the weak area of ​​side friction resistance at the soil layer boundary; Step 7: Connect the composite piles into a whole and continue construction until the composite piles are constructed to the design elevation and then stop construction. The composite pile construction is completed. At this time, the second non-connecting section (22) is located in the non-collapseable soil layer and the first non-connecting section (12) is located in the collapsible loess layer.

[0051] Through precise design in step 1 and accurate construction in the preceding steps, this invention achieves the most ideal working state for the resulting pile foundation: the continuous, uniform-diameter section of the upper pile 1 (i.e., the first non-connecting section 12) is entirely located within the collapsible loess layer, ensuring the horizontal bearing capacity of the pile top; while the concave-convex section of the lower pile 2 (i.e., the second non-connecting section 22) is entirely located within the non-collapsible, stable soil layer, and its variable-diameter structure provides a significantly enhanced positive and lateral frictional resistance. After the pile construction is completed, the subsequent pile cap construction can proceed.

[0052] It should be understood that although this specification is described according to various embodiments, not every embodiment contains only one independent technical solution. This way of describing the specification is only for clarity. Those skilled in the art should regard the specification as a whole. The technical solutions in each embodiment can also be appropriately combined to form other implementation methods that can be understood by those skilled in the art.

[0053] The detailed descriptions listed above are merely specific descriptions of feasible embodiments of the present utility model, and are not intended to limit the scope of protection of the present utility model. All equivalent embodiments or modifications made without departing from the spirit of the present utility model should be included within the scope of protection of the present utility model.

Claims

1. A composite pile for collapsible loess, characterized in that, The pile includes an upper pile (1) and a lower pile (2). The upper pile (1) has a first connecting section (11) at both ends, and the lower pile (2) has a second connecting section (21) at both ends. The first connecting section (11) of the upper pile (1) and the second connecting section (21) of the lower pile (2) are connected to form a joint groove (3) of the composite pile. The second non-connecting section (22) in the joint groove (3) is located in a non-collapsible soil layer, and the first non-connecting section (12) in the joint groove (3) is located in a collapsible loess layer.

2. The combined pile for collapsible loess according to claim 1, characterized in that, The ratio of the sum of the axial lengths of the second connecting sections (21) at both ends of the lower pile (2) to the total pile length of the lower pile (2) is less than 5%.

3. The combined pile for collapsible loess according to claim 1, characterized in that, The middle part of the upper pile (1) is the first non-connecting section (12), which is a continuous equal diameter structure; the length of the first non-connecting section (12) accounts for more than 90% of the total length of the upper pile (1).

4. The combined pile for collapsible loess according to claim 1, characterized in that, The axial length of the joint groove (3) of the combined pile is less than 500mm.

5. The combined pile for collapsible loess according to claim 1, characterized in that, The middle part of the lower pile (2) is the second non-connecting section (22), which is an irregular pile with axial diameter change. The second non-connecting section (22) includes a convex section (221) and a concave section (222), which are arranged alternately along the axial direction.

6. The combined pile for collapsible loess according to claim 2, characterized in that, The outer contour of the second connecting segment (21) is smaller than the outer contour of the protrusion (221).

7. The combined pile for collapsible loess according to claim 6, characterized in that, The outer contour of the second connecting segment (21) is at least 40 mm smaller than the outer contour of the protrusion (221).

8. The combined pile for collapsible loess according to claim 5, characterized in that, The sum of the axial length of the protruding joint (221) and the axial length of the adjacent concave joint (222) is 1.8m to 2.2m.

9. The combined pile for collapsible loess according to claim 5, characterized in that, The ratio of the net length of the convex joint (221) to the net length of the concave joint (222) is greater than 1.

2.

10. The combined pile for collapsible loess according to claim 5, characterized in that, The axial dimension of the protrusion (221) adjacent to the second connecting section (21) in the lower pile (2) is not less than 800mm; the axial dimension of the remaining protrusions (221) in the lower pile (2) is not less than 1m.

11. The combined pile for collapsible loess according to claim 5, characterized in that, The ratio of the axial length of the pile joint groove (3) to the axial length of its adjacent protrusion (221) is less than 0.5.

Images