How to design single anchor plate piles using the equivalent beam method?

Please help to complete and improve it; this is all I know! Types of failure: Summarizing the engineering accidents involving sheet piles, the main reasons for their failure are fivefold: ① The sheet piles are not buried deep enough; under the action of earth pressure, the part of the sheet piles that is buried moves, resulting in slope slumping (Figure 1-21a) ； ② Insufficient strength of the support or anchorage (Figures 1-21b, c) ； ③ Insufficient anchor length causes the anchor to lose its functionality, leading to soil sliding (Figure 1-21d) ； ④ The sheet piles lack sufficient stiffness and become unstable and bend under soil pressure (Figure 1-21e) ； ⑤ Excessive displacement of the sheet piling causes damage to the surrounding environment (Figure 1-21f). For this reason, the depth of penetration of the sheet piles into the ground, the sectional bending moment, the reaction forces at the supports, the length of the tie rods, and the displacement of the sheet piles are considered the five key design factors for sheet piles. http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/%CD%C18.GIF Figure 1-21 Engineering accidents of sheet piles a) Movement at the lower part of the sheet piles ； b) Anchor pull failure ； c) Support failure ； d) Insufficient anchor pull length ； e) Pile buckling deformation ； f) Deformation of sheet piles and settlement of the soil behind the piles – the beam method. The following describes the calculation principles and methods for single-support sheet piles: Depending on the ratio of the depth to which the sheet pile penetrates into the ground to the depth of the foundation pit, the deformation of the single-support sheet pile varies, especially in the portion that is embedded in the ground. Thus, single-pivot sheet piles are divided into freely supported single-pivot sheet piles and fixedly supported single-pivot sheet piles (Figure 1-22). http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/%CD%C19.GIF Figure 1-22 Two calculation types for single-pivot sheet piles: a) freely supported; b) fixedly supported. The earth pressure distribution, bending moments, and deformations of single-pivot sheet piles differ in these two types as well. The depth of penetration of the sheet piles into the soil is shallow; the entire sheet pile deforms towards the pit, with its bottom end rotating and experiencing slight displacement, thereby allowing the passive earth pressure at the bottom of the pit to be fully utilized. As the depth of penetration of the sheet piles increases, the passive earth pressure acting on the front of the piles also increases. When a certain equilibrium state is reached, the base of the pile C rotates only in place without any displacement. The support at the bottom ends of the above two types of sheet piles is equivalent to simply supported conditions, and is referred to as free support. If the depth of burial continues to increase, the passive earth pressure in front of the pile also increases with depth. When a certain depth, point D, is reached, there is a section at the bottom of the sheet pile where neither displacement nor rotation occurs; at this point, the sheet pile is in a fixed state within the soil. This type of sheet pile is a single-support fixed sheet pile, with zero bending moment below a certain depth D. The precise calculation of sheet piles is rather difficult, mainly because the portion embedded in the ground constitutes a statically indeterminate problem, making it hard to accurately determine the distribution of soil pressure. There are various calculation methods available, such as the \"elastic curve method\", the \"vertical elastic foundation beam method\", and the \"equivalent beam method\". Below is an introduction to the simplified calculation method for sheet piles fixed at a single support – the equivalent beam method. For the calculation of the slab wall section, the passive and active earth pressures in front of and behind the sheet piles are caused by the displacement of those sheets piles, and the displacement of the piles in turn varies depending on the magnitude of the earth pressures; it is quite complex to take into account their combined deformation. In general, earth pressure is simplified to a linear distribution for calculation. Analyze the beam shown in Figures 1-23, which has one end fixed and the other end simply supported. It is subjected to a uniformly distributed load, and the bending moment diagram and deflection curve of this beam are shown in Figure 1-22b. If beam AD is truncated at the point of reverse bending C and simply supported at that point (Figure 1-23d), then the bending moment of this beam http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%E5%F8gif.gif is the same as that of the original AC section of the beam. We refer to http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8DAgif.gif as the equivalent beam of AC. By solving for the support reactions RC of the equivalent beam http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8Dngif.gif, that is, the support reactions of the beam http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%E1%C4gif.gif, it is possible to determine the other unknown quantities of the beam http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8EMgif.gif. http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/%CD%C110.GIF Figure 1-23: Schematic diagram of a equivalent beam. http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/%CD%C111.GIF Figure 1-24: Diagram of a retaining sheet pile with fixed supports. Figure 1-24 shows the distribution of earth pressure for a single-point retaining sheet pile with fixed supports; it is difficult to calculate accurately the properties of the sheet pile and the earth pressure below point D. If the earth pressure below point D is represented by a force EP2, then there are three unknowns in this case: Tc1, EP2, and hd, while only two equilibrium equations are available: Σx=0 and ΣM=0. Thus, it remains difficult to obtain an exact solution. Treating its lower end as a fixed end, this sheet pile is similar to the beam in Figure 1-23 which has one end fixed and the other end simply supported; the only difference is that the load on the sheet pile is distributed in a triangular pattern, whereas the beam shown in Figure 1-23 is subjected to a uniformly distributed load. With such an assumption, a single-pile retaining wall with fixed supports can also be analyzed using the \"equivalent beam method\". To solve for a single-pile sheet pile with fixed supports using the aforementioned \"equivalent beam method\", it is first necessary to determine the point of reverse bending C of the sheet pile. The position of the inflection point C is related to the internal friction angle and cohesion of the soil, and is influenced by factors such as the groundwater level behind the sheet piles and surface loads. Through the study of the bending moments and deflection curves of sheet piles with different lengths and burial depths, it was found that the point of reversal C for the sheet piles is relatively close to the position where the soil pressure intensity equals zero; this point can be used as the point of reversal in calculations, resulting in minimal error while simplifying the calculations. The steps for calculating a single-pile retaining wall with fixed supports using the equivalent beam method are as follows (Figure 1-24): a. Calculate the active earth pressure and passive earth pressure acting on the pile ； b. Calculate the distance hc1 from point C, where the earth pressure on the sheet pile is zero, to the ground surface, using the following formula: http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%B2%EEgif.gif, which is also http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%B0%EFgif.gif (1-32). c. Cut the sheet pile at point C, and use Σx=0 and ΣM=0 to calculate the reaction force RC at the support of the equivalent beam AC, as well as the reaction force Tc1 from the supports or anchors ； d. Calculate the depth of penetration hd of the sheet pile into the soil: Based on the characteristics of a single-point supported sheet pile, the bending moment is zero at a certain position below the bottom of the pile; if this point is at D, then h0 can be determined using the lower section of the sheet pile, CD. Since the active and passive earth pressures on the rectangular part of the CD section pile are equal, from ΣMD＝0 we obtain http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8E%D0gif.gif; hence we have http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8C%DB.gif (1-33). As the actual passive earth pressure in front of the pile is smaller than that shown in Figure 1-24, the value obtained using equation (1-33) is lower than http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%C9%BD.gif; therefore, the depth of penetration of the pile into the ground should be increased. Taking http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8C%DC.gif and http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8C%DD.gif as 0.2 http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8C%DE.gif, the depth of penetration of the sheet pile into the ground is http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8C%DF.gif (1-34). Mmax is determined at the point where the shear force is zero ； The meanings of the symbols in equations (1-32) to (1-34) above are as follows: r – gravitational density of soil; Ka – active earth pressure coefficient, Ka = http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8C%E0.gif http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8C%E1.gif; Kp – passive earth pressure coefficient, Kp = http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%8C%E2.gif http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/13%C4%DA%D2%D9.gif. The meanings of other symbols are shown in Figure 1-24. Supports (anchors): The design of the support (anchor) system is analyzed as follows: One end of the support or anchor is fixed to the gusset beam at the top of the sheet piling, while the other end is supported on the sheet piling on the opposite side of the foundation pit or fixed to an anchor bolt or anchor plate. The support (or anchorage) reaction Tc1 per unit length of the slab wall can be determined through calculations related to the slab wall section; thereafter, the axial force acting on each support or anchorage can be calculated based on the spacing between these supports or anchors. If the support length is too large, vertical braces should be installed at the center of the support (see Figure 1-19) to prevent excessive deflection under its own weight from generating additional internal forces. The length of the anchor cable should be calculated. The length of the anchor cable should ensure that the anchor block or anchor plate is located outside the line of passive soil wedge slip caused by itself, the line of active soil wedge slip caused by sheet pile displacement, and the line of static soil wedge slip, within the shaded area shown in Figure 1-25. http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/%CD%C125.GIF Figure 1-25: Calculation of the length of the anchor cable. The minimum length of the anchor cable is calculated using the following two formulas, with the larger value being taken: L = L1 + L2 http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/14%C4%DA%8DC.gif (1-35) http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/14%C4%DA%E1%B7.gif Where: L – Minimum length of the anchor cable ； h——depth of the foundation pit ； http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/14%C4%DA%8DD.gif ——For freely supported sheet piles, this value represents the depth to which the sheet pile is embedded in the soil ； For fixed-supported sheet piles, take the distance from the bottom of the foundation pit to the point of reversal ； http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/14%C4%DA%D4%C0.gif ——Distance from the bottom of the anchor to the ground ； http://www.scutde.net/t14courses/1412-efmdljiaif/tufang/1.5/IMAGE/14%C4%DA%8DE.gif ——Internal friction angle of soil.