Timber Pole Retaining Wall? Trust The Answer

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Can timber be used as retaining wall?

Preferred in equal parts for its practicality and unique aesthetic qualities, timber retaining walls are the perfect way to go if you want a garden, house, or landscape project fence that makes the most use of its base materials.

How deep do posts need to be for retaining wall?

The theory I work on is – half the height of the wall plus 100mm. For example if your wall is going to be 800mm high, the holes for your posts should be 500mm deep. Time now to concrete your posts into position. Sleeper retaining wall posts come in two varieties – steel galvanised H Beams or a vertical sleepers.

How To Build A Timber Retaining Wall

What is the cheapest retaining wall?

The cheapest type of retaining wall is poured concrete. Prices start at $4.30 per square foot for poured concrete, $5.65 for interlocking concrete block, $6.15 for pressure-treated pine, and about $11 for stone.

How To Build A Timber Retaining Wall

How long will a timber retaining wall last?

A timber retaining wall can last a little over a decade, if treated properly. If not maintained, the lifespan of a timber wall is around 3 to 5 years. To keep its fresh look, timber requires serious maintenance. The material will hold up for so many years only if its pressure-treated with chemicals.

How To Build A Timber Retaining Wall

Does a 2 foot retaining wall need drainage?

If your retaining wall needs a drainage pipe, make sure the pipe has slots on all sides, not just one. A drainage pipe might be needed if: The retaining wall is at least four feet high or taller. Clay or other poor draining soils are behind the wall.

How To Build A Timber Retaining Wall

How much weight can a retaining wall hold?

Even small retaining walls have to contain enormous loads. A 4-foot-high, 15-foot-long wall could be holding back as much as 20 tons of saturated soil.

How To Build A Timber Retaining Wall

How thick should a timber retaining wall be?

To build your wall, dig holes and insert vertical supports using thicker sleepers, at least 75mm thick. Space the supports every 1.2m for 2.4m long sleepers, and 1.5m for 3m long sleepers. The horizontal sleepers can be 50mm thick.

How To Build A Timber Retaining Wall

How high can a timber retaining wall be?

* In New South Wales, you need permission from council to build a retaining wall higher than 600mm from ground level.

How To Build A Timber Retaining Wall

How deep should a 3 foot retaining wall be?

The general rule of thumb is to bury about one-eighth of the height of the wall. For example, if your wall will be three feet (36 inches) tall, the first course of blocks should start five inches below soil level.

How To Build A Timber Retaining Wall

What is the easiest retaining wall to build?

What is the easiest retaining wall to build? Short walls under three feet high and constructed of concrete blocks or masonry blocks are the easiest type of wall for DIYers to build. They are ideal landscape solutions for a front yard or raised flower bed.

How To Build A Timber Retaining Wall

What angle should a retaining wall be?

You will need to angle the holes so that the post is on a lean towards the bank. The lean should be 1:10 – in other words, for every 100 mm you go up, the post should angle towards the wall 10mm. A purely vertical wall will start to sag over time, so this angle is important.

How To Build A Timber Retaining Wall

How strong are timber retaining walls?

Durable. If properly treated and cared for periodically, timber retaining walls are very strong and can last for a long time (20 years or more in most cases). This means less rebuilding and construction in your future.

How To Build A Timber Retaining Wall

Timelapse of a 26 day work of building a retaining wall (in 10 minutes)

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Design of cantilever pole retaining walls

A provision is a binding decision by MBIE that provides an opportunity to resolve any dispute or question regarding building rules, building use, building accessibility, and health and safety.

If you own a building that contains a specific system such as a cable car, you must ensure that these are operated effectively for the life of the building and in accordance with the compliance timetable issued by the council.

For safe, healthy and durable buildings, all construction work in New Zealand must meet certain standards. Find out how to build within the rules.

This information is published by the Director General of the Ministry of Economy, Innovation and Employment. It is intended as a general guide only and, if used, does not relieve anyone of the obligation to examine all matters to which the information relates, having regard to the circumstances of each case. In certain circumstances, expert advice may be required. If this information relates to supporting people:

Cantilever Pole Retaining Walls – New Zealand Geotechnical Society

Cantilever Retaining Walls

abstract

Cantilevered timber stud walls are one of the most common forms of retaining wall construction used in New Zealand for low to medium support heights. Pile walls in New Zealand were mainly designed according to the isolated pile theory (Broms) or the continuous embedded wall theory.

These commonly used design analysis methods have limitations and do not accurately represent pole-soil interaction and displacement behavior. In particular, the Broms method for estimating the lateral force capacity of the pile in non-cohesive soils assumes a toe rotation of the pile rather than the correct depth at a significant height above the toe. This simplification results in a non-conservative lateral loading capacity. Simplifying the analysis by assuming that the poles behave like a continuous wall is unnecessary and can introduce errors that are difficult to quantify. Although it is unlikely that the soil strength parameters for the design of small wall structures are known precisely, it is still desirable to eliminate analysis errors as much as possible and to manage soil uncertainty by assuming moderately conservative soil strength parameters.

Pile foundation-soil interaction design should be based on experimentally verified pile theory. Pile design methods suitable for the design of cantilever walls have been presented by Guo (2008) for cohesionless soils, Motta (2013) for cohesive soils and by Zhang (2018) for the case of mixed friction and cohesion. The Guo and Motta methods are based on elastic-plastic soil reactions and provide force-displacement equations for loads from zero to the ultimate hyperbolic loads. These enable the determination of the serviceability displacements and the bearing capacity of the pile head lateral force based on the soil resilience at the base.

Although the empirical equations of Guo, Motta, and Zhang are more complex than the simple Broms equation for cohesionless soils and the Broms-theory-based equations of Pender (1997) for cohesive soils, they are not difficult to set up in a spreadsheet. Design diagrams are presented in this paper, which allow to evaluate the equations with sufficient accuracy for most design applications in both soil types as well as in the case of mixed friction and cohesion.

Limit force profiles in non-cohesive soils are a function of the effective vertical stress, which is difficult to estimate near the face of the wall. Plots of the variation with distance from the working face are presented along with recommendations for a correction factor simplification that is satisfactory for engineering applications.

The increase in strength of the concrete casing used on wooden poles embedded in the ground should be considered in the design. Information is presented in the work that allows to predict the reinforcement effect of the cladding with sufficient accuracy for wall design.

1 Introduction

Cantilevered timber stud walls are one of the most common forms of retaining wall construction used in New Zealand for low to medium height (1.0m to 3.5m) applications. Wood paneling materials are used in conjunction with the poles to form an economical wall of sustainable, locally sourced materials (See Figure 1). Construction is straightforward as the poles are typically installed in boreholes backfilled with concrete. Drilled holes range in diameter from 400mm to 600mm and can be drilled with light machinery or portable drills. The construction width of less than 1.0m is smaller than traditional concrete cantilever or gravity wall construction. This is an advantage in locations with existing buildings.

Figure 1. Typical wooden pole wall construction

For more heavily loaded or taller walls, steel section bars can be used and these can be driven rather than installed in boreholes. In the case of walls over 3.0 m, back-tied wooden posts are sometimes used.

The depth of the embedment post is usually of the same order of magnitude as the wall height above ground in front of the wall, so the ratio of the embedment length L divided by the diameter B is approximately 5. At this aspect ratio, the pile behaves as a rigid element that deforms by rotating about a point typically between 0.6L and 0.8L below the ground surface. Failure in the soil surrounding the wrapped pile is the preferred failure mechanism. The pole section near the ground is a critical location; Failure in this section can result in limited deflection before failure and should be avoided at certain design loads.

Timber design in New Zealand is performed in accordance with NZS3603:1993, Timber Structures Standard, but there is no design standard covering the geotechnical aspects of the design required for the embedded section of a pile wall. MBIE-NZGS, 2017 gives some hints based on a design example for a cantilevered wooden pole wall.

Pile walls in New Zealand were mainly designed according to the isolated pile theory of Broms, 1964a, 1964b (Pender, 1997) or the continuous embedded wall theory (McPherson and Bird, 2020; MBIE-NZGS, 2017). These two methods are summarized below.

1.1 Broms method

The best-known approach to estimating the ultimate lateral bearing capacity of a pile is that of Broms. He considers piles in cohesive soils and those in non-cohesive soils separately. In each case, Broms gives a simple method of estimating the maximum lateral pressure that the soil can mobilize and using this to assess the pile’s capacity for lateral loading applied at the top of the pile. The approach is intended to take into account the three-dimensional interaction between the pile and the surrounding soil. For a short rigid pile it is assumed that the maximum moment does not reach the bearing capacity of the pile section and failure in the surrounding soil occurs.

For piles in overconsolidated cohesive soils with undrained shear strength, s u – constant over depth, Broms assumed that the soil between the ground surface and a depth of 1.5 pile diameters offers no resistance and that the ultimate lateral soil pressure against the pile is greater downs was 9s u . Davis and Budhu, 1986 suggested that a uniform depth of 600 mm rather than 1.5 diameters might be a more appropriate depth for the no-resistance depth.

For piles in non-cohesive soils, Broms suggested a maximum lateral pressure of 3K p times the vertical effective stress in the soil adjacent to the pile (K p is the Rankine passive pressure coefficient). In the case of non-cohesive soils, in contrast to cohesive soils, the soil resistance extends to the ground surface. Broms derived a simple expression for the lateral bearing capacity of a short rigid pile in non-cohesive soils by assuming that the pile pivoted about its base.

Pender, 1997, extended the Broms analyzes to cover piles in soft clay where the undrained shear strength increases linearly with depth from zero at the soil surface. However, this case is not applicable to the short piles considered in the present application, since short piles would be unsatisfactory in this type of soil.

Broms provided design diagrams for the lateral bearing capacity of the pile in cohesive soil; However, Pender, 1997 presented a full set of equations more convenient for spreadsheets.

1.2 Continuous wall method

Pole spacing for cantilever walls is typically 1.5 to 4 times the embedded diameter of the pole including the concrete encasement. For piles spaced four times the embedment diameter or less, the subsurface soil pressure bulbs are assumed to provide continuous pressure along the wall and the calculations are performed using conventional continuous embedded wall theory (as used for sheet piling) carried out. One analysis approach used by Pender, 2000 is to assume a point of rotation at a given depth and vary this depth by iteration to satisfy both the horizontal force and moment balance equations. For non-cohesive soils, he assumed that both the active and passive pressures were calculated using Rankine’s active and passive pressure coefficients. For short-term loading of saturated clays, he assumed a passive compressive strength based on 2s u.

In the example given in MBIE-NZGS, the 2017 active and passive pressure coefficients for a long-term load (drained soil for static loads) of a clay soil were estimated using NAVFAC – DM7, 2011 charts (assuming zero cohesion). These diagrams were derived under the assumption of logarithmic spiral fracture surfaces and give coefficients for passive pressures that are significantly larger than those given by the Rankine assumption. In the case of short-term loading (earthquake load case) of the clay soil, the passive compressive strength was set at 2s u. The MBIE-NZGS analysis has been simplified by assuming a rotation around the base of the wall.

1.3 Limitations of the methods currently used

Shortcomings of the current methods commonly used to estimate the lateral capacity of tower foundations used in the construction of cantilevered tower walls are summarized below.

failure mechanism

The failure mechanism for piles with a spacing of 1.5B or more involves soil yielding at the soil-pile interface. In contrast, for a continuously embedded wall section, there is no flow mechanism and failure develops on passive pressure slip lines. Assuming continuous wall behavior for moderately spaced piles misses some of the subtleties of true behavior and this can lead to unnecessary errors in the design analysis.

Passive pressure estimation

There is limited published information on continuous wall loading tests and there is uncertainty as to how to estimate the passive pressure coefficients above and below the rotation point. On the other hand, empirical formulas for the limit pressures on individual piles were developed from a large number of pile tests. An advantage of assuming a continuous wall is that the wall can be modeled using two-dimensional numerical analysis, thereby partially overcoming the uncertainty in the estimation of passive pressures. However, the complexity of the numerical analysis makes it unsuitable for mast wall design.

rotation point

In the Broms method for cohesionless soil and in some applications of continuous wall theory, the point of rotation is assumed to be at the base of the pole. This assumption can lead to significant errors and, in the case of the Broms analysis, to an unconservative prediction of sustainability. With the availability of the automatic iteration function in spreadsheets, it is easy to solve the horizontal force and moment balance equations simultaneously to get the correct depth of the pivot point. There is no need to make the simplifying assumption of a twist about the toe.

Definition of ultimate capacity

The load-displacement curve for lateral loading of a pile has a hyperbolic shape, resulting in large deflections at peak load. Broms and the continuous wall theory assume that the confining passive pressures on the pile or wall develop over the entire length of the pile. This assumption implies that the capacity is the hyperbolic or maximum capacity. A better approach is to base the final capacity on the load required to produce a yield in the ground at the base of the pile or wall.

shifts

The present methods do not provide a load-displacement curve, although sometimes displacement estimates are based on elastic ground assumptions (Winkler or elastic continuum). In order to meet both serviceability and ultimate limit state (ULS) requirements, it is important to have a reliable method to calculate a load-deflection curve for the load going from zero to the ULS increases.

stress tests

Broms verified its resilience methods using a series of test results. In the last 50 years since the publication of the Broms methods, there have been a large number of published results on the lateral loading of rigid piles. Alternative methods using boundary pressure predictions based on this more recent test information are now available.

stack spacing

Pile spacing effects have been studied in a number of recent test and analysis studies, leading to the publication of empirical equations for estimating the lateral bearing capacity reduction resulting from the interaction of the compression balls between adjacent piles. With the availability of these equations, there is no need to assume continuous wall behavior.

gravitational stresses

The Broms and simplifications of the continuous wall methods assume that the gravity or effective vertical stress that determines the limit force profile at the pile is based on the soil surface in front of the wall. The floor depth step at the wall face increases the gravitational stresses on both the front and back of the wall face. On the reverse it is sometimes assumed that vertical stress is based on ground level behind the wall, but this assumption overestimates vertical stress near the mast.

2. Ultimate shear capacity of rigid piles in cohesive soil

In the present application it is assumed that the pile is rigid and will fail by rotation about a point on the pile below the mean embedment depth. The failure is expected to occur in the soil rather than the pile. A number of definitions of a rigid pile have been proposed. These include:

Kasch et al., 1977 proposed to use the length-diameter ratio L/B to classify the stiffness of a pile with a rigid pile with L/B < 6. Guo and Lee, 2001 defined a pile as rigid when the pile in the ground relative stiffness, E P /G s exceeds a critical ratio (E P /G s )c = 0.052(L/r 0 )4, where E P is the effective Young's modulus of the pile, defined as E P = (EI) P /(πr). 0 4/4); (EI) P is the bending stiffness of the pile; G s is the shear modulus of soil; L is the embedded pile length and r 0 is the outer radius of the pile. Poulos and Davis, 1980, considered a pile to be rigid if the stiffness ratio (EI) p /(E s L4) was greater than 10-2, where E s is the Young's modulus for the soil. Carter and Kulhawy, 1992 propose that a pile is rigid if L/B ≤ 0.07 (E P /E s ) 0.5 Poles used in low to medium height timber wall constructions (less than 3.5 m in height) are usually cast in underground with a total encasement diameter of 0.4 to 0.6 m. L/B ratios are typically < 6. The (EI) p /(E s L4) ratio is typically > 3 x 10-2 (assuming composite action between wood and concrete cladding). Concrete-encased poles generally meet all of the four criteria above.

Extensive theoretical studies, full-scale in situ tests and laboratory model tests have been carried out on side-loaded rigid piles in cohesionless soils (Brinch Hansen, 1961; Broms, 1964; Petrasovits and Awad, 1972; Meyerhof et al., 1981; Poulos and Davis 1980 Scott 1981 Fleming et al 2009 Prasad and Chari 1999 Dickin and Nazir 1999 Laman et al 1999 Guo 2008 Zhang et al 2005 Zhang 2009 Chen et al., 2011). The following methods of analysis have been proposed (Moussa and Christou, 2018).

LFP method

A ULS method based on an assumed profile of limiting soil resistance per unit length along the pile, or a limiting force profile (LFP). The analysis is reduced to a simplified two-dimensional analysis. Many of these LFP methods (Brinch Hansen, 1961; Broms, 1964; Petrasovits and Awad, 1972; Meyerhof et al., 1981; Prasad and Chari, 1999) do not account for soil deformation and therefore do not provide the associated displacements ULS. To solve this problem, Guo, 2008, established elastic-plastic solutions for the analysis of side-loaded rigid piles, assuming a subgrade reaction modulus that was either constant or linearly increasing with depth, along with an LFP that increased with depth increased linearly. His solutions enable the estimation of the non-linear response of piles and the displacement-based capacity and showed satisfactory agreement with the model pile test results presented by Prasad and Chari, 1999 and the experimental and numerical analysis results presented by Laman et al., 1999.

Winkler Spring

A Winkler spring model (Scott, 1981; Pender, 1993) can be used to estimate the displacement and bearing capacity of a rigid pile. The soil surrounding the pile is modeled as a bed from independent sources. The displacement of individual springs has no effect on the other springs, which greatly simplifies the mathematical analysis. This model neglects soil continuity or shear coupling between springs and is difficult to apply when there is significant soil nonlinearity. It is a simple method that can provide acceptable results for the serviceability limit state (SLS).

P-Y curve

A refinement of the Winkler spring method is the p-y curve analysis originally developed by McClelland and Focht in 1956. The soil response is related to the lateral movement of the pile via a non-linear load transfer function. Methods for estimating p-y curves have been developed by many authors (Reece, 1977; Scott, 1981; Murchison and O’Neill, 1984; Lam and Martin, 1986). Empirical expressions were derived from test results to express initial stiffness and final force as a function of depth, and from these values ​​a hyperbolic force versus displacement or p-y curve for nodal points on the pile can be developed. The process does not lead to closed solutions; However, software programs are available to generate the p-y curves and perform nonlinear analysis (Rollins et al., 2003). The complexity of numerical methods is not justified in many circumstances related to the present application for cantilever walls.

expansion wedge

The wedge method overcomes some of the limitations of the p-y curve method by accounting for the three-dimensional nature of the soil-pile interaction near the pile tip (Norris, 1986; Ashour et al., 1998). While traditional non-linear p-y characterization provides a reasonable assessment for a variety of loaded piles, it has been found that the p-y curve (or Winkler’s modulus of subgrade response) depends on pile properties (width, shape, bending stiffness, and pile head conditions) as well as soil properties . The wedge model allows the non-linear p-y curve response of a side-loaded pile to be evaluated based on the relationship between the three-dimensional response of a flexible pile in the ground to its one-dimensional beam on elastic foundation parameters. The model uses the stress-strain-strength behavior of the soil as determined from triaxial tests. Determining the wedge depth and the value of the bedding modulus below the wedge increases the complexity significantly. The method is more applicable to long piles than to short rigid piles.

Elastic Continuum

Continuum methods (Poulos and Davis, 1980; Pender 1983; Carter and Kulhawy, 1992) can be used to model the soil continuity, its nonlinearity and boundary conditions similar to the discrete Winkler spring or p-y spring models. The method envisions the soil as a continuous elastic medium in which stresses and displacements propagate outwards and decrease with distance from the point of application. There are closed form expressions for bearing capacities and deflections of axially and laterally loaded piles. However, few solutions are available for rigid piles that do not adequately represent the ULS of a pile in a non-linear, non-cohesive soil.

finite element

Finite element methods (FEMs) can provide rigorous results, including accounting for non-linearity and material heterogeneity (Trochanis et al., 1991; Yang and Jeremi, 2002, 2005). The results must be validated using simplified analysis methods before being used in the design. The development of an accurate FEM model is time-consuming and requires detailed soil parameters that are not typically available for the design of low- and medium-height cantilever walls.

When a pile is loaded laterally, the calculation of the soil yield stress against the pile requires the assumption of a plastic mechanism in the soil surrounding the pile. Near the surface, the mechanism is three-dimensional, since both vertical and horizontal movements of the ground occur. At depths greater than several pile diameters from the surface, the mechanism is essentially two-dimensional, with most of the displacement occurring in the horizontal plane. The solution to this plasticity problem is complex and no exact analytical solutions are available, although Reese et al., 1974 presented an approximate solution.

A rigorous estimation of the lateral load resistance of rigid piles in non-cohesive soil requires advanced modeling techniques such as the FEM method and consideration of the three-dimensional nature of the problem. However, the scale of most tower retaining wall projects, the geotechnical information available, or the budget often do not justify an advanced approach. A simplified but accurate design analysis procedure is required when advanced calculations are not justified. Of the analysis methods summarized above, the LFP method is considered the most appropriate for the present application. Simplifications in the procedure result in closed-form analytical solutions that can be quickly applied in design.

Various versions of the LFP method have been proposed and a number of these have been compared to find the most suitable for the present application. Below is a summary of the considered methods. The evaluation of their suitability was made by comparing the lateral bearing capacities calculated for each method with test results and calculated pile embedment depths for a typical pile wall.

2.1 Brinch Hansen, 1961

Brinch Hansen considered soil failure behavior at shallow, intermediate and deep depths. At shallow depths, failure was based on the difference between passive and active pressure developed on a rough, horizontally displaced wall. At moderate depths, drag was estimated by considering the balance of a passive Rankine wedge of the same width as the shank diameter. Static ground pressures acted on the sides of the wedge. At great depths, the ultimate stress was calculated assuming failure in a horizontal plane and based on the deep strip foundation solution. The following equation was developed for the soil failure pressure p u acting on the pile at any depth:

(1)

Where σ vo is the vertical overlay stress, K q is the earth pressure coefficient for overburden pressure, c is the soil cohesion and K c is the earth pressure coefficient for cohesion.

The failure pressure was assumed to act uniformly across the pile diameter and the failure force per unit length was assumed to be the failure pressure multiplied by the diameter. The depth of the point of rotation was obtained by adjusting this depth by trial and error to simultaneously satisfy force and moment balance equations. A convenient analysis method used by Brinch Hansen was to take moments about the load application point, which was assumed to be above the ground at an eccentricity of e. The failure pressure at the rotation point was assumed to change sign (see Figure 2).

Figure 2. Comparison of LFPs for methods by Brinch Hansen, Broms and Meyerhof et al.

2.2 Broms, 1964 (simple method)

Broms assumed that the face pressure developed at failure is equal to 3Kp. The accuracy of this assumption was established through comparisons with test data, and Broms stated that these comparisons showed that the assumption led to results on the safe side. For short rigid piles it was assumed that the failure pressure extends from the ground surface to the fulcrum. At a depth z below the ground surface, the assumed LFP (or boundary response of the ground per unit length) was given by:

(2)

Here B is the pile diameter, γ ‘ the effective area weight of the soil and K p the passive earth pressure coefficient according to Rankine. Broms assumed that the high negative earth pressures develop near the base of the pile and that this pressure could be replaced by a point load. Final lateral drag was determined by satisfying the moment balance assuming a toe twist. This gave the lateral capacity for a force H u applied at the top of the pile as:

(3)

where L is the pile length below ground level and e is the amount of applied lateral load above ground level (or eccentricity).

Although the Broms analysis method provides a simple closed-form equation for lateral bearing capacity, comparisons with other methods show that it overestimates the bearing capacity of short rigid piles in non-cohesive soils. A shortcoming is the assumption that the point of rotation is at the base of the pile. Other methods indicate that the rotation point is at a depth between 0.6 L and 0.8 L below the surface of the earth.

2.3 Bromine Modified

A modification of the Broms method has been proposed by others (Chen and Kulhawy, 1994). In this modified analysis, the assumption of Broms’ conventional or simple method is used that the ultimate pressure is 3 Kp, but the depth of the rotation point is obtained by satisfying both the horizontal and moment equilibrium equations. Instead of a concentrated force at the toe, the final negative pressure distribution below the rotation point is also based on the triple Rankine passive pressure assumption (see Figure 2). The simultaneous solution of the two equilibrium equations can be performed by a trial and error method. The Solver add-in for Excel can be used to speed up this process.

A closed-form solution to the modified Broms method is available from a more general analysis for cohesive and non-cohesive soils presented by Zhang, 2018. The solution for cohesionless soil is presented in terms of a general final lateral drag coefficient K. For Brom’s Modified Analysis K = 3 K p . (Zhang also pointed out that the solution is applicable to the LFP method of Petrasovits and Award, 1972, who assumed K = 3.7 Kp – Ka . where Ka is the Rankine active pressure coefficient.) The Zhang analysis involves a simultaneous solution of horizontal and moment equilibrium equations and leads to complex closed-form expressions for the ultimate lateral drag and the rotational depth. The following simplified expressions for the dimensionless final lateral bearing capacity fϕ and the depth of the center of rotation z rd were presented by Zhang as sufficiently accurate for the engineering design.

(4)

(5)

Where = depth of the center of rotation.

2.4 Meyerhof et al., 1981

Meyerhof et al. assumed that the lateral earth pressure developed at failure is equal to the Rankine passive pressure minus the Rankine active pressure, with this pressure difference increased by one pile shape factor.

At depth z below the surface, the assumed LFP was given by:

(6)

Where s bu is a pile shape factor based on the theory of pressure on a convex circular wall. It is a function of the ratio of pile length to diameter (L/B ) and the soil friction angle ϕ. Plotted values ​​are taken from Mayerhof et al. For L/B = 5 and ϕ = 35°, s bu is approximately 2.6.

The ultimate pressure was assumed to increase linearly from the surface to reach a maximum value at the point of rotation and then decrease to zero. Unterhalb des Rotationspunktes wurde angenommen, dass ein Unterdruck linear von Null bis zu einem Maximum am Pfahlfuß ansteigt. Ein Diagramm der Kraft pro Längeneinheit ist in Abbildung 2 dargestellt, wo es mit der Kraft pro Längeneinheit verglichen wird, die in den Analysen von Brinch Hansen, Broms Simple und Broms Modified angenommen wurde. Das Diagramm von Mayerhof et al. basiert auf einem L/B-Verhältnis von 5, und die Diagramme für alle Methoden gelten für ein Exzentrizitätsverhältnis von e/L = 0,1. Das L/B-Verhältnis wirkt sich auf den S-Bu-Pfahlformfaktor in der Analyse von Mayerhof et al. aus, wirkt sich jedoch nicht direkt auf die Ergebnisse der anderen Analysen aus. Die Diagramme sind in dimensionsloser Form mit der Kraft pro Längeneinheit dividiert durch K p 2 γ ‘BL dargestellt. (In anderen unten beschriebenen Methoden wird der LFP mit K p 2 in Beziehung gesetzt, und das Ausdrücken der dimensionslosen Kraft in Form dieses Faktors ist für Vergleiche bequem.)

Der seitliche Widerstand von Mayerhof et al. wurde bestimmt, indem sowohl das horizontale als auch das Momentengleichgewicht erfüllt wurden. Die endgültige Kapazität für die Seitenkraft ist ungefähr gegeben durch:

(7)

Dabei ist F b der seitliche Widerstandsfaktor für das Gewicht und hat einen Wert von 1,25, und r b ist der Reduktionsfaktor für das Moment, das sich aus der Aufbringung der Last bei einer Exzentrizität e über dem Boden ergibt, r b = 1/(1 + 1,4 e/L ).

In Meyerhof et al., 1981, werden allgemeinere seitliche Widerstandslösungen sowohl für starre Wände als auch für Pfähle in einem Boden angegeben, der aus zwei Schichten mit kohäsiven und kohäsionslosen Eigenschaften besteht.

2.5 Prasad und Chari, 1999

Prasad und Chari entwickelten einen LFP auf der Grundlage von Messungen, die sie an einem starren Modellpfahl in Sand (gleichmäßig und kohäsionslos) durchgeführt hatten, der mit verschiedenen relativen Dichten in einer Stahltrommel vorbereitet wurde. Aus den Testergebnissen wurde festgestellt, dass der maximale Erddruck in einer Tiefe von 0,6 mal der Tiefe des Rotationspunktes auftritt. Sie fanden heraus, dass der maximale Druck p m in dieser Tiefe wie folgt berechnet werden konnte:

(8th)

Wobei z r die Tiefe des Rotationspunktes unter der Bodenoberfläche ist.

Basierend auf den Testergebnissen wurde angenommen, dass der maximale Druck bei 0,6 z r am Rotationspunkt linear auf Null abfällt, dann negativ wird und linear ansteigt, um am Zeh einen Maximalwert zu erreichen, der dem 1,7-fachen des Spitzenwerts über dem Rotationspunkt entspricht (siehe Abbildung 3 ). Bei den anderen oben beschriebenen Verfahren hat der LFP am Rotationspunkt einen Wert größer Null, was theoretisch nicht korrekt ist. Bei Verdrängung Null am Drehpunkt muss auch der Druck Null sein.

Anhand von Bodendruckmessungen stellten Prasad und Chari fest, dass der durchschnittliche Druck über den Pfahlabschnitt als das 0,8-fache des gemessenen Spitzenwerts angenommen werden konnte. Unter Berücksichtigung des Kraft- und Momentgleichgewichts für den in Abbildung 3 gezeigten LFP wurde die Rotationstiefe wie folgt ermittelt:

(9)

Für e/L = 0 ergibt der obige Ausdruck die dimensionslose Rotationstiefe, z r /L = 0,79 und für e/L = 0,1, z r /L = 0,77.

Die endgültige Seitenkraftkapazität wurde angegeben als:

(10)

Abbildung 3. Vergleich von LFPs für Prasad & Chari, Zhang et al. und Guo-Methoden

Die durch die Prasad- und Chari-Gleichung angegebene endgültige Seitenkraftkapazität entsprach dem Punkt, an dem die Last gegenüber der Pfahlkopfverschiebung linear oder im Wesentlichen linear wird, nachdem sie einem gekrümmten Pfad gefolgt ist. (Diese Definition wurde von Meyerhof et al., 1981 und Chari und Meyerhof, 1983 verwendet.) Prasad und Chari wiesen darauf hin, dass es über diesen Punkt hinaus nur eine marginale Belastungszunahme gibt und aus praktischen Gründen der Boden um den Pfahl innerhalb des Versagenskeils liegen kann gelten als nachgegeben.

2.6 Zhang et al., 2005

Zhang et al., 2005, gingen davon aus, dass der Bodenwiderstand gegen seitliche Bewegung eines starren Pfahls in zwei Komponenten unterteilt werden könnte; die frontale Normalreaktion und die seitliche Reibungsreaktion. Basierend auf Druckmessungen, die in Pfahltests mit seitlicher Belastung durchgeführt wurden, die von Adams und Radhakrishna, 1973; Chari und Meyerhof, 1983; Joo, 1985; Meyerhof und Sastry, 1985, und Prasad und Chari, 1999, kamen zu dem Schluss, dass die beste Anpassung an den ultimativen Frontaldruck gegeben ist durch:

(11)

Where z is the depth below ground surface.

No measured data were available to determine the side shear resistance. They assumed that the ultimate shear stress resistance was the same as the vertical shear resistance given by API, 1991 as:

(12)

Where K r is the ratio of horizontal to vertical effective stress and δ is the interface friction between the pile and soil.

For values of K r and δ they recommended the guidelines given by Kulhawy et al, 1983, and Kulhawy, 1991. For drilled shafts and concrete surround with a rough contact surface against the soil, as generally used in pole wall construction, these guidelines give K r = (0.9 to 1.0) K o and δ = 1.0ϕ. Where K o is the at rest pressure coefficient and ϕ the soil friction angle.

Zhang et al assumed that the LFP for both the frontal soil resistance and side shear resistance followed the profile given by Prasad and Chari, 1991 (as shown in Figure 3). That is, they assumed that the maximum positive pressure occurred at a depth of 0.6 times the rotation depth and that the negative peak value at the pile toe was 1.7 times the positive peak pressure. The Zhang et al LFP is compared with those proposed by Guo and Prasad and Chari in Figure 3. Both the Prasad and Chari and Zhang et al dimensionless LFPs are a function of the soil friction angle which has been taken as 35o for the plots shown in Figure 3. (Since the LFPs are made dimensionless by dividing by K p 2 the changes in the dimensionless LFPs with friction angle are small. In the case of Zhang et al the dimensionless normal pressure force does not change with the friction angle but the shear resistance force is a function of K o which depends on the friction angle.)

Based on horizontal and moment equilibrium considerations the depth of the rotation point z r was as given by Prasad and Chari (see above) and the ultimate pile head lateral force capacity was given by:

(13)

Where η and ξ are pile shape functions. For a circular pile they are 0.8 and 1.0 respectively (Briaud and Smith, 1983).

2.7 Aguilar et al, 2019

By using the principle of minimum potential energy and assuming that the soil stiffness increased linearly with depth Aguilar et al, 2019 derived the horizontal displacement function for a rigid rotating pile as:

(14)

Where n h is the subgrade reaction modulus (FL-3 units).

By setting u = 0 in the above displacement function the dimensionless rotation depth z r /L is equal to 0.75 for a load eccentricity ratio e/L = 0, and z r /L = 0.74 for e/L = 0.1. These depths are significantly less than given by Prasad and Chari.

Aguilar et al give the pile head ultimate force capacity as:

(fifteen)

Where P m is the maximum force per unit length at the toe.

With reference to the Prasad and Chari earth pressure profile Aguilar et al give the maximum force per unit length at the toe as:

(16)

Aguilar et al indicate that the above expression comes from a statistical analysis of comparisons between the Prasad and Chari theoretical predictions and test results. However, Prasad and Chari give a significantly higher pressure force at the toe, that is a value of 1.7 times the peak value at a depth of 0.6 z r ).

At low values of e/L the Prasad and Chari ultimate lateral force capacity is approximately 10% higher than given by Aguilar et al.

Aguilar et al considered two failure criteria. The first of these was the Meyerhof criterion used by Prasad and Chari (see above) and the second was based on a hyperbolic fit to the experimental results or the nominal resistance. They suggested that the nominal resistance was a factor of 1.5 times the Meyerhof capacity. The expression for H u above is for the Meyerhof capacity.

Aguilar et al mentioned the importance of estimating displacements under serviceability conditions and proposed using their deflection equation given above for this purpose. To apply the method values of the soil subgrade reaction n h are required but for elastic response these can be estimated from published values (Terzaghi, 1955). It is a simple approach suitable for the present application.

2.8 Guo, 2008

Guo determined the soil interaction forces on laterally loaded rigid piles in cohesionless soil by assuming elastic-plastic soil behaviour as shown schematically in Figure 4. The nonlinear response of the pile was characterised by slip depths or points where the soil commenced to yield that progressed down from the ground surface and upwards from the pile-toe. Expressions for critical slip depths were developed corresponding to soil yield at the toe and at the rotation point. At toe-yield the force per unit length at the pile toe just attains the limiting yield value of P u . Prior to and at this state, the pile force profile from soil interaction follows the positive LFP (plastic soil interaction) down to a depth above the rotation depth of z o , below this depth it is governed by elastic interaction. This results in a pile force profile similar to that adopted by Prasad and Chari, 1999 (see Figures 3 and 5). Further increase in load beyond the toe-yield state results in a portion of the pile negative LFP progressing upwards from the toe (at depth L) to a depth of z 1 below the rotation point (see Figure 5). With further increase in the load the depths z o and z 1 approach each other and merge with the depth of rotation z r which is strictly unachievable. At this stage, the pile force profile follows the positive LFP down from the ground surface and the negative LFP up from the toe to the rotation point. This fully plastic or ultimate limit (although unachievable) was adopted by some investigators including Brinch Hansen, 1961 and Petrasovits and Award, 1972.

Figure 4. (a) Pile-soil system. (b) Soil load versus displacement model. Value of u* in [ ] is for Constant k. Other value is for Gibson k. From Guo 2008.

Figure 5. (a) Toe-yield state. (b) Post-toe yield state. From Guo, 2008.

Guo developed solutions for both a Constant subgrade modulus and a linearly increasing modulus with depth (Gibson k). The solutions were in reasonable agreement with data measured in tests by Prasad and Chari, 1999 and other numerical predictions (Laman et al, 1999).

Guo assumed that within the elastic soil range the pile force per unit length was given by:

P = k o z B u: for Gibson modulus (17)

P = k c B u: for Constant modulus (18)

Where k o is the soil modulus for Gibson modulus (FL-4 units), k c is the soil modulus for Constant modulus (FL-3 units), B the pile outside diameter, z the depth below the surface and u the pile lateral displacement.

Upon reaching the plastic state the net limiting force per unit length (LFP) on the pile is given by:

P u = A r z B (19)

Where A r z is the pressure on the pile surface (FL-2 units) contributed by radial and shear stresses around the pile surface.

Based on the tests carried out by Prasad and Chari, 1999 (and other experimental work), Guo assumed that A r was given by:

A r = γ ’ K p 2 (20)

Where γ ‘ is the soil effective unit weight.

The displacement of a rigid pile varies linearly with depth and is given by:

u = ω z + u o (21)

Where ω is the rotation (in radians) and u o the displacement at the ground surface. Above a depth of z o , called the slip depth, the pile displacement reaches a local threshold given by:

u* = ω z o + u o (22)

For the Gibson k assumption, the threshold displacement is given by:

u* = A r /k o (23)

The unknown rotation ω and displacement u o given in the above equations were determined by solution of the equilibrium equations for pile force and moment. Prior to toe-yield, relevant dimensionless solutions in terms of the pile head lateral force H and slip depth z o are given for the Gibson k assumption by:

(24)

(25)

(26)

(27)

The depth of the maximum moment in the pile, z m for z m < z o is given by: (28) Guo, 2008 also presents a maximum moment depth for z m > z o . This case results in a complex expression which is not relevant for the present application.

The maximum moment, M m for z m < z o is given by: (29) At toe-yield z o = z¯ o where z¯ o is given by the solution of the cubic equation: (30) This cubic equation can be solved by trial and error which can be expedited using the Excel Solver add-in. Force, displacement, and rotation solutions for Constant k are: (31) (32) (33) (34) The equations for z m and M m for z m < z o are the same as for the Gibson k equations. At toe yield z¯ o for Constant k is given by: (35) In contrast to the Gibson k case, a direct solution is obtained for z¯ o from this equation without iteration. The slip depths and rotation depths at toe-yield are plotted for both the Gibson and Constant k cases in Figures 6 and 7 respectively. For Gibson k the second order polynomial trend lines shown on the plots are sufficiently accurate for design purposes and eliminate the need to solve the cubic equation. The trend lines give the slip depth and rotation depths as: Figure 6. Slip depth ratio at toe-yield for Constant k and Gibson k. Trend line is shown for Gibson k case. Figure 7. Rotation depth ratio at toe-yield for Constant k and Gibson k. Trend line is shown for Gibson k case. Equations (23) to (37) for both the Gibson and Constant k are only valid up to toe-yield. Guo, 2008 also presents force, displacement, and rotation equations for horizontal forces greater than the toe-yield and up to the ultimate load when yield commences at the rotation point (unachievable in theory because of the large associated rotation). For the present application, the response beyond toe-yield is not of particular interest although displacements for forces beyond this point are shown in Figures 8 and 9. Force versus displacement functions (in dimensionless terms) generated from Guo’s equations for Gibson and Constant k are shown in Figures 8 and 9 respectively. For comparison, the Prasad and Chari ultimate lateral loads, H u are shown for each of the corresponding e/L curves plotted in Figure 8 for the Gibson k case. The Gibson k lateral toe-yield forces are approximately 16% higher than the Prasad and Chari ultimate forces over the e/L range of 0 to 0.4. Figure 8. Force versus displacement for Gibson k. Figure 9. Force versus displacement for Constant k Guo validated his theoretical solutions with results from tests on model piles carried out by Prasad and Chari. Figure 10 shows a comparison between the Guo predictions with results for a model steel pile with embedded depth of 612 mm, outside diameter of 102 mm and the load applied at 150 mm above the soil surface (e/L = 0.25). The soil was a well graded sand with relative density D r = 0.75 (unit weight of 18.3 kN/m3 and friction angle of 45.5o). (By curve fitting Guo derived an A r value of 739 kN/m3 which was used in the comparison. This value is about 15% higher than the A r calculated from the unit weight and friction angle.) For the comparison, the Gibson k displacements have been made dimensionless by dividing by the average k c value over the depth of the pile rather than k o which was used in Figure 8. Figure 10. Comparison of Guo theory with Prasad & Chari model pile test for sand with D r = 0.75. Figure 10 shows reasonable agreement between the theoretical load versus displacement curves and the test results, with the test results lying between the Gibson and Constant k curves. The assumption of an elastic-plastic soil rather than more realistic stress-strain behaviour is the main reason why closer agreement is not expected. The Guo dimensionless pile head lateral force at toe-yield versus the eccentricity ratio (e/L) is plotted in Figure 11 for both the Gibson and Constant k soils. A polynomial trend line for the Gibson k dimensionless pile head lateral force at toe-yield is given by: (38) Where H y is the horizontal force capacity at toe-yield. This trend line equation can be used to estimate the pile-toe yield load to sufficient accuracy for design (only accurate for e/L < 0.5). Figure 11. Pile head force at toe-yield from Guo method for Gibson and Constant k soils. Trend line is shown for Gibson k. The Guo method provides greater pile response detail than the other LFP methods investigated (including those summarised above and several others) in that it considers soils with both Constant and Gibson k stiffness properties. It also provides displacement response curves from initial pile head lateral loading up to the ULS force (yield at the rotation point). In particular, the response curve up to the toe-yield load provides the design information required for the present application. The assumption of elastic-plastic soil behaviour may limit the accuracy of the estimated displacements for some soils. Comparisons of solutions from the Guo method with other LFP methods and test results are presented in the following sections. It was concluded that the Geo method is the most satisfactory of the simplified LFP methods for the present application. Solutions can be obtained by evaluating Equations (24) to (37) on a spread sheet or by using the results plotted in Figures 6 to 9, and 11. 2.9 Comparison of LFP Methods The ultimate pile head lateral force capacity for e/L ratios from 0 to 0.5 was calculated for each of the LFP methods discussed above. Average dimensionless ultimate force capacity over this e/L range is shown in Figure 12 for soil friction angles of 30o and 35o. The toe-yield capacity for the Guo methods was used in the comparison. For the Meyerhof et al, Prasad and Chari, Zhang et al, and Aguilar et al methods the capacities shown are the Meyerhof capacities. Broms, 1964 compared the ultimate capacity from his Simple method with a number of test results and indicated that his method gave conservative results so this may indicate a capacity equivalent to the Meyerhof capacity. The Broms Modified and Brinch Hansen methods are based on a force profile that reaches yield at the rotation point and it is unclear whether they correspond to the Meyerhof capacity. The Broms Modified method is based on an LFP of 3K p 2 and this is probably a conservative assumption (see Zhang et al, 2005). Figure 12. Comparison of pile head lateral force capacities from LFP methods. For the 30o soil friction angle the dimensionless force capacities range from 0.065 (Aguilar et al) to 0.14 (Broms Simple). This is a very wide range; a factor of greater than 2 between the lowest and highest capacity estimates, and indicates that the Broms Simple method, which gives the highest capacity, is probably unconservative. With the exception of the two Broms methods, the dimensionless capacities are almost independent of the friction angle. This is because the LFPs in other than the Broms methods are essentially a function of K p 2 (with minor variations) and the capacity has been made dimensionless by dividing by K p 2. A further comparison of the LFP methods was made by calculating the pole embedment depth required by each method for a typical 3 m high vertical pole wall assumed to be loaded by gravity loads including a live load on the assumed level surface behind the wall. The parameters assumed in the analysis are summarised in Table 1. Table 1. Typical Pole Wall: Analysis Input Parameters The active pressure from the wall backfill and surface live load was assumed to act on the pole down to the rotation point but was assumed to remain constant with depth below the level of the ground in front of the wall. For calculating the passive LFP the ground was assumed to be horizontal and at the level in front of the wall. A factor of 1.2 was applied to the vertical stress to make allowance for the increase in stress resulting from the additional height of soil behind the wall (see Section 3). The lateral force capacity of the poles was approximately 64 kN. The depth of the point of rotation varied between the methods with a corresponding variation in the applied load (from the active pressure on the pole) and this changed by a small amount the ultimate lateral resistance required by each method. Main results from the wall analysis are summarised in Table 2. Table 2. Typical Pole Wall: Comparison of Pole Embedment Depths The typical wall analysis gave embedment depths between 2.4 to 3.15 m. That is, a factor of 1.3 between the smallest to greatest depth. A significant variation but perhaps not as great as the comparison of the pile head ultimate force capacities might indicate (Figure 12). 2.10 Comparison of LFP Methods with Pile Test Results Ultimate pile head lateral force capacities predicted by the LFP methods discussed above were compared with the lateral force capacities observed in reported pile tests for short rigid piles in cohesionless soils (mainly sands). For this comparison only tests with L/B and e/L ratios less than 9 and 0.5 respectively were selected. Forty-one laboratory tests, nine field tests, and two centrifuge tests satisfied these limits and were used in the comparison. A summary of the selected tests including pile dimensions and the main soil parameters is given in Table 3. (The length and eccentricity ratio limits were based on the expected range in the present application.) Table 3. Lateral Load Tests on Rigid Piles in Cohesionless Soil Ratios of the calculated lateral capacity divided by the observed capacity were computed for the LFP methods investigated. Average values of this ratio, the standard deviation of the average and the coefficient of variation are listed in Table 4. The observed capacities were thought to be based on approximately the Meyerhof capacity or approximately a factor of 1.5 less than the ULS or hyperbolic capacity. Capacities given in Chen and Kulhawy, 1994 (EPRI project) were labelled lateral or moment limits and it was stated that this limit was approximately the load at which initial failure occurred and that it did not correspond to the ULS. Ratios between the hyperbolic (ULS) and the lateral or moment limits given by Chen and Kulhawy for the tests that were considered in the present comparison were mostly between 1.2 and 2.1. There is some uncertainty in how the observed capacity was assessed so the information presented in Table 4 is illustrative of the relative magnitude of calculated/observed ratios for the methods rather than the absolute values. Table 4. Observed/Calculated Pile Test Capacities The Zhang (Barton), 2018 and the Petrasovits and Awad, 1972 methods listed in Table 4 are not discussed in detail above. Both are similar to the Broms Modified method with the LFP reaching a maximum positive value and a corresponding negative value at the rotation point. In the Zhang method, the LFP is defined by P u = z K p 2γ’B (FL-1 units) which Zhang indicated was used by Barton, 1982. In Petrasovits and Awad the LFP is defined by, P u = z (3.7K p 2 – K a ) γ’B. The average ratio of observed to calculated values shown in Table 4 indicates that all methods except the Mayerhof, Brinch Hansen and Zhang methods give satisfactory agreement with test results. The Broms Simple method gives better agreement than expected but many of the tests involved soils with friction angles greater than 35o (see Table 3). In soils with friction angles lower than this value it is expected to give unconservative capacities. Zhang, 2018 indicated that his method (using the Barton LFP) gave the hyperbolic capacity which is approximately 1.5 times greater than the Mayerhof capacity calculated by the other methods except for Guo’s methods. Therefore, it also gives capacities that are in satisfactory agreement with the test results. The Guo Gibson k method gives values that on average are approximately a factor of 1.25 times the observed test values. However, the capacity for this and the Guo Constant k methods are based on the LFP reaching the soil yield pressure at the pile-toe and this capacity is thought to be significantly higher than the Meyerhof capacity. It is unclear how the Meyerhof capacity could be determined from the Guo response curves shown in Figures 8 and 9 which have significant curvature for lateral force values greater than 0.5 times the pile-toe yield capacity. Equations given in Guo, 2008 for the case when both the positive and negative LFPs reach yield at the rotation point (force versus displacement curve approaches an asymptotic value) enable the hyperbolic capacities to be estimated and these are approximately 1.15 and 1.10 times the toe-yield capacities for Gibson and Constant k soils respectively (for e/L values between 0 to 0.4). If it is assumed that the Meyerhof capacity is approximately a factor of 0.7 times the hyperbolic capacity then the Guo Gibson k average observed/calculated test result ratio reduces to approximately 1.0 (1.25*1.1*0.7) for the Meyerhof capacity. 3. Gravity Stresses Near Wall Face The LFP in all the methods discussed above is a function of the vertical effective stress in the soil assumed to be given by z γ ‘. The lateral load capacity calculations are based on the ground having an approximately level surface near the pile with z being the depth below this level. In all the tests used for the comparison with the calculated capacities the ground surface was level. However, in the case of a pole retaining wall there is a step in the surface at the wall face with the depth of the pole toe below the ground in front of the wall being approximately the same as the height of the wall face. The wall ground surface geometry is therefore significantly different from that assumed in the pile head lateral force capacity calculation methods. In the case of a wall the passive soil resistance against the pole above the rotation point is usually based on the depth below the ground in front of the wall. In contrast, the passive resistance below the rotation point is sometimes based on the depth below the ground surface adjacent to the top of the wall. This approach results in the calculated vertical stresses near the rotation point differing by a factor of approximately two over a short horizontal distance either side of the pole. This is an approximation that needs to be investigated in more detail since an increase in the vertical stress in the soil above the assumed value in front of the wall will increase the LFP on the pole above the pole rotation point, whilst a decrease in assumed vertical stress in the soil behind the wall will decrease the LFP on the pole below the rotation point. As part of the present study the vertical stresses near the face of a 3 m high wall with level ground in front and behind the wall were calculated using an elastic plane strain finite element model. Vertical stress contours calculated by the model are shown in Figure 13. A soil unit weight of 20 kPa was used in the analysis and this resulted in the steps between the contours on the plot being approximately 13.5 kPa. The vertical wall boundary was unrestrained. This approximately simulates an active pressure state on the wall face. Figure 13. Vertical stress contours from elastic FEA analysis of 3 m high wall. At a depth of approximately the wall height below the ground surface in front of the wall there is significant variation in the vertical stresses on a horizontal plane extending both in front of the wall and into the backfill for a distance of the wall height (3 m). At a depth of 3 m below the surface in front of the wall and at distance of 3 m in front of the wall, the vertical stress is 67 kPa or a factor of 1.11 times the vertical stress at a large distance in front of the wall face (60 kPa = 3 x 20). At this same depth and at a distance of 3 m behind the wall face, the stress is 112 kPa or a factor of 0.93 the vertical stress at a large distance behind the wall face (120 kPa = 6 x 20). On a horizontal plane at a depth of 1.5 m below the surface in front of the wall these factors are lower but are still significant being 1.07 in front of the wall and 0.95 behind the wall. Figure 14 shows a plot of the vertical stresses near the wall face at a depths below the surface in front of the wall between 0.35 and 1.35 times the wall height. The horizontal distance from the face is shown in terms of pole diameters with the diameter taken as 0.6 m which is typical for a 3 m high wall. The positive direction is taken to be in the direction from the face into the backfill. The vertical stresses are plotted as the ratio of the FEA stress divided by the gravity stresses at large distances from the wall. The large distance stress is based on the depth below the ground in front of the wall and behind the wall for the stresses in front of the wall and behind the wall respectively. Figure 14. Vertical stress ratio near the wall face computed by elastic FEA. Figure 14 shows that at two pole diameters in front of the wall the vertical stress ratio is between 1.27 and 1.4 with the ratio in the shallower depths (above 0.7 of the wall depth) being approximately 1.4. At two pole diameters behind the wall face the stress ratio is approximately 0.87 for depths greater than 0.35 of the wall depth. Figure 15 is a modification of Figure 14 with the FEA vertical stresses on either side of the wall shown as the ratio of the stress divided by the gravity stress at a large distance from the front of the wall. Figure 15 is relevant for the case when the analysis of the pole lateral capacity is based on the assumption that the ground surface is horizontal and at the level in front of the wall. Both Figures 14 and 15 show that at two pole diameters in front of the wall the gravity stress ratio over the upper section of the pole is approximately 1.4. This ratio drops to about 1.2 at three pole diameters from the wall so the impact on the passive LFP is rather uncertain. Figure 15 shows that the stress ratio in the backfill region at the pole toe (depth ratio approximately 1.0) at two pole diameters from the face is between 1.5 and 2.1 and increases to between 1.6 and 2.2 at three pole diameters. These stress ratios in both directions from the wall face indicate that the pole lateral load capacity will be significantly greater than calculated assuming the ground to be horizontal at the level in front of the wall. A correction could be applied by increasing A r (or the effective unit weight of the foundation soil) by a factor of between 1.2 to 1.4. (A factor of 1.2 was used in the wall example described in Section 2.9 and Table 1.) Figure 15. Vertical stress ratio near the wall face computed by elastic FEA. In the Broms Modified analysis (effectively a trial-and-error analysis) the LFPs above and below the rotation point can independently adjusted by scaling the passive resistance in the two regions. In the 3 m wall example described in Table 1, changing the LFPs by factors of 1.4 and 0.87 above and below the rotation point respectively, increased the Broms Modified lateral load capacity compared to the same wall without LFP modifications by approximately 30%. The modified LFPs reduced the required depth of embedment of the poles by approximately 12%. (These changes were based on using the total height to the top of the wall for calculating the LFP below the rotation point.) Although it is possible to modify the Guo analyses to allow for variations in vertical stress in the soil foundation near the wall face this adds complexity to the method which is not justified for the present application. As suggested above, assuming horizontal ground at the level in front of the wall and increasing the A r by a factor of between 1.2 to 1.4 is a satisfactory approach for the present application. 4. Pile Spacing Effects For timber pole retaining walls the ratio of the pole centreline horizontal spacing, S divided by the embedded pole diameter (S/B ratio) is typically between 1.5 and 4.0. The spacing adopted is dependent on both the pole lateral resistance and the strength of the timber facing elements. For walls with a height of 3 m the spacing is likely to be at the lower end of this range. Below S/B ratios of 4.0 there is significant interaction between the stresses in the soil arising from the lateral loading of individual poles. The lateral load behaviour of piles in groups is commonly analysed using the p-y method in which pile interaction is taken into account using a p-multiplier which is applied to the ultimate lateral load resistance of a single pile. A summary of relevant research on pole spacing effects is given below. 4.1 Georgiadis et al, 2013 Georgiadis et al used lower and upper bound finite element limit analysis and analytical upper bound plasticity methods to investigate the limiting lateral resistance of piles in a single pile row embedded in an undrained cohesive soil. Numerical analyses and analytical calculations were presented for various pile spacings and pile-soil adhesion factors. The numerical results were in good agreement with each other and also with the theoretical upper bounds produced by the analytical calculations. An empirical equation was proposed for the calculation of the ultimate undrained lateral bearing capacity factor. 4.2 Pham et al, 2019 Pham et al investigated the ultimate lateral resistance for pile groups consisting of various arrangements of four piles, as well as two piles, three piles, four piles, and an infinite number of piles arranged in a row (relevant to the present application) against ground movement for various direction of the ground movement. They used a two-dimensional rigid-plastic finite element method to determine the total ultimate lateral resistance of the pile groups, but also the load bearing ratio of the piles in the group. The group effect was further investigated by considering the failure mode of the ground around the piles. Although the study was based on pile resistance to ground movement, rather than loads applied to the piles as in the Georgiadis et al study, the results were similar for both types of loadings. 4.3 Chen and Chen, 2008 Chen and Chen used elasticity theory and the concept of a fictitious pile to develop a rigorous analysis of the interaction factors between two piles subjected to horizontal loading and bending moment applied to the pile at the ground surface. By assuming the displacement compatibility between fictitious piles and the extended soil, the problem was reduced to a Fredholm integral equation of the second kind, which could be solved readily with numerical procedures. Close agreement was obtained between their results and other numerical results presented for the horizontal influence factors of single piles. They found that the conventional interaction factor approach, which ignores the pile stiffening effect, would generally yield satisfactory results, but may overestimate considerably the interaction effect when the piles are long and flexible. 4.4 Rollins et al, 2003 Static lateral load tests were conducted on three single piles and four pile groups at centre-to-centre spacings of 3.0, 3.3, 4.4 and 5.6 pile diameters. The pile groups had three to five rows with three piles in each row and the test piles consisted of steel piles of outside diameter 0.32 m and 0.61 m. Fifteen cycles of loading were applied at each deflection increment to evaluate the effect of cyclic loading and gap formation on lateral resistance. The load carried by each pile was measured along with deflection, rotation, and strain along the length of the pile to allow comparisons between the behaviour of the pile group and the single pile (Rollins et al, 1998). In addition, comparisons were made between the measured and calculated values using three computer programs based on the p-y method (Rollins et al, 2006). The lateral resistance of the piles in the group was found to be a function of row location within the group, rather than location within a row. Contrary to expectations based on elastic theory, the piles located on the edges of the group did not consistently carry more load than those located within the group. The front row piles in the groups carried the greatest load, while the second and third row piles carried successively smaller loads for a given displacement. However, the fourth and fifth row piles, when present, carried about the same load as the third-row piles. Average lateral load resistance was a function of pile spacing. Little decrease in lateral resistance was observed for the pile group spaced at 5.6 pile diameters; however, the lateral resistance consistently decreased for pile groups spaced at 4.4, 3.3 and 3.0 pile diameters. Group reduction effects typically increased as the load and deflections increased up to a given deflection but then remained relatively constant beyond this deflection. The Rollins et al pile tests were carried out in clay soils but they concluded that their p-multiplier curves for estimating pile interaction effects appeared to give reasonable estimates of the behaviour of pile groups in sand based on available full-scale and centrifuge testing (McVay et al, 1995). They also suggested that the curves were not significantly affected by the pile diameter or pile head boundary condition. 4.5 Mokwa and Duncan, 2005 In discussing a paper by IIyas et al, 2004 related to centrifuge tests on the lateral resistance of pile groups, Mokwa and Duncan presented results of their review study on the behaviour of the response of pile-groups to lateral loads (Mokwa and Duncan, 1999; Mokwa and Duncan, 2001). On the basis of the results of these studies, which summarized 29 separate field tests in varying soil conditions, Mokwa and Duncan developed a design chart to estimate values of the p-multiplier factor that were based on pile spacing and pile location within a group. 4.6 Lin et al, 2015 Lin et al carried out an experiment on a fully instrumented model to investigate the soil-structure interaction on single short, stiff pile laterally loaded at the head. The pile had a steel pipe section with diameter 102 mm, wall thickness 6.4 mm, and a length of 1.52 m. It was installed in well-graded sand and subjected to increasing lateral load. The pile and surrounding soil were fully instrumented using advanced sensors, including flexible shape acceleration arrays, thin tactile pressure sheets, and in-soil null pressure sensors. The sensors attached to the pile were used to develop the compressive soil-pile interaction pressures and the lateral displacement along the pile length. The tactile pressure sheet sensors provided the soil-pile interaction compressive pressures on the circumference of the pile at a specific depth and along the length of the pile. The null pressure sensor measurements were used to develop the distribution of horizontal stress changes in the soil around the pile as the lateral pile displacement increased. Theoretical analysis of the nonlinear interaction of piles laterally loaded in cohesionless soils is complex. Finite element analysis can be used to study this problem but numerical results for the range of pile geometries relevant to the present study have not been published. The experimental study of Lin et al is therefore informative for the present application. Figure 16 shows the pressures measured in the soil surrounding the pile at the ultimate lateral load of 3.8 kN reached at the end of a test. Contours are soil pressures with labels in kPa. Of interest for the present application is the pressure drop in the lateral direction (x-direction). At a lateral distance of 2.5 pile diameters the pressure in the soil has dropped from a peak value of greater than 50 kPa on the leading face to approximately 5 kPa. This indicates that there would be negligible interaction effect for piles spaced on centrelines of greater than 5.0 diameters in cohesionless soils. Figure 16. Contours of soil stress surrounding laterally loaded model pile at ultimate load (3,799 N and pile head displacement of 86.1 mm). From Lin et al, 2015. 4.7 Comparison of p-multiplier Interaction Curves A comparison of p-multipliers from the results of the investigations mentioned above is shown in Figure 17. The curves from Pham et al and Georgiadis et al are for piles with full adhesion (perfectly rough) and are based on theoretical finite element and plasticity theory. The Rollins et al curve and the p-multiplier proposed by Mokwa and Duncan are from pile test results and apply to the leading row of multi-row pile groups. The p-multipliers from the test results are significantly lower than the values obtained by Pham et al and Georgiadis et al indicating greater pile–soil–pile interaction effects. The test values are applicable to the entire pile length, and one of the reasons for this difference could be that the reduction in the limiting earth pressure, due to group effects, is not constant with depth, as implied by the adoption of constant p-multipliers. The theoretical p-multipliers are applicable to the lower part of piles, where the flow around failure mechanism assumed is predominant. The depth below the surface where two-dimensional plane strain failure occurs, is likely to increase with the decrease of pile spacing, which could explain the large discrepancies at small pile spacings. Other factors that may contribute to the differences are the different soil types and the different geometrical characteristics of the tests. The tests were for multiple rows and interaction between the leading and second row would result in lower p-multipliers. Chen and Chen reduction factors based on elasticity theory are shown in Figure 17 for pile to soil Young’s moduli ratios, E p /E s of 10 and 500. They are based on an assumed row of 5 piles (interaction was computed by superposition of the influence coefficients between two piles) and for L/B ≥ 10 (the only length to diameter ratio presented by Chen and Chen). Although elastic solutions are not relevant for the present application, they provide a lower bound for the p-multiplier reduction factors. For the present design application, a recommended curve is shown in Figure 17. It generally follows the Mokwa and Duncan curve (a straight line) but with the p-multiplier reducing to 1.0 at S/B = 5 instead of at a ratio of 6. Figure 17. Pile spacing reduction factors (p-multipliers). 5. Ultimate Lateral Force Capacity of Rigid Piles in Cohesive Soil Significant research effort has focused on the determination of the limiting lateral load, P u distribution with depth for single piles in clay (Matlock, 1970; Reese and Welch, 1975; Stevens and Audibert, 1980; Murff and Hamilton, 1993; Jeanjean, 2009; Georgiadis and Georgiadis, 2012). This research established that P u increases with depth in the upper part of the pile, where a wedge-type failure occurs, up to a maximum value and remains constant in the lower part of the pile. In this lower part failure take place with a flow-around mechanism. Randolph and Houlsby, 1984 developed lower and upper bound plasticity solutions for the calculation of the maximum load per unit length, and proposed the following lower bound equation, expressed in terms of the single-pile lateral bearing capacity factor, N p : (39) Where s u is the undrained shear strength, B is the pile diameter and α is the pile soil adhesion factor (limiting interface shear stress/undrained shear strength). Martin and Randolph, 2006 showed that the above expression gives the theoretically exact solution for all practical purposes. The above equation gives N p values of 9.14 and 11.94 for perfectly smooth and full adhesion on the pile-soil interface respectively. Some of the models that have been developed to describe the limiting lateral force per unit length down the full depth of the pile are described below. 5.1 Reese Model In the Reese model (Reece 1958; Reece et al 1974), the soil behaviour is divided into either shallow or deep failure. For shallow failure, a three-dimensional passive wedge is assumed to exist in front of the pile. By summing the wedge forces in the horizontal direction, the soil resistance against the shaft was obtained. This procedure gave an N p of 2 if the failure wedge occurs with a vertical inclination of 45o without shear forces between the shaft and soil. For the deep failure, at depths greater than 3B Reece undertook a simplified plasticity analysis and estimated an N p of 12. The N p between ground surface and 3B depth was obtained by linear interpolation, to give the N p profile shown in Figure 18. 5.2 Brinch Hansen Model As discussed above for cohesionless soils, Brinch Hansen, 1961, considered the soil behaviour at shallow, moderate, and large depths. He considered soils with both friction and cohesion components and developed equations and corresponding charts using simplified plasticity theory based on horizontal translation of a rough wall and deep strip foundations. For a purely cohesive soil (zero friction angle) the variation of N p with depth calculated using his equations is shown in Figure 18. The value of N p at a depth of 3B is 6.2. The yield stress was assumed to act uniformly across the shaft diameter and is multiplied by the shaft diameter to obtain the yield or lateral force per unit length. 5.3 Broms Model Broms, 1964b employed classical plasticity theory for determining values for N p and examined a number of different shaft shapes and surface roughness. The resulting N p values ranged from 8.28 for a smooth square shaft to 12.56 for a rough flat plate. The value for a smooth circular shaft was 9.14 which agrees with the value calculated from the Randolph and Houlsby, 1984, equation given above. As a simplification, he assumed that N p = 9 below a depth of 1.5B and N p = 0 above that depth (see Figure 18). 5.4 Stevens and Audibert Model By comparing available p-y curves (Matlock, 1970; API, 1977) Stevens and Audibert, 1979 back figured a profile of N p with depth from instrumented driven pile load test data. Their observed N p range versus depth is shown in Figure 18. They recommended the profile shown within the observed range. No equation was given to represent this profile. For z/B > 4, N p = 12.

5.6 Randolph and Houlsby Model

Randolph and Houlsby, 1984 considered a wedge failure at shallow depth and used plasticity theory for N p at large depths. Their N p profiles for smooth and rough shafts are shown in Figure 18. At the ground surface, a yield stress of 3s u was obtained from a passive stress of 2s u in front of the shaft and allowance for side shear. As described above, at depth they considered the soil as a perfectly plastic cohesive material. The rough shaft N p limiting value of 11.9 at depth is relevant to the present study because drilled holes backfilled with pole concrete surround are likely to be perfectly rough.

Figure 18. N p profiles. From Chen and Kulhawy, 1994.

5.7 Pender Model

Pender, 2000 found that the assumption made by Broms of an unsupported depth of 1.5B was too conservative for the short piles used in pole wall design. To determine an appropriate depth, he used the pressure distribution shown in Figure 19 that has a N p value of 5 at the ground surface increasing to 12 at a depth of 3B. His conclusion was that a depth of 250 mm of zero resistance was appropriate for short piles used in pole wall design.

Figure 19. N p profile used by Pender, 2000.

5.8 Ultimate Lateral Force Capacity in Cohesive Soil (Undrained)

The pile head lateral force capacity, H u is calculated from the cohesive LFP by limit equilibrium analysis of the horizontal forces and moment in a similar manner to that described for cohesionless soils. This requires the simultaneous solution of force and moment equilibrium equations with unknown variables the depth of rotation and the lateral force.

Solutions for the assumed LFP shown in Figure 20 are presented in Pender, 1997; Motta, 2013, and Zhang, 2018. The solution for this LFP is readily applied to the case when there is zero resistance in a top layer of depth z t below the ground surface by reducing the pile depth below the ground surface of L by z t and increasing the eccentricity e by z t (effectively an artificial ground surface at depth z t )

Figure 20. Assumed LFP for analysis. From Zhang, 2018.

Compact forms of the solution are given by Zhang, 2018. Minor rearranging of his solution gives the dimensionless ultimate lateral load capacity H ud as:

(40)

Where the limiting force on the pile per unit length,

The dimensionless rotation depth z r /L is given by:

(41)

The depth to the maximum bending moment in the pile, z m and maximum moment, M m are determined from calculating the depth of zero shear force in the pile and are given by Pender, 1997 as:

(42)

(43)

The depth z m is taken from the effective soil surface (surface level reduced by depth z t of any layer of zero resistance). If a zero-resistance surface layer is assumed, the force eccentricity e above actual ground level is increased by z t .

By assuming elastic-plastic soil response Motta, 2013 presented lateral force versus displacement curves. Defining an elastic stiffness or modulus of the soil-reaction as E s (FL-2 units) the limiting deflection is given by:

(44)

After the soil yields the soil force on the pile was assumed to be constant with increasing soil deflection (see Figure 21). Motta defined three cases of soil-pile interaction as shown in Figure 22.

Figure 21. Soil elastic-plastic response. From Motta, 2013.

Figure 22. Soil-pile interaction for limiting force on pile constant with depth. From Motta, 2013

In Case 1 the soil remains elastic over the pile depth. With increasing deflection Case 2 is reached with yield in the soil down to depth a at distance x above the rotation point. With further deflection yield in the soil is reached at the pile toe and Case 3 commences with yield progressing from the toe up to the rotation point. (The soil-pile interaction to reach limiting forces on the pile is similar to the interaction described above for cohesionless soils.)

From equations presented by Motta the pile top lateral force required to initiate soil yield at the pile toe, H y can be calculated from the following expression:

(45)

Where e d = (e + z t )/ L e and L e is the length of pile excluding the ineffective soil depth z t

The pile ground level displacement, u y at toe yield can be calculated by:

(46)

Solutions for the dimensionless ultimate lateral load, load at toe yield and the depth of the rotation point as functions of eccentricity ratio e d (e/L) are shown in Figure 23. Ultimate loads are shown for three cases; z td = 0 (no limiting force reduction in the top layer), z td = 0.1 (no resistance in top layer of depth 0.1L) and for a top layer with a limiting force profile increasing from 0.3P u at the surface to 1.0P u at a depth of z td = 0.2, where z td = z t /L. The dimensionless ultimate and yield forces, and the eccentricity ratio are plotted in terms of L the length of the pile from ground surface to the toe with the eccentricity, e taken as the height of the horizontal force above ground level. Yield loads and rotation depths are only shown for the first two cases because yield loads and rotation depths cannot be readily calculated for the case where P u varies with depth (there is no available closed form equation). The rotation depth for the z td = 0.1 case is calculated from the actual ground surface (rather than the effective surface).

Figure 23. Ultimate load, yield load capacities and rotation depths for three LFPs

For the two cases where pile toe yield capacities were calculated the ultimate load capacities are approximately 20% higher than the yield capacities over the e d range of 0 to 1.0. Reducing the resistance to zero in a top layer of dimensionless depth z td = 0.1 reduces the ultimate capacity by approximately 22%. The ultimate capacity for this case is on average 8% less than the case with the variable LFP in the z dt = 0.2 top layer so it gives a conservative estimate for the more realistic case of a variable LFP in the top layer.

Ultimate and yield capacities for values of z t > 0 can be estimated from the z t = 0 curves by using appropriate values of e and L in the dimensionless parameters. (For z t = 0.1 an e d value = (0.1+e)/0.9 and an effective length of 0.9L would be used to estimate H u and H y from the z t = 0 curve.)

Dimensionless pile head lateral force versus ground level (u o ) displacement curves, calculated from displacement equations given by Motta, 2013, for e/L ratios from 0 to 1.0 are shown in Figure 24.

Figure 24. Dimensionless force versus ground level displacement

5.9 Comparison of Capacities Using the Broms LFP with Pile Test Results

Ultimate capacities predicted by the constant LFP or Broms method discussed above for cohesive soil were compared with lateral load test results published in Chen and Kulhawy, 1994 for short rigid piles in clay soils. For this comparison only tests with L/B and e/L ratios less than 9 and 1.1 respectively were selected. Forty-three laboratory tests and seven field tests satisfied these limits and were used in the comparison. A summary of the selected tests including pile dimensions and the soil shear strength is given in Table 5.

Table 5. Lateral Load Tests on Rigid Pile Tests in Clay Soil (From Chen & Kulhawy, 1994)

Zhang, 2018 calculated ultimate capacities using the limiting force of 9s u B recommended by Broms and compared these capacities for a range of 58 laboratory and field tests, including tests on piles with L/B and e/L ratios greater than considered in the present study. However, he assumed that the soil provided resistance over the total pile length and did not use the 1.5B ineffective top layer recommended by Broms, 1964b. The average ratio of calculated ultimate capacity divided by the test capacity (H u /Q u ) over the 50 tests considered in the present study was 1.09. The test load capacities were taken as the hyperbolic (or ULS load) as presented in Chen and Kulhawy, 1994. For the present study, corresponding ultimate capacities were calculated assuming a N p value of 11 (based on full soil adhesion) instead of 9, and assuming ineffective top layer depths of zero, 0.5B and 1.0B. Results from these analyses and the analysis based on Zhang’s assumptions for the relevant 50 tests are summarised in Table 6.

Table 6. Observed/Calculated Pile Test Capacities

Overall, there is reasonable agreement between analyses based on the constant LFP (Broms method) and the test results but the level of agreement was sensitive to the assumption made regarding the depth of the top layer with ineffective resistance. The N p value of 9 and ineffective layer depth of 1.5B recommended by Broms gives load capacities less than 50% of the average of the test capacities (based on the hyperbolic, ULS). The average L/B ratio for the test piles was 4.4 so reducing the effective length by 1.5B has a significant impact on the load resistance. For longer piles the reduction in resistance would obviously be less.

For the present application, using an N p of 11 (based on a rough soil-pile interface) and assuming that the depth of ineffective layer is 0.5B appears acceptable and gives good agreement with the average of the test load capacities. A top ineffective depth of 0.5B would correspond to about 300 mm or about 0.1L for typical pole walls that have L/B ratios of about 5.

5.10 Comparison of Depths of Embedment Required for Typical Pole Wall

A further comparison of the assumptions made in application of the constant LFP assumption was made by calculating the pole embedment depth required by three of the N p and z t combinations listed in Table 6 for a typical 3 m high vertical pole wall assumed to be loaded by gravity loads including a live load on the assumed level surface behind the wall. The parameters assumed in the analysis are summarised in Table 7.

Table 7. Typical Pole Wall: Analysis Input Parameters

Active pressure from the wall backfill and surface live load were assumed to act only above ground with no load from these pressures transferred to the pole in the cohesive soil foundation. The ground in front of the wall was assumed to be horizontal.

The length of pole embedment was based on the toe yield capacity using the Motta yield equations. Using this approach ensures that deflections are unlikely to exceed acceptable limits; however, it is necessary to consider the overstrength of the foundation up to the ultimate capacity of the soil to design the pole above ground and the composite section pole with concrete encasement below ground.

The total factored load on the wall (pressures from backfill and surcharge) was 42 kN and acted at an eccentricity of 1.08 m above ground level. The analysis procedure was to determine the required total embedment of the pole to achieve the required capacity at the commencement of toe yield (42 kN) and then to calculate the ultimate capacity using the yield embedment length. In all three cases investigated the soil ultimate capacity was approximately a factor of 1.3 greater than the yield capacity. Main results from the wall analysis are summarised in Table 8.

Table 8. Typical Pole Wall: Comparison of Pole Embedment Depths

The below ground bending moment diagram and a flexural capacity curve (ULS) for the encased timber pole using N p = 11 and z t = 0.3 m is shown in Figure 25. The moments shown in dimensionless form in Table 8 and Figure 25 were calculated by dividing the moment by the limiting force per unit length and the square of the total pile length below ground (M d = M/(P u L2). The capacity curve for the pile is based on a 350 mm diameter timber pole section at ground level increasing linearly to a fully composite pile section at a depth of 600 mm (one diameter of the composite section) below the ground surface. It is assumed that the 50 mm thick concrete surround commences at ground level.

Figure 25. Bending moment and capacity curves for pile (N p = 11, z t = 0.3 m)

Figure 26. Pile top force versus displacement (N p = 11, z t = 0.3 m)

The force versus displacement plot for the example using N p = 11 and z t = 0.3 m is shown in Figure 26. The length, L e (1.7 m) used in the dimensionless factors is the embedment depth below the ineffective top layer of soil. The total eccentricity including the ineffective layer was 1.38 m giving an effective e/L e of 0.81.

6. Displacements in Cohesive and Cohesionless Soils

For most soils including clay and sand, pile head force versus displacement curves are assumed to follow a hyperbolic curve as show in Figure 27 and defined by the equation (Lam and Martin, 1986):

(47)

Where H u is the ultimate force capacity and u c the displacement at 0.5 of the ultimate force.

Figure 27. Typical pile top force versus displacement for clay.

Comparison of Figures 26 and 27 shows significant differences with the elastic portion being greater in Figure 26 which is based on an elastic-plastic assumption.

6.1 Cohesive Soils

The initial stiffness E i of the pile force versus displacement curve for a cohesive soil is given by Lam and Martin, 1986 as:

(48)

Displacement u c is given by:

(49)

Where ε c is the strain amplitude at one-half the peak deviatoric stress in an undrained compression test. It usually ranges from 0.005 to 0.025. In the absence of laboratory data, a value of 0.01 is suggested by Lam and Martin.

In the example for clay presented in the previous section the limiting force per unit length, P u is expected to vary over the top half of the pile increasing to a value of 330 kN/m (s u N p B) at the toe. Using this value gives an initial elastic modulus, E i of 22 MPa (u c = 15 mm). For the complete pile length, adopting an equivalent linear value of one-half of the initial stiffness value would appear to be appropriate for predicting the serviceability displacement (at a load of approximately toe yield capacity/1.3). From Figure 27 and making allowance for pile interaction (a reduction to stiffness of approximately 0.8) the pile top ground level displacement at toe yield was estimated as 41 mm. (The rotation depth below ground surface was estimated to be 1.31 m.) Making allowance for the load factor of 1.3 the serviceability limit state (SLS) deflection at the top of the wall was estimated to be 105 mm at a corresponding wall rotation of 1.8o. The wall top SLS displacement is significant but within acceptable limits.

6.2 Cohesionless Soils

The tangent stiffness E t of the pile force versus displacement curve for cohesionless soil is given by Lam and Martin, 1986 as:

(50)

Where E t is the force per unit length per unit deflection and varies linearly with depth z. k 1 is a coefficient for sands which varies with relative density or friction angle (FL-3 units). Values of k 1 are plotted in Figure 28. The initial tangent values are from Reese et al, 1974 and the secant values from Terzaghi, 1955.

Figure 28. Subgrade modulus values for sand.

The displacement of the typical pole wall example described in Table 1 was calculated using the Guo analysis for Gibson k. A summary of the analysis is given in Table 9.

Table 9. Deflection of 3 m High Wall in Cohesionless Soil

To obtain the displacement at toe-yield the initial tangent stiffness should be reduced by a factor of between 2 and 4. A reduction of 4 gives approximately the Terzaghi secant value (see Figures 27 and 28.) For the present example, a reduction factor of 3 was used.

Making allowance for the load factor of 1.3 the SLS deflection at the top of the wall was estimated to be 45 mm at a corresponding wall rotation of 0.5o. The wall top SLS displacement is clearly within acceptable limits.

7. Ultimate Capacity of Rigid Piles in c-ϕ Soil

Zhang, 2018 presents a solution for the ultimate capacity of a laterally loaded rigid pile in a c-ϕ soil. He expressed the ultimate capacity in dimensionless form by:

(51)

oder alternativ:

(52)

Where:

K is the ultimate lateral coefficient for cohesionless soil. in the Guo analysis.)

Complex expressions are given for z rd (z r /L) the dimensionless rotation centre. For design purposes the depth of the rotation centre can be estimated from Figure 29. This figure shows how the rotation centre varies depending on the kq ratio for the soil between the limiting cases for a purely cohesionless soil with kq = 0 and a purely cohesive soil with kq = a large value (qk = 0).

Figure 29. Dimensionless rotation centre for c-ϕ soil. Evaluated using Zhang, 2018.

Dimensionless ultimate capacities f 1 and f 2 are plotted as a function of the eccentricity ratio in Figures 30 and 31 respectively. Either of the two figures can be used but Figure 30 is more convenient for a cohesive soil and Figure 31 for cohesionless soils with small s u values. These capacities are hyperbolic values based on the assumed ultimate pressure diagram shown in Figure 32. The fϕ and f c components used in the calculation of f 1 and f 2 are computed using the z r rotation depth calculated from moment equilibrium for the combined cohesionless and cohesive components shown in

Figure 32.

Figure 30. Dimensionless ultimate capacity for c-ϕ soil based on kq ratio

Figure 31. Dimensionless ultimate capacity for c-ϕ soil based on qk ratio.

Figure 32. Assumed LFP for c-ϕ soil analysis. From Zhang, 2018.

To illustrate an application of the c-ϕ theory the analysis of a pile with a 3 m depth of embedment that might be used for a pole wall was carried out. The input parameters are summarised in Table 10. Ultimate forces per unit length for the cohesionless and cohesive soil components were assumed to be those illustrated in Figure 32. No ineffective layer below the ground surface with zero soil resistance was used. Soil strength properties used in the example would be appropriate for highly weathered greywacke rock (Pender, 1977).

Table 10. Input Parameters for Analysis of 3 m Long Pile in c-ϕ Soil

Results of the analysis are summarised in Table 11.

Table 11. Summary of Results from Analysis of 3 m Deep Pile in c-ϕ Soil

The depth of the maximum moment z m is calculated by finding the depth of zero shear in the pile and is given by:

(53)

The maximum moment is given by:

(54)

and in dimensionless form the maximum moment, M md is given by:

(55)

The c-ϕ analysis is based on the LFP shown in Figure 32. It could be modified for minor variations such as assuming there was no cohesion in an upper soil layer.

Results using the Zhang LFP assumption and analysis method suggests that a good approximation (slightly conservative) can be obtained by adding the capacities from separate zero cohesion and zero friction analyses. This finding suggests that more complex LFP’s could be analysed using the separate components although the rotation depths will be different for the two cases. (For the above example, the dimensionless rotation depths for separate analyses of the cohesive and cohesionless components were 0.64 and 0.77 respectively – see Figure 29.) Calculating the lateral force capacity at toe yield requires a more detailed analysis to obtain an exact result but reducing the hyperbolic capacity by 20% would give an approximate toe yield capacity. Alternatively, determining the pile head lateral force capacity at toe yield by adding the separate cohesion and cohesionless components at toe yield would be satisfactory for design application.

The example discussed above indicates that a small amount of cohesion can result in a significant increase in the lateral force capacity. The low cohesion of 10 kPa used in the example increased the capacity of the friction only case by approximately 40%. It is customary to neglect small amounts of cohesion in predominantly cohesionless soils because cohesion is less readily assessed and can be variable. However, in foundations in highly weathered rock making a conservative allowance for cohesion is usually considered an acceptable approach. (Pender, 1977 completed laboratory tests to determine friction and cohesion parameters for highly and completely weathered Wellington greywacke. Friction angles for the highly weathered material were in the range of 32o to 38o and the cohesion in the range of 80 to 130 kPa.)

In cohesive soils full consolidation may occur under long-term loading, so calculations in terms of drained shear strength based on both cohesion and friction might be appropriate for this case. (Where live load represents a substantial portion of the load on the wall, static design for cohesive soils should be on a short-term undrained basis.)

8. Effect of Ground Water in Cohesionless Soils

The effective unit weight of the soil should be used when calculating the lateral force capacities of poles in cohesionless soils. If the water table is below the toe of the pole an estimated bulk unit weight should be used for the effective unit weight. If the water table is at the ground surface in front of the wall the soil effective unit weight should be taken as the submerged unit weight. For typical sands and gravels the submerged unit weight will be approximately one-half of the bulk unit weight of the soil above the water table leading to an approximate 50% reduction in the lateral force capacity.

If water is expected in the backfill then it should be included as a pressure on the facing. Pole walls will often have open joints in timber facing and are well drained if good drainage practise is followed so it will seldom be necessary to include a water pressures load against the wall facing.

In higher pole walls the water table level could be located at some height between the ground surface in front of the wall and the toe of the pole. To estimate the lateral force capacity for this intermediate case an analysis was carried out on a 3 m long pole with an 0.6 m embedded diameter located in a cohesionless soil with bulk unit weight of 18 kN/m3 (buoyant unit weight of 8.2 kN/m3) and friction angle of 35o. The hyperbolic lateral force capacity was calculated assuming a Guo A r factor of γ ’ K p 2. At the hyperbolic ultimate capacity, the LFP reaches both negative and positive soil yield at the rotation point. This profile enables a simple analysis to be undertaken whereas the analysis becomes more complex for the case when yield initially commences at the pole toe (recommended capacity for design). The LFP at the hyperbolic ultimate capacity for the assumed pole and soil properties for the water table located at a depth of 0.6L is shown in Figure 33.

Figure 33. LFP for calculation of hyperbolic capacity using Guo Ar.

The ultimate hyperbolic capacity plotted as a function of the water table depth, z w below ground level for the 3 m long pole with eccentricity ratios e/L = 0, 0.2 and 0.5 is shown in Figure 34. Depths and capacities are plotted in dimensionless form to enable the results to be used for other geometries and soil properties than used in this particular example. (In the dimensionless force divisor, γ ’ is the bulk unit weight.) The capacity ratio obtained by dividing the capacity at each particular water table depth by the capacity for the case with the water table at the pole toe or deeper is shown in Figure 35. As indicated in the figure the capacity ratio is insensitive to the e/L ratio. For depths of water greater than z w /L of 0.7 the capacity is less than 5% below the case with water at the toe level or deeper. Thus, in practical design, when the water table depth is close to the pole toe depth the water table capacity reduction can be ignored.

Although the above analysis is for the hyperbolic capacity the influence of the water table depth on the Guo toe yield capacity is expected to be similar.

The influence of the water table depth on the rotation point depth is shown in Figure 36. Although the rotation depth z r is sensitive to the e/L ratio, for a particular ratio the rotation depth varies by less than 5% from the case with water at the surface or below the toe.

Figure 34. Hyperbolic capacity versus water table depth.

Figure 35. Hyperbolic capacity ratio versus water table depth.

Figure 36. Rotation point depth versus water table depth.

9. Composite Action

Cantilever timber pole retaining walls are usually constructed by drilling an oversize hole, installing the poles, and then backfilling the annulus between the pole and the soil with unreinforced concrete. In low to moderate height retaining walls the maximum bending moments in the poles occur at depths of between 0.5 to 1.5 m below ground level. The maximum moments in the embedded section are typically 1.5 times greater than the bending moments at ground level and the question arises as to whether the composite action between the timber pole and concrete encasement is effective in providing a strength increase to offset this increase. The maximum practical height that can retained using cantilever timber pole walls is increased if the timber section can be designed for the bending moment at ground level or at least for a moment intermediate between the maximum and ground level moments.

The strength of the adhesion or bond between the timber pole and concrete is a key issue in deciding whether composite action is effective. Unfortunately, this is an area where there is little published information. Observations of concrete encased timber poles removed from the ground indicate that the adhesion is generally good with the concrete surround usually intact and the concrete difficult to remove from the timber.

The adhesion might be reduced by differential shrinkage between the pole and concrete. However, the concrete effectively seals the timber at depths greater that about 300 mm below the surface so that timber shrinkage is likely to be small or at least less than the shrinkage of the concrete. Shrinkage of the concrete will depend on the moisture in the soil. In dry soil conditions concrete shrinkage will be significant. Generally, the soil will reduce the rate of shrinkage and in ideal conditions the rate of increase in the tensile strength of the concrete may be sufficient to prevent radial cracking caused by circumferential shrinkage. Even if radial cracking occurs is unlikely to be extensive at depths below the surface greater than a few hundred millimetres. In addition, moderate radial cracking is unlikely to result in a significant reduction to the adhesion between the concrete and timber.

At the point of maximum moment in the embedded pole section the shear force on the section is zero. Therefore, in the vicinity of the maximum moment the adhesion bond does not need to be large to resist the interface shearing action.

9.1 Composite Section Analysis

In the present project the enhanced strength of timber pole sections acting compositely with concrete surround was estimated based on the overriding assumptions that the concrete acts in compression but pr

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What do you use your wood retaining wall for?

Without the need for a foundation or permanence, a wooden retaining wall works well for flower beds that compete for a rustic and classic look.

Fences: Wooden retaining walls are specially designed to withstand the lateral pressure of the ground and change ground level. This is ideal for use with fences as it provides adequate underground drainage for the entire garden whilst being more attractive than concrete.

Slopes: Apart from their attractive appearance, wooden retaining walls can prevent an existing garden hill from slipping and also provide excellent support for any new landscaping.

Tools and Conditions

As with most construction work, starting your wood retaining wall without the ideal set of tools will result in a subpar end product. The presence of water plays a big part in this particular project, as areas where water tends to stand or swell will rot your wall remarkably quickly. To avoid this, you’ll need an area of ​​firm gravel, stiff clay and sand or rocky soil, or you’ll need to work in some drainage gravel or pipe to keep your wall from bending.

As a general guide for your first wood retaining wall, we recommend keeping the size at 4 feet. Most walls rely on the following set of materials:

Planed logs in type and size according to your wishes

Perforated plastic drain pipe

Carriage bolts or possibly galvanized nails

drainage gravel

cement

3lb Hammer or Mattock

tape measure

circular saw

level

drill driver

safety goggles

combination square

spade

How to build your wooden retaining wall:

There is no one set of ideal tools for the job, but for the most part the following are considered generally essential: Once you’ve selected your desired area and cleared it of clutter, cut your wood to the desired length and add an additional 18 inches added. Set aside for now.

2. Mark out the exact lengths you will be building your wall at (based on the length of your lumber) and dig two square post holes about 12 feet wide and 2 1/2 feet deep.

3. Fill each hole with approximately 100mm of drainage gravel and place a post in each end and fill to the bottom with concrete. Let absorb.

4. Make sure the other posts are aligned, lay your horizontal timber down and trim further if necessary to keep them in line.

5. Drill an undersized hole where your horizontal post will sit behind the vertical posts (so dirt would then sit behind them) and fasten the two together with the carriage bolts or galvanized nails.6. Now use the same process to stack additional wood to reach the desired height. Lay the drainage pipe behind the horizontal layer of wood and surround it with gravel. Depending on your wall’s purpose, you can now add more soil behind the two layers of wood (if you’re building on a slope) or add more walls at right angles until you have a complete square or rectangle (for flower beds).

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