Revisiting Yttrup and Abramsson’s Limit Analysis Model for Steel Screw Piles in Sand

Abstract

This work stems from the curiosity stimulated by a paper by Yttrup and Abramsson, which appeared in the journal Australian Geomechanics in 2003. Their work proposes a kinematic limit analysis method to compute the ultimate strength of steel screw piles in sand when first the bending and then the plastic collapse of the pile helix occurs. It is accompanied by insightful comments drawn from geotechnical design experience. The paper has both academic and professional impact as it is cited in scientific journals and used in engineering practice in Australia and New Zealand. However, the original paper is quite brief in its exposition. Here, Yttrup and Abramsson’s model is critically reconstructed, providing guidance that can help avoid potential pitfalls in its application. A variation of the model is proposed. Then, the calculated results are discussed and compared with experimental results, starting with those of the original paper. This work hopes to contribute to enhancing the appraisal, adoption, and utility of Yttrup and Abramsson’s model in design practice and in subsequent studies.

1. Introduction

Screw (also referred to as helical) piles are a type of foundation pile, made of steel, usually comprising a hollow circular or square shaft and one or more helical elements welded, at prescribed spacing, at the shaft base. The presence of the helix (or helices) increases the pile base capacity under axial compression loads, and provides a greater pullout capacity under tension (uplift) loads, which would otherwise be resisted by shaft friction only. The helix geometry can vary between different manufacturers, but the general concepts and installation principles of screw piles remain the same.

The first documented use of screw piles dates back to the 19th century, when Alexander Mitchell designed screw piles to support a lighthouse in England [1]. Screw piles are installed by the simultaneous application of a torque and a thrust force, using, for example, an excavator equipped with a hydraulic torque head. Torque installation depends, among other factors, on the pile configuration (helix and shaft size, the number and spacing of helices), ground conditions, the installation advancement rate and speed of rotation, and operator skill (see e.g., [2]). Installation torque logs (i.e., torque vs depth) are used as part of the quality control and assurance process during construction monitoring. Installation torque can also be correlated to the base bearing capacity and used for pile design—refer, for example, to [3] for further details, since capacity–torque correlations are outside the scope of the present work.

This article focuses on single-helix screw piles, installed in sands, subject to axial compression loads. The geotechnical axial compression capacity, $Q_c$, of a single-helix screw pile can be conceptually broken down as the sum of the base (helix), $Q_b$, and shaft, $Q_s$, capacities as follows (Figure 1):

$$Q_c=Q_b+Q_s$$

(1)

The term $Q_b$ is usually the main contributor to the screw pile axial capacity. This is recognized by some geotechnical practitioners, who, during design, conservatively ignore the resistance mobilized along the shaft, at least in part (such as in proximity of the helix). Shaft friction can however provide an important contribution to the overall pile geotechnical capacity, particularly in piles with a large shaft diameter, and its estimate is required for the correct interpretation of pile load test results. For reference, ref. [2] indicates that shaft friction is usually considered in the estimate of the capacity of piles with shaft diameters exceeding approximately 89 mm (i.e., $3.5$ inches).

It should be noted that the definition of ultimate capacity is associated with a value of limit pile displacement (settlement) corresponding to failure. Shaft friction is fully mobilized at small displacements $\delta$ (e.g., $0.3$–$1.0$% of the shaft diameter D, [4]), while the mobilization of the ultimate base capacity requires significantly greater displacements. Different criteria have been proposed in the literature and in design codes to define the ultimate bearing capacity from the results of a compression pile load test (see, e.g., [5,6,7]). For screw piles, a conventional reference value often adopted is $\delta =0.1D$ (see, e.g., [8]). For large helix diameters, this value may exceed the displacement capacity of a given structure, and a structural engineer may therefore indicate a smaller ultimate displacement value.

For the design of piles in sands, starting from an estimate of soil parameters, the term $Q_b$ can be obtained, as for other pile types, using the bearing capacity equation with an appropriate bearing capacity factor $N_q$ (see for example [3] or [4]) as follows:

$$Q_b=A_H·N_q·{\sigma }_{v,H}^{\prime }$$

(2)

where ${\sigma }_{v,H}^{\prime }$ is the vertical effective stress at the helix depth and $A_H=\pi R^2$ is the area of the helix (plus shaft), with radius R. The term $N_q·{\sigma }_{v,H}^{\prime }$ represents the average ultimate bearing pressure under the helix, $q_b$.

The shaft friction can be obtained as

$$Q_s=\sum _{i}2\pi ·s·q_{s,i}·L_i$$

(3)

where s is the pile shaft radius, subscript i refers to the i-th soil layer, $q_{s,i}$ is the shaft friction estimated in a given layer, and $L_i$ is the layer thickness. Shaft friction can be computed from estimated values of horizontal effective stress and soil internal friction angle at a given depth.

Instead of performing an indirect estimate of the shaft friction and bearing pressure, different authors have proposed direct correlations between these parameters and the results of in situ tests. Such correlations have been proposed for other pile types (see, e.g., [9,10,11,12]). Given its relevance for the present work, reference will be limited to correlations based on the cone penetration test (CPT). The correlation between CPT tip resistance, $q_c$, and ultimate (average) base resistance, $q_b$, can be written in the following form:

$$q_b={\alpha }_b·q_c$$

(4)

while the correlation with the shaft resistance can be expressed in the following form:

$$q_s={\displaystyle \frac{q_c}{{\beta }_c}}$$

(5)

where ${\alpha }_b$ and ${\beta }_c$ are empirical correlation factors. Bustamante and Gianeselli [9], following a review of field test data, suggested for piles in sand ${\alpha }_b$ = 0.30–0.40 for bored piles, and 0.40–0.50 for driven piles, with the lower values corresponding to denser soil conditions. The authors also estimated ${\beta }_c$ values ranging from 30 to 300 for piles in sand, depending on pile type and soil density. In a recent study focused solely on screw piles, Bittar et al. [8], based on the analysis of a large number of pile load tests, proposed ${\alpha }_b=0.2$ at $\delta /D=0.1$, and ${\beta }_c=230$. In their derivation of the model subject of the present paper, Yttrup and Abramsson (2003) [13] assumed ${\alpha }_b=0.3$, albeit with a non-uniform pressure distribution, as it will be discussed in the next sections.

Figure 1. Illustration of Yttrup and Abramsson’s model: (a) pile geometry showing rotation of helix around plastic hinge location relative to the pile shaft; (b) bearing pressure distribution for “strong” and “weak” helix mechanisms in the original model [13]; (c) bearing pressure distribution for “strong” helix mechanism for the proposed model; (d) displacement field (both original and proposed model).

Figure 1. Illustration of Yttrup and Abramsson’s model: (a) pile geometry showing rotation of helix around plastic hinge location relative to the pile shaft; (b) bearing pressure distribution for “strong” and “weak” helix mechanisms in the original model [13]; (c) bearing pressure distribution for “strong” helix mechanism for the proposed model; (d) displacement field (both original and proposed model).

The present work considers the analytical model proposed by Yttrup and Abramsson [13], which estimates the geotechnical compression capacity of a single-helix pile installed in sand when failure occurs due to a plastic collapse mechanism in the helix. In their work, Yttrup and Abramsson considered the influence of the deformation and yielding of the pile helix on the pressure distribution under the helix and the resulting bearing capacity.

Other design approaches, whether based on the bearing capacity Equation (2) or on correlations from in situ testing, assume a uniform bearing pressure under the helix. If not suitably tuned, the use of a uniform pressure distribution to design the helix as a cantilever against yielding can result in a very large helix thickness, inconsistent with helix thicknesses effectively and successfully employed in practice.

Assuming a trapezoidal soil pressure distribution under the helix and using a kinematic limit analysis approach, Yttrup and Abramsson [13] are able to estimate the pile base bearing capacity. The main merit of this model is that it offers a rational approach for the structural design of the pile helix and specifically, for the determination of the helix thickness. This in turn allows to avoid the instance of a “weak helix” mechanism, when the premature development of the plastic hinge in the helix prevents the full mobilization of the geotechnical base capacity of the pile.

This work began with a review of Ytrrup and Abramsson’s original study, which appeared to contain some inconsistencies in its editorial and graphical presentation, together with parts of the model derivation and its application left to the readers. This prompted a critical analysis which consisted of re-deriving the equations of the model, reconstructing aspects not explicitly mentioned, and eventually proposing an alternative, extended version of the model. Finally, the calculated axial capacities were compared with experimental results, providing the basis to discuss and assess the model’s capabilities and limitations.

2. Yttrup and Abramsson’s Model

The 2003 model by Yttrup and Abramsson [13] originated from the observation that, in many cases, steel screw piles failed before reaching the maximum strength of the foundation due to a collapse of the helix of the pile.

A model based on limit analyis is hence proposed in [13] to estimate the failure load of the helix and, consequently, of the pile.

2.1. Geometry and Initial Assumptions

A detailed illustration of Yttrup and Abramsson’s model [13] is shown in Figure 1. Each aspect is separately treated in the following sections.

Note that the notation adopted in the present work is in some instances different from that of the original publication [13].

2.1.1. Geometry

Figure 1 represents schematically a longitudinal section of a screw pile. The problem is axisymmetric. To simplify calculations, the helix is substituted by a circular plate attached to the shaft. The thickness of the plate is t. Distances $\rho$ are measured from the central axis of the pile: s is the external radius of the pile and R is the radius of the helix.

The helix fails by rotating upwards due to the formation of a circular plastic hinge line located at a distance r from the axis. In [13], Yttrup and Abramsson perform bending tests up to plastic collapse on a steel cantilever welded to a prismatic steel support. They record the position of the plastic hinge for different thicknesses of the cantilever. They report a distance of the plastic hinge from the support edge of $3t$ for $t=10$ mm, and a distance of $2t$ for $t=20$ mm.

The actual value adopted for r is not explicitly mentioned in [13], but the experimental observations can be translated as $r=s+3t$ when $t=10$ mm and $r=s+2t$ when $t=20$ mm and be used in subsequent calculations.

The length of the pile from the ground surface to the helix is L.

2.1.2. Loads

The pile is subject to a compressive force P, which at failure equals the ultimate pile capacity, $Q_c$, in vertical equilibrium with the shaft friction acting along the lateral surface of the pile and with vertical end bearing stresses, i.e., soil reaction, acting below the helix and below the bottom end of the pile.

In their work, Yttrup and Abramsson [13] assumed a non-uniform bearing stress distribution under the helix. The reference stress value f shown in Figure 1 was estimated by the authors as a function of the CPT cone tip resistance, $q_c$, as

$$f=0.3q_c$$

(6)

The bearing stress attains a value equal to $2f$ under the central section of the pile, i.e., under the shaft, and it is equal to f between the shaft and the plastic hinge, i.e., $s<\rho <r$. Yttrup and Abramsson [13] mentioned that results of finite element analyses supported their assumed stress distribution. Those analyses are not presented in [13]; however a recent work by Ho et al. [14] substantially confirms the assumptions made by Yttrup and Abramsson [13].

The bearing stress distribution beyond the plastic hinge, i.e., for $\rho >r$, is different for “weak” and “strong” helices. The distinction depends on the bending stiffness of the helix which in turn depends on the ratio $(R-s)/t$ between how much the helix plate stands out of the shaft and the thickness of the plate.

The larger deflection and premature yielding of “weak” helices reduces the bearing stresses mobilized in the underlying sand. This is captured in a simplified way in [13] by assuming a stress distribution that decreases linearly from f in $\rho =r$ to 0 in $\rho =a$, with $a<R$, and assuming that it is equal to zero for $a<\rho <R$. Stiffer, “strong” helices are instead assumed to have $a=R$ with a bearing stress distribution linearly decreasing from f in the plastic hinge at $\rho =r$ to 0 in $\rho =R$.

In Figure 1 the bearing stress distributions proposed in [13] are shown in blue below the $\rho$ axis: the one for “weak” helices is on the right side of the axis, while that for “strong” helices is shown on the left.

2.1.3. Mechanism

The mechanism, shown in green at the bottom of Figure 1, is the superposition of a vertical downward translation ${\Delta }_S$ of the whole pile, i.e., both shaft and helix, and an upward rigid rotation $\vartheta$ of the external part of the helix beyond the plastic hinge line. The vertical downward displacement at the edge of the helix is therefore ${\Delta }_S-\Delta$ with

$$\vartheta ={\displaystyle \frac{\Delta }{R-r}}$$

(7)

In [13], ${\Delta }_S$ and $\Delta$ are actually assumed to be equal. The difference is, in the end, irrelevant because what ultimately matters is the rigid rotation $\vartheta$ only and not the vertical translation ${\Delta }_S$. The distinction between ${\Delta }_S$ and $\Delta$ in Figure 1 is introduced precisely to underscore this fact.

2.2. Model Equations from 2003

Yttrup and Abramsson [13] identify four “components” of base resistance:

• $R_1$: bearing stress at the end of the shaft, i.e., 0 ≤ $\rho$ ≤ s;
• $R_2$: helix plate bearing between the pile shaft and the plastic hinge, i.e., $s<\rho$ ≤ r;
• $R_3$: bearing stress beyond the plastic hinge, i.e., $\rho >r$;
• $R_4$: helix plate bending.

All these components have the dimensions of a force. While their effective meaning can appear unclear upon reading of [13], $R_1$, $R_2$, and $R_3$ are understood to represent the resultants of the bearing stresses below the corresponding portions of the shaft and of the plate helix.

Subsequently, three equations are introduced. First the internal work $W_{\mathrm{int}}$ is calculated as

$$W_{\mathrm{int}}=R_4·\Delta ={\gamma }_{\mathrm{helix}}{\gamma }_{\mathrm{hard}}2\pi {\displaystyle \frac{R}{R-r}}{m}_y\Delta$$

(8)

with $m_y$, the limit plastic moment per unit length of the plate, given by

$$m_y=f_{sy}{\displaystyle \frac{t^2}{4}}$$

(9)

and $f_{sy}$ being the yield strength of the plate. Couples per unit length $m_y$ are indicated in red in Figure 1. ${\gamma }_{\mathrm{helix}}$ and ${\gamma }_{\mathrm{hard}}$ are two empirical correction factors introduced by Yttrup and Abramsson as follows [13]:

$$\begin{array}{cc}\hfill {\gamma }_{\mathrm{helix}}& =0.8\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\gamma }_{\mathrm{hard}}& ={\displaystyle \frac{1}{4}}{\displaystyle \frac{R-s}{10t}}+{\displaystyle \frac{3}{4}}\hfill \end{array}$$

(11)

with R being the helix radius, s being the pile shaft radius, and t being the helix thickness, as previously introduced in Section 2.1.1. Factor ${\gamma }_{\mathrm{helix}}$ is assumed to be smaller than 1 to account for the difference between the actual helix geometry and the horizontal plate to which the helix has been simplified for the purposes of this analysis. In particular, the helix has stress-free leading and trailing edges where there is no plastic strain, while the plate has no such edges. The helix is thus expected to have a lower internal work w.r.t. the one calculated for a plate in (8). Factor ${\gamma }_{\mathrm{hard}}$ accounts for the hardening observed and measured in the previously mentioned bending experiments on a cantilever welded to a prismatic steel support. The idea is that the weaker the helix, the higher the ratio $(R-s)/t$, the larger the deflection of the helix, and the more pronounced the hardening effect. From (11) it is evident that ${\gamma }_{\mathrm{hard}}$ is 1 when $(R-s)=10t$, and that it is larger than 1 for weaker helices. Equations (10) and (11) were calibrated by Yttrup and Abramsson [13] based on the aforementioned cantilever bending tests and on a back analysis of the same static load tests on steel screw piles used in [13] as validation for the proposed model.

The second equation of the model is the external work $W_{\mathrm{ext}}$, calculated as follows:

$$W_{\mathrm{ext}}=R_4·\Delta =2\pi f·{\displaystyle \frac{\Delta }{R-r}}·{\displaystyle \frac{1}{12}}·{\displaystyle \frac{a^4-2a^{3}r+2a{r}^3-r^4}{a-r}}$$

(12)

The internal and external work in Equations (8) and (12) are equalled to the product $R_4\Delta$. Component $R_4$ can then be interpreted equivalently as $W_{\mathrm{int}}/\Delta$ or $W_{\mathrm{ext}}/\Delta$.

The third equation is based, according to [13], on vertical equilibrium.

$$R_3+R_4={\displaystyle \frac{\pi }{3}}f(2r+a)(a-r)$$

(13)

As it will be shown, the right-hand side of this latter equation represents the resultant of the bearing stresses beyond the plastic hinge ($\rho >r$), so the appearance of $R_4$ in (13) is not so clear.

There are no more equations in the model in [13] and the derivation of Equations (8), (12), and (13) is not provided. Only the following points are added:

• Base resistance components $R_1$ and $R_2$ can easily be computed and added to $R_3$ and $R_4$.
• Equations (12) and (13) can be solved for the distance a.
• The authors used an iterative spreadsheet procedure to find the unknown a.

2.3. Derivation and Reconstruction

2.3.1. Derivation

Given the unclear or incomplete aspects already mentioned thusfar and to be further discussed in the following, in order to understand and assess Yttrup and Abramsson’s model [13], Equations (8), (12), and (13) are derived below. Appendix A details the functional forms of the bearing stress field $\sigma \left(\rho \right)$ and displacement field $v\left(\rho \right)$, shown in Figure 1, necessary to derive the model equations.

The internal work $W_{\mathrm{int}}$ is given by the product of the limit plastic moment per unit length $m_y$ (see (9)) for the rotation $\vartheta$ given by (7) integrated along the plastic hinge line at $\rho =r$. With the addition of correction factors ${\gamma }_{\mathrm{helix}}$ and ${\gamma }_{\mathrm{hard}}$, the internal work is given by

$$W_{\mathrm{int}}={\gamma }_{\mathrm{helix}}{\gamma }_{\mathrm{hard}}2\pi {\displaystyle \frac{r}{R-r}}{m}_y\Delta .$$

(14)

The external work is given by

$$W_{\mathrm{ext}}=2\pi {\int }_r^a{\sigma }_3\left(\rho \right)v_3\left(\rho \right)\rho \mathrm{d}\rho ={\displaystyle \frac{\pi }{6}}f{\displaystyle \frac{\Delta }{R-r}}{\left(a-r\right)}^2\left(a+r\right),$$

(15)

where the subscript 3 refers to the bearing stress and displacement fields for $\rho >r$, that is, outside the radius of the plastic hinge (refer to Appendix A). The expression of the external work $W_{\mathrm{ext}}$ in (15) can be shown after some standard manipulation to coincide with that in (12).

The resultant forces of the bearing pressure distributions under the pile shaft ($\rho <s$), between the pile shaft and the plastic hinge ($s<\rho <r$), and outside the plastic hinge ($\rho >r$), are $Q_1$, $Q_2$ and $Q_3$, respectively, and are given by

$$\begin{array}{cc}\hfill Q_1& =4\pi f{s}^2\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill Q_2& =2\pi f(r^2-s^2)\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill Q_3& =2\pi {\int }_r^a{\sigma }_3\left(\rho \right)\rho \mathrm{d}\rho ={\displaystyle \frac{\pi }{3}}f(2r+a)(a-r)\hfill \end{array}$$

(18)

In particular, the right-hand sides of Equations (13) and (18) coincide.

The equations of Yttrup and Abramsson’s model [13] are thus confirmed with some minor caveats.

First, Equations (8) and (14) do not coincide due to an evident typo: the R in the numerator in (8) should be replaced by r.

Second, it is evident from the derivation of (15) and (18), and from the fact that they coincide with (12) and (13), that a is defined as the distance from the pile axis to the point where the bearing stress is zero, as shown in Figure 1. Unfortunately, likely due to a labeling error in ([13], Figure 6b), a appears in the original work defined as the distance from the plastic hinge to the point of zero bearing stress. The only other reference to a in the text in [13], the distance “a” defines the area over which base pressure can act, does not help to dispel the ambiguity.

There is another minor graphical inaccuracy in ([13], Figure 7b) where the center of the rigid rotation appears to be at the external radius of the pile, i.e., $\rho =s$ as if $r=s$ had been assumed. In the absence of information on the choice for r or of an expression for $v_3$, this may be misleading, at least at first reading.

Finally, it is not Equations (12) and (13) that have to be solved for a as stated in ([13], Section 3.2). The solution for the unknown a comes from equating the expression of the internal virtual work in (14), namely the corrected version of (8), with the expression of the external virtual work in (12) or (15).

2.3.2. Reconstruction

This section completes the reconstruction of Yttrup and Abramsson’s model by addressing some aspects not explicitly mentioned in [13].

First, there is some latitude in the definition of geometric parameter r. Based on what is reported in Section 2.1.1,

$$r=s+t+t_0\mathrm{with}{t}_0=20\mathrm{mm},$$

(19)

is assumed. Equation (19) linearly interpolates exactly the experimental data in [13], i.e., $r-s=30$ mm for $t=10$ mm and $r-s=40$ mm for $t=20$ mm.

The other parameters f, $f_{sy}$, R, s, t being set, the unknown distance a where the bearing pressure attains zero value is determined by equating the right-hand sides of (14) and (15),

$$12{\gamma }_{\mathrm{helix}}{\gamma }_{\mathrm{hard}}r{m}_y=f{\left(a-r\right)}^2\left(a+r\right)$$

(20)

leading to a cubic equation in a. In a limited number of cases, (20) returns a solution a larger than the radius of the helix R. This is an aspect not addressed in [13], but as shown in Section 3, it occurred in one instance even when simulating Yttrup and Abramsson’s own experiments. In the present analysis, when $a>R$, it has been decided to simply enforce $a=R$, even if this violates the virtual work of Equation (20). The alternative of proceeding with $a>R$ would as well have been hardly justifiable from a physical standpoint, as it would have implied bearing stresses extending beyond the outer edge of the helix.

Once a is known, it is possible to compute the geotechnical axial compression capacity of the pile using (1). The base capacity, $Q_b$, is calculated as follows:

$$Q_b=Q_1+Q_2+Q_3$$

(21)

and is simply the sum of the bearing pressure resultants given by (16), (17), and (18). The shaft friction contribution is computed according to (5) from the CPT cone tip resistance, $q_c$, and the pile shaft length, L, and radius, s, as follows:

$$Q_s=2\pi sL{q}_s=2\pi sL{\displaystyle \frac{q_c}{{\beta }_c}}\mathrm{with}{\beta }_c=230$$

(22)

adopting the value of ${\beta }_c$ proposed by Bittar et al. [8].

2.4. A Proposed Variation

Here a variation of the reconstructed Yttrup and Abramsson [13] model of Section 2.3.2 is proposed. It comprises two small changes. The first one consists of setting

$${\gamma }_{\mathrm{helix}}=1\mathrm{and}{\gamma }_{\mathrm{hard}}=1$$

(23)

thus excluding the two correction factors (see Equations (10) and (11)), so as to assess their effect on the computed axial compression capacity $Q_c$.

The second change concerns cases in which the cubic Equation (20) returns a solution a larger than the radius of the helix R. Both alternative approaches considered in Section 2.3.2 are not physically sound, even the one eventually adopted for the reconstructed model. A possible option would be to consider a uniform (rectangular) bearing stress distribution, of magnitude f, applied to the area beyond the plastic hinge ($r<\rho \le R$). The bending moment due to such distribution is obviously greater than the bending moment due to the stress distribution for the “weak” helix when $a=R$. The exact ratio between the two bending moments depends on the values of plastic hinge radius r and helix radius R. As a reference, for the piles which will be later introduced in Section 3 to validate the models, the ratio between these two bending moments (i.e., uniform distribution over weak helix distribution when $a=R$) attains values around $3.5$. Practically speaking, using a uniform pressure distribution to design a pile helix would result in a very large helix thickness, inconsistent with helix thicknesses effectively and successfully employed in practice. The bearing stress distribution beyond the plastic hinge for the strong helix case would therefore need to be comprised between the uniform and the weak helix with $a=R$ stress distributions. As a first approximation, the stress distribution shown in Figure 1c is adopted in the following as part of a proposed model. The bearing stress decreases linearly from f at $\rho =r$ to $f_R$ at the outer edge of the helix ($\rho =R$), with $0<f_R<f$. The analytical expression of this pressure distribution is reported in Appendix A. This pressure distribution retains the conceptual simplicity of the weak helix model, being described in terms of a single parameter ($f_R$), and takes into account the numerical results of Ho et al. [14], which show a non-zero bearing stress at the outer edge of the helix. The expression of the external work changes accordingly

$$W_{\mathrm{ext}}=2\pi {\int }_r^R{\sigma }_{3R}\left(\rho \right)v_3\left(\rho \right)\rho \mathrm{d}\rho ={\displaystyle \frac{\pi }{6}}\Delta (R-r)\left(f(r+R)+f_R(r+3R)\right).$$

(24)

By equating the internal work (14) and the external work (24), a linear equation in the unknown $f_R$ is obtained. The solution is

$$f_R={\displaystyle \frac{1}{r+3R}}\left({\displaystyle \frac{12m_{y}r}{{(R-r)}^2}}-f(r+R)\right),$$

(25)

which can be used to compute the resultant of the bearing stress beyond the plastic hinge

$$Q_{3R}=2\pi {\int }_r^R{\sigma }_{3R}\left(\rho \right)\rho \mathrm{d}\rho ={\displaystyle \frac{\pi }{3}}(R-r)\left(f(2r+R)+f_R(r+2R)\right)$$

(26)

and the base capacity

$$Q_b=Q_1+Q_2+Q_{3R}$$

(27)

Finally, the axial compression capacity of the pile is evaluated using (1).

3. Comparison with Experimental Results

The first experiments to be considered are the eight tests appearing in Yttrup and Abramsson’s paper [13]. This is a necessary starting point to validate the reconstructed model and test the proposed variation against both the experimental and the computed results reported in [13]. A few other published test results on single-helix piles installed in sands are taken into account to broaden, albeit limitedly, the examination of the model’s behavior: one appeared in Gavin et al. [15] and four are taken from Li and Deng [16]. For each test, the data adopted to calculate the axial capacity of the piles are given in

Table 1. Data used for the simulation of experimental results.

#	Source	L	s	R	t	${\mathit{q}}_{\mathit{c}}$	${\mathit{f}}_{\mathit{sy}}$
		(m)	(mm)	(mm)	(mm)	(MPa)	(MPa)
1	Yttrup and Abramsson [13]	4.00	44.5	200.0	10.0	15.5	364
2	Yttrup and Abramsson [13]	4.00	44.5	200.0	12.0	15.5	355
3	Yttrup and Abramsson [13]	4.00	44.5	200.0	16.0	15.5	301
4	Yttrup and Abramsson [13]	4.00	44.5	200.0	20.0	15.5	288
5	Yttrup and Abramsson [13]	4.00	69.5	300.0	10.0	15.6	364
6	Yttrup and Abramsson [13]	4.00	69.5	300.0	12.0	15.6	355
7	Yttrup and Abramsson [13]	4.00	69.5	300.0	16.0	15.6	301
8	Yttrup and Abramsson [13]	4.00	69.5	300.0	20.0	15.6	288
9	Gavin et al. [15]	2.60	55.0	200.0	13.0	16.0	350
10	Li and Deng [16]	1.83	36.5	152.5	9.5	9.2	350
11	Li and Deng [16]	1.83	36.5	152.5	9.5	9.5	350
12	Li and Deng [16]	2.44	44.5	178.0	9.5	7.1	350
13	Li and Deng [16]	2.44	44.5	178.0	9.5	7.9	350

. The selection of data not explicitly provided in the original studies, especially $q_c$, is detailed in the following sections.

3.1. Comparison with Ytrrup and Abramsson (2003)

Two values of steel yield strength $f_{sy}$ are provided in ([13], Table 1) for different plate thicknesses: one coming from mill certificates and the other from bending tests. It is not clear which ones were used for the calculations in [13]. The values from the bending tests have been adopted here.

Four CPT profiles are shown in ([13], Figure 10). In the present work, they have been digitized using a dedicated piece of software (WebPlotDigitizer. Version 5.2) [17]. Then, for each CPT profile, the average value of $q_c$ in a zone extending a distance of $2R$ above and below the depth of the helix ($4R$ in total) has been computed. The average of the four values thus obtained is the adopted value of $q_c$ used in the calculations and shown in Table 1. The value of $q_c$ for piles with a helix radius of 200 mm is different from that of piles with a 300 mm helix radius, albeit in this case negligibly, precisely because of the different depth interval on which the $q_c$ values are integrated.

3.2. Comparison with Gavin et al. (2014)

In the absence of specific information, the steel yield strength $f_{sy}$ has been set to 350 MPa. The cone tip resistance profiles for 10 different tests are shown in ([15], Figure 2a). For this test, a digitization of the $q_c$ profile has not been performed partly due to the difficulty in discerning between the ten closely overlapping CPT traces in [15]. A value of $q_c$ of 16 MPa has therefore been assumed based on a visual assessment of the published CPT data.

3.3. Comparison with Li and Deng (2019)

Li and Deng [16] performed tests at two sites with different ground conditions. The ground profile at the first site comprised clayey soils, hence the tests for this site are not considered here. The tests performed in sandy soil at the second site are selected for the present analysis. Among the six compression tests performed at the second site, two (“P3C1” and “P3C2”) were excluded because of the high variability in $q_c$ readings around the depth of the installation of the helix ($3.96$ m), indicating the presence of a relatively weak cohesive soil layer under the helix. The CPT results for the cone tip resistance in ([16], Figure 3) were again digitized using [17] and averaged in a zone extending a distance of $2R$ above and below the depth of the helix. The values obtained in this way for each CPT test were then assigned to the screw piles, considering in each case the CPT profile or the CPT profiles closest to a given pile (see [16], Figure 5), as follows: pile 10 in Table 1 (“P1C1” in [16]) used CPT profile number 2, piles 11 and 12 (“P1C2” and “P2C1” in [16]) used average values obtained from CPT 2 and 3, while pile 13 (“P2C2”) used only CPT 3.

The steel yield strength $f_{sy}$ was again assumed equal to 350 MPa.

3.4. Results

The results are shown in

Table 2. Comparison with experimental results. Orig.: original model results in [13]. Rec.: reconstructed model. Prop.: proposed variation. Rec. model results between brackets $\left(\right)$ indicate that the virtual work in Equation (20) is not satisfied. Values highlighted in yellow indicate results for which $a>R$.

#	${\mathbf{Q}}_{\mathbf{c}}$ (kN)				Errors (%)			$\mathbf{a}/\mathbf{R}$		${\mathbf{f}}_{\mathbf{R}}/\mathbf{f}$
Num	Exp.	Orig.	Rec.	Prop.	Orig.	Rec.	Prop.	Rec.	Prop.	Prop.
1	330	325	309	316	−1.5	−6.3	−4.3	0.79	0.80
2	334	395	338	350	18.3	1.1	4.9	0.85	0.88
3	381	395	384	409	3.7	0.9	7.4	0.95	1.00
4	400	449	(417)	491	12.3	4.1	22.6	1.06	1.13	0.27
5	531	484	510	504	−8.9	−4.0	−5.0	0.64	0.63
6	568	529	546	548	−6.9	−3.8	−3.5	0.68	0.69
7	665	615	604	621	−7.5	−9.1	−6.6	0.75	0.77
8	715	773	678	711	8.1	−5.2	−0.6	0.82	0.86
9	420		405	425		−3.6	1.2	0.93	0.98
10	104		(138)	152		32.2	46.5	1.04	1.08	0.16
11	96		(142)	155		47.6	61.9	1.03	1.07	0.14
12	126		(147)	156		16.9	23.5	1.01	1.05	0.08
13	134		161	167		19.9	24.4	0.99	1.02	0.04

and in Figure 2. Appendix C summarizes the results of additional sensitivity analyses performed, for the sake of clarity, with the proposed model to assess the influence of the shaft friction coefficient ${\beta }_c$ (Equations (5) and (22)) and of the position of the plastic hinge through the parameter $t_0$ (Equation (19)). The results of these sensitivity analyses are discussed in Section 4. Some abbreviations are introduced for convenience: Exp. for the total ultimate pile capacities as determined in the load tests reported in each referenced publication; Orig. for the predicted capacities of piles 1–8 as originally reported by Yttrup and Abramsson in [13]; Rec. refers to the reconstructed Yttrup and Abramsson model of Section 2.3.2; and Prop. refers to the proposed variation of the model introduced in Section 2.4.

The information in Table 2 is separated into four groups. Columns 2 to 5 contain the values of the experimental and computed axial capacities $Q_c$. Columns 6 to 8 show the percentage errors between computed and experimental results for each model. For the reconstructed and proposed models, columns 9 and 10 provide the value of the ratio $a/R$ between the estimate of the radius of the mobilized soil reaction a and the radius of the pile helix R (refer to Figure 1). A ratio lower than 1 indicates a “weak” helix. Values of $a/R$ larger than one, corresponding to a “strong” helix, are highlighted in yellow. For “strong” helices, and for the proposed model only, column 11 gives the value of the ratio $f_R/f$ between the edge stress $f_R$ and the reference stress between the shaft and the plastic hinge f.

It is important to note that for the Rec. model, the $Q_c$ values between brackets, corresponding to the highlighted values of $a/R$, indicate instances in which Rec. is not applicable since, despite the fact that $Q_c$ is nevertheless computed by setting $a=R$, the virtual work Equation (20) is not satisfied.

In order to better assess the predictive performance of the model, the results in Table 2 are also plotted in Figure 2. The computed and experimental compression capacities of each test are in the ordinate axis, while the experimental capacities are in the abscissae. Therefore experimental values, in yellow in Figure 2, lie on the bisector. Computed values of the Orig., Rec., and Prop. models are plotted as blue, green and red marks respectively, and are scattered above and below the bisector. Light green is used to denote Rec. model results which violate the virtual work in Equation (20). Different markers are used to distinguish between experimental results: circles for Yttrup and Abramsson [13], squares for Gavin et al., [15] and triangles for Li and Deng [16].

Looking at the plot, two things stand out: first it is evident that there are no experimental results in the 130–330 kN range. Second, Li and Deng’s [16] test results are characterized by large percentage errors in Table 2, but, in fact, in absolute terms, the errors are not dissimilar to those of other tests with a larger (or much larger) compression capacity.

4. Discussion

The results of Yttrup and Abramsson’s experiments [13] are analyzed first. It is first noted that the models based on Yttrup and Abramsson’s approach (original prediction, reconstructed, and proposed models) predict that, for piles installed in similar ground conditions and with the same shaft and helix radii, the axial pile capacity depends on helix thickness. For this reason, the measured pile capacity increases from pile 1 to pile 4, and from pile 5 to pile 8. One can also note that, for the cases of piles exhibiting a “weak” helix response (1–3 and 5–8), the original prediction by Yttrup and Abramsson and the predictions by both the reconstructed and proposed models are all relatively close to the value measured in the pile load tests, with the exception of the Orig. estimate for pile 2, which exceeds the measured pile capacity by $18.3$%.

To evaluate the prediction accuracy of a specific model across multiple tests, the mean absolute percentage error (MAPE), i.e., the mean of the absolute values of the percentage errors, is used. The MAPE of the Orig. model for the first eight rows in Table 2 is $8.4$% and for Prop. is $6.9$%.

To compare the Prop. and Orig. models with the Rec. model (not applicable to the case of pile 4), the MAPEs are recomputed for all three models excluding pile 4. In this case the MAPE for Orig. is $7.8$%, $4.3$% for Rec., and $4.6$% for Prop.

These results primarily confirm the conceptual basis of the Orig. model, with any potentially misleading points or omissions in [13] confined to editorial matters and not affecting the underlying idea. At the same time, the reconstructed model is also validated and it is now possible to apply Yttrup and Abramsson’s model following the reconstruction provided here. The fact that the Prop. model has results relatively close to the Rec. model suggests that the correction factors ${\gamma }_{\mathrm{helix}}$ (for the helix geometry) and ${\gamma }_{\mathrm{hard}}$ (for the hardening observed in the bending test) proposed in [13] may be omitted. These factors, unless calibrated on a large and diverse test sample, may effectively restrict the range of application of the model. The effective influence of the empirical correction factors needs to be further scrutinized against a larger, more diverse experimental data set. In addition, the Prop. model extends the Rec. model to cases with $a>R$. Pile 4 (with $a>R$) has a greater axial capacity than piles 1, 2, and 3 because of its greater helix thickness (20 mm). Both the original model by Yttrup and Abramsson and the proposed model result in an over-prediction of the capacity $Q_c$ of pile 4 by, respectively, 12% and 23%. The information available in Yttrup and Abramsson on the pile load test setup does not allow us to identify specific details which could explain this over-prediction.

It can be noted that the MAPE values of the Rec. and Prop. models are slightly smaller than those of the Orig. model. This presumably depends on the choices made in the derivation and reconstruction phases such as the expression (19) for the position r of the plastic hinge or the expression (22) for the shaft friction. The sensitivity of the results with respect to these two parameters is presented in Appendix C and summarized here.

For all piles, varying the coefficient ${\beta }_c$ for shaft friction (Equations (5) and (22)) over the range 185–300, corresponding to a change in shaft capacity $Q_s$ of about $\pm 25$% compared with that of the adopted value of 230, and exploring a range consistent with values given in Doan and Lehane [12] and Bustamante and Gianeselli [9], resulted in variations of approximately $\pm 6$% or less in predicted capacities.

Sensitivity analyses on the location of the plastic hinge covered the range $t_0=10$–30 mm ($\pm 50$% over the reference value of 20 mm). This corresponds to a range between one and two helix thicknesses for the piles in Table 1. A smaller value of $t_0$ results in a smaller plastic hinge radius r (refer to Equation (19)) and a greater cantilevering length of the helix. In turn, this leads to a larger moment of the bearing stress and to a more conservative estimate of the unknown a and of the base capacity. The results, provided in detail in Appendix C.1, show that the sensitivity can be significant. A change in position of just 10 mm of the plastic hinge amounts to about a 16% change in axial compression capacity $Q_c$ for a helix of radius $R=150$ mm (tests 10–11 in

Table A3. Columns 4 to 7 show the axial compression capacities $Q_c$ of the proposed model for ${\beta }_c=300$ and ${\beta }_c=185$ compared with the experimental results and with the value ${\beta }_c=230$ adopted in Section 2.3.2.

#	${\mathit{Q}}_{\mathit{b}}$ (kN)	${\mathit{Q}}_{\mathit{s}}$ (kN)	${\mathit{Q}}_{\mathit{c}}$ (kN)				Change (%)
Num	${\mathit{\beta }}_{\mathit{c}}\mathbf{=}\mathbf{230}$	${\mathit{\beta }}_{\mathit{c}}\mathbf{=}\mathbf{230}$	Exp	${\mathit{\beta }}_{\mathit{c}}\mathbf{=}\mathbf{230}$	${\mathit{\beta }}_{\mathit{c}}\mathbf{=}\mathbf{300}$	${\mathit{\beta }}_{\mathit{c}}\mathbf{=}\mathbf{185}$	${\mathit{\beta }}_{\mathit{c}}\mathbf{=}\mathbf{300}$	${\mathit{\beta }}_{\mathit{c}}\mathbf{=}\mathbf{185}$
1	240	75.4	330	316	298	334	−5.58	5.81
2	275	75.4	334	350	333	369	−5.03	5.24
3	334	75.4	381	409	392	428	−4.30	4.48
4	415	75.4	400	491	473	509	−3.59	3.74
5	386	118.0	531	504	477	533	−5.46	5.70
6	430	118.0	568	548	520	577	−5.03	5.24
7	503	118.0	665	621	593	650	−4.44	4.63
8	593	118.0	715	711	683	740	−3.88	4.04
9	362	62.5	420	425	410	440	−3.43	3.58
10	136	16.8	104	152	148	156	−2.58	2.69
11	138	17.3	96	155	151	160	−2.60	2.71
12	135	21.2	126	156	151	161	−3.17	3.31
13	143	23.4	134	167	161	172	−3.27	3.41

). The effect is halved for $R=300$ mm (tests 5–8). In general, it appears that the sensitivity is inversely proportional to the helix radius R.

With respect to the remaining tests, the Rec. model is applicable only to tests 9 and 13. For these tests, the small difference between the Rec. and Prop. models is confirmed. For pile 9, the agreement of the predictions of both models with the experimental results is excellent. However, for both models for pile 13, and for the Prop. model only for piles 10–12, the predictions overestimate measured capacities by values between 20% and 60%.

For these same piles, Li and Deng in [16] reported slight overestimates, between 5 and 13% of the observed pile capacities, when applying Chin’s hyperbolic method [6] to estimate $Q_c$ at 10% $\delta /D$. Also, preliminary estimates (not reported here) of the ultimate capacity of piles 10–13 using the UWASP-22 model developed by Bittar et al. [8] turn out to be significantly larger than the experimental values.

It appears therefore that substantially different methods (Chin’s method based on curve-fitting of the pile load test curve, the proposed model based on limit analysis, and Bittar et al.’s empirical equation) all overestimate the capacity of piles 10–13, albeit with different accuracy: Chin’s method (applied by Li and Deng [16]) returns significantly better results than the Prop. model and Bittar et al.’s model [8].

There are chiefly two main reasons for the over-prediction of Li and Deng’s [16] test results. The first is the uncertainty and the variability of the specific ground conditions at the site where piles 10–13 were installed. An inspection of the CPT data presented by Li and Deng [16] indicates relatively large variations ($\pm 50$%) in measured cone tip resistance values around the depth of embedment of the helix of these piles, with large variations also between contiguous CPT profiles at a given depth. This indicates the presence of alternating dense and loose layers and lenses, laterally discontinuous, which makes it difficult to associate a representative value of cone tip resistance $q_c$ necessary both for the proposed model and for Bittar et al. ([8], Equation (6)). Chin’s method parameters, instead, are based on the effective load–displacement response observed in situ, thus introducing a constraint which, at least in this case, results in a better fit of the ultimate pile capacity. In an extreme case, the presence of a loose layer underlying a denser layer could lead to punching failure through the looser layer. The mobilized bearing pressure magnitude and distribution could then depart significantly from that of the reconstructed and proposed models, estimated from a cone tip resistance value averaged over a depth range.

The second reason for the over-prediction of Li and Deng’s [16] test results, which also serves as an essential reminder, is inherent in the nature of the reconstructed and proposed models, based on the kinematic limit analysis approach that determines an upper bound of the collapse load (see e.g., [18,19]) whenever the considered mechanism does not coincide with the actual failure mechanism. In such cases, e.g., weak soil and/or over-designed piles, the predicted pile capacities are bound to be higher than the actual axial capacities. This may also be part of the reason why the largest percentage errors are observed for strong helices ($f_R>0$) in the proposed model.

Finally, it is necessary to point out that the overall good performance of the reconstructed and proposed models observed herein for cases identified as a “weak” helix response should not be unduly generalized. Given the relatively small data set employed for validation, the future validation of these models against a broader data set covering a wider range of pile geometries and ground conditions is necessary for both “weak” and “strong” helix conditions.

5. Conclusions

The present work reviewed a model presented by Yttrup and Abramsson [13] to estimate the ultimate base capacities of screw piles in sand that fail due to a plastic mechanism at the helix. Instances of errors and inconsistencies in the notation of the original publication, likely due to mere typos, were identified and addressed following the re-derivation of the model equations. Several aspects not explicitly addressed in [13] were discussed and incorporated in the formulation of the reconstructed model and of an alternative, proposed model. The original experimental data, together with additional pile load test data available in the literature, were employed to assess the performance of the models.

The reconstructed model resolves the inconsistencies in the original publication, demonstrating that these inconsistencies do not invalidate Yttrup and Abramsson’s original conceptual framework [13].

Comparison with experimental data shows that the proposed variation offers an accuracy comparable to that of the reconstructed model, with the additional benefit of requiring fewer empirical factors. In any case, the support of further comparisons with pile load test data is required to establish practical recommendations in favor of either the proposed or the reconstructed model.

For the proposed model, the agreement with pile load test data was found to be good for piles identified as undergoing a “weak” helix failure mechanism, while it tended to over-predict the capacity of piles such as those tested in the work of Li and Deng [16], which were interpreted to undergo a “strong” helix failure mechanism.

There are two possible reasons for this over-prediction. First, it could be due to the specific ground conditions at the test site, with the presence of alternating loose and dense layers at the depth of the embedment of the pile helices. This would cause a departure from the assumed pressure distribution under the pile helix, prompting caution when applying these methods in the presence of interbedded soil profiles.

The second reason, which is important to emphasize, is that the models in the present work are based on a kinematic limit analysis approach that provides an upper bound of the collapse load whenever the considered mechanism does not coincide with the actual one. These models are therefore best complemented with failure estimates for other mechanisms. It is also important to note though, that even without being complemented, these models are useful for the preliminary sizing of features such as the thickness of a pile.

Finally, even if the comparisons between the models and most experimental results presented in this paper appear encouraging, the performance of the models needs to be further validated against a larger experimental data set. To this end, the clarifications contained in the present work should assist future users in the application and extension of the methodology originally envisaged by Yttrup and Abramsson [13].