The relationship between foliation and strain: an experimental investigation

A multilayered salt/mica specimen with embedded strain markers was shortened to produce a fold and the distribution of strain was subsequently mapped out over the profile plane. On a fine scale the initial foliation, which is parallel to the undeformed layers, is folded by tight kinks to produce two new foliations; one is defined by the preferred orientation of kink boundaries and the other by the preferred orientation of (001) of mica. In the hinge region of the fold the first of these new foliations is parallel to the local λ₁λ₂-principal plane of strain whereas the preferred orientation of mica is bimodal and is symmetrical about the λ₁λ₂-plane. Elsewhere the two new foliations are not parallel to the principal plane of strain and angular divergencies of up to 30–35° are measured. If a March model with initial random mica orientation is assumed for the development of mica preferred orientation then the correct value of strain is predicted but the orientation of the principal plane of strain can be grossly in error. A theoretical analysis of the angular relationships to be expected between kink boundaries and the λ₁λ₂-plane of strain confirms that for the type of geometries experimentally developed, large divergences of up to 35° should be common. In rocks where the foliation has developed by processes similar to those recorded here, large angular divergencies between the foliation and the λ₁λ₂-principal plane of strain should be expected as the rule.     

Seismic behavior in central Taiwan: Response to stress evolution following the 1999 Mw 7.6 Chi-Chi earthquake

Following the 1999 M_w 7.6 Chi-Chi earthquake, a large amount of seismicity occurred in the Nantou region of central Taiwan. Among the seismic activities, eight M_w ⩾ 5.8 earthquakes took place following the Chi-Chi earthquake, whereas only four earthquakes with comparable magnitudes took place from 1900 to 1998. Since the seismicity rate during the Chi-Chi postseismic period has never returned to the background level, such seismicity activation cannot simply be attributed to modified Omori’s Law decay. In this work, we attempted to associate seismic activities with stress evolution. Based on our work, it appears that the spatial distribution of the consequent seismicity can be associated with increasing coseismic stress. On the contrary, the stress changes imparted by the afterslip; lower crust–upper mantle viscoelastic relaxation; and sequent events resulted in a stress drop in most of the study region. Understanding seismogenic mechanisms in terms of stress evolution would be beneficial to seismic hazard mitigation.     

An integral-free expression for short-term changes in fault stability due to pore pressure induced when a point load is placed on the pervious boundary of a porous elastic half space containing a fault

A concentrated load with step-function time behaviour is placed normal to the planar, pervious boundary of a porous elastic half space (PEHS) with compressible constituents. A planar fault exists in the PEHS in such a way that the poroelastic behaviour of the medium is unhindered. We derive an approximate but integral-free expression for CFSCPP, i.e., changes in fault stability due to changes in pore pressure, at a point not too far off the line along which the load acts. But, in the interest of simplicity, the main discussion is focussed on a consideration of CFSCPP at a point P located on the fault at depth z directly beneath the load. It is convenient to introduce dimensionless time t _D directly proportional to real time t. The constant of proportionality is 4c/z ², where c is hydraulic diffusivity. The derived approximate expression gives results with an accuracy of greater than 99% for limited values of t _D after the load is imposed. We learn from the derived expression that, for a given z, fault stability undergoes an initial sudden decrease commensurate with the undrained pore pressure induced in the PEHS. This is followed by a more gradual decrease in fault stability with increasing t _D until a minimum is reached. The real time t to minimum fault stability increases with z. The magnitude of CFSCPP decreases with z as z ^?2 for a given t _D in the permissible range. The derived expression and the inferences based on it should be useful during earth science investigations of the possible hazards due to reactivation of a pre-existing shallow fault when a civil engineering project involving imposition of a heavy load on the earth’s surface is to be executed nearby. They should be useful also for investigations if a shallow earthquake occurs near such a project soon after its execution.     

Anisotropic growth in the olivine-spinel transformation of Mg2GeO4 under nonhydrostatic stress

The olivine-spinel phase transformation in Mg₂GeO₄ does not occur by a martensitic mechanism. The evidence, from samples transformed in a Griggs-type solid medium deformation apparatus, are:

1. (1) lack of microstructural features in the olivine phase which can be specifically associated with a martensitic mechanism

2. (2) the orientation relationship between the two phases that is predicted by the martensitic mechanism does not occur nor is there any apparent consistency of relative orientations

3. (3) application of a differential stress to the transforming sample resulted in an anisotropic growth rate for the spinel phase indicating that growth was externally controlled rather than crystallographically controlled.

Anisotropic growth of the spinel phase results in elongation of the residual olivine phase grains in the plane normal to the direction of maximum principal compressive stress. A velocity ratio of 1.7_−0.7^+5.4 has been determined for the growth rate of the spinel from measurements on residual olivine grains. The interphase grain boundary in samples transformed under stress has cusp-shaped fingers of spinel with a blunt end separated by thin spikes of olivine. Samples transformed isostatically do not exhibit this feature providing further confirmation of anisotropic growth of the spinel. The preferred growth of the spinel is consistent with a theory of phase transformation under nonhydrostatic stress. The predicted spinel finger shape based on this theory is generally consistent with observed shapes except for the blunt end. The discrepancy may be due to surface energy which has not been considered here, or to local deviations of the applied macroscopic stress.     

Microcracks in granite cores from the EPS-1 geothermal drill hole, Soultz-sous-Forêts (France): paleostress directions, paleofluids and crack-related Vp-anisotropies

For the modelling of Hot Dry Rock systems the interactions between macrofractures, prominent microcrack sets and the in situ stress field is of fundamental importance. In this study complete 3D analyses of microcrack orientations were carried out on granitoid samples from the Soultz geothermal borehole (EPS-1). The crack population is dominated by healed cracks in quartz forming three orthogonal sets (I, II, III) with strong preferred orientation, which probably result from superposition of internal thermal stresses during cooling and external tectonic stresses. The inferred orientation of paleo-σ_H is NE–SW. Based on the microthermometry of secondary fluid inclusions it can be assumed that cyclic crack/healing events occurred at P/T conditions roughly ranging between 1 and 2?kbar and 200–400?°C. A younger generation of open cracks in quartz which coincide with macroscopic fractures can be correlated with the Tertiary N–S direction of σ_H. The distribution of P-wave velocities (Vp) was measured by using the pulse transmission technique in 132 propagation directions at confining pressures up to 200?MPa. These Vp data and their variation with confining pressure are a suitable tool for detecting prominent crack patterns and to estimate the crack-controlled in situ properties. The observed bulk Vp anisotropy roughly reflects the composite microcrack fabric and the relative importance of individual sets. It can be assumed that the prominent crack sets represent planes of weakness and hence may cause a mechanically anisotropic behaviour dependent on their orientation with respect to the recent in situ stresses.     

The use of the orientation tensor for the description and statistical testing of fabrics

A measure of the degree of preferred orientation of directional data defined by Woodcock as the ratio of the maximum and minimum eigenvalues of the orientation tensor (S₁/S₃) has a disadvantage when used as the statistic for a test of uniformity. This drawback arises because the distribution of S₁/S₃ under random sampling is not independent of the shape or K value of the sample. An alternative strength parameter also based on the eigenvalues, is proposed.     

Experimental analysis of kaolinite particle orientation during triaxial path

The goal of this study was to analyze the relation between the behaviour of a clayey material at the macroscopic scale and its microfabric evolution. This may lead to a better understanding of macroscopic strain mechanisms especially the contractancy and dilatancy phenomena. The approach proposed in this paper is based on the study of clay particles orientation by SEM picture analysis after different phases of triaxial loading. In the initial state of the samples (one‐dimensional compression), the SEM observations highlight a microstructural anisotropy with a preferential orientation of the particles normal to the loading direction. During isotropic loading, densification of the clayey matrix occurs related to a random orientation of particles indicated by the term ‘depolarization’. In the earlier stages of constant σ₃ drained triaxial path on slightly overconsolidated specimens, the microstructural depolarization seems to persist inside a macroscopic domain, in which only the volumetric strains due to the isotropic part of the stress tensor evolve. Then, a rotation mechanism of the particles towards preferred directions seems to be activated. The phenomenon appears directly linked to the evolution of the deviatoric part of the stress tensor. Copyright © 2010 John Wiley & Sons, Ltd.     

The relationship between foliation and strain: an experimental investigation: reply

Contrary to the criticisms of our ideas by Treagus, we still believe that our experiments show that a secondary preferred orientation of micas or of small kink-like folds, closely resembling similar features in some cleaved rocks, can form at large angles to the λ₁λ₂ plane of strain, where the strain referred to is the strain coeval with development of these features.     

Preferred orientation in experimentally deformed limestone

Over sixty syntectonic deformation experiments in uniaxial compression have been done on fine-grained limestones in the stability fields of calcite I, calcite II and aragonite. X-ray techniques and spherical harmonic analysis of the data were used to determine preferred orientation quantitatively, and inverse pole-figures were derived for these axially symmetric specimens. They display in most cases strong preferred orientation which varies as a function of the experimental conditions, mainly temperature and pressure. At temperatures below 350° C recrystallization is lacking and flattened grains indicate that translation, twin gliding and kinking have been the dominant deformation mechanisms. The inverse pole-figure shows a maximum at c with a shoulder towards or a second maximum at e. This is in agreement with preferred orientation observed in experimentally deformed Yule marble and can be explained as the product of dominant twin gliding on e and translation gliding on r (Turner et al., 1956). At high temperatures (900–1000° C) strong grain growth (from 4 to 50 microns) indicates that the fabric recrystallized. Grains are equidimensional and clear with a marble-like texture. The inverse pole-figure shows a single maximum at r, and c-axes are oriented in a small circle around the axis of compression, ₁. Such a pattern of preferred orientation would be expected on thermodynamic grounds assuming that recrystallized grains will be oriented in such a way that the strain energy is a maximum (e.g. MacDonald, 1960). Decrease in confining pressure caused a decrease of the maximum at c and the formation of a secondary maximum at highangle positive rhombs in the inverse pole-figure. This can be interpreted as r translation dominating over e twinning. In all deformation experiments an equilibrium in preferred orientation was reached after 20 percent shortening. The strength of preferred orientation decreased with increasing temperature. Aragonite was produced within its hydrostatic stability field at temperatures above 500° C. Close to the phase boundary, coarse-grained textures showed preferred orientation with poles to (010) parallel to ₁. At higher pressures the fabric is fine-grained and [001] is aligned parallel to ₁. Evidence is given that the phase change from calcite to aragonite in these deformation experiments is a diffusive and not a martensitic transformation.Publication No. 1043, Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California.     

X射线衍射旋转撒样法分析氟金云母多型组成与含量

合成氟金云母多型种类与含量对云母的物理化学性质具有重要的影响。然而在X射线粉晶衍射(XRD)制样过程中云母00l基面极易产生择优取向,严重制约了云母多型组成和含量的分析。传统撒样法可促使晶体取向随机分布,但制备的试样表面不够平坦。本文对传统撒样法进行改进,在撒样过程中使样品架均匀旋转,从而获得表面平坦的试样。XRD测试结果表明,旋转撒样法取向指数(OI=I_(001)/I_(060))为3.9,与无择优取向的理论值4.5接近,明显优于正压法和侧装法(OI值分别为38.7和18.1),表明旋转撒样法能够显著减弱云母择优取向。这主要是由于旋转撒样法使晶体颗粒之间形成犬牙交错分布,提高了云母各晶面随机分布概率。Rietveld全图拟合分析显示,旋转撒样法获得的XRD数据精修效果较好,计算出本文合成的氟金云母样品中1M和2M_1多型含量分别为86%和14%,8个工业合成的氟金云母样品中1M和2M_1多型含量分别为57%～72%和28%～43%,并且存在较多的堆垛层错。总之,旋转撒样法减弱择优取向效果显著,为研究云母晶体生长、多型成因以及结构与性能之间的关系提供了技术支撑。     

Solar MHD turbulence in regions with various levels of flare activity

The paper continues investigations of MHD turbulence in active solar regions. The statistical distributions of the increments (structure functions) of the turbulent field are studied analytically in the context of a refined Kolmogorov theory of turbulence. Since photospheric transport of the B _z component of the magnetic field is quite similar to that of a scalar field in a turbulent flow, the theory of transport of a passive scalar can be applied. This approach enables us to show that the structure functions are determined by the competition between the dissipation of the magnetic and kinetic energies and to obtain a number of relations between the structure-function parameters and energy characteristics of the MHD turbulence. Taking into account general conclusions that can be drawn on the basis of the refined Kolmogorov turbulence theory, the structure functions of the B _z field are calculated for eight active regions (from measurements of SOHO/MDI and the Huairou Solar Observing Station, China). These calculations show that the behavior of the structure functions is different for the B _z field of each active region. The energy-dissipation index of the fluctuation spectrum (which is uniquely determined by the structure functions) is closely related to the level of flare activity: the more activity, the less steep the dissipation spectrum for a given active region. This provides a means to test and, consequently, forecast the flare activity of active regions.     

Present-day stress field and deformation in eastern Austria

 We analysed fault plane solutions and borehole breakouts in the eastern part of the Eastern Alps and found a heterogeneous stress field which we interpret as a transition zone of three different stress provinces: the western European stress province with NW to NNW S_H orientation and mainly strike-slip faulting regime; the Adriatic stress province with a radial stress pattern and thrust faulting to strike-slip faulting regime; and the Dinaric-Pannonian stress province with NE S_H orientation and strike-slip faulting regime. The western Pannonian basin seems to be a part of the transition zone with WNW to NW S_H orientation. A stress regime stimulating strike-slip faulting prevails in the Eastern Alps. The south Bohemian basement spur as a major tectonic structure with a high rheological contrast to surrounding units has a strong influence on the stress field and exhibits the highest seismicity at its tip due to stress concentration. From a constructed vertical stress orientation profile we found stress decoupling of the Northern Calcareous Alps from the underlying European foreland. Both the Molasse and the Flysch-Helvetic zone are considered as candidates for decoupling horizons due to stress orientation observations and due to their rheological behaviour. From seismological and rheological data, we suggest a horizontal stress decoupling across the Eastern Alps caused by a weakened central Alpine lithosphere. Received: 3 July 1998 / Accepted: 4 April 1999     

Pre-flare changes in the turbulence regime for the photospheric magnetic field in a solar active region

Observations of the total magnetic field in the active region NOAA 6757 have been used to study the turbulence regime from 2.5 h before the onset of a 2B/X1.5 flare until two minutes after its maximum. The curvature of the exponent ζ(q) for the structure functions of the B _z field increases monotonically before the flare (i.e., the multifractal character of the B _z field becomes more complex) but straightens at the flare maximum and coincides with a linear Kolmogorov dependence (implying a monofractal structure for the B _z field). The observed deviations of ζ(q) from a Kolmogorov line can be used for short-term forecasting of strong flares. Analysis of the power spectra of the B _z field and the dissipation of magnetic-energy fluctuations shows that the beginning of the flare is associated with the onset of a new turbulence regime, which is closer to a classical Kolmogorov regime. The scaling parameter (cancellation index) of the current helicity of the magnetic field, k _h , remains at a high level right up until the last recording of the field just before the flare but decreases considerably at the flare maximum. The variations detected in the statistical characteristics of the turbulence can be explained by the formation and amplification of small-scale flux tubes with strong fields before the flare. The dissipation of magnetic energy before the flare is primarily due to reconnection at tangential discontinuities of the field, while the dissipation after the flare maximum is due to the anomalous plasma resistance. Thus, the flare represents an avalanche dissipation of tangential discontinuities.     

In situ stress field of eastern Australia

New in situ data based on hydraulic fracturing and overcoring have been compiled for eastern Australia, increasing from 23 to 110 the number of in situ stress analyses available for the area between and including the Bowen and Sydney Basins. The Bowen Basin displays a consistent north‐northeast maximum horizontal stress (σ_H) orientation over some 500 km. Stress orientations in the Sydney Basin are more variable than in the Bowen Basin, with areas of the Sydney Basin exhibiting north‐northeast, northeast, east‐west and bimodal σ_H orientations. Most new data indicate that the overburden stress (σ_V) is the minimum principal stress in both the Bowen and Sydney Basins. The Sydney Basin is relatively seismically active, whereas the Bowen Basin is relatively aseismic. Despite the fact that in situ stress measurements sample the stress field at shallower depth than the seismogenic zone, there is a correlation between the stress measurements and seismicity in the two areas. Mohr‐Coulomb analysis of the propensity for failure in the Sydney Basin suggests 41% of the new in situ stress data are indicative of failure, as opposed to 13% in the Bowen Basin. The multiple pre‐existing structural grains in the Sydney Basin further emphasise the difference between propensity for failure in the two areas. Previous modelling of intraplate stresses due to plate boundary forces has been less successful at predicting stress orientations in eastern than in western and central Australia. Nonetheless, stress orientation in the Bowen Basin is consistent with that predicted by modelling of stresses due to plate boundary forces. Variable stress orientations in the Sydney Basin suggest that more local sources of stress, such as those associated with the continental margin and with local structure, significantly influence stress orientation. The effect of local sources of stress may be relatively pronounced because stresses due to plate boundary forces result in low horizontal stress anisotropy in the Sydney Basin.     

A Top-Cut Model for Deposits with Heavy-Tailed Grade Distribution

In some ore deposits, the grade distribution is heavy-tailed and high values make the inference of first- and second-order statistics nonrobust. In the case of gold data, high values are usually cut and the block estimation is performed using truncated grades. With this method, a part of the deposit is omitted, resulting in a potential bias on resources figures. Ad-hoc corrections can be applied on the final figures to take into account the tail of the grade distribution, but no theoretical guidelines are available. A geostatistical model has been developed to handle high values based on the assumption that for high grade zones, the only tangible information is the geometry. The grade variable z can be split into three parts: the truncated grade ( $\operatorname{Min} (z, z_{\mathrm{e}})$ where z _e is the top-cut grade), a weighted indicator above top-cut grade (1{z≥z _e}), and a residual. Within this framework, the residual is poorly structured, and in most cases is a pure nugget-effect. Moreover, it has no spatial correlation with the truncated grade and the indicator above top-cut grade. This decomposition makes the variographic study more robust because variables (indicator and truncated grade) do not keep high grade values. The underlying hypotheses can be tested on experimental indicator variograms and the top-cut grade can be justified. Finally, the estimation is based on a truncated grade and indicator cokriging. The model is applied to blast holes from a gold deposit and on a simulated example. Both cases illustrate the benefits of keeping the high values in the estimation process via an appropriate mathematical model.     

Mechanical twinning of jadeite – an indication of synseismic loading beneath the brittle–plastic transition

The orientation distribution of mechanically twinned jadeite crystals in a metagranite from the Sesia Zone, Western Alps, is evaluated to derive the orientation of the principal stress directions for a prominent stage of ductile deformation at ca. 300 to 350 °C that has been interpreted to reflect synseismic loading and postseismic creep by Küster and Stöckhert (1999). It is tested whether the orientation distribution can be used to obtain information on the magnitude of differential stress if the critical shear stress for twinning is known. Assuming that the critical shear stress of 140 MPa determined by Kollé and Blacic (1982) for mechanical twinning of clinopyroxene in the (100) [001] system holds true for jadeite, and assuming a homogeneous stress field as a first approximation, differential stresses of the order of 1-2 GPa are inferred for metagranites with ca. 15% jadeite, and of 0.5 GPa for jadeitite with 80% of jadeite. Notwithstanding the uncertainty for the critical resolved shear stress for jadeite, these unexpectedly high values are suspected to be due to a combination of the following effects: (1) an inhomogeneous stress field in the polyphase material with curved stress trajectories, (2) stress concentration in jadeite due to load transfer from the quartz matrix, and (3) minor reorientation of the jadeite crystals in the flowing quartz matrix. Hence, absolute values of differential stress cannot be derived from the orientation distribution of twinned jadeite in polyphase rocks. Notwithstanding the failure in deriving reliable absolute values, a short period with exceptionally high stress must have occurred, and is fully consistent with the microstructural record of other minerals. It is proposed that the mechanical twinning of jadeite was caused by a short-lived stress peak and flow at laboratory strain rates related to quasi-instantaneous loading of the uppermost level of the ductile lower crust during brittle failure of the brittle upper crust in a major seismic event.     

Fabric analysis of quartzites with negative magnetic susceptibility – Does AMS provide information of SPO or CPO of quartz?

We ask the question whether petrofabric data from anisotropy of magnetic susceptibility (AMS) analysis of deformed quartzites gives information about shape preferred orientation (SPO) or crystallographic preferred orientation (CPO) of quartz. Since quartz is diamagnetic and has a negative magnetic susceptibility, 11 samples of nearly pure quartzites with a negative magnetic susceptibility were chosen for this study. After performing AMS analysis, electron backscatter diffraction (EBSD) analysis was done in thin sections prepared parallel to the K₁K₃ plane of the AMS ellipsoid. Results show that in all the samples quartz SPO is sub-parallel to the orientation of the magnetic foliation. However, in most samples no clear correspondance is observed between quartz CPO and K₁ (magnetic lineation) direction. This is contrary to the parallelism observed between K₁ direction and orientation of quartz c-axis in the case of undeformed single quartz crystal. Pole figures of quartz indicate that quartz c-axis tends to be parallel to K₁ direction only in the case where intracrystalline deformation of quartz is accommodated by prism <c> slip. It is therefore established that AMS investigation of quartz from deformed rocks gives information of SPO. Thus, it is concluded that petrofabric information of quartzite obtained from AMS is a manifestation of its shape anisotropy and not crystallographic preferred orientation.     

57Fe Mössbauer spectroscopy and magnetization of cation deficient Fe2TiO4 and FeCr2O4. Part II: Magnetization data

Magnetic properties are reported for synthetic cation deficient Fe₂TiO₄ and FeCr₂O₄ particles (<1 μm). Cation deficiency, achieved by oxidation, is characterized by the oxidation parameter z which represents the fraction of Fe²⁺ ions converted to Fe³⁺ in the spinel lattice (0≤z≤1). Fe₂TiO₄ (z=0.85) has a Curie temperature T _c that can only approximately be given with a value in the range 400–700 K and it has a magnetic moment per formula unit M≈0.50 μ_B (μ_B=Bohr's magneton) at 4.2 K, for FeCr₂O₄ it is T _c≥520 K and M (4.2 K) ≈0.16 μ_B. Magnetic hysteresis parameters at various temperatures show in part characteristic features due to relaxation phenomena. In the Ti-spinel, the latter are caused by a superposition of superparamagnetism and spin relaxation and in the Cr-spinel by superparamagnetism, in agreement with Mössbauer data (part I of this paper). The cation and vacancy distribution and magnetic coupling are discussed in both compositions with respect to magnetic moment data considering magnetic dilution by incorporated vacancies, and in the Ti-spinel also by non magnetic Ti⁴⁺.     

Piezoelectric effects in quartz-rich rocks

Piezoelectricity, a polarization of charge produced by an applied stress, occurs in many minerals. It is particularly strong in quartz. Aggregates of piezoelectric grains are themselves piezoelectric if the grains are suitably aligned. Such aggregates may be said to have a piezoelectric fabric. Thus quartz-rich rocks may possess a piezoelectric fabric and this paper discusses the various possible fabrics.To test whether a piezoelectric fabric might be detected in a quartz-rich rock, apparatus was built that hydraulically applied a sinusoidal stress to cubic specimens. The three resulting orthogonal polarizations of charge were measured via a charge amplifier. A specimen of pure quartz was used to verify the experimental method and to ensure that absolute piezoelectric moduli were being measured. Rocks with and without preferred orientation were tested. Of the latter types, those containing little or no free quartz (marble, basalt) did not exhibit measurable piezoelectric effects. However, all quartz-rich rocks (quartzites, granites, gneisses, mylonites) did show piezoelectric effects when stressed. These effects were in two categories

1. (1) effects due to piezoelectric fabrics, called true piezoelectric effects

2. (2) effects due to random distributions of the piezoelectric vectors, called statistical effects.

To distinguish between these two effects, three criteria were used. Firstly, the measured effects were compared with the expected statistical effect for a rock of that grain size and composition. Secondly, where possible, multiple specimens were cut from the one rock sample, all specimens with the same orientation. Specimens from a rock with a piezoelectric fabric should show similar results. Thirdly, the optically observed c-axis distribution and orientation was compared with the piezoelectrically predicted fabric and orientation.This paper shows that while most rocks gave results consistent with statistical effects from a non-polar or random distribution, some rocks exhibited a true piezoelectric effect due to fabric. This effect may be used, with some imprecision, to locate the a-axes and c-axes of quartz in the aggregate. The polarities of the a-axes are also obtained.     

Analysis of compressible axially loaded piles in rock

Analytical solutions are developed for the prediction of the load–settlement response of a compressible axially loaded cast in situ pile in rock. The principal input parameters are derived from the τ–z curve which applies to the contact zone between the pile shaft and the embedment material. For larger diameter piles in rock, these τ–z curves can differ markedly from curves that apply to piles in clay and may include a significant strain hardening region prior to reaching peak strength. In addition to the complete analyses which would normally require the use of a computer, simplified solutions suitable for hand calculations have been derived for the peak load and the pile head displacement to peak. An example of the application of these simplified design calculations is presented.